Under a Markov model of evolution, recoding, or lumping, of the four nucleotides into fewer groups may permit analysis under simpler conditions but may unfortunately yield misleading results unless the evolutionary process of the recoded groups remains Markovian. If a Markov process is lumpable, then the evolutionary process of the recoded groups is Markovian.ad hoc test statistic, which is based on an index that is evaluated using a bootstrap approximation of its distribution; one that is based on a test proposed specifically for Markov chains; and one using a likelihood-ratio test. We show that the likelihood-ratio test is more powerful than the index test, which is more powerful than that based on the Markov chain test statistic. We also show that for stationary processes on binary trees with more than two taxa, the tests can be applied to all pairs. Finally, we show that if the process is lumpable, then estimates obtained under the recoded model agree with estimates obtained under the original model, whereas, if the process is not lumpable, then these estimates can differ substantially. We apply the new likelihood-ratio test for lumpability to two primate data sets, one with a mitochondrial origin and one with a nuclear origin.We consider stationary, reversible, and homogeneous Markov processes on two taxa and compare three tests for lumpability: one using an Recoding may result in biased phylogenetic estimates because the original evolutionary process is not lumpable. Accordingly, testing for lumpability should be done prior to phylogenetic analysis of recoded data. RY-recoded nucleotide sequences, and Gibson et al. [Y-recoded nucleotide sequences. Before these studies, other investigators had used RY-recoded nucleotide sequences to infer the evolutionary relationships among mammals [When nucleotides intentionally are recoded to a 3- or 2-state alphabet in order to focus on a subset of the possible types of substitutions  or ren et al. , who dev mammals -3 and am mammals .Recoding of nucleotides and/or amino acids has been used repeatedly in recent phylogenetic studies -31. HoweIn this paper, we take a first step by considering tests for lumpability in a Markov model of evolution for pairs of homologous nucleotide sequences  into one or two new states --in molecular phylogenetics, this procedure has been called recoding [Let recoding . Table 1e NC-IUB . The 13 n independently evolving sites, which have diverged under Markovian conditions from their common ancestor on a rooted, 2-tipped tree. Let π0 denote the initial probability vector of the nucleotide frequencies, such that A, G, C, T . Over each edge of this tree, there is a substitution process, X(t) and Y(t), respectively, described by the transition probabilitiesConsider two nucleotide sequences, A and B, each with andijf(t) denote the theoretical joint probability of a site being in state i in A and state j in B at time t:Let F(t) = {ijf(t)} denote the joint probability matrix and let PX(t) and PY(t) denote the transition probability matrices of X(t) and Y(t). In practice, the two matrices cannot be identified from F(t) without some assumptions about the evolutionary processes of the sequences A and B. We assume that the processes are globally stationary, reversible, and homogeneous (SRH)  = PY(t) = P(t) and, if X πand Y πdenote the equilibrium probability distributions of the processes X(t) and Y(t), take X π= Y π= π0 = π and write Π = diag(π). Then, from (1), we get F(t) = P(t)TΠP(t). The transition probability matrix can be expressed by an instantaneous rate matrix R, such that P(t) = eRt, where ij ≥ R0 for i ≠ j, TπR = 0T, where T πis the equilibrium distribution of R [R = SΠ, where S is a symmetric matrix with ij ≥ s0 for i ≠ j, and R and P(t) can be written in terms of the eigenvector decomposition of Π/21SΠ/21 = LΛLT. In other wordsNow, let ons, see -39, and,ons, see ). Given ion of R . Further4sijπj/πi. The matandΛ is a diagonal matrix with columns containing the eigenvalues of Π/21SΠ/21 and L is a matrix with columns containing its right eigenvectors. The joint probability matrix is then symmetric andwhere S ) and three free parameters for π ; its expected value is thus E(N) = nF(t). Because the parameters s and π are in a one-to-one relation with the elements of F, the maximum-likelihood estimates of s and π can be obtained from the eigenvector decomposition of Let andwhere q <4, such that a lumped process, The following probabilities can be defined for any given and initial probabilities π, the lumped process, defined in (7), is a Markov chain whose transition probabilities do not depend on the choice of π. A necessary and sufficient condition for X'(t) to be lumpable with respect to a partition i in X'(t) is lumpable, then the transition probabilities for X'(t) for any given pair of subsets in By definition ,42, a Mae i in Sk,42. AccoP'(t) can be expressed as a matrix function of P(t) as follows:If the Markov chain is lumpable, the lumped transition matrix V is a 4 × q matrix, where q is the number of states in the lumped process, such that the l-th column of V is a vector with 1's in the components corresponding to states in l Sand 0's otherwise, andwhere q × 4 matrix whose k-th row is a probability vector with non-zero elements corresponding to the states in is a pability ,42 isIn the case of nucleotides, the second column of Table We note that under certain conditions, such as those considered by the JC  and F81 ad hoc test based on a parametric bootstrap for an index of departure from the lumpability condition [We consider three possible tests: An ondition ; a test ondition ; and a lFrom (8), if a Markov process is lumpable, thenshould have all elements zero. Consider the index proposed in :η ≥ 0, with η being 0 only under lumpable Markovian processes. Then, the hypothesis that the Markov process is lumpable is equivalent to the hypothesis H0 : η = 0. From the observed divergence matrix, N of two homologous sequences, assuming a SRH Markovian model of evolution, an estimate H0, whereIt is clear that and B matrices can be generated by simulation under conditions of lumpability, where we take b ∈ {1, ⋯, B}, to be independent and multinomial with parameters n and The distribution of P-value is then the probability that we obtain a value as large as or larger than the observed P-value is the proportion of The true χ2 test to determine whether a Markov chain is lumpable with respect to a partition ijn}, to estimate the transition probabilities ijp, where ij nrepresents the number of transitions into state j from state i in one step, so the number of steps in the Markov chain is ••n, where the subscript • indicates summation. Now, if we start from our divergence matrix N, where ij nrepresents the number of sites that are in state i for sequence A and state j in sequence B, and the SRH assumptions are kept, either A or B can be assumed to be the original sequence at time 0, whereas the other one can be assumed to be the observed sequence at time 2 (since we took the edge lengths to be 1). Take A as the ancestral sequence, then the divergence matrix N has the same properties as a transition count matrix, and we can proceed as described in [i to A vailable . The tesribed in . A transl = 1, ..., q. From the definition of a lumpable process (7), if the Markovian process is lumpable with respect to where k γis the number of states that are part of the subset k = 1, ..., q. Therefore, H0 : i ∈ N, estimates il gand where andwhere nd Baran  obtainedwhereandil − eil oare a stack of q tables of size 4 l × γof mean-corrected multinomials with row and column sums equal to zero) that the test statistic is distributed under H0 as a χ2 variable with and showed ((1) = exp(SΠ) and {ijf} = Fπ,s)((1). These matrices are obtained as shown in (5) and (6). We also want to estimate  under the constraints imposed by the null hypothesis of lumpablity, H0. The constraints are given in the second column of Table where {A, such thatThis maximization needs a new approach. We construct an orthogonal matrix, y is the response constraint vector, defined such that two values of y are zero corresponding to the two constraints. The matrix A will, in the case of partitions into two groups of two, contain π, so to emphasise this possible dependence, write y = g(s|π). Also write s = A−1y = g−1(y|π). Thenwhere s, if dependent of π, are optimized given the original π set, then the π vector is optimized given the optimized values of s. This process is repeated until convergence is achieved.The optimization process is done in two steps: the values of LR, can be calculated withFrom these two log-likelihood values, a log likelihood-ratio, × LR is distributed as a χ2 variable with 2 degrees of freedom.Under the null hypothesis of lumpability, 2 In order to check the accuracy of the tests under the null hypothesis, Monte Carlo simulations were done from a set of parameters that meets the assumption of lumpability. The parameter vectors in this case wereandn = 1500, 5000 divergence matrices were calculated by Monte Carlo simulations assuming that Ni is multinomial with parameters ) for i = 1, . . . , 5000.The joint probability distribution was calculated by the steps given in (2), (3), and (4); then, assuming a nucleotide sequence of length PP plot displaying the distribution of observed P-values, obtained from each test, plotted against the expected P-values, obtained from the uniform distribution. The linear relationship between these two sets of P-values  and values of s that yield increasing values of η, as in (9), generated 3000 divergence matrices using Monte Carlo simulation, and then calculated the three test statistics and their corresponding P-values, using the procedures explained above, for each value of η. The power at the 5% level, is then equal to the proportion of observed P-values less than 0.05.The power of each test was compared for each recoding scheme under non-lumpable conditions. To do this, we used RY recoding--similar power curves were obtained for the other 12 recoding schemes (results not shown). All of these results indicate that the likelihood-ratio test is the most powerful of the tests considered, followed by the Index test and, finally, by the Markov Chain test.Figure P-values. A PP-plot with these P-values should yield a straight line when the data are lumpable, and deviations from this expectation when the processes are not lumpable. However, the observed P-values are not independent, so we need to show that this condition is not cause for concern for the dots in a PP-plot to be on a straight line. We give a simplified argument taken from [P-values P1, . . ., nP, which, if all the null hypotheses are true, are identically distributed as uniform random variables on . Let p be any value between 0 and 1, let I(j < pP) take the value 1 if j < p Pand 0 otherwise, and let P-values less than p. Then E(I(j < pP)) = P (j < pP) = p and so the expected number of P-values less than p is E(pN) = np. This implies that the plot of the observed P-values will lie approximately on a straight line. The dependence will cause some clustering of the observed P-values but the PP-plot will remain useful in indicating whether there is evidence against some of the hypotheses.For general cases involving more than 2 homologous sequences, we can test for lumpability in all pairs of sequences by using the methods described above under the assumption that the evolutionary process is SRH over the whole tree . For example, in the case of an alignment with seven sequences, there will be 21 ken from . ConsidePP-plots shown in Figure RY recoding, on the tree shown in Figure The RY), then we can obtain Fq(t) = VTP(t)V and from this, using  and (4), we can further obtain the transition matrix of the process X'(t),If a process is lumpable with respect to a given recoding scheme  is not a Markov process and, although we can calculate Pq(t), it is not the transition matrix of the process X'(t).where P'(t) = UP(t)V can be defined and, if the process X(t) is lumpable, then P'(t) = qP(t). On the other hand, if X(t) is not lumpable, the elements of P'(t) are still given as P'(t) is no longer a transition matrix of X'(t). Conveniently, we can compare P'(1), the true conditional probability matrix at t = 1, with Pq (1), the false transition matrix at t = 1, thus allowing us to examine the effect of non-lumpability.In either case, the matrix T π=  and T s= . As can be seen from the π- and s-vectors, the lumpable condition is met for RY-recoding but not for KM-recoding. Figures RY- and KM-recoding schemes.Figure RY-recoded data is shorter than the corresponding edge in the tree obtained from the original data. However, because the original process was lumpable with respect to RY-recoding, the relative length of each edge in the two trees is the same, the difference being equal to a scale factorEvery edge in the tree obtained from the KM-recoded data is also shorter than the corresponding edge in the tree obtained from the original data, but the relative length of each edge in the two trees differ, the reason being that the process generating the original data was not lumpable with respect to KM-recoding.Every edge in the tree obtained from the Figures PP-plots from tests for lumpability for RY recoding, indicating non-lumpability, and AGY recoding, indicating lumpability.In a previous study , a set oPP-plot in Figure PP-plots from the likelihood-ratio test for lumpability for RY-recoding, showing strong evidence against lumpability, and for the SW-recoding, which provided the least evidence against lumpability. It is evident that no recoding should be applied to these data.In a previous study , a ~9.3kBias in estimates of phylogenetic parameters can occur when recoding of nucleotides or amino acids is used to transform data associated with models of evolution, which are not lumpable with respect to the recoding scheme used. A test proposed in this paper, which is based on a likelihood-ratio test, can yield an indication of whether the same results for estimable parameters can be expected from fitting a given model of evolution and its recoded version to the data.The authors declare that they have no competing interests.LSJ and JR conceived the project. KWL carried out the pilot study, showing that the Index test was feasible. VAVR and JR developed the tests and wrote the manuscript with input from LSJ and KWL. VAVR wrote the codes and generated the numerical results. All authors have read and approved the manuscript.


One of the most puzzling facts about cuprate high-temperature superconductors in the lightly doped regime is the coexistence of uniform superconductivity and/or antiferromagnetism with many low-energy charge-ordered states in a unidirectional charge density wave or a bidirectional checkerboard structure. Recent experiments have discovered that these charge density waves exhibit different symmetries in their intra-unit-cell form factors for different cuprate families. Using a renormalized mean-field theory for a well-known, strongly correlated model of cuprates, we obtain a number of charge-ordered states with nearly degenerate energies without invoking special features of the Fermi surface. All of these self-consistent solutions have a pair density wave intertwined with a charge density wave and sometimes a spin density wave. Most of these states vanish in the underdoped regime, except for one with a large d-form factor that vanishes at approximately 19% doping of the holes, as reported by experiments. Furthermore, these states could be modified to have a global superconducting order, with a nodal-like density of states at low energy. Lax2−SrxCuO4) cuprate family. For other cuprate families, the evidence for bond-centred unidirectional domains was found via scanning tunneling spectroscopy36789121414Ever since the discovery of the high-Tc superconductivity, many low-energy charge-ordered states in the cuprate have been discovered. Neutron scattering experimentst − J model1415t − J model and the validity of replacing the oxygen hole with a Zhang-Rice singletet al.22252627The existence of these great varieties of charge-ordered states has created a great debate regarding whether the strong coupling Hubbard model or the t − J type models have revealed low-energy intertwined states appearing as stripes or bidirectional charge-ordered states, such as checkerboard (CB). However, these works usually involved different approximations and parameters, which often resulted in different types of charge-ordered patterns, and these studies were mostly concentrated at a hole concentration of 1/8, which is the most notable concentration in early experiments. Hence, it is not clear if these results were the consequence of the invoked assumption or the approximation used, or if they are a generic results in the phase diagrams of cuprates. There were attempts to produce these CDWs or PDWs using a different approach, such as using a mean field theory to study a t − J-like model but taking the strong correlation as only a renormalization effect of dispersion222340YBa2Cu3Oδ6+) compoundsFor quite some time, various calculations29303132333435363738t − J model has several stripe states, with nearly degenerate energy as the uniform state and, with coexistent superconductivity and antiferromagnetism. When the number of variational parameters is extrapolated to infinity, the authors concluded that the anti-phase stripe, which has no net pairing, has slightly higher energy than the in-phase stripe with a net pairing, which in turn, also has slightly higher energy than the uniform state. This result is very consistent with the result of variational Monte Carlo calculationst − J model30t − J model with the variational approach using GWA.Amid all this confusion, recent numerical progress achieved by using the infinite projected entangled-pair states (iPEPS) methodt − J model. We find many unidirectional and bidirectional charge-ordered states with nearly degenerate energies as the uniform state, especially in the lightly doped regime; thus, it is a much more general phenomenon than previously thought. All of these states have intertwined orders of PDW, CDW and/or SDW. One of the CDW states, denoted as AP-CDW, reveals a bond order pattern with a much larger d-form factor than s′ symmetry, as found in the experimentBi2Sr2CaCu2Ox8+) and NaCCOC (Cax2−NaxCuO2Cl2). Furthermore, just as in the experimentd-form factor. For example, a different CDW intertwined with SDW and PDW, which is the familiar stripe reported long ago for 214122931s′ form factor, as reported in the experiment2Here, we report our findings from a much more extensive examination of the renormalized mean-field theory prediction using the GWA for the hole-doped t − J model. By using GWA, we can replace the strong constraint of forbidding the double occupancy of two holes on the same site in the variational wave function using Gutzwiller factors323344As mentioned above, the variational approach has been quite effective at capturing the physics of the strong correlation present in the δi, the local spin moment i is a site position and ij is the nearest neighbour bond. An iterative method is used to self-consistently solve the mean-field Hamiltonian HMF (Eq. (S7) in the −3 between successive iterations. All the calculations are performed on a 16 by 16 square lattice. To obtain various charge orders, specific patterns of δi, In our mean field theory, there are four variational order parameters. Besides the hole density a0), as listed in a0, with two domains of size 4a0 with opposite antiferromagnetic directions joining together. The pair field has more choices. It could always be positive, with all of its x-bond pair field being positive and y-bond pair field being negative: thus, it would have a net total non-zero pair field. This is called an in-phase (IP) state, with a period of 4a0. However, just like the spin moment, the pair field could also have two domains with opposite signs and a domain wall in between: this state is known as the anti-phase (AP) state, with a period of 8a0. Thus, we could have four possible states for each unidirectional CDW or bidirectional charge CB (cCB), as we either have an IP or AP pair field with or without SDW. However, we only have three such states in a0. Later, we will show a state with a net pairing order or IP pairing state and CDW, which occurs if we do not require solutions to be commensurate with the lattice.In addition to the two uniform solutions of a dSC state and a dSC-AFM state, there are many non-uniform charge-ordered states. For simplicity, we shall first present those charge-ordered states with a period of four lattice spaces  and reciprocal lattice vectors denoted by the “+” sign, the modulation wave vector is , and the vectors are shifted by the reciprocal lattice vectors. The peaks at  are determined by AS′, while those at  and  are determined by AD. The ratio of AD to AS′, or d/s′, is approximately 7.5 in this case. This ratio is quite special for the AP-CDW state. For the IP-CDW-SDW stripe, the ratio is actually less than one. The schematic plots of the Fourier transform of IP-CDW-SDW and AP-CDW-SDW stripes are shown in d/s′ is approximately 1.2. The Fourier transform of the bond orders of the AP-cCB pattern is similar to that of AP-CDW with a dominant d-form factor, as discussed in the SM.So far, we have only discussed the pair field, hole density and spin moment; now, we shall consider more carefully the bond order d-form factor with almost zero s′. The Fourier transform of its bond order is schematically shown in d-form factors, and red (blue) represents a positive (negative) value. The wave vector with a large amplitude is at 5π/8a0 or its period is approximately 3.2a0. This length is close to the separation between the domain walls of the pair field shown in The nPDW stripe shown in et al.Ox and Oy sites. Just like π/2a0, 0) is larger than that at  and both have the same sign but opposite sign with respect to . In their sample there are two domains with density modulation in x and y directions, respectively.d-form factor actually vanishes at an approximately 19% hole concentration, as shown in et al.π/2a0, 0), which is associated with the density wave. The density wave disappeares at approximately 19% doping. Moreover, this 19% hole concentration is conspicuously close to the so-called quantum critical pointAnother interesting result regarding the AP-CDW stripe is that its t − J model. Although they do not provide extremely accurate numbers, as many sophisticated numerical methods do, our results show that they do capture the most important physics of the strong correlation. This strong correlation provides a site-dependent Gutzwiller renormalization that produces many exotic solutions of PDW stripes and/or CBs intertwined with modulations of charge density and/or spin density. These results show quantitative agreement with some of the key experiments312kF is absent. Our model does not have the second or third neighbour hopping to provide a Fermi surface with nesting vectors or “hot spots”2240a0 so far, our preliminary study also finds charge-ordered states with periods of 5a0 and even 3a0. States with a longer period should be possible, and they could also have degenerate energies34J/t away from 0.3.The results reported above are all based upon the well-established renormalized mean-field theoryt and J that will alter the preferences of these states. For example, a weak electron lattice interaction could make the IP-CDW-SDW stripe much more stable against the dSC-AFM stateAn important consequence of having all these charge-ordered states originating from the same Hamiltonian and physics is that these states are not the usual “competing states” we are familiar with. They do not stay in a deep local minima in the energy landscape. They are actually quite fragile and can easily evolve into each other, as we have already demonstrated with the nPDW stripe, which evolved from a mixture of AP-CDW and an uniform d-SC state. Other examples of the mixture of stripes listed in t − J HamiltonianWe introduce the t, as our energy unit, is set to 1, and J is set to 0.3. σ is equal to ±. Si is the spin one-half operator at site i. The Fermi surface of the uniform state is quite simple, without nesting parts, and does not intersect with the magnetic Brillouin zone boundary, thus avoiding hot spots.where nearest neighbour hopping PG) with the Gutzwiller renormalization factors. The renormalized Hamiltonian now becomesFollowing the idea of Gutzwiller32δi:where v is used to denote that these quantities are different from the real physical quantities for comparison with the experiments. Their relationship is given in Eq. (S9). As for the Gutzwiller factors, we follow the work of Yang et al.32et al.where where How to cite this article: Tu, W.-L. and Lee, T.-K. Genesis of charge orders in high temperature superconductors. Sci. Rep.6, 18675; doi: 10.1038/srep18675 (2016).


T//) and field-like (T⊥) spin-transfer torque terms in the Landau-Lifshitz-Gilbert simulation, reliable and deterministic magnetization reversal can be achieved at a significantly reduced current density of 5×109 A/m2 under the co-action of electric field and spin-polarized current, provided that the electric-field pulse duration exceeds a certain critical value τc. The required critical τc decreases with the increase of T⊥ strength because stronger T⊥ can make the finally stabilized out-of-plane component of magnetization stay in a larger negative value. The power consumption for such kind of deterministic magnetization switching is found to be two orders of magnitude lower than that of the switching driven by current only.Effective manipulation of magnetization orientation driven by electric field in a perpendicularly magnetized tunnel junction introduces technologically relevant possibility for developing low power magnetic memories. However, the bipolar orientation characteristic of toggle-like magnetization switching possesses intrinsic difficulties for practical applications. By including both the in-plane ( E-field) effect58910121314161718Jc) as low as ~109 A/m2, which becomes the most challenging issue of high density memories. Meanwhile, reducing the STT switching current is also of particular importance to lower the energy consumption192021Jc , the typical Jc of the present memory devices is still of the order of 1010–1011 A/m2. Accordingly, new switching schemes are strongly expected for further reduction of power consumption.The effective manipulation of magnetization direction by electric current or voltage has attracted much attention for potential applications in nonvolatile spintronic devices. To date, different types of electric control methods have been developed: One is based on the spin-transfer torque (STT) effect induced by spin-polarized current1E-field effect23242526272829E-field switching scheme. For the Fe/MgO or FePt/MgO-based perpendicular tunnel junctions, the application of electric field alters the Fermi level position at the interface, which changes the orbital occupancy and interface magnetic anisotropy30313233E-field control of magnetism in principle does notE-field can be reduced by a factor of 500 when compared with the STT devicesE-field have been experimentally demonstrated in MTJs: one utilizes the coercivity change to assist magnetization reversal of the free layer26273637E-field pulse strength and duration (τ), the final magnetization state of free layer in the latter scheme case can be aligned either antiparallel (AP) or parallel (P) with the magnetization of fixed layer3738E-field pulse as short as tens of picosecondsIn order to reduce energy consumption, magnetization switching driven by either spin-orbit torques45E-field effect40E-field effect. In this paper, we performed simulations to reveal the magnetization switching behaviors driven by the combined action of electric field and STT effects. We find deterministic switching occurring at a much weaker current density but a larger pulse duration region which varies with the strength of field-like STT. The parameters of E-field and STT corresponding to such deterministic switching could be used for a new switching scheme with a dramatic reduction in energy consumption.Undoubtedly, it is essential to achieve reliable deterministic magnetization switching with low power consumption for advanced perpendicular STT-MRAMs. One promising approach is to employ both the STT effect and E-field) pulse is applied between the top electrode and the magnetic free layer separated by an oxide layer for tuning the PMA of free layer. The electric current is polarized by the fixed bottom ferromagnetic layer which exerts a spin-transfer torque on the free layer. We define the positive current as the electrons flowing from the free layer to the fixed layer. In our simulations, for the case of co-action of current and E-field pulses, the pulse length of injected current is 1 ns longer than the corresponding E-field pulse. Numerous studies have shown that the current-induced spin-transfer torque includes two vector components: the in-plane torque term142444546474950aJ and bj are the torque factors (see Methods). The typical material parameters are used for the CoFeB free layer0Ms = 1.4 T, Gilbert damping factor α = 0.2, thickness d = 1.4 nm, and spin polarization P = 0.3.The model structure we considered is a MgO/CoFeB-based perpendicularly magnetized tunnel junction. As illustrated in Hext is applied with its out-of-plane component to completely compensate the stray field generated from the polarizer layer, and its in-plane component is denoted as Hy . As we know, under the action of voltage pulse, the effective magnetic field Heff will change due to the reduced PMA. We suppose that the magnetic anisotropy field E = 1.28 V/nm)In this study, we suppose that an external field E-field only, STT only, or both E-field and STT. All the simulations start from an initial parallel configuration. E-field induced magnetization switching phase diagram, for which a voltage pulse of 1.8 V with various pulse durations were used. As we mentioned, since the direction of effective magnetic field Heff moves from z-axis to y-axis due to the PMA reduction caused by E-field, the magnetization will precess around the new Heff direction [see z = + 1) or switching to the AP state (mz = −1). So, the final state of the free layer depends on the voltage pulse duration and external field, and this type of magnetization switching is termed as “toggle-like” switching modeE-field pulse width and external field) is indispensable which holds a great challenge for experiments.We have carried out simulations on the magnetization switching driven by tion see . Once thJ and in-plane external field Hy . The current pulse duration is set as 15 ns. Here both the in-plane and field-like spin-transfer torques are considered. In the simulation ξ is chosen to be 0.02, which is proportional to the strength of the field-like torque. Clearly, a critical current Jc as high as 2.5×1011 A/m2, which changes slightly with the external field, is required to achieve magnetization reversal from P to AP state. Obviously, the P-to-AP switching cannot occur if the current is below Jc, for instance, J = 1×1011 A/m2, because the driving force by the STT alone with such low current cannot provide enough power to gyrate the free layer crossing over the energy barrier to another energetically favorite state. However, for the same low current density the P-to-AP switching can be achieved with the help of an additional E-field generated by a 1.8 V voltage pulse, as shown in E-field and current, the parameter region of the first blue-stripe (AP state) is enlarged for both cases with [T⊥) of STT, implying the expected deterministic AP state can be easily realized in this parameter region. This controllable AP region can be attributed to the STT effect which favors the AP state and dominates the magnetization dynamicsT⊥, another even larger deterministic AP region appears at the top-right corner of Reliable and deterministic switching is vitally important for practical applications, which can be realized by applying an enough strong current through the STT effect. ses with  or withoses with  the fielT// and T⊥ on magnetization switching, we show in E-field and current. The only difference is that only the in-plane T// is included in T// and T⊥ torques. The simulation parameters are taken as Hy = 23 mT and τ = 8 ns (1.8 V in strength), corresponding to the parameter points marked as the red stars in 11 A/m2 to generate the STT effect. Clearly, the STT alone in both cases cannot cause the magnetization to deviate away from its initial equilibrium state since such low current is far from enough to switch the magnetization [see t = 3 ns, the E-field pulse is added, which modifies the effective field, leading to the magnetization of free layer away from the initial +z direction and precessing around the in-plane +y direction [see E-field is removed, no switching takes place and the free layer magnetization returns back to the P state if the field-like toque is not considered. On the contrary, the magnetization switches to the -z direction (AP state) when both T// and T⊥ torques are included in the simulation model.In order to distinguish the roles of the two spin torque terms of tion see . At t = tion see . As soon11 A/m2, the external field of Hy = 23 mT, and the voltage of 1.8 V, but different E-field pulse durations. T// and T⊥ torques. We can see that the free layer magnetization goes back to the parallel configuration for the pulse duration τ = 1.5, 3.0, 4.6 and 6.0 ns, whereas it switches to the AP configuration for τ = 2, 4, 5, and 6.3 ns, respectively. This is a typical toggle-like switching mode. However, as the E-field pulse is longer than 6.3 ns, the final AP state can be observed only, thus producing a deterministic magnetization switching region. This feature has been further confirmed by continuously changing the pulse duration from 0 to 30 ns. The dependence of final magnetization state on the pulse duration is given in τc of E-field is 6.3 ns. For comparison, we have also performed simulations with the in-plane torque only. As shown in E-field pulse duration smaller than 7.8 ns, above which random magnetization switching is observed. This means that the in-plane torque alone cannot generate deterministic switching even though the E-field pulse duration is long enough to stabilize the magnetization along the in-plane direction.To further clarify that, we have performed a series of simulations using the same parameters including the current density of 1.0×10τc to trigger the deterministic magnetization switching is found to be closely related to the strength of field-like STT. T⊥ strengths characterized by a parameter ξ. In these simulations, the E-field pulse duration is fixed to be 13.5 ns and the other parameters are the same as those in T// torque only, the out-of-plane magnetization component mz exhibits damped precession which finally stabilizes around the mz = 0 axis during the electric pulse duration. The final P or AP state is dominated by the fluctuated mz value at the moment when the E-field pulse is just finished, which can be either positive or negative depending on the pulse duration but not on the ξ strength (ξ changes from 0 to 0.3). For the positive mz [stay in the cyan region, see mz the PMA forces it to the final AP state. This result is responsible for the observed random switching state shown in T// on the new deterministic switching region is very weak and can be ignored. However, when the T⊥ component is included, the mz value will be shifted downwards and keeps negative when the pulse duration is over a critical value, e.g. τc = 6.3 ns for ξ = 0.02. Such shift makes the mz stabilize at a negative value, which would certainly result in a deterministic AP magnetization switching as the pulse is removed. As a result, from the oscillation curves shown in τc, over which the mz stays negative and AP state occurs. The τc value relies on the ξ, which decreases down to 4.9 and 3.4 ns as ξ increasing up to 0.05 and 0.15, respectively. For clarity, the critical pulse durations are summarized in ξ.The required critical pulse duration mz, i.e. the equilibrium position of mz during the action of E-field, can be qualitatively analysed from the LLG equation governed by ξ = 0.02. In this case since the precession vector m is close to the y-axis, the direction of T⊥ is approximately along +x direction. The directions of T⊥ (brown arrows) and mz > 0), the T⊥ has a component parallel to z/dt which is proportional to the factor ξ, as shown in T⊥, which can reduce the radius of precession, i.e. the oscillation amplitude of mz, being in agreement with the results shown in mz < 0), because the T⊥ has a component antiparallel to mz gets larger with increasing ξ due to the partially canceled damping term. Consequently, in the presence of T⊥, the mz oscillation curves are shifted downwards, which eventually stabilizes at a negative value during the E-field pulse. Therefore, deterministic magnetization switching to the antiparallel state (mz = −1) can be achieved if the pulse duration is larger than τc.The effect of field-like torque on the stabilized E-field pulse durations at a field of Hy = 5 mT (left panel) or 23 mT (right panel), where ξ is chosen as 0.02 and 0.15. Clearly, with the combined effects, the critical switching current is significantly reduced. The magnetization switching can be achieved by two different switching modes: toggle-like switching  at the case of shorter E-field pulse and weaker current density, and deterministic switching (the continuous blue area) for the E-field pulse longer than a critical value in “S2” switching region or current density higher than a critical value in “S1” switching region  although the E-field pulse duration can be reduced down to 2 ns. In contrast, in the S2 region, the reliable switching can be achieved at a current density of 0.5×1010 A/m2 and an E-field pulse of 8 ns [see ξ can reduce the critical pulse duration τc, which has been observed in and 0.15 , while ag region . Obvious8 ns see . AdditioE-field power is estimated byI is the current for STT, V is the voltage for generating E-field pulse, R is the resistance of MTJ, t is the function time of STT or E-field. We use the typical experimental parameters for MgO-based MTJE-pulse of τ = 2 ns and a switching current of E-field effect. In S2 region of E-field pulse as τ = 8 ns, a current density E-field pulse. The total energy consumption in both cases are much smaller than the switching driven by STT effect alone, which is around 312 pJ for For practical applications, the power cost for driving the magnetization switching is a key factor to be considered. The required STT power is calculated according to E-field only is toggle-like, with final state either parallel or antiparallel with the reference layer, depending on the E-field pulse duration and external magnetic field. In contrast, reliable and deterministic switching can be realized by STT only under a quite strong current, which will limit the practical applications. By using a new switching scheme combining both the E-field pulse and STT current, the power cost for deterministic magnetization switching could be significantly reduced by two orders of magnitude if both the in-plane and field-like torques are considered. The deterministic switching can occur at a reduced current density of 109 A/m2 as the E-field pulse duration is larger than a critical value, which is found to decrease with the increase of field-like STT term. The reliable and deterministic magnetization switching with low power consumption promises potential advantages for the future spintronic devices, especially for low power writing of the MRAMs.In summary, we have investigated magnetization switching properties in MgO-based magnetic tunnel junctions driven by the electric-field effect and spin-transfer torque effect. The magnetization switching of free layer driven by The roles of electric field and spin-polarized current acting on the free layer are modeled by including a change of perpendicular anisotropy and spin-transfer torque terms, respectively, into the Landau-Lifshitz-Gilbert (LLG) equation. In this study, macrospin simulations are carried out by solving the LLG equation, which is taken as:M = mMs is the magnetization vector of the free layer, m is the unit magnetization vector, Ms is the free layer saturation magnetization, Heff is the effective magnetic field that includes the external magnetic field, anisotropy field and demagnetization field. h T is an instantaneous thermal fluctuation field and in this study we set the temperature T = 300 K. The third term is the in-plane spin torque, γ is the gyromagnetic constant, J is the electric current density, d is the free layer thickness, e is the electron charge, P is the spin polarization. The simulation code is developed in-house and has been used in our previous study38How to cite this article: Zhang, X. et al. Magnetization switching by combining electric field and spin-transfer torque effects in a perpendicular magnetic tunnel junction. Sci. Rep.6, 18719; doi: 10.1038/srep18719 (2016).


For decades, research in neuroscience has supported the hypothesis that brain dynamics exhibits recurrent metastable states connected by transients, which together encode fundamental neural information processing. To understand the system's dynamics it is important to detect such recurrence domains, but it is challenging to extract them from experimental neuroscience datasets due to the large trial-to-trial variability. The proposed methodology extracts recurrent metastable states in univariate time series by transforming datasets into their time-frequency representations and computing recurrence plots based on instantaneous spectral power values in various frequency bands. Additionally, a new statistical inference analysis compares different trial recurrence plots with corresponding surrogates to obtain statistically significant recurrent structures. This combination of methods is validated by applying it to two artificial datasets. In a final study of visually-evoked Local Field Potentials in partially anesthetized ferrets, the methodology is able to reveal recurrence structures of neural responses with trial-to-trial variability. Focusing on different frequency bands, the δ-band activity is much less recurrent than α-band activity. Moreover, α-activity is susceptible to pre-stimuli, while δ-activity is much less sensitive to pre-stimuli. This difference in recurrence structures in different frequency bands indicates diverse underlying information processing steps in the brain. Investigation of metastable states (MS) and transients of complex dynamical systems has become increasingly important over the last decades. In this context, dynamical systems spend longer time intervals in MSs than in transients between MSs. The large interest in studying such states comes from the belief that a complex temporal behavior of systems may be decomposed into a simple sequence of alternating MSs and transients between them. This reduced description is a model that captures the essential dynamic elements of rather complex underlying dynamics. Applications range from spin glasses  during a visual stimulus experiment allows to extract new insights into neural information processing. For instance, we show that temporal recurrence occurs in the α-frequency band but not in the δ-frequency band. This result suggests that in the α-band the brain processes the information step-wise (state by state) while no step-wise process is performed in the δ-band.In this work we propose a new method for the detection of metastable states in univariate neural signals. To obtain statistically significant evidence of recurrence structures in signals, we conduct a statistical test over the set of novel, frequency-selective recurrence plots (RP). Below we describe methodologies for building such frequency-selective RPs and performing statistical inference tests. These tests indicate how stable the recurrence plots are with respect to trial-to-trial variability. This novel statistical evaluation is necessary in the analysis of neurophysiological data, since trial-to-trial variability is a well-known experimental finding in such signals. In our work we analyse synthetic transient oscillations and one state variable of the Lorenz attractor involving acquisition noise to validate the methodology. Finally, the study of experimental Local Field Potentials obtained in partially anesthetized ferrets  ∈ ℝn, t ∈ ℝ is sampled at times t = iΔt, i ∈ {1, 2, …, N}, where Δt is the sampling time interval and N is the total number of samples. For notation simplicity, in this paper we denote the signal sample x(iΔt) by x(i). Then, a recurrence plot (RP) is defined as the N × N matrix R, whose elements ri, j take values ri, j = 1 when two trajectory samples lay within the open ball d is a distance function and i, j ∈ {1, …, N}. Hence, recurrence plots are two-dimensional binary matrices obtained by distance based thresholding and its elements take values ri, j ∈ {0, 1}. In this work, pixels in RPs will be color-coded white for values ri, j = 0 and black otherwise.Deterministic dynamical systems are described by their trajectory. A trajectory * which maximizes the symbolic entropy for a given distance function, as proposed in beim Graben and Hutt  in EEG recordings. To take into account the distinct signatures of spectral bands present in neural signals, we propose a novel concept for building recurrence plots from time-frequency signal representations, instead of building them directly from univariate data or constructing them by employing delay-embedding techniques. Such representations, in general adapted for non-stationary signal analysis, give insights into frequency bands of importance and provide additional flexibility to recurrence plot analysis that is not present in time-domain, for e.g., the possibility to weight the importance of some frequency bands. In the literature there are several ways to choose values of the frequency bands. We use the following frequency interval definitions: the δ-frequency band denotes the interval , the θ-frequency band the interval , α-band , β-band , and the γ-band denotes the interval .T univariate trials {x1, x2, …, xT} to their corresponding time-frequency domains; (ii) we compute the mean power of the spectrum over certain sets of frequencies. These mean power time series may be called S is the number of frequency bands; (iii) we compute recurrence plots by computing distances between vectors y(i) and y(j), i, j ∈ {1, …, N} as in Equation (1). In the following paragraphs we describe these blocks in more detail.We build novel recurrence plots in three steps, as shown in Figure The first block in Figure The second block in Figure Finally, in the third processing block in the figure we compute recurrence plots from the obtained time-frequency dataset as in Equation (1). If we do not explicitly mention otherwise, we use features from all the frequency bands to compute recurrence plots. In the experimental ferret dataset, we additionally present cases when recurrence plots are calculated from the single frequency band features such as δ- or α-frequency bands, since these bands play an important role in the loss of consciousness under anesthesia.To summarize, the proposed method for building recurrence plots from time-frequency representations grasps band-related features and allows flexibility in the analysis of particular frequency bands, which is not possible in the classical RP analysis. Our approach however requires additional computations of the synchrosqueezing transform and mean power of the spectrum.f(t) can be well approximated with K signal components, Ak(t) and e(t) represents a small error. We assume that the components fk have slowly time-varying amplitudes Ak(t) and sufficiently smooth IFs. These conditions assure that signal components are well separated in frequencies and the complete definition is available in Thakur et al.  be a square integrable and normalized function. Then, its scaled and time-shifted variants f at scale a and time shift b is defined by a. For example, for the first signal component with frequency ω1, the value of the wavelet coefficient Wf spreads around the scale factor ψ is the central wavelet frequency. Therefore, the estimated IF in the neighborhood of this value of the scale is equal to the frequency ω1. The synchrosqueezing transform T uses estimates of the instantaneous frequency ωf, computed for each scale-time pair  by ap (Δω) denote resolution steps in scale (frequency). Then, this transform, defined by q, b) in time-frequency plots, see Figures 5A,E, 6A,C.Let a wavelet ψ is defined in Equation (3), ωq are frequencies of one frequency band and t is time.For each frequency band with We study statistical properties of frequency-selective RPs obtained from time-frequency trial representations. By virtue of noise effects and an expected trial-to-trial variability, recurrence plot structures are expected to vary from trial to trial. To evaluate the recurrence plots statistically, we perform a statistical inference analysis based on a classical chi-squared test Yates, . To thisClassically, surrogate sets of univariate signals . To perform a chi-square test for categorical data, we build a two-by-two contingency table. For explanation, this tables first row takes values from the original RPs and the second row contains values from surrogate RP. The first table column marks the number of values ri, j = 1 and the second column the number of elements ri, j = 0. The elements of this table (two rows and two columns) have the coordinates , l, m ∈ {1, 2}. Then, the chi-square statistics for the pixel , i, j ∈ {1, …, N} is computed byl, m) for the pixel  and nr(l) is the total number of elements in the row l, nc(m) is the number of elements in the column m and q is the total number of elements in the two-by-two table. The calculated chi-square value is compared with the result in the chi-square table for predefined values of the degree of freedom df = 1 and the significance level αs = 0.05. If the calculated chi-square value is larger than the value in the table, the hypothesis that signals share the same distribution is rejected, see Yates  ≥ 0, i ∈ {1, 2, 3} is the activity rate of the element i, σn is the growth rate of the n-th population and ρi define interactions between elements. In our setup, σ1 = 1, σ2 = 1.2 and σ3 = 1.6, ρii = 1, ρ12 = 1.33, ρ13 = 1.125, ρ21 = 0.7, ρ23 = 1.25, ρ31 = 2.1, and ρ32 = 0.83. The output signal s(t) is a linear superposition of transient oscillations with frequencies ν1 = 170 Hz, ν2 = 20 Hz, ν3 = 75 Hz, where at one time instance, only one of these three components is dominant, see more details below. We point out that these frequencies are chosen rather arbitrarily for an optimal illustration. The activity rate xi defines the amplitude of the component ai with frequency νi and the output signal obeys1 = 0.5, η2 = 0.33, η3 = 0.4. By this construction, the amplitudes ai increase and decrease in a certain time window outside of which they almost vanish. These windows of the three oscillation modes i = 1, 2, 3 do not overlap and the transitions between them are rather rapid. The variable ξ(t) represents measurement noise and its random values are i.i.d. Gaussian noise with zero mean and variance 0.5. The sampling rate is 450 Hz. We generate 10 trials which are time-jittered by shifting the trials by 1 sample to later instances, while each trial is subject to additive noise different in each trial. A single trial is given in Figure A modified Lotka-Volterra model with x(t), which is the solution of the above given system of equations. This time series may represent a macroscopic measured signal such as EEG recording  at 47 cm distance from the animal. Each trial was 30 s long and consisted of three parts: (i) recording interval . Hence the system remains close to the resting state (t < 0) during the first stimulus. Conversely, Figure t = 2 s resembles the activity just after t = 2.5 s. These different findings for two trials are surprising since the experimental presentation of the visual stimulus is well-controlled and the stimulus is simple enough to expect almost identical neural responses.Figure To reveal the recurrent structure that is common in all trials, we now study the trial-to-trial variability of recurrence plots and aim to reveal whether the signal trials preserve the same dynamical behavior, cf. Figure t-test results). The figure reveals that there is no statistically significant recurrent structure in the δ frequency band in signals under study. Conversely, the α-frequency band exhibits significant recurrent structures in the single granular electrode in both datasets, cf. Figures session one the first response to the stimulus at t = 0 s returns at t = 1 s. Results for all frequency bands differ to results obtained in the α frequency band. The differences are dependent on the experimental sessions suggesting the presence of strong recurrences in bands different to α and δ or strong noise artifacts. To gain further insights into the dependence on frequency bands, we consider single trials which represent spatial averages of time series from adjacent granular layers. This average denoises the time series. Figures t = 0.5 s indicates a MS common to all data with t ≥ 0.5 s. Additional recurrences occur in the time intervals , , and  ms. These results are consistent in the chi-square significance tests and the t-test involving corrections for multiple comparison. At last, we mention the prominent lack of recurrence in the baseline time interval observed in a single electrode in session one, cf. Figure session one nor in the spatially averaged data shown in Figures Figure subset one and the subset of “strongly spatially filtered version of foxes” denoted by subset two. Subset one includes the trials {4, 6, 9}, while subset two is composed of the trials {1, 5, 10} of datasets session one and session two. The comparison of various pre-stimuli data is done by the chi-squared difference measure based on recurrence plots of both subsets. Figure t = 0 s) and during the stimuli, for t ∈  s. In α-band, the figure shows more prominent recurrences, such as the patterns around t = 1.5 and 2.7 s. We point out that recurrences within δ- and α-bands do not overlap, except in the pre-stimuli period, for t ∈ {−0.5, 0} s. Finally, considering all the frequency bands together does not reveal significant similarities of two pre-stimuli. Results from the chi-square test and the t-test involving multiple comparison correction are similar. However, it is interesting to note that the t-test reveals more significant common recurrences than obtained with the naive chi-square test.The experimental paradigm includes visual presentations of stimulus types in a randomized order. The previous paragraphs show neurophysiological responses to the sine-wave stimulus only. To gain further insight into the trial-to-trial variability subject to various pre-stimuli, we have selected two subsets of sine-wave trials that have two different preceding stimuli, namely the “black screen” subset denoted by The present work introduces a new recurrence analysis methodology for univariate time series. The first new element is the transformation to a time-frequency representation leading to a multivariate time series of spectral power. This new technique generates a new high-dimensional phase space in which the instantaneous power of the signal evolves. This high-dimensional phase space is mandatory to apply recurrence analysis. In addition, it permits to compute recurrence plots for specific frequency bands. The second new element is the statistical analysis of recurrence plots that takes into account spurious recurrence structures and allows to suppress them. The combination of the two methods permits to extract temporal recurrence structures in data which may reflect underlying transient dynamics in a certain range of frequencies that would have been hidden in conventional methods. To our best knowledge these two techniques have not been considered before.The first results for two artificial datasets illustrate the methodology and indicate that method detects recurrences in a variable dataset  by the statistical analysis as seen in Figure The subsequent analysis of single Local Field Potentials measured experimentally in ferret visual cortex reveals a high trial-to-trial variability, cf. Figure The effects of pre-stimuli have been hypothesized . It turns out that this conventional method does not provide high-quality extraction of transient recurrent structures, given by the reassignment method. This may result from the worse time-frequency resolution of conventional Morlet wavelets. Future work will further investigate the best choice of multi-resolution time-frequency methods. Moreover, the methodology considers surrogate data generated by a temporal random shuffling of data and hence destructing all temporal structure. Future work may include the destruction of the recurrence structure by phase randomization in certain frequency bands .We provide the time-series of the transient oscillation dataset and Lorenz dataset on the webpage of the corresponding author (The majority of the analysis steps and the implementations have been performed by TT. KS and FF have provided the experimental data and neurophysiological insights. MF has contributed additional time-frequency analysis results and PB and AH have conceived the study. All authors wrote the manuscript together.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Reserving time slots for urgent data, such as life-critical information, seems to be very attractive to guarantee their deadline requirements in wireless body area sensor networks (WBASNs). On the other hand, this reservation imposes a negative impact on performance for the utilization of a channel. This paper proposes a new channel access scheme referred to as the contention over reservation MAC (CoR-MAC) protocol for time-critical services in wireless body area sensor networks. CoR-MAC uses the dual reservation; if the reserved time slots are known to be vacant, other nodes can access the time slots by contention-based reservation to maximize the utilization of a channel and decrease the delay of the data. To measure the effectiveness of the proposed scheme against IEEE 802.15.4 and IEEE 802.15.6, we evaluated their performances with various performance indexes. The CoR-MAC showed 50% to 850% performance improvement in terms of the delay of urgent and time-critical data according to the number of nodes. To display human physiological status, these biosensors continuously monitor and transmit health conditions to a coordinating device wirelessly, forming a WBASN.Spurred by the rapid convergence of key technologies, such as physiological sensors, wireless communication and low power micro-electro-mechanical systems (MEMS), a new generation of wireless body area sensor networks (WBASNs) is emerging. These WBASNs exploit special-purpose implantable biosensors that aim to provide a large variety of applications, including computer-assisted rehabilitation, early detection of medical conditions, priority. There have been a large number of studies on priority-based mechanisms [Although the applications of these WBASNs are attractive, there are still a large number of challenges to be tackled for providing reliable and efficient WBASN systems. One of the important challenges is to provide urgent data transmissions for life-critical applications. Techniques for urgent data transmissions have traditionally been based on chanisms ,3,4,5,6 In addition to the above schemes, a number of studies related to WBASN are based on the IEEE 802.15.4 technology ,10,11. Tparametric [channelization-based [hybrid [For WBASNs, the few proposals for urgent data transmission in terms of priority-based MAC can be classified into three categories: rametric ,14,15, con-based ,17,18,19 [hybrid ,21 appro [hybrid . The chaIn this paper, we propose a hybrid priority-based MAC protocol for WBASN, referred to as contention over reservation MAC (CoR-MAC). In the CoR-MAC, each time slot is dedicated to a node that might transmit urgent data. However, dedicated time slots waste wireless resources. When the dedicated time slots are not used, the other nodes cannot use those time slots. To solve this problem, in the CoR-MAC, data can be transmitted over the time slot, which is dedicated to another node and not in use.The rest of this paper is organized as follows. We survey the existing related work in et al. [i.e., the AC traffic has the highest priority). To differentiate AC and CD traffic, a unique backoff exponent (BE) is configured for each. In other words, the AC has more transmission to opportunities than the CD, since the AC has a smaller BE. Nevertheless, there is no way that the AC can preempt the CD when the CD has already decreased its backoff counter to be smaller than that of the AC. Zhang et al. [et al. [According to the taxonomy mentioned in et al.  proposedg et al.  proposed [et al.  suggesteet al. [et al. [et al. [et al. [Zhang et al.  proposed [et al.  proposed [et al.  proposed [et al.  proposed [et al. , the priet al. [opportunistically. In other words, non-urgent data perform clear channel assessment (CCA) in the ETS and check for any urgent data transmission with shorter inter-frame space (IFS) and, thus, higher priority than the non-urgent data. Although an emergency alarm can occur during the DTSs, priority is given to non-urgent data in the PNP-MAC. Zhou et al. [As mentioned in et al.  suggesteu et al.  proposedu et al. . Since Iu et al. ,20,22,24u et al. . IEEE 80As we can conclude from the above data, research for WBASNs concentrates on urgent data transmission with a guaranteed time slot for urgent data. However, these research works are not considered to be wasting network utilization caused by the non-transmission of urgent data on the slot. The CoR-MAC, as we mentioned, can resolve this problem with the dual reservation scheme that allows transmitting within unused slots for urgent data.N sensor nodes  consisting of one coordinator , electroencephalogram (EEG), electromyographic (EMG) and medical video data. The non-time-critical data include messaging and traffic service, as in . SimilarNo non-emergency traffic can be transmitted during an inactive phase that is designed for energy reduction. Although transmission is not allowed, this phase can be utilized for emergency data transmission, as in . TransmiThe beacon phase is the time in which only the coordinator is allowed to transmit a beacon frame, and it is used for network synchronization and advertising information, such as time slot allocation. The beacon frame includes superframe length, subsequent superframe start time, beacon length, CFP length, CAP length, length of the CFP time slot, network ID, the number of time slots in the CFP and the CFP allocation map, including slot number and node ID. The size of the beacon is varied according to the size of the CFP allocation map. If the number of nodes is 10∼15, the size of the beacon will be around 228∼278 bits when the size of the CFP allocation information per node is 10 bits (five bits for the number of slots and five bits for the node’s ID).Urgent data can be transmitted during either the CFP or CAP. A detailed description of urgent data transmission is described below.The time-critical data can be transmitted during CAP or the reserved time slot in CFP, as long as urgent data are not being transmitted.Non-time-critical data can be transmitted during either CFP or CAP, as long as urgent, time-critical or other non-time-critical data are not being transmitted.To accommodate all three traffic types in …, M is the maximum allowable number of time slots in the CFP and is selected based on the number of nodes that generate urgent data. We call this phase the self CFP (SCFP). Further, to increase the transmission opportunities, urgent data of opportunistic CFP (OCFP). To prioritize Generally, transmitting urgent data over a CAP is not desirable, since timely transmission is not guaranteed due to contentions. Therefore, it would be better to transmit urgent data over a CFP by reserving a dedicated slot. However, the dedicated time slot might not be frequently utilized since urgent traffic is generated sporadically in general, resulting in degraded throughput. Therefore, in the CoR-MAC, the CFP is divided into time slots i.e., i.e., a reservation request command for time-critical data is transmitted over the CAP. To prioritize urgent data against time-critical data, time-critical data wait for the medium inter-frame space (MIFS) before transmitting a frame, where MIFS > SIFS. In other words, even if time-critical data are reserved at Moreover, in the CoR-MAC, In addition to the CFP, the CAP is used for the transmission of urgent data and non-time-critical data. In this way, urgent data have ample opportunity to access a channel. Using this mechanism, our scheme can transmit the urgent data significantly faster than the method in .Initially, there is no CFP, and the CAP is extended to the end of the beacon phase, as shown in  network c. S2, S3M, the coordinator allocates The CoR-MAC algorithm for the coordinator is shown in Algorithm 1. When Algorithm 1 The CoR-MAC algorithm for the coordinator. 1: 2: 3:M← The maximum allowable number of time slots in CFP; 4:i’-th position; 5:i; 6:loop 7:if Current time = start of beacon phase then  8:  Send beacon with CFP allocation map 9:end if 10:ifthen 11:if # of time slots in the CFP < Mthen  12:   Allocate 13:   Update CFP allocation map14:else  15:   Reject the join request16:end if  17:end if 18:if DATA is received from then 19:  Send ACK to 20:elseif RTS is received from then 21:  Send CTS to 22:end if 23:end loopAlgorithm 2 The CoR-MAC algorithm for the node. 1:i’-th position; 2:Φ ← The current time; 3: 4: 5: 6:loop 7:if BEACON is received from the coordinator then  8:  Synchronize itself to the coordinator; 9:  Record this 10:elseif The node has urgent data to transmit then 11:if Φ ⊂ then  12:   Send urgent DATA immediately;13:elseif Φ ⊂ then  14:   Send DATA with CSMA/CA after SIFS;15:elseif Φ ⊂ then  16:   Send DATA with CSMA/CA;17:end if  18:elseif The node has time-critical data to transmit then 19:if Φ ⊂ then  20:   Send DATA after MIFS;21:elseif Φ ⊂ then  22:   Send DATA with CSMA/CA;23:end if  24:elseif The node has non-time-critical data to transmit then 25:if Φ ⊂ then  26:   Send DATA with CSMA/CA after LIFS;27:elseif Φ ⊂ then  28:   Send DATA with CSMA/CA;29:end if  30:end if 31:end loopThe CoR-MAC algorithm for the sensor node is shown in Algorithm 2. When a beacon is received from the coordinator, nodes synchronize themselves to the coordinator and record this In this section, we analyze an urgent data delay of the proposed scheme with a theoretical approach. We show that our scheme guarantees the delay requirement of the urgent data. In the following, we describe our assumption and system model to obtain the expected delay in M) is less than or equal to the number of nodes N. In the CoR-MAC protocol, non-urgent data can be transmitted when no node has urgent data in CFP. Therefore, we do not consider non-urgent data in CFP. Denoting We assume that each node generates both urgent data and non-urgent data. Note that the arrival rates of urgent data follow exponential distributions with L as the number of G as the rate of attempted transmissions, including retransmission, new arrivals, urgent data and non-urgent data. Although G depends on time in reality, we assume that G is a constant value in our analysis.We assume that CAP consists of a number of system time slots . Therefore, each node has an opportunity to transmit if the slot owner does not have urgent data. If the slot owner has urgent data, the other nodes wait N’-th slot in If the slot owner of the slot does not have urgent data, each node with urgent data vies with other nodes. If a node wins contention and transmissions are successful in the slot, then the delay is Equation , where PL is the number of CAP consists of a number of Equation .(6)D, res, resD; and the energy consumption.versus the arrival rate of the number of nodes. The delay of IEEE 802.15.6 increases from 11–12 nodes since IEEE 802.15.6 has an exclusive access phase (EAP) for the processing of urgent data and time-critical data. The EAP is efficient for processing urgent data and time-critical data, but cannot process non-time-critical data. In addition, the CoR-MAC and IEEE 802.15.4 can process non-time-critical data during the entire superframe. With a high arrival rate, the non-time-critical data delay of all schemes increases extremely, since those schemes have lower priority of non-time-critical data, and those MACs are unable to process non-time-critical data.versus the number of nodes, respectively. The delay of IEEE 802.15.4 is higher than that of the other schemes because IEEE 802.15.6 has EAP. Furthermore, CoR-MAC has a dual reservation scheme for urgent data and time-critical data. In any environment, CoR-MAC has a shorter delay than the other schemes, since the dual reservation scheme allows use of a slot even if the slot is owned by another node. IEEE 802.15.6 and IEEE 802.15.4 also have reserved slots; however, in these schemes, a slot cannot be used if it is owned by another node.For urgent data, the goal is to guarantee a delay that is below a given deadline (20 ms), not that the delay is the minimum possible value. Therefore, we measure the percentage of packets that exceeds the required deadline. The throughput is measured by the sensor nodes. When a node receives an ACK message from a data frame, it records the throughput with the size of the data frame. The utilization of a phase is measured by a node. When the node successfully transmits a data frame, the node records a utilization in that phase. Remark 1. As previously mentioned, the contention window of CoR-MAC is divided into SIFS, MIFS and LIFS. This will lead to overhead, such as the increasing of each time slot’s size and extra energy consumption. As shown in i.e., the slots are dually reserved) in order to increase the overall throughput.In this paper, we addressed the problem of the delay of urgent data and time-critical data. We also addressed the utilization of dedicated slots. Existing WBASN schemes cannot properly resolve the above problems and do not address heavy traffic load situations, since they do not consider urgent data transmission. To reduce the delay of urgent data transmission, we proposed a contention over reservation MAC protocol for WBASNs. Our protocol exploits CFP reservation among nodes. For the reservation, traffic is divided into urgent, time-critical and non-time critical data, in descending order of priority. Each time slot in the CFP is dedicated to a node that might transmit urgent data. To increase transmission opportunities, urgent data can be transmitted over the unused CFP time slots. Since the urgent data are sporadically generated, the slots of CFP can be further reserved for time-critical data (The simulation results show that our protocol is very effective and provides significantly shorter delay by providing opportunistic transmission chances. In the simulation, CoR-MAC shows an approximate 50%–850% better performance in the delay of urgent and time-critical data. Furthermore, our protocol provides flexible power consumption compared to the existing protocols.


A common problem in phylogenetic analysis is to identify frequent patterns in a collection of phylogenetic trees. The goal is, roughly, to find a subset of the species (taxa) on which all or some significant subset of the trees agree. One popular method to do so is through maximum agreement subtrees (MASTs). MASTs are also used, among other things, as a metric for comparing phylogenetic trees, computing congruence indices and to identify horizontal gene transfer events.https://code.google.com/p/mfst-miner/.We give algorithms and experimental results for two approaches to identify common patterns in a collection of phylogenetic trees, one based on agreement subtrees, called maximal agreement subtrees, the other on frequent subtrees, called maximal frequent subtrees. These approaches can return subtrees on larger sets of taxa than MASTs, and can reveal new common phylogenetic relationships not present in either MASTs or the majority rule tree (a popular consensus method). Our current implementation is available on the web at Our computational results confirm that maximal agreement subtrees and all maximal frequent subtrees can reveal a more complete phylogenetic picture of the common patterns in collections of phylogenetic trees than maximum agreement subtrees; they are also often more resolved than the majority rule tree. Further, our experiments show that enumerating maximal frequent subtrees is considerably more practical than enumerating ordinary  frequent subtrees. A phylogenetic tree is an unordered rooted tree whose leaves are in one-to-one correspondence with a set of species ; its topology represents the hypothetical evolutionary relationships among these species.agreement subtree (AST) for a collection of phylogenetic trees on a common leaf set is a minimal subtree connecting a fixed set of leaves that is homeomorphically included in all of the input trees. A maximal agreement subtree (MXST) is an agreement subtree that is not a subtree of any other agreement subtree. An MXST is a maximum agreement subtree (MAST) if it has the largest number of leaves  is an agreement triplet if all the input trees display the same topology over the leaf set {i,j,k}. For an ordered pair of trees  in an equivalence class, having a and b as their respective rightmost leaves, we say tree Tabexists if  join as Tab across all the input trees. Let I, J and K be three trees belonging to a common equivalence class; let i, j and k be their respective rightmost leaves. Suppose that Tij and Tik exist. If the ordered pair  exhibits a common join across all the input trees, we denote the join as Ti−jk. Theorem 2 characterizes pruning among “siblings” and “first cousins” in the enumeration tree.For the next set of results, let X prunes Y if either of the following holds: 1. Txy exists and is not of join type 2.(a) Txy exists as a join of type 2 and for every child Z of E such that Tyz exists,  is an agreement triplet.(b) Txy and Tyz exist, and Txy is of join type 2, then Tyz is not of join type 2.2. If Part 1 of Theorem 2 deals with pruning among siblings; part 2 deals with pruning among first cousins. The proof of Theorem 2 relies on Lemmas 1, 2, 3 and 4, which we present next.Lemma 1. Suppose that Txy and Tyzexist. Then, the following holds:1. If Txyis not a result of a type-2 join:(a) Txzexists and is not a result of a type-2 join.(b) There exists an AST T on leaf setandis a descendant of X.2. If both Txy and Tyzare results of join type 2, and  is an agreement triplet:Txzexists.(a) (b) There exists an AST T on leaf setandis a descendant of X.3. If Txyis a result of a type-2 join and Tyzis not a result of a type-2 join:(a) Txzexists and is not a result of a type-2 join.(b) There exists an AST T on leaf setandis a descendant ofTxy−z exists.Y, X has a descendant  whose core tree displays Tyx. Intuitively, this is an intermediate step in proving conditions under which X either prunes Y or Y or Txy and Tyz can be a result of one of the four types of join. Thus, considering Txy and Tyz together, there are 16 possibilities. Out of these, the case where each Txy and Tyz is result of a type-2 join is the only case where there can be multiple topologies that display both Txy and Tyz. That is why type-2 joins have a special significance in Lemma 1. The remaining 15 cases guarantee the existence of a single topology that displays both Txy and Tyz.Lemma 1 gives conditions under which for a given child Lemma 2. If Txyand Txzexist, and  is an agreement triplet, either Tx−yz or Tx−zy exists.X with y and z as the rightmost leaves of their respective core trees, the existence of agreement triplet  is a sufficient condition for the children to join in one of the two possible ways. That is, there exists only one topology that displays both the children; further, the topology must be a descendant of X.Lemma 2 states that for any two children of Lemma 3. Suppose A and B are two ASTs with a and b as their respective rightmost leaves such thatThen,is a descendant ofif and only if for everyTaℓexists.Lemma 4. Suppose Txyexists as a result of a type-2 join and D is descendant of Y. Then, for anyx,a,b] is an agreement triplet if  and  are agreement triplets.Txy. Intuitively, it states that for any two children of Y with a and b as the rightmost leaves of their respective core trees, agreement triplets  and  together form a sufficient condition for the existence of a descendant of X whose core tree displays the core trees of both children of Y. This is a stepping stone in proving that for every descendant D of Y, there exists a descendant of X whose core tree displays the core tree of D, thus, X prunes Y.Lemma 4 deals with the special case of type-2 joins for http://evolution.genetics.washington.edu/phylip/newicktree.html). E.g., ) represents the tree with leaf set {a,b,c} where the LCA of b and c is a proper descendant of the LCA of a, b and c, and  represents unresolved (star) tree on {a,b,c}.The proofs of the above Lemmas are given in Appendix: Proofs. We present next the proof of Theorem 2. In the proof, and in the rest of the paper, we represent trees in parenthesized Newick format  Txa∈X and Txb∈X exist, and each is also not a result of a type 2-join, and (ii) there exists ASTs on leaf sets x,y,a] and  are agreement triplets. Let A and B denote the children of E with a and b as their respective rightmost leaves. Since AST S exists, there exists an AST on leaf set Tab∈A or Tba∈B exists. Without loss of generality, let Tab exist. Since, Txa exists and is not a result of a type-2 join, and Tab exists, as per Lemma 1.1, there exists an AST on leaf set 1. (a) Consider any descendant Txℓ exists,• for every leaf x1,x2} in• thus, for every pair of leaves {Xc,  is an agreement triplet.• • for every pair of leaves•• on leaf set ℓ in Xc,  is an agreement triplet.• leaf T on leaf set T displays Sc. Further, for every Txℓ exists. Thus, by Lemma 3, T is descendant of X. Hence, X prunes Y, as claimed.Thus, there exists an AST S of Y. Consider any S is a descendant of Y, by Lemma 3, Tya and Tyb exists. Thus, as per the if condition of the claim to be proved,  and  are agreement triplets. Since Txy exists as a result of type-2 join and,  and  are agreement triplets, by Lemma 4,  is an agreement triplet, and, by Lemma 1 (part 2 and 3), Txa and Txb exist. Thus, as per Lemma 2, there exists an AST on leaf set (b) Consider any descendant Txℓ exists,• for every leaf x1,x2} in• thus, for every pair of leaves {Xc,  is an agreement triplet.• • for every pair of leavesℓ in Xc, there exists and AST on leaf set ℓ,a,b] is an agreement triplet.•T on leaf set T displays Sc. Further, for every Txℓ exists, thus, by Lemma 3, T is descendant of X. Hence, X prunes Y, as claimed.Thus, there exists an AST Tyz is not a result of a type-2 join, by Lemma 1.3, Txz exists and is not a result of a type-2 join, and Txy−z exists. Consider any descendant S of S is a descendant of Y, by Lemma 3, Tyℓ exists. We show that Tyℓ is not a result of a type-2 join by contradiction. Let Tyℓ be a result of a type-2 join with triplet  of type ). Since S is a descendant of Tyz,Tyℓ) exists. Let Tyz be of type 1 join with triplet  of type . Since Ty−zℓ displays both ) and , it must also display ,z). However, ,z) cannot exist in Ty−zℓ because ℓ and z cannot be the last and second-to-last leaves respectively in the IDFT. Similarly, for Tyz of join type 3 or type 4, we can show that ℓ cannot be the rightmost leaf in Ty−zℓ. Thus, Tyℓ is not of join type 2. Thus, by Lemma 1.3, Txℓ exists and is not a result of type 2 join, and Txy−ℓ exists. Consider any Txa and Txb exist and each is not a result of join type 2. Let A and B denote the ASTs in E with a and b as their rightmost leaves. Without loss of generality, let the AST on leaf set Ac,Bc), i.e., Tab exists. Thus, by Lemma 1.1, there exists an AST on leaf set 2. Since Txy−ℓ exists, thus, for every pair of leaves {x1,x2} in Txy,  is an agreement triplet.• for every ℓ in Txy, there exists an AST on leaf set ℓ,a,b] is an agreement triplet.• for every T on leaf set T displays Sc. Further, since for every Txy−ℓ exists, by Lemma 3, T is descendant of Thus, there exists an AST □Y. However, there are cases when Y or a descendant of Y is pruned by X but it cannot be identified using Theorem 2. In this case, the branch at Y must be enumerated and potential pruning by X verified. For this, we maintain a “pruner-list” for every child of Y. We explain this idea next.Note that conditions in part (1a) and part (2) of Theorem 2 can be evaluated in constant time while testing the condition in part (1b) of Theorem 2 takes time linear in the size of the tree being pruned. Neither of these require enumerating the pruned branch at X,Y,Z be children of an equivalence class E in the enumeration tree. Let x,y,z be the rightmost leaves of Xc,Yc,Zc respectively. Let joins Txy,Tyz exist such that none of the cases of Theorem 2 hold. Then the pruner-list of Let Now, using arguments similar to the proof of Theorem 2, we can show that:Y is pruned by an equivalence class A with a as the rightmost leaf of Ac if either of the following holds: Y has no children and has a in its pruner-list.1. Y have a in their pruner-list.2. All children of MFSTMINER for the special case of enumerating all MXSTs for a collection C of input trees. MFSTMINER first invokes ENUMERATEAST_TRIPLETS (whose details are omitted) to enumerate all AST triplets and to partition them into equivalence classes. The set of all such classes is denoted by EC3. Note that each equivalence class in EC3 is a child of the root of the enumeration tree. After this, MFSTMINER invokes subroutine ENUMERATENODE, explained next, to enumerate the elements of the branch at each equivalence class in EC3. Algorithm 1 is a high-level description of ENUMERATENODE, which accepts an equivalence class E as input and enumerates the branch at E in the enumeration tree. In the pseudocode, Tx denotes an AST that has x as its rightmost leaf. Lines 3-5 perform pair-wise joining among members of an equivalence class in the enumeration tree. Comments in braces indicate where the algorithm performs assignment to pruner-lists, as per conditions (2) and (3), and pruning, according to the different cases of Theorems 2 and 3. In line 12, the core tree of an empty equivalence class — representing a leaf node — is produced as output if it is found to be not pruned. Algorithm 2 shows the details of ENUMERATENODE calls itself recursively to enumerate the children of a non-empty equivalence class In line 17, MFSTMINER handles the general case of mining MFSTs. The main difference between mining for MXSTs and mining for MFSTs is that the former is (as we have seen) based on enumerating ASTs, while the latter is based on enumerating FSTs. While an ASTs must be supported by all the input trees , an FSTs need only be supported by some fraction We now explain how Tx and Ty in an equivalence class, a join Txy on ordered pair  is an FST if it is supported by at least a fraction f of the input trees . Note that any such Txy is supported only by those trees that support both Tx and Ty as well. Motivated by this, for each FST Tx we maintain a support list, denoted by Tx.supList, that contains all trees in the input collection that support Tx. To estimate if the join on  results in an FST, we apply Theorem 1 only on trees in Tx.supList∩Ty.supList. We store the support list as a bitmap  is a frequent triplet if at least fraction f of the input trees display the same triplet over the leaf set {x,y,z}. Let .supList denote the support list of such a frequent triplet. Based on this, we can restate Theorem 2 for the case of enumerating MFSTs as follows.To verify whether an equivalence class X prunes Y if either 1. Txy exists, Yc.supList⊆Xc.supList and Txy is not of join type 2, or(a) Txy exists as a type-2 join, Yc.supList⊆Xc.supList and for every Z∈E such that Tyz exists,  is a frequent triplet with Yc.supList⊆.supList.(b) Txy and Tyz, exist and Txy is of join type 2, then Tyz is not of join type 2 and Tyz.supList⊆Txy.supList.2. If Tx,Ty,Tz be FSTs in an equivalence class E with x,y,z as their respective rightmost leaves, such that joins Txy and Tyz exist, and none of the cases of Theorem 4 hold. For an equivalence class A, let A.prunerList denote its pruner-list. Then, the next set of conditions describe pruner-lists for enumerating MFSTs. Pruning cases not identified by Theorem 4 require the use of pruner-list. In the case of MFSTs, along with leaf label the pruner-list also contains the support list of the core tree of the equivalence class that is claiming to prune. To explain further, let x,Txy.supList) if Txy is not of join type 2 and |Txy.supList∩Tyz.supList|≥f.1.x,S∩=Txy.supList∩.supList) if Txy is of join type 2,  is a frequent triplet and |S∩∩Tyz.supList|≥f.2.w such that S∩∩Tyz.supList|≥f.3. For every leaf label Condition 1 and 2 describe addition of new labels to the pruner-list of A with a as the rightmost leaf of Ac, Given an equivalence class A if (i) for every a,Sb(a)), and (ii) for 1.A if (i) a in its pruner list, or (ii) every child A as per part 1 of this Theorem.2.MFSTMINER for the general case of mining MFSTs. The overall framework is the same as the special case of mining all MXSTs. The difference lies in the finer details of incorporating support lists in the support estimation and the pruning step. These details were discussed in this section and are easy to incorporate in Algorithms 1 and 2.This completes the description of MFSTMINER algorithm. In the following discussion, let the input collection consist of n trees on a common leaf set f denote the input fraction for computing f-frequent subtrees.Here we discuss the runtime complexity of the preprocessing step, the data structures used in the algorithm implementation and the memory requirements in each step of T in the input collection, and every pair {u,v} of leaves of T, step (1) computes LCAT and stores it in a three-dimensional array indexed by triplet , thus, requires f=1. We should point out that it is well-known that one can pre-process a tree in linear time and space to produce a data structure that can answer any LCA query on that tree in constant time  is of type . We have four possibilities to consider. 1. Suppose Tyz is a result of a type-1 join <ψ(z) and agreement triplet  is of type . Potential AST T must be obtained by grafting z in Txy. Since T must display , there is only one possibility of grafting z in Txy: z should be grafted on the common parent of x and y. Thus, AST T exists. Further, since, ψ(x)<ψ(y)<ψ(z), there is only one possible canonical topology for potential AST T: see Figure T has x, y, z, as the third-to-last, second-to-last and last leaf respectively in the IDFT, pruning y will result in a tree that is (a) canonical, (b) has z as the last leaf in the IDFT, and (c) has x as the second-to-last leaf in the IDFT. By definition, the resulting tree is Txz  Tyz is a result of a type-2 join <ψ(z) and agreement triplet  is of type ). Potential AST T must be obtained by grafting z in Txy. Since T must display ), there is only one possibility of grafting z in Txy: z should be grafted on the edge . Thus, AST T exists. Further, since, ψ(x)<ψ(y)<ψ(z), there is only one possible canonical topology for potential AST T; see Figure T has x, y, z as the third-to-last, second-to-last and last leaf respectively in the IDFT, pruning y will result in a tree that is (a) canonical, (b) has z as the last leaf in the IDFT, and (c) has x as the second-to-last leaf in the IDFT. By definition, the resulting tree is Txz  Tyz is a result of a type-3 join , there is only one possibility of grafting x in Tyz: x should be grafted on the parent of y. Thus, AST T exists. Further, since, ψ(x)<ψ(y), there is only one possible canonical topology for potential AST T; see Figure T has x, y, z as the third-to-last, second-to-last and last leaf respectively in the IDFT, pruning y will result in a tree that is (a) canonical, (b) has z as the last leaf in the IDFT, and (c) has x as the second-to-last leaf in the IDFT. By definition, the resulting tree is Txz  Tyz is a result of a type-4 join , there is only one possibility of grafting x in Tyz: x should be grafted on the parent of y. Thus, AST T exists. Further, since, ψ(x)<ψ(y), there is only one possible canonical topology for potential AST T; see Figure T has x, y, z as the third-to-last, second-to-last and last leaf respectively in the IDFT, pruning y will result in a tree that is (a) canonical, (b) has z as the last leaf in the IDFT, and (c) has x as the second-to-last leaf in the IDFT. By definition, the resulting tree is Txz  in T will result in Txy. Thus, X.(d) Txy is a result of a join of type 3 or 4, irrespective of the type of join Tyz is a result of, ASTs T and Txz exist, Txz is not a result of a type-2 join, and X.Similarly, one can show that if Txy and Tyz is a result of a type-2 join ) and ψ(y)<ψ(z). Potential AST T must be obtained by grafting z in Txy. Since T must display ) and ψ(x)<ψ(y)<ψ(z), there are four possible canonical topologies for potential AST T: see Figures x,y,z] is an agreement triplet, only one of the four topologies exists across all input trees. Thus, AST T exists.2. Suppose each of T are y and z (either y comes before z or vice-versa), while x is the third-to-last leaf. Thus, pruning y in T will result in a canonical tree where x and z are the second-to-last and last leaves in the IDFT. By definition, the resulting tree is Txz. Thus, Txz exists. Further, pruning the last leaf in T will result in either Txy (if z is the last leaf in T) or Txz (if y is the last leaf in T). In either case, X, as claimed.Further, in all the cases discussed above, the last two leaves in the IDFT of Txy is a result of a type-2 join. Thus, ψ(x)<ψ(y). Consider the following possibilities: 3. Suppose Tyz is a result of a type-1 join <ψ(z). Potential AST T must be obtained by grafting x in Tyz. Since T must display ), there is only one possibility of grafting x in Tyz: x should be grafted on the edge . Thus, AST T exists. Further, since, ψ(x)<ψ(y)<ψ(z), there is only one possible canonical topology for potential AST T: see Figure T has x, y, z as the third-to-last, second-to-last and last leaf respectively in the IDFT, pruning y will result in a tree that is (a) canonical, (b) has z as the last leaf in the IDFT, and (c) has x as the second-to-last leaf in the IDFT. By definition, the resulting tree is Txz  Tyz is a result of a type-3 join ), there is only one possibility of grafting x in Tyz: x should be grafted on the edge . Thus, AST T exists. Further, since, ψ(x)<ψ(y), there is only one possible canonical topology for potential AST T: see Figure T has x, y, z as the third-to-last, second-to-last and last leaf respectively in the IDFT, pruning y will result in a tree that is (a) canonical, (b) has z as the last leaf in the IDFT, and (c) has x as the second-to-last leaf in the IDFT. By definition, the resulting tree is Txz  Tyz is a result of a type-4 join ), there is only one possibility of grafting x in Tyz: x should be grafted on the edge . Thus, AST T exists. Further, since, ψ(x)<ψ(y), there is only one possible canonical topology for potential AST T: see Figure T has x, y, z as the third-to-last, second-to-last and last leaf respectively in the IDFT, pruning y will result in a tree that is (a) canonical, (b) has z as the last leaf in the IDFT, and (c) has x as the second-to-last leaf in the IDFT. By definition, the resulting tree is Txz  y and z are the second-to-last and the last leaves in T. Thus, pruning z — the rightmost leaf — in T will result in a tree that is (a) canonical, (b) has y as the last leaf in the IDFT, and (c) has x as the second-to-last leaf in the IDFT. By definition, the resulting tree is Txy. Hence, Further, in all the above cases, □ψ(y)<ψ(z). Let px, py and pz denote the parent of x, y and z respectively. Consider the following cases: Without loss of generality, let Tx−yz exists as a result of a type-4 join.1.Tx−zy exists as a result of a type-4 join.2.3.x,y,z] is of type . Thus, depth)=depth)=depth) across all input trees. Thus, as per Theorem 1, Tx−yz exists as a result of a type-1 join.(a) Agreement triplet  is of type ). Thus, depth)=depth) and depth)<depth) across all input trees. Thus, as per Theorem 1, Tx−yz exist as a result of a type-2 join.(b) Agreement triplet  is of type ,z). Thus, depth)>depth) across all input trees. Thus, as per Theorem 1, Tx−yz exists as a result of a type-3 join.(c) Agreement triplet  is of type ,y). Thus, depth)>depth) across all input trees. Thus, as per Theorem 1, Tx−zy exists as a result of a type-3 join.(d) Agreement triplet  is an agreement triplet , where T1=A, such that B. Thus, Txy is a result of a type-2 join, agreement triplet  is of type ) and ψ(x)<ψ(y). Consider any x,y,ℓ] is an agreement triplet. Since D is a descendant of Y, by Lemma 3, Tyℓ exists. Since,  is an agreement triplet, by Lemma 1, there exists an AST T on the leaf set Txy and Tyℓ, and X. Consider the possible topologies for T.  Since Txy is a result of a type-2 join, the topology of T depends on the type of join Tyℓ is a result of. Consider the following cases. Since Tyℓ is a result of a type-1 join. There exists only one possible canonical topology for T: the one corresponding to Figure z with ℓ in tree T).1. Tyℓ is a result of a type-2 join. There exists four possible canonical topologies for T: the ones corresponding to Figure z with ℓ in tree T).2. Tyℓ is a result of a type-3 join. There exists only one possible canonical topology for T: the one corresponding to Figure z with ℓ in tree T).3. Tyℓ is a result of a type-4 join. There exists only one possible canonical topology for T: the one corresponding to Figure z with ℓ in tree T).4. T, only one has ℓ as the second-to-last leaf in the IDFT: the one corresponding to Figure z with ℓ in tree T). Thus, this topology belongs to belongs to y as the second-to-last leaf in the IDFT, thus belong to x,y,a] and  are agreement triplets. Thus, by our earlier discussion in this proof, there exist ASTs Ta on leaf set Tb on leaf set X. Consider the following cases: Note that out of the possible topologies for Ta and Tb belong to y,a,b] is an agreement triplet, by Lemma 2, there exists an AST on leaf set x,a,b] is an agreement triplet.1. Both Ta and Tb do not belong to Ta belong to z with a in tree T). Thus, agreement triplet  is of type ,y) and ψ(x)<ψ(a). Consider potential AST T’ on leaf set Ta and Tb. Since, the topology Ta is known, the topology of T’ depends on the topology of Tb. Consider the following cases. In the subsequent discussion, let px, py, pa and pb denote the parent of x, y, a and b respectively. 2. Both Tb corresponds to the topology in Figure z with b in tree T). Potential AST T’ must be obtained by grafting a in Tb. Since T’ must display ,y), there is only one possibility of grafting a in Tb: a should be grafted on the edge . Thus, T’ exists. Considering ψ(x)<ψ(a), the canonical topology for AST T’ is shown in Figure (a) Tb corresponds to the topology in Figure z with b in tree T). Potential AST T’ must be obtained by grafting a in Tb. Since T’ must display ,y), there is only one possibility of grafting a in Tb: a should be grafted on the edge . Thus, T’ exists. Considering ψ(x)<ψ(a), the canonical topology for AST T’ is shown in Figure (b) Tb corresponds to the topology in Figure z with b in tree T). Potential AST T’ must be obtained by grafting a in Tb. Since T’ must display ,y), there is only one possibility of grafting a in Tb: a should be grafted on the edge . Thus, T’ exists. Considering ψ(x)<ψ(a), the canonical topology for AST T’ is shown in Figure (c) Tb corresponds to the topology in Figure z with b in tree T). Potential AST T’ must be obtained by grafting a in Tb. Since T’ must display ,y), there is only one possibility of grafting a in Tb: a should be grafted on the edge . Thus, T’ exists. Considering ψ(x)<ψ(a), the canonical topology for AST T’ is shown in Figure (d) Tb corresponds to the topology in Figure z with b in tree T). Thus, ψ(x)<ψ(b). Since, both Ta and Tb correspond to the topology in Figure ψ(a)<ψ(b). Potential AST T’ must be obtained by grafting a in Tb and must display ,y). Since ψ(x)<ψ(a)<ψ(b), there are four possible canonical topologies for T’; see Figure y,a,b] is an agreement triplet, thus, only one of the four possible topologies exits across all input trees Thus, T’ exists.(e) Tb corresponds to the topology in Figure z with b in tree T). Potential AST T’ must be obtained by grafting a in Tb. Since T’ must display ,y), there is only one possibility of grafting a in Tb: a should be grafted on the edge . Thus, T’ exists. Considering ψ(x)<ψ(a), the canonical topology for AST T’ is shown in Figure (f) Tb corresponds to the topology in Figure z with b in tree T). Potential AST T’ must be obtained by grafting a in Tb. Since T’ must display ,y), there is only one possibility of grafting a in Tb: a should be grafted on the edge . Thus, T’ exists. Considering ψ(x)<ψ(x), the canonical topology for AST T’ is shown in Figure (g) T’ exists. Thus,  is an agreement triplet.Thus, in each of the above cases This completes the proof. □The authors declare that they have no competing interests.AD and DFB conceived the problem, designed the experiments and drafted the manuscript. AD designed and implemented the algorithms, and implemented the experiments. DFB coordinated the project. All authors read and approved the final manuscript.