control and synchronization of large - scale dynamical systems have received much attention in recent years - . in some cases ,it is desired to control a complex network to a homogeneous trajectory of the uncoupled system , and many control strategies are taken into account to stabilize the system . among them , pinning control is an effective scheme . due to the interaction of the network ,it is not necessary to impose controllers on all nodes .the general idea behind pinning control is to apply some local feedback controllers only to a fraction of nodes while the rest of nodes can be affected through the interactions among nodes - .  in most existing works onlinearly coupled dynamical systems , each node needs to gather information of its own state and neighborhood s states and update them continuously or in a fixed sampling rate - .however , as pointed out in , the event - based sampling technique showed better performance than sampling periodically in time for some simple systems .hereafter , a number of researchers suggested that the event - based control algorithms can be utilized to reduce communication and computation load in networked systems but still maintain control performance - .therefore , the event - based control is particularly suitable for networked systems with limited resources and has attracted wide interests in the scope of distributed control of networked systems .  in some recent papers - , the authors addressed event - triggered algorithms for pinning control of networks . gave an exponentially decreasing threshold function , hence , the convergence rates of algorithms are predesigned .the event - triggering threshold in was prescribed by the distance among states of nodes and target . however , the coupling topology of the network was static . employed event - triggered configurations and pinning control terms to realize stability in linearly coupled dynamical systems with markovian switching in both coupling matrix and pinned node set .however , to realize stability of the switching system , there must exist at least one stable subsystem .if stability can not be achieved under any subsystem , the switching sequence need to be designed to assure the stability .especially , in the fast switching theory , by constructing a stable time - average system , the dynamics of switched system can be stable when the switching is fast enough - .  in the real world, the graph topology of a network may change very quickly by jumps or switches , due to link failures or new creation in a network .so , it is inevitable to study the stability of fast switching systems .motivated by these works as well as our previous work , in this paper , we employ the event - triggered strategy in both coupling configuration and pinning control terms to realize stability in dynamical systems with fast markovian switching couplings and pinned node set .hence , all the subsystems among switching can be unstable in this paper . noticing the significance of the average system in the analysis of stability of fast switching system - . in this paper ,one triggered strategy is given on the average coupling matrix and average pinned node set , the other is given on the time - varying coupling matrix and pinned node set .for each strategy , it is proved that the proposed event - triggered strategy guarantees the stability of the coupled dynamical systems .this paper is organized as follows . in sec .[ sec - pre ] , the underlying problem is formulated . in sec .[ sec - con ] , we propose the event - triggering schemes of diffusion configuration and pinning terms to pin the coupled systems to a homogenous pre - assigned trajectory of the uncoupled node system .numerical simulations are given in sec .[ sec - sim ] to verify the theoretical results .finally , this paper is concluded in sec .[ sec - conclu ] ._ notations _ : for a matrix , denote the elements of on the -th row and -th column . denotes the symmetry part of a square matrix . for a vector ,denote by that every element of is positive . denotes the identity matrix with dimension . for a matrix ,denote by and the largest and smallest eigenvalues in module .for a symmetric matrix , denote its -th largest eigenvalue by .the symbol represents the kronecker product . denotes the matrix norm of induced by the vector norm . for a matrix , .in particular , without special notes , -vector norm is used in this paper and denote it by , i.e. .in this paper , we consider a network of linearly coupled dynamical systems with discontinuous diffusions and feedback pinning terms as follows : \\ -c\epsilon d_i(\sigma_t)\gamma[x_{i}(t_{k}^{i})-s(t_k^i ) ] , ~ t_{k}^{i}\le t < t_{k+1}^{i } \label{cds}\end{aligned}\ ] ] where denotes the state vector of node , the continuous map denotes the identical node dynamics . is the uniform coupling strength at each node . denotes the switching rule . if is linked to otherwise , and .l=1}^n\in\mathbb r^{n\times n} ] is given by : where , , is the transition probability from state to if , while .let for be the successive sojourn time between jumps of .therefore , the sojourn time in state is exponentially distributed with parameter .clearly , \le \max_j\frac{1}{-q_{jj}} ] the probability transition matrix of the embedded discrete - time markov process of .  denote  ] satisfying , where for all . from , there exist positive numbers and such that .then , and  ] is negative semi - definite .let and . pick a constant satisfying .set as the time point defined by the following rule  [ event1 - 1 ] _k+1^i & = & _ > t_k^i\ { : e_i ( ) _ i()-s ( ) } , + [ event1 - 2 ] t_k+1^i & = & \{_k+1^i , t_k^i + t}.  if satisfies  [ switch - speed-4-rule-1 ] -(-)_j e^_1 _ j+[2_2(-)+k_1 ] + _jm_1(t,_j ) + k_2_je^_1 _ j + < 0  where satisfies and   then under the updating rule ( [ event1 - 1 ] ) and ( [ event1 - 2 ] ) , system ( [ cds ] ) is stable at the homogeneous trajectory in the mean square sense .  _brief proof ._ let , then the dynamics of in is   -c\epsilon d_i(\sigma_t)\gamma\hat{x}_{i}(t)+\hat{e}_i(t)\end{aligned}\ ] ]  with  \\ \label{error-2 } + \epsilon { d}_i(\sigma_t)\gamma\left[x_i(t)-s(t)-x_i(t_{k_i(t)}^i)+s(t_{k_i(t)}^i)\right]\end{aligned}\ ] ]  let ^{\top},~~\hat{e}(t ) = [ \hat{e}^{\top}_1(t),\cdots,\hat{e}^{\top}_m(t)]^{\top}\end{aligned}\ ] ] and . by , ,cauchy- schwarz inequality , inequalities ( [ est - ave - l ] ) , ( [ est - ave - d ] ) and lemma [ lemma-1 ] , we have  - v(\tau_r)\\ \nonumber \le & -2(\beta-\beta')\mathbb e\left[\delta_r\cdot e^{-\rho_1\delta_r}\right ] v(\tau_{r})\\ \nonumber & + 2(\beta-\beta')\rho_2\cdot \mathbb e\left [ \delta_r^2 \cdot m_1(t,\delta_r ) \right]v(\tau_{r})\\ \nonumber & + [ m_2 + m_3\cdot \mathbb e\left[m_1(t,\delta_r)\right ] ] \cdot c_0\cdot \kappa^r  v(\tau_{r})\\ \nonumber & + k_1 \cdot \mathbb e\left[\delta_r^2\cdot  m_1(t,\delta_r)\right ] v(\tau_r )  \\ \nonumber & + k_2\cdot \mathbb e\big[\delta_r^2\cdot e^{\rho_1 \delta_r}+\rho_2\delta_r^3 \cdot e^{\rho_1 \delta_r } m_1(t,\delta_r)\big ] v(\tau_r)\end{aligned}\ ] ]  with    the property of markov process gives   = \sum_{j=1}^m \pi_{j}(r ) \frac{1}{-q_{jj } } \le \max_j\frac{1}{-q_{jj}},\\\label{exp - sojourn - time-1 }   & \mathbb e\left[\delta_r\cdot e^{-\rho_1\delta_r }  \right]\ge \min_j\frac{1}{-q_{jj}}e^{-\rho_1\max_j  \frac{1}{-q_{jj}}},\\ \label{exp - sojourn - time-2 }  & \mathbb e\left[\delta_r^2\cdot m_1(t,\delta_r)\right ]  \le \max_j \frac{1}{q^2_{jj}}m_1(t,\max_j\frac{1}{-  q_{jj } } ) ,  \end{aligned}\ ] ]  and  \\ \nonumber \le & \max_j\frac{1}{q^2_{jj}}\cdot e^{\rho_1 \max_j\frac{1}{-q_{jj}}}\left[1+\rho_2 \max_{j}\frac{1}{-q_{jj}}\cdot  m_1(t,\max_{j}\frac{1}{-q_{jj}})\right].\end{aligned}\ ] ]  by , we have that for any , there exists , such that for any ,  \cdot \kappa^r \le { \delta}.\ ] ]  apply assumption ( [ switch - speed-4-rule-1 ] ) and estimations ( [ exp - sojourn - time-1])-([est - switch - times ] ) to ( [ est - v ] ) , we get that there exists some such that for all ,  - v(\tau_r ) \\\nonumber \le & -(\beta-\beta')\min_j\frac{1}{-q_{jj}}\cdot e^{-\rho_1\max_j \frac{1}{-q_{jj}}}v(\tau_r).\end{aligned}\ ] ]  take expectation on both sides of ( [ con - est - v-5 ] ) ,  \\ \le & \left[1-(\beta-\beta')\min_j\frac{1}{-q_{jj}}\cdot e^{-\rho_1\max_j \frac{1}{-q_{jj}}}\right]\mathbb e\left[v(\tau_r)\right].\end{aligned}\ ] ]  which implies for ,  ] , we have  ], an upper bound of the inter - event interval , denoted by , is imposed to all nodes .practically , if the time calculated from rule ( [ event1 - 1 ] ) is time longer than current triggered time , an externally triggering will be applied to the node to guarantee hold for all and .  in the above theorem , the updating rule ( [ event1 - 1 ] ) and ( [ event1 - 2 ] )is defined by in ( [ error-1 ] ) .next , we propose another updating rule based on in ( [ error-2 ] ) to stabilize the system .[ thm2 ] suppose all assumptions in theorem [ thm1 ] hold . for ,set as the time point defined by the following rule  [ event2 ] t_k+1^i = _> t_k^i\ { : _ i ( ) _ i()-s ( ) }  where is defined in ( [ error-2 ] ) .if satisfies where  \frac{1}{{\underline{\lambda } } } + \frac{\beta'^2{\underline{\lambda}}}{c{\overline{\lambda}}^2},\\ \rho_1 ' = & 2(\beta+\beta')+2\max_{i , j}\big|\lambda_j\big(\{p(\alpha i_m + cl(i)\\ \nonumber & + c\epsilon d(i))\otimes g\gamma\}^{s}\big)\big|\frac{1}{{\underline{\lambda}}}\end{aligned}\ ] ]  then under the updating rule ( [ event2 ] ) , system ( [ cds ] ) is stable at the homogeneous trajectory in the mean square sense .  the proof of this theorem is similar to that of theorem [ thm1 ] .here it is omitted for saving space .  comparing the updating rule ( [ event1 - 1 ] ) and ( [ event1 - 2 ] ) and rule ( [ event2 ] ) , it can be seen that the upper bound of triggered time intervals of each node , denoted by in ( [ event1 - 2 ] ) , is no longer required in the updating rule ( [ event2 ] ) defined by . however , as a trade off , the triggering events under rule ( [ event2 ] ) happen more frequently than rule ( [ event1 - 1 ] ) and ( [ event1 - 2 ] ) if the switching among topologies is sufficiently fast . since is decided by the varying coupling matrix and pinned matrix , which jumps at the time point when the topology switches .therefore , for systems with updating rule ( [ event2 ] ) , each node has to update its state information at the switching time points of the topologies .if the switching among topologies is sufficiently fast , the triggered times of all nodes will substantially increase .  similar to work , underthe above two updating rules ( [ event1 - 1 ] ) , ( [ event1 - 2 ] ) and ( [ event2 ] ) , there exists at least one node with its next inter - event interval being strictly positive . under some special hypothesis ,the zeno behavior can be excluded for all nodes .[ proposition-1 ] suppose all hypotheses of theorem 1 hold .  1 .  under either of the updating rule ( [ event1 - 1 ] ) and ( [ event1 - 2 ] ) and rule ( [ event2 ] ) , if the system does not converge , there exists at least one node such that its next inter - event interval is strictly positive .suppose there exists some ( possibly negative ) such that holds for all .if there exists a constant such that holds for some , then the next inter - event interval of node is strictly positive and lower bounded by a common constant .the proof of this proposition is omitted here for saving space .in this section , we present numerical examples to illustrate the theoretical results . in these examples , we consider three - dimensional neural network as the uncoupled node dynamics : with , \end{aligned}\ ] ] , , and . this system has a double - scrolling chaotic attractor with initial value  . noting that has jacobin matrices and is the upper bound of the matrices of .hence , we estimate and .( -1 , 1 ) node(1)[rectangle , scale=0.7 , fill = red!30!white]1 ( 0 , 1 ) node(2)[rectangle , scale=0.7,fill = red!30!white]2 ( 1 , 1 ) node(3)[circle , scale=0.6,fill = blue!30!white]3 ( -1 , 0 ) node(4)[circle , scale=0.6,fill = blue!30!white]4 ( 0 , 0 ) node(5)[circle , scale=0.6,fill = blue!30!white]5 ( 1 , 0 ) node(6)[circle , scale=0.6,fill = blue!30!white]6 ( -1 , -1 ) node(7)[circle , scale=0.6,fill = blue!30!white]7 ( 0 , -1 ) node(8)[circle , scale=0.6,fill = blue!30!white]8 ( 1 , -1 ) node(9)[circle , scale=0.6,fill = blue!30!white]9 ( 0.5 , -0.5 ) node(10)[circle , scale=0.6,fill = blue!30!white]10  ( 0,-1.5 ) node(tt ) [ ] ( a ) ; ( 1 ) ( 4 ) ( 1 ) ( 5 ) ( 5 ) ( 2 ) ( 5 ) ( 3 ) ( 4 ) ( 7 ) ( 5 ) ( 7 ) ( 6 ) ( 10 ) ( 6 ) ( 9 ) ( 9 ) ( 10 ) ( 8) ( 9 ) ;  ( 2 , 1 ) node(11)[rectangle , scale=0.7,fill = red!30!white]1 ( 3 , 1 ) node(12)[circle , scale=0.6,fill = blue!30!white]2 ( 4 , 1 ) node(13)[circle , scale=0.6,fill = blue!30!white]3 ( 2 , 0 ) node(14)[circle , scale=0.6,fill = blue!30!white]4 ( 3 , 0 ) node(15)[circle , scale=0.6,fill = blue!30!white]5 ( 4 , 0 ) node(16)[circle , scale=0.6,fill = blue!30!white]6 ( 2 , -1 ) node(17)[circle , scale=0.6,fill = blue!30!white]7 ( 3 , -1 ) node(18)[circle , scale=0.6,fill = blue!30!white]8 ( 4 , -1 ) node(19)[rectangle , scale=0.7,fill = red!30!white]9 ( 3.5,-0.5)node(20)[circle , scale=0.6,fill = blue!30!white]10 ( 3,-1.5 ) node(tt ) [ ] ( b ) ; ( 11 ) ( 14 ) ( 11 ) ( 15 ) ( 15 ) ( 20 ) ( 14 ) ( 20 ) ( 12 ) ( 13 ) ( 13 ) ( 16 ) ( 12 ) ( 16 ) ( 17 ) ( 20 ) ( 19 ) ( 20 ) ( 18 ) ( 19 ) ( 17 ) ( 18 ) ;  ( -1 , -2 ) node(21)[circle , scale=0.6,fill = blue!30!white]1 ( 0 , -2 ) node(22)[rectangle , scale=0.7,fill = red!30!white]2 ( 1 , -2 ) node(23)[circle , scale=0.6,fill = blue!30!white]3 ( -1 , -3 ) node(24)[circle , scale=0.6,fill = blue!30!white]4 ( 0 , -3 ) node(25)[circle , scale=0.6,fill = blue!30!white]5 ( 1 , -3 ) node(26)[circle , scale=0.6,fill = blue!30!white]6 ( -1 , -4 ) node(27)[circle , scale=0.6,fill = blue!30!white]7 ( 0 , -4 ) node(28)[circle , scale=0.6,fill = blue!30!white]8 ( 1 , -4 ) node(29)[circle , scale=0.6,fill = blue!30!white]9 ( -0.5,-3.5)node(30)[circle , scale=0.6,fill = blue!30!white]10 ( 0,-4.5 ) node(tt ) [ ] ( c ) ; ( 21 ) ( 22 ) ( 21 ) ( 25 ) ( 22 ) ( 23 ) ( 23 ) ( 26 ) ( 26 ) ( 29 ) ( 22 ) ( 29 ) ( 25 ) ( 29 ) ( 24 ) ( 30 ) ( 28 ) ( 30 ) ( 27 ) ( 28 ) ( 24 ) ( 27 ) ;  ( 2 , -2 ) node(31)[circle , scale=0.6,fill = blue!30!white]1 ( 3 , -2 ) node(32)[circle , scale=0.6,fill = blue!30!white]2 ( 4 , -2 ) node(33)[rectangle , scale=0.7,fill = red!30!white]3 ( 2 , -3 ) node(34)[circle , scale=0.6,fill = blue!30!white]4 ( 3 , -3 ) node(35)[circle , scale=0.6,fill = blue!30!white]5 ( 4 , -3 ) node(36)[circle , scale=0.6,fill = blue!30!white]6 ( 2 , -4 ) node(37)[circle , scale=0.6,fill = blue!30!white]7 ( 3 , -4 ) node(38)[rectangle , scale=0.7,fill = red!30!white]8 ( 4 , -4 ) node(39)[circle , scale=0.6,fill = blue!30!white]9 ( 3.5,-3.5)node(40)[circle , scale=0.6,fill = blue!30!white]10 ( 3,-4.5 ) node(tt ) [ ] ( d ) ; ( 31 ) ( 34 ) ( 31 ) ( 35 ) ( 33 ) ( 36 ) ( 34 ) ( 35 ) ( 34 ) ( 37 ) ( 35 ) ( 37 ) ( 32 ) ( 33 ) ( 32 ) ( 36 ) ( 38 ) ( 40 ) ( 39 ) ( 40 ) ;  the possible coupling graph topologies and pinned node sets are shown in fig .[ topology ] , here .noting that subsystems with every possible network topology and pinned nodes can not be stabilized to the target trajectory .theorems [ thm1],[thm2 ] indicate that if the average system can be stabilized and the switching among subsystems is sufficiently fast , then under the event - triggered strategies , the switching system can be stabilized . via the following numerical simulations, it can be seen that the switching system can be stabilized .suppose the generator of the markov chain is then the sojourn time in each topology follows the exponential distribution with parameter .in the following examples , suppose the inner coupling matrix , , , .the ordinary differential equation ( [ cds ] ) is numerically solved by the euler method with a time step 0.001(seconds ) and the time duration of the numerical simulations is  ] s under updating rule ( [ event1 - 1 ] ) and ( [ event1 - 2 ] ) and rule ( [ event2]).,scaledwidth=50.0% ]  furthermore , it can be seen from fig .[ event - times ] that the events of updating the diffusion and pinning terms under rule ( [ event2 ] ) happen more than rule ( [ event1 - 1 ] ) and ( [ event1 - 2 ] ) , as a consequence of fast switching among network topologies .in this paper , event - triggered configurations and feedback pinning are employed to realize stability in linearly coupled dynamical systems with fast markovian switching , which reduces communication and computation loads .once an event for a node is triggered , the diffusion coupling term and feedback control ( if pinned ) of this node will be updated .event triggering criteria are derived for each node that can be computed in a parallel way .two event - triggered rules are proposed and proved to perform well and can exclude zeno behaviors in some cases .simulations are given to verify these theoretical results .this work was jointly supported by the national natural sciences foundation of china under grant nos .61273211 and 61673119 and the program for new century excellent talents in university ( ncet-13 - 0139 ) .t. chen , x. liu , and w. lu , pinning complex networks by a single controller , _ ieee transactions on circuits and systems - i : regular papers _ , 54(6 ) , 2007 , 1317 - 1326 w. lu , x. li , and z. rong , global stabilization of complex networks with digraph topologies via a local pinning algorithm , _ automatica _ , 46(1 ) : 116 - 121 , 2010 .l. gao , x. liao , and h. li , pinning controllability analysis of complex networks with a distributed event - triggered mechanism , _ ieee trans . on circuits and systemsii : express briefs _ , 61(7 ) : 541 - 545 , 2014 .

Abstract: in this paper , stability of linearly coupled dynamical systems with feedback pinning is studied . event - triggered rules are employed on both diffusion coupling and feedback pinning to reduce the updating load of the coupled system . here , both the coupling matrix and the set of pinned - nodes vary with time are induced by a homogeneous markov chain . for each node , the diffusion coupling is set up from the state information of its neighbors at their latest triggered time and the feedback pinning uses the target s ( if pinned ) information at the node s latest event time . the next event time is triggered by some specified criteria . two event - triggering rules are proposed and it is proved that if the system with time - average coupling and pinning gains are stable , the event - triggered strategies can stabilize the system if the switching is sufficiently fast . moreover , zeno behaviors are excluded in some cases . finally , numerical examples are presented to illustrate the theoretical results .


consider multiple hypotheses to be tested by individual tests using a procedure which corrects for multiplicity , such as the procedure or the correction .the procedure of has recently received much attention , resulting in various generalizations .standard procedures require knowledge of the ideal p - values of all tests .we consider the case where p - values are not known exactly and can only be computed by simulation . for example , this occurs for bootstrap or permutation tests .we will call such tests monte carlo tests .safe implementations of individual monte carlo tests and computation of their power have been proposed .recent studies involving monte carlo tests use a variety of data sources such as data from a genome data archive , brain activity data and microarray data . as an example , we consider microarray data of gene expressions for yeast chemostat cultivations in the present article .this article introduces ` mmctest ` , an algorithm to implement the multiplicity correction for multiple monte carlo tests .the algorithm gives , with pre - specified probability , the same classification ( rejected and non - rejected hypotheses ) as the classification based on the ideal p - values .for permutation tests , the ideal p - values can in principle be obtained by running through all permutations . for bootstrap tests ,the ideal p - value is the probability that a bootstrapped test statistic is at least as extreme as the observed test statistic .  the motivation for trying to achieve the same classification as the one obtained with the ideal p - values is mainly repeatability and objectivity of the results , which called _ first law of applied statistics _ : `` two individuals using the same statistical method on the same data should arrive at the same conclusion . ''our algorithm achieves this up to a guaranteed pre - specified error probability .another reason for comparing to the ideal p - values is that all the theoretical results of the multiple testing procedure based on the ideal p - values still hold ( again up to the guaranteed error probability ) .our proposed algorithm is sequential : it starts with all hypotheses being unclassified and then takes samples and classifies hypotheses until all but a certain number of hypotheses have been classified or until a certain effort is reached .the proposed algorithm can be stopped earlier while having the same guarantee on the probability of misclassifications . when stopped beforeall hypotheses have been classified , the algorithm returns three sets : the rejected , the non - rejected and the not yet classified hypotheses .a literature review detailing existing approaches used to classify multiple hypotheses without knowledge of the ideal p - values is presented in section [ section_literature ] .the basic ` mmctest ` algorithm is described in section [ section_algorithm ]. moreover , section [ section_algorithm ] states conditions which bound the probability of classification errors and which guarantee the convergence of the testing result of ` mmctest ` to the classification based on the ideal p - values .the multiple testing procedure of and the correction satisfy these conditions ( appendix [ appendix_properties ] ) .  in section [ section_empirical_comparison ], we first present an application of ` mmctest ` motivated by real biological data , given by a microarray dataset of gene expressions for yeast chemostat cultivations .afterwards , we conduct simulation studies motivated by this real data to compare the performance of a naive approach and of ` mcfdr ` to ` mmctest ` . furthermore , we investigate the dependance of ` mmctest ` on certain parameters .we conclude with a discussion in section [ section_discussion ] .all proofs can be found in the appendix .the ` mmctest ` algorithm is implemented in an r - package ( ` simctest ` , available on cran , the comprehensive r archive network ) .the supporting information includes an evaluation of a second dataset and further simulation studies with different testing thresholds .several algorithms in the literature are related to our approach in that they aim to stop drawing samples for certain hypotheses .some of them aim to give guarantees on their result , but these guarantees are usually much weaker than our guarantee .superficially , our algorithm is close to the algorithm proposed by .both algorithms maintain confidence intervals for the p - values corresponding to each hypothesis , and stop generating samples for hypotheses for which a decision can be reached . for thisboth algorithms rely on the monotonicity property of the procedure .  however , there are crucial differences .these mainly come from the different aim of the algorithms : aim to reduce the effort compared to the naive approach with a fixed number of samples per hypothesis .we aim to give the same classification as the classification using the ideal p - values . as a consequence, their algorithm imposes an upper bound on the number of samples generated per hypothesis , whereas our algorithm is open - ended .their algorithm does not aim to ensure repeatability , whereas we aim to do so . to be able to do thiswe judiciously control the joint coverage probability of the intervals .  to be specific , the main results in ( * ?* proposition 1 , theorem 1 ) are related to lemma [ lemma_monotonicity_sets ] in the present article , with the difference that compare the classification to the naive approach with a fixed number of samples , whereas we compare the classification to the one based on the ideal p - values .furthermore , the article of has no equivalence to our theorem [ theorem_convergence ] in which we prove that the classification returned by our algorithm converges to the one based on the ideal p - values . present an early stopping procedure with a bound on its computational savings . as , aim at designing a procedure which gives the same result as the naive approach with a fixed number of samples .moreover , the authors show that their procedure only controls the false discovery rate ( fdr ) up to an error term .the ad - hoc method of stops generating samples for hypotheses for which a lower confidence level exceeds a pre - specified threshold ( leading to a non - rejection ) . no early stopping for rejections is proposed .being an ad - hoc method for a specific application , no explicit theoretical results are given .the algorithm ` mcfdr ` of is a modification of the algorithm of , which , for a single hypothesis , stops drawing further samples when a fixed number of exceedances has been observed .the main idea of ` mcfdr ` is to use the criterion of to obtain quick non - rejections and to stop the entire algorithm once all remaining hypotheses are rejected based on their current estimated p - values . although ` mcfdr ` gives quick results , the algorithm does not give any guarantees on how its result relates to the result obtained with the ideal p - values .in contrast to the above approaches , the focus of our algorithm lies on the computation of the same classification as the one obtained if the ideal p - value of each test had been available .there are other related methods which are not necessarily designed to apply the procedure or the correction : the method proposed by uses ordinary permutation p - values if sufficiently many exceedances can be observed ; otherwise , the authors approximate the p - values using a fitted extreme value distribution .the aim is to efficiently compute an estimate of all p - values , without giving any theoretical guarantees .several specialized resampling - based testing procedures for various sampling methods and various statistics can be found in .all above methods do not try to take the ( unknown ) dependance between the test statistics into account .using permutation methods this can be attempted .consider testing null hypotheses having corresponding test statistics and observed values .a large value of shall indicate evidence against . moreover , let denote the ideal p - value corresponding to the hypothesis .we assume that are not available analytically , but have to be obtained through simulations .we assume that for every hypothesis , where , we can obtain independent samples from the test statistic under the null hypothesis .we will denote these by , and the corresponding exceedance indicators will be denoted by , , where is the indicator function . in the case of a permutation test ,computing involves generating permutations without replacement .suppose that ^m \rightarrow \mathcal{p}(\ { 1,\ldots , m \}) ] , i.e. if lower p - values lead to more rejections .the following generic algorithm is designed for monotonic multiple testing procedures .it iteratively controls the set of hypotheses for which further samples need to be drawn by refining confidence intervals for every through monte carlo sampling . at iteration , the confidence interval for the p - value denoted by .the upper confidence limit of a confidence interval is denoted by and the lower confidence limit is denoted by .the following variables and functions control the behavior of the algorithm .the variable controls how many additional samples are drawn in each iteration .it is increased geometrically by a constant in each step of the algorithm , starting at . in the examples of this article we use and .two vectors keep track of counts .the function computes a confidence interval for the ideal p - value of a hypothesis based on the number of exceedances and the number of samples drawn for this hypothesis .the dependance on the current value of is needed to be able to guarantee a joint coverage probability of all confidence intervals produced in the algorithm . for simplicity, we will assume that returns closed confidence intervals .in appendix [ appendix_clopper_pearson ] we give an example for such an which computes confidence intervals and uses a spending sequence to guarantee an overall coverage probability .the algorithm runs until at most hypotheses are classified or until the total number of samples drawn reaches a pre - specified limit .the following pseudo - code uses and , thereby computing a classification of all hypotheses .  in the remainder of this article, denotes the number of elements in a finite set and the length of an interval .moreover , denotes the euclidean norm of a vector .[ algorithm_basic ] } ] ,  . * while * * .* * . * * for all : * * * . * * * . * * * . * * for all : . * * set , . *return .the algorithm works as follows : the number of additional samples drawn in every step is increased geometrically .the total number of samples drawn up to iteration for a hypothesis is stored in and the total number of observed exceedances is stored in . for all hypotheses which are still under consideration , i.e. those in , an additional batch of samplesis drawn and new confidence intervals are computed .the confidence intervals remain unchanged for the other hypotheses .new classifications are then computed based on the updated upper and lower confidence limits .the confidence intervals computed in algorithm [ algorithm_basic ] are nested by construction .hypotheses using the benjamini - hochberg procedure : after the second iteration ( left ) , after a few additional iterations ( center ) and after the last iteration ( right ) .bold confidence intervals denote elements of in the upper row and elements of in the lower row .the lower ( upper ) confidence limits used to compute ( ) are marked with a cross . ]  [ example_visualization ] an example run of ` mmctest ` ( with hypotheses and ) is shown in figure [ figure_visualisation ] .we use the fdr controlling procedure ( see appendix [ appendix_benjamini_hochberg ] ) with threshold as the multiple testing function . the function given in appendix [ appendix_clopper_pearson ]is used to compute confidence intervals .columns show different iterations , the upper row shows the computation of , the lower row shows the computation of .the indices contained in and are visualized with bold confidence intervals .additionally , the lower ( upper ) confidence limits used to compute ( ) are marked with a cross .only the lower ( upper ) end of the confidence interval matters for the computation of ( ) , thus the hypotheses are ordered by their lower ( upper ) confidence limit in the upper ( lower ) row . in this examplethis turns out to be the same ordering .after the second iteration ( left column ) , ` mmctest ` has already classified the last hypothesis as being non - rejected as the lower confidence limit of its p - value lies above the line connecting the points and which we call the benjamini - hochberg line .all other hypotheses are still undecided and thus their confidence intervals will be refined . after a few additional iterations ( middle column ), the seven smallest values can be classified as rejected as the upper confidence limit of the seventh value is below the line .likewise , the confidence interval of the ninth value has now been shrunk to be entirely above the line which classifies this value as non - rejected .the eighth p - value is still unclassified as its confidence interval overlaps with the line .after refining the confidence interval further , the algorithm stops in the situation depicted in the right column with a complete classification ( ) .the monotonicity of implies immediately that the sequence of sets is increasing , that the sequence of sets is decreasing and , on an additional assumption , that each ( ) is a subset ( superset ) of the ideal set of rejections .[ lemma_monotonicity_sets ] assume that is monotonic .1 .  and .2 .  if  , then  .   in this sectionwe show that under certain conditions the classification of ` mmctest ` is correct with high probability , meaning that all classifications are identical to the classifications based on the ideal p - values .furthermore , we show that all hypotheses will be classified .the first condition pertains to the multiple testing procedure . besides asking for monotonicity, it ensures that lowering the p - value of a rejected hypothesis or increasing the p - value of a non - rejected hypothesis does not change the result of .[ condition_invariance ]  1 . is monotonic .2 .  let ^m ] and imply . then , on the event , both sequences and converge to , i.e. there exists such that  .the condition on in theorem [ theorem_convergence ] ensures that has a neighborhood on which is constant .  the fdr controlling procedure of ( see appendix [ appendix_benjamini_hochberg ] ) and the correction ( see appendix [ appendix_bonferroni_correction ] ) both satisfy condition [ condition_invariance ] ( see corollary [ corollary_benjamini_hochberg ] and corollary [ corollary_bonferroni ] in appendix [ appendix_properties ] ) and the condition on in theorem [ theorem_convergence ] ( see lemma [ lemma_invariance_delta ] and lemma [ lemma_bonferroni_invariance_delta ] in appendix [ appendix_properties ] ) for almost all .the following third condition ensures that the confidence intervals computed by the function in algorithm [ algorithm_basic ] have a guaranteed joint coverage probability . the choice of given in appendix [ appendix_clopper_pearson ] satisfies condition [ condition_intervals ] and condition [ condition_probability ] ( see lemma [ lemma_clopper_pearson_satisfy_condition ] in appendix [ appendix_clopper_pearson ] ) .[ condition_probability ] for a given , the function computes confidence intervals in such a way that .  the main theorem and condition [ condition_probability ] togetherimmediately give a bound on the probability of misclassifications .[ corollary_convergence ] under the conditions of theorem [ theorem_convergence ] and under condition [ condition_probability ] , i.e. the probability that all classifications are correct is at least .the first aim of this section is to demonstrate that ` mmctest ` can be used to classify thousands of hypotheses commonly encountered in real data studies ( section [ subsection_real_application ] ) .moreover , this section shows that when matching the effort , ` mmctest ` computes classifications containing a number of unclassified hypotheses which is comparable to the number of misclassifications incurred by current approaches like the naive method or the ` mcfdr ` algorithm even though ` mmctest ` is able to guarantee the correctness of all its classified hypotheses while for the two other methods , misclassified hypotheses typically remain unidentified in the testing result ( section [ subsection_comparison_naive ] and section [ subsection_comparison_mcfdr ] ) .an ad - hoc variant of ` mmctest ` computing a complete classification yields less misclassifications and random classifications than the other methods , demonstrating that ` mmctest ` is the superior method for practical applications .the aim of the last two sections is to empirically investigate the dependance of ` mmctest ` on certain parameters .section [ subsection_effort_dependence_number_hypotheses ] studies the dependance of the computational effort of ` mmctest ` on the number of hypotheses .we conclude by empirically assessing the runtime of ` mmctest ` in section [ subsection_effort_dependence_unclassified_hypotheses ] , demonstrating that whilst a complete classification can be computationally very expensive , most hypotheses can be classified with a reasonable effort .the following parameters were used throughout section [ section_empirical_comparison ] .the batch size in algorithm [ algorithm_basic ] is increased by in every iteration , starting with .confidence intervals are computed using the function with confidence intervals and parameters and ( see appendix [ appendix_clopper_pearson ] ) . the benjamini - hochberg procedure ( at threshold )as defined in appendix [ appendix_benjamini_hochberg ] always serves as multiple testing procedure .we use a yeast chemostat cultivation dataset of .this dataset consists of microarrays of yeast cultivations .the first microarrays correspond to yeast which was grown aerobically , the second microarrays correspond to yeast which was grown anaerobically .every microarray reacts to genes , thus giving rise to null hypotheses ( no effect of the gene onto the response ) .  to speed up the computation of the simulation studies in this and the following sections as well as to have an underlying `` truth '' for the dataset, we estimated each of the p - values once by generating permutations per hypothesis as outlined in the supporting information .such a number of permutations is far more than what would commonly be used in practice .we then define these approximated p - values to be the ideal p - values we are interested in , although they do not necessarily have to be equal to the p - values underlying each hypothesis .a plot of the ideal p - values is given in the supporting information .in the following sections , we draw bernoulli samples with success probabilities instead of generating actual permutations . the classification obtained by applying the benjamini - hochberg procedure directly to the ideal p - values is used to compute misclassifications .  the supporting information contains similar simulations as the ones which are about to follow for two additional testing thresholds ( 1% and 5% ) .the behavior of ` mmctest ` is qualitatively similar .furthermore , the supporting information contains another comparison of ` mmctest ` to the naive method and to ` mcfdr ` on a simulated dataset with a larger proportion of true null hypotheses than the one of the dataset of , broadly confirming the qualitative results of sections [ subsection_comparison_naive ] and [ subsection_comparison_mcfdr ] .    after having drawn samples all but hypotheses are classified .this corresponds to only around samples per hypothesis , thus making a classification with such a precision fairly easy to compute . drawing roughly the same number of samples again ( a total number of samples )classifies all but hypotheses .  `mmctest ` can be stopped whenever the user s desired number of classifications is achieved . all but hypotheses are classified after samples and all but hypotheses after samples .a classification of all but hypotheses is obtained after having drawn a total number of samples .this is , of course , extremely computationally intensive .the total number of samples drawn for a classification of all but hypotheses corresponds to roughly samples per hypothesis .a comparison to the classification result obtained by applying the procedure to the ideal p - values shows that in all the classifications previously reported , none of the decided hypotheses was wrongly classified .         by theorem [ theorem_convergence ]we have , as conditional on . under condition[ condition_probability ] , this event occurs with probability , hence .finner , h. , gontscharuk , v. & dickhaus , t. ( 2012 ) . false discovery rate control of step - up - down tests with special emphasis on the asymptotically optimal rejection curve .j. stat . _* 39 * , 382397 .knijnenburg , t. , daran , j .-, van den broek , m. , daran - lapujade , p. , de winde , j. , pronk , j. , reinders , m. & wessels , l. ( 2009 ) .combinatorial effects of environmental parameters on transcriptional regulation in saccharomyces cerevisiae : a quantitative analysis of a compendium of chemostat - based transcriptome data . _bmc genomics _ * 10*.     nusinow , d. , kiezun , a. , oconnell , d. , chick , j. , yue , y. , maas , r. , gygi , s. & sunyaev , s. ( 2012 ) .network - based inference from complex proteomic mixtures using snipe ._ bioinformatics _ * 28 * , 31153122 .

Abstract: consider testing multiple hypotheses using tests that can only be evaluated by simulation , such as permutation tests or bootstrap tests . this article introduces ` mmctest ` , a sequential algorithm which gives , with arbitrarily high probability , the same classification as a specific multiple testing procedure applied to ideal p - values . the method can be used with a class of multiple testing procedures which includes the benjamini & hochberg false discovery rate ( fdr ) procedure and the bonferroni correction controlling the familywise error rate . one of the key features of the algorithm is that it stops sampling for all the hypotheses which can already be decided as being rejected or non - rejected . ` mmctest ` can be interrupted at any stage and then returns three sets of hypotheses : the rejected , the non - rejected and the undecided hypotheses . a simulation study motivated by actual biological data shows that ` mmctest ` is usable in practice and that , despite the additional guarantee , it can be computationally more efficient than other methods .  _ key words : _ benjamini hochberg , bonferroni correction , bootstrap , false discovery rate , multiple comparisons , resampling , sequential algorithm


ever since it was proposed more than 80 years ago , quantum mechanics has raised great challenges both in foundations and in applications .the latter have been developed at a very rapid pace , opening up new vistas in most branches of physics as well as in much of chemistry and engineering .substantial progress and important discoveries have also been made in foundations , though at a much slower rate .the measurement problem , long - distance correlations , and the meaning of the state vector are three of the foundational problems on which there has been and still is lively debate .it is fair to say that foundational studies have largely contributed to the burgeoning of quantum information theory , one of the most active areas of development of quantum mechanics in the past 25 years .quantum information is dependent on entanglement , whose significance was brought to light through the einstein - podolsky - rosen argument  .the realization that transfer protocols based on quantum entanglement may be absolutely secure has opened new windows in the field of cryptography  .and the development of quantum algorithms thought to be exponentially faster than their best classical counterparts has drawn great interest in the construction of quantum computers  .these face up extraordinary challenges on the experimental side . butattempts to build them are likely to throw much light on the fundamental process of decoherence and perhaps on the limits of quantum mechanics itself  .along with quantum information theory came also a reemphasis of the view that the wave function ( or state vector , or density matrix ) properly represents knowledge , or information  .this is often called the _ epistemic view _ of quantum states . onwhat the wave function is knowledge of , proponents of the epistemic view do not necessarily agree .the variant most relevant to the present discussion is that rather than referring to objective properties of microscopic objects ( such as electrons , photons , etc . ) , the wave function encapsulates probabilities of results of eventual macroscopic measurements . the hilbert space formalism of quantum mechanics is taken as complete , and its objects in no need of a realistic interpretation .additional constructs , like value assignments  , multiple worlds  , or bohmian trajectories  are viewed as superfluous at best .just like foundational studies have contributed to the development of quantum information theory , many investigators think that the latter can help in solving the foundational and interpretative problems of quantum mechanics .a number of proponents of the epistemic view believe that it considerably attenuates , or even completely solves , the problems of quantum measurement , of long - distance correlations , and of the meaning of the state vector .their arguments will be briefly summarized in sec .2 . i will then argue , in sec . 3 , that the epistemic view and related approaches fail to take into consideration that quantum mechanics has two very different explanatory functions , that of accounting for classically unexplainable correlations between classical phenomena , and that of explaining the microscopic structure of classical objects .in sec . 4 , i will ask the question of what it means to interpret quantum mechanics , or any scientific theory for that matter .drawing from the so - called semantic view of theories , i will argue than interpreting quantum mechanics means answering the question , `` how can the world be for quantum mechanics to be true ? ''i will conclude that so construed , an interpretation can hardly be dispensed with .let us first examine the arguments that advocates of the epistemic view offer to solve the foundational and interpretative problems of quantum mechanics .i should point out that they do not all attribute the same strength and generality to these arguments .some advocates believe that the problems are completely solved by the epistemic view , while others are of the opinion that they are just attenuated .this distinction , however , is not crucial to our purpose , and i will simply give the arguments as they are typically formulated .  the first problem that is addressed by the epistemic viewis the one of the interpretation of the state vector ( or state operator , or wave function ) .as the name suggests , the state vector is normally interpreted as representing the state of quantum systems .it is a matter of debate whether the state it represents pertains to an individual system or to statistical ensembles of systems  .but the epistemic view , which goes back at least to writings of heisenberg  , claims that it represents neither .it denies that the ( in this context utterly misnamed ) state vector represents the state of a microscopic system .rather , it represents knowledge about the probabilities of results of measurements performed in a given context with a macroscopic apparatus , in other words , information about `` the potential consequences of our experimental interventions into nature ''  .this is often set in the framework of a bayesian approach , where probability is interpreted in a subjective way .the epistemic view also addresses the notorious measurement problem . broadly speaking ,the problem is the following .suppose we want to describe , in a completely quantum - mechanical way , the process of measuring a physical quantity pertaining to a microscopic system . for simplicity ,assume that the spectrum of is discrete and nondegenerate , and that the normalized eigenvector corresponds to the eigenvalue .the measurement apparatus should then also be considered as a quantum system , which comes to interact with the microscopic system .let denote the initial state of the apparatus .the interaction will represent a faithful measurement of if the combined system evolves like represents a state of the apparatus wherein the pointer shows the value ( with if ) .if the schrdinger equation is universally valid , the combined evolution of the microscopic system and macroscopic apparatus is unitary ( assuming , unrealistically , that they form together a closed system ) .but then , an initial state involving the superposition of several eigenstates of an observable of the microscopic system evolves into a final state involving a superposition of macroscopically distinct states of the apparatus ( or of the apparatus and environment in more realistic situations ) .explicitly , one solution to this problem appeals to the collapse of the state vector  , in which the schrdinger equation breaks down and only one term of the macroscopic superposition ( e.g. ) remains .how does the epistemic view deal with the measurement problem ?it does so by construing the collapse of the state vector not as a physical process , but as a change of knowledge  .insofar as the state vector is interpreted as objectively describing the state of a physical system , its abrupt change in a measurement implies a similar change in the system , which calls for explanation .if , on the other hand , and in line with a bayesian view , the state vector describes knowledge of conditional probabilities ( i.e. probabilities of future macroscopic events conditional on past macroscopic events ) , then as long as what is conditionalized upon remains the same , the state vector evolves unitarily .it collapses when the knowledge base changes ( this is bayesian updating ) , thereby simply reflecting the change in the conditions being held fixed in the specification of probabilities .a third problem which is addressed by the epistemic view is the one of long - distance correlations  .consider the realization of the einstein - podolsky - rosen setup in terms of two spin 1/2 particles ( labelled and ) , where the state vector of the compound system is an eigenstate of the total spin operator with eigenvalue zero . in this case latexmath:[\[|\chi\rangle= \frac{1}{\sqrt{2 } } \left\ { |+ ; \mathbf{n } \rangle \otimes |- ; \mathbf{n } \rangle -  the first vector in a tensor product refers to particle and the second vector to particle .the vector , for instance , stands for an eigenvector of the -component of the particle s spin operator , with eigenvalue ( in units of ) .the unit vector can point in any direction , a freedom which corresponds to the rotational symmetry of  .suppose alice measures the -component of s spin and obtains the value .then she can predict with certainty that if bob measures the same component of s spin , he will obtain the value .if the state vector represents the objective state of a quantum system , it then seems that s state changes immediately upon alice s obtaining her result , and this no matter how far apart and are . since the word `` immediately '' , when referring to spatially separated events , is not a relativistically invariant concept , such a mechanism is not easy to reconcile with the theory of special relativity .  in the epistemic view , what changes when alice performs a measurement is alice s knowledge .bob s knowledge will change either if he himself performs a measurement , or if alice sends him the result of her measurement by conventional means .hence no information is transmitted instantaneously , and there is no physical collapse on an equal time or spacelike hypersurface .  related tothe epistemic view is the idea of _ genuine fortuitousness _ , a radically instrumentalist view of quantum mechanics .the idea `` implies that the basic event , a click in a counter , comes without any cause and thus as a discontinuity in spacetime ''  .indeed  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ i]t is a hallmark of the theory based on genuine fortuitousness that it does not admit physical variables .it is , therefore , of a novel kind that does not deal with things ( objects in space ) , or measurements , and may be referred to as the theory of no things .( p. 410 ) __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  such approaches to the interpretation of quantum mechanics are to be contrasted with realist views like bohmian mechanics  . hereparticles are taken to exist and they follow deterministic trajectories . for illustration , consider a set of nonrelativistic spin - zero particles interacting through a potential .the system s wave function is a solution of the schrdinger equation .we can write it in polar form as the equation of motion of particle is given by in general , is a nonadditive and nonseparable function of all particle coordinates .this can be shown to account for nonlocal effects like long - distance correlations .  in bohmian mechanics ,the statistical properties of quantum mechanics arise from an incomplete knowledge of the system s initial conditions .indeed the particles initial positions , although well - defined , are `` hidden '' to any observer .what is known is their statistical distribution , taken to be proportional to the absolute square of the wave function .it can be shown that bohmian mechanics exactly reproduces the statistical results of quantum mechanics .in particular  , interference fringes in young s two - slit setup follow directly from ( [ motion ] ) .if bohmian mechanics exactly reproduces the statistical results of quantum mechanics , are nt the trajectories superfluous , and should nt they be discarded ?the analogy has been made between such trajectories and the concept of the ether prevalent at the turn of the twentieth century  .h. a. lorentz and his contemporaries viewed electromagnetic phenomena as taking place in a hypothetical medium called the ether . from this , lorentz developed a description of electromagnetism in moving reference frames , and he found that the motion is undetectable . following einstein s formulation of the electrodynamics of moving bodies , the ether was recognized as playing no role , and was henceforth discarded .so should it be , according to most proponents of the epistemic view of quantum states , with interpretations of quantum mechanics that posit observer - independent elements of reality like bohmian trajectories .they predict no empirical differences with the hilbert space formalism , and therefore should be discarded .to examine how appropriate are the epistemic and related views of quantum mechanics , it is important to properly understand the explanatory role of quantum mechanics as a physical theory . although all measurements are made by means of macroscopic apparatus , quantum mechanics is used , as an explanatory theory , in two different ways : it is meant to explain ( i ) nonclassical correlations between macroscopic objects and ( ii ) the small - scale structure of macroscopic objects . that these two functions are distinct is best shown by contrasting the world in which we live with a hypothetical , closely related one .roughly speaking , the hypothetical world is defined so that ( a ) for all practical purposes , all macroscopic experiments give results that coincide with what we find in the real world , and ( b ) its microscopic structure , if applicable , is different from the one of the real world .let us spell this out in more detail .  in the hypothetical worldlarge scale objects , i.e. objects much larger than atomic sizes , behave just like large scale objects in the real world .the trajectories of baseballs and airplanes can be computed accurately by means of classical mechanics with the use of a uniform downward force , air friction , and an appropriate propelling force .waveguides and antennas obey maxwell s equations .steam engines and heat pumps work according to the laws of classical thermodynamics .the motion of planets , comets , and asteroids is well described by newton s laws of gravitation and of motion , slightly corrected by the equations of general relativity .close to atomic scales , however , these laws may no longer hold . except for one restrictionsoon to be spelled out , i shall not be specific about the changes that macroscopic laws may or may not undergo in the microscopic realm .matter , for instance , could either be continuous down to the smallest scales , or made of a small number of constituent particles like our atoms. the laws of particles and fields could be the same at all scales , or else they could undergo significant changes as we probed smaller and smaller distances .  in the hypothetical world one can perform experiments with pieces of equipment like young s two - slit setup , stern - gerlach devices , or mach - zehnder interferometers .let us focus on the young type experiment .it makes use of two macroscopic objects which we label and .these symbols could stand for `` emittor '' and `` detector '' if it were not that , as we shall see , they may not emit or detect anything . at any rate , and both have on and off states and work in the following way .whenever is suitably oriented with respect to ( say , roughly along the axis ) and both are in the on state , clicks in a more or less random way .the average time interval between clicks depends on the distance between and , and falls roughly as .the clicking stops if , as shown in fig . 1, a shield of a suitable material is placed perpendicularly to the axis , between and .if holes are pierced through the shield , however , the clicking resumes .in particular , with two small holes of appropriate size and separation , differences in the clicking rate are observed for small transverse displacements of behind the shield . a plot of the clicking rate against s transverse coordinate displays maxima and minima just as in a wave interference pattern .no such maxima and minima are observed , however , if just one hole is open or if both holes are open alternately .  at this stageeverything happens as if emitted some kind of particles and detected them , and the particles behaved according to the rules of quantum mechanics .nevertheless , we shall nor commit ourselves to the existence or nonexistence of these particles , except on one count .such particles , if they exist , are not in any way related to hypothetical constituents of the material making up , , or the shield , or of any macroscopic object whatsoever. whatever the microscopic structure of macroscopic objects is , it has nothing to do with what is responsible for the correlations between and .  in a similar way , we can perform in the hypothetical world experiments with stern - gerlach devices , mach - zehnder interferometers , or other setups used in the typical quantum - mechanical investigations carried out in the real world .correlations are observed between initial states of `` emittors '' and final states of `` detectors '' which are unexplainable by classical mechanics but follow the rules of quantum mechanics .we assume again that , if these correlations have something to do with the emission and absorption of particles , these are in no way related to eventual microscopic constituents of the macroscopic devices .  in the experimentsjust described that relate to the hypothetical world , quantum mechanics correctly predicts the correlations between and ( or other `` emittors '' and `` absorbers '' ) when suitable experimental configurations are set up . in these situations, the theory can be interpreted in ( at least ) two broadly different ways . in the first one ,the theory is understood as applying to genuine microscopic objects , emitted by and detected by .perhaps these objects follow bohmian - like trajectories , or behave between and in some other way compatible with quantum mechanics . in the other interpretation ,there are no microscopic objects whatsoever going from to . there may be something like an action at a distance . at any rate the theory is in that case interpreted instrumentally , for the purpose of quantitatively accounting for correlations in the stochastic behavior of and .  in the hypothetical worldwe are considering , i believe that both interpretations are logically consistent and adequate .of course , each investigator can find more satisfaction in one interpretation than in the other .the epistemic view of quantum mechanics corresponds to the instrumentalist interpretation .it simply rejects the existence of microscopic objects that have no other use than the one of predicting observed correlations between macroscopic objects .  in the worldin which we live , however , the situation is crucially different .the electrons , neutrons , photons , and other particles that diffract or interfere are the same that one appeals to in order to explain the structure of macroscopic objects .denying their existence , as is done in the approach of genuine fortuitousness , dissolves such explanatory power .denying that they have states , as is done in the epistemic view , leaves one to explain the state of a macroscopic object on the basis of entities that have no state .the epistemic and related views therefore fail to account for the second explanatory role of quantum mechanics . to reinforce this conclusion ,it is instructive to investigate what it means to interpret a theory .  with most physical theories ,interpretation is rather straightforward . but this should not blind us to the fact that even very familiar theories can in general be interpreted in more than one way . a simple example is classical mechanics .classical mechanics is based on a well - defined mathematical structure .this consists of constants , functions , and vector fields ( understood as masses , positions , and forces ) , together with the system of second - order differential equations .a specific realization of this structure consists in a system of ten point masses interacting through the gravitational force .a hypothesis may then assert that the solar system corresponds to this realization , if the sun and nine planets are considered pointlike and all other objects neglected .predictions made on the basis of this model correspond rather well with reality .but obviously the model can be made much more sophisticated , taking into account for instance the shape of the sun and planets , the planets satellites , interplanetary matter , and so on .  now what does the theory have to say about how a world of interacting masses is really like ?it turns out that such a world can be viewed in ( at least ) two empirically equivalent but conceptually very different ways .the first one consists in asserting that the world is made only of small ( or extended ) masses that interact by instantaneous action at a distance .the second way asserts that the masses produce everywhere in space a gravitational field , which then locally exerts forces on the masses .these two ways constitute two different interpretations of the theory .each one expresses a possible way of making the theory true ( assuming empirical adequacy ) . whether the world is such that masses instantaneously interact at a distance in a vacuum , or a genuine gravitational fieldis produced throughout space , the theory can be held as truly realized .similar remarks apply to classical electromagnetism .the mathematical equations can be interpreted as referring to charges and currents interacting locally through the mediation of electric and magnetic fields .alternatively , they can be viewed as referring to charges and currents only , interacting by means of ( delayed ) action at a distance  .  in thisrespect , quantum mechanics seems different from all other physical theories .there appears to be no straightforward way to visualize , so to speak , the behavior of microscopic objects .this was vividly pointed out by feynman  who , after a discussion of young s two - slit experiment with electrons , concluded that `` it is safe to say that no one understands quantum mechanics . [... ] nobody knows how it can be like that . '' but the process of interpreting quantum mechanics lies precisely in taking up feynman s challenge .it is to answer the question , `` how can the world be for quantum mechanics to be true ? ''if we adopt this point of view ( known as the semantic view of theories  ) , we can understand the function of bohmian trajectories or , for that matter , of other interpretative schemes of quantum mechanics .each provides us with one clear way that the microscopic objects can behave so as to reproduce the quantum - mechanical rules and , therefore , the observable behavior of macroscopic objects .it is true that , just like the ether in special relativity , they do nt lead to specific empirical consequences .but although they could be dispensed with in the hypothetical world of sec .3 , they can not in the real world unless , just like the ether was eventually replaced by the free - standing electromagnetic field , they are replaced by something that can account for the structure of macroscopic objects .in all physical theories other than quantum mechanics , there are straightforward and credible answers to the question raised above , of `` how can the world be for the theory to be true ? '' . in quantum mechanicsthere are a number of answers , for instance bohmian trajectories , multiple worlds , modal approaches , etc .none is straightforward , and none gains universal credibility .should we then adopt the attitude of the epistemic or related views , which decide not to answer the question ?i believe that , from a foundational point of view , this is not tenable . for how can we believe in a theory , if we are not prepared to believe in any of the ways it can be true , or worse , if we do not know any way that it can be true ?the epistemic view of quantum mechanics is an attempt to solve or attenuate the foundational problems of the theory .we have seen that it would succeed if quantum mechanics were used only to explain nonclassical correlations between macroscopic objects .but it is also used to explain the microscopic structure of macroscopic objects . interpreting the theory means finding ways that it can be intelligible .a number of proposals go a long way towards this , but much work remains to be done to make some of them sufficiently clear and precise .this work was supported by the natural sciences and engineering research council of canada .99 a. einstein , b. podolsky , and n. rosen , `` can quantum - mechanical description of physical reality be considered complete ? 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Abstract: since the beginning , quantum mechanics has raised major foundational and interpretative problems . foundational research has been an important factor in the development of quantum cryptography , quantum information theory and , perhaps one day , practical quantum computers . many believe that , in turn , quantum information theory has bearing on foundational research . this is largely related to the so - called epistemic view of quantum states , which maintains that the state vector represents information on a system and has led to the suggestion that quantum theory needs no interpretation . i will argue that this and related approaches fail to take into consideration two different explanatory functions of quantum mechanics , namely that of accounting for classically unexplainable correlations between classical phenomena and that of explaining the microscopic structure of classical objects . if interpreting quantum mechanics means answering the question , `` how can the world be for quantum mechanics to be true ? '' , there seems to be no way around it .   * does quantum mechanics need interpretation ? * + * louis marchildon * + dpartement de physique , universit du qubec , + trois - rivires , qc . canada g9a 5h7 + email : marchild.ca +


the anisotropic models occup an important place among various realistic models for study of the dynamical evolution of globular clusters . in frame of this modelsit have been work out the many evolution theories of the kinematics and dynamics of spherical self - gravitating systems ( e.g. , bettwieser & spurzem 1986 ; spurzem 1996 ; louis 1990 ; solanes & salvador - sole 1990 ; takahashi 1996 ) . at the same timehave been accumlated many observational data .these data mostly concerned star counts and surface brightness profiles , proper motions , stellar radial velocity dispersion , etc .( meylan & heggie 1997 ) . despite of the presence such great quantity information ,still there are difficulties in comparision problem the observational data with results of numerical simulations in the frame work of anisotropic models .one of causes of this phenomenon perhaps is concluded in that for their performe we must have at least the value of the anisotropy parameter , where and are radial and tangential components of velocity dispersion respectively ( ) . in general case it depends on radial distance and time .the aim of this work is to determine the value of anisotropy parameter for 10 globular clusters , to try find out the its dependence on limiting magnitude and on radial distance .we have used the method that was proposed by agekian and petrovskaya ( 1962 ) .let us assume : 1 ) the star ensemble has an equal - mass spectrum ; 2 ) the cluster is spherically symmetric and is in stationary state in the regular force - field ; 3 ) the cluster consists of single stars ; 4 ) and along radius ( so ) .then the cluster is described by means of the equation where and are undimensional radial distance and undimensional spatial density .a theoretical surface mass density profile is determined as where and are scale factors , is constant taking into account the infinity radius of the model and is the solution of ( 1 ) . from assumption 1 ) it follows that , where n is spatial number star density .then is theoretical surface number density of stars . hereconstants , and are determined by minimization of the function ^ 2 $ ] where n is number of star counts zones and is observed surface number density of stars .that value of that gives minimum is adopted as most probable value of a.we determined the value of anisotropy parameter for following 10 globular clusters : m3 , m12 , m13 , m15 , m55 , m56 , palomar 3 , palomar 14 , ngc5139 ( cen ) and ngc6535 .the results are given in table 1 .for most clusters we have , i.e. they have almost isotropic velocity distribution .it is interesting that for all clusters theoretical number density profiles are in good agreement with observational ones .but the theoretical density profile in the halo almost in all cases is lower than observational one .perhaps the cause is in the assumption about equal - mass .the fact is , as a result of mass segregation the low - mass stars prevail in the halo and consequently we overestimated mass density in this region . in realitythe mass density less than we suppose .for m56 at limiting magnitude we found .hence in this cluster the velocity distribution is strongly anisotropic yet .besides in order to find out an influence of limiting magnitude to value of , we calculated it at and found , i.e. if in this case the limiting magnitude decreases , then the anisotropy parameter decreases also .perhaps it is because , in first case ( ) it is expected the influence of low - mass stars to value of that generally are in halo and move along elongated orbits .  in order to find out the possible dependence of the anisotropy parameter on the radial distance we performed for ngc5139 ( cen ) additional calculations in three separate regions .the results show that in the central region , in middle region and in the halo . in this casethe agreement of the theoretical and observational density profiles essentially was improved than one for whole cluster .ccccc cluster & a & limit .& source of & note + & & or exp .time & observational data & + m3 & 0 & 10 min & king et al .1968 & + m12 & 0 & 9 min & king et al .1968 & + m13 & 0 & 10 min & king et al .1968 & + m15 & 0 & 30 min & king et al .1968 & + m55 & 0 & 2 min & king et al .1968 & + pal 3 & 0 & 30 min & king et al .1968 & + pal 14 & 0 & v=224 & harris & & + & & & van den berg 1984 & + ngc6535 & 0 & b=200 & peykov & & + & & & roussev 1988 & + m56 & 0.8 & b=215 & peykov & & + & & & roussev 1986 & + -//- & 0.3 & b=190 & peykov & & + & & & roussev 1986 & + ngc5139 & 0 & 30 min & king et al .1968 & + -//- & 0 & -//- & king et al .1968 & + -//- & 0.1 & -//- & king et al .1968 & + -//- & 0.3 & -//- & king et al .1968 & +  we think this way allowes to solve partly the problem of dependence of the anisotropy parameter on radial distance for any globular cluster .if we will make use of the more precise observational data , e.g. surface brightness profile recevied from hubble space telescope , then we can find the more precisely dependence of anisotropy parameter on radial distance in the concrete observed globular cluster .further we plan to study nostationary effects .

Abstract: the problem of determination of the velocity anisotropy parameter from star density profiles in globular clusters is considered . the calculations are performed for 10 globular clusters . the simple way for study of the dependence of anisotropy parameter on radial distance is used .   # 1_#1 _ # 1_#1 _ =  # 1 1.25 in .125 in .25 in


stochastic methods are lately being profusely used in quantum mechanics in general and in quantum optics in particular to better study and analyze the physical processes taking place during the interaction between the system and the measuring apparatus .the conceptual framework is very clear , consisting in an open quantum system which is being continuously monitorized by an appropiate device .this situation is very well described using stochastic unravellings for the lindblad equation of the system density operator , the most general expression of which for the diffusive case has been recently given in .needless to say , the presence of an observer is compulsory for orthodox quantum principles to be applied .on the other hand , in the early years of quantum mechanics , the role of the observer was crucial to both justify the indeterministic nature of the results of a measurement upon quantum systems and to understand the lack of quantum interference in a , say , double - slit experiment .nowadays it is understood that this lack of interference is not strictly due to the presence of the observer , but to the possibility of knowing which path the particle followed .keeping orthodox principles as far as possible , the role of the observer is then left as the only justification to set the indeterminism in the measurement process of quantum systems .  in this article , we take under consideration the possibility of reducing once more the role of the observer and study the consequences of letting quantum systems have an intrinsic stochastic nature . in the same fashion , this hypothesis has already given rise to different stochastic evolution models ( cf . and references therein to previous works ) , the main purpose of which was to reproduce faithfully all nonrelativistic quantum predictions with the sole exception of quantum superpositions among macroscopically distinguishable states .by concentrating upon a very simple quantum system , namely a two - level system , we will show how this stochastic assumption allows us to make predictions only restricted to relativistic extensions of quantum mechanics ( qed ) or only achievable by resorting to the theory of open quantum systems .we will restrict ourselves to the nonrelativistic domain , i.e. we will keep on using state vectors to describe physical systems , something which is impossible to do in an attempt to merge relativistic principles with quantum theory .the main physical hypothesis we address here is the possibility of endowing the evolution operator of a quantum system with stochastic nature . to be concretewe will deal with a two - level quantum system in particular .this assumption has already been done in another places ( cf .e.g. for a general overview ) and can be alternatively motivated in two ways . on one hand any system , whether quantum or classical , is continuously subjected to the external influence of its environment , however small the said influence may be .this standpoint is assumed in the program of decoherence , in fact it is its conceptual starting point .such external influences are , needless to say , uncontrollable , so we may encode them using stochastic methods .this way of proceeding is similar to for instance the modelling of the evolution of a lorentz particle . in van kampen s terminologyit corresponds to what he calls _ external noise _ which is due essentially to the huge amount of uncontrollable external factors that affect the system evolution .  alternatively , we may adopt the complementary point of view , stating that the stochasticity arises just as _ internal _ noise ( cf . againthis second standpoint is subtler and , in our opinion , much less intuitive but hopefully more accurate .let us use a canonical example ( which will be extensively studied below ) , a two - level quantum system in electromagnetic vacuum . within the non - relativistic quantum - mechanical standpoint the evolution of this systemis completely determined by its hamiltonian and since the latter is time - independent , the system is stationary , i.e. the energy does not change . within qed framework ,the description varies to include the effect of _ vacuum fluctuations_. these fluctuations , quantum in nature , drive the system to its ground energy level , thus producing _ spontaneous _ decay .this situation can also be understood letting the system be modelled by stochastic evolution , rooting the stochasticity in this quantum vacuum . since these fluctuations do not occur in real ordinary space , but in hilbert space we use stochastic evolution operators to describe them .the term _ internal _ is in this situation a bit dubious , since the electromagnetic vacuum can hardly be thought as internal to the two - level system .we should understand then _ internal _ as making reference to the essence itself of the system . as a matter of factthis vacuum is only detectable through its effects on a quantum system , so in certain sense the evolution of the system should also contain such effects .  from a mathematical point of view the question of the origin ofthe stochasticity is secondary and eveything reduces to find the form of such a stochastic evolution operator .so we proceed by substituting and then investigating the form of . to do this we resort to a general decomposition theorem of real random variables the generalization to ( bounded ) operator - valued random variables of which we take for granted .[ decompth ] let be an operator - valued random variable acting upon a n - dimensional hilbert space .then there exist operator - valued processes such that where denotes the expectation value with respect to the probability measure and are complex wiener processes .expressing the latter integral as a function of the upper interval limit ( ) we may write as the more general form for a stochastic evolution operator    where in general the wiener processes will have the covariance matrix given by  =a_{kk'}t.\ ] ]  the connection with ordinary quantum mechanics is made by stating    where is the system density operator induced by the previous stochastic evolution operator , namely    henceforward , we adopt the same assumptions as in the orthodox formalism , in particular we force trace - preserving evolution. this immediately imposes conditions on the absolute value of , as the following proposition shows :  [ expval ] if is defined as ( [ densop ] ) and , then where is a selfadjoint operator .appendix a.  a few comments should be included .firstly the selfadjoint operator is to be identified with the hamiltonian of the system .secondly note , as expected , that if , then we recover the usual quantum - mechanical formalism , thus reinforcing the idea that new effects are contained only in the stochastic part of the evolution operator , leaving the hamiltonian partial evolution unchanged .thirdly the square root imposes restriction upon since    must hold for all .finally is not an unitary operator , unitarity is only obtained in stochastic average , i.e.  =i.\ ] ]  this last feature is the crucial difference with other stochastic evolution schemes showing the same philosophy ( specially those assumed in and ) .there , as in the scheme we propose here , it is assumed a basic stochastic evolution described by two operators , namely the hamiltonian ( for the deterministic part ) and an _ ad hoc _ operator which gives rise to the lindblad operator after the whole calculation is carried on .nevertheless the stochastic nature appears only as a modification of the evolution operator generator .for instance example 30.1 of proposes the _ ansatz _  =\mathbb{e}[e^{iw_{t}l}\rho(0)e^{-iw_{t}l}],\ ] ]  thus obtaining a lindblad - type generator  =\frac{1}{2}\left\ { [ lx , l]+[l , xl ] \right\}.\ ] ]  this is straightforwardly generalized to more than one lindblad operator . in a simpler and more intuitive form much the same it is arrived at in .they only confer stochastic nature to the evolution operator by adding random parts to its generator , thus obtaining some further restrictions on the final evolution , such as only selfadjoint lindblad operators and consequently not achieving the desirable most general situation ( any lindblad operator ) .parthasarathy has however provided a scheme to obtain general lindblad evolution , namely he has proposed the evolution  =\mathbb{e}[u^{\dagger n(t)}xu^{n(t)}],\ ] ]  where is poisson process with intensity and .needless to say , though he arrives at a lindblad generator +[l,\cdot l^{\dagger } ] \right\} ] which spoils the markovianity .this on the other hand is also obtained in the more general quantum evolution of a subsystem in the orthodox formalism ( cf . or ) . the physical conditions to be met to guarantee markovianity must be found .the would - be lindblad operators are time - dependent .though time - dependent lindblad operators can be found in the literature ( cf . and references therein ) , this is not the usual case upon which we will focus .thus we must find the conditions under which on one hand =0 ] is shown to be negligible :  both =0 ] are independent sufficient conditions on their own to get energy conservation , i.e.    its elementary using ( [ difevoldensop ] ) with =0 ] for any election of , somarkovianity is straightforwardly achieved . to obtain time - independent lindblad operators ,we insert sucessively each in the definition of ( eq . ( [ lindblads ] ) ) . forcing time - independency sets an ordinary differential equation for whose solution leaves as the only option for the expression    the lindblad operators produced in this way are of the form    the physical interpretation is rather intuitive .if for instance we choose , then we obtain an energy - preserving decohering evolution denote the entry of the density matrix .] , i.e.  +\frac{\gamma}{2}{\left(\begin{array}{cc }                \hspace*{-1.5 mm } 0 & -\rho_{12}(t)\hspace*{-1.5 mm } \\                \hspace*{-1.5 mm } -\rho_{21}(t ) & 0\hspace*{-1.5 mm }                 \end{array } \right)}.\ ] ]  this means that represents a vacuum - system interaction which does not change the energy of the system , but that introduces decohering factors .notice that exactly the same equation ( [ onlydecoh ] ) would have been obtained had we chosen .the difference stems from the fact that whereas represents an interaction through coupling to the excited energy level , represents a similar interaction but coupled to the ground energy level .graphically the situation can be represented as in fig .[ decohfactors ] .  ,width=321 ]  this same interpretation can be carried over to the other two terms .for instance , for the final evolution equation is  +\frac{\gamma}{2}{\left(\begin{array}{cc }                \hspace*{-1.5 mm } -2\rho_{22}(t ) & -\rho_{12}(t)\hspace*{-1.5 mm } \\\hspace*{-1.5 mm } -\rho_{21}(t ) & 2\rho_{22}(t)\hspace*{-1.5 mm }                 \end{array } \right)}.\ ] ]  , width=226,height=188 ]  now besides the off - diagonal decohering terms a new effect appears : the system _ spontaneously _ decays into the ground state .this behaviour is indeed exactly what is obtained by orthodox means ( cf .e.g. and fig .[ spondecay ] ) .the reverse behaviour , as expected , is achieved if we choose . in these two casesthen the vacuum - system interaction is understood as an energy - non - preserving ( decaying or pumping ) interaction , depending upon whether the coupling takes place through the excited or the ground state .pictorially we may represent these effects as in fig .[ decpumpfactors ] .the whole results are collected in table [ onestpart ] for each of the previous stochastic parts .  ,width=321 ]  [ cols="^,^,^",options="header " , ]   now forcing trace preservation and using the cyclic property of the trace we may write  &=&{\textrm{tr}}\rho(0),{\nonumber}\\ & & \end{aligned}\ ] ]  which if it is to be valid for all implies    from this and making use of the polar decomposition of , we conclude    where is an arbitrary unitary operator which is expressed in the usual exponential form .the idea of the proof is to use the linearity of the space .we will firstly focus upon the canonical basis which we shall rename respectively as .then we calculate  ] transforms into    now we must restrict the form of , since it must be of the type , where is a density matrix ( positive selfadjoint matrix with unity trace ) .using again the linearity of , we may express then may be written as with    these show the following immediate properties  1 . 2 . 3 . 4 .  . in the first casewe get whereas in the second we get .then , in matricial form , may show the following aspects :  * if , then . * if , then . *if with at least three s not equal to , then .carrying these s to ( [ nullcond ] ) and taking into account the expression for s we obtain the conditions announced in prop .[ prop4 ] .the jaynes - cummings hamiltonian can be straightforwardly diagonalized by noticing that in the basis it adopts the matricial representation + from which we extract the eigenvalues ( )     these are the eigenvalues and eigenvectors of the dressed atom , which is commonly used in quantum optics ( cf . ) . notice that the set constitutes an orthonormal basis for the joint hilbert space of atom+field .

Abstract: we consider the simple hypothesis of letting quantum systems have an inherent random nature . using well - known stochastic methods we thus derive a stochastic evolution operator which let us define a stochastic density operator whose expectation value under certain conditions satisfies a lindblad equation . as natural consequences of the former assumption decoherence and spontaneous emission processes are obtained under the same conceptual scheme . a temptative solution for the preferred basis problem is suggested . all this is illustrated with a comprehensive study of a two - level quantum system evolution .