An operational method to construct UMEBs containing d(d′ − 1) maximally entangled vectors is established, and two UMEBs in d(d′ − r) maximally entangled vectors in r = 1, 2, …, d − 1. Correspondingly, two UMEBs in We study unextendible maximally entangled bases (UMEBs) in  Quantum teleportation, which can be used for distributed quantum learning5 and even in organisms6, is a essential element in quantum information processing. Maximally entangled states attract much attention due to their importance in ensuring the highest fidelity and efficiency in quantum teleportation7. A pure state |ψ〉 is said to be a d ⊗ d′ (d < d′) maximally entangled state if and only if for an arbitrary given orthonormal basis {|iA〉} of subsystem A, there exists an orthonormal basis {|iB〉} of subsystem B such that |ψ〉 can be written as 8.Quantum entanglement lies in the heart of the quantum information processing. It plays important roles in many fields such as quantum teleportation, quantum coding, quantum key distribution protocol, quantum non-locality13 and plays an important role in Van der Waals interaction in transformation optics14. It is tightly related to entanglement. While, it is proven that the unextendible product bases (UPBs) reveal some nolocality without entanglement16. The UPB is a set of incomplete orthogonal product states in bipartite quantum system dd′ vectors which have no additional product states are orthogonal to each element of the set17.Nonlocality is a very useful concept in quantum mechanics17:A UPB in Obviously, they are all product states.ψ〉, which is orthogonal to |ϕi〉. If |ψ〉 is a product state, it can be expressed as |ψ〉 = (a|0〉 + b|1〉 + c|2〉) ⊗ (a′|0〉 + b′|1〉 + c′|2〉), where a2 + b2 + c2 = a′2 + b′2 + c′2 = 1. No loss of generalization, we assume that a, a′ ≠ 0. From |ψ〉 is orthogonal to |ϕ0〉, we have b′ = a′ ≠ 0. Due to |ψ〉 is orthogonal to |ϕ4〉, we can conclude that c′ ≠ 0. Because of that |ψ〉 is orthogonal to |ϕ1〉, we have b = a ≠ 0. Owing to that |ψ〉 is orthogonal to |ϕ3〉, we obtain that c = b ≠ 0. And |ψ〉 is orthogonal to |ϕ2〉, we get that c′ = b′ ≠ 0. That is to say, a = b = c = a′ = b′ = c′ = ψ〉 is equal to |ϕ4〉, instead of being orthogonal to |ϕ4〉. Therefore, |ψ〉 can not be a product state. That’s why the set There exist a nonzero pure state |18 generalized the notion of UPB to unextendible maximally entangled bases (UMEB): a set of incomplete orthogonal maximally entangled states in bipartite quantum system dd′ vectors which have no additional maximally entangled vectors orthogonal to all of them. They state that UMEBs can be used to construct examples of states for which 1-copy entanglement of assistance (EoA) is strictly smaller than the asymptotic EoA and find quantum channels that are unital but not convex mixtures of unitary operations18.Bravyi and Smolin19:Let {|0〉, |1〉} be a orthogonal base of Obviously, they are all maximally entangled states.ψ〉 is orthogonal to |ϕi〉, it’s sure that |ψ〉 = (a|0〉 + b|1〉) ⊗ |2′〉, where a2 + b2 = 1. In other words, |ψ〉 must be a product state, rather than maximally entangled states. Hence If a nonzero pure state |n maximally entangled vectors is usually expressed as a n-number UMBE, when n is smaller than dd′. Chen and Fei19 provided a way to construct d2-member UMEBs in et al.20 and Li et al.21 constructed two sets of UMEBs in d < d′) independently. Wang et al.22 put forward a method of constructing UMEBs in d = p or 2p, where p is a prime and p = 3 mod 4. They also presented a 23-member UMEB in 23. Then Guo26 proposed a scenario of constructing UMEBs via the space decomposition, which improves the previous work about UMEBs.The number of the vectors in a UMEB is less than the dimension of the bipartite system space. Therefore a UMEB in d(d′ − 1)-number UMEB and then present explicit constructions of UMEBs in d(d′ − r)-member UMEBs in r = 1, 2, …, d − 1, and give two examples in In this paper, we give two methods of constructing UMEBs in d < d′. Let us first recall some basic notions and lemmas24. Let {|k〉} and Md×d′ be the Hilbert space of all d × d′ complex matrices equipped with the inner product defined by 〈A|B〉 = Tr(A†B) for any A, B ∈ Md×d′. If Md×d′, where 〈Ai|Aj〉 = dδij, then there is a one-to-one correspondence between {|ϕi〉} and {Ai} as follows26:Sr(|ϕi〉) denotes the Schmidt number of |ϕi〉. Obviously, |ϕi〉 is a maximally entangled pure state in Cd ⊗ Cd′ iff (d)1/2Ai is a d × d′ singular-value-1 matrix .Throughout this paper, we assume that Cd ⊗ Cd′ is called a maximally entangled basis (MEB) of Cd ⊗ Cd′. A set of pure states 19:(i)ϕi〉, i = 1, 2, 3 ... n are all maximally entangled states.|(ii)(iii)n < dd′, and if a pure state |ψ〉 satisfies that 〈ϕi|ψ〉 = 0, i = 1, 2, 3... n, then |ψ〉 can not be maximally entangled.A basis Md×d′ is called single-value-1 Hilbert-Schmidt basis (SV1B) of Md×d′. A set of d × d′ matrices Md×d′24:(i)Ai, i = 1, 2, 3 ... n are all single-value-1 matrices.(ii)i, j = 1, 2, 3... n.(iii)n < dd′, and if a matrix X satisfies that Tr(X†Ai) = 0, i = 1, 2, 3... n, then X can not be a single-value-1 matrix.A Hilbert-Schmidt basis Md×d′ iff Cd ⊗ Cd′, and Md × d′ iff Cd ⊗ Cd′. Therefore, for convenience, we may just call an SV1B Md × d′ an MEB Md×d′ a UMEB It is obvious that 24.In deriving our main results, we need the following lemma in ref.ϕi〉} is a MEB in M1 and {|ψi〉} is a UMEB in ϕi〉}∪{|ψi〉} is a UMEB in M24. If {|ϕi〉} is a MEB in M1 and ϕi〉} is a UMEB in M.Let d(d′ − 1)-member UMEBs in In this section, we will establish a flexible method to construct Md×d′ be the Hilbert space of all d × d′ complex matrices. If V is a subspace of Md×d′ such that each matrix in V is a d × d′ matrix ignoring d entries which occupy different rows and N columns with N < d, then there exists a d(d′ − 1)-member MEB in V, as well as a d(d′ − 1)-member UMEB in Md×d′.Let d entries in V only occupy the former N columns. Let bi, i = 0, 1, ..., d − 1, denote the column number of the ignored element in the i-th row. Obviously, bi+1 − bi = 0 or 1.Without loss of generality, we can always assume the ignored Denoted(d′ − 1) pure states in p ⊕ d′m denotes (p + m) mod d′.We can construct V.Next, we prove that all the states in  constitu(i) Maximally entangled.C = 0 for any m, it is obvious that tmj ≠ tm′j for m ≠ m′.If C = 1 for some m ≠ 0, from the definition of tmj one has tm−1, j ≠ bm−1. Note that bm − bm−1 = 0 or 1, then tm−1, j = bm−1 ⊕d′ 1.If tmj, we also have C = 0 for k ≠ m − 1. Hencep − m) mod d′. In particular,From the definition of Thentmj ≠ tm′j for m ≠ m′. Namely, the states |ϕ′j,n〉 in  Orthogonality.tmj〉 = |tmj′〉 if and only if j = j′.We first show that |tmj = tmj′ for j = j′. If j ≠ j′, without loss of generality, let tmj ≠ tmj′ when tm−1,j ≠ tm−1,j′. Otherwise, from the definition of tmj we have C = 1. Note that C = 1, as proved in (i). Therefore, tm−1, j′ = bm−1, which contradicts to the definition of tmj. Furthermore, tmj ≠ tmj′ when tj0 ≠ tj′0. Therefore,Obviously, d(d′ − 1) states {|ϕj, n〉} in |ϕj′〉, we get the following MEB V, i.e., another UMEB in C5 ⊗ C6:By inverse unitary transformation |Actually both  and 10)10) are UConstructing a UMEB in V1 and V2 by elementary transformation respectively:One can easily get the following simple formations Then following Theorem 1 we can construct the following UMEBs V1 and V2, respectively,By inverse transformation ϕj〉} ∪ {|ψj〉} constitutes a UMEB in V in -member UMEBs in r = 1, 2, …, d − 1, that is to say, it presents d − 1 constructions of UMEB in In this section, we construct UMEBs consisting of fewer elements in d(d′ − r)-member UMEB in Cd ⊗ Cd′:j = 0, 1,…, s − 1; n = 0, 1 …, d − 1.Let ϕl,j,n〉 in ((i) It is obvious that |,j,n〉 in  are all (ii) Orthogonality,M1 the d ⊗ (d′ − n) matrix space, a subspace of Md×d′. Since the number of {|ϕl,j,n〉} in  Denote j,n〉} in  equals tUMEBs in Obviously, 10 = 4 + 5 + 1 or 10 = 4 + 4 + 2. According to Theorem 2, we can construct the following 27-number UMEB  and 24-nCd ⊗ Cd′, which is more than all the previous numbers. For example, the 27-number UMEB -member UMEB in d(d − r)-number UMEBs in r = 1, 2, …, d − 1. Namely, we have presented more than d − 1 constructions of UMEBs in 21 and ref.20. We have also shown 27-number UMEB and 24-number UMEB in We have provided new constructions of unextendible maximally entangled bases in arbitrary bipartite spaces


Science China Information Sciences, 2011, 1596-1607 by Pan et al.Spiking neural P systems are a new candidate in spiking neural network models. By using neuron division and budding, such systems can generate/produce exponential working space in linear computational steps, thus provide a way to solve computational hard problems in feasible  time with a “time-space trade-off” strategy. In this work, a new mechanism called neuron dissolution is introduced, by which redundant neurons produced during the computation can be removed. As applications, uniform solutions to two NP-hard problems: SAT problem and Subset Sum problem are constructed in linear time, working in a deterministic way. The neuron dissolution strategy is used to eliminate invalid solutions, and all answers to these two problems are encoded as indices of output neurons. Our results improve the one obtained in  Each neuron has a certain number of spikes and rules. Spikes can evolve through application of rules. Since SN P systems were proposed, they become a rapid developing area of membrane computing i, where, i ∈ H, E is a regular expression over a, c ≥ 1, p ≥ 1, c ≥ p, d ≥ 0. If E = ac, the firing rule is simply written as i. If d = 0, the firing rule is simply written as [E/ac → ap]i. If E = ac and d = 0, the firing rule is simply written as [ac → ap]i;forgetting rule [E/as → λ]i, where, i ∈ H, E is a regular expression over a, s ≥ 1. If E = as, the forgetting rule is simply written as [as → λ]i;neuron division rule [E]i → j || k, where, i, j, k ∈ H, E is a regular expression over a;neuron dissolution rule [E]i → δ, where, i ∈ H, E is a regular expression over a, object δ represents that neuron σi is dissolved;in, out ⊆ H represent the input and output neurons of Π, respectively.syn shows the initial structure of the system and guides how to establish new synapses when new neurons are established.The synapse dictionary σi has h spikes, and ah ∈ L(E), h ≥ c, the firing rule i can be applied. c spikes are consumed (h − c spikes remain in neuron σi.), and p spikes are emitted after d time units (steps). If d = 0, p spikes are emitted immediately; if d = 1, p spikes are emitted at the next step; if this firing rule is applied at step t and d ≥ 1, p spikes are emitted at step t + d. Neuron σi is closed at steps t, t + 1, t + 2, …, t + d − 1, which means no rule will be applied and no spike will be received in this period. At step t + d, neuron σi becomes open again, and can receive new spikes. Once these p spikes are emitted from neuron σi, they reach each neuron σj which has a synapse going from neuron σi to neuron σj and is open. The spikes sent to a closed neuron are lost.If neuron σi has h spikes, and ah ∈ L(E), h ≥ s, the forgetting rule [E/as → λ]i can be applied. s spikes are consumed immediately.If neuron σi has h spikes, and ah ∈ L(E), and (2). no synapse , , ,  exists in the system, the neuron division rule [E]i → j || k can be applied. All h spikes in neuron σi are consumed and neuron σi is divided into two neurons σj and σk. No spike is in neurons σj and σk at this moment. The labels of the two generated neurons can be different or the same, and the labels of the two generated neurons can be different from or the same with the label of their father neuron σi, too. The new generated neurons inherit the synapses of their father neuron σi. That is to say, if there is a synapse  going from neuron σi to neuron σg, two synapses  and  are established after the division rule is applied; if there is a synapse  going from neuron σg to neuron σi, two synapses  and  are established after the division rule is applied. In addition to inheritance of synapses, new generated neurons also have synapses provided by the synapse dictionary syn. Synapses not existing in the synapse dictionary syn may appear because of inheritance of synapses. The condition (2) avoids the situation that the start and the end of a synapse are the same neuron. For example, if synapse  exists in the system, synapses ,  will appear which is not permitted.If (1). neuron σ3. Considering the two conditions mentioned in the above paragraph, 1). Neuron σ3 has one spike a and the regular expressions of both two division rules are exactly {a}, where a ∈ {a}. Therefore, both of these two division rules meet the condition (1). 2). For rule [a]3 → 2 || 3, the label 3 of the father neuron σ3 corresponds to i in the normalization rule, and the label 2 and 3 of the two new neurons σ2 and σ3 corresponds to j and k in the normalization rule. Synapses , , ,  cannot exist in the system. That is to say, , , ,  cannot exist in this system. However, a synapse  is in this system, therefore rule [a]3 → 2 || 3 cannot be applied. Only rule [a]3 → 3 || 4 meet the two conditions. The spike a in neuron σ3 is consumed, neuron σ3 is divided into two neurons σ3 and σ4, and two synapses ,  going from neuron σ2 to these two new neurons are established because there is a synapse going from neuron σ2 to the father neuron σ3 of the two new neurons (the inheritance of synapses). Because rules in this system are related to the labels of neurons, the new neuron σ3 contains these two rules. The system is changed to a]3 → 3 || 4.A simple example shown in σi has h spikes, and ah ∈ L(E), the neuron dissolution rule [E]i → δ can be applied. All h spikes in neuron σi are consumed and neuron σi is dissolved. All synapses going from/to neuron σi are dissolved, too.If neuron σ1 has one spike a and the regular expression of the dissolution rule is exactly {a}, where a ∈ {a}. Then rule [a]1 → δ is applied and the system is changed to σ1 is dissolved, and synapse  connected with neuron σ1 is also dissolved.).A simple example shown in σi can be applied, this rule must be applied; if two or more rules in neuron σi can be applied, one of these rules is applied non-deterministically. Rules are applied in a sequential manner in each neuron and in parallel between neurons.At each step, if only one rule in neuron configuration of the system is described by the synapses connections, the spikes number in each neuron, and the state of each neuron (open or closed). By applying rules, the configuration is transformed from one to the next one. The transition sequence starting from the initial configuration is called a computation, and a computation halts if it reaches a configuration where all neurons are open and no rule can be applied.The IX, ΘX both in a semi-uniform way and in a uniform way, where IX is a language over a finite alphabet and ΘX is a total boolean function over IX (The elements in IX are instances.). In the semi-uniform way, a specified SN P system is constructed for each instance of a decision problem, in which the instance parameters are embedded in the SN P system. In the uniform way, a SN P system is constructed for all instances of a decision problem, in which the different instances parameters enter the SN P system as input spikes. The uniform solutions are preferred because they only relate to the structure of a problem.SN P systems can used to solve the decision problem r ≥ 1, ij ≥ 0 for each 1 ≤ j ≤ r, which means ij spikes enter the system through input neuron σin at step j. Specially, ij = 0 means no spike enters the system at step j.The input of a SN P system is a spike train X = {x1, x2, …, xn}, a literal li is xi or ¬xi for 1 ≤ i ≤ n. A clause Ci is a disjunction of literals Ci = ln1 ∨ ln2 ∨ … ∨ lnr, 1 ≤ r ≤ n. A conjunctive normal form  is a conjunction of clauses C1 ∧ C2 ∧ … ∧ Cm. An assignment is a mapping X → {0, 1} from each variable xi to its value . For example, X = {x1, x2, x3}, the conjunctive normal form is (x1 ∨ ¬x2) ∧ (x1 ∨ x3). The x1 ∨ ¬x2 and x1 ∨ x3 are the two clauses. The first clause contains two literals x1 and ¬x2, and the second clause contains two literals x1 and x3. If an assignment of x1, x2, …, xn can be found, which makes at least one literal true in each clause and then makes all m clauses true, this SAT problem is satisfiable. Otherwise, this SAT problem is unsatisfiable i, i = 1, 23[a(a2)+/a2 → a]d[a → a]inxi, i = 1, 2, …, n[a2 → a2]inxi, i = 1, 2, …, n[a3 → a3]inxi, i = 1, 2, …, n[a → a]Cxi1, i = 1, 2, …, n[a3 → a]Cxi1, i = 1, 2, …, n[a → a]Cxi0, i = 1, 2, …, n[a2 → a]Cxi0, i = 1, 2, …, n[forgetting rule:a2 → λ]2[a2 → λ]Cxi1, i = 1, 2, …, n[a3 → λ]Cxi0, i = 1, 2, …, n[a → λ]ot1t2…tn, t1, t2, …, tn = 0, 1[a2 → λ]ot1t2…tn, t1, t2, …, tn = 0, 1[…an −1 → λ]ot1t2…tn, t1, t2, …, tn = 0, 1[neuron division rule:a]0 → o1 || o0[a]ot1 → ot11 || ot10, t1 = 0, 1[a]ot1t2 → ot1t21 || ot1t20, t1, t2 = 0, 1[…a]ot1t2…tn−1 → ot1t2…tn−11 || ot1t2…tn−10, t1, t2, …, tn−1 = 0, 1[neuron dissolution rule:an]ot1t2…tn → δ, t1, t2, …, tn = 0, 1.0 → o1 || o0 is applied to generate neurons σo1 and σo0, which means an assignment in regard to x1 has two choices: 1 or 0. Synapses  and  are established through the inheritance of synapse , and synapses  and  are established through synapse dictionary syn. Synapse  establishes a channel between the input and the assignment including x1 = 1; synapse  establishes a channel between the input and the assignment including x1 = 0. At the same time, auxiliary neuron σ2 has one spike, rule a → a is applied and one spike is emitted to neuron σ1; auxiliary neuron σ3 has one spike, rule a → a; 2n − 1 is applied and one spike will be emitted to neurons σ2 and σd at step 2n. The system after step one is shown in σ1 has one spike, the firing rule a → a is applied, and one spike is emitted to neurons σ2, σo1 and σo0.At step two, neuron σo1 and σo0 has one spike, the division rule [a]ot1 → ot11 || ot10 is applied to generate neurons σo11, σo10, σo01 and σo00, which means an assignment in regard to x1 and x2 has four choices: 11, 10, 01, 00. Synapses , ,  and  are established through the inheritance of synapses , ; synapses , ,  and  are established through the inheritance of synapses , ; synapses , , S and  are established through synapse dictionary syn. Synapses  and  establish channels between the input and the assignments including x1 = 1; synapses  and  establish channels between the input and the assignments including x1 = 0; synapses  and  establish channels between the input and the assignments including x2 = 1; synapses  and  establish channels between the input and the assignments including x2 = 0. At the same time, auxiliary neuron σ2 has one spike, rule a → a is applied and one spike is emitted to neuron σ1. The system after step three is shown in At step three, each of neurons n − 1, 2n neurons labeled ot1t2…tn are generated. The system after step 2n − 1 is shown in Similar process repeats. At step 2n, each neuron σot1t2…tn receives one spike emitted from neuron σ1 which will be deleted at the next step by the forgetting rule [a → λ]ot1t2…tn. Neuron σ2 receives two spikes , the forgetting rule a2 → λ is applied at step 2n + 1, and no spike will be emitted to neuron σ1 later. At the same time, neuron σd receives one spike emitted from σ3. The system after step 2n is shown in At step 2Input Stage: At step 2n + 1, the first clause of the conjunctive normal form expression enters the system through input neurons σinxi, i = 1, 2, …, n. The literal in regard to x1 enters neuron σinx1; the literal in regard to x2 enters neuron σinx2; … the literal in regard to xn enters neuron σinxn. At the same time, one spike is emitted to neuron σinxi from neuron σd.n + 2, the spikes in neuron σinxi are replicated, and are emitted to neurons σCxi1 and σCxi0.At step 2n + 3, different rules are applied according to the number of spikes in neurons σCxi1 and σCxi0.At step 2σCxi1:For neuron σCxi1, which means neither xi nor ¬xi is in the clause, rule a → a is applied. One spike is emitted to neurons having synapses going from neuron σCxi1 to them. It aims to show that xi = 1 makes no contribution to let the clause true.If one spike is in neuron σCxi1, which means xi is in the clause, rule a2 → λ is applied. These two spikes are deleted. It aims to show that xi = 1 makes contribution to let the clause true.If two spikes are in neuron σCxi1, which means ¬xi is in the clause, rule a3 → a is applied. One spike is emitted to neurons having synapses going from neuron σCxi1 to them. It aims to show that xi = 1 makes no contribution to let the clause true.If three spikes are in neuron σCxi0:For neuron σCxi0, which means neither xi nor ¬xi is in the clause, rule a → a is applied. One spike is emitted to neurons having synapses going from neuron σCxi0 to them. It aims to show that xi = 0 makes no contribution to let the clause true.If one spike is in neuron σCxi0, which means xi is in the clause, rule a2 → a is applied. One spike is emitted to neurons having synapses going from neuron σCxi0 to them. It aims to show that xi = 0 makes no contribution to let the clause true.If two spikes are in neuron σCxi0, which means ¬xi is in the clause, rule a3 → λ is applied. These three spikes are deleted. It aims to show that xi = 0 makes contribution to let the clause true.If three spikes are in neuron Satisfiability Stage: Each neuron σot1t2…tn receives zero or more spikes at step 2n + 3. If one neuron σot1t2…tn receives n spikes which means the clause contains n literals that make no contribution to let the clause true, the dissolution rule [an]ot1t2…tn → δ is applied at step 2n + 4 to dissolve this neuron . Otherwise, at least one literal is true in this assignment and this assignment is reserved to check the next clause.m clauses are in a SAT problem, the satisfiability checking stage lasts for m + 3 steps. If some neurons σot1t2…tn are still in the system at step 2n + m + 3, the labels of these neurons σot1t2…tn are all solutions to this SAT problem, i.e., this SAT problem is satisfiable. Otherwise, this SAT problem is unsatisfiable.Due to SAT problem can be solved in linear time, and all solutions can be obtained through this system.It can be seen that any Some steps comparison results between our solution and other solutions, which use the neuron division to solve the NP-complete problems, are shown in SAT : (x1 ⋁ x2) ⋀ (¬x2 ⋁ x3) ⋀ (¬x1 ⋁ x2 ⋁ x3), the DDSN P system Π3,3 is used to solve it. After 12 computational steps, neurons σo111, σo101 and σo011 are remaining which shows that {x1 = true, x2 = true, x3 = true}, {x1 = true, x2 = false, x3 = true} and {x1 = false, x2 = true, x3 = true} are all solutions to this SAT problem.Considering a SAT problem The SN P system with neuron division and budding and the SN P system with neuron division need 21 steps and 26 steps to judge this problem has solutions, respectively, while our DDSN P system need only 12 steps.n, m ≤ 50) are solved using the three systems in SAT problems with different sizes  denotes the set of all instances of the Subset Sum problem having n integers. In this section, a uniform solution working in a deterministic way is constructed by DDSN P system, which can solve all SubsetSumproblem (n) problems in linear time.a0 ⋅)n2 is introduced into the front of each spike train.An integer is represented by corresponding number of spikes. In order to generate necessary workspace before computing, a spike train (SubsetSumproblem (n) ;syn = {, , , , , }ini, di1), , |i = 1, 2, …, n}⋃{}⋃{(1)|i = 2, 3, …, n};⋃{;firing rule:a → a]i, i = 1, 23[a3(a3)+/a3 → a3]ini, i = 1, 2, …, n[a3 → a]ini, i = 1, 2, …, n[a → a]di1, i = 1, 2, …, n[a3 → a3]di2, i = 1, 2, …, n[an → a]4[a(a2)+/a2 → a2]s[a → a]s[forgetting rule:a2 → λ]2[a3 → λ]di1, i = 1, 2, …, n[a → λ]di2, i = 1, 2, …, n[a2(a3)+/a5 → λ]ot1t2…tn, t1, t2, …, tn = 0, 1[a → λ]ot1t2…tn, t1, t2, …, tn = 0, 1[neuron division rule:a]0 → o1 || o0[a]ot1 → ot11 || ot10, t1 = 0, 1[a]ot1t2 → ot1t21 || ot1t20, t1, t2 = 0, 1[…a]ot1t2…tn−1 → ot1t2…tn−11 || ot1t2…tn−10, t1, t2, …, tn−1 = 0, 1[neuron dissolution rule:a(a3)+]ot1t2…tn → δ, t1, t2, …, tn = 0, 1[a2]ot1t2…tn → δ, t1, t2, …, tn = 0, 1.0 → o1 || o0 is applied to generate neurons σo1 and σo0, which means one subset in regard to x1 has two choices: x1 is included in this subset (represent by 1) and x1 is not included in this subset (represent by 0). Synapses  and  are established through the inheritance of synapse ; synapses  and  are established through the inheritance of synapse ; the synapse  is established through synapse dictionary syn. The synapse between neurons d12 and o1 establishes a channel between the input and the subset having x1. At the same time, auxiliary neuron σ2 has one spike, rule a → a is applied and one spike is emitted to neuron σ1; auxiliary neuron σ3 has one spike, rulea → a; 2n − 1 is applied and one spike will be emitted to neurons σ2 at step 2n. The system after step one is shown in σ1 has one spike, the firing rule a → a is applied, and one spike is emitted to neurons σ2, σo1 and σo0.At step two, neuron σo1 and σo0 has one spike, the division rule [a]ot1 → ot11 || ot10 is applied to generate neurons σo11, σo10, σo01 and σo00, which means one subset in regard to x1 and x2 has four choices: x1x2 are included in this subset (represent by 11), x1 is included in this subset and x2 is not included in this subset (represent by 10), x1 is not included in this subset and x2 is included in this subset (represent by 01), and x1x2 are not included in this subset (represent by 00). Synapses , , , , , , , ,  and  are established through the inheritance of synapse , , ,  and ; synapses  and  are established through synapse dictionary syn. Synapses  and  establish channels between the input and the subsets having x1; synapses  and  establish channels between the input and the subsets having x2. At the same time, auxiliary neuron σ2 has one spike, rule a → a is applied and one spike is emitted to neuron σ1. The system after step three is shown in At step three, each of neurons n − 1, 2n neurons labeled ot1t2…tn are generated. The system after step 2n − 1 is shown in Similar process repeats. At step 2n, each neuron σot1t2…tn receives one spike emitted from neuron σ1 which will be deleted at the next step by the forgetting rule [a → λ]ot1t2…tn. Neuron σ2 receives two spikes , the forgetting rule a2 → λ is applied at step 2n + 1, and no spike will be emitted to neuron σ1 later. The system after step 2n is shown in At step 2Input Stage: At step 2n + 1, x1, x2, …, xn enter the system through input neurons σini. 3x1 + 3 spikes (ax1+33) enter neuron σin1; 3x2 + 3 spikes (ax2+33) enter neuron σin2;… 3xn + 3 spikes (axn+33) enter neuron σinn.n + 2, the firing rule a3(a3)+/a3 → a3 is applied, and three spikes are replicated and are emitted to neurons σdi1 and σdi2. Spikes in neuron σdi1 are forgotten, and spikes in neuron σdi2 are emitted to these σot1t2…,tn having synapses going from neuron σdi2 to them (These neurons represent the subsets having the integer xi.) at step 2n + 3. This process repeats until only 3 spikes are in neuron σini.At step 2n + xi + 2, the firing rule a3 → a is applied, and one spike is replicated and is emitted to neurons σdi1 and σdi2.At step 2n + xi + 3, the spike in neuron σdi1 is emitted to neuron σ4 showing that all spikes in neuron σini have been passed to neurons σot1t2…,tn having synapses going from neuron σdi2 to them. The spike in neuron σdi2 is forgotten. Up to this step, 3xi spikes are emitted to neurons σot1t2…,tn which represent the subsets having the integer xi.At step 2σini are passed to neurons σot1t2…tn at step 2n + xmax + 3, neuron σ4 receives n spikes, and one spike is emitted to neuron σs at step 2n + xmax + 4. Up to this step, the number of spikes in neurons σot1t2…,tn is When all input spikes in neurons Checking Stage: At step 2n + xmax + 5, 2s + 1 spikes are in neuron σs, the firing rule a(a2) + /a2 → a2 is applied, two spikes are emitted to neurons σot1t2…tn. This process lasts for S circles.n + xmax + s + 4, only one spike is in neuron s, and this spike is emitted to neurons σot1t2…tn.At step 2σot1t2…tn.There are three rule execution situations in neurons σot1t2…tn initially. 2 spikes are emitted to this neuron from neuron σS, then forgetting rule a2(a3)+/a5 → λ can be applied with 5 spikes consumed. The number of spikes decreases to S times, and all spikes in neuron σot1t2…tn are consumed. At this step, the last one spike is emitted to this neuron from neuron σS, forgetting rule a → λ can be applied to consume this spike.σot1t2…tn initially. 2 spikes are emitted to this neuron from neuron σS, then forgetting rule a2(a3)+/a5 → λ can be applied with 5 spikes consumed. The number of spikes decreases to σot1t2…tn are consumed. At this step, two spikes are emitted to this neuron from neuron σS, and neuron dissolution rule [a2]ot1t2…tn → δ is applied to dissolve this neuron.σot1t2…tn initially. 2 spikes are emitted to this neuron from neuron σS, then forgetting rule a2(a3)+/a5 → λ can be applied with 5 spikes consumed. The number of spikes decreases to S times, and σS is emitted to this neuron, and dissolution rule [a(a3)+]ot1t2…tn → δ is applied to dissolve this neuron.σot1t2…tn are still in the system after step 2n + xmax + s + 5, the labels of these neurons σot1t2…tn are all solutions to this Subset Sum problem.If some neurons SubsetSumproblem (n) can be solved in linear time, and all solutions can be obtained through this system.It can be seen that any k means that all x1, …, xn, S can be transformed into k-bit binary numbers.Some steps comparison results between our solution and other solutions, which use the non-deterministic method to solve the NP-complete problems, are shown in n − 1)-times computations should be processed. Although the time complexity of each computation is a constant, the whole time complexity cannot be a polynomial of n. The proposed DDSN P system can solve the Subset Sum problem in a linear time, which improves the computational efficiency.The conventional methods use the nondeterminism of SN P systems to solve Subset Sum problem, which means that whether a random combination of integers is one of the solutions or not can be checked by one computational process. These SN P systems can only judge whether a certain subset is the answer or not, but cannot search all solution space to judge whether a Subset Sum problem has solutions. Even if we let all combinations be traversed artificially to determine whether a Subset Sum problem has solutions or not, (2Subset Sum problem (4): X = {1, 2, 3, 4}, S = 5, the DDSN P system Π4 is used to solve it. After 22 computational steps, neurons σo0110 and σo1001 are remaining which shows that {2, 3} and {1, 4} are all solutions to this Subset Sum problem. Methods proposed in [Considering a Subset Sum problem posed in , 20, 23 X = {1, 2, …, n}, S = 5 are solved using the five systems in A series of Subset Sum problems: The new mechanism called neuron dissolution is introduced into the framework of SN P systems in this work. By this mechanism, redundant neurons can be dissolved immediately. The computational resources can be saved, which means more work can be done using the same resources, or the same work can be done using less resources. We also proved that this new variant of SN P system can obtain all solutions to NP-complete problems , such as SAT problem and the Subset Sum problem, in linear time, which enhances the application fields of SN P systems such as the register allocation problem.This work provides a new thought of storing information in SN P systems, which can be used to store other information. The dissolution rule can be used to many situations to decrease the space complexity of a SN P system. This variant of SN P system can be used to solve other NP-complete problems and application problems. It is also an attractive direction to introduce other biological phenomena into SN P systems to reduce computational resources and enhance computational space efficiency.


Scientific Reports3: Article number: 1627; 10.1038/srep01627 published online: 04092013; updated: 02022018.The original version of this Article contained an error in the spelling of the author Yangchao Shen, which was incorrectly given as Shen Yangchao.This error has now been corrected in the PDF and HTML versions of the Article.ViVj〉 terms in Table 1 were omitted from the calculation of In addition, the 〈should read:As a result, in the Abstract,5 random numbers that are guaranteed to have 5.2 × 104 bits of minimum entropy with a 99% confidence level.”“In our experiment, we generate 1 × 10should read:5 random numbers that are guaranteed to have 2.4 × 104 bits of minimum entropy with a 99% confidence level.”“In our experiment, we generate 1 × 10In the Results section, under subheading ‘Random number results’,δ does not have any noticable influence on the bound of min-entropy. Here we used the thresholds of KCBS violations “As shown in Table 1, we observe the expectation should read:δ does not have any noticeable influence on the bound of min-entropy. Here we used the thresholds of KCBS violations “As shown in Table 1, we observe the expectation In the title of Table 1,“Our experimental test clearly shows the violation of the extended inequality (3) with 31 σ”should read:“Our experimental test clearly shows the violation of the extended inequality (3) with 18 σ”ViVj〉 in Table 1 for the n = 2 × 105 with the new biased distribution parameter α = 12 in order to observe the net randomness. Therefore, the contents of the paper related to the biased choice of measurement settings should be corrected as follows.Moreover, the presented data for the biased choice of measurement settings does not show the net randomness after including the terms 〈In the Results section, under subheading ‘Random number results’,P (V1) = 1 − 4q, P (V2) = P (V3) = P (V4) = P (V5) = q, and q = αn−1/2 with α = 6 and n = 105. We observe basically the same behavior of the min-entropy for the generated stream except for a slightly smaller bound due to the non-uniform setting. We get the min-entropy bound 5 rounds with violation of 4) exceeds the input entropy (1.14 × 104), and we obtain 2.1 × 103 net random bits.”“We also generate random bits with a biased choice of measurement settings, where should read:P (V1) = 1 − 4q, P (V2) = P (V3) = P (V4) = P (V5) = q, and q = αn−1/2 with α = 12 and n = 2 × 105. We observe basically the same behavior of the min-entropy for the generated stream except for a slightly smaller bound due to the non-uniform setting. We get the min-entropy bound 5 rounds with violation of 4) exceeds the input entropy (3.28 × 104), and we obtain 6.8 × 103 net random bits.”“We also generate random bits with a biased choice of measurement settings, where In the legend of Figure 4,P(Vi) = 1/5 and (c) a biased distribution with P (V1) = 1 − 4q, P (V2) = P (V3) = P (V4) = P (V5) = q, where q = 6(100000)−1/2 with the probablity of errors “(a)(c)The min-entropy should read:P(Vi) = 1/5 and (c) a biased distribution with P (V1) = 1 − 4q, P (V2) = P (V3) = P (V4) = P (V5) = q, where q = 12(200000)−1/2 with the probability of errors “(a)(c)The min-entropy Figures 4 and 5 based on the corrections of the In addition, this Article contains typographical errors in the Results section, under subheading ‘The KCBS inequality’.v1〉 = |1〉, |v2〉 = |2〉, |v3〉 = R1  |v1〉, |v4〉 = R2  |v2〉, |v5〉 = R1  |v3〉 and γ = 51.83° and R1,2 denote the rotation operations between |1〉 to |3〉 and between |2〉 to |3〉, respectively.”“Here |should read:v1〉 = |1〉, |v2〉 = |2〉, |v3〉 = R1−1  |v1〉, |v4〉 = R1−1  R2  |v2〉, |v5〉 = R1−1  |v3〉R2−1  |v3〉 and γ = 103.68° and R1,2 denote the rotation operations between |1〉 to |3〉 and between |2〉 to |3〉, respectively.”“Here |In the legend of Figure 1,R1 and R2 represent the coherent rotations between |1〉 to |3〉 and between |2〉 to |3〉, respectively, where θ = 41.97° and ϕ = 64.09°. The sequence starts from |3〉 state (black filled circle) after optical pumping. (c)–(g) The pulse sequences for the measurement configurations (c) A1A2, (d) A2A3, (e) A3A4, (f) A4A5, (g) A5A’1, where γ = 51.84°.”“(b) The pulse sequence to prepare should read:R1 and R2 represent the coherent rotations between |1〉 to |3〉 and between |2〉 to |3〉, respectively, where θ = 83.94° and ϕ = 128.18°. The sequence starts from |3〉 state (black filled circle) after optical pumping. (c)–(g) The pulse sequences for the measurement configurations (c) A1A2, (d) A2A3, (e) A3A4, (f) A4A5, (g) A5A’1, where γ = 103.68°.”“(b) The pulse sequence to prepare


Residence” should have appeared as “3.1.3. Residence.” “3.1.1. Socioeconomic risk factors” should have appeared as “3.1.4. Socioeconomic risk factors.”In the article, “Is rotavirus still a major cause for diarrheal illness in hospitalized pediatric patients after rotavirus vaccine introduction in the Saudi national immunization program?”,


A novel aspect of the work is the introduction of a recommended distribution, incrementally learned from the data, to optimally refine the inferred network. Unlike existing system identification techniques, this “active learning” method automatically focuses its attention on key undiscovered areas of the network, instead of targeting global uncertainty indicators like parameter variance. We show how active learning leads to faster inference while, at the same time, provides confidence intervals for the network parameters. We present simulations on artificial small-world networks to validate the methods and apply the method to real data. Analysis of frequency of motifs recovered show that cortical networks are consistent with a small-world topology model.Understanding how groups of neurons interact within a network is a fundamental question in system neuroscience. Instead of passively observing the ongoing activity of a network, we can typically perturb its activity, either by external sensory stimulation or directly via techniques such as two-photon optogenetics. A natural question is how to use such perturbations to identify the connectivity of the network efficiently. Here we introduce a method to infer sparse connectivity graphs from  A fundamental question of system neuroscience is how large groups of neurons interact, within a network to perform computations that go beyond the individual ability of each one. One hypothesis is that the emergent behavior in neural networks results from their organization into a hierarchy of modular sub-networks, or motifs, each performing simpler computations than the network as a whole .To test this hypothesis and to understand brain networks in general we need to develop methods that can reliably measure network connectivity, detect recurring motifs, elucidate the computations they perform, and understand how these smaller modules are combined into larger networks capable of performing increasingly complex computations.in-vivo, two-photon imaging data. Advances in two-photon imaging are giving us the first look at how large ensembles of neurons behave in-vivo during complex behavioral tasks  and E[yc|Ri = 0]. The dependence on the κ parameter is not easy to discern at a glance, but we can observe how the Z-score behaves for various combinations of spiking rate parameters and κ values.Using this approximation, we can gain some intuition on how the Z-score will behave under certain conditions. In particular, we see that the Z-score depends approximately linearly on the number of observed interactions, it also depends on what is essentially a ratio between κ as a function of the fractional rate κ. We decided to choose κ = 10 for all experimental results since it seemed to provide a nice middle ground between dampening the Z-score for low spiking rate inhibitions and not giving disproportionately large Z-scores for small excitatory rates.Numerical issues arise from picking very large values of We analyze the behaviour of ill-conditioned but non-singular Fisher matrices, where there is a potentially large difference between the largest and smallest eigenvalues present in the Fisher information matrix, and show how the Wald test present in the model selection strategy copes with this issue. We use this analysis as we take the limit of the smallest eigenvalue going to 0 to illustrate what this test does for singular matrices.Overall, we provide a link between the smallest eigenvalue of the Fisher information matrix, and the largest observed parameter variance we obtain from the diagonal elements of the inverse Fisher information matrix. From this we finally conclude that regressor subsets that produce nearly singular matrices are consistently rejected from our model selection process.A, if λ is an eigenvalue of A, then λ−1 is an eigenvalue of A−1. So the inverse of a matrix with a very small eigenvalue has a correspondingly large eigenvalue.We first note that, for any non-singular matrix 1 ≥ λ2 ≥ … ≥ λr ≥ 0, where r is the number of rows (regressors) of the matrix. The corresponding eigenvalues of the inverse Fisher matrix are ordered: We also note that the Fisher information matrix is positive (semi) definite, we can thus sort its eigenvalues in descending order and write λxi is the i-th largest entry in the diagonal elements in the inverse Fisher matrix, the equality holds exactly for i = r. These values are the fisher variances of the regressors in Since the inverse Fisher information is symmetric, we can use the Schur-Horn theorem to see that its diagonal elements are majorized by its eigenvalues. By definition of majorization, this means that:From this, we can easily see that the largest observed fisher variance is at leastPAc yielding an ill-conditioned Fisher information matrix will be rejected.From this we can conclude that as the smallest value of the Fisher information matrix goes to 0, at least one regressor’s Fisher variance grows arbitrarily large. Relating this back into Eqs ν used for the random subset selection step of the elastic-forward model selection method. The ν parameter is tested in the 0.5 to 0.9 ranges. This is compared against the baseline oracle lasso method and model selection where no random subset selection is performed (no-subset).We evaluate the effect of varying the sample fraction F1, precision, and recall performance metrics. Results are shown in All methods are evaluated on samples drawn from simulated network SW1CL using the ν parameters perform similarly for all but the smallest sample sizes. Additionally, the use of no-subset model selection had a large performance decrease for small sample sizes when compared to all ν parameters, but similar performance on larger sample sizes.We can observe from A natural question that may arise is if the Poisson GLM model can still capture directed interactions between neurons when the data does not come from a Poisson distribution.To that effect, we repeated the simulated experiments using the same connectivity matrices defined for networks SW1CL and SW3CL, but this time, the spiking activity was obtained using a Leaky Integrate and Fire model .vc is the membrane potential of neuron c, and sc its corresponding spike train.The equations of the model can be summarized asW and H are the inter-neuron and stimuli-neuron connectivity matrices respectively. The parameter h(t) represent the influence kernel, and depends on the synaptic density (a) and kernel delay (tD). b(t) is the direct current parameter.b(t) from a uniform distribution (b(t) ∼ U), where b was chosen such that the average spiking rate of each neuron is the same as in the original Poisson GLM simulations. The W and H connectivity matrices were linearly scaled with respect to the original SW1CL network to preserve the conditional spiking rates. For these simulations, we set a = 1.5 and tD = 2.For our simulations we sampled From Algorithm 1 Elastic-forward selectionRequire:Xc sequence of observations of neuron c                       ▷                      ▷ k maximum number of regressors to add per step                    ▷ γ minimum allowed regressor p-value                         ▷ r′ = {}   ▷ Initialize set of active regressors to empty set (only bias is included). repeat                       ▷ Both sub routines are explained in algorithms 3 and 4if {ji} = ∅ then                              ▷ No suitable candidate foundreturn   end if    success = Falsen = |{ji}|  rbest = r′  best = BICr′  BICwhile not sucess do  r‡ = r′ + {ji}[: n]                              ▷ Update best set found so far using Algorithm 3ifmax ≤ γ and BICr‡ ≤ BICbestthen   rbest = r‡          end if          ▷ Already found best set in the descending sequence, update and exit loopif BICr‡ ≥ BICbest and rbest ≠ r′ then        success = Truer′ = rbest    end if   n = n − 1   end while  until True Algorithm 2 Iterative Active LearningRequire:X, I, k, γ, βX Initial observations for all neurons                           ▷ I Initial applied stimuli                                 ▷ k maximum number of regressors to add per step                    ▷ γ minimum allowed regressor p-value                         ▷ β active learning smoothing parameter                       ▷ forl ∈ interventions do X and I as defined for Eq (3)  forc ∈ neurons do                        ▷ Perform model selection using Algorithm 4   c   ▷ Get log-likelihood difference of considering all regressors that are not parent edges of neuron    forr ∈ Rnonparent(c)do   Lr, c ← Compute Loglikelihood difference    ▷ Use     end for   end for   ▷ Compute the impact and score of each stimuli according to previous observationsfors ∈ stimulido  forc ∈ neuronsdo   IMs,c ← Compute impact of stimuli s on neuron c             ▷ Use Eq (14)    end for   forsi ∈ stimulido   s on stimuli si          ▷ Use Eq (17)    end for   SCs ← Compute score of stimuli s                 ▷ Use Eqs  ← Acquire Samples            ▷Acquire samples of X and I drawing I with probability Pl+1  [0: k] {returnr† = r + {ji}, {ji}


It is desirable to implement an efficient quantum information process demanding fewer quantum resources. We designed two compact quantum circuits for determinately implementing four-qubit Toffoli and Fredkin gates on single-photon systems in both the polarization and spatial degrees of freedom (DoFs) via diamond nitrogen-vacancy (NV) centers in resonators. The gates are heralded by the electron spins associated with the diamond NV centers. In contrast to the ones with one DoF, our implementations reduce the quantum resource and are robust against the decoherence. Evaluations of fidelities and efficiencies of our gates show that our schemes may be implemented with current technology. Encoding qubits in multiple degrees of freedom (DoFs) increases the capacity of the quantum channel, reduces the quantum resources, and are robust against the decoherence caused by the environments. Single photon is nowadays recognized as an excellent candidate for quantum information process in multiple DoFs as it carries many qubit-like DoFsms) even at room temperature32384345et al.39314849Isolated electron-spin in diamond nitrogen-vacancy (NV) center has been recognized as an unique and promising candidate for solid-state quantum computing. The appeal feature of the diamond NV center is its urtralong coherence time  gates acting on the target qubits, i.e., UToffoli = H. CCCPF. H. Here, we employ the NV-cavity entangled platform to overcome the weak interactions between two single photons involved in the Toffoli gate. As shown in ms〉 = |±1〉 and |ms〉 = |0〉  are the ground triple states of the diamond NV center. In absence of a magnetic field, |±1〉 and |0〉 are split with 2.87 GHz due to the spin-spin interaction, and the corresponding transitions, |0〉 ↔ |±1〉, are in the microwave regime. The six electron excited states includeE1〉 = |E−〉|−1〉 − |E+〉|+1〉, |E2〉 = |E−〉|−1〉 + |E+〉|+1〉, |Ex〉 = |X〉|0〉, |Ey〉 = |Y〉|0〉, |A1〉 = |E−〉|+1〉 − |E+〉|−1〉, and |A2〉 = |E−〉|+1〉 + |E+〉|−1〉. |E±〉, and |X〉, |Y〉 are orbital states with angular momentum projections ±1 and 0, respectively. The energy gap arising from the spin-spin and spin-orbit interactions protects state |A2〉 against small strain and magnetic filed. The transitions, |±〉 → |A2〉, are driven by the resonator optical pluses with σ− and σ+-polarized at 637 nm, respectively  gate up to two Hadamard (vely see . Therefoσ+-polarized photon via port a (port c) senses a cold cavity, and departs through port b (d) with phase shift σ−-polarized photon via port a (port c) senses a hot cavity, and leaves from the cavity via port d (b) with phase shift σ−-polarized photon senses a cold cavity and it is transmitted, while the σ+-polarized photon senses the hot cavity and it is reflected. The reflection and transmission coefficients of the NV-cavity system can be obtained by solving the Heisenberg equations for the cavity mode a with frequency ωc and the diamond NV lowering operator σ− with frequency ω0, that isCompared with the coupling of an unconfined diamond NV center, trapping a diamond NV center into an optical cavity enhances the coupling between the incident photon and the diamond NV center. Nowadays, various of optical microcavities, such as photonic crystal cavityκ and γ/2 are the decay rates of the cavity field and the diamond NV dipole, respectively. κs/2 is the cavity intrinsic loss rate (side leakage). g is the coupling rate between the diamond NV center and the resonator. σz is the pauli operator. h and f are the noise operators. The input field is connected with the output filed by aking the weak excitation limitation, i.e., <σz> ≈ −1, through the operation, the transmission and reflection coefficients of the NV-cavity system can be respectively written as5758Here, ωc = ω0 = ωp, if the Purcell factor g2/(κγ) ≫ 1 with κs ≈ 0 is ignored, the cavity mode of the hot cavity is reflected, i.e., r(ωp) → 1 and t(ωp) → 0. If κs/κ ≪ 1, the cavity mode of the bare cavity is transmission, i.e., r0(ωp) → 0 and t0(ωp) → −1. That is, the input-output relations between the single photon and NV center can be summarized asUnder the resonator condition R, a〉 and |L, d〉 with sz = −1, |L, a〉 and |R, d〉 with sz = +1, the polarization of the photon is defined with respect to the z axis (the diamond NV center axis).Here, |The quantum circuit we designed, as shown in α1, α2, β1, β2, γ1, γ2, δ1, and δ2 are arbitrary complex parameters satisfying |α1|2 + |α2|2 = 1, |β1|2 + |β2|2 = 1, |γ1|2 + |γ2|2 = 1, and |δ1|2 + |δ2|2 = 1. The indices 1 and 2 of R (or L) denote the R- (or L-) polarized wave-packets of the photon 1 and photon 2, respectively. a1 (b1) and a2 (b2) represent photon 1 (photon 2) emitted from the spatial modes 1 and 2, respectively.where 1, to depart the wave-packets, R and L, of photon 1 emitted from the spatial mode a2., i.e., the R1-component is transmitted and the L1-component is reflected. The R1-component does not interact with the diamond NV center and arrives at CPBS2 directly. Before and after the L1-component interacts with the block composed of CPBS3, NV, and CPBS4, the two single-qubit rotations Ry(−π/2) and Ry(π/2) are respectively applied on photon 1 simultaneously with two Hadamard operations Hes applied on the diamond NV center. Subsequently, the R1-component and L1-component mix at CPBS2. The operations (CPBS1 → Ry(−π/2), He → CPBS3 → NV → CPBS4 → Ry(π/2), He → CPBS2) transform the system composed of the photon 1, photon 2, and diamond NV center from the initial state |ψ0〉 into |ψ1〉. Here,In He can be achieved by employing π/2 pulse and it completes the following transformationsThe Hadamard transformation Ry(±π/2) can be achieved by employing two half-wave plates oriented at 45° and 22.5°, respectively. In the standard basis {|R〉, |L〉}, Ry(±π/2) can be written asThe single-qubit rotations b2 is injected into the block. Before and after photon 2 interacts with the block, Ry(−π/2) → CPBS3 → NV → CPBS4 → Ry(π/2)) transform |ψ1〉 intoNext, photon 2 in the spatial mode π on the photon emitted from the spatial mode b2, i.e., |R2〉|b2〉 → −|R2〉|b2〉 and |L2〉|b2〉 → −|L2〉|b2〉. If the diamond NV in the state π and σz = |R〉〈R| − |L〉〈L|, are performed on the photons emitted from the spatial modes b2 and a2, respectively. After the above operations, the state of the system composed the two photons becomesThird, we measure the electron spin state of the diamond NV center in the basis L1〉|a2〉|L2〉|b2〉, and has no change otherwise. Therefore, a two-photon four-qubit Toffoli gate which performs a NOT operation on the spatial (or polarization) state of the photon 2 depending on the states of other qubits can be achieved by the scheme depicted by It is obvious that the quantum circuit shown in The Fredkin gate swaps the states of the two target qubits conditional on the states of the control qubits. The quantum circuit depicted by a2 into the block composed of CPBS3, phase shifters 4. Before and after the photon interacts with such block, Hadamard operations, Hes applied on the diamond NV centers. The above operations  part with coefficient r0 (t) of incident photon in the cold (hot) cavity. Therefore, in the realistic case, the optical transition rules of the incident photon described by R, a〉|+1〉 → r|L, d〉|+1〉 + t|R, b〉|+1〉, |R, a〉|−1〉 → t0|R, b〉|−1〉 + r0|L, d〉|−1〉, |L, c〉|+1〉 → r|R, b〉|+1〉 + t|L, d〉|+1〉, |L, c〉|−1〉 → t0|L, d〉|−1〉 + r0|R, b〉|−1〉, |R, c〉|+1〉 → t0|R, d〉|+1〉 + r0|L, b〉|+1〉, |R, c〉|−1〉 → r|L, b〉|−1〉 + t|R, d〉|−1〉, |L, a〉|+1〉 → t0|L, b〉|+1〉 + r0|R, d〉|+1〉, and |L, a〉|−1〉 → r|R, d〉|−1〉 + t|L, d〉|−1〉. It is note that the above undesirable parts will degrade the performance of the emitter. Hence, in order to quality the performances of our schemes, we should investigate the average fidelity, which is defined as ψr〉 and |ψi〉 are the normalized output states of the system in the realistic and idealistic cases, respectively. Therefore, the average fidelity of our Toffoli or Fredkin gate, The key element of our schemes is the NV-cavity emitter. In the idealistic case, the optical transition rules of the emitter is perfect, however, the phase and amplitude of the incident photon are not perfect in experiment, and they are the functions of the α = α1, sin α = α2, cos β = β1, sin β = β2, cos γ = γ1, sin γ = γ2, and cos δ = δ1, sin δ = δ2 are taken for g2/κγ and κs/κ are plotted in where cos noutput/ninput, i.e., the yield of the incident photons. noutput and nintput are the numbers of gate’s output and input photons, respectively. The average efficiencies of our Toffoli and Fredkin gates, Efficiency is another powerful and practical tool for qualifying the performance of the setup. In experiment, the efficiency is defined as α = α1, sin α = α2, cos β = β1, sin β = β2, cos γ = γ1, sin γ = γ2, and cos δ = δ1, sin δ = δ2. By calculating, one can find thatwhere cos g2/κγ and κs/κ are presented in The date shows g2/κγ and the lower κs/κ, the higher fidelity and efficiency. The high fidelities and efficiencies of our Toffoli and Fredkin gates can be achieved. For example, in the case κs/κ = 0.1 and g2/(κγ) = 2.4, κs/κ = 1 and g2/(κγ) = 2.4, From A2〉 and other excited states, signal to noise in the ZPL channel, imperfect spin initialization and detection. These imperfections degrade the fidelities and efficiencies of our two gates, and the imperfections which are due to the technical imperfection will be improved with further technical advances. Given the current technology, our schemes are more efficient than the traditional ones with one DoF, the synthesis-based ones which are rely on CNOT gates and single rotations, and the parity-check-based ones, and they may be possible in experiment.Our schemes are deterministic if the unavoidable effect of photon loss and imperfections are neglected. Our schemes in practice working with photon loss, non-ideal single-photon sources, unbalanced CPBSs and BSs, timing jitter, spin flips during the optical excitation, the mix between state |How to cite this article: Wei, H.-R. and Zhu, P.-J. Implementations of two-photon four-qubit Toffoli and Fredkin gates assisted by nitrogen-vacancy centers. Sci. Rep. 6, 35529; doi: 10.1038/srep35529 (2016).


The appendices are provided below.In the article “Can we reduce negative blood cultures with clinical scores and blood markers? Results from an observational cohort study”,Appendix 1: Clinical scores for prediction of positive blood culturesShapiro score . This score includes “major criteria”  and “minor criteria” . According to the original study, a blood culture is indicated if at least one major criterion or two minor criteria are present. In addition to this cut-off, we categorized each patient on the basis of a sum score of 0–15 points derived by assigning one point per “minor criterion” and 2 points per “major criterion”.Lee score [5]. Lee and colleagues found systolic blood pressure < 90 mm Hg, heart rate > 125 /minute, body temperature < 35°C or > 40°C, WBC < 4 × 109/L or > 12 × 109/L, platelets < 130 × 109/L, albumin < 3.3 g/dL or 33 g/L, and CRP > 17 mg/dL or 170 mg/L to accurately predict bacteremia in patients with CAP. As proposed by the authors, we assigned five points for body temperature <35°C or >40°C, three points each for hypotension, tachycardia, or CRP > 17 mg/dL, and 2 points each for WBC < 4 × 109/L or > 12 × 109/L, or albumin < 3.3 g/dL. The overall risk score was regarded as the sum of the calculated points, with the following risk categories being recognized: low risk: ≤ 5 points, moderate risk: ≥6–10 points, and high risk: ≥11 points.SIRS criteria [6]. As suggested by Jones and Lowes, who proposed that systemic inflammatory response syndrome (SIRS) could serve as a predictor of bacteremia, one point was assigned to each SIRS criterion: temperature >38.3°C or <36°C, heart rate ≥ 90 /min, tachypnea with a respiratory rate ≥ 20 breaths/min, WBC > 12 × 109/L, or normal WBC with ≥10% immature (band) forms.Metersky score [7]. This score includes antibiotic treatment, liver disease, systolic blood pressure < 90 mm Hg, temperature < 35°C or ≥40°C, pulse ≥ 125 /min, BUN ≥ 30 mg/dL or urea ≥ 10.7 mmol/L, sodium < 130 mmol/L, and WBC < 5 × 109/L or >20 × 109/L. As recommended in the original publication, the risk of bacteremia was evaluated according to the number of predictors present: low-risk-group with zero predictors and prior antibiotics, moderate-risk group with zero predictors and no prior antibiotics or with one predictor and prior antibiotics, and high-risk group with one predictor and no prior antibiotics or with two or more predictors.Tokuda score I and II [8]. For Tokuda score I, the presence of chills, pulse > 120 /min, and high risk infective site was evaluated and patients were classified as follows with regard to their risk for bacteremia: low risk (no chills and pulse < 120 /min or chills and physicians’ diagnosis of low risk infective site), intermediate risk (no chills and pulse > 120 /min), and high risk (chills and physicians’ diagnosis of high risk infective site) as proposed in the original article. The Tokuda score II was calculated similarly, except that the variable pulse was replaced by CRP and the following risk groups were recognized: low risk (no chills and CRP < 10 mg/dL or 100 mg/L or chills and physicians’ diagnosis of low risk infective site), intermediate risk (no chills and CRP > 10 mg/dL or 100 mg/L), and high risk (chills and physicians’ diagnosis of high risk infective site).Appendix 1: Clinical scores for prediction of positive blood culturesAppendix 2: Microorganisms isolated from blood culturesAppendix 3: Characteristics of patients with true positive blood cultures and PCT < 0.25 μg/L


In the article titled “Epstein-Barr Virus-Induced Gene 3 (EBI3) Blocking Leads to Induce Antitumor Cytotoxic T Lymphocyte Response and Suppress Tumor Growth in Colorectal Cancer by Bidirectional Reciprocal-Regulation STAT3 Signaling Pathway” , there wThe title should be corrected to “Epstein-Barr Virus-Induced Gene 3 (EBI3) Blocking Leads to Induction of Antitumor Cytotoxic T Lymphocyte Response and Suppression of Tumor Growth in Colorectal Cancer by Bidirectional Reciprocal-Regulation STAT3 Signaling Pathway.”In the last sentence of the Introduction, “suppress” should be corrected to “promote.”In the Materials and Methods, in Section  2.3 “Balb/c” should be corrected to “BALB/c,” in Section  2.4 “Balb/c” should be corrected to “BALB/c” and “RPMI1640” should be corrected to “RPMI 1640,” and in Section  2.9 “1x PBS” should be corrected to “1 × PBS.”In the legend of Figure  2 “Balb/c” should be corrected to “BALB/c” and “RPMI1640” to “RPMI 1640.”


L2][BF4]2⋅Me2CO  is high‐spin at room temperature, and undergoes an abrupt, hysteretic spin‐crossover at T1/2=137 K (ΔT1/2=14 K) that proceeds to about 50 % completeness. This is associated with a crystallographic phase transition, from phase 1  to phase 2 . The cations associate into chains in the crystal through weak intermolecular π⋅⋅⋅π interactions. Phase 2 contains a mixture of high‐spin and low‐spin molecules, which are grouped into triads along these chains. The perchlorate salt [FeL2][ClO4]2⋅Me2CO also adopts phase 1 at room temperature but undergoes a different phase transition near 135 K to phase 3  without a change in spin state.Crystalline [Fe The structural chemistry of spin‐crossover (SCO) compoundsCrystallographic symmetry breaking during SCO is observed in a number of materials.2]2+ ,Z′) observed in a molecular compound.As part of our continuing investigations of complexes derived from 2⋅6 H2O or Fe[ClO4]2⋅6 H2O in acetone afforded crystalline [FeL2][BF4]2⋅Me2CO (4]2⋅1[BFMe2CO) and [FeL2][ClO4]2⋅Me2CO (4]2⋅12⋅1[BF4]2} assemblies, through N−H⋅⋅⋅F hydrogen bonds between the acetamido substituents and BF4− ions . The loss of the crystallographic c glide and inversion center in phase 2, together with its unit cell volume expansion, generates 24 unique molecules in its asymmetric unit which are labelled ‘A′–‘X′  spin‐state patterning. The four mixed‐spin molecules are well‐separated from each other in the lattice  and φ ‐Fe(1)‐N(22) in Figure Dd2 symmetric complex gives θ=90° and φ=180°. Most low‐spin [Fe(bpp)2]2+ derivatives approach these values, but high‐spin complexes show much more variation. In practice, high‐spin complexes deviating more strongly from the ideal values of θ and φ are less likely to transform to their low‐spin state upon cooling.The presence or absence of SCO in solid, high‐spin [Fe(bpp)s N(2)‐Fe‐N22) in in2]2+ dθ versus φ relationship, which is not usual in plots of this type.Notably, nine of the ten high‐spin cations in phase 2 have a more distorted coordination geometry than the high‐spin molecule in phase 1, which could explain why they remain high‐spin at low temperatures Figure . Interes4]2⋅12⋅12⋅1[ClOMe2CO indeed remains predominantly high‐spin between 5–300 K. However, an abrupt reduction of χMT from 3.3 to 3.0 cm3 mol−1 K occurs reproducibly near 145 K, close to the crystallographic phase transition temperature  should be addressed to the authors.SupplementaryClick here for additional data file.


It also studies k-conjugacy of partial arrays.Research in combinatorics on words goes back a century. Berstel and Boasson introduced the partial words in the context of gene comparison. Alignment of two genes can be viewed as a construction of two partial words that are said to be compatible. In this paper, we examine to which extent the fundamental properties of partial words such as compatbility and conjugacy remain true for partial arrays. This paper studies a relaxation of the compatibility relation called  The stimulus for recent works on combinatorics is the study of biological sequences such as DNA and protein that play an important role in molecular biology [The genetic information in almost all organisms is carried by molecules of DNA. A DNA molecule is a quite long but finite string of nucleotides of 4 possible types:  biology –3. Seque biology , 4 in or biology  considerA of size  over Σ, a finite alphabet, is partial function A : Z+2 → Σ, where Z+ is the set of all positive integers. In this paper, we extend the combinatorial properties of partial words to partial arrays. Also, this paper studies a relation called k-compatibility where a number of insertions and deletions are allowed as well as k-mismatches. The conjugacy result [k-Conjugacy of partial arrays is discussed.Partial array y result  which waIn this section, we give a brief overview of partial words .u of length n over A, a nonempty finite alphabet, is partial map u : {1,2,…, n} → A. If 1 ≤ i ≤ n, then i belongs to the domain of u (denoted by Domain(u)) in the case where u(i) is defined, and i belongs to the set of holes of u (denoted by Hole(u)), otherwise.Partial word A with an empty set of holes.A word –10 is a u be a partial word of length n over A. The companion of u (denoted by u◊) is map u◊ : {1,2,…, n} → A ∪ {◊} defined byLet u◊ = ba◊ab◊ is the companion of the partial word. The length of the partial word is 6. D(u) = {1,2, 4,5}. H(u) = {3,6}.The symbol ◊ is viewed as a “do not know” symbol. Word u and v be two partial words of length n. Partial word u is said to be contained in partial word v (denoted by u ⊂ v), if Domain(u) ⊂ Domain(v) and u(i) = v(i) for all i ∈ Domain(u). Partial words u and v are called compatible (denoted by u↑v), if there exists partial word w such that u ⊂ w and v ⊂ w (in which case we define u∨v by u ⊂ u∨v and v ⊂ u∨v and Domain(u∨v) = Domain(u) ∪ Domain(v)). As an example, u◊ = aba◊◊a and v◊ = abab◊a.Let (i)Multiplication: If u↑v and x↑y, then ux↑vy.(ii)Simplification: If ux↑vy and |u| = |v|, then u↑v and x↑y.(iii)Weakening: If u↑v and w ⊂ u, then w↑v.The following rules are useful for computing with partial words:u, v, x, y be partial words such that ux↑vy. (i)u | ≥|v|, then there exist partial words w, z such that u = wz, v↑w, and y↑zx.If |(ii)u | ≤|v|, then there exist partial words w, z such that b = wz, v↑w, and x↑zy.If |Let u and v are called conjugate, if there exist partial words x and y such that u ⊂ xy and v ⊂ yx.Two partial words u and v are called k-conjugate, if there exist nonnegative integers k1, k2 whose sum is k and partial words x and y such that u⊂k1xy and v⊂k2yx.Two partial words This section is devoted to review the basic concepts on partial arrays .A of size  over Σ, a nonempty set or an alphabet, is partial function A : Z+2 → Σ, where Z+ is the set of all positive integers. For 1 ≤ i ≤ m, 1 ≤ j ≤ n, and if A is defined, then we say that  belongs to the domain of A  ∈ D(A)). Otherwise, we say that  belongs to the set of holes of A  ∈ H(A)).Partial array An array  over Σ iA is a partial array of size  over Σ, then the companion of A (denoted by A◊) is total function A◊ : Z+2 → Σ ∪ {◊} defined byIf A → A◊ allows defining the catenation of two partial arrays in a trivial way.The bijectivity of map A of size , where D(A) = {, , , , , , }, H(A) = {, }.Partial array LetA and B are two partial arrays of equal size, then A is contained in B denoted by A ⊂ B if D(A)⊆D(B) andIf A and B are said to be compatible denoted by A↑B, if there exists partial array C such that A ⊂ C and B ⊂ C.Partial arrays The rules mentioned for partial words are also true for partial arrays.A,  B,  X,  Y be partial arrays.(i)Multiplication: If A↑B and X↑Y, then AX↑BY either by column catenation or by row catenation.(ii)Simplification: If AX↑BY either by column catenation or by row catenation with A and B being of same size, then A↑B and X↑Y.(iii)Weakening: If A↑B and C ⊂ A, then C↑B.Let A,  B,  X,  Y be partial arrays such that AX↑BY, either by column catenation or by row catenation. (i)A≥ order of B, then there exist partial arrays C, Z such that A = CZ, B↑C, and Y↑ZX.If order of (ii)A≤ order of B, then there exist partial arrays C, Z such that B = CZ, A↑C, and X↑ZY.If order of Let A and B are two partial arrays of same size and k is nonnegative integer, then A is said to be k-contained in B denoted by A⊂kB if D(A) ⊂ D(B) and there exists subset E of D(A) of cardinality k called the error set such that If A and B are two partial arrays of same order and k is a nonnegative integer, then A and B are called k-compatible denoted by A↑kB if there exist partial array Z and nonnegative integers k1,  k2 such that(i)A⊂k1Z with error set E1;(ii)B⊂k2Z with error set E2;(iii)E1∩E2 = ϕ;(iv)k1 + k2 = k.If E1 = {, }, E2 = {} and k1 = 2, k2 = 1⇒k = 3; that is, A↑3B.A and B are k-compatible, if there exists subset E of D(A)∩D(B) of cardinality k called the error set such that (i)A = B∀ ∈ D(A)∩D(B)∖E;(ii)A ≠ B∀ ∈ E.If A and B are arrays, then A↑∘B means A = B. We sometimes use notation A↑k≤B, if set E has cardinality ≤k.Equivalently, Multiplication. If A↑k1B and X↑k2Y, then AX↑k1+k2BY where A,  B,  X, and Y are partial arrays and k1,  k2 are nonnegative integers, using column catenation.AX↑6+7BY.Simplification. If AX↑kBY and order of A is equal to order of B, then A↑k1B and X↑k2Y for some k1,  k2, satisfying k1 + k2 = k.AX↑8BY⇒A↑5B and X↑3Y with 5 + 3 = 8.Weakening. If A↑kB and Z ⊂ A, then Z↑k≤B.Z↑≤7B with k = 7.A and B be partial arrays of orders a × b and a × c, respectively. If there exist array Z of order a × d and integers k1,  k2,  m, and n such that A⊂k1Zm with error set E1 and B⊂k2Zn with error set E2, then there exist integers p and q such that Ap↑k≤Bq withE1∩E2 = ϕ, then Ap↑kBq.Let A and B be partial arrays of a × b and a × c, respectively. Let array z of order a × d exist such that, by using column catenation, A⊂k1Zm and B⊂k2Zn for some integers k1,  k2,  m, and n. Let E1 be the error set of cardinality k1 such that A = Zm for all  ∈ D(A)∖E1 and A ≠ Zm for all  ∈ E1 and E2 be the error set of cardinality k2 such that B = Zn for all  ∈ D(B)∖E2 and B ≠ Zn for all  ∈ E2. We have An⊂nk1Zmn with error set E1 of cardinality nk1 and Bm⊂mk2Zmn with error set E2 of cardinality mk2.Let i, j)≤ and Zmn = a for some letter a. There are 4 possibilities.Let ≤ ∉ E1 and  ∉ E2, then An∈{◊, a} and Bm∈{◊, a}. It does not give any error, when we align An with Bm.Case 2. If  ∉ E1 and  ∈ E2, then An∈{◊, a} and Bm = b for some b ≠ a. It gives an error in the alignment of An with Bm only when An = a or when  ∈ D(A).Case 3. If  ∈ E1 and  ∈ E2, then Bm∈{◊, a} and An = b for some b ≠ a. It gives an error in the alignment of An with Bm only when Bm = a or when  ∈ D(B).Case 4. If  ∈ E1 and  ∈ E2, then An = b for some b ≠ a and Bm = c for some c ≠ a. It gives an error in the alignment of An with Bm only when b ≠ c.E1∩E2 = ϕ then An↑kBm with k = ‖(D(a)∩E2)∪(D(B)∩E1‖ and E1∩E2 ≠ ϕ then An↑k≤Bm.Therefore, if A⊂4Z3 with error set E1 = {, , , }, and  B⊂2Z2 with error set E2 = {, }. k = 6:(i)D(A) = {, , , , , ,}. D(B) = {, , , , }(ii)D(A) = {, , , , , , , , , , , , , }. D(B) = {, , , , , , , , , , , , , , }(iii)E1 = {, , , , , , , }. E2 = {, , , , , }. E1∩E2 ≠ ϕ. k = ‖(D(A)∩E2) ∪ (D(B)∩E1)‖ = ‖, , , , , , , , , , , , , )∩, , , , , ))∪, , , , , , , , , , , , , , )∩, , , , , , , ))‖ = ‖, , , , , , , , ‖. k = 9:We have A2↑≤9B3(A2↑6B3).A and B of same order are called conjugate if there exist partial arrays X and Y such that A ⊂ XY and B ⊂ YX using row catenation or column catenation.Two partial arrays 0-conjugacy on partial arrays with same order is trivially reflexive and symmetric but not transitive.A ⊂ XY and B ⊂ YX showing that A and B are conjugate similarly and, by taking B ⊂ X′Y′ and C ⊂ Y′X′ showing that B and C are conjugate. But A and C are not conjugate.By taking That is, conjugate relation is not transitive.A and B be nonempty partial arrays of same size. If A and B are conjugate, then there exists partial array C such that AC↑CB, either by column catenation or by row catenation.Let A and B be nonempty partial arrays of same order. Suppose A and B are conjugate and let X,  Y be partial arrays such that A ⊂ XY and B ⊂ YX either by column catenation or by row catenation; then AX ⊂ XYX and XB ⊂ XYX. So, for C = X, we have AC↑CB.Let A and B of same order are k-conjugate, if there exist nonnegative integers k1k2 whose sum is k and partial arrays X and Y such that A⊂k1XY and B⊂k2YX with row or column catenation.Two partial arrays A and B be nonempty partial arrays of same order. If A and B are k-conjugate, then there exists partial array Z such that AZ↑k≤ZB.Let A,  B be two partial arrays of same order. Supposing that A and B are k-conjugate, then, by definition, there exist nonnegative integers k1, k2 whose sum is k and partial arrays X and Y such that A⊂k1XY with error set E1 and B⊂k2YX with error set E2 using row catenation or column catenation accordingly.Let AX⊂k1XYX with error set E1 and XB⊂k2XYX with error set E2′ = {/ ∈ E2} or E2′ = {/ ∈ E2} according to row or column catenation and so, for Z = X, we have AZ↑k≤ZB.Then, Given A⊂3XY and B⊂2YX, k = k1 + k2 = 5.There exist AZ↑≤5ZB.There exist k-compatibility and k-conjugacy of partial arrays. We prove that, given partial arrays A,  B and integers p,  q satisfying |A|p = |B|q, we find k such that Ap↑kBq. Also, there exists partial array Z such that AZ↑k≤ZB.Motivated by compatibility and conjugacy properties of partial words, we define the conjugacy of partial array and derive the compatibility properties of partial arrays. By giving relaxation to the compatibility relation, we discuss


Scientific Reports7: Article number: 4005410.1038/srep40054; published online: 01102017; updated: 03142017This Article contains typographical errors in the Results section under subheading ‘ISs fragment the up states in the opposite hemisphere’.“The presence of ISs in Hem 2 was associated with a change of the statistics of the slow-wave oscillations in Hem 1; in particular, as shown in Fig. 3c,d, the frequency of up states increased from 0.70 ± 0.06 Hz to 0.81 ± 0.06 Hz , while their duration decreased from 0.74 ± 0.04 s to 0.64 ± 0.03 s ”.should read:“The presence of ISs in Hem 2 was associated with a change of the statistics of the slow-wave oscillations in Hem 1; in particular, as shown in Fig. 3c,d, the frequency of up states increased from 0.57 ± 0.03 Hz to 0.64 ± 0.02 Hz , while their duration decreased from 0.74 ± 0.04 s to 0.64 ± 0.03 s ”.


Its general properties have been used to study the behavior of phylogenetic tree shape indices under the probability distribution it defines. But the explicit formulas provided by Ford for the probabilities of unlabeled trees and phylogenetic trees fail in some cases. In this paper we give correct explicit formulas for these probabilities.Ford's  In particular, we shall call the image of a cladogram under v in a tree T, κT(v) is its number of descendant leaves. For every internal node v in an ordered tree T, with children v1≺vv2, its numerical split is the ordered pair NST(v) = (κT(v1), κT(v2)). If, instead, T is unordered and if child(v) = {v1, v2} with κT(v1) ⩽ κT(v2), then NST(v) = (κT(v1), κT(v2)). In both cases, the multiset of numerical splits of T is NS(T) = {NST(v)∣v ∈ Vint(T)}. For instance, if T is the cladogram depicted in Let us introduce some more notations. For every node  symmetric branch point in a tree T is an internal node v such that if v1 and v2 are its children, then the subtrees Tv1 and Tv2 of T rooted at them have the same shape. For instance, the symmetric branch points in the cladogram depicted in AT and T′ on Σ and Σ′, respectively, with Σ∩Σ′ = ∅, their root join is the cladogram T⋆T′ on Σ ∪ Σ′ obtained by connecting the roots of T and T′ to a (new) common root r; see T, T′ are ordered cladograms, T⋆T′ is ordered by inheriting the orderings on T and T′ and ordering the children of the new root r as rT≺rrT′. If T and T′ are tree shapes, a similar construction yields a tree shape T⋆T′; if they are moreover ordered, then T⋆T′ becomes an ordered tree shape as explained above.Given two cladograms α-model [n⩾1, a family of probability density functions Pα,n∗) into two arcs  and  and then adding a new arc from the inserted node w to a new leaf) or to a new root . The value of Pα,n∗)((T∗) for T∗ ∈ 𝒯n∗ is determined through all possible ways of constructing cladograms with shape T∗ in this way. More specifically,(1)T1 and T2 denote, respectively, the only cladograms in 𝒯1 and 𝒯2, let Pα,1′(T1) = Pα,2′(T2) = 1;if (2)m = 3,…, n, let Tm ∈ 𝒯m be obtained by adding a new leaf labeled m to Tm−1. Then for every (3)n of leaves is reached, the probability of every tree shape Tn∗ ∈ 𝒯n∗ is defined as When the desired number It is well known  that eve𝒯5 with the same shape and how their probability Pα,5′ is built using the recursion in Step (2). If we generate all cladograms in 𝒯5 with this shape, we compute their probabilities Pα,5′, and then we add up all these probabilities, we obtain the probability Pα,5∗)/(4 − α); cf. [For instance, 2(1 − α)/ − α; cf.Pα,n∗)T ∈ 𝒯n, if π(T) = T∗ ∈ 𝒯n∗ and it has k symmetric branch points, thenFor every π−1(T∗)| = n!/2k To ∈ 𝒪𝒯n, if πo(To) = T ∈ 𝒯n, thenFor every πo−1(T)| = 2n−1 , v ∈ Vint(T∗)).because |(iii)To∗ ∈ 𝒪𝒯n∗, if πo,∗(To∗) = T∗ ∈ 𝒯n∗ and it has k symmetric branch points, thenFor every πo,∗−1(T∗)| = 2n−1−k , v ∈ Vint(T∗), those differing only on the orderings on the children of the k symmetric branch points are actually the same ordered tree shape).because |Once Pα,no,∗)n of probabilities of ordered tree shapes satisfies the useful Markov branching recurrence  = 1 and, for every n⩾2, Γα(n) = (n − 1 − α) · Γα(n − 1).The family  givm < n and for every Tm∗ ∈ 𝒪𝒯m∗ and Tn−m∗ ∈ 𝒪𝒯n−m∗, For every 0 < Pα,1o,∗), for every n⩾1 and T ∈ 𝒯n.Our first result is an explicit formula for T ∈ 𝒯n, its probability under the α-model is For every T ∈ 𝒯n, let To be any ordered cladogram such that πo(To) = T, and let To∗ = π∗(To) ∈ 𝒪𝒯n∗ and T∗ = π(T) = πo,∗(To∗). If T∗ has k symmetric branch points, then, by , we denote its children by v1 and v2, then v ∈ Vint(T)∖{rT}, the term Γα(κT(v)) appears twice in this product: in the denominator of the factor corresponding to v itself and in the numerator of the factor corresponding to its parent. Therefore, all terms Γα(κT(v)) in this product vanish except Γα(κT(rT)) = Γα(n) (that appears in the denominator of its factor) and every Γα(κT(v)) = Γα(1) = 1 with v, a leaf. Thus, Given then, by , 6), an, anT ∈ 𝒯then, by , (12)Pα,T ∈ 𝒯n, then k is the number of symmetric branching points in T and Pα,n(T) becomesm is the number of internal nodes whose children have different numbers of descendant leaves. This formula does not agree with the one given in k + m < n − 1. The first example of a cladogram with this property  is the cladogram ProblsAlpha.pdf in https://github.com/biocom-uib/prob-alpha)α = 1/2, Ford's model is equivalent to the uniform model, where every cladogram in 𝒯n has the same probabilityα = 0, Ford's model gives rise to the Yule model [T ∈ 𝒯n is Ford states (see [) becomesPα,nT=2k+stly, by , §3.12, le model , where t𝒯8 the probabilities computed with ProblsAlpha.pdf), these probabilities do not add up 1.As a second reason, which can be checked using a symbolic computation program, let us mention that if we take expression  as the pCombining T∗ ∈ 𝒯n∗ with k symmetric branch points, For every This formula does not agree, either, with the one given in [Pα,n)n satisfies the following Markov branching recurrence.The family of density mappings n does not satisfy any Markov branching recurrence; that is, there does not exist any symmetric mapping Q : ℤ+ × ℤ+ → ℝ such that, for every k, l⩾1 and for every Tk ∈ 𝒯k∗ and Tl ∈ 𝒯l∗, m leaves and k symmetric branch points, for instance, the tree shapes in 𝒯6∗ depicted in Tm∗⋆Tm∗ ∈ 𝒯m2∗ has 2k + 1 symmetric branch points and therefore k symmetric branch points and therefore qα ≠ 2qα. This shows that there does not exist any well-defined, single real number Q such that Tm1,∗, Tm2,∗ ∈ 𝒯m∗.Against what is stated in , (Pα,n


Scientific Reports7:12636; doi:10.1038/s41598-017-13049-0; Article published online 03 October 2017The original HTML version of this Article contained errors in Table 1, where text was inadvertently inserted on three occasions.AGATCTYCACGAGCCCTATCAGGTGTCTTT Bgl II” and now reads “GAAGATCTYCACGAGCCCTATCAGGTGTCTTT Bgl II”.For the Target gene ‘CP hairpin sense strand’ with Primer name ‘CP-RNAi-P2’, the Primer sequence (5′-3′) incorrectly read “GAcomment=” Underline the selected text “GTCGACTCAACGCCGGAACTAGTGGAACTT Sal I” and now reads “GCGTCGACTCAACGCCGGAACTAGTGGAACTT Sal I”.For the Target gene ‘CP hairpin reverse strand’ with Primer name ‘CP-RNAi-P3’, the Primer sequence (5′-3′) incorrectly read “GCcomment=” Underline the selected text “comment=” Underline the selected text “GGATCCTCACGAGCCCTATCAGGTGTCTTT BamH I” and now reads “CGGGATCCTCACGAGCCCTATCAGGTGTCTTT BamH I”.For the Target gene ‘CP hairpin reverse strand’ with Primer name ‘CP-RNAi-P4’, the Primer sequence (5′-3′) incorrectly read “CGcomment =” Underline the selected text “comment =” the selected text“comment =” Underline the selected text “These errors have been corrected in the HTML version of the Article; the PDF version was correct from the time of publication.


In the article titled “Intraoperative Contrast Enhanced Ultrasound Evaluates the Grade of Glioma” , there wIn Section  2.7.3 (Expression of VEGF), “41% to 75% was (+)” should be corrected to “41% to 75% was (+ +).”The age ranges were incorrectly reported in Section  3.1 (The Basic Condition of the Patients) as “20 to 69 years with a mean age of 47.9 ± 11.4” and should be corrected to “18 to 69 years with a mean age of 45 ± 12.8 years.”


The statement in the Funding section is incorrect. The correct funding information is as follows: This work was supported by grants from the Hungarian Scientific Research Fund OTKA PD104878 and K109249, and by TÁMOP 4.2.2.A‐11/1/KONV‐2012‐0025, TÁMOP‐4.2.2/B‐10/1‐2010‐0024, and TÁMOP‐4.2.4.A/2‐11‐1‐2012‐0001 projects, and by the University of Debrecen (RH/751/2015).


Scientific Reports6: Article number: 23946; 10.1038/srep23946 published online: 04042016; updated: 01192017.−10 Pa−1)’ is incorrectly given as ‘Acoustic compressibility (TPa−1)’. The correct This Article contains errors. In The legend of Figure 3 contains errors.−1 for MCF7 and 4.3 ± 0.2  TPa−1 for MDA’.‘Cellular acoustic compressibility versus cell size: MCF7 and MDA show a very similar cell size, 17.3 ± 1.0 μm, but different acoustic compressibility, 3.8 ± 0.3 TPashould read:−10 Pa−1 for MCF7 and 4.3 ± 0.2 10−10 Pa−1 for MDA’.‘Cellular acoustic compressibility versus cell size: MCF7 and MDA show a very similar cell size, 17.3 ± 1.0 μm, but different acoustic compressibility, 3.8 ± 0.3 10−10 Pa−1)’ is incorrectly given as ‘Acoustic compressibility (TPa−1)’. The correct In In the Results section under the subheading ‘Determination of cellular acoustic compressibility and optical deformability on the same cell’,−1)”.“The results obtained here indicate that if one uses an OD threshold-value of ≈11%, the cell identification based on OD introduces an error of 25%; while this error drops to 12% by using AC (threshold ≈ 4.07 TPashould read:−10 Pa−1).”“The results obtained here indicate that if one uses an OD threshold-value of ≈11%, the cell identification based on OD introduces an error of 25%; while this error drops to 12% by using AC (threshold ≈ 4.07 10


Upon publication of the original article , it was The second line of the subsection “Read alignment” currently reads “--outSJfilterOverhangMin 12 12 12” and should instead be “--outSJfilterOverhangMin 12 12 12 12”.


Scientific Reports6: Article number: 21259; 10.1038/srep21259 published online: 02162016; updated: 01292018.This Article contains typographical errors. In Table 1, the ‘Yield per plant (g)’ for PAT11 (NE) “23.09 ± 0.92” should read “23.09 ± 8.11”. In the Methods section under subheading ‘PCR analysis of transgenic rice’,2+ plus), 0.4 μL 10 mM dNTP, 0.3 μL 10 μM RePAT-F (5′-GGATCCAGACTCACTCTGAG-3′), 0.3 μL 10 μM RePAT-R (5′-GCATGCGGTGGACACGCTGG-3′) and 1 U Taq DNA polymerase in a total volume of 20 μL, and under the conditions of 94 °C for 5 min, then 30 cycles of 94 °C for 30 s, 58 °C for 30 s, 72 °C for 30 s, and finally 72 °C for 8 min.”“PCR assay was performed in a mixture containing 50 ng rice genomic DNA or 1 ng plasmid DNA, 2 μL 10 × PCR buffer (Mgshould read:2+ plus), 0.4 μL 10 mM dNTP, 0.3 μL 10 μM RePAT-F (5′-TTCCTTAAAGCGAAAACCCC-3′), 0.3 μL 10 μM RePAT-R (5′-ACGATCCTGAAGCCGAACCT-3′) and 1 U Taq DNA polymerase in a total volume of 20 μL, and under the conditions of 94 °C for 5 min, then 30 cycles of 94 °C for 30 s, 58 °C for 30 s, 72 °C for 30 s, and finally 72 °C for 8 min.”“PCR assay was performed in a mixture containing 50 ng rice genomic DNA or 1 ng plasmid DNA, 2 μL 10 × PCR buffer (Mg


AbstractHemiptera, Phylloxeridae) is presented. Six family-group names are listed, three being synonyms. Thirty-five genus-group names, of which six are subjectively valid, are presented with their type species, etymology, and grammatical gender. Ninety-four species-group names are listed, of which 73 are considered subjectively valid. This is the last group of Aphidomorpha to be catalogued, bringing the list of valid extant species to 5,218.A taxonomic and nomenclatural catalog of the phylloxerids ( Phylloxeridae is a small family of Hemiptera, closely related to Adelgidae and Aphididae. Little is known of the biology of most of the family’s 69 species, although that of the economically important grape phylloxeran, Daktulosphairavitifoliae (Fitch), has been studied in detail. Most species of phylloxerid feed on species of Juglandaceae or Fagaceae, with a large number forming galls on North American hickories (Carya spp.). Host alternation exists within the family  .rhyncha) . Whereasal times  and the recently , the Physil taxa , the entIn this catalog, we present six family-group, 35 genus-group, and 94 species-group names of extant phylloxerids. The family-group names include two valid subfamilies and two valid tribes and three subjective synonyms. The genus-group names include six valid names, 21 junior subjective synonyms, three junior objective synonyms, three junior homonyms, and two unavailable names. The species-group names include four subspecies , 14 subjective synonyms, one junior primary homonym, two nomina dubia, and four unavailable names.Phylloxeridae in English is usually pronounced with the accent on the third syllable. However, the name of its type genus, Phylloxera, is often pronounced with an accent on the second. Because the e of xērós is an eta, the word made from it, once written in Roman letters and given Latin endings, must be considered to have a long e. The penultimate syllable of a Latin word must be accented when it contains such a long vowel and it is a fixed principle that the accentuation of Latin words is to be kept when they are borrowed into English. Therefore, strictly-speaking, only accentuation of the third syllable of Phylloxera is historically justified.The name DaktulosphairaDactylosphaera .Also as with other recent catalogs of groups of included . Where oPageBreakPHYLLOXERIDAEHerrich-Schaeffer 1854PHYLLOXERINAEHerrich-Schaeffer 1854Subfamily ACANTHOCHERMESINITribe AcanthochermesiniOriginal spelling. AcanthochermesKollar 1848Type genus. ACANTHOCHERMESAcanthochermesquercusType species. Chermes [Hemiptera]Etymology. Greek ákantha ‘thorn’ + Gender. MasculinequercusAcanthochermes)balbianii (Phylloxera)=albianii : 782 (PhsimiliquercusAcanthochermes)PHYLLOXERINITribe Original spelling. PhylloxeridenPhylloxeraBoyer de Fonscolombe, 1834Type genus. =DACTYLOSPHAER– DactylosphaeridaeOriginal spelling. DactylosphaeraShimer, 1867Type genus. =MORITZIELL– MoritzielliniOriginal spelling. MoritziellaBörner 1908bType genus. =VACUN– Original spelling. VacunidenVacunavon Heyden 1837Type genus. APHANOSTIGMAPhylloxerapiriType species. Etymology. Greek aphanḗs ‘invisible’ + -o + Greek stigma ‘spot’ [pterostigma]Gender. NeuterCINACIUM=CinaciumiaksuienseType species. Etymology. Japanese Kinako ‘soybean flour’ + -iumGender. Neuteriaksuiense (Cinacium)ksuiense : 473 (Cipiri (Phylloxera)piri : 119 tifoliae : 862 (Pepemphigoides (Phylloxera)=higoides : 1246 (Ppervastatrix (Peritymbiavitifoliae (Fitch))=astatrix : 4 (subsvastatrix (Planchon in Rhizaphis)=vitisana (Peritymbia)=vitisana : 109 (Pevitisviniferae (Phylloxera) nomen nudum=iniferae : 337 (Phvulpinae (Viteusvitifoliae (Fitch))=vulpinae : 213 (suFOAIELLAPhylloxeradanesiiType species. Etymology. (Anna) Foà [Italian entomologist] + -i + ella [diminutive suffix]Gender. FeminineBoerneriaPeritymbiaWestwood 1869Note. Replacement name for PageBreakBOERNERIA=PhylloxeradanesiiType species. Etymology. (Carl) Börner [German entomologist] + -iaGender. FeminineBoerneriaCollembola) and BoerneriaCollembola). Replaced by FoaiellaBörner 1909bNote. Junior homonym of danesii (Phylloxera)danesii : 431  [Russian entomologist] + -iaGender. Feminineulmifoliae (Aphanostigma)mifoliae : 144  Moritz [German entomologist] + -i + ella [diminutive suffix]Gender. FeminineNOTABILIA=PhylloxeranotabilisType species. notabilis ‘remarkable, sizeable’, inflected in the neuter pluralEtymology. Latin Gender. NeuterPARAMORITZIELLA Grassi in =PhylloxeracaryaefoliaeType species. MoritziellaEtymology. Greek pará ‘beside’ + Gender. FemininePARAPERGANDEA=PhylloxeracaryaevenaeType species. PergandeaEtymology. Greek pará ‘beside’ + Gender. FemininePARAPHYLLOXERA Grassi in =VacunaglabraType species. PhylloxeraEtymology. Greek pará ‘beside’ + Gender. FemininePARTHENOPHYLLOXERA Grassi in =ParthenophylloxerailicisType species. PhylloxeraEtymology. Greek parthénos ‘girl, virgin’ + Gender. FemininePERGANDEA=DactylosphaeraconicaType species. Etymology. (Theodore) Pergande [American entomologist] + -aGender. FemininePergandeaHymenoptera). Described as subgenus of DactylosphaeraShimer 1867Note. Junior homonym of PHYLLOXERELLA Grassi in =PhylloxerellaconfusaType species. Phylloxera + -ella [diminutive suffix]Etymology. Gender. FemininePHYLLOXEROIDES Grassi in =PhylloxeraitalicaType species. Phylloxera + Greek –ō(i)dēs ‘resembling’Etymology. Gender. MasculinePSYLLOPTERA=PsyllopteraquercinaType species. Psylla [Hemiptera: Psyllidae] + Greek pterá ‘wings’Etymology. Gender. FeminineRHANIS=Type species. NonePageBreakEtymology. Greek rhanís ‘drop (of a liquid)’Gender. FeminineVacunaRhanisColeoptera)Note. Unavailable, described in synonymy with TROITZKYA=DactylosphaeracaryaesemenType species. Etymology. (Nikolay Nikolaevich) Troitzky [Russian entomologist] + -aGender. FeminineVACUNA=VacunacoccineaType species. Vacuna [minor goddess of ancient Italy]Etymology. Latin Gender. FeminineXEROPHYLLA=PemphiguscaryaecaulisType species. Etymology. Greek xērós ‘dry’ + Greek phýllon ‘leaf’Gender. FemininecaryaeavellanaPhylloxera)caryaecaulis (Pemphigus)aecaulis : 859 (Pecaryaemagna (Dactylosphaera)=yaemagna : 391 caryaefoliaePhylloxera)caryaeglobuliPhylloxera)hemisphericum (Dactylosphaera)=phericum : 387 (DacaryaegummosaPhylloxera)caryaepilula (Xerophylla) nomen nudumaepilula : 283 (XecaryaerenPhylloxera) original spelling caryaereniformis but caryaeren in prevailing usage (ICZN Article 33.3.1)caryaescissaPhylloxera)caryaesemen (Dactylosphaera) specific epithet first used by yaesemen : 392 (Dacaryaesepta (yaesepta caryaesepta (Dactylosphaera)subspecies yaesepta : 389 (DaperforansPhylloxeracaryaesepta (subspecies yaesepta )caryaevenae (Pemphigus)yaevenae : 444 (Pecastaneae (Chermes)astaneae : 106 (Chcastaneivora (Moritziella)aneivora : 400 (Mococcinea (Vacuna)coccinea : 289  nomen nudum=globifera (Rhanis) unavailable, described in synonymy with Vacunacoccineavon Heyden 1837=lobifera : 289 (RhPageBreakrutilaPhylloxera)=confusa Grassi in Phylloxera)conica (Dactylosphaera)conica : 390 ibericaPhylloxera)=lichtensteiniiPhylloxera)=davidsoniPhylloxera)deplanataPhylloxera)depressa (Dactylosphaera)depressa : 390 (DadevastatrixPhylloxera)foaePhylloxera)foveata (Dactylosphaera)foveata : 393 (DafoveolaPhylloxera)fraxiniPhylloxera) nomen dubium, only the gall was described and it is probably not a phylloxeridgeorgianaPhylloxera)glabra (Vacuna)glabra : 291 (VapunctataPhylloxera) original name bipunctatum but punctata in prevailing usage (ICZN Article 33.3.1)=globosa (globosa coniferum (Dactylosphaera)subspecies oniferum : 397 (Daglobosa (Dactylosphaera)subspecies globosa : 2 (Dactilicis (Grassi in Parthenophylloxera)intermediaPhylloxera)italica (Grassi in Phylloxeroides)kunugiPhylloxera)minima (Dactylosphaera)minima : 391 (DanotabilisPhylloxera)perniciosaPhylloxera)pictaPhylloxera)pilosulaPhylloxera)quercetiPhylloxera)quercina (Psylloptera)quercina : 85 (PsyspinulosaPhylloxera)=quercusPhylloxera)florentinaPhylloxera)=scutiferaPhylloxera) nomen dubium; Phylloxeraquercus Boyer de Fonscolombe except that scutifera was “slightly larger and darker”; he also drew a scale-like structure  that is not of phylloxerid origin, suggesting his description included a mixture of species=signoretiPhylloxera)=reticulataPhylloxera)PageBreakrileyiPhylloxera)rimosalisPhylloxera)russellaePhylloxera)similansPhylloxera)spiniferaPhylloxera)spinosa (Dactylosphaera)spinosa : 397 (DaspinuloidesPhylloxera)stanfordianaPhylloxera)stellataPhylloxera)subelliptica (Dactylosphaera)lliptica : 389 (DasymmetricaPergande 1904purpureaPhylloxerasymmetricasubspecies symmetricaPhylloxera)subspecies vasculosaPhylloxerasymmetricasubspecies texanaPhylloxera)tuberculiferaPhylloxera)PHYLLOXERININAESubfamily PhylloxerininiOriginal spelling. PhylloxerinaBörner 1908aType genus. PHYLLOXERINAPhylloxerasalicisType species. Phylloxera + Latin -ina ‘in relation to’Etymology. Gender. FeminineGUERCIOJA=ChermespopuliType species. Etymology. (Giacomo Del) Guercio [Italian entomologist] + -jaGender. FeminineLAUFFERELLA=ChermespopuliType species. Etymology. (Jorge) Lauffer [German entomologist] + -ella [diminutive suffix]Gender. FemininePseudochermesGuerciojaMordvilko 1909Note. Replacement name for PSEUDOCHERMES=ChermespopuliType species. Chermes [Hemiptera]Etymology. Greek pseudo- ‘untrue’ + Gender. MasculinePseudochermes Nitsche in Hemiptera: Cryptococcidae). Replaced by LauferellaLindinger 1933Note. Junior homonym of PageBreakcapreaePhylloxerina)daphnoidisPhylloxerina)moniliferae (Guercioja) new name for ChermespopuliPhylloxerinapopularia (Pergande)iliferae : 696 (Gupopuli (Chermes) junior primary homonym of Phylloxerinapopuli (=populi : 269 (Chapopuli nyssae (Phylloxera)nyssae : 269 (Phpopularia (Phylloxera)opularia : 266 (Phpopuli (Chermes)populi : 81,83 (prolifera (Phylloxera)rolifera : 16 salicis : 616 salicola : 267 (PhACANTHAPHISPhylloxeraACANTHOCHERMESPhylloxerinae, AcanthochermesiniAPHANOSTIGMAPhylloxerinae, PhylloxerinibalbianiiAcanthochermesquercusbipunctatapunctataBOERNERIAFoaiellacapreaePhylloxerinacaryaeavellanaPhylloxeracaryaecaulisPhylloxeracaryaefallaxPhylloxeracaryaefoliaePhylloxeracaryaeglobuliPhylloxeracaryaegummosaPhylloxeracaryaemagnaPhylloxeracaryaecauliscaryaepilulaPhylloxeracaryaerenPhylloxeracaryaereniformiscaryaerencaryaescissaPhylloxeracaryaesemenPhylloxeracaryaeseptaPhylloxeracaryaevenaePhylloxeracastaneaePhylloxeracastaneivoraPhylloxeraCINACIUMAphanostigmacoccineaPhylloxeraconfusa Grassi in PhylloxeraconicaPhylloxeraPageBreakconiferumPhylloxeraglobosacorticalisPhylloxeraDACTYLOSPHAERAPhylloxeraDAKTULOSPHAIRAPhylloxerinae, PhylloxerinidanesiiFoaielladaphnoidisPhylloxerinadavidsoniPhylloxeradeplanataPhylloxeradepressaPhylloxeradevastatrixPhylloxeraescorialensisPhylloxeracoccineaEUPHYLLOXERAPhylloxeraflorentinaPhylloxeraquercusfoaePhylloxeraFOAIELLAPhylloxerinae, PhylloxerinifoveataPhylloxerafoveolaPhylloxerafraxiniPhylloxerageorgianaPhylloxeraglabraPhylloxeraglobiferaPhylloxeracoccineaglobosaPhylloxeraGUERCIOJAPhylloxerinahemisphericumPhylloxeracaryaeglobuliHYSTRICHIELLAPhylloxeraiaksuienseAphanostigmaibericaPhylloxeracorticalisilicis Grassi in PhylloxeraintermediaPhylloxeraitalica Grassi in PhylloxerakunugiPhylloxeraLAUFFERELLAPhylloxerinalichtensteiniiPhylloxeracorticalisMICRACANTHAPHIS Grassi in PhylloxeraminimaPhylloxeramoniliferaePhylloxerinaMORITZIELLAPhylloxeraNOTABILIAPhylloxeranotabilisPhylloxeranyssaePhylloxerinaOLEGIAPhylloxerinae, PhylloxeriniPARAMORITZIELLA Grassi in PhylloxeraPARAPERGANDEAPhylloxeraPARAPHYLLOXERA Grassi in PhylloxeraPageBreakPARTHENOPHYLLOXERA Grassi in PhylloxerapemphigoidesDaktulosphairavitifoliaeperforansPhylloxeracaryaeseptaPERGANDEAPhylloxeraPERITYMBIADaktulosphairaperniciosaPhylloxerapervastatrixDaktulosphairavitifoliaePHYLLOXERAPhylloxerinae, PhylloxeriniPHYLLOXERELLA Grassi in PhylloxeraPHYLLOXERINAPhylloxerininaePHYLLOXEROIDES Grassi in PhylloxerapictaPhylloxerapilosulaPhylloxerapiriAphanostigmapopulariaPhylloxerinapopuliPhylloxerinapopuliPhylloxerinamoniliferaeproliferaPhylloxerinaPSEUDOCHERMESPhylloxerinaPSYLLOPTERAPhylloxerapunctataPhylloxeraglabrapurpureaPhylloxerasymmetricaquercetiPhylloxeraquercinaPhylloxeraquercusPhylloxeraquercusAcanthochermesreticulataPhylloxeraRHANISPhylloxeraRHIZAPHIS Planchon in DaktulosphairaRHIZOCERADaktulosphairarileyiPhylloxerarimosalisPhylloxerarussellaePhylloxerarutilaPhylloxeracoccineasalicisPhylloxerinasalicolaPhylloxerinascutiferaPhylloxeraquercussignoretiPhylloxeraquercussimilansPhylloxerasimiliquercusAcanthochermesspiniferaPhylloxeraspinosaPhylloxeraspinuloidesPhylloxeraspinulosaPhylloxeraquercinaPageBreakstanfordianaPhylloxerastellataPhylloxerasubellipticaPhylloxerasymmetricaPhylloxeratexanaPhylloxeraTROITZKYAPhylloxeratuberculiferaPhylloxeraulmifoliaeOlegiaVACUNAPhylloxeravasculosaPhylloxerasymmetricavastatrix Planchon in DaktulosphairavitifoliaeVITEUSDaktulosphairavitifoliaeDaktulosphairavitisviniferaeDaktulosphairavitifoliaevitisanaDaktulosphairavitifoliaevulpinaeDaktulosphairavitifoliaeXERAMPELUSDaktulosphairaXEROPHYLLAPhylloxera