To be exact, we employ entangled states prepared by m-bonacci sequences to detect eavesdropping. Meanwhile, we encode m-bonacci sequences in Lagrange interpolation polynomials to generate the shares of a secret with reverse Huffman-Fibonacci-tree coding. The advantages of the proposed scheme is that it can detect eavesdropping without joint quantum operations, and permits secret sharing for an arbitrary but no less than threshold-value number of classical participants with much lower bandwidth. Also, in comparison with existing quantum secret sharing schemes, it still works when there are dynamic changes, such as the unavailability of some quantum channel, the arrival of new participants and the departure of participants. Finally, we provide security analysis of the new hybrid quantum secret sharing scheme and discuss its useful features for modern applications.With prevalent attacks in communication, sharing a secret between communicating parties is an ongoing challenge. Moreover, it is important to integrate quantum solutions with classical secret sharing schemes with low computational cost for the real world use. This paper proposes a novel hybrid threshold adaptable quantum secret sharing scheme, using an  Secret sharing is an important and powerful tool for protecting confidentiality and integrity of sensitive information, such as missile launch codes, bank account information, medical information and encryption keys. Secret sharing can be categorized into two broad classes: classical and quantum. Secret sharing was invented in its classical form simultaneously by Shamir3456et al.m, n)-threshold quantum secret sharing (QSS), which is the generation of quantum key distribution (QKD)910et al.’s work, many quantum secret sharing schemes14151617181920212223242526et al.m and n is restricted while in CSS, the parameters can be arbitrarily selected as long as n ≥ m. In particular, in a -threshold QSS, the parameters m, n must satisfy the condition, 2m − 1 > n, which is imposed by the quantum no-cloning theoremm and n to be flexible and scalable. Therefore, it is very challenging to develop a new -threshold QSS scheme, whose parameters m and n are not restricted.To address the eavesdropping problem, Hillery m and n and therefore it is suitable for many real-world applications. We employ entangled states prepared by m-bonacci sequences to detect eavesdropping, which can be done by any subset of participants that contains at least x) members. That is to say, not all participants are required to reach a consensus in order to reveal eavesdropping. We use m-bonacci sequences encoded in Lagrange interpolation polynomials to generate the secret, with no restrictions imposed on the parameters m, n. Given that m-bonacci numbers can be represented by Fibonacci numbers, we use the structure of the Huffman-Fibonacci tree with the greedy algorithm to encode m-bonacci sequences, i.e., the higher the frequency at which a Fibonacci number appears in m-bonacci sequences, the longer the block of binary codes. Therefore, our scheme can greatly improve the coding capacity, thus reducing the use of entangled photons, which are expensive and difficult to prepare. In real-world applications, some changes may occur when a new participant joins or alternatively, an existing participant leaves or there is a sudden disruption of some quantum channels. The new scheme has the capability to deal with such changes since the m-bonacci-number coding can be easily modified to reflect changes in secret sharing.In this paper, we propose a hybrid quantum secret sharing scheme to address this challenge. The new scheme is free from any restrictions on the parameters m-bonacci sequences, as shown in m + 1 participants to recover the sequence. The one-time-pad encryption is done by a collection of m + 1 classical participants. The decryption can be done by any other collection of m + 1 classical participants. There are three phases in the proposed scheme:share generation and distribution – the dealer phaseeavesdropping detection phasesecret reconstruction phaseIn this section, we first describe a new hybrid quantum secret sharing based on Then its security is analyzed and compared with other related QKD protocols.m-bonacci sequence which is used in our later scheme. The m-bonacci sequence of order m ≥ 2 denoted by First, we introduce the generalized Fibonacci sequencem − 2 initial terms set to 0 and the (m − 1)th initial term set to 1. In particular, the 2-bonacci sequence is the usual Fibonacci sequencem-bonacci numbers when m = 2, 3, 4, 5, 6.with the first a dealer,m (which is the same as m in m-bonacci numbers) participants who hold quantum shares (quantum participants),Adversaries.Then, we present the entities used in our secret sharing, which are as follows:Dealer is a party who is trusted by all quantum and classical participants. It is responsible for the initialization of the secret sharing. It generates shares and distributes them to all the participants. It is assumed that after finishing its tasks, the dealer “forgets” all the parameters of the scheme together with the secret.Quantum participants hold their quantum shares. Each quantum participant owns one quantum share. Their task is to detect eavesdropping. This guarantees unconditional security of the scheme.Classical participants hold their classical shares. There are q (q ≥ m + 1) classical participants can recover the key (secret) using their classical shares (after eavesdropping detection).Adversaries includes the outsider and at most m insiders. The former has no valid share, while the latter is actually a legal participant with a valid share.Finally, the proposed hybrid Definition 1 (Hybridthreshold QSS). There are m  Dealer first prepares entangled states using the m-bonacci number source as shown in et al.’s work31m entangled photons with smaller m-bonacci values. Each red circle dot represents an orbital angular momentum (OAM) shifter that takes the OAM in that branch to zero. Each place where lines split or cross is implied to have a beam splitter. The portion shown has OAM values from m, which is as follows:An OAM-entangled outgoing state depends on m runs through the allowed m-bonacci numbers in the pump beam: m quantum participants receives one entangled photon from the entangled states. In the lab of each quantum participant, there are two types of detection sorters , directing the entangled photon to one of them at random. The Ei sorter is made up of an OAM sorterFi sorter is used to distinguish different superpositions of the form. The states obtained in the Ei- and Fi- type measurements are nonorthogonal to each other. Therefore, the security of our proposed scheme is based on the fact that nonorthogonal states are indistinguishableEi-type measurement form a mutually orthogonal set among themselves, while those in the Fi-type measurements are not all orthogonal to each other but form a chain, where each state is nonorthogonal to the two adjacent states in the chain. Moreover, for orbital angular momentum, it is not necessary for quantum states that are orthogonal in Hilbert space to associate to orthogonal vectors in the physical space. Likewise, it is not necessary for quantum states that are nonorthogonal in Hilbert space to associate to nonorthogonal vectors in the physical space. That is the second fact for our scheme’s security.where the index Ei or the sorter Fi at random, where i ∈ {1, 2, ···, m}. Quantum participants B1, B2, ···, Bm record the sorter to which the photon goes and the detected OAM value.(2) The beam splitter in the quantum participant’s laboratory, sends the entangled photon to either the sorter m-bonacci numbers to produce new secret shares in terms of the mentioned flow of participants. This is a novel feature of our threshold adaptable secret sharing scheme.(3) Dealer allocates Next the dealer uses the following algorithm to encode the secret:Algorithmp, Choose a prime Randomly and uniformly generate a number f (x) at A1 = f(1), A2 = f (2), …, i′, Ai′), for Sample i′, Ai′) to appropriate classical participants, where Finally, the dealer communicates Pe is larger than the preset threshold t between Dealer and participants, they abort this communication and return to the Dealer’s phase. Otherwise, the communication continues to obtain a secure key for encrypting the shared secret, until the dealer sends an error notification or stops sending secret shares. The details are given below.m + 1 classical participants use their shares Any m > 2) of B1, B2, ···, Bm quantum participants can detect eavesdropping, by comparing the detected values with the values recovered by the m + 1 classical participants. We take m = 2 for example to illustrate how eavesdropping is detected. Suppose that an adversary Eve is eavesdropping on the quantum channel between B1, B2. Clearly, she does not know which type of a detection measurement (the sorter Ei or the sorter Fi) took place in Bi’s laboratory (1 ≤ i ≤ m). So, Eve has to guess. If the entangled photon goes to the Ei sorter in Eve’s laboratory when going to the Fi sorter in Bi’s laboratory, or the entangled photon goes to the Fi sorter in Eve’s laboratory when going to the Ei sorter in Bi’s laboratory, then the Bi measurement is going to be erroneous with the probability of B1, B2, ···, Bm when they compare their scheme transcripts. More precisely, we have the following two cases to consider:Ei-type measurement on a photon, which is actually in the eigenstate Bi. If Bi receives one of these superpositions and makes an Fi-type measurement, she will read out one of the values Eve makes an Fi-type measurement on a photon, which is actually in the superposition state Bi. If Bi receives one of these eigenstates and makes an Ei-type measurement, she will obtain one of the superpositions Eve makes an Any During the process of secret share distribution, when there is a mismatch in the entangled state photons, and if the error rate η (fraction of times Eve interferes) of the trials, the fraction of times Eve guesses wrong basis is B1 compares her results with B2’s, they will find that their outcomes are inconsistent a fraction f of the time, which isIn both cases, if Eve eavesdrops with a fraction m > 2. This is because we can apply the recovered m-bonacci numbers with classical shares, to verify its consistency with the m-bonacci numbers carried by entangled states. Furthermore, with the detected values from the entangled photons, we can assess the security of the quantum channel .Note that in our scheme, each entangled photon binds the classical share together. Moreover, not all quantum participants are needed to detect eavesdropping when t, they abort the communication and postpone the secret reconstruction phase. Otherwise, any m + 1 classical participants can use their classical shares to recover values m + 1 classical participants can recover and further share the secret.If the participants detect eavesdropping and the error rate is higher than the preset threshold value m-bonacci sequences can be used to encode the final key. That is, m is changeable for the key, which can address the problems of restricted quantum sources in certain settings and the membership change of participants, such as the joining of a new participant or departure of existing participants. Therefore, our scheme is more practical compared the other quantum secret sharing schemes.In this scheme, we use quantum coding to generate the key for encrypting messages. Moreover, various et al.3132First, using the technique of Simon Theorem 1 (Insiders attacks). Given the hybrid k < m + 1 insiders  is P(⋅) on the secret space S. That is, for any p0 such that for any p > p0 with H(s) is the entropy of s, the conditional entropy s conditioned on s generated by the knowledge of where Proof. It is worth noting that our proposed hybrid d (d = |Kd| < m + 1) insiders. d classical shares i1 is the public information of d = m, i.e., the upper bound of si′ obtained by the recovered Lagrange polynomial is less than p, then we can have a0 is chosen at random in Fp and thus Next, we only need to consider the case of p → +∞, k < m + 1 classical participants, our proposed hybrid P(⋅) on the secret space S. □That is to say, when Theorem 2 (Outsider attacks). Given the hybrid Q with |Q| = m, and an outsider has already known any subset P(⋅) on the secret space S. That is, for any p0 such that for any p > p0 with H(s) is the entropy of s, the conditional entropy H(s|Qk) denotes the entropy of s conditioned on Qk, and Δ denotes the entropy loss of s generated by the knowledge of Qk.where Proof. First, there are m + 1 (m > t′) classical participants {C1, C2, ···, Cm+1} and t′ quantum participants {B1, B2, ···, Bt′}. Suppose that an outsider has already known m-bonacci number, there is still an m-fold uncertainty about the m-bonacci number other parties detect in ref. s′ and s are actually independent of each other.On the one hand, for the party who detects a particular et al.’s QSSs′ = s.On the other hand, the quantum participants’ shares are independent of the classical participants’ shares. Moreover, the quantum participants’ shares are used for eavesdropping detection rather than key generation. Consequently, outsiders cannot use their quantum shares directly as they do in Cleve Second, as we know, the classical shares are allocated in Shamir’s SSp → +∞, Δ → 0. Hence, our proposed hybrid S. Here, That is to say, when m-bonacci-value-entangled photon, according to the quantum no-cloning theorem, she is unable to clone any other undetected m-bonacci-value-entangled photon.Our scheme is based on the fact that the nonorthogonal states are indistinguishable, so it is immune against cloning and impersonation attacks. Even if Eve successfully detects the value of an m-bonacci-value-entangled photon and send a fake m-bonacci-value-entangled photon, say m-bonacci-value-entangled photon from the fake state Bi. However, due to the particular encoding used in our scheme, the m-bonacci-value-entangled photon varies and the impersonation attack is easy to be detected. For example, if she detects that the value of entangled photon is 2 from Let us consider impersonation attacks. A possible strategy Eve can use is to capture the original m-bonacci sequences for preparing m-bonacci-sequence entangled states, the quantum and classical shares change accordingly. Moreover, due to the use of varying m-bonacci numbers to prepare entangled states, Eve cannot know which m-bonacci sequence is really used every time. For example, suppose that for the fourth subkey, 3-bonacci sequences are used, however, 6-bonacci sequences are used for the fifth subkey. As a result, it is impossible to launch an impersonation attack by inserting the used m-bonacci sequences for the subkey. Therefore, our scheme is immune to replay attacks.Our scheme is immune against replay attacks. A replay attack is such an attack that a valid data transmission is maliciously or fraudulently repeated or delayed. Because we use adaptable m-bonacci number, there is still a m-fold uncertainty about the m-bonacci number other parties detect. Suppose that the eavesdropper Eve, is in possession of an entangled state analyzer for the m-bonacci-value entangled states. If so, she will be able to distinguish m-bonacci-value entangled state according to the result of her m-bonacci-value entangled state analysis. In order to detect the eavesdropper, we should consider what happens if, for instance, the state m-bonacci-value entangled state perfectly. In the remaining B1, B2, ··· Bm together with any m + 1 classical participants. By adding all the probability of causing an error becomes A man-in-the-middle attack is an attack, in which Eve intercepts the transmitted entangled photons and replays other entangled photons. We now show that our scheme provides resistance against the man-in-the-middle attacks. First, quantum channels are authenticated; second, for the party who detects a particular et al.m-bonacci sequences into both quantum and classical coding, with reverse Huffman-Fibonacci-tree coding, to achieve higher-capacity and lower-bandwidth hybrid threshold adaptable QSS scheme.Simon m-bonacci values can be chosen to achieve the lower error rates and longer distances. Hence, we propose to use Huffman coding tree to encode m-bonacci numbers with the greedy algorithm, which can greatly improve the coding capacity, thus reducing the use of entangled photons. To be exact, for fixed m-bonacci number sets, we use binary representations of m-bonacci numbers based on Fibonacci numbers of order m ≥ 2  the m-bonacci values are encoded in Lagrange polynominal, and as a result, any t′ quantum participants and any m + 1 classical participants can recover the secret. Due to the no-cloning theorem and its impact on parameters, for QSS schemes142426t′ must satisfy the requirement of t′ quantum participants are fixed, other quantum participants are unable to participate in the recovery of the secret. However, in our scheme, the parameters t′ can be arbitrary, and our scheme is robust when a new participant joins or an existing participant leaves.Besides the information capacity, 1424m-bonacci sequences, the detection is possible by any subset that contains at least m-bonacci number are significantly improved, from four bits at most to more than ten bits using a similar experimental setup. Consequently, our scheme can greatly improve the coding capacity, thus reducing the use of entangled photons which are expensive and difficult to prepare. Moreover, to generate the secret, the m-bonacci sequences encoded in Lagrange interpolation polynomials and the Huffman-Fibonacci coding are applied. So, compared with the CSS in ref. m-bonacci sequences such as m = 2, 3, 4, 5, 6 can be used in our proposed scheme, the pump values in Unlike eavesdropping detection in QSS schemes1424m quantum and P and p (P ≫ p) be primes, then the communication overhead in ref. n|P| bits, where |P| and |p| denote the bit number of P and p. It can be known that the size of the key is |P|, so, |P| photons are prepared. In ref. n|P| qubits. The ref. For the communication overhead, there are m + 1 provided that m-bonacci numbers, aiming at improving the coding capacity greatly, and thus incurring a low communication overhead. When compared to the existing QSS schemes, there is an improvement in sharing the secret without joint quantum operations. Meanwhile, our scheme still works when there are dynamic changes in comparison with existing quantum secret sharing, such as the unavailability of some quantum channel, the arrival of new participants and the departure of participants.In conclusion, we combine the Huffman-Fibonacci quantum coding with Lagrange polynomials to achieve threshold adaptable QSS. The key point of our scheme is that it does not suffer from the restriction derived from the quantum no-cloning theorem, because it permits secret sharing for arbitrary values of parameters m-bonacci number can be represented by a binary string of length r, cr, cr−1, …, c2, c1 such thatFraenkel and Kleinm-bonacci values.When one uses the following procedure to produce it, the m + 1 classical participants encode the reconstructed m-bonacci value based on the fact that the higher the frequency at which Fibonacci numbers appear in m-bonacci values, the longer the set of binary codes, which is opposite to the idea of Huffman coding. Let the binary codes of m-bonacci values decrease, the coding is as follows .


A frequent bottleneck in interpreting phylogenomic output is the need to screen often thousands of trees for features of interest, particularly robust clades of specific taxa, as evidence of monophyletic relationship and/or reticulated evolution. Here we present PhySortR, a fast, flexible R package for classifying phylogenetic trees. Unlike existing utilities, PhySortR allows for identification of both exclusive and non-exclusive clades uniting the target taxa based on tip labels  on a tree, with customisable options to assess clades within the context of the whole tree. Using simulated and empirical datasets, we demonstrate the potential and scalability of PhySortR in analysis of thousands of phylogenetic trees without a priori assumption of tree-rooting, and in yielding readily interpretable trees that unambiguously satisfy the query. PhySortR is a command-line tool that is freely available and easily automatable. Here, phylogenetic trees in flat files  are sorted  based on text-pattern matching. This principle is not to be confused with the tructure . Currenttructure  and SICLtructure  screen ay; SICLE  specificy; SICLE  do not asortTrees function, allowing for screening (within a tree) for “Exclusive” clades that contain only the target leaves and/or “Non-Exclusive” clades that include a defined portion of non-target leaves. Using simulated data, we assess the runtime of PhySortR based on the number of trees and the number of leaves within a tree, and demonstrate the potential of PhySortR in the analysis of multiple, large-scale empirical datasets.Here we present PhySortR, a fast, flexible R package for screening and sorting phylogenetic trees. The command-line package provides the quick and highly flexible Z; the first tree  are named consistently across all input trees; this is a simple string-matching exercise, i.e., the terms specified here determine taxon-level resolution of targets. In the examples shown in target.groups = “Rhodophyta,Viridiplantae,Stramenopiles.” The minimum support for a clade (min.support) can refer to bootstrap, Bayesian posterior probability, or any other measure of support. Here, no prior assumption of tree root is made , thus clade membership on both sides of each node is considered. For instance, the node support for Clade Z (bootstrap 99%) in rst tree  is an emrst tree , on whicrst tree –1D are bZ in Z in Z  present in a clade relative to the total number of target(s) found in a tree. The default value of min.prop.target is set at 0.7 as guidance. At 1.0, a strict monophyletic relationship of the target group is enforced . When the value is set too low , one would identify a target clade that is more-narrowly defined  a robust 24-member clade , the minimum proportion of target leaves within a “Non-Exclusive” clade is 0.9, thus the maximum proportion of non-target leaves allowed in the clade is 0.1 . For instance, the proportion of target leaves within Clade Z (0.79) in clade.exclusivity of 0.9; this clade is therefore not considered as “Non-Exclusive” at the default setting. This option accepts any value <1.0, and is applicable only for sorting “Non-Exclusive” clades (see below); at 1.0 , the clade is considered “Exclusive”.Alternatively, the user may wish to screen for a robust clade that contains the target groups and a small proportion of “interrupting” non-target leaves, e.g., a 24-member clade consisting of 23 leaves from the targets Rhodophyta, Viridiplantae, and Stramenopiles, as well as one from a non-target leaf, Haptophyceae  and must be sufficiently specific in the dataset for the purpose of the screening. For instance, “plantae” and “Viridiplantae” might not be appropriate in a single query because all tips that are identified by “Viridiplantae” will also be identified by “plantae.”The arameter  is the omode argument, the function will always return a list of the trees that have been identified as containing clades that meet the specified criteria. If the move (mode = “m”) or copy (mode = “c”) command is given, subdirectories will be created in out.dir that contain trees with a particular clade, i.e., the directory out.dir/Exclusive/ will be created for the trees with “Exclusive” clades and out.dir/Non_Exclusive/ for trees with “Non-Exclusive” clades. If the function is instructed to search for “Exclusive” trees it will also return trees that contain only target leaves, termed “All Exclusive” trees. These trees are a subset of “Exclusive” trees and will be transferred to a subdirectory (if the move/copy parameter is given) within the “Exclusive” directory i.e., out.dir/Exclusive/All_Exclusive.Regardless of which parameters are passed to the clades.sorted parameter can be used to change the types of clades that the function will search for. For example if clades.sorted = “E” is given, the function will only search for trees that have “Exclusive” clades, but if the default value of clades.sorted = “NE,E” is given, the function will search for both “Exclusive” and “Non-Exclusive” clades. During each run the function will create a log file, called “out.dir.log,” in the in.dir directory. This file will contain information about each identified clade, e.g., the names of the leaves in the clade, the support for the clade, the proportion of “interrupting” leaves, and so forth.The http://evolution.genetics.washington.edu/phylip/newick_doc.html) that has been widely adopted in phylogenetic software. This format was later modified to incorporate more-complex network information such as hybrid nodes, in the form of extended Newick (eNewick) format; see convert.eNewick function that takes a single phylogenetic tree in eNewick format and returns the same tree in Newick format. This function in isolation can be used as a general-purpose tool for converting phylogenetic trees in eNewick format into a format that is usable by the popular phylogenetic packages in R, ape  and leaves per tree (X); see X = 100, we used a base phylogenetic tree with 1.05X tips, i.e., 105 tips. An “Exclusive” 20-leaf target clade  is defined, tip labels of other non-target leaves are swapped (at random), following which 0.05X  of the overall tree branches  chosen at random were removed using phytools, resulting in the final tree of size X. This tree was then replicated up to N number of trees as per our experimental design below.To test the scalability of the PhySortR package we simulated benchmarking datasets composed of a given number of trees  at tree size X = 100, 200, 300, 400 and 500 . The R script for simulating these topologies and the corresponding templates are available as Simulation of trees at different eplicate . All benhttp://phylosort.sourceforge.net/), we screened for trees that contain a strongly supported (bootstrap ≥ 90%) exclusive clade of Viridiplantae + Stramenopiles. The closest equivalent parameter settings were used for PhySortR  and PhyloSort (Taxa regexp =([∧_] + _[∧_] + ).∗, query taxa group 1 = all Viridiplantae leaves, query taxa group 2 = all Stramenopiles leaves, Min bootstrap = 90, Exclusive mode). For PhyloSort, independent analysis was done with the option Root by outgroup (PhyloSort-root hereinafter) and without (PhyloSort-no-root).As input, here we used 897 empirical protein trees from an earlier study . Using Pape  to be sorted and the number of leaves (X) within a tree. As with any utility , and sorting through 1,000 trees of X = 500 took <350 s (∼5.8 min). We observed negligible differences in runtime with negative controls (trees containing no identifiable clades) as input, compared to the test set in  utility , the runy with N  and supeith N (X . In the Unlike PhyloSort, no prior assumption of tree-rooting is made in PhySortR. To illustrate the impact of tree-rooting assumption on the sorting process, we performed sorting among 897 empirical protein trees  using PhFurthermore, the algorithm of PhySortR  is distiape (phytools (ape and phytools accept only Newick as input, PhySortR accepts tree files in both Newick and eNewick (PhySortR is an R implementation based on the basic sorting principles of ape  and phytphytools , allowin eNewick  formats. eNewick  is open 10.7717/peerj.2038/supp-1Data S1Click here for additional data file.10.7717/peerj.2038/supp-2Data S2Click here for additional data file.10.7717/peerj.2038/supp-3Data S3Click here for additional data file.10.7717/peerj.2038/supp-4Text S1Click here for additional data file.10.7717/peerj.2038/supp-5Figure S1Click here for additional data file.


The purpose of this paper is to present the definition of  In fact the study of such spaces received an impetus with the pioneering work of Schweizer and Sklar  is a binary operation T on  which satisfies the following conditions:T is associative and commutative;T is continuous;A triangular norm  where E is a nonempty set, T is a continuous t-norm and F is a mapping from F at the pair , the following conditions hold:(MPM-1) (MPM-2) (MPM-3) A Menger probabilistic metric space is a triple  be a Menger probabilistic metric space.E is said to converge to A sequence E is called a Cauchy sequence if for any A sequence E, F, T) is said to be M-complete if each Cauchy sequence in E converges to some point in E. is said to be G-complete if each G-Cauchy sequence is convergent in E.(ii)(iii)(iv) into it-self:M-complete Menger probabilistic spaces, they have proved a generalized F is triangular:Kutbi et al.  proved tThe purpose of this paper is to present the definition of We denote by E, F, T) be a Menger probabilistic space and f satisfying condition  be a Menger probabilistic space and f satisfying condition  be aG-complete Menger probabilistic space andbe acontractive mapping. Assume thatThenfhas a unique fixed point.p, we getG-Cauchy sequence. Since  is G-complete, there exists a point v such that For any mentclass2pt{minimG-complete Menger probabilistic space. Meanwhile, they need to add the uniqueness condition of fixed point  be aG-complete Menger probabilistic space andbe acontractive mapping. Assume thatThenfhas a fixed point. denotes the set of all fixed points of a mapping f.In order to get the uniqueness of fixed point, authors added the following condition:Adding conditionto the hypotheses ofTheorem we obtain uniqueness of the fixed point. be aG-complete Menger probabilistic space andbe acontractive mapping. Assume thatThenfhas a unique fixed point, whereT is also a From Theorem Is the following property right?Let  be aG-complete Menger probabilistic space andbe acontractive mapping. Assume thatThenfhas a unique fixed point.The property  is rightIt is not hard to show that, the property  is equivNow we proveThis completes the proof. Let  be a metric space,be a mapping satisfying the following condition:whereis a constant. LetThenis a Menger probabilistic metric space;Tis acontractive mapping, whereTis also acontractive mapping, where(1) We prove (2) From  we have\T is also a (3) By using Theorem Let  be a metric space,be a nonexpansive mapping. Letis a Menger probabilistic metric space;Tis acontractive mapping, whereTis also acontractive mapping, whereT is nonexpansive, let T is also a (1) It is a conclusion of Theorem


Objective: Prepulse inhibition (PPI) is an operational measure of sensory gating. PPI of cortical response to a startling pulse is known to be modulated by attention. With a time-frequency analysis, we sought to determine whether goal-directed and stimulus-driven attention differentially modulate inhibition of cortical oscillations elicited by a startling pulse.Methods: An electroencephalogram (EEG) was recorded in 26 healthy controls performing an active acoustic PPI paradigm. Startling stimuli were presented alone or either 400 or 1000 ms after one of three types of visual prepulse: to-be-attended , unexpected (stimulus-driven attention) or to-be-ignored (non-focused attention). We calculated the percentage PPI for the auditory event-related spectral perturbation (ERSP) of theta (4–7 Hz), alpha (8–12 Hz), beta1 (13–20 Hz) and beta2 (20–30 Hz) oscillations and changes in inter-trial coherence (ITC), a measure of phase synchronization of electroencephalographic activity.Results: At 400 ms: (i) PPI of the ERSP of alpha, theta and beta1 oscillation was greater after an unexpected and a to-be-attended prepulse than after a to-be-ignored prepulse; and (ii) PPI of beta2 oscillations was greater after a to-be-attended than a to-be-ignored prepulse. At 1000 ms: (i) PPI of alpha oscillations was greater after an unexpected and a to-be-attended prepulse than after a to-be-ignored prepulse; and (ii) PPI of beta1 oscillations was greater after a to-be-attended than a to-be-ignored prepulse. The ITC values did not vary according to the type of prepulse.Conclusions: In an active PPI paradigm, stimulus-driven and goal-directed attention each have differential effects on the modulation of cortical oscillations. Filtering out irrelevant information is a crucial way of protecting the cognitive resources required for goal-directed activities. One of the physiological indices of these protective neural processes is referred to as prepulse inhibition . It corresponds to the attenuation of the amplitude of a startle reflex to an intense stimulus  when a weaker, non-startling stimulus (the prepulse) precedes the pulse by approximately 30–500 ms. The prepulse attenuates not only motor responses  but also cortical responses to a sound pulse, such as the N100 and P200 components of the auditory evoked potential  activity can be described in a “time-frequency domain” using two measures: event-related spectral perturbation (ERSP) and inter-trial coherence  have been linked to focused attention  years). All underwent a semi-structured interview and those with a history of neurological or psychiatric disorders were excluded, as well as those taking psycho-active drugs, including tobacco or cannabis. Subjects with a history of visual or auditory impairments were excluded from the study. All participants gave their written informed consent to participation. The study protocol was approved by the local institutional review board .The study population was the same as in our previous study  presented during 29 ms, with a mean 1400 ms inter-stimulus interval (range: 1300–1500 ms). Ten pulses were randomly delivered during the presentation, with an inter-pulse interval ranging from 18 to 30 s (mean: 24 s). The pulses were not temporally related to the visual stimuli.After the control task, participants performed two blocks of 360 trials of the startle-CPT task Figure . During The mean response time (in ms), the number of hits and the number of false alarms were recorded.® 128 AgCl electrode cap (ANT Software BV) placed according to the 10/05 international system  software  for data acquisition. The EEG and EOG signals were digitized at a sampling rate of 1024 Hz. Electrode impedances were kept below 5 KΩ.An EEG was recorded continuously from 128 scalp locations, using a DC amplifier  and a Quick-cap® software. The EEG signal was band-pass filtered between 0.1 and 30 Hz, and ocular artifacts were detected and removed off-line using the software’s principal component analysis procedure.The EEG data were analyzed with ASAf and latency t relative to the pulse onset. Typically, for n trials, if Fk  is the spectral estimate of trial k at frequency f and time t, then:ERSP was measured using the EEGLab toolbox for MATLAB software , calculation was based on Morlet wavelet decomposition. Using the settings in EEGLab, cycles were set to , i.e., 2.5 cycles at the lowest frequency (4 Hz) with and 9.375 cycles at the highest frequency (30 Hz). A baseline-normalized ERSP was obtained during the control task and for each lead interval (400 and 1000 ms) after each type of prepulse  during the startle-CPT task. Ten trials per condition were considered.Following calculation of the ERSP, the ITC was computed in order to: (i) evaluate the effect of the prepulse on intertrial phase variability; and (ii) determine whether any abnormalities in ERSP power were due to impairments in phase variability. To compute ITC, the complex output of the baseline-normalized ERSP was divided by its complex norm , which was then averaged across trials. The complex norm of this averaged value yields the ITC for different time and frequency points. Using the same notation as above, the ITC is defined as:where ∣ · ∣ represents the complex norm.ITC was obtained during the control task and for each lead interval (400 and 1000 ms) after each type of prepulse  during the startle-CPT task. Ten trials per condition were considered.2/Hz). Indeed, to calculate the percentage PPI of the power for a given frequency domain, we chose to use the absolute value of the power in mV2/Hz  because the baseline period  differed between the control session and the startle-CPT session. The percentage PPI of the absolute power for the frequency domains was calculated according to the following equation:Apart from the baseline-normalized spectral power value (expressed in dB), the spectral power was also expressed as an absolute value , for each type of prepulse.The following frequency domains were assessed: theta (4–7 Hz), alpha (8–12 Hz), beta1 (13–20 Hz), and beta2 (20–30 Hz).p < 0.05. Thereafter, we analyzed the signal on the location where the amplitude of the response was maximum.For time-frequency mapping, we used EEGLab to perform permutation tests with four conditions . The threshold for statistical significance (after Holm’s correction) was set to ® SPSS®, Armonk, New York, NY, USA).All other analyses were performed with SPSS 16.0 for Windows software  were performed separately for each lead interval and each frequency domain, with the prepulse  as the within-subject factor for the PPI of ERSP and  for the ITC. When required, p < 0.05 for all analyses.The significance threshold was set to p < 0.01, for all analyses). By consequence, further analyses were not adjusted.All data were normally distributed and are presented as the mean and SD. Mean %PPI values did not significantly differ according to sex (The mean reaction time was 369 (43) ms, the mean percentage of correct answers was 99% (2.5) and the mean number of false alarms was 0.15 (0.37).Time-frequency values of the ERSP are displayed as scalp topographies maps during the control session  and during the startle-CPT task [when the pulse is preceded by an “A”, “O” or “symbol” prepulse with a lead interval of 400 ms Figure  and 1000As displayed on the maps, an occurrence of a pulse was followed by a higher synchronization of cortical rhythms that predominated on the central area and was mainly visible in the theta and alpha frequency bands and to a lesser extent, in the beta1 and beta2 frequency bands. When a prepulse preceded occurrence of the pulse, this synchronization was clearly attenuated but much more when the prepulse was a to-be-attended (“A”) or an unexpected (symbol) stimulus than a to-be-ignored (“O”) stimulus. This phenomenon was observed for both lead intervals, but in an attenuated manner at long-lead interval (1000 ms). Moreover, at short-lead interval (400 ms), when the prepulse was a to-be-attended (“A”) stimulus, a desynchronization was observed 200 ms after the pulse occurrence, mainly in the alpha frequency band and, to a lesser extent, in the beta1 frequency band.As activity largely predominated on the central scalp area, further analyses and PPI measures were performed only on the Cz location  = 13.42, p < 0.001). Further comparisons (Bonferroni-corrected) revealed that the PPI of the theta rhythm was greater after a prepulse “A” than “O” (t(25) = 5.18, p < 0.001), and greater after a “symbol” than after a “O” (t(25) = 3.84, p = 0.001). There was no significant difference between the “A” and “symbol” prepulses (t(25) = 0.96, p = 0.34).For the theta band, an ANOVA revealed a significant main effect of prepulse type  = 10.7, p < 0.001). Further comparisons (Bonferroni-corrected) revealed that the PPI was greater after a prepulse “A” than “O” (t(25) = 5.7, p < 0.001) and greater after a “symbol” than “O” (t(25) = 2.7, p = 0.01). There was no significant difference between the “A” and “symbol” prepulses (t(25) = 1.23, p = 0.22).For the alpha band, an ANOVA revealed a significant main effect of prepulse type  = 13, p < 0.001). Further comparisons (Bonferroni-corrected) revealed than the PPI was greater after a prepulse “A” than “O” (t(25) = 5, p < 0.001) and greater after a “symbol” than “O” (t(25) = 3.2, p = 0.003). There was no significant difference between the “A” and symbol prepulses (t(25) = 1.47, p = 0.15).For the beta1 band, an ANOVA revealed a significant main effect of prepulse type  = 3.89, p = 0.02). Further comparisons (Bonferroni-corrected) revealed that the PPI was greater after a prepulse “A” than “O” (t(25) = 2.8, p = 0.009). There was no significant difference between the “A” and “symbol” prepulses (t(25) = 1.92, p = 0.06) nor between the “O” and “symbol” prepulses (t(25) = 0.8, p = 0.42).For the beta2 band, an ANOVA revealed a significant main effect of prepulse type  = 5.43, p = 0.008). Further comparisons (Bonferroni-corrected) revealed that the PPI was greater after a prepulse “A” than “O” (t(25) = 2.63, p = 0.014), and greater after a “symbol” than “O” (t(25) = 3.1, p = 0.005). There was no significant difference between the “A” and “symbol” prepulses (t(25) = 0.8, p = 0.43).For the alpha band, an ANOVA revealed a significant main effect of prepulse type  = 3.2, p = 0.05). Further comparisons (Bonferroni-corrected) revealed that the PPI was greater after a prepulse “A” than “O” (t(25) = 2.7, p = 0.01). There was no significant difference between the “A” and “symbol” prepulses (t(25) = 1.3, p = 0.2) nor between the “O” and “symbol” prepulses (t(25) = 1.14, p = 0.26).For the beta1 band, an ANOVA revealed a significant main effect of prepulse type  = 7.75, p < 0.001). Further comparisons revealed than the ITC was greater when the pulse was presented alone (during the control session) than after a “A” (t(25) = 3.2, p = 0.003), “O” (t(25) = 3.8, p < 0.001) or “symbol” prepulse (t(25) = 3.97, p < 0.001). The other differences were not significant (p > 0.05).For the 400 ms prepulse-pulse interval, there was no significant effect of the condition for the theta, beta1 and beta2 bands, although an ANOVA revealed a significant main effect of condition type for the alpha band  = 9.3, p < 0.001). Further comparisons revealed than the ITC was greater when the pulse was presented alone (during the control session) than after an “A” (t(25) = 3.67, p = 0.001), “O” (t(25) = 4.35, p < 0.001) or “symbol” prepulse (t(25) = 3.82, p < 0.001). The other differences were not significant (p > 0.05).For the 1000 ms prepulse-pulse interval, there was no significant effect of the condition for the theta, beta1 and beta2 bands, although an ANOVA revealed a significant main effect of condition type for the alpha band  changed when the said information was preceded by an attention-focusing prepulse.Our results show that at a prepulse-pulse interval of 400 ms, inhibition of spectral power for the theta, alpha, and beta1 bands was greater after a to-be-attended prepulse and an unexpected prepulse than after a to-be-ignored prepulse. Moreover, inhibition of spectral power in the beta2 band was greater after a to-be-attended prepulse than after a to-be-ignored prepulse, with a trend toward a greater inhibition after a to-be-attended prepulse than after an unexpected prepulse. At a prepulse-pulse interval of 1000 ms, inhibition of spectral power for the alpha band was greater after a to-be-attended prepulse and an unexpected prepulse than after a to-be-ignored prepulse. Inhibition of spectral power for the beta1 was also greater after a to-be-attended prepulse than after a to-be-ignored prepulse. The ITC values did not differ significantly as a function of the type of prepulse.Hence, by studying brain oscillations, we showed that inhibition of cortical responses produced by an irrelevant stimulus is modulated by attention. Our first starting hypothesis was thus confirmed: inhibition of spectral power in various frequency domains is dependent on the type of prepulse, with a greater inhibition after a prepulse on which attention is focused than after a to-be-ignored prepulse. Regarding our second hypothesis (that inhibition would be greater after a to-be-attended prepulse than after an unexpected prepulse), we only observed a trend towards such an effect for the beta2 frequency band, without significant difference for the other bands. As phase synchrony did not change with respect to the type of prepulse, it can be argued that PPI here is due to a reduction of the number of neurons discharging synchronously.One of the strengths of our study was its novel design, which enabled us to control for the respective effects of goal-directed attention and stimulus-driven attention.At a prepulse-pulse interval of 400 ms, we found that the effects of goal-directed attention and stimulus-driven attention on PPI of the brain oscillations were the same for the theta, alpha and beta1 bands, whereas inhibition of brain oscillations for the beta2 band was greater after a to-be-attended prepulse  than after an unexpected prepulse (involving stimulus-driven attention). Given that spectral power reflects the number of neurons that discharge synchronously, our results suggest that both stimulus-driven attention and goal-directed attention reduce the number of synchronous discharges in response to an irrelevant stimulus for the theta, alpha and beta1 frequency bands. Moreover, goal-directed attention may reduce the number of neurons that discharge synchronously for the beta2 band. Hence, by using a time-frequency analysis, we confirmed the previous reports  or preceded by a prepulse (of whatever type). An effect of attention on ITC was only observed for the alpha band (regardless of the prepulse type). This partially counter our hypothesis in which the prepulse would impair phase synchronization. The lack of attentional modulation of ITC for the theta, beta1 and beta2 bands is not in agreement with the spectral power results—suggesting that inhibition of power is not due to an impairment in intertrial phase variability. This is in line with the fact that the ITC is independent of changes in EEG power . At a short-lead interval, stimulus-driven attention and goal-directed attention both modulate auditory-elicited theta, alpha and beta1 oscillations, whereas beta2 oscillations are preferentially inhibited when goal-directed attention is engaged. Our results also highlighted the impact of selective attention processes on sensory gating at long-lead intervals. Overall, it comes out of our results that auditory-elicited oscillations are globally modulated by attention whatever the frequency band, but that goal-directed attention has a specific effect on alpha and beta oscillations, with even a late desynchronization in the alpha frequency band. Our present findings do not suggest that phase synchronization of EEG activity is specifically modulated by attention. Given that PPI has been shown to be a useful index of sensory gating impairments in basal ganglia disorders [e.g., schizophrenia and Parkinson’s disease (Perriol et al., Conception and design of the work, acquisition, analysis or interpretation for the work: AA, J-LB, CT, KD. Drafting the work or revising it critically for important intellectual content: AA, J-LB, AD, PB, CT, PD, KD. Final approval of the version to be published: AA, J-LB, AD, PB, CT, PD, KD. Agreement to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved: AA, J-LB, AD, PB, CT, PD, KD.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


High-entropy alloys (HEAs) are new alloys that contain five or more elements in roughly-equal proportion. We present new experiments and theory on the deformation behavior of HEAs under slow stretching (straining), and observe differences, compared to conventional alloys with fewer elements. For a specific range of temperatures and strain-rates, HEAs deform in a jerky way, with sudden slips that make it difficult to precisely control the deformation. An analytic model explains these slips as avalanches of slipping weak spots and predicts the observed slip statistics, stress-strain curves, and their dependence on temperature, strain-rate, and material composition. The ratio of the weak spots’ healing rate to the strain-rate is the main tuning parameter, reminiscent of the Portevin-LeChatellier effect and time-temperature superposition in polymers. Our model predictions agree with the experimental results. The proposed widely-applicable deformation mechanism is useful for deformation control and alloy design. A series of new tensile deformation experiments is presented, which are performed at slow strain-rates on metals ranging from conventional alloys  to high-entropy alloys (HEAs)12345678910111213141516171234567891011121314151617Here we present new experimental results on alloys exposed to slow tensile deformation. We compare the results to the predictions of an analytic model that explains the observed serration statistics and the time-series properties of the slips. The model predicts how the scaling behavior of the slip-size distribution and related statistical properties can be changed by tuning temperature, strain-rate, or alloy composition. The experimental results agree with the model predictions. As explained below, both the experiments and the model show that the deformation mechanism of HEAs differs from that of conventional alloys. In the following sections, we discuss (1) the experiments, (2) the model, its predictions and applicability range, and (3) the comparison of the experimental results to the model predictions. Potential applications of the newly-proposed deformation mechanisms, include materials testing, evaluation, and the design of new alloys with improved deformation properties.We perform experiments on HEAs123456789101112131415161719201023HEAs are defined as alloys that have a mixing entropy higher than 1.5R, where R is the gas constant. In addition, medium-entropy alloys (MEAs) are defined as having the mixing entropy between 1R and 1.5R, and low-entropy alloys (LEAs) as being below 1R. Most conventional alloys are LEAs. However, some stainless steels, superalloys, and bulk metallic glasses are MEAs because of their high concentrations of alloying elements.In our experiments, a series of metals, containing Ni, CoNi, CoFeNi, CoCrFeNi, and CoCrFeMnNi, were designed specifically because they all have a solo face-centered-cubic (FCC) structure, thus providing a common basis for the comparison of serration phenomena. Their mixing entropies, approximately represented by the configurational entropy, are 0R, 0.69R, 1.1R, 1.39R, and 1.61R, respectively. With this range of entropies, the alloy series covers LEAs, MEAs, and HEAs. We observe changes in the serration phenomena from FCC-LEAs to FCC-HEAs, as temperature is tuned. As we will show, the measurements confirm the predicted effects on the serration statistics, of tuning temperature, strain-rate, and alloy composition.−5/s to 1 × 10−2/s and temperatures from 300 °C to 700 °C. Each experiment was repeated 3 to 4 times to ensure the reliability of the test results. The slip statistics were extracted from the measured stress-time curves and, then, compared with the new predictions of the above statistical model. Details on the experiments are given in the Methods Section.In our experiments, the five metals, Ni, CoNi, CoFeNi, CoCrFeNi, and CoCrFeMnNi, were slowly deformed under tension at different strain-rates from 1 × 10−4/s, and indicates the occurrence or nonoccurrence of serration behavior. Among the 31-analyzed samples at different temperatures, 14 samples display sudden slips, or stress-drops, called “serrations”, which are reminiscent of the Portevin-LeChatellier (PLC) effect that is well studied for conventional materials25262728293031323334353637383940large) serrations that occur less regularly than in the type-C PLC-bands, along with smaller slips that may have a fairly-broad size distribution, and type-A PLC-bands252627282930313233343536373839252627282930313233343536Tensile test results have shown that the present FCC alloy series of CoCrFeMnNi have good elongation between 15% and 50% in the range of test temperatures and strain-rates. 252627282930313233343536−4/s, the Ni metal and CoNi alloy do not display serrations from 300 °C to 700 °C. The CoFeNi alloy has serrations from 400 °C to 500 °C. CoCrFeNi has serrations from 300 °C to 600 °C. CoCrFeMnNi has serrations from 300 °C to 620 °C. Note that over the temperature ranges we explored, we observed serrations only in the ternary, quaternary, and quinary alloys. Additionally, the temperature range of serrations broadens, as the number of elements increases. As for the evolution of serration types, the quinary alloy displays the whole span evolving from type A at low temperatures to type B at medium temperatures and to type C at high temperatures, which is consistent with the general observation from conventional alloysIn C(S), which gives the fraction of serrations that are larger than size, S. Here S is defined as the total stress drop observed during a serration. In the following sections, we first describe a simple model that explains (1) the observed time series properties of the serrations  and (2) the observed scaling behavior of the serration-size distributions and their dependence on temperature, strain-rate, and composition.To quantify the serration statistics, The simple coarse-grained modelN lattice points, the local stress, τl, at a lattice point, l, is given by42If the material has F is the applied stress, J is the elastic-coupling constant, and um is the cumulative slip of the weak spot at the site, m. Each point fails, when the local stress, τl, exceeds its local failure threshold (slip stress), τf,l. When site, l, fails, it slips by an amount, Δul, thereby reducing the stress to a sticking stress, τa,l. The associated stress drop is τf,l − τa,l ~ 2G Δul, where G ~ J is the elastic shear modulus.where Λ. Specifically, after a weak spot has slipped, the local failure stress, τf,l, drops from its initial value, τs,l, to a lower value, τd,l, by an amount given by the weakening parameter, ε τs,l − τd,l)/τs,l. It remains at this weakened value until the slip avalanche is completed. Afterwards, τf,l is increased to its initial value, τs,l, at the temperature-dependent healing rate, Λ, until the weak spot slips again during a new avalanche, and the process repeats.For HEAs, the model assumes that weak spots are weakened even further upon slipping to model how dislocations break free from the surrounding pinning atoms. This assumption is reminiscent of the Portevin-LeChatellier (PLC) effect or dynamic strain aging2530383940The re-strengthening of the weak spots models the rearrangement of atoms around arrested dislocations, leading to the (re-)“pinning” or anchoring of stationary dislocations in place. The weakening process models the dislocation breaking free from the pinning atoms around it. While the dislocation is free, it can slip more easily; when it meets obstacles, it is pinned again.As we discuss below, for the alloys discussed here, the weakening, ε, depends on temperature and strain-rate. ε, that grows with the ratio, Λ/Ω, of the healing rate, Λ, relative to the strain-rate, Ω. For sufficiently-low temperatures, it is reasonable to assume that pinning of the dislocations involves activated processes. Consequently, the healing rate, Λ, follows an Arrhenius-like temperature (T) dependence, Λ ~ exp(−B/T), where B is a material-dependent constant. In close analogy to a similar mechanism in polymers, known as time-temperature superpositionΛ/Ω ~ exp(−B/T)/Ω, i.e., similar slip statistics are expected for temperatures, T, and strain-rates, Ω, for which exp(−B/T)/Ω = constant. As discussed in the experimental section, this prediction is consistent with the experimental data  for the PLC effect2530383940As a result, the strengthening  rate, Λ, in the PLC effect for MEAs and HEAs is much higher than that in the conventional PLC effect.However, there is a key difference between our proposed strengthening mechanism for medium-entropy alloys (MEAs) and high-entropy alloys (HEAs) and the conventional DSA mechanism in the PLC effect. In MEAs and HEAs, the “pinning” of dislocations originates from abundant sources of solutes in the whole-solute matrix. Ω, either by tensile deformation or by compression. The model can be solved in the mean-field theory (MFT) approximation4244i.e., in a particular temperature and strain-rate regime, the material produces small and large avalanches. The small avalanches are much smaller in extent than the sample size, so that they are not affected by sample boundaries. In contrast, the large avalanches span the system, touching opposite sample boundaries. The large avalanches recur almost periodically, while the small avalanches occur at random times between the large avalanches. The large avalanches are predicted to show crack-like scalingS ~ R3, of the avalanche size, S, with its radius, R. The reason for the crack-like scaling behavior is that for the large slips, the model predicts that the total amount of slip at a point grows with the radius R of the slipping area, so that the total slip integrated over the entire area scales as R3. The large avalanches typically have a Gaussian size-distribution. Their average avalanche size grows with increasing weakening, ε. In contrast, for small avalanches, the model predicts S ~ R2. The reason is that small avalanches are more pulse-like, where the amount of slip in each slipping patch is fixed, i.e., it does not grow with the radius, R, of the slipping areaD(S), of the small slip-avalanche sizes, S, scales asThe model predicts the serration statistics for experiments in which the material is sheared at a slow strain-rate, κ = 1.5, g is a material-specific constant, and Smax is the expected cutoff size of the power-law scaling regime of D(S)Smax ~ 1/ε2, decreases with the inverse square of the weakening parameter, ε. The complementary cumulative distribution function (CCDF), defined as S. For the smaller slips that occur between the almost periodically-recurring largest slips, C(S) scales as4245where κ = 1.5, andwhere again, A as a material-dependent parameter.is another quickly-decaying scaling function42ε, in the model:For sufficiently-slow imposed strain-rates , the weak spots have sufficient time to fully re-heal after a slip avalanche. Consequently, the weakening, ε, upon slipping again is large in this case, and the small avalanches may be almost undetectably small, because Smax ~ 1/ε2 is small. Strong weakening also implies almost regularly-recurring large slips with a narrowly-peaked size distribution. In this strain-rate regime, only the periodically-recurring large slips are large enough to be detectable2530383940In the intermediate strain-rate regime , there is less time to pin the dislocations between avalanches. Consequently, the dislocations cannot be pinned as strongly before they move again, making the weakening, ε, smaller than in the first case. As a result, the largest small avalanche, Smax ~ 1/ε2, is larger than in the previous case. In contrast, the almost periodically-recurring large avalanches are smaller than before. Additionally, these large avalanches occur less regularly than in case (1). Because the small avalanches are now larger, they are more easily detectable in this regime than in the first case2530383940large) slips that occur less regularly than in the above type-C PLC-bands, and along with many broadly-distributed small slips , there is very little time for the dislocations to be re-pinned after they slip. Consequently, the model assumes that in this regime, the effective weakening, ε, is small. The reason is that a slipping weak spot frees a dislocation from a much weaker pinning potential than in the other cases, because the weak spots have effectively no time to re-strengthen between slips.  In this case, one may expect Smax ~ 1/ε2 to be large. As we show below, in the experiments presented here, this regime is observed at the lowest tested temperatures, where the pinning atoms move more slowly than in the other two regimes, and, therefore, do not have sufficient time to strongly pin the dislocations. The resulting reduced pinning of the dislocations can reduce Smax to a smaller value46Smax the distribution will show power law scaling in the range S < Smax  the deformation rate is slow, compared to the healing rate of the weak spots , (2) the deformation rate is comparable to the healing rate [in this case the pinning of the dislocations by solute atoms is not as strong as in case (1)], and (3) the deformation rate is fast, compared to the healing rate . As explained below, these three cases correspond to different values of the weakening parameter, ε, is the key parameter that determines whether the serrations in the model are of type-A, B, or C PLC-bands. The weakening, ε, is determined by how much time the atoms have to pin the dislocations, before they slip again in the next avalanche. Consequently , Λ ~ exp(−B/T), where B is a material-dependent constant.  Larger numbers of elements are also expected to lead to faster healing rates, as they provide more mobile atoms that can move to stick to the dislocations to pin them more strongly in place.The healing rate, In summary, by tuning the temperature, strain-rate, and/or the number of elements in the alloy, the serration behavior can be tuned between the three different regimes discussed above.Λ, first increases and, then, decreases again. The weakening, ε ~ Λ/Ω, follows the same trend. As a result, the model predicts that at fixed, low strain-rates, Ω, smaller serrations with a broad power-law size distribution are observed only in a specific temperature regime. This temperature range is set by the condition that that ε must be sufficiently small so that the power-law-distributed events of the maximum size, Smax ~ 1/ε2, are large enough to be detected in the experiments (such as in types-A and B PLC-bands). As a consequence, the model predicts that for increasing temperature, the serrated stress-strain curves are expected to move from type-A to B to C PLC-bands. At temperatures far above the serration regime, all serrations are washed out by thermal fluctuations, and the stress-strain curves look smooth. At temperatures far below this regime, serrations are not observed either, because of the lack of sufficient pinning of the dislocations by the surrounding atoms on experimental time scales in this regime. The range of temperatures that shows a broad range of serration sizes is predicted to be material specific, since the healing rate is determined by the material-specific diffusion constants of atoms.For increasing temperature, the healing rate, The model also predicts the dependence on the strain-rate and composition of the temperature window in which serrations are observed: The ratio, Λ/Ω, of the healing rate to the strain-rate is the key parameter determining the temperature range where serrations are observed. We already saw that Λ decreases with decreasing temperatures. The model, then, predicts that for smaller strain-rates, serrations are observed for a smaller window of healing rates, and, thus, for a smaller range of temperatures. The dependence of the serration statistics on Λ/Ω implies that at fixed temperatures and increased strain-rates, HEAs are predicted to move from type-C to B to A PLC-bands, to no serrations at all.Λ/Ω), thereby widening the temperature range over which serrations are observed. In the following section, we compare the theoretical predictions to the experimental results.Increasing the number of components also increases the healing rate and, thus, the ratio . cted see .S, follows a power-law, Sκ−, over a wide range of sizes, S, consistent with the model predictions of κ = 1.5, predicted by the mean-field model for type-A bands is consistent with the observations   S, folloions see .−2/s for higher temperatures, except for the experiments at 300 °C. This fact demonstrates that the pinning of dislocations is fast. Thus, the pinning mechanism of such alloys differs from that in conventional alloys with low solute contents, for which the pinning of dislocation is much slower. The difference in the pinning mechanisms will be quantified in future work.Furthermore, we find that the PLC effect of CoCrFeMnNi remains at greater strain-rates up to 10in-situ pin) at a relatively-fast healing rate between avalanches. In addition, as temperature is increased, the healing rate, Λ, of the weak spots increases to a maximum at a certain temperature, and, then, quickly decreases. As a result, six phenomena are observed that are consistent with the proposed model:i.e., the healing rate, Λ, of the weak spots increases with temperature). However, a limiting temperature is reached, when the thermal vibration of atoms is too large to settle down the effective locking . The experiments show that the working temperature range to observe serration behavior at a strain-rate of 10−4/s covers from around 300 °C to 620 °C for the present quinary alloy (CoCrFeMnNi). At temperatures above 650 °C, no serrations are observed prior to the critical failure Λ, increases and decreases again with temperature). This trend explains why the quinary alloy has the largest range, while the ternary alloy has the narrowest range. Ni and CoNi alloys are lacking sufficient sources and size differences and, accordingly, show no PLC effect , compared to conventional PLC-pinning mechanisms.The PLC effect of CoCrFeMnNi remains at higher strain-rates up to 10S, of the small stress drops follows a power law, Sκ−, with an exponent, κ, that is consistent with the theoretical value of κ = 1.5. The range over which a power law can be observed depends on temperature, strain-rate, and material composition. The model predicts that the power law distribution is multiplied by an exponentially-quickly decaying roll-off function, which cuts off the power law at a largest slip size, Smax for the largest slip sizesSmax, where the exponential roll-off is observed depends on the temperature and the number of elements in the material. For type-B bands in a certain temperature range, Smax, is so small that the power law range is not visible and only the exponential roll-off is seen because it dominates over the power law decay for the measured slip sizes  are expected for temperatures, T, and strain-rates, Ω, chosen such that Λ/Ω ~ exp(−B/T)/ Ω = constant, or equivalentlyAs argued in the Model Section, for a given HEA material, The striking agreement of the serration statistics and the observed PLC-bands with the predictions of the model for slip avalanches shows that the serrations can be explained as avalanches of slipping weak spots. In MEAs and HEAs, the weak spots  weaken upon slipping and re-strengthen ) versus 1/T for the temperatures, T, in Ω, T) = ,  = , and  = . As predicted, within the accuracy of the temperature step size in the experiments, the data points in Here −4/s. The effect of varying the strain-rates from 1 × 10−5 to 1 × 10−2/s on the serration behavior was examined. The PLC effect is observed in a specific range of temperatures and strain-rates for MEAs and HEAs, and not in LEAs representing the conventional substitutional alloys.The effect of changing temperatures and adding elements to alloys ranging from FCC-LEAs to FCC-HEAs has been studied to investigate the PLC effect at a strain-rate of 10in-situ pin) at a relatively fast healing rate between avalanches, compared to conventional alloys. In addition, the PLC effect is more pronounced in HEAs than in MEAs.As shown in the above six observations, the experimental results agree with the model introduced above. Although the pinning mechanism of such alloys is different from that in conventional alloys with low solute contents, their serrations can be explained as avalanches of slipping weak spots. The main difference is that in MEAs and HEAs, the weak spots (or dislocations) weaken upon slipping and re-strengthen  trigger other weak spots to slip, resulting in slip avalanches typically detected as sudden stress drops. During slip avalanches, the participating weak spots are weakened from their original strength to a lower strength, but they are re-strengthened during the time intervals between slip avalanches. Re-strengthening or healing of weak spots between slip avalanches is temperature-, strain-rate-, and composition-dependent. More abundant sources of solute atoms give higher healing rates. The healing rate reaches a maximum at a material-specific temperature and strain-rate. For the temperature regime in which type-A PLC-bands are observed, the healing rate has an Arrhenius-type temperature dependence. For a given HEA material, similar slip statistics are observed for experiments at similar ratios of the healing rate to strain-rate, thereby revealing an underlying deformation mechanism that is analogous to the time-temperature superposition in polymersIn the present series of experiments on LEAs (Ni and CoNi), MEAs (CoFeNi and CoCrFeNi), and HEA (CoCrFeMnNi) with a single FCC structure, serration phenomena are only found in MEAs and HEA. The experimental results are in remarkable agreement with the predictions of a simple MFT model. The model predicts that the critical exponent, The results provide us with a new understanding of the deformation mechanisms of LEAs, MEAs, and HEAs and a simple mechanism for the PLC effect in general. The observed serration statistics are related to the more general phenomenon of crackling noise and/or avalanches in many other systems, ranging from HEAs, conventional alloys, bulk metallic glasses, nanomaterials and magnets to earthquakes and other systems3642444546−4/s and a data-acquisition frequency of 20 Hz. Each alloy was tested at different temperatures:, 300 °C, 400 °C, 500 °C, 600 °C, and 700 °C. The CoCrFeMnNi alloy had additional samples tested at 275 °C, 325 °C, 350 °C, 375 °C, 620 °C, and 650 °C, with additional strain-rates of 10−2/s, 10−3/s, and 10−5/s for detailed investigations. For each condition, 3 or 4 test specimens were used. The stress versus time data was recorded during the experiment. The complementary cumulative distribution function, C(S), of stress-drop sizes for each condition was extracted from a representative curve of three or four measured curves45The five experimental alloys were vacuum arc-melted from the starting elements with purities higher than 99.9 weight percent (wt. %). Each sample was re-melted and re-solidified at least four times to assure the uniform composition under the protection of argon. The alloys were, then, cast into a slab of dimensions, 10 mm × 20 mm × 40 mm. After surface machining on cast slabs, a cold-rolling reduction of 65% was performed from the initial thickness of 8 mm to a thickness of 3.2 mm. The alloys were subsequently subjected to a homogenization treatment at 1,100 °C for 6 hours and follow-on quenching in water. The sheets were wire cut by the electrical discharge machine into tensile specimens with dimensions shown in How to cite this article: Carroll, R. et al. Experiments and Model for Serration Statistics in Low-Entropy, Medium-Entropy, and High-Entropy Alloys. Sci. Rep.5, 16997; doi: 10.1038/srep16997 (2015).


A number of alignment tools have been developed to align sequencing reads to the human reference genome. The scale of information from next-generation sequencing (NGS) experiments, however, is increasing rapidly. Recent studies based on NGS technology have routinely produced exome or whole-genome sequences from several hundreds or thousands of samples. To accommodate the increasing need of analyzing very large NGS data sets, it is necessary to develop faster, more sensitive and accurate mapping tools.HIA uses two indices, a hash table index and a suffix array index. The hash table performs direct lookup of a q-gram, and the suffix array performs very fast lookup of variable-length strings by exploiting binary search. We observed that combining hash table and suffix array (hybrid index) is much faster than the suffix array method for finding a substring in the reference sequence. Here, we defined the matching region (MR) is a longest common substring between a reference and a read. And, we also defined the candidate alignment regions (CARs) as a list of MRs that is close to each other. The hybrid index is used to find candidate alignment regions (CARs) between a reference and a read. We found that aligning only the unmatched regions in the CAR is much faster than aligning the whole CAR. In benchmark analysis, HIA outperformed in mapping speed compared with the other aligners, without significant loss of mapping accuracy.Our experiments show that the hybrid of hash table and suffix array is useful in terms of speed for mapping NGS sequencing reads to the human reference genome sequence. In conclusion, our tool is appropriate for aligning massive data sets generated by NGS sequencing.The online version of this article (doi:10.1186/s13015-015-0062-4) contains supplementary material, which is available to authorized users. Recent studies based on next-generation sequencing (NGS) technology have produced hundreds or thousands of exome or whole genome sequences with decreasing cost of NGS experiments . As the All HT-based alignment tools use the same seed-and-extend strategy , 12, whiSupport of long-read alignment, high speed, accuracy, and sensitivity are essential features of the current NGS mapping tools. Here, we tried to merge the advantages of HT-based and suffix tree-based alignment in a tool that satisfies these requirements. To this end, we developed a genome mapper using hybrid index-based sequence alignment (HIA).In this article, we describe the HIA tool, and show the results of comparisons of performance on simulated and real read data between HIA and the other four popular alignment tools including BWA, Bowtie2, SOAP2 and SeqAlto. The results of the benchmark analysis demonstrate that HIA outperforms the other aligners, especially in speed.0s1 … sm−1 be a string over Σ. Let |S| = m be the length of S, S[i] = si be the i-th symbol of S, S = si … sj be a substring, and Si = S be a suffix of S. We define a q-gram as a substring of S with length q. In the context of DNA sequence, the alphabet Σ consists of the four nucleotides A, C, G, and T, i.e., Σ = {A, C, G, T}. We assign A, C, G, and T to the numbers 0, 1, 2, and 3, respectively. Thus, each q-gram is encoded as an unsigned integer with two bits per nucleotide. However, most reference genome sequences contain a nucleotide other than {A, C, G, T}, such as ‘N’. This occasionally happens with NGS read sequences as well. We replace ‘N’ with a uniform random nucleotide such as BWA and many other tools do.Let Σ be an alphabet and S = sq + 1. Given a q-gram, HT[x] is the first position of the q-gram in SA, where x is the numeric value of the q-gram. We define the range (R) of the q-gram in SA by  is empty.In terms of a hybrid index, SOAP2 implements a hash table on the FM-index which is a compressed SA. On the other hand, our hybrid index consists of a reference sequence, a suffix array (SA), and a hash table (HT), as in Fig. The procedure for constructing the hybrid index consists of four processes, as follows: 1) packing the reference sequence in a 2-bits per base format (Sequence); (2) counting each q-gram in the sequence and set the range of q-grams in SA (HT); (3) inserting the sequence positions of each q-gram into its range of SA; and (4) sorting the range for each q-gram (SA) lexicographically. Based on the heuristic determining the final positions of most of the suffixes using only the first few symbols of each suffix , the SA  packing Retrieving positions of a query Q in the sequence is implemented in two steps: HT lookup and binary search of the SA. If the prefix of Q of length q (q-gram) is a substring of the sequence, we find the range (R) in which the q-gram belongs in the SA, using Eq. . OtherwiHybrid mapping follows the same seed-and-extend approach used by all HT-based tools. A MR (matching region) is a common substring between the reference sequence and the read. Let ‘sp’ be the starting position in the reference sequence where the MR occurs. We will indicate each MR as 3-tuple <dv, ro, L>, where ‘ro’ (read offset) is the starting position in the read where the MR occurs, ‘dv’  is defined as dv = sp − ro, and ‘L’ is the length of the MR. Given a length-m′ MR, there are (m′ − q + 1) q-grams having the same diagonal values and consecutive read offsets. Diagonal values having same values implies that the corresponding MRs are close each other in the reference sequence. A CAR  is a list of MRs, which are close each other and ordered by ‘ro’. We define a CAR as a seed and align only the unmatched regions in CAR.I.Retrieve range of SA of each q-gram using HT and SAThe procedure for finding MRs and CARs of a read is as follows: 1) retrieving range of SA of each q-gram using HT and SA; (2) computing diagonal values; (3) sorting by diagonal value and offset; (4) grouping MRs with same diagonal value and successive offsets; (5) merging the adjacent MRs into CARs; (6) sorting the CARs by matched bases in descending order. For example, given a read (r = GCCATG) and q-gram length (q = 2) and the hybrid index constructed in Fig.  retrievi0-th q-gram positions: 1-th q-gram positions: (CC: SA[8])2-th q-gram positions: 3-th q-gram positions: II.Compute diagonal values4-th q-gram positions: (TG: SA[15])III.Sort by diagonal value and offset, , , , , , , IV.Group MRs with same diagonal value and successive offsets, , , , , , , MR0:  ← MR1:  ← , , MR2:  ← V.Merge the adjacent MRs into CARs and set matched basesMR3:  ← , , CAR0: CAR1: VI.Sort the CARs by matched basesCAR2: CAR1: CAR0: CAR2: .Although diagonal values of two adjacent MRs are different, they could be located in a same CAR if there were inserted or deleted bases between them. In the case of CAR1, the difference value between diagonal value of MR1 and diagonal value of MR2 is 1 and there is one inserted base (C) between MR1 and MR2. We refer to this value as adjacency and we use the value in order to set the permitted size of insertion and deletion between MRs.In order to find MRs and CARs efficiently, we apply three heuristics. Let the read length and the error rate be m and ε, respectively. The first heuristic is that there is a common substring of length at least m/(k + 1) between two reads of length m with k differences . Let λ =The formula  is the cSecondly, since a q-gram that occurs in many regions in the sequence is not a good discriminator, such q-gram was given less weight than one that occurs in few regions. This heuristic is based on the inverse document frequency (IDF) commonly used in the field of information retrieval. The IDF is a measure of whether or not the term is common across all documents . ApplyinFinally, given a read (r) and two strings, S1 and S2, both of length m, if the number of matched bases between r and S1 is greater than the number of matched bases between r and S2, then the number of differences between r and S1 is smaller than the number of differences between r and S2. This heuristic can be used to rank CARs by the number of matched bases and to filter out the lower-ranked CARs.To align the top-ranked CARs, we simply align the unmatched regions in each CAR, because the MRs are already aligned (matched). We classify the unmatched regions into three groups: leftmost unmatched region (LMUR), which is the left unmatched region of the first MR in a CAR; rightmost unmatched region (RMUR), which is the right unmatched region of the last MR in a CAR; and unmatched regions between two MRs (MRURs). The matched and unmatched region can be separated because of mismatch, insertion and deletion. We analyze these split causes such as Fig. There are also many split causes of MRUR like as Fig. Hybrid mapping might miss some CARs and result in the failure of mapping (unmapped). The main reasons for missed CARs might be one of following reasons: (1) too many errors in a read; (2) too many highly frequent q-grams in a read; and (3) many high-ranked CARs. In the first case, a long q-gram based on the first heuristic approach results in empty q-gram ranges. In the second case, the second heuristic approach causes highly frequent informative q-grams to be missed. In the third case, the lower-ranked but informative CARs are lost because of the third heuristic approach. We applied extended hybrid mapping to the unmapped, which uses shorter q-grams including some highly frequent q-grams, and extends ranked CARs further. Together, these modifications increase the sensitivity of the technique.We implemented HIA on Java to support multiple platforms , Unique (%), Q10 (%) and Time (s). The Aligned (%) denotes the percent of aligned reads over total reads and indicates the overall mapping rate. The Unique (%) measures the percent of uniquely aligned reads over total reads and refers to MAPQ ≥ 1. The Q10 (%) measures the fraction of mapped reads MAPQ ≥ 10. The Time is the elapsed time (seconds) including both the index loading time and the alignment time. In the case of the simulated datasets, the %Err measures the percent of wrong aligned reads over the reads satisfying Unique (%) or Q10 (%). We adopt the concept of a correct alignment from Langmead and Salzberg , who detTo evaluate the performance of HIA, we compared HIA to BWA, Bowtie2, SOAP2 and SeqAlto on six simulated datasets and two real datasets. In all tests, we used the GRCH37 build of the human genome as the reference sequence used in alignment. We performed the alignments using a computer with two Intel Xeon 6-Core X5670 2.93-GHz processors and 48 GB RAM. All alignment tools were run with a single thread for alignment except for multi-thread tests.Table In the perspective of SA construction, we performed several tests of index generation and compared the results from our index generation algorithm and the divsufsort that is one of the best SA construction algorithms . It was We ran six of the aligners with various parameter settings for 100, 150, 250 and 400 bp single-end reads. All results can be found in the Additional file Table We also ran six of the aligners with the same parameter settings as in the single-end reads for 100 and 150 bp paired-end reads. All results can be found in the Additional file Table We ran six of the aligners with various parameter settings for one paired-end reads and two single-end reads. All results can be found in the Additional file Table We ran six of the aligners with 6 threads and 12 threads modes for the whole of the paired-end HiSeq reads. All results can be found in the Table We developed a new sequence alignment tool for aligning short and long reads to a reference genome. HIA has two indexes, a HT index and a SA index. The HT is capable of direct lookup of a q-gram, and the SA can very rapidly look up q-grams of variable length. Our experiments show that the hybrid of HT and SA is useful, from the perspective of speed, for mapping NGS sequencing reads to a reference genome sequence. HIA also supports the multithreading of mapping. In particular, HIA is much faster than all of the other aligners; therefore, our tool is appropriate for aligning massive data sets generated by NGS sequencing.The accuracy of alignment is very important in re-sequencing because the main purpose of alignment is to discover the variants relative to a reference genome. Although these variants (or sequencing errors) cause sequencing reads within them to match inexactly to the reference, alignment tools should nonetheless correctly map these reads to the reference. Considering the results of the benchmark analysis, we can conclude that HIA performs comparably to four popular alignment tools.Project name: HIAhttp://biomi.cdc.go.kr/hia/Project home page: Operating systems: Platform independentProgramming languageLicense: GNU GPLAny restrictions to use by non-academics: none.


PMS receives as input n strings and two integers ℓ and d. It returns all sequences M of length ℓ that occur in each input string, where each occurrence differs from M in at most d positions. Another formulation is quorum PMS (qPMS), where the motif appears in at least q% of the strings. We introduce qPMS9, a parallel exact qPMS algorithm that offers significant runtime improvements on DNA and protein datasets. qPMS9 solves the challenging DNA -instances  and . The source code is available at https://code.google.com/p/qpms9/.Discovering patterns in biological sequences is a crucial problem. For example, the identification of patterns in DNA sequences has resulted in the determination of open reading frames, identification of gene promoter elements, intron/exon splicing sites, and SH RNAs, location of RNA degradation signals, identification of alternative splicing sites, etc. In protein sequences, patterns have led to domain identification, location of protease cleavage sites, identification of signal peptides, protein interactions, determination of protein degradation elements, identification of protein trafficking elements, discovery of short functional motifs, etc. In this paper we focus on the identification of an important class of patterns, namely, motifs. We study the ( It returns all possible biological sequences M of length ℓ such that M occurs in each of the input strings, and each occurrence differs from M in at most d positions. Any such M is called a motif.The Planted Motif Search (PMS) problem, also known as the  motif search is the Closest Substring problem. The Closest Substring problem is essentially the PMS problem where the aim is to find the smallest d for which there exists at least one motif. These two problems have applications in PCR primer design, genetic probe design, discovering potential drug targets, antisense drug design, finding unbiased consensus of a protein family, creating diagnostic probes and motif finding . In qPMS we are interested in motifs that appear in at least ℓ-mer) M is generated and “planted” at a random location in each input string (or in q% of the input strings for qPMS). Every planted instance of the motif is mutated in exactly d positions.The Closest Substring problem is NP-HardDefinition 1.An  instance is defined to be achallenging instanceif d is the largest integer for which the expected number of motifs of length ℓ that would occur in the input by random chance does not exceed a constant . According to the above definition, the challenging instances for PMS are , , , , , , , , , , etc.Intuitively the more we increase ℓ-mers coming from distinct input strings. In the pattern driven stage, the algorithm generates the common d-neighborhood of the ℓ-mers in the tuple. Each such ℓ-mer becomes a motif candidate. The size of the tuple is usually fixed to a value such as 1 . For any string T, we denote the substring of T starting at position i and ending at position j by T[i..j].We start by defining the PMS and qPMS problems more formally. A string of length Definition 2.1, s2, …, sn, over an alphabet Σ, and two integers ℓ and d, identify all ℓ-mers M, M ∈ Σl, such that ∀i, 1 ≤ i ≤ n, ∃ji, 1 ≤ ji ≤ |si| − l + 1, such that Hd ≤ dThe PMS problem: Given n sequences s.Definition 3.The qPMS problem: same as the PMS problem, however the motif appears in at least q% of the strings, instead of all of them. PMS is a special case of qPMS for which q = 100%.d-neighborhood. Given a tuple of ℓ-mers T = , the common d-neighborhood of T includes all the ℓ-mers r such that Hd ≤ d, µ1 ≤ i ≤ s.Another useful notion is that of a ℓ-mer and the consensus total distance for a tuple of ℓ-mers. The consensus ℓ-mer for a tuple of ℓ-mers T =  is an ℓ-mer u where u[i] is the most common character among  for each 1 ≤ i ≤ ℓ. If p is the consensus ℓ-mer for T then the consensus total distance of T is defined as ℓ-mers.We now define the consensus ℓ-mers  are generated, where it is an ℓ-mer in iS. Then, in the pattern driven part, for each tuple, its common d-neighborhood is generated. Every ℓ-mer in the neighborhood is a candidate motif. In PMS8k is variable. By default, a good value for k is estimated automatically based on the input parameters , where it is an ℓ-mer from string is, ∀i = 1..k, are generated based on the following principles. First, if T has a common d-neighborhood, then every subset of T has a common d-neighborhood. Second, for a motif to exist, there has to be at least one ℓ-mer u in each of the remaining strings ks + 1, ks + 2, …, ns such that T ∪ {u} has a common d-neighborhood. We call such ℓ-mers u “alive” with respect to tuple T. As we add ℓ-mers to T we update the alive ℓ-mers and reorder the strings in increasing order of the number of alive ℓ-mers. This reordering reduces the running time because it leads to generating fewer tuples overall.In the sample driven part of PMS8, tuples T be the current tuple of ℓ-mers and let u be an alive ℓ-mer with respect to T. If we add u to T, then the consensus total distance of T increases. We compute this additional distance Cd(T∪{u}) − Cd(T). For each of the remaining strings, we compute the minimum additional distance for any alive ℓ-mer in that string. Then we sort the strings decreasingly by the minimum additional distance. Therefore, we give priority to the string with the largest minimum additional distance. If two strings have the same minimum additional distance, we give priority to the string with fewer alive ℓ-mers. The intuition is that larger additional distance could indicate more “diversity” among the ℓ-mers in the tuple, which means smaller common d-neighborhoods. The pseudocode for generating tuples T is given in GenTuples where the matrix R contains all the ℓ-mers in all the input strings, grouped as one row per string.In qPMS9 we change the criteria by which the strings are reordered, as follows. Let T, obtained as described in the previous section, we generate the common d-neighbors of the ℓ-mers in the tuple. In qPMS9, the neighbor generation uses the same process as in PMS8For every tuple T =  of ℓ-mers, we want to generate all ℓ-mers M such that Hd ≤ d, ∀i = 1..k. We traverse the tree of all possible ℓ-mers. A node at depth r, which represents an r-mer, is not explored deeper if certain pruning conditions are met. Necessary and sufficient conditions for 2 and 3 ℓ-mers to have a common neighbor are given in Ref. ℓ-mers to have a common neighbor. The interested reader is referred to the PMS8 paperGiven a tuple ℓ-mer in a given string by adding it to the tuple T and recursively calling the algorithm for the remaining strings. For the qPMS version we have an additional step where, if the value of q permits, we skip the current string and try ℓ-mers from the next string. At all times we keep track of how many strings we have skipped. The pseudocode for this algorithm is given in QGenerateTuples where R contains all the ℓ-mers in all the strings.We extend the algorithm to solve the qPMS problem. In the qPMS problem, when we generate tuples, we may “skip” some of the strings entirely. This translates to the implementation as follows: in the PMS version we successively try every alive m = |s1| − ℓ + 1 independent subproblems P1, P2, …, mP, where iP explores the d-neighborhood of ℓ-mer s1[i..i + ℓ − 1]. In the parallel implementation, processor 0 acts as both a master and a worker, the other processors are workers. Each worker requests a subproblem from the master, solves it, then repeats until all subproblems have been solved. Communication between processors is done using the Message Passing Interface (MPI).In PMS8q version. We split the problem into subproblems P1,1, P1,2, …, P2,1, P2,2, …, rP,1, rP,2, …, r = n − Q + 1 and iPj, explores the d-neighborhood of the j-th ℓ-lmer in string is and searches for ℓ-mers M such that there are Q − 1 instances of M in strings is+1, …, ns. Notice that Q is fixed, therefore subproblems iPj, get progressively easier as i increases.In qPMS9, we extend the previous idea to the ℓ-mer M as a motif and plant modified versions of it in q% of the n strings. Each planted instance is modified in d random positions.As mentioned in the introduction, PMS algorithms are typically tested on datasets generated as follows. 20 strings of length 600 each are generated from the i.i.d. We choose an ℓ-mer u is within distance at most d from another ℓ-mer v isIt is useful to estimate how many “spurious” motifs (motifs expected by random chance) will be found in a random sample. For that, we make the following observations. The probability that a random ℓ-mer is within distance d from any of the ℓ-mers in a string S of length m is:The probability that an ℓ-mer is within distance d from at least q out of n strings of length m each is:The probability that an ℓQ. Based on these formulas, we compute for every ℓ the largest value of d such that the number of spurious motifs does not exceed 500. These values are presented in Therefore, the expected number of motifs for a given qPMS instance is: |Σ|q = 100% we compare all three algorithms, for q = 50% we compare only the algorithms that solve the quorum PMS problem: TraverStringRef and qPMS9. All programs were executed on the Hornet cluster at the University of Connecticut, which is a highend, 104-node, 1408-core High Performance Computing cluster. For our experiments we used Intel Xeon X5650 Westmere cores. Most results refer to single core execution, unless specified otherwise.In this section we analyze the running times of PMS8q = 100%. In In q = 50%. In q = 50%.In ℓ and d with ℓ up to 50 and d up to 25, with q = 100%. In ℓ and d with ℓ up to 30 and d up to 21, with q = 100%.In ℓ and d such as  for DNA data or  for protein data.We have presented qPMS9, an efficient algorithm for Quorum Planted Motif Search. The algorithm is based on the PMS8 algorithmℓ-mers for Hamming Distances could benefit for the work in Ref. For future work, one of our reviewers kindly pointed out that our approach of filtering