This paper investigates the problem of synchronization of fractional-order complex-variable chaotic systems (FOCCS) with unknown complex parameters. Based on the complex-variable inequality and stability theory for fractional-order complex-valued system, a new scheme is presented for adaptive synchronization of FOCCS with unknown complex parameters. The proposed scheme not only provides a new method to analyze fractional-order complex-valued system but also significantly reduces the complexity of computation and analysis. Theoretical proof and simulation results substantiate the effectiveness of the presented synchronization scheme. In the past 20 years, fractional-order chaotic systems have been extensively studied due to their wide applications in the fields of secure communication, control engineering, finance, physical and mathematical science, entropy, encryption and signal processing ,2,3,4. MThe aforementioned works mainly investigated the fractional-order systems with real variables, not involving complex variables. Because complex variables that double the number of variables can generate complicated dynamical behaviors, enhance anti-attack ability and achieve higher transmission efficiency ,19,20, mInspired by the above discussions, the synchronization problem of FOCCS with unknown complex parameters was investigated in this paper. Using the inequality of the fractional derivative containing complex variable and the stability theory for fractional-order complex-valued system, we realized synchronization of such systems by constructing a suitable response system. It should be noted that we deal with the synchronization problem of fractional-order uncertain complex-variable system in complex-valued domain. That is to say, it is not necessary to separate the complex-variable system into its real and imaginary parts. This greatly reduces the complexity of computation and the difficulty of theoretical analysis.n-dimensional space. For l2-norm of Notation: Definition 1 .The fractional integral of order α for a function f is defined as:0 and α > 0.where t ≥ t.The fraDefinition 2 .Caputo’s fractional derivative of orderfor a functionis defined by:0 and n is a positive integer such that n − < α < 1where t ≥ t..Caputo’Lemma 1 .Let0 andbe a continuous and derivable vector function. Then, for any time instant t ≥ t.Letx and letandare continuously differentiable functions. If:and:whereis a positive constant. Then z(t) = 0 is asymptotically stable.Proof. See the Remark 1.Using Lemmas 2.2–2.3, one can directly analyze fractional order complex-variable system in the complex space.We considered a kind of FOCCS described by:Proposition 1.1 such that the following inequality holds:There exists a positive constant lProof. Given that ectively ,37.Let Proposition 2 . For the Lipschitz continuous function l2 such that the following inequality holds: . For thProof. For Remark 2.It is easy to check that many typical FOCCSs, such as the fractional-order complex-variable Chen system , T systeChoose system (11) as the master system, then the controlled response system is given by:Theorem 1.Asymptotically synchronization and parameter identification of systems (13) and (11) can be achieved under adaptive controller:and the complex update laws:whereis the error vector,is the parameter error, σ, η are two arbitrary positive constants.Proof. From the error vector and systems (11) and (13), it yields:Using Lemma 2.1, Corollary 2.1 and Lemma 2.2, we have:Substitute Equations (15) and (16) into the inequality above, we further have:According to Lemma 2.3, one has rem 1 in , parametRemark 3.In previous work ,27,28,29Remark 4.If the system parameters are known, the update law will be reduced to (15) only.In this section, in order to show the effectiveness of the proposed scheme in preceding section, numerical example on fractional-order complex chaotic system will be provided. When numerically solving such systems, we first adopt the predictor–corrector method  by MATLAa, b, c are system parameters; let Consider the Lorenz-like fractional-order complex chaotic system with commensurate order:Recently, ref.  describeia (the imaginary of a) is depicted in Field-programmable gate array (FPGA)-based implementation of chaotic oscillators has demonstrated its usefulness in the development of engineering applications in a wide variety of fields, such as: random number generators, robotics and chaotic secure communication systems, signal processing. Very recently, Pano-Azucena et al.  implemena, b and c are unknown, the response system is given as follows:Taking the system (19) as master system, and assuming the parameters According to Theorem 1, the controllers and the update rules are selected as:a, b, c) = , the initial conditions In the simulation, let We studied the adaptive synchronization of FOCCS with unknown complex parameters, and proposed a method for analyzing FOCCS without separating system into real and imaginary parts. By this method, the constructed response system can be asymptotically synchronized to an uncertain drive system with a desired complex scaling diagonal matrix. The proposed synchronization scheme retains the complex nature of fractional-order complex chaotic system. It not only provides a new method of analyzing FOCCS, but also significantly decreases the complexity of computation and analysis. We hope that the work performed will be helpful to further research of nonlinear fractional order complex-variable systems.


For any k-CNF to d-regular -CNF is given. Here regular -CNF is a subclass of CNF, where each clause of the formula has exactly k distinct variables, and each variable occurs in exactly s clauses. A d-regular -CNF formula is a regular -CNF formula, in which the absolute value of the difference between positive and negative occurrences of every variable is at most a nonnegative integer d. We prove that for all s, such that every d-regular -CNF formula is satisfiable. In this study, s such that there exists a uniquely satisfiable d-regular -CNF formula. We further show that for d-regular Unique  The r k≥3 in . That isr k≥3 in , Glucoser k≥3 in . The conk-SAT denotes the promise search problem of k-SAT where the number of solutions is either 0 or 1. The harder instances should have fewer solutions. But Calabro and Paturi in [k-CNF formula has a solution is the same as that of deciding whether it has exactly one solution, both when it is promised and when it is not promised that the input formula has a solution. Thus, the research of uniquely satisfiable SAT instances is a very significant work.A natural measure of the solution space is the number of solutions. Unique aturi in  proved tk distinct variables per clause and at most s occurrences of each variable. Regular -SAT problem in [k-SAT problem showed that the constrained density k-SAT problem such that(i)k-CNF instances with all random (ii)k-CNF instances with all random The  of each variable have its own characteristics.In ,14, M. Wuses. In , Johannsd-regular , if F is unsatisfiable and F, a minimal unsatisfiable formula can be obtained by removing some clauses from F.If the formulas Definition 1.For each Definition 2.A k-CNF formula F is called a k-forced-once d-regular ((i) there exist k variables (ii) except for the k variables, every variable occurs in exactly s clauses, and the absolute value of the difference between positive and negative occurrences of every variable is no more than the nonnegative integer d.(iii) F is satisfiable and for any truth assignment τ satisfying F, it holds thatWe can represent a CNF formula as a matrix. Each variable F is a CNF formula with 15 variables F isLet F is a 3-forced 0-regular -CNF formula. Each of the three variables F and is forced to be Clearly, Definition 3.In the context of SAT, a reduction M is identified to be parsimonious if x and Lemma 1.Let , and each new variable occurs F and the absolute value of the difference between positive and negative occurrences of each variable is no more than d. Therefor, F is turned into an unsatisfiable d-regular Z occurs positively in Every variable of (ii)All literals of (iii)Z.Every clause of Introduce a new boolean variable set Define Z does not occur in Z be x and all literals of x, every variable of s clause, and meet the d-regularity  and (ii) of Definition 2 hold in d-regular . Therefore, Next, we will assess the feasibility of the construction of Lemma 4.For Proof. k-forced-once d-regular X and Y occurs in exactly every variable of (ii)Every clause of We construct a Define s clauses, and the absolute value of the difference between positive and negative occurrences of every variable is at most d. The number of unforced variables in n. So d-regular Let (ii)LetFor simplicity, let Define Step 3 We will make up the gap of the number of occurrences of every variable. Using the variables in sets X and (i)For (ii)For (iii)x in Each variable (iv)x in Each variable (v)Every clause of Step 4 Let d-regular -CNF formula is unsatisfiable, then d-regular -CNF formula is unsatisfiable, then d-regular (By 7)≤17 in , if a -CNF formula is unsatisfiable, then d-regular (By 8)≤29 in , all +2 in , we obtaCorollary 1.For  Ψ that has exactly one satisfying assignment.Proof. The statement follows directly from Lemmas 5 and 6. □k-CNF formula into a d-regular . The variables in Z are sorted by their subscripts.Step 4 We construct a k-CNF formula X and Z, satisfying the following conditions.(i)Z occurs in exactly Every variable Otherwise(ii)X occurs in exactly For (iii)X occurs in exactly For (iv)X.Every clause of Step 5 We construct the formula s clauses, and the absolute value of the difference between positive and negative occurrences of every variable of d. Therefore, d-regular (Obviously, every variable of X in First, we focus on the feasibility of X and Z. The variable set X generates Z generates The variables of For X in For For X is more than the number of clauses in Obviously, the number of positive literals of Second, we will prove that the formula It is assumed that X. As a result, Obvious, the truth assignment It is assumed that We substitute Equation  into Ψ3,Obviously, the truth assignment Therefore, X are forced to be Z that replaced the same variable of Z. Due to only one solution of Finally, we will explain why the polynomial-time reduction is parsimonious. If k-SAT problem is a NP-complete problem. From Theorem 5, it demonstrates that there exists a polynomial time reduction from k-SAT to d-regular (d-regular (For d-regular (F in which the positive and negative occurrences number of every variable do not exceed F is a (F must be satisfiable for k, can be satisfiable for From Lemma 6, this suggests that for d-regular (d-regular (We present the construction method of a uniquely satisfiable


Correction to: Trialshttps://doi.org/10.1186/s13063-019-3351-2After publication of our article  we have n = 1013 per group), instead of 6000. Moreover, degrees of freedom should equal to 1, since there are two study groups.According to the calculation described in the article, sample size should be 2026 in total  would be required for our study. If the attrition rate was set at 10%, a total of 2026 patients (1013 in each group) would be required.Figure


A symmetric block cipher employing a substitution–permutation duo is an effective technique for the provision of information security. For substitution, modern block ciphers use one or more substitution boxes (S-Boxes). Certain criteria and design principles are fulfilled and followed for the construction of a good S-Box. In this paper, an innovative technique to construct substitution-boxes using our cubic fractional transformation (CFT) is presented. The cryptographic strength of the proposed S-box is critically evaluated against the state of the art performance criteria of strong S-boxes, including bijection, nonlinearity, bit independence criterion, strict avalanche effect, and linear and differential approximation probabilities. The performance results of the proposed S-Box are compared with recently investigated S-Boxes to prove its cryptographic strength. The simulation and comparison analyses validate that the proposed S-Box construction method has adequate efficacy to generate efficient candidate S-Boxes for usage in block ciphers. Cryptography helps individuals and organizations to protect their data. For this purpose, different symmetric and asymmetric ciphers have been designed. Symmetric ciphers possess simplicity and efficiency, and consume fewer computational resources as compared to asymmetric ciphers. Symmetric ciphers have two major categories as stream and block ciphers . A streaA block cipher is particularly useful to achieve data confidentiality, which is one of the cryptography goals. It is considered as one of the most widely used tools for the provision of data security . The mosAn S-Box is a crucial part of modern-day block ciphers and is used to create a muddled ciphertext from the given plaintext. An S-Box is one of the fundamental techniques used to provide candid confusion. Confusion is the complex relationship that must be established between the plaintext and the ciphertext . The strIn , the autGenerally, a block cipher consists of many parts. An S-Box, being the lone non-linear part of a block cipher, is very useful for enhancing the security of the plaintext by creating confusion in the ciphertext. The non-linearity provided by an S-Box offers defense against linear cryptanalysis . Block cAES is one of the popular block ciphers which use S-Boxes in the encryption and decryption processes. Sahmoud et al.  proposedThe Feistel structure has been used as the main construct in many of the symmetric ciphers, like DES, GOST, RC5, etc. DES and GOST each uses eight S-Boxes with sizes 6 × 4 and 4 × 4, respectively. The authors in  used theOne of the desirable properties of modern block ciphers is the avalanche effect . This prChaotic cryptography is among the most interesting areas in the field of information security in the recent era as the chaotic systems possess the property of randomness . Many reAnother popular area in cryptography is DNA computing, which is being considered as a possible solution to the design of resilient ciphers. Kadhim et al.  and Al-WCiphers using S-Boxes highly depend on the security of the S-Boxes. Thus, the identification of a tool to evaluate and find an S-Box with high security that can also assist in the design of efficient S-Boxes is considered critical. Wang et al.  developeLinear fractional transformation (LFT) is another area which helps in the generation of better S-Boxes. Farwa et al.  proposedThe techniques and methods for the generation of S-Boxes presented in the literature are either suitable for the creation of static S-boxes or are very complicated and time-consuming. Static S-Boxes have their own limitations and weaknesses. These S-Boxes may help attackers in the cryptanalysis of the captured ciphertext and hence they may reach the original plaintext. On the other hand, the methods presented in the literature that generate dynamic and key-dependent S-boxes are very complex and less efficient. Thus, the need for a simple and efficient method to generate dynamic S-Boxes exists.An S-box that helps in the security enhancement of the block cipher and resists cryptanalysis;An S-Box that is simple to construct;An S-Box that is generated dynamically using sub-keys;An S-Box that fulfills the most needed S-Box criteria, like NL, SAC, BIC, LP, DP, etc.In this paper, a novel design method for the construction of efficient S-Boxes for block ciphers is proposed. The following considerations were kept in mind while designing the proposed S-Box:The method proposed in this paper for the construction of an S-Box is an innovative one and is quite different from the approaches presented in the literature. A cubic fractional transformation is proposed for the construction of strong S-Boxes. After the S-Box was designed, a performance analysis was performed to show its strength. The proposed S-Box demonstrated a very good cryptographic strength when compared with other recently designed S-Boxes. The results indicated that the proposed S-Box is a good choice for block ciphers.The structure of the rest of the paper is as follows. Modern block ciphers employ byte substitution to replace a complete byte (one element) of a matrix with another complete byte using the substitution box (S-Box). Generally, the design of an S-Box involves nonlinear mapping, which results in bijection. Many researchers have designed S-Boxes that are cryptographically strong using such mappings. One such mapping is linear fractional transformation (LFT), which was exhaustively explored for the construction of the S-boxes ,50,51,52Z = {0, 1, ……, 2n − 1}, both α and β are not 0 at the same time, and α(z)3 + β ≠ 0 is used to construct the n × n S-box. The nonlinear nature of CFT stimulates its usage in byte substitution. The procedure to generate the proposed S-Box for n = 8 is illustrated in In this paper, we extend the idea of LFT and construct our new transformation to generate an S-Box using another nonlinear mapping method in a simple and efficient way. We call this extended transformation cubic fractional transformation (CFT). A cubic fractional transformation is a function of the form:Z = {0, 1, …, 2n − 1} = {0, 1, …, 28 − 1} = {0, 1, 2, …, 254, 255} for n = 8. Any values can be chosen for α and β  that gratify the condition of α(z)3 + β ≠ 0. For the sake of the calculations here, we have chosen α = 95 and β = 15. The CFT function, C(z), given in Equation (2) generates values of Z – {0, 106} when z ∈ Z – {176, 184}. When z = 176, C(z) evaluates to 256 ∉ Z. When z = 184, the denominator of Equation (2) evaluates to 0. To keep the function, C(z), bijective, we explicitly define C(z) for z ∈ {176, 184} as conditioned in Equation (2). An example S-Box of a size of 8 × 8 is generated using a CFT function, c: To elaborate the construction of the proposed S-Box using Equation (1), let us have a specific type of cubic fractional transformation as given in Equation (2). Let This particular cubic fractional transformation of Equation (2) generates the elements of our proposed S-Box, which are organized in a 16 × 16 matrix as shown in α and β  can be used in Equation (1) to generate an S-Box. One can choose sub-keys as the values for α and β to generate dynamic and key-dependent S-boxes.As mentioned above, any values for In this section, we analyze our method and S-Box given in f: yY∈, ∃ a unique x ∈ X, such that f(x) = y. For n-bit inputs, this property maps all possible 2n input values to distinct output values. In other words, when x1 ≠ x2, then f(x1) ≠ f(x2). All component Boolean functions (f1 to f8) of the proposed S-box are balanced (number of 1’s = number of 0’s). Further, all 28 output values of the S-box are distinctive where each output value ∈ Z = {0, 1, …., 255}.A function, n-bit Boolean function, f, is calculated as [fW(z) = Walsh spectrum of the coordinate Boolean function, f, which is measured as:t.z is the dot product of t and z in bit-by-bit fashion and z ∈ {0, 1}n. The nonlinearity values, NL(f), of the Boolean functions of our S-Box are given in An S-Box operation should not be a linear mapping of an input to an output as it weakens the strength of any cipher. A high value of non-linearity provides resistance against linear cryptanalysis. The nonlinearity of an lated as ,56:(3)NLWebster et al.  introducx, is inverted; this changes the output bits, y and z, independently. For greater security, efforts are made to decrease the dependence between output bits. If a given S-Box satisfies the BIC, all the component Boolean functions possess high nonlinearity and meet the SAC [The authors in  introduc the SAC . Table 5Considering the nonlinearities and SAC, the average BIC values are 103.5 and 0.5, respectively. If a given S-Box is non-linear and demonstrates the SAC, it fulfills BIC . These vModern block ciphers are designed to create as much diffusion and confusion of the bits as possible for the security of data and provide a shield against different approaches that cryptanalysts adopt to obtain the plaintext. Mostly, this is achieved by S-Boxes, which provide nonlinear transformations. If an S-Box is designed with a low linear probability (LP), it is a very good cryptographic tool against linear cryptanalysis.xA and xB represent the input and output masks, respectively and Z = {0, 1, …., 255}. The linear probability of an S-Box is calculated using the following equation :(4)LP=maThe maximum value of LP of our S-box is only 0.156, and thus our S-Box provides good resistance against linear cryptanalysis.x and Δy are the input and output differentials, respectively. An S-box with smaller differentials is better at repelling differential cryptanalysis. Differential cryptanalysis is one of the most commonly used methods to reach the plaintext. Here, the differences in the original message (plaintext) and the differences in the ciphertext are obtained. The pairing of these differences may help reach some of the key values. To defy differential cryptanalysis, a small value of differential uniformity (DU) for a given S-Box is required. Differential uniformity is calculated as :(5)DU=maA high value of non-linearity provides resistance against linear cryptanalysis . The aveAn SAC value near 0.5  is the ultimate goal of every S-Box designer. Similarly, the BIC value of our S-box is better than the BIC values of more than half of the other S-boxes. Any S-Box with a lesser value of differential probability is more resilient against differential cryptanalysis. The DP value of our S-Box is 0.039, which is better than the DP values of nine other S-Boxes and equal to the DP values of two other S-Boxes as shown in To defy linear cryptanalysis, a smaller value of LP for a given S-Box is desired by S-Box designers. The LP value of our S-Box is 0.156. Due to this small value, we can say that our S-box is resistant to linear cryptanalysis.From the above comparison it is evident that our S-Box fulfills the most needed S-Box criteria and benchmarks, like SAC, BIC, NL, LP, DP, etc., and hence possesses better cryptographic strength.In this paper, we proposed a new transformation and suggested a novel method to construct efficient S-Boxes using cubic fractional transformation. The security strength of the proposed S-Box was studied using different standard criteria. The simulation results were in accordance with other relevant S-Boxes, rationalizing the performance of our S-Box method. The performance of our S-Box was good in most of the cases when compared with other recent S-Boxes. In particular, the scores of the SAC, BIC, nonlinearity, LP, and DP of the proposed S-Box provide evidence for it as a new alternative in the S-Box design domain. The promising results of the proposed S-Box analysis make it a potential candidate for usage in modern-day block ciphers. It is worth mentioning that our method is the first to explore the cubic fractional transformation for S-Box construction. Stronger S-boxes using cubic fractional transformation, like the proposed S-Box, are expected to emerge for usage in practical systems for secure communication.


Since reachability is already known to be undecidable in the fragment of PS 2.0 with only release-acquire accesses (PS 2.0-ra), we consider the fragment with only relaxed accesses and promises (PS 2.0-rlx). We show that reachability under PS 2.0-rlx is undecidable in general and that it becomes decidable, albeit non-primitive recursive, if we bound the number of promises.We consider the reachability problem for finite-state multi-threaded programs under the view-switches”, i.e., the number of times the processes may switch their local views of the global memory. We provide a code-to-code translation from an input program under PS 2.0  to a program under SC, thereby reducing the bounded reachability problem under PS 2.0 to the bounded context-switching problem under SC. We have implemented a tool and tested it on a set of benchmarks, demonstrating that typical bugs in programs can be found with a small bound.Given these results, we consider a bounded version of the reachability problem. To this end, we bound both the number of promises and of “


This paper investigates the problem of complex modified projective synchronization (CMPS) of fractional-order complex-variable chaotic systems (FOCCS) with unknown complex parameters. By a complex-variable inequality and a stability theory for fractional-order nonlinear systems, a new scheme is presented for constructing CMPS of FOCCS with unknown complex parameters. The proposed scheme not only provides a new method to analyze fractional-order complex-valued systems but also significantly reduces the complexity of computation and analysis. Theoretical proof and simulation results substantiate the effectiveness of the presented synchronization scheme. In the past twenty years, the application of fractional calculus has become a focus of attention, since fractional derivatives can more accurately describe the actual physical model. So, it has become an efficient and an excellent tool in physics, mathematical science, chemistry, control engineering, finance, signal processing and other fields ,3,4,5,6.The above-mentioned works mainly investigated the fractional-order systems with real variables, not involving complex variables. It is well known that complex variables, which double the number of variables, can generate complicated dynamical behaviors, enhance anti-attack ability and achieve higher transmission efficiency ,24,25. TInspired by the above discussions, the CMPS problem of FOCCS with unknown complex parameters is investigated in this paper. First, we present a stability theory for fractional-order uncertain nonlinear systems. Then, using this theory, the inequality proposed by Xu et al.  and compn-dimensional space. l2-norm of Notation: Definition 1The fractional integral of order α for a function f is defined as0 and where t . The fraDefinition 2Caputo’s fractional derivative of order 0 and n is a positive integer such that where t . Caputo’Lemma 1.0 and Let  Let x(t)Corollary 1.For a scalar derivable function Lemma 2.Let where  Let z ∈ℂLemma 3. Let andwhere Proof. By Next, we adopt contradiction to prove rem 1 in .k = 1, 2, 3, …, from (7), we have:d is bounded). Obviously, Suppose that Remark 1.1(t) and V2(t).Lemma 3 provides a stability criterion for the fractional-order nonlinear uncertain system by choosing a Lyapunov function that includes two parts VRemark 2.Lemma 3 provides a Lyapunov-based adaptive control method for stability analysis and synchronization of fractional-order systems (FOS).Remark 3.Lemma 3 is suitable for verifying the stability and stabilization controller design of FOS with unknown parameters and external disturbances.Remark 4.If the FOS has no uncertainties, then Lemma 3 is still valid.Lemma 4.For the fractional-order complex-variable systemswhere andwhere  For the Lemma 5.1 such that the following inequality holds:For all where ,46 For aLemma 6.2 such that the following inequality holds:For the Lipschitz continuous function ,46 For tWe consider a kind of fractional-order complex-variable chaotic drive and response systemsDefinition 3.For given drive system (8) and response system (9), it is said to achieve CMPS ifwhere  For giveRemark 5.Obviously, some known synchronization ways are the special cases of CMPS, such as CS, AS, PS and MPS.We consider a kind of FOCCS with unknown complex parameters asSystem (13) was chosen as the master system. In this case, we constructed the slave system as follows:Remark 2. From (13), it follows that Theorem 1.Asymptotically CMPS of systems (13) and (14) can be achieved under adaptive controllerand the complex update laws:where Proof. From the error vector and systems (13) and (14), it follows:Using Lemma 1, Corollary 1 and Lemma 2, we have:L1, l2 and L2 are three positive constants. Then, one has:Substituting Equations (16) and(17) into the inequality above, we further have:By en, from , V1(t)=eTherefore, the systems (13) and (14) can reach asymptotically CMPS under the adaptive control strategy (15–17). ☐Remark 6.Unlike previous works, in our proposed method, the entire analysis process is performed in the complex-valued domain, and the complex function theory is used to derive synchronization conditions without separating the original complex-valued chaotic system into two real-valued systems, which reduces the complexity of analysis and computation.Remark 7.If the system parameters are known, the update law will be reduced to (16) only.In this section, in order to show the effectiveness of the proposed scheme in preceding section, numerical example on fractional-order complex chaotic system will be provided. When numerically solving such systems, we adopted the Grünwald–Letnikov (G-L) method  using MAa, b, c are system parameters, let ia (the imaginary of a) is depicted in Consider the fractional-order Chen complex chaotic system with commensurate order:Assuming the parameters The error vector a, b, c) = , the initial conditions In the simulation, let We study the CMPS of FOCCS with unknown complex parameters, and propose a method for analyzing FOCCS without separating systems into real and imaginary parts. By this method, the constructed response system can be asymptotically synchronized to an uncertain drive system with a desired complex scaling diagonal matrix. The proposed synchronization scheme retains the complex nature of fractional-order complex chaotic systems. It not only provides a new method of analyzing FOCCS but also significantly decreases the complexity of computation and analysis. In future works, we will further investigate the synchronization of FOCCS, and generalize the obtained results to more general cases, i.e., FOCCS with time delay and external disturbances.


They form complex communities and collectively affect host health. Recently, the advances in next-generation sequencing technology enable the high-throughput profiling of the human microbiome. This calls for a statistical model to construct microbial networks from the microbiome sequencing count data. As microbiome count data are high-dimensional and suffer from uneven sampling depth, over-dispersion, and zero-inflation, these characteristics can bias the network estimation and require specialized analytical tools. Here we propose a general framework, HARMONIES, Hybrid Approach foR MicrobiOme Network Inferences via Exploiting Sparsity, to infer a sparse microbiome network. HARMONIES first utilizes a zero-inflated negative binomial (ZINB) distribution to model the skewness and excess zeros in the microbiome data, as well as incorporates a stochastic process prior for sample-wise normalization. This approach infers a sparse and stable network by imposing non-trivial regularizations based on the Gaussian graphical model. In comprehensive simulation studies, HARMONIES outperformed four other commonly used methods. When using published microbiome data from a colorectal cancer study, it discovered a novel community with disease-enriched bacteria. In summary, HARMONIES is a novel and useful statistical framework for microbiome network inference, and it is available at  Microbiota form complex community structures and collectively affect human health. Studying their relationship as a network can provide key insights into their biological mechanisms. The exponentially growing large datasets made available by next-generation sequencing (NGS) technology Metzker, , such asIn sequencing-based microbial association studies, the enormous amount of NGS data can be summarized in a sample-by-taxon count table where each entry is a proxy to the underlying true abundance. However, there is no simple relationship between the true abundances and the observed counts. Additionally, microbiome sequencing data usually have an inflated amount of zeros, uneven sequencing depths across samples, and over-dispersion. Initial attempts at constructing microbial association networks with this type of data , to infer the microbiome networks. It consists of two major steps: (1) normalization of the microbiome count data by fitting a zero-inflated negative binomial (ZINB) model with the Dirichlet process prior (DPP), (2) application of Glasso to ensure sparsity and using a stability-based approach to select the tuning parameter in Glasso. The estimated network contains the information of both the degree and the direction of associations between taxa, which facilitates the biological interpretation. We demonstrated that HARMONIES could outperform other state-of-the-art tools on extensive simulated and synthetic data. Further, we used HARMONIES to uncover unique associations between disease-specific genera from microbiome profiling data generated from a colorectal cancer study. Based on these results, HARMONIES will be a valuable statistical model to understand the complex microbial associations in microbiome studies. The Y denote the n-by-p taxonomic count matrix obtained from either the 16S rRNA or the metagenomic shotgun sequencing (MSS) technology. Each entry yij, i = 1, …, n, j = 1, …, p is a non-negative integer, indicating the total reads related to taxon j observed in sample i. It is recommended that all chosen taxa should be at the same taxonomic level  since that mixing different taxonomic levels in the proposed model could lead to improper biological interpretation. As the real microbiome data are characterized by zero-inflation and over-dispersion, we model yij through a zero-inflated negative binomial (ZINB) model asLet i representing the proportion of “extra” zeros in sample i. The second component, NB, models the “true” zeros and all the nonzero observed counts. i.e., counts generated from a negative binomial (NB) distribution with the expectation of λij and dispersion 1/ϕj. Here, “true” zero refers to a taxon that is truly absent in the corresponding sample. The variance of the random variable from NB distribution, under the current parameterization, equals to j can lead to over-dispersion.The first component in the Equation (1) models whether zeros come from a degenerate distribution with a point mass at zero. It can be interpreted as the “extra” zeros due to insufficient sequencing effort. We can assume there exists a true underlying abundance for the taxon in its sample, but we fail to observe it with the mixture probability πi's and ϕj's, we use a Bayesian hierarchical model for parameter inference. First, we rewrite the model (1) by introducing a binary indicator variable ηij ~ Bernoulli(πi), such that yij = 0 if ηij = 1, and yij ~ NB if ηij = 0. Then, we formulate a beta-Bernoulli prior of ηij by assuming πi ~ Beta, and we let aπ = bπ = 1 to obtain a non-informative prior on ηij. We specify independent Gamma prior Ga for each dispersion parameter ϕj. Letting aϕ = bϕ = 0.001 results in a weakly informative gamma prior.To avoid explicitly fixing the value of πij, contains the key information of the true underlying abundance of the corresponding count. As λij is affected by the varying sequencing effort across samples, we use a multiplicative characterization of the NB mean to justify the latent heterogeneity in microbiome sequencing data. Specifically, we assume λij = siαij. Here, si is the sample-specific size factor that captures the variation in sequencing depth across samples, and αij is the normalized abundance of taxon j in sample i.The mean parameter of the NB distribution, λsi and αij. For example, si can be the reciprocal of the total number of reads in sample i. The resulted αij is often called relative abundance, which represents the proportion of taxon j in sample i. In this setting, the relative abundances of all the taxa in one sample always sum up to 1. Similarly, other methods have been proposed with different constraints for normalizing the sequencing data  = 0. For the outer mixtures, M is an arbitrarily large positive integer. Letting M → ∞ and defining the weight ψm by the stick-breaking procedure  such that νm ~ N, tm ~ Beta, and Vm ~ Beta. We further set where ψA = {αij} represents the true underlying abundance of the original count matrix. We further assume ij not only reduces the skewness of the normalized abundance, but converts the non-negative αij to a real number. We apply the following conjugate setting to specify the priors for μj and j and j follows a non-standardized Student's t-distribution, i.e.,In our model, the normalized abundance matrix a0, b0, h0, and a0 = 2, b0 = 1 to obtain a weakly informative prior for h0 = 10 such that the normal prior on μj is fairly flat. We adopt the following prior specification for the rest model parameters. First, we assume an noninformative prior for each πi by letting aπ = bπ = 1. Next, we specify aϕ = bϕ = 0.001 in the Gamma prior distribution for all ϕj's. Then, we apply the following prior setting for the DPP: M = n/2, σs = 1, τν = 1, at = bt = 1, and am = bm = 1.As for the fixed parameters i et al.  and set A, denoted as Z = log A, represents the normalized microbiome abundances on the log scale. We use Markov chain Monte Carlo (MCMC) algorithm for model parameter estimation  based on the estimated ηij for each observed yij = 0 in the data. In particular, suppose that we observe L zeros in total. We calculate the marginal posterior probability of being 1 for each ηl, l = 1, …, L as B is the number of MCMC iteration after burn-in. This marginal posterior probability pl represents the proportion of MCMC iterations in which the lth 0 is essentially a missing value rather than the lowest count in the corresponding sample. Then, the observed zeros can be dichotomized by thresholding the L probabilities. The zeros with pl greater than the threshold are considered as “true" zeros in the data, whereas the rest are imputed by the corresponding posterior mean of logαj·. We used the method proposed by Newton et al. , which guarantees the imputed zeros have a Bayesian FDR to be smaller than cη,The logarithmic scale of n et al.  to detercη = 0.01 guarantees that the Bayesian FDR to be at most 0.01. We set cη = 0.05 for the simulation study and cη = 0.01 for the real data analysis.In practice, a choice of G =  is used to illustrate the associations among vertices V = {1, …, p}, representing the p microbial taxa. E = {emk} is the collection of (undirected) edges, which is equivalently represented via a p-by-p adjacency matrix with emk = 1 or 0 according to whether vertices m and k are directly connected in G or not. GGM assumes that the joint distribution of p vertices is multivariate Gaussian N, yielding the following relationship between the dependency structure and the network: a zero entry in the precision matrix Ω = Σ−1 indicates the corresponding vertices are conditional independent, and there is no edge between them in the graph G. Hence, a GGM can be defined in terms of the pairwise conditional independence. If X ~ N, thenBased on the normalized microbial abundances, we estimate their partial correlation matrix in order to construct the microbiome network under the Gaussian graphical model (GGM) framework. An undirected graph m and k, representing the degree and direction of association between two vertices, conditional on the rest variables. Consequently, learning the network is equivalent to estimating the precision matrix Ω. For real microbiome data, we set the taxa (on the same taxonomic level) as vertices. Hence, a zero partial correlation in the precision matrix can be interpreted as no association between the corresponding pair of taxa, while a nonzero partial correlation can be interpreted as cooperative or competing associations between that taxa pair.where Ω. The sparsity can be achieved by imposing l1-penalized log-likelihood,In biological applications, we often require a sparse and stable estimation of the precision matrix S is the sample covariance matrix. The coordinate descent algorithm can iteratively solve p. The estimated precision matrix is sparsistent   to simulate the count table Yn × p from a DM model, with its parameters being exp(D); (5) to mimic the zero-inflation in real microbiome data by randomly setting part of entries in the count table to zeros. Note that the data generative scheme is different from the model assumption, which is given in Equation (1). The detailed generative models are described below.We generated the simulated datasets from a Dirichlet-multinomial (DM) model using the following steps: (1) to generate the binary adjacency matrix; (2) to simulate the precision matrix and the corresponding covariance matrix; (3) to generate p-by-p adjacency matrix for the p taxa in the network. Here, the adjacency matrix was generated according to an Erdős–Rényi (ER) model. An ER model ER generates each edge in a graph G with probability ρ independently from every other edge. Therefore, all graphs with p nodes and M edges have an equal probability of G correspond to the 1's in the resulted binary adjacency matrix. Next, we simulated the precision matrix Ω following Peng et al. . To ensure positive definiteness of the precision matrix, we followed Peng et al.  to represent the true underlying abundances D. To obtain a count matrix that fully mimics the microbiome sequencing data, we generated counts from a DM model with parameter exp(D). Specifically, we first sampled the underlying fractional abundances for the ith sample from a Dirichlet distribution. The ith underlying fractional abundance was then denoted as ψi ~ Dirichlet(exp(Di·)). Next, the counts in the ith sample were generated from Multinomial. Finally, we randomly selected π0% out of n × p counts and set them to zeros to mimic the zero-inflation observed in the real microbiome data. In general, the generative process had different assumptions from the proposed method. Under the appropriate choice of parameters, the simulated count data was zero-inflated, overdispersed, and the total reads varied largely between samples. In practice, we let ρ = 0.1 in the ER model. The mean parameter μ of the underlying multivariate Gaussian variable was randomly sampled from a uniform distribution Unif. The number of total counts across samples Ni, i = 1, …, n was sampled from a discrete uniform distribution with range . Under each combination of n, p, and π0, we generated 50 replicated datasets by repeating the process above.Next, we simulated p taxa from a real microbiome dataset, the NorTA generates the synthetic data with n samples as follows: (1) to calculate the p-by-p covariance matrix Σ0 from the input real dataset; (2) to generate an n-by-p matrix, denoted by Z0, from a multivariate Gaussian distribution with a mean of 0p1 ×  and the covariance matrix of Σ0; (3) to use standard normal cumulative distribution function to scale values in each column of Z0 within ; (4) to apply the quantile function of a ZINB distribution to generate count data from those scaled values in each column of Z0. In practice, we used R package SPIEC-EASI to implement the above data generative scheme, where the real data were from those healthy control subjects in our case study presented in section 3.2. Under each combination of n and p, we generated 50 replicated datasets.We generated synthetic data following the Normal-to-Anything (NorTA) approach proposed in Kurtz et al. . NorTA wl1 regularization. However, as discussed in section 1, representing the dependency structure by the correlation matrix may lead to the detection of spurious associations.We considered the four commonly used network learning methods. The first two methods, SPIEC-EASI-Glasso and SPIEC-EASI-mb, use the transformed microbiome abundances which are different from the normalized abundances estimated by HARMONIES. Both infer the microbial network by estimating a sparse precision matrix. The former method (SPIEC-EASI-Glasso) measures the dependency among microbiota by their partial correlation coefficients, and the latter method (SPIEC-EASI-mb) uses the “neighborhood selection” introduced by Meinshausen and Bühlmann  to constm and taxon k to be true positive if ωmk ≠ 0, mk. We calculated the number of true negative (TN), false positive (FP), and false negative (FN) in a similar manner. Therefore, each tuning parameter defined a point on a ROC curve. As for the correlation-based methods, we started with ranking the absolute values in the estimated correlation matrices, denoted as We quantified the model performances on the simulated data by computing their receiver operating characteristic (ROC) curves and area under the ROC curve (AUC). For the HARMONIES or SPIEC-EASI, the network inference was based on the precision matrix. Hence, under each tuning parameter of Glasso, we calculated the number of edges being true positive (TP) by directly comparing the estimated precision matrix against the true one. More specifically, we considered an edge between taxon We further used the Matthew's correlation coefficient (MCC) to evaluate results from the simulated data. The MCC is defined asHere, the MCC was particularly suitable for evaluating network models. As the number of conditionally independent taxa pairs was assumed to be much greater than the number of dependent pairs in a sparse network, MCC was preferable to quantify the performances under such an imbalanced situation. Note that MCC ranges from , with a value close to 1 suggesting a better performance. Since each value of MCC was calculated using a given set of TP, TN, FP, and FN, we adopted the optimal choice of tuning parameter for the HARMONIES or SPIEC-EASI (with either Glasso or MB for network inference), given by StARS. As for the correlation-based methods, CClasso outputted a sparse correlation matrix. We used the result to calculate TP, TN, FP, and FN directly. For Pearson-corr, we set the threshold such that the resulted number of nonzero entries in the sparse correlation matrix was the same as the number of non-zero entries in the true sparse partial correlation matrix. In fact, this choice could favor the performance of Pearson-corr for larger sample size, as shown in section 3.1.p-values were used.To assess model performances on the synthetic datasets, we followed Kurtz et al.  to use an = 60, 100, 200, or 500), total numbers of taxa (p = 40 or 60), extra percentages of zeros added . In each subfigure, the HARMONIES outperformed the alternative methods in terms of both AUC and MCC, and it maintained this advantage even with the number of sample size greatly increases. Further, a smaller sample size, a larger proportion of extra zeros added (π0 = 20%), as well as a larger number of taxa in the network (p = 60), would hamper the performance of all the methods, as we expected. Two modes of SPIEC-EASI, SPIEC-EASI-Glasso, and SPIEC-EASI-mb, showed very similar performances under all the scenarios, with SPIEC-EASI-Glasso having only a marginal advantage over the other. Further, we observed that the Pearson-corr method yielded higher AUCs even than the precision matrix based methods, especially when there was a lager proportion of extra zeros or larger number of taxa in the network. This result suggested that the Pearson-corr could capture the overall rank of the signal strength in the actual network. However, under a fixed cut-off value that gave a sparse correlation network, the MCCs from the Pearson-corr were always smaller than the precision matrix based methods. Note that the cut-off value we specified for Pearson's correlation method indeed favored its performance. In general, the alternative methods considered here were able to reflect the overall rank of the signal strength by showing reasonable AUCs. However, they failed to give an accurate estimation of the network under a fixed cut-off value.n or decreasing the number of features p would improve the performance of all methods and lead to greater disparity between partial and pairwise correlation-based methods. In general, our HARMONIES maintained the best in all simulation and evaluation settings except for one case, where the SPIEC-EASI-mb only showed a marginal advantage  is the third most common cancer diagnosed in both men and women in the United States .The datasets generated for the simulation study can be found in the author's Github page—SJ performed the experiments. GX and AK provided resources and helpful discussions. SJ, QL, and XZ designed the experiment, performed data analysis, wrote the software, and the manuscript. SJ, YC, BY, and XZ developed the website for online implementation of HARMONIES.AK is a consultant for Merck and the principal investigator on a Norvatis sponsored study. The remaining 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.


Scientific Reports 10.1038/s41598-019-41394-9, published online 22 March 2019Correction to: The original version of this Article contained errors.The title included the “≤” symbol at the beginning as a result of typesetting error.In addition, as a result of figure assembly errors, panels B and F of Figure 2 were duplicated from author’s previous research. This figure is now correct, and the raw data underlying it is now also included in the Supplementary Information file.These have now been corrected in the HTML and PDF versions of this Article, and in the accompanying Supplemental Material.


Scientific Reports 10.1038/s41598-020-65076-z, published online 25 May 2020Correction to: The original version of this Article contained an error in the title of the paper, where the word “cooperation” was incorrectly given as “cooperaion”. This has now been corrected in the PDF and HTML versions of the Article and in the accompanying Supplementary Information file.


While equilibrium phase transitions are easily described by order parameters and free-energy landscapes, for their non-stationary counterparts these quantities are usually ill-defined. Here, we probe transient non-equilibrium dynamics of an optically pumped, dye-filled microcavity. We quench the system to a far-from-equilibrium state and find delayed condensation close to a critical excitation energy, a transient equivalent of critical slowing down. Besides number fluctuations near the critical excitation energy, we show that transient phase transitions exhibit timing jitter in the condensate formation. This jitter is a manifestation of the randomness associated with spontaneous emission, showing that condensation is a stochastic, rather than deterministic process. Despite the non-equilibrium character of this phase transition, we construct an effective free-energy landscape that describes the formation jitter and allows, in principle, its generalization to a wider class of processes. Description of non-equilibrium phase transitions is problematic, due to the absence of suitable free energy landscapes. Here, the authors experimentally show delayed photon condensation and timing jitter in a dye-filled microcavity, modelled by a non-equilibrium extension of the free-energy landscape. This relation is particularly relevant near a second-order phase transition, where all the relevant macroscopic details are described by an emergent order parameter. This is, on average, located at the minimum of the free-energy landscape3, with its neighbourhood being locally probed as the system is driven through configuration space by fluctuations . While we still lack a universal generalization of these ideas to non-equilibrium systems, Jaynes has suggested its possibility4. Some well-established, near-equilibrium stochastic descriptions of relaxation involving free-energy surfaces are known5, but are not necessarily valid far from equilibrium. A particular example in this direction has been constructed for the laser6, a fundamentally non-equilibrium system whose steady state can be described as the minimum of a properly defined effective free energy, corresponding to a detailed balance between driving and dissipation.The connection between the properties of a system at thermal equilibrium and the geometry of its free-energy landscape is a powerful concept that dates back to original ideas developed by Gibbs8 or the build-up of anti-ferromagnetic correlations in Ising models9. Non-Hamiltonian quenches contain a more general class of processes. In cold atoms, for instance, the Kibble–Zurek mechanism11 is observed by evaporatively cooling the system at a finite rate, quenching the system through a BEC phase transition. We shall refer to a quench as a sudden change in one of the system parameters that brings it to a far-from-equilibrium state, without affecting its Hamiltonian.While the previous arguments are relevant for systems close to a steady state, a sudden parameter change, often called a quench, necessarily brings the system sufficiently far from equilibrium to question the validity of such approaches. The meaning of a quench depends on context and, in particular, one can distinguish between Hamiltonian and non-Hamiltonian cases. The former consist of time-dependent variations in some sort of interaction term, involved, for instance, in the Mott insulator-superfluid transition15. Following the initial observation of Bose–Einstein condensation of photons16, a number of experiments on grand-canonical fluctuations17, spontaneous symmetry breaking18, emergence of long-range order19, among other aspects of equilibrium physics21 have been described. Their non-equilibrium counterparts, however, remain greatly unexplored.Photon condensates are ideal platforms to explore both equilibrium and non-equilibrium physics. A thermalizing medium, typically a dye solution, is placed inside an incoherently pumped optical microcavity. The combined rates of thermalization, pumping and cavity loss enable such a driven-dissipative system to be tuned between in- and out-of-equilibrium regimesg(2). It provides access to the statistical properties of the photon condensation transition, and is particularly relevant in non-stationary systems, when the full knowledge of individual realizations is inaccessible. While the usual stationary correlation function, g(2)(τ), accurately accounts for fluctuations in steady state, g(2) is the appropriate quantity to describe the evolution of transient, non-equilibrium systems. The averaged condensate intensity as a function of time shows width broadening, a manifestation of diverging jitter in the condensate formation time upon approaching the critical excitation energy. This effect is directly witnessed by distinctive off-diagonal anti-correlations in g(2) and originates from quantum fluctuations associated with spontaneous emission. By properly defining an effective (non-equilibrium) free energy, we suggest that jitter may be a universal feature of transient phase transitions in systems obeying relatively general conditions on the convexity of their free-energy landscape.Here, we study the transient dynamics of photon condensation that follows a quench in a dye-filled optical microcavity. Besides measuring the ensemble-averaged photon number dynamics, we introduce the non-stationary, two-time, second-order correlation function κ and Γ↓, respectively. The essentials of the cavity dynamics are described by the density operator ρ, for both photons and molecules, which obeys the master equation23E and A the dye emission and absorption rates, respectively, Γ↑ the incoherent  pumping rate and Nmol the total number of molecules inside the cavity. In general, Γ↑ is a time-dependent quantity, such as in the case of pulsed pumping. Due to the high collision rate between dye and solvent molecules, all the relevant cavity processes, including light–matter interactions, are incoherent.Despite the fundamentally multi-mode character of our optical cavity, the phenomenology described here is essentially that of a single-mode system. Cavity excitations (photons and excited molecules) can be lost by two processes: mirror transmission and molecular spontaneous emission into free space, at rates f = fc, and in the absence of pumping and losses (κ = Γ↓ = Γ↑ = 0), an equilibrium (steady state) between molecular excitations and photons is established by a principle of detailed balance13. The photon number in this equilibrium state would show a phase transition as the total number of cavity excitations, Nex = n + f Nmol, which is the control parameter, is increased. The photon number, or order parameter, ranges from a disordered phase (n ≲ 1) dominated by spontaneous emission to an ordered phase (n ≫ 1) dominated by stimulated emission. While there is, in principle, a U(1) symmetry breaking upon crossing the condensation phase transition, the full dynamics can be described simply through photon number n12. By exciting a large number of dye molecules over a short period of time, the cavity can be quenched through this phase transition to a far-from-equilibrium state. The subsequent relaxation dynamics correspond to a non-stationary, transient counterpart of the equilibrium phase transition described above. In this way, we define a transient phase transition as the evolution in configuration space after a jump across a phase transition in parameter space (a quench), where “phase transition” has its usual time-independent, thermodynamic meaning. This is distinct from the recently introduced concept of dynamical phase transitions25. The non-linear coupling between photons and molecular excitations occurring during this transient relaxation process gives rise to non-trivial fluctuation and correlation properties. Finally, given its lossy character, the light will transition back to the phase dominated by spontaneous emission before all excitations are lost.Mean-field rate equations are obtained by taking expectation values and neglecting correlation terms in Eq. . The num↓ accounts for emission both into free space and cavity modes that do not reach the regime of stimulated emission (do not condense), which will be discussed in more detail later. Also, and despite not being relevant for the results discussed here, effects associated with the multi-mode character as well as spatially resolved molecular reservoirs have been appreciated in the context of gain clamping26 and decondensation mechanisms14.A few notes are in order regarding the multi-mode nature of our cavity. Within the single-mode approximation, the rate term Γμm. A 40 ps laser pulse at 532 nm, typically ranging from 0.5 to 2 nJ in energy, is used to rapidly excite the molecules, quenching the cavity to a far-from-equilibrium state. In response, a much longer pulse (≳1 ns) of light leaks from the cavity mirrors, the exact temporal shape of which depends both on the cavity parameters  as well as the number of molecular excitations that follow the pump pulse, which is the control parameter used to select the different dynamical phases. A portion of pump light is directed onto two saturated avalanche single-photon detectors (APDs), O1 and O2, where a coincident detection is used as a time stamp for the beginning of the experiment, with a measured uncertainty of about 10 ps. The cavity output light is directed onto two unsaturated APDs, B1 and B2, with an average of 0.1 detections per pulse, on each detector. The experiment is conducted at a repetition rate of 11 kHz. Such a low repetition rate ensures a complete decay of all excitations and statistical independence between different realizations. We describe the experimental results in the form of following three sets:zero-time statistics: full time-averaged cavity output;one-time statistics: time-resolved, but averaged over all forms of fluctuations and correlations in the cavity output;two-time statistics: unequal time, cross-correlated signal from detectors B1 and B2, providing access to fluctuations in the cavity output.The experimental configuration is sketched in Fig. P is increased beyond a critical value Pth, as shown in Fig. 14. Consequently, and despite the absence of a full thermal distribution, parallels may be drawn with Bose–Einstein condensation of photons28. While BEC is only strictly defined in thermal equilibrium as the macroscopic occupation of the ground state, we are assuming here a broader concept of condensation, as discussed in such diverse fields as physics, ecology, network theory or social sciences31. This can be thought of as the process where a particular, or small set of modes, in a multi-mode system becomes macroscopically occupied while the remaining ones saturate or become depleted.We begin by demonstrating the existence of a (condensation) phase transition in the total amount of light emitted by the cavity as the excitation energy, or pump energy, α is an empirical parameter set by fitting the light-yield curve to the mean-field rate equations. Loosely speaking, α determines the fraction of spontaneous emission into non-modelled cavity modes and is expected to be a small contribution, which will be verified in “One-time statistics” section.By counting the rate of detection events in B1 and B2, we measure the total cavity output as a function of input pulse energy , as shown in the inset of Fig. P < Pth) display a simple exponential decay on a time scale of about τ0 ~ 4 ns, the molecular excited-state lifetime. Above threshold, stimulated emission becomes important, leading to a large increase in photon number, followed by rapid depopulation of the condensate before a final decay at the slower time scale of the molecular excited-state decay.Here, we expand the time-averaged results of the previous section to the time-dependent cavity output pulse shape, as shown in Fig. f, and the number of cavity photons, n. In equilibrium, the molecular excitation fraction, f, cannot exceed its critical value, fc. Under non-equilibrium conditions, however, if at any instant f > fc (e.g. after a quench), the photon population will grow exponentially until f drops below fc. This exponential increase in the number of photons, resulting from the onset of stimulated emission, is accompanied by rapid de-excitation of molecules, as shown in Fig. f Nmol, and the photon number n. We choose experimental parameters that make the number of photons comparable with f Nmol, such that the effects of this two-way coupling become more prominent. On the opposite limit of f Nmol ≫ n, which is approached for lower values of λ0 (the cavity cutoff wavelength), the larger molecular reservoir becomes insensitive to photon number fluctuations, approaching the limit of a Markovian bath. This limit is at the origin of the observation of grand-canonical number statistics in photon BECs17.It is instructive at this point to reflect upon the interplay and coupled dynamics of the molecular excitation fraction, ↓ and κ to the results in Fig. κ = 1010 s−1, corresponding to a cavity lifetime of 100 ps, and Γ↓ = 0.998Γ0, with Γ0 = 1∕τ0 the molecular fluorescence decay rate. Within the single-mode approximation, this means that only 0.2% of the total molecular emission goes into the condensing mode while 99.8% goes both into free space and excited modes that do not reach the regime of stimulated emission. Together with the light-yield curves in Fig. α = 0.13; small as expected. The emission and absorption rates are not taken as fitting parameters but rather calculated from experimental absorption and emission data for rhodamine-6G32. The total number of molecules is calculated from the dye concentration and cavity volume to be Nmol = 1.9 × 108.We fit Γf approaches fc from above, the cavity dynamics become slow, as dictated by Eq. . Second-order correlations are typically described by the single-time g(2)(τ) function, with τ = t1 − t2, due to time-translation symmetry in steady-state conditions. In transient systems, however, the absence of this symmetry means that the full two-time, t1 and t2, dependence must be retained. We can then defineP is the joint probability of photon detection at times t1 and t2 in detectors B1 and B2, respectively. By marginalizing over the second detector, P(t1) and P(t2) are obtained as the single-detector probabilities. The approximation in Eq. (a†(t1), a(t2)] ≈ 0 or 〈a†(t)a(t)〉 ≫ 1. The former is satisfied when ∣t1 − t2∣ is larger than the coherence time , and the latter is true for large photon numbers, as verified in Fig. Correlations and fluctuations of the cavity output can now be investigated by retaining the labelling of detection timestamps in B1 and B2. We then construct the two-time, non-stationary, second-order correlation function g(2) > 1) and the off-diagonal anti-correlation (g(2) < 1) lobes. These features are mainly a manifestation of the same kind of fluctuations—jitter, or shot-to-shot timing fluctuations, in the condensate formation—which become amplified near the critical excitation energy. In the remainder of this section, we discuss this effect associated with transient phase transitions.The second-order correlation function is shown in Fig. g(2) provides immediate information on number, or intensity, fluctuations, namely g(2) ≃ 1 + 〈Δn(t)2〉∕〈n(t)〉2. As such, periods of larger fluctuations coincide with the inflection point of the average pulse shape, consistent with a condensate forming at slightly different instants in each realization of the experiment. In a microscopic picture of the cavity dynamics, spontaneously emitted photons are required to seed the condensate growth. The randomness associated with the quantum nature of spontaneous emission then leads to such shot-to-shot time fluctuations, or jitter in the condensate formation. As we shall demonstrate in the next section, these periods of larger fluctuations correspond to a passage through the convex part of an effective free-energy landscape.Let us proceed by separately analysing diagonal and anti-diagonal correlations, as shown in Fig. g(2) < 1 are then an immediate witness of fluctuations in formation time. Also, it is further evidence of the two-way coupling between photons and molecules, as discussed previously.The above interpretation is further supported by the off-diagonal anti-correlation lobes, as seen in Figs. 37, the two-time correlation function can be obtained via the quantum regression theorem38, as shown in Fig. 42. Different classes of events  are defined and drawn at random given their respective rates. These are dynamically calculated as the number of photons and molecular excitations are updated at each step. Full details of this model can be found in “Methods” section. In the limit of a large number of realizations, this approach is equivalent to evolving the density matrix according to Eq. (By retaining correlations up to second-order in Eq.  using a g to Eq. . By retag(2). However, the quantum trajectories method allows us to easily appreciate the effect of formation jitter, depicted in Fig. g(2) depends on both the individual pulse shapes and their uncertainty in formation time. As it turns out, the earlier forming pulses (relatively far above threshold) are of shorter duration than later forming pulses . This effect competes with the larger fluctuations in formation time closer to threshold, such that a diverging behaviour may not be extractable from the g(2) maps alone.The experiment does not allow direct access to individual trajectories, only the effect of their relative fluctuations on 17, which predicts g(2)(0) = 2. In the latter, a steady state is achieved by a detailed balance between cavity loss and continuous pumping, with the fluctuations being related to the coupling between the photons and the molecular grand-canonical reservoir in conditions of thermal equilibrium. There is time-translation symmetry and number fluctuations are damped within a 2 ns time scale17. In contrast, the g(2) structure we identify here reflects the propagation of the initial fluctuations associated with the spontaneous emission events that trigger the growth of the condensate pulse. The system never reaches a steady state and all the dynamics are fundamentally transient. In principle, g(2), which depends on both the individual pulse shapes and their formation jitter, can even be larger than 2.Note that the fluctuations described here are of a different origin than those arising from the grand-canonical nature of a photon BECψ(t) towards its equilibrium value ψ0 can be generically modelled by the time-dependent Landau equation45F = F(ψ) the near-equilibrium free energy and η a generic Langevin stochastic force. This defines a universal class of dissipative relaxation processes5, typically valid near thermal equilibrium, where F = F(ψ) can be expanded in Taylor serious around ψ0. We shall undertake here a different approach that will extend the validity of the model above into far-from-equilibrium conditions.We now develop a general treatment of the relaxation process described in “Results” section, thus highlighting its universal features and applicability outside the particular case of photon condensates. Relaxation of an order parameter κ = Γ↓ = Γ↑ ≡ 0), the full non-equilibrium dynamics described by Eqs. (47ψ → n) and total number of excitations, Nex = n + f Nmol, the control parameter, as defined earlier. By retaining the full dynamical information contained in the mean-field rate Eqs.   κ = Γ↓ = by Eqs. , which dentclass1pt{minimaη accounts for spontaneous emission, such that 〈η(t)η(0)〉 = f 2E 2δ(t)50. This allows us to include fluctuations and beyond-mean field effects in an universal model of relaxation. An alternative, and formally equivalent, approach would be the construction of a Fokker–Planck equation5 for the probability density function (PDF) Pn(t), with a drift term given by the derivative of the effective free energy, as in Eq. , and shown in Fig. F. While the free-energy landscape is more convex for larger values of the control parameter, which increases number fluctuations, these are longer-lived close to threshold (larger jitter). This effect witnesses the complex interplay between number and time fluctuations associated with the stochastic transient relaxation process determined by Eq.  curvature in the free energy tends to localise it. At a second-order phase transition, the curvature at the minimum of the free energy (defining an equilibrium order parameter) vanishes and the PDF shows diverging fluctuations that persist for long times, giving rise to critical slowing down. In transient, non-equilibrium systems dramatic features also occur, with regions of negative (convex) curvature acting to amplify fluctuations. A PDF evolving through these regions while relaxing towards the free-energy minimum experiences a short-lived but large increase of fluctuations, as shown in Fig. d by Eq. . Even inκ, as in the case of the experiment described in “Results” section, the rate of change of the free-energy landscape depends on the photon number n, which is itself described by a given PDF. The landscape is now neither constant, nor a simple function of time, but rather coupled to the photon number history, such that for trajectories where the condensate forms early, it also decays early, leading to the anti-correlation lobes seen in g(2). This is essentially the same result as depicted by the quantum trajectories simulation in Fig. In the presence of loss by cavity transmission, the free energy and photon number PDF are coupled in a non-trivial way. For sufficiently large Nex, the control parameter, to be fixed, such that all molecular excitations are converted into cavity photons, corresponding to the limit of negligible losses. As a final remark, Eq. (F(ψ), from the observed average dynamics of the order parameter, ψ(t), although in practice the results are not very informative.The free-energy description assumes the total number of cavity excitations ark, Eq.  allows fIn this work, we have described the transient non-equilibrium dynamics of light in a dye-filled optical cavity quenched through a condensation phase transition. By rapidly exciting a large number of dye molecules, the system is brought to a far-from-equilibrium state. By averaging over all forms of fluctuations, we observed a delayed formation of the condensed phase, interpreted as a transient equivalent of critical slowing down. When quenched above the condensation threshold excitation energy, the quantum fluctuations associated with spontaneous emission seed the growth of the order parameter as the system relaxes into equilibrium. The relaxation dynamics is slower close to the critical point, a feature easily interpreted under the geometrical properties of the effective free-energy landscape, which becomes flat. The same mechanism is responsible for the usual critical slowing down in the relaxation rate of the ordered phase that follows a second-order phase transition. Also, despite the absence of latent heat and the fact that we are dealing with second-order and not first-order phase transitions, analogies can be drawn with the precipitation in supercooled, or supersaturated, liquids. Even quenched above the critical point, a seed of spontaneously emitted photons is needed to nucleate condensation, playing the role of the seeding crystals in supercooled, or supersaturated, liquids. Also, once seeded, crystallization across the entire liquid is faster for liquids quenched further across their critical parameters, with temperature playing the same role as the excitation fraction that follows the quench, in the optical cavity context.g(2). More precisely, we demonstrated that while the diagonal of g(2) is a powerful probe of the geometrical properties of the free-energy landscape, its off-diagonal elements reflect the relevant dissipation processes, with the anti-correlation lobes a joint effect of jitter and cavity loss. Fluctuations, arising from spontaneous emission, are highly amplified as the order parameter goes through the convex part of the free-energy landscape towards its equilibrium point.By measuring the statistical properties of this transient condensation, we describe a novel form of diverging fluctuations around the critical point, jitter in the formation of the ordered phase. These are witnessed by strong diagonal correlations and off-diagonal anti-correlations in the non-stationary, second-order correlation function, 54 may now benefit from being re-examined. Despite some recent efforts in describing time fluctuations and other non-equilibrium features of micro- and nano-lasers57, and to the best of our knowledge, we present here for the first time a generic and comprehensive description of the relation between temporal and number fluctuations in the non-stationary dynamics of systems undergoing second-order phase transitions. Finally, the system studied in this work, as well as the related examples stated above can be described by single-value order parameters. One may wonder on the generalization of these effects in spatially extended systems, where the order parameter is a function of both space and time. In the context of the Kibble–Zurek mechanism11, for instance, most studies are simply concerned with the defect number scaling after the system relaxes to some steady state, with the intrinsic relaxation dynamics often ignored. As such, although we cannot anticipate specific effects, one wonders about the correspondence between transient fluctuation dynamics of the zero-dimensional system described in our paper and that of spatially extended systems.The description in terms of the geometric properties of the effective free-energy landscape, being independent of the microscopical details of our particular system, allows us to generalize our observations. In particular, both the transient critical slowing down and the jitter in the formation of the order parameter are expected to be universal features of the dynamics that follows a quench through a second-order phase transition. In micro- and nano-lasers, in particular, the full two-time, non-stationary analysis of the relaxation process has been greatly overlooked and previous resultsFrom the non-equilibrium model introduced in Eq. , one cannm〉 depends on the estimation of 〈n2m〉, 〈nm2〉, 〈n3〉, … , which requires solving a large number of ordinary differential equations. These can be reduced with an hierarchical set of approximations. For instance, in the semi-classical limit, the expectation values for n and m are factorized, 〈mn〉 ≈ 〈m〉〈n〉, reducing Eq. ..3). Here37 given byIn order to account for correlations and fluctuations, one needs to go beyond the semi-classical approximation. In particular, the expectation values can be expanded in a hierarchical mannerx2〉 − 〈x〉2, with x = {n, m}, we explicitly writeThese represent the second, third, and fourth order cumulants, with the summation referring to all possible combination of variables. A minimal description of correlations is constructed by truncating the hierarchy at second order. In this way, and by defining g(2)(t), follow immediately asThe second-order photon correlation function at zero-time delay, 38, which allows us to calculate any quantity of the form 〈X(t + τ)Y(t)〉 using two single-time evolutions. Let the initial state of the system be χ(0), and the evolution be given by the map, χ(t) = The two-time second-order correlation function can be obtained by invoking the quantum regression theoremχ(0) from 0 to t, followed by the conditional state Yχ(t) from t to t + τ. For our cavity model, we begin by first evolving the density operator, ρ, from t = 0 to t = t1, using the second-order rate Eqs. (g(2)(t1). Second, the first-order rate Eqs. χ(0) frome Eqs. (g(t1). Secate Eqs.  are usedwing Eq. , one the42. Here, the Lindblad dynamics of the density operator ρ is replaced by a wavefunction whose evolution is given by a non-Hermitian effective Hamiltonian, interspersed with stochastic quantum jumps. Subsequently, evolution of ρ is approximated by an ensemble average of wavefunctions, or trajectories, say z, the average of any observable is then given byThe second-order approach described above corresponds to a first-level approximation to the description of correlations and fluctuations in the cavity dynamics. Moving to higher-order expansions increases the number of ordinary differential equations needed to resolve the dynamics, which soon becomes cumbersome and impractical. An alternative approach to solve the master Eq.  is to usJk are the jump operators defining the stochastic dynamics. In the non-equilibrium cavity model, coherences cannot be created by Jk, occuring at rates Rk:The effective non-Hermitian Hamiltonian for the non-equilibrium cavity model in Eq.  is givenRk. The time between consecutive events is drawn from an exponential distribution, whose mean is the inverse of total rate of events. From a large ensemble of trajectories, we can calculate the non-stationary second-order correlation function, g(2), asA particular quantum trajectory is constructed by drawing a series of stochastic events, with their individual probabilities proportional to the rates Peer Review File


The support of a patch is the union of the supports of the tiles that are in it. The translate of a patch P by translationally equivalent if tiling of patch. Recall that a tiling repetitive if every finite local complexity (FLC) if, for every R > 0, there are finitely many equivalence classes of R.We let 2.2.set in Delone κ-set in A κ-Meyer set in F  of points for Delone κ-sets have a meaning analogous to the colours of tiles for tilings. We define repetitivity and FLC for a Delone κ-set in the same way as for tilings. A Delone set Λ is called a 2.3.expansive if there is a constant c > 1 with d on We say that a linear map prototiles. Denote by tile-substitution (or simply substitution) with an expansive map ϕ if there exist finite sets digit set  (b)  consists of a collection of spaces and mappings as follows: H is a locally compact Abelian group, i.e. a discrete subgroup for which the quotient group H. For a subset model set in regular if the boundary of Wmodel κ-set  if each H is a Euclidean space, we call the model set Λ a Euclidean model set rigid if A tiling 3.1.J, and et al. For any Note that □<!?tpb=-12pt>4. here.et al., 2007(Schlottmann, 2000From the assumption of pure discrete spectrum and Remark 5.5Let Under the assumption of pure discrete spectrum, we know that myak 2019 and ϕ fumyak 2019. From Th□Let It is known that any regular model sets have pure discrete spectrum in quite a general setting (Schlottmann, 2000□The next example shows that the unimodularity of ϕ is necessary.et al. (1998et al. (1998Let us consider an example of non-unimodular substitution tiling which is studied by Baake  al. 1998. This ex al. 1998 with thea to the right-hand side by the substitution and the letter b to the left-hand side. So we can get a bi-infinite sequence fixed under the substitution. A geometric substitution tiling arising from this substitution can be obtained by replacing symbols a and b in this sequence by the intervals of length a, and b, we can check The substitution matrix of the primitive two-letter substitution 6.et al. (1998et al. (1998We have mainly considered unimodular substitution tilings in this paper. Example 5.11 al. 1998. It cann al. 1998, which s


The standard tool to classify ceers is provided by the computable reducibility  Computable reducibility is a longstanding notion that allows classifying equivalence relations on natural numbers according to their complexity.R, S be equivalence relations with domain R is computably reducible to S, denoted f such that, for all Let f is a computable function that reduces R to S; c-degrees are introduced in the standard way.We write universal ceers, i.e., ceers to which all other ceers are computably reducible. The degree of universal ceers is by now significantly explored: for instance, in , where s. Except for R[0], which is the empty set, for every R[s] is an equivalence relation in which almost all equivalence classes are singletons. Notice that we do not define here R with an accompanying R so that f must start with x.Let us define the Strategies for the requirements and their interactions The strategy for Q-requirement i. The reason we choose P-requirements. The requirement u, v, both even, or both odd, which avoid the finitely many classes restrained by higher priority requirements. When found, it simply R-collapses u, v : this ensures that R. Notice that if R-collapsing two even numbers or two odd numbers, since the construction will ensure that a necessary condition, at any stage s, for which we may have x, y have different parity.For the R, i.e., R being in When at a stage The construction The construction is in stages: at stage s we define the approximation R[s] to R, and the approximations to the various parameters initialize a Q-requirement at stage s means to set as undefined at that stage the current value of its witness. A pair active witness for s if the pair has been appointed as a witness for inactive at the end of stage s if there are already distinct numbers s; it is active otherwise.Trequires attention at stageT is initialized; ore: s, and at the current stage there is a pair of distinct numbers u, v bigger than all numbers in the union of all current equivalence classes of numbers u, v are respectful of the restraint imposed by higher priority requirements);T at the end of stage s.one of the following holds, for some Stage 0 Initialize all Q-requirements. Define x; consequently A requirement Stage 1 Define x, leaving f and StageT that requires attention.  Action: T is initialized, then e: choose a fresh witness T, i.e. i is bigger than all numbers so far used in the construction ; in turn, R is not re-initialized: this string automatically becomes R in the priority ordering. The strings s.The strategy for U-equivalence classes of i and The strategy for P-requirement U, V so that there are distinct u, v can be V-collapsed without injuring higher priority restraints. It sets up a restraint asking that the initial segment U, sufficient to keep Let us now consider a P- and Q-requirements may have to be (re-)initialized as in the proof of Theorem V-collapses a pair of distinct numbers, then this collapse is forever, reflected in the fact that we define U and V are interchanged.Due to the action of higher priority requirements, Rrequires attention at stageR is initialized and R is a Q-requirement; orU and V): R is active at the end of s  of distinct numbers of the same parity such that: (i)U[s]), and (ii)V[s] give the same equivalence classes relatively to the V-equivalence classes restrained by higher priority V-collapse of the equivalence classes of u, v does not alter R at the end of stage s.one of the following holds, with A requirement The construction At stage s we define approximations to U[s] and V[s] will be the equivalence relations having U[s] and V[s] are equivalence relations with only finite equivalence classes and such that almost all equivalence classes are singletons; on the other hand we define Stage 0 Initialize all P- and Q-requirements; for x; consequently Stage 1 For x, keeping the values already defined at 0 for the other values of both StageR that requires attention.  Action: R is initialized, then e and U and V. As in the proof of Theorem R, i.e. i is bigger than all numbers so far used in the construction, bigger than the length of s; consequently If e and U and V. Then pick the least (by code) triple V-collapsed following the V-collapse of u and v performed by s. Consequently U and V. Define U against s. Consequently U[s] on all other pairs.  and thuc-incomparable dark equivalence relations R and S have an infimum.There are E and F have an infimum.For every m-degrees. We choose c.e. sets Q.(1) Let X and T is a lower bound for R and S, and R and S are Q ensures that R and S are dark.We will show that the equivalence relations E is a lower bound of R and S. Consider a reduction E to Q, and we have E to There is only one class g(x) must be an odd number.There are two different classes S contains only one non-computable co-c.e. class, namely, the S-class X and Y implies that the set m-reducible to both X and Y, and thus would be computable), which gives a contradiction.We distinguish two cases. If the class In each of the cases above, we showed that T is the greatest lower bound of R and S.Assume that X and Y such that m-degrees a), there are T-degrees that form a minimal pair and do not contain any set of a lower (2) The proof of the second part is similar to the first one, modulo the following key modification: One needs to choose X, Y such that For every notation This theorem is a combination of the two following results. The first one is Selivanov’s result  about prN-requirement, see . Moreover, R-part and the S-part of R-part of B is guaranteed by the fact that each equivalence class of the R-part of U with an equivalence class of the S-part of T to U that hits only the S-part of U,Assume that there is P-requirements are satisfied. The last lemma guarantees that, given T to U or by providing two equivalence classes that can be collapsed in U to diagonalize against Q-requirements are also satisfied because we carefully avoid, within the construction, to collapse classes of the same parity. We are in the position now to show that all By modifying the last proof, we can obtain something stronger.R is S-dark or S is R-dark, then R, S have no If S is R-dark. Then the proof of Proposition Suppose that T is a U by employing precisely the same construction as in Theorem Assume that the function R-part or the S-part of the relation Thus, one of the following two cases holds.Suppose that for some Case 1. The function R-part and infinitely many classes in the S-part. Then the same argument as in the proof of Lemma R-darkness of S.Case 2. Assume that R-part and only finitely many classes in the S-part. Then choose a computable function S to U, the c.e. set R-classes. Therefore, one can choose an infinite R-c.e. set A with the following properties: A is an R-c.e. transversal of S, which contradicts the R-darkness of S.Therefore, one can re-prove Lemma R, S are mutually dark equivalence relations of any complexity, then they have no sup.The above result contrasts with the fact that R is essentiallyn-dimensional if R has exactly n noncomputable equivalence classesWe finally turn to the problem of whether there are equivalence relations now from  that thiR and S such that both R and S properly belong to R, S have no Suppose that A properly belonging to the class U and V such that U and V are Q, R, and S as follows:Q, R, S is light and properly belongs to Fix a set T is the supremum of R and S, we have T must be essentially 1-dimensional. Let T-class. The conditions T should be reducible to F, we have either U and V. Thus, R and S have no supremum. Assume that mentclass2pt{minimWe conclude the paper with the following open question.For which


Nature Communications 10.1038/s41467-020-18005-7, published online 24 August 2020.Correction to: The original version of this Article contained an error in Fig. 1c. The length of the silicon core at the center of the panel was incorrectly labelled as ‘4 mm’, rather than the correct ‘4 µm’. This has been corrected in both the PDF and HTML versions of the Article.


Nature Communications 10.1038/s41467-020-20265-2, published online 4 January 2021.Correction to: The original version of this Article contained an error in the title, which was previously incorrectly given as ‘Three-phase electric power driven electoluminescent devices’. The correct version states ‘electroluminescent’ in place of ‘electoluminescent’.This has been corrected in both the PDF and HTML versions of the Article.