Recently found zero-determinant (ZD) strategies have substantially been investigated in evolutionary game theory. The original memory-one ZD strategies unilaterally enforce linear relationships between average pay-offs of players. Here, we extend the concept of ZD strategies to memory-two strategies in repeated games. Memory-two ZD strategies unilaterally enforce linear relationships between correlation functions of pay-offs and pay-offs of the previous round. Examples of memory-two ZD strategy in the repeated Prisoner’s Dilemma game are provided, some of which generalize the tit-for-tat strategy to a memory-two case. Extension of ZD strategies to memory- SinceIn 2012, Press and Dyson discovered a novel class of memory-one strategies called zero-determinant (ZD) strategies . ZD straRecently, properties of longer memory strategies have been investigated in the context of repeated games with implementation errors –45. In gIn this paper, we extend the concept of ZD strategies to memory-two strategies in repeated games. Memory-two ZD strategies unilaterally enforce linear relationships between correlation functions of pay-offs at the present round and pay-offs at the previous round. We provide examples of memory-two ZD strategy in a repeated Prisoner’s Dilemma game. Particularly, one of the examples can be regarded as an extension of TFT strategy to a memory-two case.n case with n ≥ 2 is discussed. Section 6 is devoted to concluding remarks.The paper is organized as follows. In §2, we introduce a model of repeated game with memory-two strategies. In §3, we extend ZD strategies to the memory-two strategy class. In §4, we provide examples of memory-two ZD strategies in a repeated Prisoner’s Dilemma game. In §5, extension of ZD strategies to a memory-. 2N-player repeated game. Action of player a ∈ {1,…, N} is described as σa ∈ {1,…, M}. We collectively denote state n with n ≥ 2.) Strategy of player a is described as the conditional probability σa when states at last round and second-to-last round are a when the state is t, and we have defined the transition probabilityδ = 1 .  Actions of both players at t = 0 and t = 1 are sampled from uniform distribution. In t = 100000. We can see that the linear relation . Then, we find that her PD matrix iss1〉(st) = 〈s2〉(st) [Here, we introduce a memory-two ZD strategy which can be called an extended tit-for-tat (ETFT) strategy. We consider the situation when player 1 takes the following memory-two strategy:〈s2〉(st) , equatioq.We provide numerical results about the ETFT strategy. Parameters and strategies of player 2 are set to the same values as those in the previous subsection. In We can understand feelings of the ETFT player as follows. We find that, when the previous state is  or , ETFT behaves as TFT. When the previous state and the second-to-last state are TFT-ATFT :4.11T1, because T1 = 1. Therefore, it is expected that ETFT-2 is less successful than ETFT.It should be noted that a slightly modified version of ETFT:. 4.3R > T + S, which corresponds to the condition that mutual cooperation is favourable to the period-two sequence  →  →  → · · · [Here, we introduce another memory-two ZD strategy which can be called a fickle tit-for-tat (FTFT) strategy. In this subsection, we assume 2 → · · · . We consequation ). FTFT cq.We provide numerical results about the FTFT strategy. Parameters and strategies of player 2 are set to the same values as those in the previous subsections. In . 4.4R > T + S and 2P < T + S. In memory-one strategies, there exists the following memory-one ZD strategy:Here we assume that 2As an extension of ZSS, we can consider the following memory-two strategy:q.In . 4.5As in memory-one cases , possibl. 5n (n ≥ 2) case. By using the technique of this paper, extension of Akin’s lemma  case is straightforward.In this paper, we extended the concept of ZD strategies in repeated games to memory-two strategies. Memory-two ZD strategies unilaterally enforce linear relationships between correlation functions of pay-offs and pay-offs at the previous round. We provided examples of memory-two ZD strategies in the Prisoner’s Dilemma game. Some of them can be regarded as variants of TFT strategy. We also discussed that extension of ZD strategies to a memory-Before ending this paper, we make two remarks. First, in our numerical simulations, we investigated only simple situations which correspond to well-mixed populations and without evolutionary behaviour. It has been known that evolutionary behaviour can drastically change when populations are structured –52. TherThe second remark is related to the length of memory. Recently, it has been found that long memory can promote cooperation in the Prisoner’s Dilemma game ,54. Our Click here for additional data file.


Scientific Reports 10.1038/s41598-021-94505-w, published online 22 July 2021Correction to: In the original PDF version of this Article a previous rendition of Figure 1 was published. The original Figure This error has now been corrected in the PDF version of the Article; the HTML version was correct from the time of publication.


Correction to: Translational Psychiatry 10.1038/s41398-022-01795-3, published online 26 January 2022The original version of this article unfortunately contained a mistake in Figure 2. Unfortunately, the direction of the label is incorrect. Currently it reads “Significant effects of Group: TS – NTS”. The correct title is in fact “Significant Effects of Group: NTS – TS”. All other details in the paper, including the reporting of results and discussion of findings are correct as published. The original article has been corrected.


GW as post-Hartree-Fock methods. We found GW to be the most accurate of these three approximations for the ionization potential, by far. Going beyond GW by adding more diagrams is a tedious and dangerous activity: We tried to complement GW with second-order exchange (SOX), with second-order screened exchange (SOSEX), with interacting electron-hole pairs (WTDHF), and with a GW density-matrix (γGW). Only the γGW result has a positive impact. Finally using an improved hybrid functional for the non-interacting Green’s function, considering it as a cheap way to approximate self-consistency, the accuracy of the simplest GW approximation improves even more. We conclude that GW is a miracle: Its subtle balance makes GW both accurate and fast.We use the GW100 benchmark set to systematically judge the quality of several perturbation theories against high-level quantum chemistry methods. First of all, we revisit the reference CCSD(T) ionization potentials for this popular benchmark set and establish a revised set of CCSD(T) results. Then, for all of these 100 molecules, we calculate the HOMO energy within second and third-order perturbation theory (PT2 and PT3), and,  GW approximation   is curreximation  for elecximation  for neutv. Here we name the second-order perturbation theory, PT2, and the third-order perturbation theory, PT3. Going beyond the third-order has seldom been attempted . With GW we denote here the one-shot GW appoximation that does not include self-consistency; in the literature it is sometimes denoted as 0W0G. 2 IP was more than 1 eV off the trend. We present here a complete recalculation of the CCSD(T) reference IPs for the GW100 benchmarkHowever, before doing so, we will revisit the CCSD(T) reference IPs. We observed that the set from .The GW100 benchmark is hence to be the boxing ring in which we want to scrutinize the quality of the different MBPT approximations  ionization potentials for GW100, we set up new CCSD(T) reference values of the IPs for the GW100 benchmark set. Benchmarking the MBPT Strategies compares the performance of the different approximations based on a standard Hartree-Fock starting point. MBPT From an Improved Mean-Field Starting Point shows an attempt to approach MBPT self-consistency with tuned hybrid functionals. Finally the conclusions are drawn in Conclusion. Hartree atomic units are used throughout this work. The numerical values are made available as Supplemental Material, under the wide-spread machine- and human-readable JSON file format.The article is organized as follows: In In MBPT, the central quantity is the one-electron Green’s function. The Green’s function describes the time-propagation of an additional particle in the electronic system: an extra electron for propagation forward in time, or a hole for propagation backward in time. The Green’s function contains a great deal of information. For instance its diagonal is the electronic density, and, most interesting for us, its poles are the ionization energies .G0 is known, the exact Green’s function G can be obtained thanks to the Dyson equation:p and q, have been dropped for simplicity.Once an approximate Green’s function G0 to G. If the Hartree-Fock approximation (HF) is used for G0, then ΔΣ coincides with the missing correlation part of the self-energy Σc.The operator ΔΣ stands for the self-energy difference. It performs the humongous task of connecting G0, it can be expressed analytically:p and q are molecular orbital (MO) indices, ϵp and ± iη is a vanishing imaginary number that ensures the correct analytic behavior of G0. G0 is diagonal in the corresponding MO basis.When a mean-field approximation is selected for In practice, we make the further approximation that the self-energy difference is also diagonal in the MO basis:is field .I is the identity operator, it becomes clear that the diagonal approximation of ΔΣ will induce a diagonal approximation to G, since G and therefore also Recasting the Dyson G correspond to the zeroes of the term in between the brackets in p index that corresponds to occupied states is ϵHOMO = − IP. The HOMO energies reported in this work are obtained with this procedure, which is often referred to as the “graphical solution” of the quasiparticle equation .Let us start gradually and begin with the formulation of the HF approximation in terms of Goldstone-Feynman diagrams. In the upper panel of r) = G is the electronic density. From the Hartree Goldstone-Feynman diagram, we can immediately see that the Hartree potential is local in space and in time, since the entry and exit points are identical. The exchange diagram , labeled WTDHF.Adding more diagrams in l effect . We willW together with an unscreened interaction v. The complete second-order diagrams with two W wiggly lines has been very recently considered in Ref.  reference set.In their work providing CCSD(T) reference values for the GW100 molecules, Krause asis set . Close iThe CCSD(T) benchmark values for the ionization potentials, which have been used by all GW100 studies up to now, were done by the authors of Ref. . using aconvenient choice, it may not lead to the most accurate CCSD(T) energy. For example, in the case of in this case. Indeed, GW calculation matched the experimental ionization energy better than the “benchmark” CCSD(T) calculations did.While using the lowest energy UHF solution determined via such a stability analysis, can be a very evel see . While ievel see , so the  of Ref. , in termet al. accidentally used sulfur instead of selenium in their calculation. The most noticeable updates are SO2, MgO, cytosine, and uracil with changes larger than 0.4 eV.In this work we have re-calculated the frozen-core CCSD(T) energies for the entire GW100 set, however we chose to use GAUSSIAN 16  with de, includiet al. with an MAE of 0.09 eV. However, this correct MAE is hiding a few terrible outliers, such as SO2, MgO, cytosine, and uracil.Our revised IPs improve very much the consistency of CCSD(T) with the related method named equation-of-motion coupled-cluster (EOM-IP-CCSD). Indeed, Lange and Berkelbach  have evaNow, comparing our updated CCSD(T) to Lange’s EOM-IP-CCSD yields not only an improved MAE of 0.06 eV, but also fixes all the mentioned outliers. The deviations between the updated CCSD(T) and EOM-IP-CCSD never exceed 0.30 eV.As our updated CCSD(T) set very much improves the consistency across the methods and the comparison to experiment when experimental data are available, we have confidence that our updated values are a genuine improvement. We remind the Reader that all the numerical values are reported in the Supplemental Material.G0 in G0 is maybe not the optimal Green’s function.A noticeable source of misunderstanding between the different MBPT flavors is the starting mean-field approximation used for the non-interacting Green’s function G, best W”, is very effective for periodic systems  or on a semi-local approximation yields very decent results . In ∂Σc/∂ω is sometimes positive, which is not allowed for the exact self-energy. Remember that Z introduced in GW only contain single poles as shown for GW in Eq. 47 of Ref. ), MgO is more intriguing. It was very good with GW (−0.08 eV compared to CCSD(T)) and deteriorates very much with GW + γGW. BeO, which is chemically similar to MgO, is quite different in terms of its deviation, with a deviation of only 0.01 eV for GW + γGW with respect to CCSD(T).Finally, we evaluate the effect of the first-order correction to the Hartree and Fock exchange terms, as depicted in GW. However, we can state that with GW being already very good, it is a difficult task to improve over it. Adding diagrams may destroy the subtle balance, which makes GW so successful. Among all the additions we considered, only GW + γGW can be considered as a systematic improvement.Of course, we did not explore all the possible combinations of diagrams beyond It is attractive to calculate Green’s functions self-consistently for several theoretical reasons. First, this is a systematic way to include more diagrams . The GreHowever for practical reasons, self-consistent calculations are rarely carried out and one rather uses a one-shot approximation on top of a mean-field calculation. In the previous Section, we only used an HF mean-field for comparison reasons.G0. For molecules, it has been identified , a global hybrid functional which mixes the PBE exchange energy and the Hartree-Fock exchange energy in a 1:3 ratio . In the box plot reported in At this point, there is a cross-road between chemistry and physics methods again. When performing a perturbation theory based on a mean-field different from HF, the Brillouin theorem breaks down  and firsG0 is an approximation to the self-consistent G. As a consequence, no first-order terms appear in PT2 and some Goldstone-Feynman diagrams should be removed from the original PT3. The static diagrams (“A” diagrams in Cederbaum’s notation) are corrections to the Hartree and Fock exchange terms (See Then we consider that the PBEh(0.75) Green’s function erms See . If PBEhHence, GW@PBEh(0.75) in GW calculations of Caruso and coworkers  Green’s function. The GW + SOSEX somewhat improves compared to GW + SOSEX@HF. But it is still deteriorating the results compared to the simpler GW approximation. Next, we test GWTDDFT where W was obtained from time-dependent DFT using the same functional as for G0. Again the results are disappointing.Finally, we make an attempt at combining a better non-interacting Green’s function with the additional diagrams we tested in Section IVB. In GW + γGW. If G0 was the self-consistent GW Green’s function, the γGW diagrams would vanish. Remember that the γGW diagrams are not present in Hedin’s equations, which are obtained for a self-consistent G. GW + γGW is very similar to GW. Besides MgO, which behaves badly again, the similarity between the error distribution of GW and GW + γGW is compelling.Last, we consider In this study, we have conducted a comprehensive benchmark of the MBPT performance for the calculation of the IP of molecules. Our boxing ring was the GW100 set introduced by one of us  a few yeGW, and several methods beyond GW. Among the classical approximations, GW is clearly the winner. Then our attempts to improve over GW by adding more diagrams have been unsuccessful, besides the GW + γGW diagrams that add corrections to the Hartree and Fock exchange expectation values.Based on the same HF starting point, we evaluated the 100 HOMO energies of GW100 for PT2, PT3, GW@PBEh(0.75) with a claimed MAE of 0.15 eV.Then starting from an improved mean-field (here we chose PBEh(0.75)), deteriorates the classical approximations, PT2 and PT3. Contrarily, GW improves with a more realistic starting mean-field. Our champion is then GW is the fact that its in a featherweight class: GW, when combined with the resolution-of-the-identify, has an attractive N4 scaling. GW now routinely runs on molecular systems with several hundreds of atoms   or equatof atoms .


Zinkiewicz et al. was published as a Letter in NanoLetters.On March 8th, 2021, the manuscriptentitled “Excitonic Complexes in n-Doped WSgnexp in Numerical values placed in the column


We also give a general method for transforming any such family to an ε-almost-strongly universal family of hash functions. The latter families can then, among other applications, be used in the Wegman–Carter MAC construction which has been shown to provide a universally composable authentication method in QKD protocols. As Polynomial Hash has found many applications, our constructions and results are potentially of interest in various areas.Authentication plays a critical role in the security of quantum key distribution (QKD) protocols. We propose using Polynomial Hash and its variants for authentication of variable length messages in QKD protocols. Since universal hashing is used not only for authentication in QKD but also in other steps in QKD like error correction and privacy amplification, and also in several other areas of quantum cryptography, Polynomial Hash and its variants as the most efficient universal hash function families can be used in these important steps and areas, as well. We introduce and analyze several efficient variants of Polynomial Hash and, using deep results from number theory, prove that each variant gives an  Key establishment protocols, in which cryptographic keys are securely exchanged between parties over a public channel, usually use methods from public-key cryptography, like Diffie–Hellman key exchange (DH) and elliptic-curve Diffie–Hellman (ECDH); see  for a coε-almost-strongly universal hash function and then encrypted with a one-time pad. The application of the Wegman–Carter paradigm in QKD was originally proposed by Bennett and Brassard [ε-almost-strongly universal (ε-ASU) family of hash functions, and a secret encryption key (a sequence of random one-time pads). A message is authenticated by first hashing it with the shared hash function and then encrypting the resulting hash value with the shared encryption key (shared one-time pad). The resulting encrypted hash value, called an authentication tag, is transmitted together with the message (as a pair). Upon receiving this pair, the legitimate party recomputes and validates it. Such a MAC algorithm is information-theoretically  secure, that is, even an adversary who has unbounded computational power cannot forge the MAC with probability greater than the collision probability of the hash function family [Studying the security of QKD has become a topic of great importance .growing (rather than quantum key distribution) protocol. The Wegman–Carter MAC construction has been shown in [ε-ASU hash function families because these families are the main ingredient in the Wegman–Carter construction .Because, in the authentication of the classical channel, the legitimate parties need to share some initial small secret information in advance as described above, QKD is sometimes called a quantum key shown in  to be unshown in –21, and Polynomial Hash (PH) and its variants for authentication of variable length messages in QKD systems. Since universal hashing is used not only for authentication in QKD but also in other steps in QKD like error correction and privacy amplification [ε-almost-Δ-universal (ε-AΔU) family of hash functions which has found various important applications, for example, Galois/Counter Mode (GCM) [ε-AΔU and so can be used in various applications. Furthermore, we propose a general method by which any ε-AΔU hash function family can be transformed to an ε-ASU family. Therefore, the Polynomial Hash variants constructed in this paper can all be transformed to ε-ASU families which makes them useful for various applications including authentication of variable length messages in QKD.In this paper, following –24 we prfication , 25–30, de (GCM)  (which ide (GCM)  (which ide (GCM) –43 for vn, ε-AΔU families to ε-ASU families. In Sect. The rest of this paper is organized as follows. In Sect. n is a positive integer, p is a prime, n defined as q elements, where q is a prime power. Also, Throughout the paper, Finding (the number of) solutions of univariate and multivariate polynomial equations over fields and rings is a fundamental problem in mathematics, computer science, and related areas with many applications in various domains. In this paper, by a polynomial we mean a univariate polynomial. As a classical example, one can mention the Fundamental Theorem of Algebra which gives the exact number of solutions of polynomial equations over the field of complex numbers.Letbe a non-zero polynomial of degreewith complex coefficients. Then the equationhas, counting multiplicities, exactlydcomplex solutions. Equivalently, the field of complex numbers is algebraically closed.There are about a hundred proofs(!) of the Fundamental Theorem of Algebra . See 4646 for “osolution of the polynomial congruencen has at most n solutions. Similarly, every multivariate polynomial congruence in k variables modulo n has at most By a n)? The answer is no; there is no direct analog of the Fundamental Theorem of Algebra for polynomial congruences. Let us see some examples. The following result, proved by D. N. Lehmer [A natural question is whether the Fundamental Theorem of Algebra can be applied to the ring . Lehmer , gives a(Lehmer’s Theorem)LetThe linear congruencehas a solutionif and only ifwhereFurthermore, if this condition is satisfied, then there aresolutions.Note that the generalization of Lehmer’s Theorem to higher degree multivariate polynomial congruences is a challenging problem. In fact, even the quadratic version addressed by Cohen  has mucha and b are integers, has zero, one, or more solutions .(Lagrange’s Theorem)Given a primep, letbe a polynomial with integer coefficients such thatsaid to be of degreed). Then the polynomial congruencehas at mostdsolutions.Lagrange’s Theorem can be extended from the prime field e, e.g., ):Letbe a field andbe a non-zero polynomial of degreewith coefficients inThen the polynomial equationhas, counting multiplicities, at mostdsolutions inTherefore, it has at mostddistinct solutions inIt would be useful to compare the above results:Setting exact number of complex solutions of polynomial equations over the field of complex numbers, but Lagrange’s Theorem and Theorem upper bounds for the number of solutions over the prime field and arbitrary fields, respectively.While Theorem The proof of the Fundamental Theorem of Algebra is totally different from the proof of Lagrange’s Theorem and Theorem The following observations are useful, specially when discussing the Polynomial Hash and its variants: absolutely distinct if Note that Lagrange’s Theorem does not hold for composite moduli. For example, the quadratic congruence Vandiver  obtained(Vandiver’s Theorem)Letbe a commutative ring with identity. Letbe a polynomial with coefficients insuch thatThen the polynomial equationhas at mostdabsolutely distinct solutions.a and b are said to be absolutely incongruent modulo n if n.Taking Vandiver  derived . For generalization to prime power moduli, an upper bound for the number of solutions can be obtained using the following result .Supposeis an integer andsis a solution of the polynomial congruenceThen we have the following cases: Ifthenscan be lifted in a unique way fromtoThat is, there is a uniquewhich generatessand which satisfies the polynomial congruenceIfthen: Ifscan be lifted fromtoinpdistinct ways.Ifscannot be lifted fromton, letd). Denote by Given a positive integer IfthenThe proof easily follows from the basic properties of congruences. □Therefore, by Lemma Using Lagrange’s Theorem and Theorem Letbe an integer. Thenp there will be 0, 1, or p solutions modulo p solutions modulo Clearly, if Is there a better upper bound for  Stewart , and to  Stewart . But we  Stewart , 45 (in (Konyagin’s Theorem)Letbe an integer. ThenFurthermore, ifandthenSo far, we have very good upper bounds for the number of solutions of polynomial congruences modulo prime powers. Now, we generalize these upper bounds to arbitrary moduli. For this we need the following tool .Letbe a polynomial with integer coefficients. Also, letbe positive integers, pairwise coprime, and letThen the polynomial congruencehas a solution if and only if each of the polynomial congruenceshas a solution. Moreover, ifanddenote the number of solutions of  is at most D do not collide too often). Formally, H is universal if for any two distinct H is an ε-almost universal (ε-AU) family of hash functions if for any two distinct ε-AU family, for a sufficiently small ε, is close to being universal.The family R is a finite additive Abelian group. The family H is a Δ-universal family of hash functions if, given a randomly chosen hash function from H, the difference of the hash values of any two distinct elements of D is uniformly distributed in R. Formally, H is Δ-universal if for any two distinct H is an ε-almost-Δ-universal (ε-AΔU) family of hash functions if for any two distinct k, the operation ‘−’ can be replaced by ‘⊕’ (XOR), and H is also called ε-almost XOR universal (ε-AXU) or ε-otp-secure.Suppose H is a strongly universal (or 2-wise-independent) family of hash functions if, given a randomly chosen hash function from H, the hash values of any two distinct elements of D are independent and uniformly distributed in R. Formally, H is strongly universal if for any two distinct H is an ε-almost-strongly universal (ε-ASU) family of hash functions if for any two distinct h from H, R), andh from H, hard to guess even if The family H is ε-ASU if for any two distinct Equivalently, Let H be an ε-AU family of hash functions, which maps blocks of length 2l to blocks of length l. At each round of tree hash, the message is split into blocks of length 2l and each block is hashed with some n rounds of tree hash, the collision probability is Because many universal hash functions only work on fixed length messages, it is often necessary to extend the domain of the hash function to work on longer messages. Wegman and Carter  introduc1−(1−ε)n . HoweverForletbe almost-universal families of hash functions whereand defineThenHhas the following property: IfisAU andisAU, thenHisAU.IfisAU andisAΔU, thenHisAΔU.IfisAU andisASU, thenHisASU.ε-AU hash family with an ε-AΔU or ε-ASU hash family to create an efficient ε-AΔU or ε-ASU family. We can also use this result to create a Merkle–Damgård like paradigm for universal hash functions. Let ε-AU and let l for any positive l, and is The last two parts of this result can be used to pair an efficient d Tsunoo , and a md Tsunoo .H is an ε-AU family of hash functions, then by hashing a message with two independent keys and concatenating the results, the probability of collision is lowered to n hash values to get a collision probability of n, the size of the key only increases by n values. Not only does this save key material, it reduces memory accesses, thus potentially improving performance.Often the collision probability of a hash function may be larger than desired. For this reason, there are several techniques for reducing the collision probability of a hash function family. If ms c.f. , 64), re, reH is ε-AΔU families to ε-ASU families. Because for authentication in QKD systems we need efficient ε-ASU families, our result implies that constructing such families boils down to constructing efficient ε-AΔU families. Our result is a generalization of the following result by Etzel et al. [Now, we prove a general result for transforming l et al.  which seLet the familybe a Δ-universal family of hash functions, whereKis the key space andRis a finite additive Abelian group. Then the familywhereand ‘+’ denotes the group addition operation, is strongly universal.In order to generalize the above result, we also need the following result (see ):LetGbe an Abelian group, and letbe independent random variables which take on values inG. If one ofis uniformly distributed inG, then the sumis also uniformly distributed inG.G, under which the sum G.More generally, Sherstnev  gave necNow, we are ready to prove our result.Let the familybe anε-almost-Δ-universal family of hash functions, whereKis the key space andRis a finite additive Abelian group. Then the familywhereand ‘+’ denotes the group addition operation, isε-almost-strongly universal.H is ε-almost-Δ-universal, we haveFor any two distinct ε-AΔU family of hash functions which has received much attention is Polynomial Hash (PH), which is used for hashing variable length messages. The idea is that we put the message blocks as the coefficients of a polynomial and then evaluate the polynomial at the secret key, where all operations are done in a specific field or ring. In this section, we introduce and analyze several efficient variants of Polynomial Hash and then compare our results with available results. As Polynomial Hash has found many applications, our constructions and results might be of interest in various areas.An Here we introduce five variants of Polynomial Hash  and analyze their universality using results from Sect. Polynomial Hash Over Ring of Integers Modulon(PH-IM): In this family, message blocks n, the key x is in (PH-IM)Given an integer Lethas the prime factorizationwhere’s are prime andfor alli. Then we have the following cases: The family PH-IM isalmost-Δ-universal.Ifnis square-free, then the family PH-IM isalmost-Δ-universal.Ifandfor alli, then the family PH-IM isalmost-Δ-universal.i, then the polynomial congruenceWe only prove the last part; the proofs for other parts are similar. Let Polynomial Hash With Probability At LeastZero Collision (PH-ZC): Let p be an odd prime and k be a positive even integer not divisible by p. Denote the set of even elements of E and the set of odd elements of O. In this family, message blocks E or are all in O, the key x is in E; the definition and the result over O are similar. Formally,(PH-ZC)p and a positive even integer k not divisible by p, we defineGiven an odd prime The family PH-ZC isalmost-Δ-universal. Furthermore, with probability at leastthe collision probability is exactly zero.b is odd then the polynomial congruenceb is even then it has at most k solutions. On the other hand, by Lagrange’s Theorem, for every d solutions. Now, using Theorem b is odd, and has at most kd solutions if b is even.Let b is odd and possibly in other cases : In this family, each message block x are in (PH-PF)p,Given a prime The family PH-PF isalmost-Δ-universal.Same as above, just use Lagrange’s Theorem or Theorem arbitrary non-negative integers as long as no two messages have all their corresponding blocks congruent modulo p or modulo n. See Theorem It is important to note that we do not have to restrict the message blocks to be in Polynomial Hash Over Prime Fields With Arbitrary Message Blocks (PH-PA): Let A be a subset of A have all their corresponding coordinates congruent modulo p. In this family, each message m is in A, the key x is in (PH-PA)p,Given a prime The family PH-PA isalmost-Δ-universal.p (by the definition of the set A) so the assumption of Lagrange’s Theorem is satisfied. Also, note that Theorem Same as above, just use Lagrange’s Theorem. Note that when we find the difference of the two polynomials, at least one of the coefficients is non-zero modulo Polynomial Hash Over Finite Fields (PH-FF): In this family, each message block x are in (PH-FF)q elements, where q is a prime power,Given the finite field The family PH-FF isalmost-Δ-universal.Same as above, just use Theorem Using Theorem anyε-AΔU family, in particular the families studied in this paper, can be transformed toε-ASU families which makes them useful for various applications including authentication of variable length messages in QKD. This can be done by adding a uniform valueto the hash functions, whereRis the range of the corresponding hash functions.Polynomial Hash is widely attributed to Wegman and Carter , Dietzfeε-almost-Δ-universality of these families. As discussed in detail in Remark So far, only the families PH-PF and PH-FF have been introduced in the literature but, unfortunately, there is a growing number of papers that explicitly or implicitly have used the Fundamental Theorem of Algebra to prove the d,d multiplications and d additions since, by Horner’s rule, d multiplications and d additions, while hashing the same message using most other universal hash function families needs more computations, for example, hashing it using MMH∗ [The evaluation of a polynomial ing MMH∗ , 72, 73 ∗ and its variants) which hash fixed length messages  Polynomial Hash can be used for hashing variable length messages because each message block becomes the coefficient of the polynomial, and so is independent of the key.Unlike most other universal hash function families   . See Remark We do not have to restrict the message blocks to be in In some of these variants with probability at least We do not need large prime numbers or finite field arithmetic anymore  using results from Sect. Although in QKD the legitimate parties need to share some initial small secret information in advance for the authentication of the classical channel, each round of QKD provides substantially larger fresh key materials, part of which can be used for authentication in the next round of QKD. Furthermore, keys generated in each round of QKD are completely independent of all prior keys and messages , 6, 74. We connected Polynomial Hash and QKD with deep results in number theory. This may motivate more work in these areas.Our study of Polynomial Hash over The above techniques and results on the Polynomial Hash and its variants and comparing them with what were known before, reveals some remarks:


As a result, we obtained the p-type thermoelectric figure of merit up to 0.96 at 300 K, indicating that the boron monolayer is a promising p-type thermoelectric material.A heavy element is a special character for high thermoelectric performance since it generally guarantees a low lattice thermal conductivity. Here, we unexpectedly found a promising thermoelectric performance in a two-dimensional semiconducting monolayer consisting of a light boron element. Using first-principles combined with the Boltzmann transport theory, we have shown that in contrast to graphene or black phosphorus, the boron monolayer has a low lattice thermal conductivity arising from its complex crystal of hexagonal vacancies. The conduction band with an intrinsic camelback shape leads to the high DOS and a high  ZT = S2σT/κ, where S is the Seebeck coefficient, σ is the electrical conductivity, κ is the thermal conductivity including both electronic and lattice contributions, and T is the absolute temperature, respectively  light elements with high vibration frequency and 2) large atomic weight difference forbids the anharmonic scattering. In this regard, we intended to think that is there possibility to achieve promising thermoelectric transport in other single-layer crystal consisting of light elements? In recent years, boron, one of the carbon’s nearest neighbors, demonstrated the polymorphism in two-dimensional crystals, which are called borophene. However, most of the boron monolayers were found to be metallic by experiment or theory. Interestingly, Hu et al. recently proposed a series of semiconducting boron monolayers formulated by the connected network of hexagonal vacancies   using projector-augmented wave (PAW)  pseudopoon (mBJ)  functionn-type doping while down for the p-type. However, within this approximation, the Seebeck coefficient can be calculated independent of carrier relaxation time τ, while the evaluation of electrical conductivity still requires the knowledge of τ. In this regard, we employed deformation potential theory based on effective mass approximation to calculate τ  under a constant relaxation time approximation as implemented in BoltzTraP . A rigidculate τ . At lastculate τ  package culate τ , and a cet al  is located in the interval between Γ and Y points. In consistent with previous calculation, the VBM is dominated by the out-of-plane pz orbitals while the CBM is attributed to the in-plane s+px,y orbitals   . Thus, tow 1ps−1 . Figure ZT values for both the n- and p-doped boron monolayer as a function of the carrier concentration at room temperature. Obviously, the p-type thermoelectric performance is superior to n-type due to the excellent p-type power factor. Combined with the relatively low lattice thermal conductivity, the optimal p-type ZT value of the boron monolayer reaches 0.96 at an optimal carrier concentration of about 1 × 1012 cm−2, which is a recorded value among single-layer materials consisting of light elements. Our results indicate that the semiconducting Combining the electronic and phonon transport properties, we evaluated the thermoelectric performance of the n-type Seebeck coefficient arises from the camelback shape of the lowest conduction band, while the highly degenerate valence band with small effective mass leads to the high hole mobility and long relaxation time, which contributes to the superior hole transport performance. Importantly, we found relatively low lattice thermal conductivity in the boron monolayer, ∼20 W/mK at 300 K, as compared with graphene or phosphorene also consisting of a light element. This is primarily ascribed to the complex unit cell with the hexagonal vacancy. Finally, we obtained an optimal p-type ZT of about 0.96 at 300 K in this boron monolayer, indicating its potential as p-type thermoelectric materials.We have investigated the thermoelectric performance of a semiconducting


Communications Biology 10.1038/s42003-021-02449-8, published online 29 July 2021.Correction to: In the original version of the Article, the title contained an error and incorrectly spelt “polyploid” as “polypoid”. This has now been corrected in the PDF and HTML versions of the Article.


Nature Communications 10.1038/s41467-020-19264-0, published online 02 November 2020.Correction to: y-axis was incorrectly labelled as ‘ApoE in RIPA’ instead of ‘ApoE in FA’. This has been corrected in both the PDF and HTML versions of the Article.The original version of this Article contained an error in Fig. 5b, in which the


Translational Psychiatry 10.1038/s41398-021-01257-2, published online 18 February 2021Correction to: The original version of this article unfortunately contained an error. The authors recommend adding a phrase at the end of the section “Gene–environment interactions, epigenetic factors, and demographic characteristics”.In doing so, the phrase “for instance, the aforementioned interaction between 2R- and 3R-repeat MAO-A variants  and child abuse was associated with violence and antisocial behavior only in Caucasians but not in individuals from other ethnic group” changes to “for instance, the aforementioned interaction between 2R- and 3R-repeat MAO-A variants  and child abuse was associated with violence and antisocial behavior only in Caucasians but not in individuals from other ethnic group; however, as this study did not control for gender, it is likely its results could be pertinent to non-Caucasians, as well”. The original article has been corrected.


A number of methods have been proposed for face reconstruction from single/multiple image(s). However, it is still a challenge to do reconstruction for limited number of wild images, in which there exists complex different imaging conditions, various face appearance, and limited number of high-quality images. And most current mesh model based methods cannot generate high-quality face model because of the local mapping deviation in geometric optics and distortion error brought by discrete differential operation. In this paper, accurate geometrical consistency modeling on B-spline parameter domain is proposed to reconstruct high-quality face surface from the various images. The modeling is completely consistent with the law of geometric optics, and B-spline reduces the distortion during surface deformation. In our method, 0th- and 1st-order consistency of stereo are formulated based on low-rank texture structures and local normals, respectively, to approach the pinpoint geometric modeling for face reconstruction. A practical solution combining the two consistency as well as an iterative algorithm is proposed to optimize high-detailed B-spline face effectively. Extensive empirical evaluations on synthetic data and unconstrained data are conducted, and the experimental results demonstrate the effectiveness of our method on challenging scenario, e.g., limited number of images with different head poses, illuminations, and expressions. Point cloud, skeleton, and mesh grid are the widely used man-made shape type for face reconstruction. Lu et al.  present Ii from a camera is −1 denotes the inverse operator of Π. The camera operator i = 1, 2, …, n. And i of BP imaging from si, Ri, and ti denote scale, rotation, and translation parameter in projection Πi. The ith projection operation is simplyOur problem modeling is illustrated in Ri,  expresses the first two rows of Ri.F−1 denote the inverse operator of F. It is thus clear that different embedding F determine different face shapes. According to the geometric optics of BP imaging, a image point via the operatorIi in the i-th view is mapped to surface S, and then is mapped to texture space byi, i = 1, 2, …, n generate discrete and inconsistent rays mapping in texture space because of the discrete and different images domains, as well as the noises, seen in Let F according to from multiple images. If all images are the captures of a same Generally, the problem is how to determine # is a composition operator of fitting and sampling, to handle the inconsistency. It firstly fits a texture function based on the discrete texture and parameters mapped from one image, and then samples texture intensity values at unified parameter points {uj}j=1 : Np.First, that satisfiesn and 1st-order partial derivative in the first formulation, and the equivalence relation among albedo ρ, normal n, light direction l, and image intensity Second, it satisfiesWe refer to Equations (5) and (6) as 0th- and 1st- order consistence equations in 3D surface reconstruction respectively. Generally, researchers solve any one of the two consistence problem to reconstruct 3D surface, classically, by multi-view stereo . The surface function isRm,n(u) = Nm,4(u) · Nn,4(v) andThe human face is assumed to be a uniform B-spline surface F is C2, meaning that it can approximate the true shape in arbitrary uv precision with deterministic k-ordered partial derivative k = 1, 2, and b denotes a 3MN × 1 vector storing B-spline control points, and T|u denotes a sparse 3 × 3MN matrix stacking the 0th-order coefficients at parameter We give a more brief formulation of 0th-order representation as follows:f landmark points on human face as shown in u(li) is the parameter point of the i-th feature point, i = 1, 2, …, f. The landmarks cover a sparse structure of face.In fact, we needn't consider all 3D points mapping to 2D images when estimating a operator Π. Instead, we only consider F w.r.t u and v areThe 1st-order partial derivatives of T1|u and T2|u denote the matrixes stacking the 1st-order coefficients w.r.t u and v, respectively.Similarly, we give a more brief formulation of 1st-order partial derivative as follows:u can be computed by the cross productTherefore, the surface normal vector at u and v, respectively areT11|u and T22|u denote the matrixes stacking the 2nd-order coefficients w.r.t u and v, respectively. The 2nd-order information can be used for smooth control during optimization.And similarly, the 2nd-order partial derivatives w.r.t. Based on face surface embedded with B-spline function, we present the pinpoint 0th- and 1st-order geometric consistency conditions in the following section.F by solving 0th-order consistence of Equation (5) or 1st-order consistence of Equation (6). Generally, two consistency conditions are combined for face reconstruction considering that estimating abundant consistent points in images is limited and that the estimated normals are unfaithful. Furthermore, how to obtain the accurate registration of 0th- and 1st-order information is the most important to high-detailed B-spline reconstruction.Reconstruction problem is to compute The well-registered textures are low-rank structures of the back projection texture charts. But in practice, they can be easily violated due to the presence of partial occlusions or expressions in the images captured. Since these errors typically affect only a small fraction of all pixels in an chart, they can be modeled as sparse errors whose nonzero entries can have arbitrarily large magnitude.ei represent the error corresponding to image Ii such that the back projection texture charts F, and free of any corruptions or expressions. Also combining with 0th-order representation of B-spline face in Equation (7), the formulation (5) can be modified as follows:E = .Let However, the solution # to sample Np dense parameter points {uj}j=1 : Np on the domain The resolution of reconstruction is determined by the density of correctly estimated normals. To enhance the resolution of B-spline surface, we use operator (·)i = 1, 2, …, n and j = 1, 2, …, Np.Then the well-registered and dense texture are obtained bynj as well as light li can be computed from the shading (intensity) of charts According to Lambertian illumination model seen in Equation (6), dense normals Finally, the high detailed reconstruction must satisfyThe problems of both 0th-order consistence and 1st-order consistence are difficult to solve. For , Jacobian matrices w.r.t. i}i=1 : n, surface parameters F (or b), and texture parameters D) need to be computed, but they are difficult to be solved simultaneously. We adopt to optimize them by turns, instead.In Equation (6), three kind of parameters including camera parameters {Πsi, rotation Ri and translationti of landmarks for each image Ii, i = 1, 2, …, n based on the and SVD method .According to linear transformation from 3D to 2D in Equation (1), we can estimate scale f denote a 2nf × 1 vector stacking f landmarks of n images, and P denote a 2nf × 3f projection matrix stacking n views of parameters siRi, , and t denote a 2nf × 1 vector stacking f translation. The update of b can be implemented by solving:#l is a sampling operator that selects B-spline coefficients of landmarks at parameters {u(li)}i=1 : f, and (·)# selects B-spline coefficients at {uj}i=1 : Np. In fact, T#l is a 3f × 3MN matrix that stacks T|u(li), i = 1, 2, …, f, and f × 3MN matrix that stacks T11|uj (or T22|uj), j = 1, 2, …, Np.Let b and b0. It helps eliminate affect of geometric rotation brought by 0st-order warping, and guarantee a smoothness changing during optimization. Particularly, ζ cannot be too small, otherwise a fast changing may bring a local optimal.The second item also work as a regularization measuring the distance of local information between faces i is determined by Equation (2) when Πi and b is known. Then texture chart with noise is obtained by applying consistent parameter sampling i = 1, 2, …, n, and j = 1, 2, …, Np.Firstly, texture charts based photometric stereo method is used to estimate the local normals. Secondly, a normals driven optimization strategy is proposed to optimize the B-spline face.n texture charts are input into Mn×Np for estimating the initial shape M = LSvia SVD  can be rewritten asNp × 3Np diagonal matrix that stores 3Np reciprocals of lengths of the normals {nj}j=1 : Np; and (·)# is a selection operator that selects 3Np rows of 1st-order coefficients at parameter {uj}j=1 : Np; and b0 represent the control points of a B-spline template face. Particularly, symbol ⊗ denotes a composite operator of cross product, which makes w ⊗ v = , where w and v are 3Np × 1 vectors containing Np normals.We normalize h may not guarantee an unique solution of high-dimension b; and (2) the system is not simply linear, which is difficult to be solved. Therefore, a frontal constraint based on template b0 is applied to make a unique solution; And a strategy of approximating to linearization is also proposed to make a linear solution.However, there exists two issues: (1) the low-dimension x- and y-component:T#xy stacks 0th-order coefficients at parameter {uj}j=1 : Np corresponding to x- and y- components. Operator (·)#sxy also sets the coefficients corresponding to z- components to zeros.The frontal constraint is a distance measurement condition between surface O1 is not a simple linear form, for which an approximating to linearization is proposed.Particularly, the first item O1 can be rewritten as a linear-like formulation:According to the characteristics of the cross-product ⊗, the first item in ⊗ makes a 3Np × 1 vector Np × 3Np sparse matrix [w]⊗ = diag, where Particularly, the operation [·]b is a known parameter, e.g., as b0, for L|b, the minimization of ||h − L|b0 · b|| will be a linear system. That is also true for R|b.If h − L|b0 · b|| to optimize the control points in parameter space of v by fixing u, and use ||h − R|b0 · b|| to optimize in parameter space of u by fixing v.In fact, we can use formulation ||u and v parameter spaces by turns. The two iteration items are rewritten as1|b0 (or Λ2|b0) is a 3Np × 3Np diagonal matrix that stores 3Np reciprocals of lengths of tangent vector uj}j=1 : Np. During this procedure b0 is updated step-by-step. As shown in u, v) are updated until n.A practical skill is to optimize the control points on O2, the final formulation of optimization consists of two items as follows:b0 is initialized by value of b0. Then we can solve b and update b0 orderly by minimizing (a) and (b) in Equation (18) iteratively until convergence.By integrating with An iterative algorithm is presented for this practical solution in Algorithm 1. Processes of 0th-order consistence and 1st-order consistence are separately conducted in the inner loop. And the outer loop guarantees a global convergence on two consistence problem.n2f2MN), and that of both Equations (18.a) and (18.b) are O, (18.a), and (18.b), SVD for estimating camera parameter, and Robust PCA for Equation (17). In detail, the computational complexity for solving Equation (16) is O(In this section experiments are presented to verify our automatic free-form surface modeling method. We first describe the pipeline to prepare a collection of face images of a person for B-spline face reconstruction. And then we demonstrate the quantitative and qualitative comparisons with recent baseline methods on projected standard images from ground truth 3D data (Zhang et al., The ground truth data are from the space-times faces (Zhang et al., The wild data (Huang et al., To verify the accuracy of automatic surface reconstruction, discrete points are sampled from the generated continuous free-form shape, and are compared to the traditional discrete reconstructions, e.g., work by Kemelmacher and Seitz  and RothS1, S2, S3, and S4 by using their corresponding images, and the fifth session S5 is based on images with different expressions. Each session contains 40 images with various illumination and different poses. Reconstruction results are compared with the re-implemented method Kemel_meth by Kemelmacher and Seitz (We conduct five sessions of reconstructions: the first four are used to reconstruct expression nd Seitz  and Rothnd Seitz . Kemel_mTo compare the approaches numerically, we compute the shortest point-to-point distance from ground truth to reconstruction. Point clouds are sampled from B-spline face and aligned according to absolute orientation problem. As done in work of Roth et al. , mean EuThe visual results in C2 differentiable parametric surface while the other two are discrete model. Conclusively, 0th- and 1st-order consistency modeling using B-spline surface is efficient to reconstruct parametric surface of individual face.We give statistics of characteristics of the results generated by the three methods in Our method is also tested based on real unconstrained data. Unconstrained data mean that the images are captured under uncertain condition, and the faces in the images are different in expression, pose and illumination condition. It is difficult to build the geometrical consistency for reconstruction using such data. Unlike the experiments in the work by Kemelmacher and Seitz  using huA label the results generated by the reimplemented Kemel_meth, and let B label the results generated by the reimplemented Roth_meth, and let C label the method Seta_meth of deep learning by Sela et al. (D label our results. Particularly, the input for Seta_meth is one image selected from the 35 images. Images in column 1, 5, and 8 are corresponding mean textures and two views of images respectively. By comparing these results, we observe some phenomena as follows:A and D show more vivid details than B, e.g., eyes and nose of Colin Powell. But in an other viewpoint, D shows more credible shape than A, e.g., the eyes and the forehead of Colin Powell, and the forehead and the mouth of Donald Rumsfeld.In frontal viewpoint, A loses the local information completely, but B, C, and D still maintain general geometrical shape of face. For all methods, reconstructing nose is a challenge because the geometric curvature of the nose varies greatly. When the images are not enough, the noise could be amplified. So B shows bad results at nose being limited by number of input images.When the normals are incorrectly estimated from a limited number of images, e.g., for Gloria Macapagal Arroyo, C is a approximately frontal face image selected. As the model of C is learning on a set of 3D face data, it may not handle the uncertain noise and identity of inputs. So the details in reconstruction by C don't look real, although their global shapes are stable and like human faces.The input of By comparison, our method steadily produces better looking results than others from different viewpoints in the dataset. Clear and vivid details can be seen at key components such as eyes, nose and mouth, forehead, and cheek.Visual face reconstructions for Colin Powell, Donald Rumsfeld, George W. Bush, Hugo Chavez, and Gloria Macapagal Arroyo are compared with other two methods, as shown in a et al.  and let B looks better. It could deal with 3D reconstruction with limited number of images. Because we may not obtain large amount of images for reconstruction as done by Roth et al. (A are mainly resulted from the inauthentic depth generated by integration method. And the bad results of B are caused by that the mesh template cannot build correct geometric consistency of number limited of unconstrained images and that the discrete differential operating on estimated noisy normal brings distortion errors. In contrast, we build pinpoint geometric consistency using B-spline surface. B-spline can smooth the noise in estimated normal better. So D can reconstruct correct face shape with little distortion, showing better result as a whole.All the above experiments prove that our method can build pinpoint geometrical consistency on the limited number of real unconstrained data. Our method may not be best method in area of 3D reconstruction from multiple images, as the results in the original work by h et al. , for somIn the comparison, we don't consider other deep learning methods based methods appeared in recent years (Dou et al., This study set out to present high-detailed face reconstruction from multiple images based on pinpoint 0th- and 1st-order geometric consistence using B-spline embedding. Based on the good consistence modeling in geometric optics, the method works well for data with different poses and expressions in the wild. The key contribution of this study is that surface modeling adapts the correct rays in geometric optics by using B-spline embedding. This makes the high-detailed B-spline modeling from a number limited of face images captured under wild condition become reality. The method could also be applied to expression tracking and assisting face recognition in a monitoring or robot system.The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.WP and ZF has contributed equally to the core idea as well as the experiment design and results analysis. YS, KT, and CX has provided assistance in experiments and analysis, under ZF's supervision. Besides, KT and MF provided the research group with financial support and experimental equipments. KT and ZF are supportive corresponding authors. All authors contributed to the article and approved the submitted version.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.


Formulae for the c-subdifferential and biconjugate of the objective function of a general optimization problem are provided, too. We also characterize the total duality by means of the saddle-point theory for a notion of Lagrangian adapted to the considered framework.We present new results on optimization problems where the involved functions are evenly convex. By means of a generalized conjugation scheme and the perturbation theory introduced by Rockafellar, we propose an alternative dual problem for a general optimization one defined on a separated locally convex topological space. Sufficient conditions for converse and total duality involving the even convexity of the perturbation function and  Another related interesting problem in conjugate duality theory is the notion of converse duality. It corresponds to the situation where there is no duality gap and the primal problem is solvable. This issue was investigated in the convex setting in [An important part of mathematical programming from both theoretical and computational points of view is the duality theory. Rockafellar (cf. ) developtting in  for Fenctting in .c-conjugation scheme and it was developed in [c-conjugation scheme is associated with the even convexity of such a function. Evenly convex sets (functions) are a generalization of closed and convex sets (functions), see, for instance, [evenly convex, e-convex in brief, if it is the intersection of an arbitrary family, possibly empty, of open halfspaces. Such sets have been employed to study the solvability of semi-infinite linear systems containing infinitely many strict inequalities in [e-convex if its epigraph is e-convex. According to [c-conjugation scheme is suitable for this class of functions in the sense that the double conjugate function equals the original one if it is proper and e-convex. This result can be seen as the e-convex counterpart of the celebrated Fenchel–Moreau theorem. Concerning the usage of e-convexity in finance mathematics and consumer theory, we refer to [c-conjugate scheme in the direction of formulas for asymptotic functions and multi-marginal monotone sets can be found in [In this paper we propose a new dual problem to a general primal one defined on locally convex spaces by means of a generalized conjugation scheme and study converse duality for this primal–dual pair. This pattern, which is inspired by a survey done by Martínez-Legaz where generalized convex duality theory is applied to quasiconvex programming, is called loped in . In the loped in  – the usnstance, . Fenchelnstance,  a set evities in , whereasities in . Due to ities in , an exterding to , the c-crefer to ,12. Somefound in ,14, respc-conjugate duality theory, and the counterparts of some regularity conditions, i.e. conditions ensuring strong duality, from the Fenchel conjugate setting were obtained. Moreover, in [The theory developed in  motivateover, in  regulariover, in  we analyover, in  the Fencover, in  we used over, in  to derivtotal duality  from this general perspective. Moreover and extending in this way results from the convex setting [c-subdifferential and biconjugate of the objective function of a given general optimization problem. Motivated by [The purpose of the paper is twofold. First, we analyse the fulfillment of converse duality for a primal–dual pair expressed via the perturbation function, where its even convexity will play a fundamental role. In order to avoid repetitive arguments, we have decided to derive the aforementioned converse duality in the general form, leaving as an application the natural particularizations into the previously mentioned special cases. As a second target, we address the problem of  setting , we provvated by , we highc-subdifferentials which allow to characterize total duality for a general primal–dual pair. Moreover, the ϵ-c-subdifferential of the objective function of the considered problem is expressed via the ϵ-c-subdifferential of the considered perturbation function. Last but not least in Section The layout of this work is as follows. Section 2.X be a nontrivial separated locally convex space, lcs in brief, equipped with the Y be another lcs. By Let evenly convex, e-convex in short, if for every point e-convex hull of C, X containing C. For a convex subset X is a separated lcs, X is e-convex and every closed or open convex set is e-convex as well.According to , a set Ceffective domain of f and by epigraph and its graph, respectively. We say that f is proper if lower semicontinuous hull of f, which is the function whose epigraph equals f is lower semicontinuous, lsc in brief, if for all e-convex if For a function n see in , Ex. 2.1e-convex hull of a function f. Based on the generalized convex conjugation theory introduced by Moreau [the coupling functionsc-conjugate of f, c of g, The y Moreau , a suitay Moreau . Let us c-elementary, and, in a similar way, functions of the form . In [X to c-elementary functions. Similarly, a function e if it is the pointwise supremum of sets of e of an extended real function g defined on W, denoted by g such that D.Functions of the form tary. In  it is shThe following counterpart of the Fenchel–Moreau theorem for e-convex and eTheorem 2.1f is e-convex if and only if g is eLet recession cone of a nonempty convex subset The following lemma and proposition were shown in the finitely dimensional case in ,26, respLemma 2.2Let Proposition 2.3Let For convenience in Section Definition 2.4functionally representable if A non-empty e-convex set Remark 2.1A non-empty e-convex set 3.c-conjugate Let lines of , we consinfimum value functionGDc). In , via theIn order to be able to provide a counterpart of (C5) for the pair ms – see ,18,20. RTheorem 3.1It always holds Proof.Let us take W. Then,)ction in )cNext we present the counterparts of Lemma 5.1, 5.2 and 5.3 from  that areLemma 3.2Proof.Fixing It holdsWe continue with a Moreau–Rockafellar-type result involving the Lemma 3.3Proof.We have, for all Lemma 3.4If the e-convex hull of Proof.p,C is functionally representable by assumption, due to Remark First, we see that  Theorem epiΨ+fcc′(x)Remark 4.1f is e-convex, f is not necessary in one sense of the equivalence.As a consequence of [total duality for We apply these results to ing (see ), total Proposition 4.6Let Proof.If  Furtherv(GP)≤Φ.Proposition 5.4One always hasProposition 5.5L if and only if Let us assume that Proof.L. From [Let  L. From , Sec.3 iL. From [−Φc) andΦ=suL.For the converse statement, since n, using  and 19)v(GP)=Φ=in6.ϵ-duality statements from [In this paper we present new results regarding evenly convex (e-convex) functions, in particular converse and total duality statements for e-convex problems that extend their counterparts from the  convex case. Other results can be generalized to the current setting as well, for instance the nts from , howevernts from , Ex. 2.1c-subdifferentials and c-subdifferential and biconjugate of the objective function of a given general optimization problem are provided, too. On the other hand, we extend the definition of the classical Lagrangian towards the e-convex setting by means of the c-conjugation scheme and relate the corresponding saddle-points to total duality. The results for general optimization problems can be specialized for constrained and unconstrained optimization problems as well.We extend in this article the notion of converse duality from the convex setting to e-convex optimization problems, providing a sufficient closedness-type regularity condition for it and an alternative formulation using the infimum value function. In order to prove the mentioned results we introduced the notion of functionally representable functions and we also gave a new Moreau–Rockafellar type result for ec- and c-conjugation, other paths can be followed along this direction. On the other hand, the investigations from [We have restricted ourselves to closedness type regularity conditions, but taking into consideration , Sect. 3ons from  on subdi


We study the problem of communicating over a discrete memoryless two-way channel using non-adaptive schemes, under a zero probability of error criterion. We derive single-letter inner and outer bounds for the zero-error capacity region, based on random coding, linear programming, linear codes, and the asymptotic spectrum of graphs. Among others, we provide a single-letter outer bound based on a combination of Shannon’s vanishing-error capacity region and a two-way analogue of the linear programming bound for point-to-point channels, which, in contrast to the one-way case, is generally better than both. Moreover, we establish an outer bound for the zero-error capacity region of a two-way channel via the asymptotic spectrum of graphs, and show that this bound can be achieved in certain cases. Shannond DM-TWC ,4,5, andd DM-TWC ,7. Howevd DM-TWC ,9,10.In yet another seminal work, Shannon proposed and studied the zero-error capacity of the point-to-point discrete memoryless channel , also knn uses of the DM-TWC codebook pair. Note that the encoding functions depend only on the messages, and not on the observed outputs during the transmission, hence the name non-adaptive. When transmissions end, Alice and Bob observe the resulting (random) channel outputs . A rate pair achievable for the DM-TWC if an n. The non-adaptive zero-error capacity region of a DM-TWC non-adaptive zero-error sum-capacity of a DM-TWC The problem of non-adaptive communication over a DM-TWC can be formulated as follows. Alice and Bob would like to simultaneously convey messages The main objective of this paper is to provide several single-letter outer and inner bounds on the non-adaptive zero-error capacity region of the DM-TWC. The remainder of this paper is organized as follows. In V and edge set E. Two vertices adjacent, denoted as independent set in G is a subset of pairwise non-adjacent vertices. A maximum independent set is an independent set with the largest possible number of vertices. The size of a maximum independent set in G is called the independence number of G, denoted by complement of a graph G, denoted by G. We write n vertices, respectively.Let strong product  G and H is a graph such that(1)the vertex set of (2)two vertices The n-fold strong product of graph G with itself is denoted as Shannon capacity of graph G was defined in [Let (G)×V(H);two vertidisjoint union G and H is a graph such that graph homomorphism from G to H, denoted by G, then H. We write G to the complement of H.The In , ZuiddamTheorem 1. ⊔ and the strong product ⊠, and also contains the graph with a single vertex Let (1) if (2) (3) (4) Then, . Let G As remarked in , 2Θ(G) i of Alon , and als of Alon . In SectIn this subsection, we characterize the zero-error capacity of a discrete memoryless point-to-point channel, as well as the zero-error capacity region of a DM-TWC, in terms of suitably defined graphs. The point-to-point characterization is well known and goes back to Shannon , and thememoryless in the sense that ith channel use. Suppose that a transmitter would like to convey a message , and such a code is called uniquely decodable if R is called achievable if an n. The zero-error capacity of the channel is defined as the supremum of all achievable rates.A discrete memoryless point-to-point channel consists of a finite input alphabet confusion graph G, whose vertex set is the input alphabet n-fold strong product of graph G. Consequently, the zero-error capacity of a point-to-point channel is equal to the Shannon capacity of its confusion graph G. Note that there are infinitely many distinct channels with the same confusion graph, and all of these channels have the same zero-error capacity.A channel We now proceed to similarly associate a DM-TWC with a collection of confusion graphs, which would then be shown to characterize its zero-error capacity region. To that end, note that when Alice sends a letter Proposition 1.Consider a DM-TWC In particular, we see that the capacity region Proposition 2.If This further immediately implies:Proposition 3.The strong product of two DM-TWCs Lemma 1.Proof. To prove this lemma, it is sufficient to prove that, for any dual graph homomorphism from (1)if (2)if It is easy to see that the dual graph homomorphism is a natural generalization of the standard graph homomorphism of two graphs, in the sense that they are both adjacency preserving. We write In this subsection we study the behavior of the zero-error capacity region of a DM-TWC under graph homomorphisms, generalizing a similar analysis from the point-to-point channel case . Let [{GLemma 2.If Proof. n for the DM-TWC Suppose φ(an)=)one-shot zero-error capacity region, and the associated sum-rate as the one-shot zero-error sum-capacity. Recall that the one-shot zero-error capacity of a point-to-point channel is simply the logarithm of the independence number of its confusion graph; this quantity yields a lower bound on the zero-error capacity of the channel, and also provides an infinite-letter expression for the capacity when evaluated over the product graph. It is therefore interesting to study the analogue of the independence number in the two-way case, which in particular would yield an inner bound on the zero-error capacity region of the DM-TWC. For simplicity of exposition, we will focus here on the one-shot zero-error sum-capacity only.In this subsection, we consider the problem of zero-error communication over a DM-TWC with only a single channel use by the two parties  If (2) (3) S is an independent set of Proof. The results follow directly from the definition of dual independent pairs. □Lemma 3.Let (1) (2) (3) (4) Proof. S and T. From Proposition 4 (2), we have(1) The lower bound follows from Proposition 4 (1) and the symmetry of S and T as the maximum independent sets of H and G respectively.(2) From claim (1) above, we have (3) From claims (1) and (2) above, we have(4) is a direct consequence of claim (1) above. The lemma follows. □By graph homomorphisms we immediately have:Proposition 5.If J be an A and B, denote A. Now define Next, we shall provide an upper bound for a number . Let Γ bLemma 4.Proof. m and a set S, denote Suppose X, where Y similarly. The Shannon entropy of X is denoted by binary entropy function is written as conditional entropy of X given Y is written as mutual information between X and Y is conditional mutual information of Z is We recall some standard information-theoretic quantities that will be used in the sequel. Let Proposition 6.(1) (2) (3) In this section, we provide single-letter outer bounds for the non-adaptive zero-error capacity region of the DM-TWC. First in It is trivial to see that Shannon’s vanishing-error non-adaptive capacity region of the DM-TWC  and ande. Hence|Ai(S)|=∑ l(λ) in . This coThe following is a trivial corollary of Lemmas 6 and 7.Corollary 1.whereWe now provide a single-letter outer bound, in which the order of the minimum and the maximum in  is swappTheorem 2.whereThe first intersection is taken over all DM-TWCs Proof. The intersection over all n, we will show that:n, where n. The base case of n. Suppose that n. Following the same steps of  in  in teps of , we, we(22)I|Ai,Bi)2≤∑i=1nH(Yondition .By the definition of logl(λ)}≥maxPX1,P we have−logl(λ)>mbining , weclose to , as desiD is a set of letters, n, and n. Let:I, and denoteIn this subsection, we present a construction of uniquely decodable codes via linear codes, which generalizes a known result for the binary multiplying channel . Let us detecting symbol, if detecting vector for detecting vector for Let Proposition 7.Let Proposition 7 provides a sufficient condition for a uniquely decodable code, which is not always necessary (see Example 2). Nevertheless, this sufficient condition furnishes us with a way of constructing uniquely decodable codes by employing linear codes.Example 2.Suppose that Assume that  channel .Theorem 4.Let where To prove this theorem, we need the following lemma. The case that oved in  (2) (3) for each Proof. A be a q-ary A. Recall that for every A with columns indexed by Let D} as in . Denote:On the one hand, the number of vectors Fq, see . Le. LeA . First we briefly review the probabilistic refinement of the Shannon capacity of a graph in type n. For every type classP is the set of sequences of type P, that is, ϵ-typical set of P isjoint type conditional type We first recall some basic notions and results from the method-of-types. Let Lemma 11. . |Pn|≤ (2) . The liAccording to Lemma 11, it is easily seen that:arton in  via a noarton in .In this subsection, we derive an outer bound for the case when all Theorem 5.Proof. n. For any Suppose that n tends to infinity; (56) follows from  follows from Theorem 1 that each Now we are going to upper bound the cardinality of og|A||B|≤lim supn),where ; (55) foely have|Jn could outperform Theorem 2, see the following example.Example 3.Consider the channel K¯5,C5;K¯,…,K¯2 whown from  that Θ, we, we64), her with , the proIn this paper, we investigated the non-adaptive zero-error capacity region of the DM-TWC and provided several single-letter inner and outer bounds, some of which coincide in certain special cases. Determining the exact zero-error capacity region of a general DM-TWC remains an open problem, and clearly a difficult one, since it includes the notorious Shannon capacity of a graph as a special case. Despite this inherent difficulty, the problem is richer than the graph capacity setting, and we believe it deserves further study in order to obtain tighter bounds and smarter constructions.A. We have also looked at some variants of the orthonormal representation. For example, we considered the case where each graph vertex a is labelled by a unit vector a and F, then the vector projections of F are orthogonal. However, proving sub-multiplicativity in any of these settings has so far resisted our best efforts.One appealing direction is to extend the Lovász’s semi-definite relaxation approach in order to obtain tighter outer bounds, mimicking the graph capacity case. This, however, does not seem to be a simple task. In particular, one may ask whether the natural quantity fined in , which ufined in  is not afined in . In partIt would be also of much interest to consider the adaptive zero-error capacity of the DM-TWC. Allowing Alice and Bob to adapt their transmissions on the fly can in general enlarge the zero-error capacity region. As a simple example, note that a point-to-point channel with noiseless feedback is a special case of the DM-TWC (where Bob has no information to send). In , Shannon


Nature Communications 10.1038/s41467-019-11558-2, published online 14 August 2019.Correction to: The original version of this Article contained an error in Fig. 1c, in which the top SNP was incorrectly labelled ‘rs4293358’. The correct label is ‘rs429358’. This has been corrected in both the PDF and HTML versions of the Article.


In their study of the effect of an LCHF-diet on blood lipids, Burén et al. became worried because, after a few weeks, the participants’ LDL-cholesterol (LDL-C) increased significantly . HoweverThe reason why high LDL-C is beneficial is most likely that LDL participates in our immune system by adhering to and inactivating all types of microorganisms and their toxic products, a little-known fact that has been documented in many ways by more than a dozen research groups ,6.


No additional rigidity decline is observed when d_seq ≥  ~ 30 and this upper limit is very modest, close to 0.015 Å.It was observed that the rigidity degree diminishes if the number of covalent bonds intercalated between the two atoms (This suggests that protein flexibility is not fully described by B-factors that capture only partially the wide range of distortions that proteins can afford.The online version contains supplementary material available at 10.1186/s12859-021-04393-0. Molecule flexibility is inherent in thermodynamic stability and chemical reactivity . In globStudies on protein flexibility have addressed numerous molecular features by means of several methodological approaches. Atomic resolution crystallography allowed the characterization of conformationally disordered atoms –10. TimeAnother source of information about protein flexibility is provided by the atomic displacement parameters—usually referred to as B-factor (B)—that monitor the positional displacements of the atoms around their equilibrium positions , 17. B-fHere a new and insofar unexplored aspect is considered: how does flexibility decrease when the separation between atoms increases. It can be expected that flexibility is minimal for covalently bound atoms and, more in general, for atoms close to each other, since close interatomic contacts tend to be rigid —this is X and uY) must be identical. This means that their difference (Delta-u) must be equal to zero Å:Flexibility degree can be monitored by means of the Hirshfeld test , which eOn the contrary, Delta-u far from zero Å is expected for atoms that do not behave as a rigid body and have displacements and dispersions around their average locations independent of each other.Atom pair separation is defined in two different ways. On the one hand, it is the Euclidean distance (d) between the atoms and, on the other, it is the number of covalent bonds intercalated between the atoms .It is observed that Delta-u values increase if d or d_seq increase. However, the dependence of Delta-u on d is likely to be due to the fact that d is proportional to d_seq. In fact, for a given value of d_seq, Delta-u does not depend on d.Moreover, it is observed that Delta-u tends to rich its maximal value at d_seq ≈ 30 and to be nearly constant for d_seq > 30. This maximal value is considerably smaller if the Delta-u values are computed with anisotropic B-factors than with isotropic B-factors, suggesting that the isotropic B-factors overestimate protein flexibility.The maximal Delta-u values are however very modest, close to 0.015 Å, indicating that B-factors are rather unrelated, on average, to the stereochemical rearrangements, which are known to confer high flexibility to proteins, for example for exchanging buried water molecules with the external solvent.Delta-u values, Euclidean distances and covalent separations were computed for 6,794,404 pairs of atoms in 30 crystal structures, with covalent separation up to 50.The relationships between Delta-u and Euclidean distance or covalent separation are shown in Fig. First, the flexibility of atom pairs is clearly overestimated by isotropic Delta-u. This is not unexpected, since anisotropically refined B-factors represent better the positional scatter of the atoms. It is however surprising that the difference between isotropic and anisotropic Delta-u is so large: for atoms 30–35 Å apart, the isotropic Delta-u (ca. 0.08–0.09 Å) is about 4 times larger than its anisotropic counterpart (ca. 0.02 Å); and for atoms separated by 30 covalent bonds it (ca. 0.065 Å) is about 4 times larger than the anisotropic Delta-u (ca. 0.015 Å).Second, a difference between Euclidean distances and covalent separation appears too. The Delta-us, both isotropic and anisotropic, tend to increase with Euclidean distance and the increase is rather linear for Euclidean distances larger than 10 Å . They adopt different fold types, a TIM-barrel for 1us0, essentially a β-barrel for 2b97, and a α-β-α roll for 3ui4, and one of them, 1us0, is much larger than the others. These proteins show similar trends and there are no enormous differences between them; furthermore, the difference between the two enzymes is comparable to their difference from hydrophoibin, and the largest protein (1us0) is intermediate between the other two.Interestingly, this trend seems to be independent of protein dimension, type of fold, secondary structure composition or biochemical function. As an example, Fig. Crystallographic B-factors are largely unable to monitor transitions amongst conformational sub-states. This has been observed, implicitly, in some previous studies. For example, according to a recent study, protein conformational entropy, defined as the movements of certain groups in proteins, is not monitored quantitatively by crystallographic B-factors . Also, iThese observations can be explicated as it follows. Crystal structures show the dominating and most stable protein conformation while alternative sub-states remain undetected, especially at low resolution. Some conformational disorder can be observed and refined experimentally only at high resolution –10. B-faAs a consequence, B-factors can be computed quite successfully in—very—small molecule crystals, independently of diffraction data, where B-factors monitor quite effectively atomic fluctuations. The vibrational component of the atomic displacement parameter can be computed with quantum chemistry computations in crystals with very small asymmetric units. For example, density functional theory (DFT)-based methods were used for crystalline l-alanine and crystalline urea , and denOn the contrary, protein crystallographic B-factors are affected by too many non-vibrational components and cannot be predicted by computing the energy of the environment of the atoms by means of quantum chemistry approaches, though it has been shown that protein B-factors are somehow correlated to packing density . At thisIt must be remembered too that most of protein crystal structure information is being produced at low temperature—100 K—and that a different flexibility might be detected at room temperature or at physiological temperature . HoweverThe above discussion does not imply that crystallographic B-factors are of limited value and disconnected from the physicochemical nature of proteins. For example, information about local flexibility can be extracted from B-factor analyses, for example for protein-DNA complexes , cold adWhile covalently bound atoms form a rigid structural unit, this rigidity, monitored through the Hirshfeld Delta-u , is prog30 crystal structures were extracted from the Protein Data Bank , 49 accoThe Delta-u values were computed with anisotropic B-factors (U)n is the unit vector from atom X to atom Y. These values are referred to as anisotropic Delta-u, to distinguish them from the isotropic Delta-u, computed with the isotropic B-factor equivalent, defined asaccording toby means of the following expression.All computations were performed with locally written software.Additional file 1. Table S1: List of the entries of the Protein Data Bank examined in the present article. Figure S1: Relationship between Delta-u and covalent separation in three equally populated subsets of the structures examined in the present communication.


However, the binding of a TF to a gene does not necessarily imply regulation. Thus, it is important to develop methods to identify regulatory targets of TFs from ChIP-chip data.ChIP-chip data, which indicate binding of transcription factors (TFs) to DNA regions We developed a method, called Temporal Relationship Identification Algorithm (TRIA), which uses gene expression data to identify a TF's regulatory targets among its binding targets inferred from ChIP-chip data. We applied TRIA to yeast cell cycle microarray data and identified many plausible regulatory targets of cell cycle TFs. We validated our predictions by checking the enrichments for functional annotation and known cell cycle genes. Moreover, we showed that TRIA performs better than two published methods (MA-Network and MFA). It is known that co-regulated genes may not be co-expressed. TRIA has the ability to identify subsets of highly co-expressed genes among the regulatory targets of a TF. Different functional roles are found for different subsets, indicating the diverse functions a TF could have. Finally, for a control, we showed that TRIA also performs well for cell-cycle irrelevant TFs.Finding the regulatory targets of TFs is important for understanding how cells change their transcription program to adapt to environmental stimuli. Our algorithm TRIA is helpful for achieving this purpose. By organizing the genes in a genome into transcriptional regulatory modules (TRMs), a living cell can coordinate the activities of many genes and carry out complex functions. Therefore, identifying TRMs is useful for understanding cellular responses to internal and external signals. Advances in high-throughput tools such as DNA microarray ,2 and chGenome-wide gene expression analysis has been used to investigate TRMs controlling a variety of cellular processes in yeast -9. Clustet al. [et al. [et al. [On the other hand, the ChIP-chip technique was developed to identify physical interactions between TFs and DNA regions. Using ChIP-chip data, Simon et al.  investig [et al.  construc [et al.  construcSynechocystis sp [To overcome this problem, several algorithms have been developed to combine gene expression and ChIP-chip data to infer the regulatory targets of a TF. For instance, NCA  and MA-Nystis sp . Therefop-value to the binding relationship of a TF-gene pair. They found that if p-value ≤ 0.001, the binding relationship of a TF-gene pair is of high confidence and can usually be confirmed by gene-specific PCR. Therefore, we include a gene in the set B+ if the TF-gene binding p-value in the ChIP-chip data is ≤ 0.001, i.e. B+ consists of genes that are significantly bound by a TF. Further, a gene in B+ is assigned into B+R+ if it has a temporal relationship with the TF but into B+R- otherwise. A TF-gene pair is said to have a temporal relationship if the gene's expression profile is significantly correlated with the TF's regulatory profile possibly with time lags (see Methods). Our hypothesis is that the genes in B+R+ are more likely to be the regulatory targets of a TF than are the genes in B+R-. TRIA is developed to classify B+ into B+R+ and B+R-.Two previous papers ,19 used B+R+ and B+R- ≥ 25 (i.e. at least 25 genes in each group). The number of genes in each group (B+R+ and B+R-) is listed in Table We considered nine cell cycle TFs that have both sizes of B+R+ is shown to be more enriched than B+R- for specific MIPS functional categories with adjusted p-value < 0.05 (after the Bonferroni correction for multiple tests) using the cumulative hypergeometric distribution . In most cases (7/9), except for Abf1 and Ace2, the cell cycle genes are more enriched in B+R+ than in B+R-  from the binding targets (B+) of a TF. Then, we classify B+R+ into subsets Ai and Ri, where Ai (Ri) contains all genes whose expression profiles are positively (negatively) correlated with the TF's regulatory profile with a lag of i time points. Finally, we test whether the expression coherence of Xi is statistically higher than that of B+R+, where Xi = Ai or Ri. The expression coherence of genes in a set G (i.e. EC(G)) is defined as the fraction of gene pairs in G with a correlation in expression level higher than a threshold T [T was determined to be the 95th percentile correlation value of all pairwise correlations between 2000 randomly chosen genes in the yeast genome. Note that 0 ≤ EC(G) ≤ 1. The cumulative hypergeometric distribution is used to assign a p-value for rejecting the null hypothesis EC(Xi) = EC(B+R+), where Xi = Ai or Ri (see the Appendix for details).TRIA has the ability to identify subsets of highly co-expressed genes among the regulatory targets of a TF. First, we use TRIA to identify the plausible regulatory targets  be the set of regulatory targets of a TF that are identified by TRIA but not by MA-Network (MFA) and S2 (T2) be the set of regulatory targets of a TF that are identified by MA-Network (MFA) but not by TRIA. We tested over-representation of the high-confidence TF binding motifs in S1 and S2 (T1 and T2). The cumulative hypergeometric distribution is used to assign a p-value to the motif enrichment (see the Appendix for details). We found that in four of the five (4/5) cases the high-confidence TF binding motifs are enriched in S1 with p-value < 0.001 but only two of the five (2/5) cases in S2 are enriched  cases in T2 is enriched  using the cumulative hypergeometric distribution. This result suggests that TRIA performed well for cell-cycle irrelevant TFs.In this paper, TRIA is used to identify regulatory targets of cell cycle TFs. For a control, we show that TRIA can also perform well for cell-cycle irrelevant regulators. In this regard, we apply TRIA to identify regulatory targets of TFs that are activated by amino acid starved stress. The genome-wide gene expression and ChIP-chip data under amino acid starved growth condition are download from ,19. As sThe development of TRIA was motivated by two biological observations. First, it is known that TF binding affects gene expression in a nonlinear fashion: below some level it has no effect, and above some level the effect may saturate. This type of behavior can be modeled using a sigmoid function. Therefore, we define a TF's regulatory profile as a sigmoid function of its expression profile as in previous studies -35. AlthIn this study, we use time-lagged correlation analysis between a TF and its binding targets to identify its regulatory targets. However, in some cases, TFs may interact with each other and together regulate a group of target genes. This issue will be addressed in the future. We will try to define an overall regulatory profile of a TF complex and apply TRIA to identify target genes that are co-regulated by the same TF complex.An algorithm called TRIA is developed to identify the plausible regulatory targets of a TF from its binding targets. Since the binding of a TF to a gene does not necessarily imply regulation, TRIA is used to solve this ambiguity. We validated the effectiveness of TRIA by checking the enrichments for functional annotation and known cell cycle genes. Moreover, the performance of TRIA was shown to be better than two published methods (MA-Network and MFA). Moreover, TRIA has the ability to identify subsets of highly co-expressed genes among the regulatory targets of a TF. In addition, TRIA has been successfully applied to identify high-confidence cell cycle TF binding sites  and to rThree types of data are used in this study. First, the ChIP-chip data of the cell cycle TFs under the rich media are downloaded from . Second,Temporal Relationship Identification Algorithm (TRIA) is developed to identify TF-gene pairs that have a temporal relationship. A cell cycle TF and its binding target are said to have a positively (negatively) temporal relationship if the target gene's expression profile is significantly positively (negatively) correlated with the TF's regulatory profile possibly with a time lag. It is known that TF binding affects gene expression in a nonlinear fashion: below some level it has no effect, and above some level the effect may become saturated. This type of behavior can be modeled using a sigmoid function. Therefore, we define a TF's regulatory profile as a sigmoid function of its expression profile as in previous studies -35.x1, ..., xN) be the gene expression time profile of cell cycle TF x and y1, ..., yN) be the expression profile of gene y. The regulatory profile RP(f(x1), ..., f(xN)) of TF x is defined as a sigmoid function:Let s is the sample standard deviation of RP = 0 and the alternative hypothesis H1: r(k) ≠ 0 by the bootstrap method (see the Appendix) and get a p-value p(k). The time-lagged correlation (TlC) of RP(r(j) that has the smallest p-value (i.e. TlC(RP(r(j) if p(j) ≤ p(k) ∀k ≠ j). Note that -1 ≤ TlC(RP(RP(RP(j time points if TlC(RP(r(j) > 0 &p(j) ≤ pThreshold and (2) negatively correlated with a lag of j time points if TlC(RP(r(j) < 0 &p(j) ≤ pThreshold. The pThreshold is chosen to ensure that we have at most a 5% false discovery rate (FDR) [x, after a lag of j time points, activates (represses) gene y if RP(j time points.Then we test the null hypothesis Hte (FDR) . We may B+R+ is statistically higher than that in B+R-. Following Banerjee and Zhang [We want to test whether the enrichment of the known cell cycle genes (identified in ) in B+R+nd Zhang , a modelWe calculate:N = na + nb, M = ma + mb, na(nb) is the number of genes in B+R+ (B+R-), ma(mb) is the number of the known cell cycle genes in B+R+ (B+R-), andwhere xa, xb such that xa ≥ ma, which is taken as the p-value for rejecting the null hypothesis that enrichment of the known cell cycle genes in B+R+ is not statistically higher than that in B+R-.Then, we consider all possible combinations of EC) of sets B+R+, Ai and Ri are calculated, where Ai (Ri) contains all genes in B+R+ whose expression profiles are positively (negatively) correlated with the TF's regulatory profile with a lag of i time points.The expression coherence  = EC(B+R+)  > EC(B+R+)) is defined as in Equation (A1). N is the number of gene pairs in B+R+, M is the number of gene pairs in Xi, where Xi = Ai or Ri, na is the number of gene pairs in B+R+ that have correlations higher than the threshold T, and ma is the number of gene pairs in Xi that have correlations higher than the threshold T.We want to test whether the expression coherence of S1 (T1) and S2 (T2), where S1 (T1) is the set of regulatory targets of a TF that are identified by TRIA but not by MA-Network (MFA) and S2(T2) is the set of regulatory targets of a TF that are identified by MA-Network (MFA) but not by TRIA. Only 5 TFs  are studied for both TRIA and MA-Network. Only 8 TFs  are studied for both TRIA and MFA.The proportions of genes whose promoter regions contain the high-confidence TF binding motif are calculated for The high-confidence TF binding motifs were derived by using six motif discovery methods, under the requirement for conservation across at least three of the four related yeast species . The yeaS1 (T1) and S2 (T2). The cumulative hypergeometric distribution is used to determine the statistical significance. The p-value is defined as in Equation (A1), where N = 6229 is the number of genes in the yeast genome, M is the number of genes in G, where G = S1 (T1) or S2(T2) , na is the number of genes in the yeast genome that contain binding sites of a TF under study  and ma is the number of genes in G that contain binding sites of a TF under study .We tested over-representation of the high-confidence TF binding motif in N-k pairs of observations, Z = {zi: i = 1, ..., N-k and zi = (f(xi), yi+k)}. The correlation coefficient from the sample is calculated and denoted asWe observed r(k) ≤ 1. It is aimed to use these observations to test if r(k) is different from 0 significantly. Suppose the null hypothesis is H0: r(k)= 0 and the alternative hypothesis is H1: r(k) ≠ 0. We will apply the bootstrap method to perform this hypothesis testing based on the observations. Keeping the pair relationship of these N-k pairs to maintain the dependence between (f(xi), yi+k), zi are sampled with replacement N-k times to form a bootstrap sample, Z* = {zi*: i = 1, ..., N-k and zi* belongs to Z}. The correlation coefficient from the bootstrap sample Z* is computed and denoted as r*(k), -1 ≤ r*(k) ≤ 1. Repeat the resampling procedure B times, we will observed r1*(k), r2*(k), ..., rB*(k). These bootstrap correlation coefficients are sorted to be p-value is the smallest value of α for which the null hypothesis will be rejected based on the observation, the p-value for this test is estimated by the following:where I{·} is the indicator function whose value is one when the event is true and zero otherwise.where WSW developed the algorithm, performed the simulation and wrote the manuscript. WHL and BSC gave the research topic, provided essential guidance and revised the manuscript. All authors read and approved the final manuscript.Supplementary Table 1Click here for fileSupplementary Table 2Click here for file