As in the standard complex setting, states and observables of a p-adic quantum system are implemented by suitable linear operators in a p-adic Hilbert space. In particular, owing to the distinguishing features of p-adic probability theory, the states of an p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)—a suitable p-adic counterpart of a POVM in a complex Hilbert space—as a convenient mathematical tool describing the physical observables of a p-adic quantum system. Eventually, we focus on the special case where p-adic qubit states and 2-dimensional SOVMs. The analogies—but also the non-trivial differences—with respect to the qubit states of standard quantum mechanics are then analyzed.We propose a model of a quantum  It came as a complete surprise when, at the end of the past century, some concrete applications of p-adic numbers to physical theories began to appear. Indeed, in the late 1980s, Vladimirov, Volovich and Zelenov [non-Riemannian model, that is, a model in which the Archimedean property is no more valid at very small distances. In particular, they proposed a model of quantum mechanics based on the non-Archimedean field of p-adic numbers. Later on, different p-adic quantum mechanical models were studied [p-adic numbers and theoretical physics were revealed. e.g., it was argued that the natural fractal-like structure of this field makes it suitable for the description of the dynamics of chaotic and disordered systems. In particular, it was proved that the ground state of spin glasses exhibits a natural (non-Archimedean) ultrametric structure [The field of  Zelenov  argued t studied ,12, and  studied ,16,17,18tructure ,21.p-adic numbers, not necessarily related to foundational physics, have begun to appear. Indeed, p-adic numbers have found a fertile ground of application in the context of algebraic dynamical systems, also in connection with problems from computer science, image analysis, compression of information, image recognition and cryptography [pseudorandom numbers and uniform distribution of sequences [More recently, new and interesting applications of tography ,23,24,25equences ,27,28.p-adic quantum information theory is two-fold: On the one hand, the peculiarities of p-adic numbers may provide a new line of attack for notoriously hard problems in quantum information theory; e.g., it has been recently argued that p-adic numbers  can be profitably used in the construction of mutually unbiased bases (MUBs), in any given Hilbert space dimension [p-adic model of quantum information may provide useful tools for the study of fundamental physical theories.A very recent research trend involves applications to quantum information theory , as wellimension . On the p-adic numbers has been formulated. We believe that the main reason for this is the lack of a well established theory of quantum states in the p-adic setting. Recently, this issue has been addressed in [p-adic framework, by suitable p-adic (trace class) operators in a p-adic Hilbert space. In the present contribution, on the base of the ideas presented in the aforementioned reference, we propose a p-adic model of a quNit. In particular, in these finite dimensional systems, we will show that the set of p-adic linear operators has itself a structure of a p-adic Hilbert space. This turns out to be useful for describing observables and the measurement process. The case of p-adic qubit will be worked out explicitly.Until today, it seems, however, that no general model of quantum information based on essed in , where ip-adic numbers and its quadratic extensions. In p-adic Hilbert spaces and the associated linear operators. We devote p-adic states in a finite-dimensional quantum system. Finally, by focusing on the case of a two-dimensional p-adic Hilbert space, we obtain an explicit realization of a p-adic qubit. The structure of the paper is as follows. In p-adic numbers that are relevant for p-adic quantum mechanics [In this section, we remind the reader some basic notions and results concerning echanics ,6,7,9,15valuation  on By a (V1)(V2)(V3)(V3))In particular, in the case where (V4)non-Archimedean valuation; otherwise, the valuation is called Archimedean.we say that distance function (or metric) defined by: non-Archimedean, instead, we still have a metric on (V4)) entails that the distance ultrametricity and, accordingly, a metric function satisfying it is called an ultrametric.An Archimedean valuation ined by: d|·| namely,d|·|Example 1 Recall that, according to the unique factorization theorem, every rational number ble by p ,35. The (V4). Therefore, if we consider the associated metric functionIt is easily shown that we obtain an ultrametric on . It is a metric space that, by means of a standard procedure, can be completed [p-adic numbersp-adic numbers. A more concrete characterization is given as follows. Let x admits a unique decomposition of the formp-adic number by means of a suitable converging series. In particular, this is reminiscent, to some extent, of the usual decimal expansion of a real number Consider the pair ompleted . The resnt of Qp . Therefothe formx=∑i=0∞xip-adic valuation on p-adic numbers when describing physics on length scales comparable to Planck’s length (P1)Every point in an open (closed) ball is a centre.(P2)Two open (closed) balls are either disjoint or one is contained in the other.(P3)Every ball in (P4)All triangles are isosceles in .As a topological space, completely regular (being a metric space) and totally disconnected; namely, the only connected subsets of The rametric  can be eength lP ,5,10,11.ength lP ,38, we lngletons .quadratic extensions of We devote the last part of this section to a brief discussion of the on on Qp ,33. Thisimaginary unit in conjugation, namely, the mappingp-adic absolute value (1)non-quadratic unit, i.e., If (2)if The definition of a quadratic extension of  we have :(1)If p≠Example 2. For p-adic Hilbert space and the associated p-adic linear operators [This section is devoted to introduce a suitable notion of a perators  iff;(N2);(N3),for all Remark 1. (N3) differs significantly w.r.t. the standard  case, where the usual triangle inequality holds.The explicit definition of a p-adic Banach space is motivated by the fact that the strong triangle inequality As is well known, complex Hilbert spaces are defined as (complex) Banach spaces endowed with a suitable inner product, namely, the one inducing the relevant norm. It turns out that this familiar picture keeps some of its main futures—but also requires some essential modification—when switching to the field of p-adic normed space. Our first concern is to provide a suitable notion of a basis for this space [p-adic normed space X are said to be  norm-orthogonal if, for any X is norm-orthogonal if any finite subset of X is normal, if it is norm-orthogonal and, additionally, p-adic Banach space, and let countable subset of X  basis, if dimension of X—in symbols, X, i.e., we set p-adic Banach space X admitting a normal basis a normal p-adic Banach space.Let is space ,34,37,42Example 3. Let us consider the space (b)(c)We call the triple As in the standard complex case, also in the (a) and (b) of Definition 2, it is clear that the inner product sesquilinear form. Also note that, from the Hermitianity condition (b), and the sesquilinearity of non-degenerate if the condition From conditions Example 4. Let where Remark 2. The notion of non-Archimedean inner product naturally leads us to a notion of inner-product orthogonality, which is distinct from the—previously introduced—norm orthogonality. Explicitly, we say that two vectors The notion of inner-product orthogonality, introduced in Remark 2, entails the following natural extension of the notion of normal basis:Definition 3. Ψ is a normal basis in X, and its elements are mutually IP-orthogonal, namely, Let p-adic Banach space X is, in general, not guaranteed. On the other hand, when X is a normal p-adic Banach space—where the existence of a normal basis is assumed—it is always possible to turn any given normal basis into an orthonormal one by making a suitable choice of the inner product. Indeed, it suffices to consider the canonical inner product associated with this normal basis in X . Therefore, we have the following natural definition of Hilbert space in the p-adic setting:We stress that the existence of an orthonormal basis in an inner-product Definition 4. A p-adic Hilbert space is a triple p-adic Hilbert space may be thought of as a normal p-adic Banach space endowed with a distinguished normal basis and with the associated canonical inner product. It is then not difficult to check the following two properties of a p-adic Hilbert space:(H1)Every vector (H2)non-Archimedean Parseval identity holds true:The From the previous definition, it is clear that a Example 5. Let us consider the set rd basis . Then, iterature ,43, thisI)); see .p-adic case one can define a convenient notion of isomorphism of Hilbert spaces  [p-adic Hilbert space, where Hilbert space automorphism of two Hilbert spaces over different, but mutually isomorphic, p-adic Hilbert spaces. The p-adic Hilbert spaces stemming from the same p-adic Banach space As in the complex setting, also in the another) . Let us Remark 3. It is worth stressing that the analogies between complex and p-adic Hilbert spaces cannot be pursued too far. Indeed, quite generally, in a p-adic Hilbert space, p-adic Hilbert space p-adic quNit.Hereafter, borrowing the terminology from the standard (complex) quantum mechanics, we shall call a quantum system with associated p-adic framework as well, with some non-trivial differences w.r.t. the standard complex setting.In , it is dfinite-dimensional p-adic Hilbert space. In this case, we only need to consider the space Since our main concern is to consider applications to quantum information theory, we will actually focus our attention to linear operators acting in a p-adic Hilbert space, with L and the fixed orthonormal basis operator norm—which is given byp-adic Banach space.Then, let 6.2.1 in ), it is given byL≔sup0≠ϕ∈Remark 4. Let us explicitly note that, by using the Dirac notation, the operator adjointL is given byinvolutive automorphism of p-adic Banach space p-adic Banach ∗-algebra. In fact, in the next section, the set physical observables of a  p-adic quantum system.For every given byL*=∑i=1N∑Remark 5. As in the complex setting, also in the p-adic case it is possible to single out the subset is verified.p-adic Hilbert space. Indeed, let us first observe that given trace, linear functional—namely, the trace functional—on Hermitian, becausep-adic Hilbert-Schmidt product. Next, note that, for all non-Archimedean inner product, and inner-product p-adic Banach space.To conclude this section, we will now argue that Now, let Remark 6. Let From this fact, we deduce that the product orthonormal basis in We now prove that the set —as (cf. )(36)L=∑Summarizing, we have the following result:Theorem 1. Given an algebraic formulation. This formulation essentially relies on two fundamental assumptions; namely, that every quantum system can be described by means of two main classes of objects—i.e., states and observables—mutually related by means of a natural pairing map. Specifically, the observables can be identified with the selfadjoint elements of an abstract non-commutative unital density or statistical operators [As is well known, the most general and abstract description of quantum mechanics is provided by the so-called perators ,47,48,49p-adic quantum mechanics, physical states should be defined as  involution-preserving bounded functionals on the unital Banach ∗-algebra p-adic setting, by the class of the so-called p-adic statistical operators, defined as a suitable subclass of the selfadjoint trace class operators in a p-adic Hilbert space p-adic probability theory [Following the same route, it has been recently argued that, in perators . It has  space H . The proy theory ,7,50,51.Remark 7. Since p-adic probability theory is a rather non standard topic, for reader’s convenience, we now briefly sketch its main features. Let us first observe that both p-adic and classical probability theory arise in a natural way from a common conceptual background. In fact, in both theories, one starts by considering the set ibutions ,7,50,51.Definition 5. Let the set of p-adic statistical operators in p-adic quantum mechanics. To begin with, we need to introduce a suitable notion of observable in the p-adic setting. In particular, as argued in [p-adic setting is provided by the so-called (discrete) selfadjoint operator valued measures (SOVMs) in p-adic counterpart of a POVM in a complex Hilbert space. In the finite-dimensional setting we are considering, a discrete SOVM can be defined as a family I is a finite index set—of selfadjoint operators in It is convenient, at this point, to better clarify the statistical interpretation of rgued in , a conveRemark 8. We remark that, as for a discrete POVM, a discrete SOVM should actually be defined as an additive measure on the algebra of subsets of the index set I (the power set of I), by putting, e.g., for Remark 9. We stress that, in p-adic quantum mechanics, there is no straightforward counterpart of a POVM of standard (complex) quantum mechanics. In fact, the field of p-adic numbers pairing between states  and observables. To this end, similarly to the standard complex case, by means of the trace functional normalized involution-preserving linear functional on pairing between p-adic statistical operators and observables.The trace functional p-adic probability distribution.For every elations  guaranteAs a further step, we need to specify the We have then obtained a complete description of the statistical content of the theory.p-adic counterpart of the complex qubit. To this end, let us first note that, since trace-one p-adic Hermitian . Let us focus on the particular case where traceless p-adic Hermitian matrix. In particular, the conditions Q:p-adic Hermitian matrix, with zero trace, is given by:p-adic Pauli matricesQ in a more compact form:p-adic Hermitian matrix in p-adic quantum systems isp-adic Pauli matrices defined in  is not algebraically closed, namely, not every non-constant polynomial admits a root in formal eigenvaluesdoes not admit any root in the quadratic extension of not diagonalizable. Actually, it is not difficult to construct examples of non-diagonalizable qubit states also for all other quadratic extensions of p-adic qubit states that cannot be diagonalized, for every quadratic extension As a first point, note that ic qubit . As is eQ2,2 see . Now, coe matrix  does notExample 6. We now provide an explicit example of a SOVM for a two-dimensional p-adic quantum system. Let us consider the family of linear (matrix) operators It is clear that the matrices Therefore, it state , iswith p-adic numbers, we have proposed a model of QuNit on the field As a first step towards a quantum information theory based on a quadratic extension of the non-Archimedean field of p-adic Hilbert space and, restricting to the case where p-adic Banach ∗-algebra. Then, we have proved that the linear space p-adic Hilbert space, once it is endowed with the p-adic Hilbert-Schmidt inner product.We started by introducing a notion of p-adic probability theory, we have argued that the states of an p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. In particular, it turns out that the set of p-adic statistical operators,  subset of p-adic probability theory—hence, it is an unbounded subset of Owing to the distinguishing features of p-adic counterpart of a POVM in a complex Hilbert space—as a convenient mathematical tool describing the physical observables of a p-adic quantum system.We have next introduced the notion of (discrete) selfadjoint-operator-valued measure (SOVM)—a suitable p-adic qubit states and of two-dimensional SOVMs.Eventually, focusing on the special case where p-adic model of quantum information theory. Tensor products and entanglement play a central role in quantum information theory, and we expect that they will play a central role in the p-adic setting too. Therefore, as a first step, we plan to investigate tensor products of p-adic Hilbert spaces and the associated classes of entangled states. Our next concern is the description of dynamical maps and dynamical (semi-)groups in p-adic quantum mechanics. This will provide a suitable p-adic counterpart to quantum channels. Another interesting prospect concerns the possibility of defining typical entropic quantities, such as the von Neumann and the Rényi entropies—which are relevant in standard quantum information theory—in the p-adic framework too.We close by outlining some potential extensions of this work, especially focusing on those ones that are relevant for our  program aimed at developing a


Correction to: Pharmaceutical Research10.1007/s11095-022-03249-3This article was updated to correct the phrase “in vitro” in the Abstract and in three instances in the text.


Nature Communications 10.1038/s41467-022-33453-z, published online 30 September 2022Correction to: The original version of this Article contained an error in a sentence in the last paragraph of the Introduction, which incorrectly read ‘The shuttling-mode exchange switching allows us to efficiently control exchange coupling with an on/off ratio above 1000,000.’ The correct version states ‘1000’ in place of ‘1000,000’.This has been corrected in the PDF version of the Article.


We prove that for antisymmetric vector field  GL(N)-gauge, namely a matrix-valued function GL(N)-gauges have the advantage that they can transform equations of the formH-surfaces, and more generally the Euler–Lagrange equations of a large class of conformally invariant variational functionals. The GL(N)-gauge transform allows for regularity theory and the study of weak convergence [In the celebrated work  Rivière  gauges, , namely howed in , the GL-Coulomb gauge [Applying a gauge  the Eq. , we find the Eq. ,\documenmb gauge , namely There exists a number If As an immediate corollary, we obtainA such that for (Conservation law) Assume Theorem With the methods of Theorem f in Assume Here, as usual, we denoteod" stands for “off diagonal”.We follow the notation of  for the s-gradient For two vector fields s-divergence in the distributional wayLet With this notation we have Let Moreover, for any We haveombining . The proombining  is similWe also denoteLet For the proof see Appendix C.We will also need the following Wente’s inequality from .p. Assume moreover that R be a linear operator such that for some ,,3.23), aly, from , 3.21),,3.25\docproof of .In order to prove  we obserstimates  and R.For the last term of , we simixdy=0.By , 4.11),,\documenxdy=0.By , and , , \documeombining .Step  2. We claim that (up to a subsequence)Indeed, we writeAs for the second term on the right-hand side of , we begiWe verify the convergence of the first term on the right-hand side of . First w term of  follows  term of .Now  follows Step  3. We claim thatStep  4. We claim that (up to a subsequence)R, whenever As for the second term of the right-hand side of ,,\document}ℓ→∞ in , and (4.t}ℓ→∞ in , we obtaLet We follow in spirit the covering argument of Sacks–Uhlenbeck [By assumptions there is a number Let In order to conclude we will need a removability of singularities lemma, compare with [Let For simplicity of presentation let us assume that Now let Thus, by  and by tSecondly,Thus, passing with t}ℓ→∞ in  we get fause, by , we haveThis finishes the proof. Combining Corollary


Scientific Reports 10.1038/s41598-022-22856-z, published online 28 October 2022.Correction to: The original HTML version of this Article contained an error in Figure The original Figure This error has now been corrected in the HTML version of the Article; the PDF version was correct from the time of publication.


This paper proposed an image algorithm based on a cascaded chaotic system to improve the performance of the encryption algorithm. Firstly, this paper proposed an improved cascaded two-dimensional map 2D-Cosine-Logistic-Sine map (2D-CLSM). Cascade chaotic system offers good advantages in terms of key space, complexity and sensitivity to initial conditions. By using the control parameters and initial values associated with the plaintext, the system generates two chaotic sequences associated with the plaintext image. Then, an S-box construction method is proposed, and an encryption method is designed based on the S-box. Encryption is divided into bit-level encryption and pixel-level encryption, and a diffusion method was devised to improve security and efficiency in bit-level encryption. Performance analysis shows that the encryption algorithm has good security and is easily resistant to various attacks. After decades of development, digital information has grown extensively in capacity, and various electronic devices are changing gradually, accompanied by a large amount of data for transmission and communication , which bDue to chaotic sensitivity to initial state and control value, good pseudo-randomness, ergodicity and unpredictable trajectory , the comTwo types of cryptosystems can be classified: stream ciphers, which are converted bit-by-bit, and packet ciphers, which convert n-bit inputs into m-bit outputs. At the core of this conversion is the static S-box, which gives the cryptosystem the obfuscation properties described by Shannon . An S-boAs the security strength of the entire encryption algorithm is determined by the cryptographic strength of the S-box, the researcher has proposed many methods to enhance the S-box’s performance, such as using the high-dimensional chaotic system, and multiple chaotic systems to generate S-boxes have become solutions. Liu et al.  used a hBased on the above analysis, a cascade chaotic map 2D-Cosine-Logistic-Sine map (2D-CLSM) is proposed in this paper. The initial values and parameters of the chaotic map are combined with the original image to resist known plaintext attacks. Then, an S-box construction method is proposed, and an S-box-based encryption method is designed. The encryption method is divided into four stages: key generation, S-box generation, bit-level encryption and pixel-level encryption. In bit-level encryption, a bit-level operation is performed on the plaintext pixel values, converting the original decimal plaintext image pixels to eight-bit binary, then permuting the lower four bits and using a proposed new diagonal diffusion method for the higher four bits. In the pixel-level encryption part, one diffusion is completed by a three-number XOR , and the resulting value calculates the row and column index of the S-box for pixel value replacement. Through experimental analysis, the encryption algorithm in this paper has good security performance.The remainder of this essay is structured as follows: In nonlinear dynamic systems, chaos is a stochastic process that is frequently employed in cryptography research . One-dimThe chaotic behavior of a chaotic system can be measured by the bifurcation diagram and the Lyapunov exponent. A cascade map is a form of a chaotic map; the use of cascades can effectively improve the performance of chaotic systems. In order to solve the problem of small chaotic intervals and uneven distribution, we designed a two-dimensional cascade chaotic map, which has a more complex chaotic behavior than one-dimensional chaos and a faster iteration speed than two-dimensional chaos. The cascade system is shown in The essence of cascading is that the output of a certain initial value after the iteration of system 1 is taken as the iterative output of system 2, and the iterative output of system 2 is taken as the iterative output of system 1, thus forming a circular iteration between two subsystems. Chaotic systems are generated by the cascade method, where the Lyapunov value of the system is the sum of the Lyapunov exponents of the cascaded subsystems, and the sequential trajectory of the system output deviates sharply as the number of iterations increases .Using cosine functions is proposed and demonstrated in , where eIn Equation (3), to increase the chaotic complexity of the cosine function-based system, a new control parameter Further, controlling the value of the parameter k2=1−19θ , and wheIn order to analyze the 2D-CLSM’s performance, we compared it with another existing 2D chaotic map for image encryption, i.e., the 2D Logistic-Sine-Coupling Map (2D-LSCM) .(8)xi+1The trajectory of a 2D-CLSM shows how motion increases over time from a specific initial state. In the case of periodic motion, the trajectory would be a closed curve, whereas the trajectory of chaotic behavior would theoretically never close or repeat. Therefore, chaotic trajectories usually occupy a part of the phase space and can reflect the randomness of the chaotic system output. If a chaotic trajectory can occupy a larger portion of the phase space, the chaotic system has a better stochastic output.From We set initial value as The LE (Lyapunov exponent) describes the sensitivity of a chaotic mapping to initial values. In general, a chaotic map is in a chaotic state when Its Jacobian matrix at the point Let The larger the value of p-values to show the test result. The sequence is considered random when the p-value is over 0.01. In this paper, 15 NIST tests were used to test the randomness of the generated sequences, and 15 correspond to In information theory, information entropy is used to quantify the uncertainty of the information content. It can be used to evaluate how random a set of data is. We transformed the chaotic sequence obtained by iteration into values between 0 and 255 and obtain the information entropy according to Equation (12):Better randomness is correlated with larger information entropy values, and There are four main stages in our encryption algorithm: key generation, S-box generation, bit-level encryption and pixel-level encryption. The size of the plaintext image Two different chaotic maps are used in our algorithm: Logistic and 2D-CLSM, which are used in different stages of encryption. In the S-box generation stage, a Logistic map is used to generate the chaotic sequence needed to generate the S-box; a 2D-CLSM map is used in the bit-level operation and pixel-level operation phases. By using the different chaotic maps in different stages to expand the key space, a more complex encryption algorithm is generated.If the key stream used for encryption is only related to the key and not related to the plaintext image, the designed algorithm is not safe for chosen/known plaintext attack. In order to ensure the security of encryption, the encryption key is generated from the initial key and the plaintext image.Algorithm 1 Generation of the encryption keysOutput: The encryption keys   1: Read the size:   2: Obtain the sum of all pixels and add scramble number        3: Calculate the mean of the all pixels   4:   5:   6:   7:   8:   9:Input: Plain image Initial encryption keys contain six decimal numbers, In the construction method of a chaotic S-box, there is the problem of generating useless chaotic sequences that affect efficiency. Wang et al. proposed in  to use aStep 1: Obtain the length Step 2: Generate a random number Step 3: Shuffle the values of the two positions according to Step 4: Subtract 1 from Step 5: Repeat step1~step4 until The Fisher–Yates shuffling algorithm is an algorithm proposed by R. Fisher and F. Yates for generating random permutations of finite linear arrays . The mosAlgorithm 2 Generation of S-boxOutput: S-box   1: for       Substituting       if          obtain the 512-length chaotic sequences   2: Divide    sort index of the    3: Read the size:   4: while       Obtain a random number       Swap the value of       Set   5: If   6: Obtain S-boxInput: encryption keysAlgorithm 2 describes the generation process of the S-box. In order to verify the performance and strength of the generated S-boxes, we used the general criteria for S-box performance evaluation to be able to test them . In this(I) Nonlinearity In the process of encryption, if the given S-box makes a linear mapping between the input (plaintext) and the output (ciphertext) [NLf is the 8-bit Boolean function, hertext) , then a (II) Strict Avalanche Criterion (SAC) Webster and Tavares used the strict avalanche criterion as an important feature of the performance of an S-box [an S-box . The str(III) Output Bits Independence Criterion (BIC) This criterion was introduced by Webster et al. as one of the important properties of S-box evaluation, a property that ensures that there is no dependence on the change of any two output bits when a single input bit is changed, a property that makes any Boolean function must be independent and highly nonlinear. For two output bit Boolean functions Through experimental tests, the average BIC nonlinearity value of the proposed S-box is 104.5, and the BIC-SAC test result is 0.5009, which satisfies the BIC criterion.(IV) Difference Approximation Probability (DP) differential cryptanalysis, introduced by Biham and Shamir [d Shamir , is ableIn comparison with the other four methods, the results show in Traditional image encryption generally falls into one of three categories: permutation-only, diffusion-only, or combined forms. Due to its simpler computational complexity, the permutation-only type of these is more efficient, but the security is not very strong. Due to the substantial computational load required for real arithmetic operations, the diffusion type is a time-consuming process. The pixel values and random numbers are used to perform an XOR operation, and the calculated numbers are used to determine the ranks of the S-boxes and substitute the original pixels with the values in the S-boxes, but the overall encryption efficiency is low due to the lack of permutation and diffusion operations on the plaintext image and the low efficiency of S-box generation .In this stage, by separating the eight-bit pixel values into upper four bits and lower four bits, different operations are performed on the two parts, respectively, and a new diagonal diffusion method for irregular matrices is proposed to be applied to the high four-bit matrix. The bit-level encryption process is shown in We converted the original pixel value of the plaintext image into a binary representation and extracted it into two parts for different operations. The upper four bits P1 of the pixel value and the lower four bits P2 of the pixel value are shown in In order to make the information of pixel points have a good diffusion effect, many diffusion methods, such as V-diffusion, zigzag diffusion and chunk diffusion, were proposed. A zigzag algorithm was used in  and imprHowever, the security of the zigzag is not guaranteed due to the fact that it is extremely easy to be broken by a single variation.In order to obtain faster diffusion speed and diffusion effect, this paper proposes a new diagonal diffusion method by randomly and irregularly rearranging the image matrix through a chaotic sequence. The number of pixel values in each row is controlled by the random sequence, there may be no pixel values at some positions of the diagonal, and the matrix presents an irregular arrangement. Through the proposed new diagonal diffusion, it can complete the diagonal of diffusion. Through the comparative analysis of the relevant experiments in Algorithm 3 Matrix rearrangeOutput: irregular matrices   1: Obtain the size of   2: Set an empty list   3: While                                   4:Input: upper 4-bits sequenceIn this paper, after obtaining the upper four-bit matrix, we changed its alignment through a random sequence to generate different irregular matrices, performed diagonal diffusion on the matrix according to the changed alignment and changed its alignment, as shown in Where Based on the rearranged matrices, we proposed a diagonal diffusion to perform diffusion operations on upper 4-bit matrices capable of new diagonal diffusion operations based on different irregular matrices, as shown in Algorithm 4 Diagonal diffusion orderOutput: order sequence   1: Obtain the size of   2: Set tow empty list   3: for       Obtain the size of       for             4: Obtain the size of   5: for       Obtain the size of       for            Input: irregular matricesAccording to the irregular matrix obtained by Algorithm 3, we need to diffuse according to the diagonal, but the shape of the matrix is irregular, so given Algorithm 4 used to obtain the order of the pixel values of the diagonal of the irregular matrix, sequentially traversing the diagonal to obtain its diffusion order matrix for diffusion, according to different irregular matrices for the order of diagonal diffusions, such as the first: (1- > 3- > 2- > 7- > 4- > 8- > 5- > 9- > 6), the second: (1- > 2- > 4- > 3- > 5- > 6- > 7- > 8- > 9) and the third: (1- > 5- > 2- > 6- > 3- > 9- > 7- > 4- > 8), if we use the normal zigzag, then the order of each diffusion is fixed in a 3 × 3 matrix, and the result is (1- > 2- > 4- > 7- > 5- > 3- > 6- > 8- > 9) each time, which is not guaranteed in terms of security.Where Step 1: Generate S-boxes Step 2: Generate the encryption key Step 3: Iterate over Step 4: Process the chaotic sequence to obtain the random numbers for array rearrangement;Step 5: Based on the resulting upper 4-bit matrix Step 6: Perform a diffusion operation on the irregular matrix Step 7: Diffusion operation based on the 1D matrix Step 8: Process the chaotic sequence to obtain chaotic values for the lower four positions;Step 9: Based on the resulting At the end of the bit-level encryption, pixel-level encryption is performed by the resulting pixel values using the S-box. It mainly includes pixel value diffusion and substitution. The main process is shown in Generate an empty matrix Finally, The simulation results of the proposed method are presented in this section by running PyCharm software under Windows 10 64-bit system. By using the method in this paper, 10 images were both encrypted and decrypted. The results of encrypting and decrypting several grey-scale images such as Baboon.png, House.png, Cameraman.png and Peppers.png pixels are shown here, respectively. The initial keys used for encryption/decryption are shown in From Assuming that the accuracy of the computer is In a secure encryption system, a mirror change in the key can cause a complete change in the ciphertext obtained from the encryption. Through testing, the encryption key As shown in The distribution of the image’s pixel values can be directly described by the histogram. A secure encryption system should ensure that pixel values of the obtained ciphertext image are uniformly distributed in order to reduce readability, improve security and provide effective protection against statistical attacks.The variance of the histogram can be used to specifically describe the distribution of pixel values. The calculation equation is shown in (21), and the result shows in Where A differential attack is a common form of attack in cryptography, in which an attacker usually selects to encrypt the plaintext and a slightly altered plaintext and analyses them to find a specific relationship. In order to be effective against differential attacks, the encryption algorithm should be sensitive enough to the plaintext that a diffusion operation during encryption can cause small changes in the plaintext to affect all pixels, and we usually evaluate the sensitivity of the algorithm using the pixel change rate NPCR and the normalized pixel average change intensity UACI, their ideal expectations are 99.6094% and 33.4635%, respectively. We obtained NPCR and UACI values by randomly modifying two pixel points of the plaintext image. In this paper, we choose to modify pixel points  and . In When the image is not encrypted, there is a strong correlation between the pixel values of the images; the encryption operation on the plaintext allows us to break this correlation. We calculated the correlation coefficient by randomly selecting The closer the calculated correlation coefficient is to 1, the stronger the correlation is, and the closer it is to 0, the weaker the correlation is. As shown in In addition, from Information entropy is another metric for evaluating the security of a ciphertext and represents the strength of the uncertainty of the image information, which is expressed in Equation (12).For a grey-scale image with 256 states, it has a maximum expectation value of 8, During the transmission of information, information may be lost due to humans or some uncontrollable factors. If the encryption algorithm is able to restore the plaintext image in this case, then the encryption algorithm is valid. In Running speed is an important index to evaluate the encryption algorithm; we simulated it in Python 3.7 environment on a PC with a 2.2 GHz CPU and 8 G RAM. As can be seen from In this paper, we propose an improved cascaded chaotic map 2D-CLSM and design a novel encryption scheme based on it and S-box. By generating a key associated with the plaintext, the whole encryption process is associated with the plaintext, effectively resisting the chosen/known plaintext attack. An S-box encryption scheme is designed and applied to our encryption, increasing the overall security of the algorithm. Dividing the encryption into bit-level encryption and pixel-level encryption improves the complexity and security of the encryption to a certain extent but reduces its efficiency. The encryption method in this paper can be applied to different image types, such as grey-scale images, grey-scale medical images, etc., while the algorithm is able to meet everyday encryption requirements, both in terms of efficiency and security. The experimental results show that the algorithm has sufficient key space, is resistant to brute force attacks and performs well against statistical analysis attacks, clipping attacks and differential attacks. However, although we improved the efficiency of S-box generation, we used two rounds of encryption, one at the bit level and one at the pixel level, which makes us less efficient overall, so the next step is to improve the efficiency of encryption in both rounds.


Nature Communications 10.1038/s41467-022-30773-y, published online 27 July 2022Correction to: The original version of this Article contained an error in Fig. 3, in which the case and control colors were swapped in panels c and f. The correct version of Fig. 3 is:which replaces the previous incorrect version:b, e, h, j) Bar plot of multiple P-value thresholds (PT) for PRS prediction of Sjögren’s”. The correct version replaces this sentence with “ Bar plot of multiple P-value thresholds (PT) for PRS prediction of Sjögren’s”. This has been corrected in both the PDF and HTML versions of the Article.In addition, the original version of this Article contained an error in the legend for Fig. 3, which incorrectly read “(


Nature Materials 10.1038/s43588-01431-x. Published online 29 December 2022.Correction to: In the version of this article initially published, the first name of Flora Ascione was listed as “Floriana”; the error has been corrected in the HTML and PDF versions of the article.


Although the continuous-variable position-momentum entanglement of photon pairs produced by parametric down-conversion has applicability in several quantum information applications, it is not suitable for applications involving long-distance propagation. This is because entanglement in the position-momentum bases, as seen through Einstein-Podolsky-Rosen (EPR)–correlation measurements, decays very rapidly with photons propagating away from the source. In contrast, in this article, we show that in the continuous-variable bases of angle–orbital angular momentum (OAM), the entanglement, as seen through EPR-correlation measurements, exhibits a remarkably different behavior. As with the position-momentum bases, initially, the entanglement in the angle-OAM bases also decays with propagation, and after a few centimeters of propagation, there is no angle-OAM entanglement left. However, as the photons continue to travel further away from the source, the entanglement in the angle-OAM bases revives. We theoretically and experimentally demonstrate this behavior and show that angle-OAM entanglement revives even in the presence of strong turbulence. Entanglement in the angle–orbital angular momentum bases survives at large distances even in the presence of a turbulent medium. Quantum entanglement  and ρi ≡  are the transverse positions of the signal and idler photons, respectively at z, and where k = π/λp. In addition, w0 is the pump beam waist at z = 0, L is the length of the crystal, λp is the wavelength of the pump field, and P = ∣ψ2∣ at z is therefore given byWe present a quantitative analysis of the propagation of conditional position and angle uncertainties of the signal photon in SPDC. For a Gaussian pump with beam waist at the crystal plane y direction, the two-photon position probability distribution is referred to as the conditional position probability distribution of the signal photon at z and is denoted by P. The standard deviation of P is referred to as the conditional position uncertainty Δ of the signal photon  in the transverse momentum basis, one can calculate conditional momentum uncertainty Δ of the signal photon at z. According to the EPR criterion of entanglement, if the product ΔΔ < 0.5 ħ, then the two photons are entangled in the position-momentum bases  and ρi = , we can write the two-photon angle probability distribution function P asrs, θs) and  are the polar coordinates of the signal and idler photons. Using P, one can obtain the conditional angle probability distribution function P and thereby the conditional angle uncertainty Δ of the signal photon , we write the EPR criterion for entanglement in the angle-OAM bases as ΔΔ < 0.5 ħ[1 − 2πP]  represents the conditional probability of detecting the signal photon at the boundary θs = θ0.In addition to the position-momentum bases, the down-converted photons are rendered entangled in the angle-OAM bases as well. Using w0 = 507 μm, L = 5 mm, and λp = 355 nm, we numerically evaluate P and P at different propagation distances z and plot them in P and P, we scale them to make their maximum values equal to one. Next, by fixing yi = 0 in P and θi = 0 in P, we calculate P and P and thereby the conditional position uncertainty Δ and the conditional angle uncertainty Δ, and we plot them in Now, using z, we derive expressions for how the conditional uncertainties scale with z in the near- and far-field regions. The two dotted lines in ys∣yi; z) scales with z in the near and far fields. We find that Δ increases as σ(z) in the near field, while it increases as w(z) in the far field. The two dotted lines in s∣θi; z) scales with z. We find that while Δ increases as z in the near-field regions, it decreases as 1/z in the far-field regions. .Although it is very difficult to derive the general analytical expressions for the conditional position and angle uncertainties as a function of ys∣yi; z) and conditional angle uncertainty Δ change as a function of z. We note that the conditional momentum and OAM uncertainties Δ and Δ remain constant as a function of z because of the conservations of momentum and OAM, respectively, in SPDC. As a result, the functional dependence of ΔΔ and ΔΔ on z is same as that of Δ and Δ, respectively. Therefore, the entanglement in the position-momentum bases as seen through EPR correlation product was found to decay very rapidly as a function of z, and after a few centimeters of propagation, there is no entanglement left in the position-momentum bases. On the other hand, although there is no entanglement left in the angle-OAM bases after a few centimeters of propagation just as in the case of position-momentum bases, the entanglement in the angle-OAM bases comes back after further propagation. We refer to this as the propagation-induced revival of entanglement in the angle-OAM bases. Although in our analysis, so far, we have used EPR uncertainty product as an entanglement witness, we observe qualitatively the same features in the propagation of angle-OAM entanglement with another witness, entanglement of formation , which is the conditional position uncertainty of the signal photon at z. Therefore, if we have a line of idler positions, shown by green line, the signal photon has the appreciable probability of being found in the blue shaded region. This green line represents the angular position θi = 0 of the idler. The corresponding conditional angle uncertainty Δ of the signal photon can be estimated by averaging the angle over the radial extent of the down-converted field from the center up to beam width ws(z) of the field. However, to a very good approximation, we can take the conditional angle uncertainty Δ to be of the order of Δ/ws. ys∣yi; z) changes as a function of z, and fig. S3 in the Supplementary Materials shows how ws(z) changes as a function of z. We find that in the near field, Δ increases much faster compared to ws(z), while in the far field, it increases much slower compared to ws(z). Because of this, as a function of z, the conditional angle uncertainty Δ increases in the near field, while it decreases in the far field , P, P, and P through coincidence measurements of the two photons. An ultraviolet (UV) continuous wave (CW) Gaussian pump (Coherent Genesis STM UV laser) of wavelength λp = 355 nm, beam waist w0 = 507 μm is incident on a 5 mm–by–5 mm–by–5 mm β-barium borate (BBO) crystal. The crystal is cut in a manner that it produces signal and idler photons with collinear type I phase-matching condition. A long-pass filter is placed after the crystal to block the UV pump. We use an Andor iXon Ultra-897 electron-multiplied charge-coupled device (EMCCD) camera that has a 512 × 512 pixel grid with each pixel being 16 μm by 16 μm in size. A 10-nm bandpass filter centered at 710 nm is used to detect the down-converted photons. The blower heater (BH) produces turbulence by blowing hot air, and it is switched on during our experiments involving turbulence.P, P, and P, we use an EMCCD camera  and P and thereby the corresponding uncertainties Δ and Δ, we image the transverse plane at z onto the EMCCD camera plane using a 4f-imaging system, as depicted in P at z, we use a 2f imaging system and keep the EMCCD camera plane at the Fourier plane of the transverse plane at z, as depicted in P at z. The conditional momentum uncertainty Δ is obtained by multiplying the conditional position uncertainty at the EMCCD plane by kħ/f, where f is the focal length of the lens. . For the coincidence measurements of the two-photon OAM probability distribution P, we make use of two electronically gated single-photon avalanche diode (SPAD) detectors  kept in the signal and idler arms. Specific holograms are displayed onto both the SLMs, and then, the signal and idler SLM planes are imaged onto the input facets of single-mode fibers (SMFs) kept in the signal and idler arms. The combination of the hologram and SMF in each arm projects the input field into a particular OAM mode, which then gets detected by the SPAD detector through the SMF. An electronic coincidence circuit then yields the coincidence counts. By displaying different holograms on the SLMs, we measure the two-photon OAM probability distribution. Next, we report our measurements of the condition uncertainty products ΔΔ and ΔΔ at various z values in the absence of turbulence. In our experiments, we measure Δ to be 2.13ħ ± 0.1ħ mm−1, which is in good agreement with the value 1.97 ħ mm−1 calculated using ls∣li; z) should be zero. However, because of the imperfection in the spatial profile of the beam and other background issues, Δ is always finite in realistic experimental situations. We measure Δ to be 0.72 ± 0.04 ħ radian−1 and use this in our experiments . Last, we measure Δ and Δ at various z values and plot the conditional uncertainty products ΔΔ and ΔΔ as a function of z in For the coincidence measurements of z = 15 cm to introduce turbulence Δ at different propagation distances ranging from z = 15 cm to z = 60 cm and plot them in s∣θi; z)Δ in the presence of turbulence, see sections S6 and S7 of the Supplementary Materials. Our numerical simulations show that in the presence of turbulence, the angle-OAM entanglement revives at z = 35 cm; experimentally, we find this distance to be about 45 cm. Therefore, we find that although turbulence does affect entanglement in the angle-OAM bases in an adverse manner, its effect can be completely undone by just propagating the photons further away by some distance.We next investigate whether the propagation-induced entanglement revival takes place in turbulent environments, which is quite often the limiting factor in the practical implementations of many entanglement-based applications. For this, we repeat our experiments depicted in ys∣yi; z)Δ and ΔΔ close to the crystal (z < 3 cm) or in the far field (z > 10 cm). However, because of the signal-to-noise limitations of the EMCCD camera, it is not possible to make measurements in the intermediate regions. We find a very good qualitative agreement between our experiment results and numerical simulations reported inz = 0 plane of the down-conversion crystal onto a suitable z plane in the far field (In conclusion, using the two-photon field produced by SPDC, we have reported experimental observations of propagation-induced revival of entanglement in the angle-OAM bases. We have demonstrated this entanglement revival even in the presence of turbulence, the only effect of which is to increase the propagation distance for revival. Once revived, the two photons remain entangled up to an arbitrary propagation distance in the angle-OAM bases. We note that the entanglement revival strategies in turbulence or random media are usually based on adaptive optics techniques (


Lyapunov's first method, and Matlab 2020b is used for simulation analysis to verify the influence of each decision variable on different players' strategic choices. The results show that, firstly, new media's true reporting can make up for the lack of supervision of drug enterprises by local government, and the greater the impact of new media reporting, the more active drug enterprises will be to produce high-quality drugs. Secondly, non-collusion of third-party testing agencies can improve the self-discipline ability of drug enterprises, encourage new media to report truthfully, and play the role of co-regulation supervision. Furthermore, the greater the probability of new media's true reporting, the more local government tend to be stricter, and the probability of strict supervision is positively related to the central government's accountability. Finally, increasing penalty for producing low-quality drugs and collusion will help standardize the behavior of drug enterprises and third-party testing agencies. This article enriches and expands the theoretical basis of the drug quality co-regulation supervision and proposes corresponding countermeasures and suggestions.The efficiency and level of drug quality supervision are highly related to the distorted or true reporting of new media, and the collusion or non-collusion of third-party testing agencies. Therefore, based on the co-regulation information platform, considering the strategic choices of local government, drug enterprises, third-party testing agencies and new media, this article constructs a four-party evolutionary game model of co-regulation supervision. The stable equilibrium points of each participant's strategic choices are solved. The stability of the strategic combination is analyzed by  Drug quality supervision is an important part of ensuring the safety of public using drug and the national medical level. The development of the internet and the rise of new media have brought new opportunities and challenges to countries in drug quality supervision. In order to effectively reduce the uncertainty of regulatory decision-making, the US Food and Drug Administration utilizes new media platform, through social supervision and industry self-discipline, and after considerable development, has formed a set of effective regulatory model. The European Directorate for the Quality of Medicines is responsible for the drafting of policies on drug quality and risk prevention management of counterfeit drug, and incorporates drug safety supervision into daily management. The Pharmaceuticals and Medical Devices Agency uses non-governmental organizations, such as societies, industry associations, and the media, to supervise the self-discipline of the pharmaceutical industry, which has been an effective monitoring method for drug safety in Japan in recent years. In order to promote scientific drug supervision, the Chinese government is establishing a sound scientific and reasonable drug quality co-regulation supervision system .With the development of economic globalization, drug supervision is facing more problems. The development of regulatory science in the field of drug is not only between regulatory agencies and the industry, but also between patients, third-party testing agencies and society. Therefore, countries need to establish and improve drug supervision technology and information platforms, and make full use of emerging information technologies such as big data and artificial intelligence to establish drug information collection and tracking platform based on the concept of drug life cycle risk management. At the same time, there must be a standardized and authoritative new media platform to screen and release information to accelerate the development of scientific supervision and ensure the quality and safety of drug.Therefore, this article considers new media reporting and third-party testing agency behavior, based on the co-regulation information platform, constructs an evolutionary game model involving the participation of local government, drug enterprises, third-party testing agencies and new media. The stable equilibrium point of each game player's strategic choice is solved. The stability of the strategy combination and the influence of the changes of decision variables on the strategic choice are analyzed. The following three problems are solved, firstly, how does the co-regulation system affect the strategic choices of local government, drug enterprises, third-party testing agencies and new media? Secondly, how to play the role of the co-regulation information platform to promote the drug quality co-regulation supervision? Thirdly, how does the participation of local government and new media affect the strategic choices of drug enterprises and third-party testing agencies?Lyapunov's first method; The sixth part is the simulation analysis with Matlab 2020b; The seventh part discusses and outlines related suggestions; The last part suggests the conclusions.The remaining parts of this article are arranged as follows. The second part combs and reviews the relevant literatures; The third part makes hypotheses and constructs an evolutionary game model which involves local government, drug enterprises, third-party testing agencies and new media; The fourth part analyzes the stability of the strategic choices of the four participants; The fifth part analyzes the stability of strategy combination according to The problems and potential hidden dangers of drug quality shouldn't be underestimated. Once a drug quality accident occurs, a strong data drive is needed to support and enhance supervision decision-making . The outWith the increasingly wide range of supervision channels , the sinThe emergence of new media has made the transmission of information fast and extensive . When neFeedback from participants outside the government can strengthen the role of a national supervision system . Third-pIn summary, there is still a lack of consideration of four participants from the perspective of co-regulation supervision. In addition, the influence of new media reporting on the different strategic choices of each participant in the co-regulation supervision is not taken into account.Matlab 2020b simulation analysis, the validity of the model is verified, and countermeasures and suggestions are put forward for co-regulation supervision.Therefore, the research contributions of this article have the following three points. Firstly, considering new media participation and collusion behavior, based on the co-regulation information platform, this article builds a game model of quartet participation involving local government, drug enterprises, third-party testing agencies and new media. Secondly, considering the influence of true or distorted reports by new media, collusion or non-collusion of third-party testing agencies on the strategic choices of other participants, the evolutionary stable strategic combination under different conditions is solved. Finally, the influence of each decision variable on the strategic choices of four participants is solved and analyzed. And through This article chooses evolutionary game as the research method because it can explain the dynamic process of the evolution of each stakeholder's strategic choice and explain why this state has been reached and how to reach. The analysis paradigm of evolutionary game has a wide range of applications in social governance. Using the evolutionary perspective can more clearly see the final optimal decision of all parties in the game. Of course, when building a game model, the assumptions made for various relationships often deviate slightly from the reality, and need to be considered repeatedly, not only to meet the actual situation, but also to ensure the feasibility of the model. Therefore, considering new media participation and third-party testing agencies' collusion behavior, this article constructs the drug quality supervision structure relationship involving local government, drug enterprises, third-party testing agencies, and new media, as shown in H1 the local government chooses strict supervision with probability x, and loose supervision with probability 1 − x; drug enterprise chooses to produce high-quality drug with probability y, and low-quality drug with probability 1 − y; third-party testing agency chooses non-collusion with probability z, and collusion with probability 1 − z; new media chooses true reporting with probability p, and distorted reporting with probability 1 − p;x, y, z, p ∈ . The four players are bounded rational participants.H2 the income from the operation of drug enterprises is Re, the cost of producing high-quality drugs is Ch, and the cost of producing low-quality drugs is Cl(Cl < Ch). The cost of seeking collusion is B.H3 the cost of strict supervision is Gh, and the cost of loose supervision is Gl (Gl < Gh). When local government strictly supervises, the drug enterprises that produce low-quality drugs will be fined as Fe; the third-party testing agencies that participate in collusion will also be fined as Fp.H4 the income of third-party testing agencies is V, and their detection cost is Cp. If third-party testing agencies choose collusion, they need to forge test records and issue false test reports. Let the speculative cost of collusion be Ct (Ct < B).H5 the cost of new media reporting is Cm, and the revenue from the true reporting is Rml. In reality, new media will distort reports in order to pursue economic benefits and obtain additional page views. Distorted reports will obtain a profit of Rmh. According to the “spiral effect” of media communication, negative news has a magnifying effect. The news that drug enterprises produce low-quality drug will get more clicks and benefits than news that produce high-quality drug, so there are Rml < Rmh. When drug enterprises produce low-quality drug, local government loosely supervises but third-party testing agencies are not colluding, the true reports of new media will cause local government to be held accountable by the central government, setting the accountability as Fg.H6 new media reports will affect the reputation of drug enterprises, which will bring economic gains or losses to them. Set the influence effect of new media reports as δ, δ ∈ . When drug enterprises produce high-quality drug, new media true reporting will bring a reputation value to them δI, and distorted reports will cause negative impacts and economic losses δL. At this time, drug enterprises can sue new media for rights protection and receive compensation Fm. When drug enterprises produce low-quality drug, new media reporting will damage the reputation of drug enterprises and cause loss of reputation value δN.The parameters and descriptions of this article are shown in Based on the above hypotheses, this article constructs a four-party mixed strategy game matrix for local government, drug enterprises, third-party testing agencies and new media, as shown in The expected benefit of local government choosing the “strict supervision” strategy is:The expected benefit of local government choosing the “loose supervision” strategy is:The replicated dynamic equation and the first derivative of the local government's strategic choice are:F(x) < 0 and F′(x) < 0.According to the stability theorem of differential equations, if the probability of local government choosing the “strict supervision” strategy is to be in a stable state, the following conditions must be met: Proposition 1 When y < y0, z < z0 or p > p0, the local government's stabilization strategy is “strict supervision.” When y > y0, z > z0 or p < p0, its stabilization strategy is “loose supervision.” When y = y0,z = z0 or p = p0, we are unable to determine its stabilization strategy. The thresholds areProof Make G = Gl − Gh + Fg − yFg + (1 − y)(1 − z)p(Fe + Fp − Fg), when G = 0,G/∂y < 0, ∂G/∂z < 0 and ∂G/∂p > 0, G is a decreasing function of y and z, and an increasing function of p. When, z < z0 or p > p0,G > 0,F(x)|x = 1 = 0 can be calculated, so x = 1 is stable. When y > y0, or p < p0 „ F(x)|x = 0 = 0 can be calculated, so x = 0 is stable. When, z = z0 or p = p0,G = 0 and F′(x) = 0 can be calculated, we are unable to determine a stable strategy.y or z increases, the local government's stabilization strategy will change from strict supervision to loose supervision. If increases, the local government's stabilization strategy will change from loose supervision to strict supervision. Therefore, in drug quality co-regulation supervision, true reporting from new media can urge local government to perform strict supervision duty. When drug enterprises consciously produce high-quality drugs, local government will choose loose supervision in order to maximize social benefits and save supervision cost.Proposition 1 shows that if x isIn summary, the response function of According to Proposition 1, the phase diagram of local government strategic choice is shown in Vg1 is the probability that local government chooses the “strict supervision” strategy, and the volume of Vg0 is the probability that local government chooses the “loose supervision” strategy. And,It can be seen from Corollary 1.1 when local government imposes more fines on drug enterprises and third-party testing agencies, it tends more to choose the “strict supervision” strategy.Proof According to the probability Vg1, the first-order partial derivatives of Fe and Fp can be calculated:Fe and Fp. With the increase of Fe and, local government will conduct “strict supervision” in order to obtain more revenue.Corollary 1.1 shows that the probability of local government choosing “strict supervision” strategy is an increasing function of Corollary 1.2 When local government pays more additional cost for strict supervision, it tends more to choose the “loose supervision” strategy.Proof According to the probability Vg0, the first-order partial derivatives of Gh − Gl can be calculated:Gh − Gl. The higher the additional cost of strict supervision, local government will tend to choose loose supervision in order to save cost.Corollary 1.2 shows that the probability of local government choosing the “loose supervision” strategy is an increasing function of The expected benefit of drug enterprises choosing the “producing high-quality drug” strategy is:The expected benefit of drug enterprises choosing the “producing low-quality drug” strategy is:The replicated dynamic equation and the first derivative of drug enterprises' strategic choice are:F(y) = 0 and F′(y) = 0.According to the stability theorem of differential equations, if the probability of drug enterprises choosing the “producing high-quality drug” strategy is to be in a stable state, the following conditions must be met: Proposition 2 When x < X1, z > z1 or p > p1, drug enterprises' stabilization strategy is “producing high-quality drug.” When x < x1,z < z1 or p < p1, their stabilization strategy is “producing low-quality drug.” When x = x1,z = z1 or p = p1, we are unable to determine their stabilization strategy. The thresholds areProof Make, when K = 0, K/∂x > 0,∂K/∂z > 0 and ∂K/∂p > 0, K is an increasing function of x, z and p. When x > x1, z > z1 or p > p1, K > 0, F(y)|y = 1 = 0 can be calculated, so y = 1 is stable. When x < x1,z < z1 or p < p1,K < 0,F(y)|y = 0 = 0 can be calculated, so y = 0 is stable. When x = x1,z = z1 or p = p1,K = 0 and can be calculated, we are unable to determine a stable strategy.x, or p increases, drug enterprises' stabilization strategy will change from producing low-quality drug to producing high-quality drug. Therefore, strict supervision by local government, urging third-party testing agencies to perform non-collusive social responsibility, and encouraging true reporting by new media can all prompt drug enterprises to proactively produce high-quality drug.Proposition 2 shows that if In summary, the response function of y is:According to Proposition 2, the phase diagram of drug enterprises strategic choice is shown in Ve1 is the probability that drug enterprises choose the “producing high-quality drug” strategy, and the volume of Ve0 is the probability that drug enterprises choose the “producing low-quality drug” strategy. And,It can be seen from Corollary 2.1 The higher the reputation value of high-quality drugs produced by drug enterprises and the greater the reputation loss of low-quality drugs, the greater the probability of producing high-quality drugs.Proof Make A = Ch − Cl − Fm − Fe−δ(N − L)−(1 − z)B, T = δI + δN + Fe. According to the probability Ve1, the first-order partial derivatives of N and I can be calculated:Corollary 2.1 shows that increasing the price that drug enterprises need to pay for producing low-quality drugs or increasing the reputation benefits they earn from producing high-quality drugs can increase the probability that drug enterprises choose the strategy of “producing high-quality drug.”Corollary 2.2 The higher the additional cost of producing high-quality drug by drug enterprises, the higher the probability of producing low-quality drug; the greater the influence effect of new media reporting, the lower the probability of drug enterprises producing low-quality drug.Proof Make A = Ch − Cl − Fm − Fe − δ(N − L)−(1 − z)B, T = δI + δN + Fe, U = δI+δL − Fm. According to the probability, the first-order partial derivatives of (Ch − Cl) and δ can be calculated:Ve0 is an increasing function of (Ch − Cl), and a decreasing function of δ. When the additional cost of producing high-quality drug increases, drug enterprises tend to produce low-quality drug for economic benefits; when the influence effect of new media reporting is greater, in order to reduce reputation loss and ensure economic benefits, they are less likely to choose the strategy of “producing low-quality drug.”Corollary 2.2 shows that the probability The expected benefit of third-party testing agencies choosing the “non-collusion” strategy is:The expected benefit of third-party testing agencies choosing the “collusion” strategy is:The replicated dynamic equation and the first derivative of third-party testing agencies' strategic choice are:F(z) = 0 and F′(z) = 0.According to the stability theorem of differential equations, if the probability of third-party testing agencies choosing the “non-collusion” strategy is to be in a stable state, the following conditions must be met: Proposition 3 When x > x2, or p < p2, third-party testing agencies' stabilization strategy is “non-collusion.” When x < x2,y < y2 or p > p2, their stabilization strategy is “collusion.” When x = x2, y = y2 or p = p2 we are unable to determine their stabilization strategy. The thresholds are Proof Make J = y(V − Cp) + (1 − y)(Fp + Ct − B) − (1 − y)(1 − x)pFp, when J = 0,J/∂x > 0,∂J/∂y > 0 and, J is an increasing function of x and y, and a decreasing function of p. When x > x2, y > y2 or p < p2,J > 0,F(z)|z = 1 = 0 can be calculated, so z = 1 is stable. When x < x2,y < y2 or p > p2,J < 0,F(z)|z = 0 = 0 can be calculated, so is stable. When x = x2, y = y2 or p = p2, J = 0 and F′(z) = 0 can be calculated, we are unable to determine a stable strategy.x, yincreases, or p decreases, the stability strategy of third-party testing agencies changes from collusion to non-collusion. Therefore, in the drug quality co-regulation supervision, the government should strictly restrict the behavior of third-party testing agencies and promote the work pattern of co-regulation supervision.Proposition 3 shows that when z isIn summary, the response function of According to Proposition 3, the phase diagram of third-party testing agencies strategic choice is shown in Vp1 is the probability that the third-party testing agencies choose the “non-collusion” strategy, and the volume of Vp0 is the probability that the third-party testing agencies choose the “collusion” strategy. And,It can be seen from Corollary 3.1 The probability of non-collusion by third-party testing agencies is positively related to the speculative cost of participating in collusion.Proof According to the probability, the first-order partial derivatives of can be calculated:Corollary 3.1 shows that when the speculative cost of collusion by third-party testing agencies is higher, their motivation for collusion is weakened and the probability of choosing “non-collusion” is higher.Corollary 3.2 The probability of collusion by third-party testing agencies is positively related to the cost of drug enterprises seeking to collude.Proof According to the probability, the first-order partial derivatives of B can be calculated:Corollary 3.2 shows that when the cost of seeking collusion increases, in order to obtain economic benefits, the higher the probability that third-party testing agencies choose to “collusion.”The expected benefit of new media choosing the “true reporting” strategy is:The expected benefit of new media choosing the “distorted reporting” strategy is:The replicated dynamic equation and the first derivative of new media's strategic choice are:F(p) = 0 and F′(p) < 0.According to the stability theorem of differential equations, if the probability of new media choosing the “true reporting” strategy is to be in a stable state, the following conditions must be met: Proposition 3 When x > x3, y < y3 or z > z3, new media's stabilization strategy is “true reporting.” When, y > y3 or z < z3, its stabilization strategy is “distorted reporting.” When, y = y3 or z = z3, we are unable to determine its stabilization strategy. The thresholds areProof Make H = y(Rml − Rmh + Fm) + (1 − y)(1 − x)(1 − z)(Rml − Rmh), when H = 0„,H/∂x > 0, ∂H/∂y < 0 and ∂H/∂z > 0, H is an increasing function of x and z, and a decreasing function of y. When x > x3, y < y3 or z > z3H > 0,p = 1 is stable. When x < x3, y > y3 or, H < 0,F(p)|p=0 = 0 can be calculated, so is stable. When x = x3, y = y3 or z = z3, H = 0 and F′(p) = 0 can be calculated, we are unable to determine a stable strategy.x, z increases, or y decreases, the stability strategy of new media changes from distorted reporting to true reporting. Therefore, in drug quality supervision, new media can urge local government to strictly supervise, drug enterprises' integrity production, and third-party testing agencies to fulfill their social responsibility by reporting on drug quality information.Proposition 4 shows that whenIn summary, the response function of is:According to Proposition 4, the phase diagram of new media strategic choice is shown in Vm1 is the probability that new media chooses the “true reporting” strategy, and the volume of Vm0is the probability that new media chooses the “distorted reporting” strategy. And,It can be seen from Corollary 4.1 The probability of true reporting by new media is positively correlated with the compensation to drug enterprises.Proof According to the probability Vm1, the first-order partial derivatives of can be calculated:Corollary 4.1 shows that as the cost of distorted reports increases, new media tend to report truthfully, which will help promote the social pattern of co-regulation supervision.Corollary 4.2 The probability of new media “distorted reporting” is positively correlated with the additional revenue gained from distorted reporting.Proof According to the probability Vm0, the first-order partial derivatives of Rmh − Rml can be calculated:Corollary 4.2 shows that when the additional revenue of distorted reporting increases, new media will be more inclined to choose “distorted reporting.” Therefore, local government should pay attention to the construction of new media platform, strengthen the management and regulation of it, and build new media into a healthy and upward platform.Corollary 4.3 When Fm > Fm0, new media will choose the “true reporting” strategy. When Fm < Fm0, it will choose the “distorted reporting” strategy. The threshold is Proof According to Proposition 4, when H = 0, can be calculated. Because ∂H/∂Fm > 0, H is an increasing function of Fm. When Fm > Fm0, H > 0, F(p)|p = 1 = 0 can be calculated. When Fm < Fm0, H < 0, F(p)|p = 0 = 0 can be calculated.Corollary 4.3 shows that when new media chooses to report distortedly, it needs to pay compensation to drug enterprises. If, it is possible to ensure that it will choose the “true reporting” strategy. Therefore, increasing compensation for distorted reporting can prompt new media to report truthfully, while at the same time making up for the losses caused by distorted reporting to drug enterprises, so as to avoid damaging the enthusiasm of them to produce high-quality drugs.Lyapunov's first method. If all the eigenvalues of the Jacobian matrix are negative, the equilibrium point is an evolutionary stable strategy (ESS). If at least one eigenvalue is positive, the equilibrium point is an unstable point. If the eigenvalues are all negative except for zero, the equilibrium point is in a critical state and the stability is uncertain. This article analyzes the stability of the 16 pure strategy Nash equilibrium points, the Jacobian matrix of the replication dynamic system isThe stability of each strategic combination can be judged according to When the stability strategy of new media is “true reporting,” that is, when condition is satisfied, the asymptotic stability analysis of the equilibrium point of the replicated dynamic system is shown in Cp < V, the equilibrium point  of the replicated dynamic system is ESS.It can be seen from The production of high-quality drugs by drug enterprises, non-collusion by third-party testing agencies, and true reporting by new media are the most ideal situation for local government. Therefore, when drug enterprises produce low-quality drug, local government must actively perform its supervision responsibility and impose severe penalty on them; encourage new media to report truthfully in a timely manner, so as to urge drug enterprises consciously produce high-quality drug.Rmh − Rml > Fm is satisfied, the asymptotic stability analysis of the equilibrium point of the replicated dynamic system is shown in When the stability strategy of new media is “distorted reporting,” that is, when condition Cp < V, the equilibrium point  of the replicated dynamic system is ESS. When Gh − Gl < Fg and Cp < V, the equilibrium point  of the replicated dynamic system is ESS. When Cp > V, the equilibrium point  of the replicated dynamic system is ESS.It can be seen from When drug enterprises produce high-quality drug, in order to obtain economic benefits, new media will have distorted reporting, which will undermine the enthusiasm of drug enterprises to produce high-quality drug. Therefore, it is necessary to encourage all parties to perform their social responsibilities and play their respective roles. Standardize the behavior of new media and impose legal sanctions on distortions, rumors and so on. In addition, it is necessary to guide it to build a positive platform to better reflect the authority of the co-regulation information platform.Matlab 2020b for numerical simulation to verify the validity of the evolutionary stability analysis. Based on the actual conditions and the research of other scholars, this article makes assumptions about the numerical values in This article uses I = {0.0, 2.0, 4.0}, N = {0.0, 3.0, 6.0}, and the evolution process and results of players' strategy of the quartet game are shown in Set I and N are zero, the stable strategy of drug enterprises is to produce low-quality drugs. At this time, local government chooses loose supervision, new media chooses distorted reporting, and the probability of third-party testing agencies choosing non-collusion is stable at a relatively high level. With the increase of and, local government's stable strategy is loose supervision, the strategic choices of others continue to fluctuate. When I and N further increase, drug enterprises produce high-quality drug, new media reports truthfully, and third-party testing agencies are not colluding. At this time, local government doesn't need to conduct strict supervision. New media must actively play a supervisory and reporting role, urge drug enterprises to produce high-quality drugs, and establish a good image. Therefore, local government should improve drug quality public opinion supervision mechanism, use the co-regulation information platform to share information with new media, and give full play to the advantages of new media in drug quality supervision.Rmh − Rml = {0.0, 2.0, 4.0}, and the evolution process and results of players' strategy of the quartet game are shown in Set δ = {0.0, 0.4, 0.8}, Rmh − Rml are zero, the stability strategy of third-party testing agencies is collusion, the stability strategy of drug enterprises is the producing low-quality drugs, and the stability strategy of local government is loose supervision. The probability of new media choosing true reporting strategy is stable at a higher probability. When δ and Rmh − Rml increase, local government loosely supervises, and the strategic choices of drug enterprises, third-party testing agencies and new media are in a state of fluctuation. With the further increase in δ and, the strategic choices of drug enterprises and third-party testing agencies are in a state of fluctuation, local government loosely supervises, and new media distortedly reports. Therefore, we must pay attention to the construction of new media platforms, standardize the dissemination of drug quality information, and realize the active governance function of new media.p = 0.5. Suppose Gh − Gl = {0.0, 7.0, 11} and Ch − Cl = {0.0, 7.0, 11}. The tripartite game evolutionary process and results of local government, drug enterprises, and third-party testing agencies are shown in Let the probability that new media chooses true reporting is It can be seen from p = 0.5. Let Fg = {0.0, 4.5, 6.0}, Fe = {0.0, 4.5, 6.0}, and Fp = {0.0, 4.5, 6.0}. The tripartite game evolutionary process and results of local government, drug enterprises, and third-party testing agencies are shown in Suppose the probability of new media true reporting is It can be seen from Considering new media participation and third-party testing agencies' collusion behavior, this article constructs a quartet evolutionary game model. This article analyzes the influence of the change of each decision variable on the strategy evolution and puts forward the following suggestions.First of all, complete a co-regulation information platform, promote the establishment of a good co-regulation supervision system, and emphasize the joint participation of local government, drug enterprises, new media, and third-party testing agencies. It is necessary to mobilize the forces of all sectors of society to promote the formation of a drug quality supervision pattern involving government supervision, drug enterprises operation, new media reporting, and third-party testing agencies participation.Secondly, fully implement the main responsibility of drug enterprises, establish a “blacklist” system for them, notify and criticize enterprises that produce low-quality drugs, increase the scale and frequency of drug sampling, and give play to their self-discipline capability. In addition, drug enterprises should actively use new media platform to release authoritative information, track drug quality events and provide scientific professional knowledge explanations. Improve the public's ability to discriminate, and increase interactive communication through platforms such as Facebook, Twitter, and WeChat. Guide the public to use drug safely, and avoid losses caused by negative effects.Besides, we must attach importance to the construction of new media platforms, strengthen the management of new media, and impose legal sanctions on unscrupulous media that distortedly reports. Standardize the dissemination of drug quality information, improve the quality of new media practitioners, and build it into a healthy and upward platform. Give full play to the assisting role of media supervision to government supervision, further win the trust of the public, and realize the positive governance function of new media.Eventually, standardize the behavior of third-party testing agencies. Encourage them to fulfill their social responsibilities, feedback real drug quality test reports to the co-regulation information platform, and promote a good situation in drug quality co-regulation supervision.The sharing and interactive nature of new media makes it play an important role in drug quality supervision. Promoting the modernization of drug quality supervision requires the joint participation of multiple parties. Improve the social responsibility awareness of drug enterprises, increase policy support, reduce their production costs, and encourage them to consciously produce high-quality drugs. Restrict the behavior of third-party testing agencies, urge them not to participate in collusion, and assist local government to strictly supervise drug enterprises. Give full play to the advantages and functions of new media and encourage it to actively participate in co-regulation. According to the information provided by the co-regulation information platform, true reporting by new media are of great significance to enhancing the dissemination, guidance, and credibility of local government news releases.Considering the strategic choices of local government, drug enterprises, third-party testing agencies, and new media, this article constructs a four-party evolutionary game model. However, the constructed game model is fully informative and single-stage under bounded rationality, and the game sequence is not considered. Therefore, considering the condition of incomplete information, constructing a multi-stage, repetitive and dynamic game model with multi-agent participation is the next research direction.The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.SZ wrote the manuscript and analyzed the data. LZ revised the manuscript and solved the model. All authors contributed to the article and approved the submitted version.This work was supported by the National Social Science Fund of China under grant Nos. 20BGL272 and 21ZDA024.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.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Nature Communications 10.1038/s41467-022-32852-6, published online 31 August 2022Correction to: The original version of this article contained errors in Eq. (1) and in Fig. 6b.T2. The equation incorrectly read:In Eq. (1), the values of the constant and coefficients were incorrectly given and the third term within the parentheses was missing the quadratic variable The correct form of Eq. (1) is:Furthermore, the original version of this article contained an error in Fig. 6b, in which the ‘x’ and ‘y’ axes were incorrectly labelled as ‘Time (min)’ and ‘Resonant frequency (GHz)’, respectively. The correct version of Fig. 6b is:These have been corrected in both the PDF and HTML versions of the article.


Recently, deep learning methods have allowed the community to address both problems simultaneously, especially in the non-Cartesian acquisition setting. This work aims to contribute to this field by tackling some major concerns in existing approaches. Particularly, current state-of-the-art learning methods seek hardware compliant k-space sampling trajectories by enforcing the hardware constraints through additional penalty terms in the training loss. Through ablation studies, we rather show the benefit of using a projection step to enforce these constraints and demonstrate that the resulting k-space trajectories are more flexible under a projection-based scheme, which results in superior performance in reconstructed image quality. In 2D studies, our novel PROjection for Jointly lEarning non-Cartesian Trajectories while Optimizing Reconstructor (PROJeCTOR) trajectories present an improved image reconstruction quality at a 20-fold acceleration factor on the fastMRI data set with SSIM scores of nearly 0.92–0.95 in our retrospective studies as compared to the corresponding Cartesian reference and also see a 3–4 dB gain in PSNR as compared to earlier state-of-the-art methods. Finally, we extend the algorithm to 3D and by comparing optimization as learning-based projection schemes, we show that data-driven joint learning-based PROJeCTOR trajectories outperform model-based methods such as SPARKLING through a 2 dB gain in PSNR and 0.02 gain in SSIM.Compressed sensing in magnetic resonance imaging essentially involves the optimization of (1) the sampling pattern in  Using Compressed Sensing (CS) theories [k-space according to Variable Density Sampling (VDS) [k-space, as this type of sampling, which relies on curves, is more flexible and efficient compared to straight lines used in traditional Cartesian acquisitions. While conventional non-Cartesian sampling patterns such as spiral, radial, rosette, etc. [k-space according to VDS, they do not sample at a well-defined user specified Target Sampling Density (TSD). Tailoring such non-Cartesian trajectories according to a MR imaging protocol and a given TSD is hard, as these k-space sampling curves or trajectories are constrained by the MR hardware limits, notably on the maximum gradient magnitude A major challenge limiting the use of Magnetic Resonance Imaging (MRI) is long acquisition times, arising due to short decay of the MR signal, which is used to sample multi-dimensional theories , signifing (VDS) ,3,4,5,6,te, etc. ,10,11,12k-space sampling pattern according to a prescribed TSD. Such patterns are typically segmented into multiple shots or k-space trajectories, each of them being compliant with the above-mentioned MR hardware constraints. Further, the algorithm results in locally uniform sampling patterns, and thus avoids holes and clusters in k-space. However, the SPARKLING is a model-driven framework, which is characterized by a TSD that needs to be known in advance to feed the optimization process. Previously, in [To meet these constraints in a safe manner, the Spreading Projection Algorithm for Rapid K-space sampLING (SPARKLING) was introduced in ,14, and usly, in , to addrusly, in  as an acusly, in , griddedk-space sampling trajectories and MR image reconstruction in a data-driven manner on the fastMRI dataset [Recently, new methods ,19,20 ha dataset . In [18, dataset , resultik-space. Finally, both methods do not make use of Density Compensators (DCp), which plays a key role in obtaining clearer MR images in the non-Cartesian deep learning setting [In B-spline parameterized Joint Optimization of Reconstruction and K-space trajectories (BJORK) , the aut setting .k-space trajectories in a data-driven manner while embedding a projected gradient descent algorithm [k-space sampling trajectories and use multi-resolution [k-space trajectories and demonstrate its superiority over penalty-based methods like PILOT and BJORK to enforce hardware constraints. Finally, we compare and show the superiority of data-driven PROJeCTOR trajectories compared to model-based non-Cartesian SPARKLING trajectories.In this work, we first develop a generic model for PROJeCTOR. More precisely, we introduce a method that learns the lgorithm  to fulfisolution  similar solution  and BJORsolution  in 2D MRk-space trajectories and deep neural networks for MR image reconstruction. Particularly, we discuss 2 sub-models; namely, (1) an acquisition model parameterized by k-space trajectory, and (2) a reconstruction model parameterized by a deep neural network. Later, we discuss in detail how to handle the MR hardware constraints and which approach seems the most efficient within the sampling pattern optimization process to end up with hardware compliant k-space trajectories.In this section, we present a generic and modular framework  for leark-space data as input in the pipeline shown in In order to reduce the memory footprint and the training time, we did not process multicoil N. This assumption is purely for notational convenience and does not limit the applicability of our framework to isotropic data. If D is the imaging dimension, we denote a MR image or volume as k-space of this acquisition is defined in k-space to TObs) = 5.12 ms (readout time), giving us Overall, we rely on notations developed in , and we For our experiments in 2D imaging, we used the fastMRI brain MR data set , which cdata set , which c 20 (see ), resultk-space sampling pattern The acquisition model is parameterized by a k-space trajectories are constrained in speed and acceleration by the maximum gradient strength k-space is sampled at Echo Time (TE) in every shot, resulting in a stable and required target contrast of reconstructed MR images. From [The es. From ,24, we m Ω (from ):(2a)α=A and v in  at every dwell time k-space data k-space trajectory to have With the  (NUFFT)  operatork-space trajectories are non-Cartesian, this creates a variable density sampling in k-space, due to which a simple adjoint of NUFFT operator k-space samples associated with the low and high frequencies during the iterative reconstruction process. Following this principle, we obtained DS(K) for the linearly interpolated k-space trajectory As the 20 years . It alloribed in . As noteribed in , DCp is k-space data k-space trajectory The reconstruction network k-space and convolutional neural network (CNN) based denoising in the image domain with kernel size A simple parameter-free reconstruction would be the density compensated adjoint, i.e., NC-PDNet ) as the re as in  except tx and its reconstruction S) [k-space trajectory K as follows: The reconstruction error used as loss function pired by  and is dndex (S) :LrK, we derived the gradient of the loss function K: For optimizing the trajectory For ease of mathematical derivation, here we take the case of a parameter-free reconstruction as described in k-space trajectory K, we used [k-space region (as noted in [In order to compute the gradient of NUFFT operators  we used  to obtainoted in ), we usek-space trajectory were extremely noisy for this low batch size in 3D, we relied on a smaller learning rate of During training, the gradient descent was carried out stochastically with a batch size of 64 in 2D, while due to memory limitations, it was limited to 1 in 3D. However, as the gradients with respect to k-space sampling trajectories was performed using a multi-resolution strategy [K at the maximal Inspired by SPARKLING , the leastrategy  which stk-space trajectory over a reduced number of locations LNeed for hyper-parameter tuning: Under the penalty based formulation, the hyper-parameters Equation  as an auEquation , the corInfluence of gradients and convergence: With the addition of penalty terms k-space trajectories. Gradient updates with these additional gradient terms can no longer guarantee optimal image reconstruction by minimizing the reconstruction loss Guarantee of admissibility: Finally, the optimization of the augmented Lagrangian form does not guarantee that the final optimized k-space trajectory K satisfies the constraints Equation  generatPractical implementations involved 50 iterations of the projection algorithm from , which wtensorflow-nufft [cuFINUFFT [All our implementations in 2D were carried out on a V100 GPU with 32GB memory, while our 3D implementations needed the next generation A100 GPUs with 80 GB of memory. Most of the memory in 3D was occupied by the activations from the 3D convolutional neural networks used in the image denoising step in NC-PDNet. Memory efficient implementations of NUFFT was carried out by using ow-nufft , which iuFINUFFT .k-space sampling pattern and reconstruction network in 3D by comparing it to SPARKLING trajectories with a learned neural network for image reconstruction.In this section, we first compare our results with state-of-the-art methods, particularly BJORK  and PILOk-space trajectories with We learned ng LOUPE  as obtaing LOUPE  as a recWe compared our results with PILOT and BJORK trajectories , which wk-space is under-sampled with respect to the fully sampled Cartesian k-space, while AF reflects on how fast the scan is with respect to the Cartesian reference scan.As the BJORK trajectory was learned for mparison  was donek-space trajectories as compared to those yielded by BJORK and PILOT. Then, we compare the reconstruction results of the learned trajectories with BJORK and PILOT.We first proceed to analyze the k-space (cf. the white spot shown in the bottom inset), while BJORK samples the k-space densely slightly off the center (cf. bottom inset), which is suboptimal. In contrast, PROJeCTOR and SPARKLING methods sample the central region of k-space more densely, which could help obtain improved image quality, notably the contrast.When looking at the zoomed portions of optimized trajectories in We also observe at the bottom of each panel in k-space data using PILOT /Peak Signal-to-Noise Ratio (PSNR) scores between PILOT and PROJeCTOR on one hand and BJORK and PROJeCTOR on the other hand and obtained p-values Next, we compared the results of image reconstruction from retrospectively under-sampled ng PILOT  and BJORng PILOT  trajectok-space sampling trajectories through joint optimization with a reconstruction network using a penalty term instead of a projector.In the above section, we showed how our method outperforms PILOT and BJORK in terms of reconstructed image quality. We assume that these results are due to the different manners that the hardware constraints on the gradients are enforced in the learning process (projector vs. regularizer). To validate this hypothesis, we learned 3D hardware compliant k-space sampling trajectories using the projection and penalty-based methods, and then in In k-space trajectories get more constrained. Interestingly, as k-space trajectories are getting hardware compliant  and (A.c)). This type of imperfection is not present in the learned PROJeCTOR k-space sampling pattern ((B.b) and (B.c)). Further, as the trajectories and reconstruction network were learned on partial Fourier k-space data, PROJeCTOR trajectories also learned to exploit this by not sampling these regions (see the dark areas pointed out by green arrows in (B.b) and (B.d)).From the mid-slice cuts of gridded sampling patterns in Finally, comparing the actual reconstructed MR images in In this work, we present a generic framework for jointly learning the trajectory and image reconstruction neural network. We embedded the projection step from  and learAlthough the learned neural networks in PILOT  and BJORThis improvement over state-of-the-art methods can be attributed to the embedded projection step as compared to penalty to ensure hardware compliance. We carried out an ablation study and showed that the projection step is crucial for having significantly improved performance of the learned trajectories, as compared to penalty-based approaches.Finally, in 3D, we compared the model-driven method SPARKLING with the data-driven method PROJeCTOR and showed a gain of 2 dB in PSNR and 0.02 gain in SSIM in favor of the latter.Future prospects of this work include prospective implementations through modifications of A limitation of current work is that our training paradigm was set-up in an emulated single coil setting as we were limited by memory constraints on GPU. A more realistic implementation would involve a multi-coil imaging setting, which is mandatory to efficiently utilize parallel imaging and become closer to the real data acquisition context, allowing us to reach higher AF. However, this memory bottleneck can be alleviated through efficient transfers between CPU and GPU or multi-GPU implementations. Further, the network can be improved by extending the currently implemented simple forward acquisition model NUFFT to a more realistic and complex model, which takes off-resonance effects due to eneities  and grad


Originally, the information in row Availability of work and column Comparison category/base category appeared as 0: Working, 1: Not working. This has been corrected to 0: Not Working, 1: Working.In the article, “Factors related to preparedness for emergency hemodialysis in the event of a natural disaster”,