More precisely, given a multifurcated (non-binary) gene tree with leaves labeled 0 or 1 depending on whether the corresponding genes belong to the mitochondrial or nuclear genome of the corresponding species, we investigate the problem of inferring a most parsimonious Duplication, Loss and EGT (DLE) Reconciliation of any binary refinement of the tree. We present a general two-steps method: ignoring the 0–1 labeling of leaves, output a binary resolution minimizing the Duplication and Loss (DL) Reconciliation and then, for such resolution, assign a known number of 0s and 1s to the leaves in a way minimizing EGT events. While the first step corresponds to the well studied non-binary DL-Reconciliation problem, the complexity of the label assignment problem corresponding to the second step is unknown. We show that this problem is NP-complete, even when the tree is restricted to a single polytomy, and even if transfers can occur in only one direction. We present a general algorithm solving each polytomy separately, which is shown optimal for a unitary cost of operation, and a polynomial-time algorithm for solving a polytomy in the special case where genes are specific to a single genome  in all but one species. This work represents the first algorithmic study for reconciliation with endosymbiotic gene transfers in the case of a multifurcated gene tree.Reconciling a non-binary gene tree with a binary species tree can be done efficiently in the absence of horizontal gene transfers, but becomes NP-hard in the presence of gene transfers. Here, we focus on the special case of  Reconciliation is the process of embedding a gene family tree into a species tree (i.e. reconstructing a mapping between the gene tree and the species tree) to explain how the gene family evolved inside the species tree according to the gene tree model, through evolutionary events modifying gene contents in genomes, such as losses, duplications or horizontal gene transfers (HGTs). This allows deciphering the orthology (divergence through speciation), paralogy (divergence through duplication) or xenology (divergence through HGT) relation between genes, which has important implications on understanding functional specificity of gene copies. For this purpose, the most critical part is the construction of a “good” gene tree, i.e. a gene tree reflecting the true evolution of the nucleotide or amino acid sequences of genes. In fact, as shown in many studies , the resUnfortunately, for many reasons related to sequence alignment, limitations of the considered phylogenetic method or issues with the sequence dataset , gene trees are almost never inferred with absolute certainty. As phylogenetic reconstruction methods are usually accompanied with statistical evaluations on branches, a solution for removing ambiguities in a tree is collapsing its weakly supported branches, leading to a non-binary tree . The problem then becomes one of simultaneously finding a binary refinement and optimal reconciliation of the multifurcated tree, more precisely, inferring an optimal evolutionary scenario leading to a binary refinement of the tree. This strategy has been applied, for example, to infer the evolution of the gene families responsible for alkaloid accumulation in plants .Reconciling a non-binary gene tree with a binary species tree can be done efficiently in the absence of HGTs (a review can be found in ). As farIn this paper, we focus on the particular case of DTL non-binary gene tree reconciliation, where transfers can only move genes between the mitochondrial and nuclear genome of the same species – called endosymbiotic gene transfers. In fact, it is well known that episodes of such gene transfers, mainly from the mitochondria to the nucleus, have marked the eukaryote evolution since an initial endosymbiotic event integrating an From a computational point of view, EndoRex  is the fHere, we explore the case of a non-binary gene tree. More precisely, given a multifurcated gene tree for a gene family with 0–1 labeled genes (leaflabels of the gene tree), the problem consists in inferring a most parsimonious duplication, loss and EGT scenario leading to a binary refinement of the tree. Our method is in two steps: ignoring the 0–1 labeling of the gene tree leaves, output all resolutions minimizing the DL-Reconciliation cost and then, for each resolution (i.e. binary tree), assign a known number of 0s and 1s to the leaves in a way minimizing EGT events. Step one can be done efficiently as recalled above. Therefore, we focus on the second step which consists in assigning a 0–1 labeling to the nodes of a binary tree, in a way minimizing the considered evolutionary distance. We show in "b-labeling problem in the special case where, in each polytomy, genes are specific to a single genome  in all but one species. We first begin, in the next section, by formally defining our problems.Except for species conserving the traces of an ancestral eukaryotic origin, few genes are expected to reflect an intermediate endosymbiotic integration of the mitochondrial gene content to the nucleus, with gene copies in both the nuclear and mitochondrial genome. This is the case of the eukaryotes with complete mitochondrial genomes explored in   its root, by V(T) its set of nodes and by size of T. A node x is a descendant of y if x is on the path from y to a leaf of T and an ancestor of y if x is on the path from r(T) to y; x is a strict descendant (respect. strict ancestor) of x is the parent of p(y), if it directly precedes y on this path. In this latter case, y is a child of x. We denote by E(T) the set of edges of T, where an edge is represented by its two terminal nodes , with x being the parent of y. More generally, if x is an ancestor of y,  denotes the path between x and y. The subtree of T rooted at x  is denoted T[x]. The lowest common ancestor (LCA) in T of a subset L(T), denoted All trees are considered rooted. Given a tree unary if it has a single child, binary if it has two children, and a polytomy if it has at least two children. Moreover, a star-tree is a tree with a single internal node. We will denote by the sibling of An internal node (a node which is not a leaf) is said to be R is an extension of a tree T if it is obtained from T by grafting unary or binary nodes in T, where grafting a unary node x on an edge  consists in creating a new node x, removing the edge  and creating two edges  and , and in the case of grafting a binary node, also creating a new leaf y and an edge . In the latter case, we say that y is a grafted leaf. Moreover, given a function f defined from U to V, an extensionf is a function defined from A tree species tree for a set S with a bijection between L(S) and S for a given set of species S in Fig. gene family is a set G is a gene tree for a gene family G is meant to be fully identified by its species labeling, i.e. the species A mentclass2pt{minimmentclass2pt{minimb-labeling for a gene x: x belongs to the mitochondrial genome of x belongs to the nuclear genome of G is fully identified by these two labels (e.g. trees (2) and (4) in Fig. G, G are identified. Later, we will need to define labeling for internal nodes of G.In this paper, we will consider an additional mentclass2pt{minimbinary tree is a tree with all internal nodes being binary. If internal nodes have one or two children, then the tree is said partially binary. A multifurcated tree is a tree containing at least one polytomy. For example, in Fig. A binary refinement of (binary refinement) Let b-labeling.As for a multifurcated tree In Fig. count matrixCount(X) for X is a We need a final notation. Let Speciation (Spe) when the species to which they belong do, but also Duplication (Dup) i.e. the creation of a new gene copy, Loss of a gene copy, and transfer when a gene is transmitted from a source to a target genome. In this paper, we only consider endosymbiotic gene transfers, denoted EGT, i.e. the special case of transfers only allowing the transmission of genes from the mitochondrial genome to the nuclear genome of the same species, or vice-versa. If the transmission of a gene from a genome A to a genome B is accompanied by the loss of the gene in A, we refer to the event as an EGTL for  is a quadruplet R is a partially binary extension of G, s is an extension of V(R) to V(S), b is an extension of V(R) to e is an event labeling of the internal nodes of R, such that: x with a single child y is such that x is an EGTL event with source genome s(y)).Each unary node x of R with two children s(x) in S and y be the element of e(x) is an EGT with source genome For each binary node Grafted leaves in the extension R correspond to gene losses.(DLE-Reconciliation) Let R is as an extension of G, each node in G has a corresponding node in R. In particular, the s, b and e labeling on R induce an s, b and e labeling on the nodes of G. The difference between G and R are additional binary nodes with a child being a grafted leaf (a loss), and unary nodes corresponding to EGTL events.As DL-reconciliation of b-labeling, i.e. it is a tuple R is an extension of G. For example, in Fig. A entclass1pt{minimaOptimal reconciliation: Let c be a function attributing a cost to each event in R plus the sum of costs of the losses, i.e. optimal reconciliation. We denote by DLE ) the cost of an optimal DLE-Reconciliation (respec. DL-Reconciliation).Given a DLE-Reconciliation unitary when From now on, we denote by The following lemma makes the link between an optimal DLE-Reconciliation and the optimal DL-Reconciliation.Any optimal DLE-Reconciliation x be a duplication of x is either a duplication or an EGT node in x is a duplication in x is the root then its other child becomes the new root) would result in a DLE-Reconciliation If x is an EGT in If Therefore, G. Let G with S. Note that G, it has also a corresponding node in x in x in x would be a speciation. Similarly to the previous case, x is either a duplication or an EGT in x transformed into a speciation, and thus Let’s consider, by contradiction, an optimal DLE-Reconciliation iliation . As eachx of G, x of G, C.Recall that the optimal DL-Reconciliation is unique and -mapping , i.e. fomentclass2pt{minimGiven a DLE-Reconciliation compressed DLE-Reconciliation of e restricted to the nodes of G and G indicating the presence or absence of an EGTL node on that edge, i.e. obtained as follows: Let R by removing grafted leaves and their parental nodes (i.e. ignoring losses). For each edge  of G, let G only by unary nodes). Then:(Compressed reconciliation) A compressed DL-Reconciliation of b and the A mentclass2pt{minimP, i.e. the number of EGTL events, and define the cost of For a compressed DLE-Reconciliation compressed DLE-ReconciliationFrom a Let G with S. We construct a DLE-Reconciliation p(x), x)   in Let b-labeling of M(x) for each I is the subset of M, I) is called the b-constraint of G  for G, a labeling consistent with  if, for any Given a binary tree s labeling of an optimal DLE-Reconciliation should be the LCA-Mapping. We denote it Moreover, recall from Lemma DLE-BinL problem and (2) is its output.The main problem Step 2) can thus be defined as follows. See an example in Fig.  can thusDLE-BinL Problem:Input: A binary tree M, I) and a species tree S;Output: An optimal DLE-Reconciliation S, where b-labeling consistent with .Notice that, from Lemma DLE-BinL1 the DLE-BinL problem where I is restricted to the root of G .We call DLE-BinL problem in its decision version is defined bellow; the decision version of DLE-BinL1 is defined similarly.In this section, the considered cost is unitary; the complexity results are then naturally extendable to a general cost. The DLE-BinL decision version:Input: A binary tree M, I), a species tree S and an integer Cost;Question: Is there a DLE-Reconciliation S where b-labeling consistent with  for which DLE-BinL decision problem is in NP. In fact, given a DLE-Reconciliation Cost) and verify if the b-labeling M, I) in polynomial time by traversing the tree G.First observe that the DLE-BinL problem is NP-complete event with this requirement, in all but one very constrained version of the problem.According to the considered Resolution method presented in "S. The DL-DLE-BinL (respec. DL-DLE-BinL1) decision problem is defined as the DLE-BinL (respec. DLE-BinL1) decision problem with the additional restriction that the binary tree given as input is in For a multifurcated tree Weighted Monotone 1-in-3-SAT Problem), that the DL-DLE-BinL1 decision problem is NP-complete. We can then deduce that DL-DLE-BinL is also NP-complete, as well as the more general DLE-BinL problem.We first show, by reduction from Weighted Monotone one-in-three-satisfiability problem .As the Weighted Monotone 1-in-3-SAT:Instance: A set of clauses n  is defined as follows:For each species n of the m leaves mapped to For each species Finally, Cost is set to DL.The b-constraint  of G.Let j, TA, we set x of the subtree TA . If x on the path from the parent of y on the path from the parent of r(G). Else (when x on the path from the parent of r(G).For all b-label. Notice that all nodes on the the path from the parent of r(G) are duplication nodes in TA, we set TA, we set There are no EGTL events in the subtrees In both case, it is easy to see that this mapping is valid and that no EGTL events are required in DL and thus As there are no EGTL events in x of G, we set n variables are set to true in TA and as one variable per clause is set to True in TA, we know, by construction, that for each species n of the m leaves mapped to b-labeling b we constructed is thus consistent with .For each leaf b-labeling consistent with  for which DL-DLE-BinL1 decision problem admits a DLE-Reconciliation of cost lower than or equal to Cost. We then obtain a DLE-Reconciliation Weighted Monotone 1-in-3-SAT problem. Then its corresponding instance DL-DLE-BinL1 decision problem does not admit a DLE-Reconciliation of cost equal or lower than Cost.Let Weighted Monotone 1-in-3-SAT problem, its corresponding instance DL-DLE-BinL1 decision problem does admit an optimal DLE-Reconciliation Cost. In that case, By contradiction, let us suppose that for an unsatisfiable instance mentclass2pt{minimS, we know from Lemma b-label.As there are no duplication nodes in the DL-Reconciliation of the subtrees mentclass2pt{minimTA as follows: for all b-label of the nodes in We now define a truth assignment G. Therefore the truth assignment TA satisfies TA. We know that exactly n variables are set to True in TA, as exactly n subtrees M, I) requiring exactly n of the m leaves mapped to For each species Weighted Monotone 1-in-3-SAT problem, then its corresponding instance DL-DLE-BinL1 decision problem does not admit a DLE-Reconciliation of cost equal or lower than Cost. Weighted Monotone 1-in-3-SAT is NP-complete, Lemmas Since DL-DLE-BinL1 decision problem is NP-complete.The DL-DLE-BinL and DLE-BinL decision problems are NP-complete.The multiplicitymultiplicity factorGiven S, and the multiplicity factor DL-DLE-BinL1 Problem is fixed parameter tractable with respect to the multiplicity factor The two following lemmas make the link between the maximum number of non-loss nodes in an optimal DL-Reconciliation S.Let We consider for this proof non-compressed reconciliations.V(S), there are more than k speciation nodes of R that are mapped to separated, i.e. for any node v of R, v belongs to at most one of these subtrees.Let Consider Algorithm 1 above. We show that it transforms R in a position respecting definition 2 . By Lemma p(x) is at most x is at most From Lemma S and b-labeling for the non-loss nodes of R. The optimal DLE-Reconciliation O(n) time where Let EGTcopy holds for an EGT event and EGTcut holds for an EGTL event. We can do so by using Algorithm 1 in . Note thS. For the proof of the next Theorem, given a node S, we denote by b-labeling for all non-loss nodes of R mapped to k such nodes, then the number of possible S, we define R which roots are mapped to S.Let DL-DLE-BinL1 decision problem is fixed-parameter tractable with respect to the multiplicity factor The Here, we consider non-compressed reconciliations.DL-DLE-BinL1 decision problem using Algorithm 3.We can solve the S and all possible b-labelings We show by induction that Algorithm 3 computes the correct cost S, then Algorithm 3 computes the correct If the node b-labelings for the two children S. Let show that Algorithm 3 is correct for R, the two children of a non-loss and non-leaf node mapped to the node We may suppose now by the induction hypothesis that Algorithm 3 computes the correct cost for all possible costOp computed in line 24 of Algorithm 3 is the cost of an optimal DLE-Reconciliation of S.Note that, S. The number of possible b-labelings for the nodes mapped to S, then all nodes of the subtrees in b-labeling As for the complexity of the algorithm, from Lemma CurrentCost in line 13 can be computed in time Now for internal nodes, in line 13, The problem can thus be solved in time DL-DLE-BinL1 and DL-DLE-BinL, the general problems DLE-BinL1 and DLE-BinL remain NP-complete even if the multiplicity factor of Finally, the next theorem states that, in contrast to DLE-BinL1 decision problem is NP-complete, even for The The proof, given in DLE-BinL decision problem is NP-complete, even for The b-labels can only switch from 0 to 1, or only from 1 to 0. In the following definition, with no loss of generality, we assume transitions from 0 to 1.As endosymbiotic transfer events often move genes from the mitochondrial to the nuclear genome, and rarely in the opposite direction, we address the specific case where transfers are only allowed in one direction, i.e. when One-direction DLE-Reconciliation for x, y) of G, if (One-direction DLE-Reconciliation) Let One-DLE-BinL Problem:Input: A binary tree M, I) and a species tree S;Output: An optimal One-direction DLE-Reconciliation S where b-labeling consistent with .One-DLE-BinL1 problem where I is restricted to the root of G, and the corresponding decision problems. We next show that even this very restricted version of our initial problem is intractable. Moreover, the One-DL-DLE-BinL (respec.One-DL-DLE-BinL1) problem is defined as the One-DLE-BinL (respec. One-DLE-BinL1) problem with the additional restriction that the binary tree given as input is in We also define, in a similar way as before, the One-DL-DLE-BinL1 and One-DL-DLE-BinL are NP-hard but fixed parameter tractable with the multiplicity factor, while One-DLE-BinL1 and One-DLE-BinL are NP-hard even with a multiplicity factor of two.We show that One-DL-DLE-BinL decision problem is NP-complete.The One-DL-DLE-BinL1 is the same as that of Theorem One-DL-DLE-BinL follows. The proof for NP-completeness of One-DL-DLE-BinL1 is fixed parameter tractable with respect to the multiplicity factor The One-DL-DLE-BinL1 Problem using the algorithm in the proof of Theorem Note that the proof of Lemma One-DLE-BinL is NP-complete. However, as for DLE-BinL1 and DLE-BinL, One-DLE-BinL1 and One-DLE-BinL remain NP-complete even if the multiplicity factor of It follows from Theorem One-DLE-BinL1 and One-DLE-BinL decision problems are NP-complete, even for The DLE-BinL problem for G is a binary resolution of an initial multifurcated tree with initial polytomies reflected by a b-Constraint , would be to solve each polytomy, i.e. each subtree rooted at a node x of I, individually, in a post-order traversal of the tree. In fact, this strategy leads to an exact algorithm for the DL Non-binary Reconciliation Problem  and replaces it by a single leaf, with an appropriate b-label.Algorithm 4 traverses the tree G has been completely traversed, the subtrees are put back in the tree. Notice that on line 13, the algorithm adds a new intermediate species to b-label. Such nodes are technically ignored for the rest of the traversal of G and just used to re-graft the corresponding subtrees at the end (line 27).Once the tree DLE-BinL1 Problem such that DLE-BinL1 Problem is also NP-complete. In the next section, we will present DLEBinL1OneSpecies which can be substituted to DLEBinL1 in Algorithm 4 for a restriction of the problem, where, for each polytomy, genes belonging to the same species have the same b-label for all but one species.Algorithm 4 calls a function M, I) be a b-Constraint for S be a species tree. Then, with the input b-labeling consistent with .Let The proof is by induction on the number of node b is a b-labeling consistent with  by construction.Notice that the DLE-Reconciliation x is the root of G by definition. The algorithm then returns an optimal solution, as we assume that we can solve If there is only one node G is in I by definition. By induction, we may assume that for each node G[x] computed by the algorithm is exact. For each of those subtrees G[x], we then know the possible b-label(s) at the root leading to an optimal reconciliation of G[x] and the corresponding optimal reconciliation of G[x]. We now give the index 1 to G and replace it by a new leaf without a b-label. It solves G by the solved G is solved). b-label of the parent of G we can obtain an optimal reconciliation of b-label as its parent .G and replace it by a new leaf with b-label by Then, Algorithm 4 solves If there is more than one node Let’s show that this solution is optimal. By contradiction, suppose that there is Thus, Algorithm 4 returns an optimal solution for the input We conclude, by induction, that the solution returned by Algorithm 4 is optimal. DLE-BinL1 Problem where genes are specific to a single genome  in all but one species. We call it the DLE-BinL1-OneSpecies problem. In its simplest version where a single species is present, the problem reduces to assigning a multiset of two labels (a given number of 0 s and a given number of 1 s) to the leaves of a tree-shape (i.e. a tree with no leaf labels), in a way minimizing 0–1 transitions in the tree. Similar problems on assigning leaves to tree-shapes or to multilabeled trees (MUL-trees) have been considered in the context of other tree distances , most of them being NP-complete  with S. The proof is by induction.The key point is therefore showing that the arrays computed by Algorithm 6 are exact, i.e., for each node x is a leaf (either free or fixed), it is easy to see that x.array is correct.If x is an internal node, we may assume that i, j) such that j free leaves labeled by 0 under x, the sum of the numbers of free leaves labeled by 0 under j. If the genome labels of the children of x are not the same as i, x is converted as an EGT event if x is a duplication node in the DL-Reconciliation (and possibly an EGTL event is added) and if x is not a duplication node then some EGTL events may be added on the edges between x and its children. As the algorithm considers all possibilities of genome labels for j free leaves under x labeled to 0  needed from x to its children), the particular possibility leading to i, j) and x.array is exact.Now, if We conclude, by induction, that the arrays computed by Algorithm 6 are exact. DLE-BinL1-OneSpecies problem in Algorithm 5 computes the solution of the G, the associated array is computed in time O(k). For each internal node of G, the associated array is computed in time For each leaves of G in time O(n)Once all the arrays are computed, the algorithm finds the optimal assignation of the internal nodes with a preorder traversal of We conclude that the time complexity of Algorithm 5 is b-labeling  to the nodes of the tree in a way minimizing a total DLE  cost.In this paper, we present the first method for DLE-Reconciliation, that is a reconciliation accounting for duplications, losses, but also EGTs, for a multifurcated gene tree. It is a natural extension of the DL-Reconciliation of a multifurcated tree, where we first consider a solution for this problem, i.e. an optimal DL-Reconciliation, and then appropriately assign the binary b-labeling assignment step is NP-complete even if the gene tree in input is a binary refinement of a star-tree, and even when genes are present in only two copies in each species. Moreover, the problem remains NP-complete when the transfers are allowed in a single direction (e.g. only from 0 to 1) and even if the gene tree in input is an optimal resolution for the DL-Reconciliation. In this latter case, the problem is shown fixed-parameter tractable with respect to the gene tree’s multiplicity factor. We then present a greedy heuristic for the general version of the problem solving each polytomy independently in a bottom-up traversal of the tree. This heuristic is shown to be exact for a unitary cost of operations. Moreover, we give a polynomial-time algorithm for the resolution of a single polytomy in the case where genes are specific to a single genome in all but one species. We did not explore the case where genes are specific to a single genome in all but a fixed number of species, but we believe Algorithm 5 can be extended to solve this problem in polynomial time.We show that the optimal From a biological point of view, the next step will be to apply our method to the orthologous mitochondrial protein-coding genes (MitoCOGs) dataset , 10.DLE-BinL and DLE-BinL1 problems in the general case, a more fundamental question is whether an exact one-step method, considering all the events at once, can be developed. In fact, the complexity results obtained here do not allow to conclude on the complexity of the DLE Non-binary Reconciliation problem. It is indeed not excluded that the polynomial-time PolytomySolver algorithm [b-labeling of leaves, at least in special cases. In the near future, we will first explore the extension of PolytomySolver to the one species restriction of the model, before considering generalization to an arbitrary number of species.From a theoretical and algorithmic point of view, many open questions remain. Apart from the fact that a heuristic combining accuracy and time-efficiency should be developed for both the lgorithm  can be e


Nature Communications 10.1038/s41467-023-35836-2, published online 14 January 2023Correction to: The original version of this Article contained errors in Fig. 4d and Supplementary Fig. 4e, which showed plots of erroneous cumulative distance values, which arose from an erroneous handling of NaN values in the underlying code for distance calculations.The correct version of Fig. 4d is:which replaces the previous incorrect version:This has been corrected in both the PDF and HTML versions of the Article.The correct version of Supplementary Fig. 4e is:which replaces the previous incorrect version:The HTML has been updated to include a corrected version of the The Is to increase along the rostral-caudal axis  as IS values decrease across the forebrain regions ” incorrectly referred to Fig. 5b, g–i, instead of Fig. 5c, g–i. These errors have been corrected in both the PDF and the HTML versions of the Article.In addition, in the results section “Integrated microfluidics-light sheet fluorescence microscopy system for imaging neuronal activities and behavior” the sentence “Simultaneously, the tail flipping behaviors were monitored using high-speed IR camera imaging at 200 fps ” incorrectly referred to Fig. 3b, instead of Fig. 3c, and in the results section “Revealing brainwide representation of cadaverine sensing using the integrated optofluidic system” the sentence “Interestingly, apart from observing a supralinear gain of information for the majority of neurons , there is a trend for FUpdated Supplementary InformationUpdated Source Data


Scientific Reportshttps://doi.org/10.1038/s41598-023-38966-1, published online 24 July 2023Correction to: The original version of this Article contained a formatting error in Equation 1. As a result, the subsequent paragraph,“Yit is the logit binomial plague occurrence at site i in year t. Parameter a is the overall intercept, f is the thin-plate spline function, and ε is the uncorrelated random error term. 1 is the smooth function of marmot density at site i in year t,”now reads:“Yi,t is the logit binomial plague occurrence at site i in year t. Parameter a is the overall intercept, f is the thin-plate spline function, and ε is the uncorrelated random error term. 1 is the smooth function of marmot density at site i in year t,”The original Article has been corrected.


In the published article, there was a mistake in the keyword “raw sugarcane vinegar”.The correct keyword is “sugarcane original vinegar”.In the published article “Chen et al. 2015b” was cited in the Introduction, first-paragraph, line 5 as “It is an advanced type of vinegar product ”.The correct citation is “It is an advanced type of vinegar product . Research on the production technology of sugarcane vinegar and fruit vinegar beverages. The correct reference is: “Chen, G.-L., Zheng, F.-J., Lin, B., Wang, T.-S., and Li, Y.-R. (2013). Preparation and characteristics of sugarcane low alcoholic drink by submerged alcoholic fermentation. Sugar Tech. 15, 412–416. doi: 10.1007/s12355-013-0248-3”.The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.


Empirical studies have demonstrated that self-relevant information affects temporal order perception. Therefore, the question arises of whether personal values—which are the core components of the self—influence temporal order perception. To explore this problem, we chose harmony, one of the most common values in Chinese culture, as the starting point. First, the harmony scale was used to measure the harmony values of the participants, and the participants were divided into high- and low-harmony groups. The validity of the grouping was then verified using an implicit-association test. Furthermore, two temporal order judgment (TOJ) tasks were used to explore the impact of harmony values on temporal order perception. The results revealed that in both TOJ tasks, participants in the high-harmony group tended to perceive harmonious stimuli before non-harmonious stimuli, while the effect was not found in the low-harmony group. We conclude that harmony values affect temporal order perception, and only if the values are important to the individual. Personal values are broadly desirable goals that motivate people’s actions and serve as guiding principles in their lives. Values affect people’s perception, cognition, and behavior over time and across situations . So, do Values are closely linked to the self. Verplanken (2002)  affirmedGallagher (2000)  collatedA few studies based on the SoA found that in the TOJ task, participants reported that the stimuli caused by them appeared earlier than the stimuli caused by others. For example, in the study of Haering and Kiesel (2012) , red andChinese people hold a deep-rooted desire to pursue “harmony” when cultivating one’s self, handling interpersonal matters, and confronting the universe and nature. In China, many proverbs show Chinese people’s emphasis on harmony values. For example, “家和万事兴” (harmony in a family makes everything successful), “和气生财” , and so on. A few empirical studies have demonstrated that harmony values rank highly in the Chinese personal values system ,17. The The stimuli presented to the participants in the TOJ tasks were usually simple graphics. In Chinese, the writing of harmony- and disharmony-related words is quite complicated, so we believed that it might not be appropriate to use Chinese characters as stimuli for order perception. The perceptual matching task can help to solve this problem. In the perceptual matching task used in a previous study, participants were asked to associate an abstract graph with a certain label, and then use the graph as a stimulus for subsequent research . TherefoM = 21.80 years). All participants were right-handed. Participants had normal or corrected-to-normal vision.This study was approved by the Ethics Committee of Southwest University. A group of 53 undergraduates were initially recruited to participate in the experiment. Four participants performed unsatisfactorily in the TOJ tasks and were excluded, and 49 valid participants were retained , “和睦” (concord), “融洽” (in tune), “协调” (coordinate), “友好” (friendly), and “祥和” (auspicious and harmonious). Words related to non-harmony were: “冲突” (conflict), “矛盾” (contradiction), “对立” (oppose), “摩擦” (friction), “争执” (dispute), and “对抗” (confrontation). The attribute words included six agreement-related words, and six disagreement-related words. The agreement-related words were: “赞同” (agree), “同意” (consent), “赞成” (approve), “满意” (satisfaction), “接受” (accept), and “支持” (support), while the disagreement-related words included “反对” (oppose), “拒绝” (reject), “排斥” (repulsion), “不悦” (dissatisfaction), “否认” (deny), and “抗拒” (resist).Materials for TOJ tasks. In the shape-matching TOJ task, the targets were a black cir-cle and a black triangle of 1.5° in width and 1.5° high, presented randomly at two different positions, at a 4.8° visual angle from the fixation on a white background. The fixation point consisted of a centrally presented black fixation cross . In the color-matching TOJ task, the targets were a red circle and a green circle of 1.5° in width and 1.5° high.The experiment was divided into two parts. Part A involved the measurement of harmony values, and included explicit questionnaire measurement and IAT procedure measurement. Half of the participants performed explicit measurements first, and implicit measurements later, while the other half did the opposite. Part B included TOJ tasks, including the shape-matching TOJ task and the color-matching TOJ task. Half of the participants completed the shape-matching TOJ first, and then the color-matching TOJ, while the other half did the opposite. Some participants completed Part A first and then Part B, while others did the opposite. The E-Prime 1.0 software was used for programming and timing the experimental operations.Procedure of implicit harmony value measures. The participants were instructed to classify a series of stimulus words into the target category, and two attributive categories. The IAT procedure consisted of seven stages. In the first stage, the F key was pressed when the agreement-related word appeared, and the J key was pressed when the disagreement-related word appeared . In the second stage, the participants pressed the F key when they saw the harmonious stimulus, but pressed the J key when they saw the non-harmonious stimulus . In the third stage, the F key was pressed when the harmony words or agreement-related words appeared, while the J key was pressed when the non-harmony words or disagreement-related words appeared . The first three stages were practice stages, and the fourth stage was the formal test of the third stage . The fifth stage was the practice stage, in which the participants were asked to press the F key when the non-harmony words appeared, and press the J key when the harmony words appeared. The sixth stage was also the practice stage. At this stage, the F key was pressed when the non-harmony words or agreement-related words appeared, while the J key was pressed when the harmony words or disagreement-related words appeared . The seventh stage was the formal test of the sixth stage . Each incorrect response was signaled by a red X centered below the stimulus word for 150 ms, while a correct response was signaled by a green O centered below the stimulus word for 150 ms.Procedure of TOJs. Before the TOJ task, the participants were asked to complete the perceptual-matching task. Taking the shape-matching task as an example, the participants were first told that the circle represented the harmonious stimulus, while the triangle represented the non-harmonious stimulus, and then the participants carried out the shape-matching task. In each trial, a fixation point of 500–1000 ms was first presented on the screen. Subsequently, a pairing of a shape and label (harmony or non-harmony) was presented for 2000 ms. Two geometric shapes  were presented above the fixation at the center of the screen. The word of harmony or non-harmony (3.1°/3.6° × 1.6°) was displayed below the fixation. The angular distance between the center of the shape or the word, and the fixation cross, was 3.5°. All stimuli were shown on a white background. Participants judged whether the pairings of shape and label matched. Feedback (correct or incorrect) was given on the screen for 500 ms at the end of each trial. Each participant performed 120 trials. The color-matching task was the same procedure, except that in the color-matching task, the green circle represented the harmonious stimulus, while the red circle represented the non-harmonious stimulus.In each trial of the TOJ tasks, a fixation point was presented at the center of the screen for 500–1000 ms, and then a stimulus appeared on one side of the fixation point (left or right). After a variable interval, the second stimulus appeared on the other side of the fixation point. Once the second stimulus had appeared, the image remained rendered for 1000 ms, before disappearing. A question mark then appeared at the center of the screen, asking the participants to judge which stimulus appeared first. In the shape-matching TOJ, participants were asked to judge which shape stimulus appeared first, while in the color-matching TOJ, they were asked to judge which color stimulus appeared first. Stimulus-onset asynchrony (SOA) had nine levels at ±120 ms, ±90 ms, ±60 ms, ±30 ms, and 0 ms. The SOA was positive for the first circle (or green circle), and negative for the first triangle (or red circle). The SOA was 0 when the stimulus appeared simultaneously on both sides. Half of the participants attempted the shape-matching TOJ, followed by color-matching TOJ. Meanwhile, the other half of the participants attempted the TOJ in reverse order. The response key was balanced between the different participants. Half of the participants pressed “D” when the circle (or green circle) side stimulus appeared earlier, and pressed “K” when the triangle (or red circle) side stimulus appeared earlier. Each condition was repeated 15 times. The total number of trials was 270. Participants had some practice before the official experiment. The TOJ task process is depicted in t-test was conducted for the score , and the median value 3. The results revealed a significant difference between the score and the median value (t(48) = 13.00, p < 0.001, d = 1.86), indicating that participants regarded harmony as an important value. Each participant’s level of harmonious values was then assessed by comparing the difference between the harmony scores of the participants and the median value 3. It was found that 33 participants had scores significantly higher than 3, and these participants were classified in the high-harmony group . The other 14 participants whose scores were not significantly different from 3 were placed in the low-harmony group . An independent samples t-test was conducted on the harmony values scores of the two groups, and the results revealed a significant difference in the scores of the two groups (t(47) = 6.72, p < 0.001, d = 2.13). This indicated that participants in the high-harmony group hold harmony values in higher esteem than those in the low-harmony group do.The harmony values questionnaire was analyzed, and the one-sample t-test was performed on this value  of all participants, and the results showed significant difference from 0 (t(48) = 30.60, p < 0.001, d = 4.37), which indicated that participants had implicit harmony values. A one-sample t-test was conducted on the implicit effects of the high-harmony group  and the low-harmony group . The results demonstrated that both the high-harmony group (t(34) = 37.20, p < 0.001, d = 0.28) and the low-harmony group (t(13) = 11.60, p < 0.001, d = 3.11) were significantly different from 0. An independent samples t-test was conducted on the d-prime of the implicit effect of the two groups. The results showed that the implicit effect size of the participants in the high-harmony group was significantly larger than that of the low-harmony group (t(47) = 3.88, p < 0.001, d = 1.23), which indicated that the classification of high- and low-harmony groups was effective.In accordance with the recommendations of Greenwald et al. (1998) , first, t-test was conducted on the accuracy of participants in the high- and low-harmony groups. The results revealed that there was no significant difference between the performance of the two groups in either the shape-matching task (t(47) = −1.45, p = 0.16, d = −0.46) or the color-matching task (t(47) = −0.48, p = 0.63, d = −0.15).The participants’ accuracy reached 0.95 in both perceptual-matching tasks, which indicated that the participants indeed combined the shape with the value words. An in-dependent samples R2 value was less than 0.75 [For the shape-matching TOJ task-data analysis, first, E-prime was used to calculate the correct responses under each SOA. Then, MATLAB software was used to calculate the point of subjective simultaneity (PSS). The psychometric function was obtained by calculating the percentages of “harmony first” (circle first) responses as a function of SOA , and a chan 0.75 , the datt-test on the participants’ PSS against a test value of 0 to determine the PSS shift between stimuli [M = −3.13, SD = 7.77) were significantly different from 0 ms (t(48) = −2.82, p < 0.01, d = −0.40, BF10 = 5.19 (moderate evidence for H1)). We then explored the relationship between participants’ harmony scores and the PSS; the correlation between the D score and PSS was significant (r (49) = −0.41, p < 0.01), and the correlation between the Harmony Scale score and PSS was marginally significant (r (49) = −0.27, p = 0.06). The PSS of participants in the high-harmony and low-harmony groups were then analyzed separately. The results showed that the PSS of the high-harmony group  was significantly different from 0 ms (t(34) = −3.87, p < 0.001, d = −0.66, BF10 = 63.00 (very strong evidence for H1)), while the PSS of the low-harmony group  was not significantly different from 0 ms (t(13) = 1.24, p = 0.24, d = 0.33, BF10 = 0.52 ).We conducted a one-sample  stimuli ,26. The The color-matching TOJ task-data analysis method was consistent with the shape-matching TOJ task. The psychometric function was obtained by calculating the percentages of “green first” responses as a function of SOA , and thet-test was performed on the PSS. The results showed that the PSS of all the participants  were significantly different from 0 ms (t(48) = −2.77, p < 0.01, d = −0.40, BF10 = 4.59 (moderate evidence for H1)). We then explored the relationship between participants’ harmony scores and the PSS; the correlation between d-prime and PSS was significant (r (49) = −0.51, p < 0.001), and the correlation between the Harmony Scale score and the PSS was significant (r (49) = −0.32, p < 0.05). The PSS of participants in the high-harmony and low-harmony groups were then analyzed separately. The results revealed that the PSS of the participants with high-harmony values  was significantly different from 0 ms (t(34) = −3.79, p <0.001, d = −0.64, BF10 = 50.56 (very strong evidence for H1)), while the PSS of the participants with low-harmony values  had no significant difference from 0 ms (t(13) = 1.04, p = 0.32, d = 0.28, BF10 = 0.43 ).A single-sample F = 0.03, p = 0.86, F = 13.42, p < 0.001,F = 0.06, p = 0.81,We conducted a mixed-factor ANOVA with the perceptual-matching type (shape-matching or color-matching) as a within-subject factor and the harmony group (high-harmony group or low-harmony group) as a between-subject factor. The results revealed that the main effect of the matching type was not significant   found thThis study has a few shortcomings, which can be improved on in the follow-up research. For example, in the perceptual-matching task, we only associated green with harmonious information and red with disharmonious information. A more rigorous operation should consider the opposite condition. In addition, only harmony, a common value in Chinese culture, was selected in this study, and several more values should be selected in subsequent studies, to further verify the influence of values on temporal order perception.This study used two TOJ tasks to explore the effect of harmony values on temporal order perception; the result revealed that values do affect temporal order perception, and only if the values are important to the individual.


An additional detail regarding the computation of the weighted average 2.1 Performance metric design”, paragraph 7. This sentence previously stated:A correction has been made to Section 2. Materials and methods, subsection ““The intra-class dispersion is computed using:2 ∈  is a constant, ϕk−1, c is the intra-class dispersion for the class c trials of the (k-1)th block, and ϕk, c is the intra-class dispersion of class c trials computed only during the current (kth) block.”where αThe corrected sentences appear below:classDistinct and classStability metrics, we made the following modification to the calculation of the intra-class dispersion. We split the set of trials, T, into Ns subsets of Nt trials, Tj, such that“For the weighted average Nt trials performed during a block would be grouped into subset T1. Using these subsets, we computed a modified intra-class dispersion as:Subsets were formed by splitting trials according to the chronological order in which they were performed; for example, the first Ns is the number of trial subsets, Nt is the number of trials in each subset, jth subset of trials, ΓTj, i is the covariance matrix of the ith trial within subset Tj, and δR denotes the Riemannian distance. The motivation behind this modification was to reduce the impact of signal non-stationarities that may artificially increase the intra-class dispersion when considering a large number of trials. For our analysis, we set Nt = 5. Trial subsets were disjoint save for when computing within-block post-trial intra-class dispersion values. If the number of trials completed within the block was not divisible by Nt, subset TNs was formed using the most recently completed Nt trials; consequently, this subset could share up to Nt−1 trials with subset TNs − 1.where The post-trial intra-class dispersion was computed using this modified intra-class dispersion:2 ∈  is a constant, c trials of the (k−1)th block, and c trials completed only during the current (kth) block.”where αThe authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.


Subject clustering  is a problem of significant interest. In recent years, many approaches have been proposed, among which unsupervised deep learning (UDL) has received much attention. Two interesting questions are 1) how to combine the strengths of UDL and other approaches and 2) how these approaches compare to each other. We combine the variational auto-encoder (VAE), a popular UDL approach, with the recent idea of influential feature-principal component analysis (IF-PCA) and propose IF-VAE as a new method for subject clustering. We study IF-VAE and compare it with several other methods  on 10 gene microarray data sets and eight single-cell RNA-seq data sets. We find that IF-VAE shows significant improvement over VAE, but still underperforms compared to IF-PCA. We also find that IF-PCA is quite competitive, slightly outperforming Seurat and SC3 over the eight single-cell data sets. IF-PCA is conceptually simple and permits delicate analysis. We demonstrate that IF-PCA is capable of achieving phase transition in a rare/weak model. Comparatively, Seurat and SC3 are more complex and theoretically difficult to analyze . Suppose we have a group of n subjects  measured on the same set of p features . The subjects come from K different classes or groups , but unfortunately, the class labels are unknown. In such a case, we say the data are unlabeled. For 1 ≤ i ≤ n, we denote the class label of subject i by Yi and denote the p-dimensional measured feature vector of subject i by Xi. It is important to note that Yi takes values from {1, 2, …, K}. The class labels are unknown, and the goal is to predict them using the measured features X1, X2, …, Xn.We are interested in the problem of big data era: although the volume of available scientific data increases rapidly, a significant fraction of them are unlabeled. In some cases, it is simply hard to label each individual sample , where k. We assumeA reasonable model is as follows. We encode the class label Let In addition, it is reasonable to assume that out of many measured features, only a small fraction of them are useful in the clustering decision. Therefore, letting n × p signal matrix K.It follows that the W is the normalized data matrix. Similar to W as the sum of a signal matrix and a noise matrix. However, due to the normalization, the rank of the signal matrix is reduced to (K − 1).We recall that πi. For this reason, we may view Models In Models f  belongs to a class of non-linear functions. For example, we may assume f  belongs to the class of functions :A1, A2, … , AL are matrices of certain sizes and s1, s2, … , sL are some nonlinear functions. Similar to Models The recent idea of neural network modeling provides a wide class of non-linear models, which may be useful for our setting. As an alternative to Models X and number of clusters K. Output: predicted class label vector n × K matrix kth left singular vector of X .• We obtain the n rows of K groups by applying the classical k-means assuming there are i. Output • We cluster the PCA is a classical spectral clustering approach, which is especially appropriate for linear models like those in W. As explained before, we can similarly write W as the sum of a “signal” matrix and a “noise” matrix as in K to (K − 1). In such a case, we replace the n × K matrix n × (K − 1) matrixkth left singular vector of W.From time to time, we choose to apply the PCA clustering approach to the normalized data matrix X (or W) directly, Spectral-Gem constructs an n × n symmetric matrix M, where M can be viewed as a similarity metric between subject i and subject j. The remaining part of the algorithm has many small steps, but the essence is to apply the PCA clustering approach to the Laplacian normalized graph induced by M.The PCA clustering approach has many modern variants, including, but not limited to, Spectral-Gem  and SCOR• The signal matrix p ≪ n) or how the sparsity can be exploited is unclear.• It is hard to exploit sparsity in the data: either the data are non-sparse , the essence of the PCA spectral clustering approach is to obtain a rank-d approximation of X to use singular value decomposition (SVD),σk is the kth smallest singular value of X, and uk and vk are the corresponding left and right singular vectors of X, respectively. The VAE can be viewed as an extension of SVD, which obtains a rank-d approximation of X from training a neural network. The classical SVD is a linear method, but the neural network approach can be highly non-linear.Given an X into a matrix d ≪ p), and the decoder maps d approximation of X. Different from classical SVD, X and The VAE was first introduced by K, and dimension of the latent space d . Output: predicted class label vector n × d matrix • (Dimension reduction by the VAE). We train the VAE and use the trained encoder to obtain an n subjects into K classes by applying k-means to the rows of • (Clustering). We cluster all A popular way to use the VAE for subject clustering is as follows . Input: W or the unnormalized data matrix X. We call them VAE(W) and VAE(X), respectively. In the context of using these notations, it is unnecessary to use (W) and (X) at the same time, so we write VAE(W) as VAE for short  as VAE(X)).Except for using a non-linear approach to dimension reduction, the VAE is similar to the PCA approach in clustering. We can apply the VAE either to the normalized data matrix Q have a large ℓ2-norm; the ℓ2-norm of other columns is small or 0). In such a setting, it is appropriate to conduct feature selection, which removes a substantial amount of noise while keeping most non-zero columns of Q.For many genomic and genetic data, Models Such observations motivate the (orthodox) IF-PCA. The IF-PCA was first proposed in n (univariate) samples z1, z2, … , zn from a cumulative distribution function (CDF) denoted by F. We introduce the empirical CDF byThe first component of the IF step is the use of the Kolmogorov–Smirnov (KS) test for feature selection. Suppose we have z = . The KS testing score is thenLet F to be the theoretical CDF of i ≤ n; and z1, z2, … , zn, respectively.In the IF-PCA given as follows, we take p marginal p-values, one for a feature, and outputs a threshold for feature selection. Suppose we have p-values π1, π2, … , πp. We sort them in the ascending order:The second component of the IF step is the higher criticism threshold (HCT). Higher criticism (HC) was initially introduced by We define the feature-wise HC score by K. Output: predicted class label vector j ≤ p, we compute a KS score for feature j by applying z = wj. We denote the KS scores by ϕn (w1), …, ϕn (wp) and let μ* and σ* be their empirical mean and standard deviation, respectively. Let p-values by F is the same CDF used in π1, π2, … , πp. We retain feature j if • (IF step). For each 1 ≤ WIF be the n × m sub-matrix of W consisting of columns of W corresponding to the retained features only [m is the number of retained features in (a)]. For any 1 ≤ k ≤ min{m, n}, let WIF corresponding to the kth largest singular value of WIF. Let n subjects by applying the k-means to the n rows of K clusters. Let • (Clustering step). Let IF-PCA runs as follows. Input: normalized feature vectors In the IF step, the normalization of rrection , a simplrrection . We remaW. Seemingly, for the IF step, applying the algorithm to W instead of the un-normalized data matrix X is preferred. However, for the clustering step, whether we should apply the algorithm to W or X remains unclear. We propose a small variant of IF-PCA by applying the IF step and the clustering step to W and X, respectively.W as in the (orthodox) IF-PCA previously mentioned.• (IF step). We apply exactly the same IF step to XIF be the n × m sub-matrix of X consisting of columns of X corresponding to the retained features in the IF step only. For any 1 ≤ k ≤ min{m, n}, let XIF corresponding to the kth largest singular value of XIF. Let n subjects by applying the k-means to the n rows of K clusters. Let • (Clustering step). Let In the (orthodox) IF-PCA, we apply both the IF step and the clustering step to the normalized data matrix To differentiate from the (orthodox) IF-PCA , we call the aforementioned variant IF-PCA(X). See Near the end of K, and dimension of the latent space in the VAE (denoted by d). Output: predicted class label vector IF step). We run the same IF step as in m is the number of retained features).• (Clustering step). We apply the VAE with n × d matrix WIF. We cluster the n samples into K clusters by applying the classical k-means to K classes. Let • . See In the clustering step, we apply the VAE to the normalized data matrix We now introduce Seurat and SC3, two recent algorithms that are especially popular for subject clustering with single-cell RNA-seq data. We discuss them separately.m, N, k0, δ, where m is the number of selected features in the feature selection step, and N, k0, δ are the clustering step, corresponding to the PCA part, the k-nearest neighborhood algorithm part, and the modularity optimization part, respectively.Seurat was proposed in n × p data matrix X, number of clusters K, and tuning parameters m, N, k0, δ. Output: predicted class label vectors m features that are mostly variable. We obtain the n × m post-selection data matrix.• (IF step). We select the N left singular vectors. For each pair of subjects, we compute how many neighbors  they share with each other and use the results to construct a shared nearest neighborhood (SNN) graph. We cluster the class labels by applying a modularity optimization algorithm to the SNN graph, where we need a resolution parameter δ.• (Clustering step). We normalize the post-selection data matrix and obtain the first A high-level sketch for Seurat is mentioned as follows  = , but it remains unclear how to select .An apparent limitation of Seurat is that it needs four tuning parameters. Following the recommendations by K groups by applying the classical hierarchical clustering method to the consensus matrix. SC3 uses one tuning parameter x0 in the gene-filtering step and two tuning parameters, d0 and k0, in the clustering step, corresponding to the PCA part and the hierarchical clustering part, respectively.SC3 was first presented by n × p data matrix X, true number of clusters K, and tuning parameters x0, d0, k0. Output: predicted class label vectors x0% of cells or expressed  in at least (100 − x0)% of cells. This step may reduce a significant fraction of features, and we consider it to be more like a feature selection step than a preprocessing step.• (Gene-filtering step). Removes genes/transcripts that are either expressed  in less than n × n matrix M, where M is some kind of distance  between subject i and j. Second, let kth singular vector of M . Third, for d = 1, 2, … , d0, we cluster all n subjects to K classes by applying the k-means to the rows of the n × d sub-matrix of d columns and use the results to build a consensus matrix using the Cluster-based Similarity Partitioning Algorithm (CSPA) (k0 levels of hierarchy.• (Clustering step). First, we take a log-transformation of the post-filtering data matrix and construct an m (CSPA) . FinallyA high-level sketch for SC3 is given as follows  =  and take k0 to be the true number of clusters K. Such a tuning parameter choice may work effectively in some cases, but for more general cases, we may , IF-VAE, and IF-VAE(X)) have never never proposed before. Among these methods, the VAE is a popular UDL approach, Seurat and SC3 are especially popular in clustering with single-cell data, and IF-PCA is a conceptually simple method that has been shown to be effective in clustering with gene microarray data. It is important to note that some of the methods are conceptually similar to each other, with some small differences . For example, many of these methods are two-stage methods, containing an IF step and a clustering step. In the IF step, we usually use the normalized data matrix https://github.com/ZhengTracyKe/IFPCA. The 10 microarray data sets can be downloaded at https://data.mendeley.com/datasets/cdsz2ddv3t, and the eight single-cell RNA-seq data sets can be downloaded at https://data.mendeley.com/drafts/nv2x6kf5rd.Our study consists of two parts. In X or the normalized data matrix W. The version of IF-PCA associated with X is called IF-PCA(X), and the version associated with W is still called IF-PCA; similar rules apply to the VAE and IF-VAE. A total of eight different algorithms were obtained.First, we compare the IF-VAE approach introduced in K clusters) of these methods. The results of SpecGem and IF-PCA are copied from d = 25 and use a traditional architecture in which both the encoder and decoder have one hidden layer; the encoder uses the ReLU activation, and the decoder uses the sigmoid activation; when training the encoder and decoder, we use a mini-batch stochastic gradient descent with 50 batches, 100 epochs, and a learning rate of 0.0005. The same neural network architecture and tuning parameters are applied to the VAE. We note that the outputs of these methods may have randomness due to the initialization in the k-means step or in the VAE step. For the VAE, IF-VAE, and IF-VAE(X), we repeat the algorithm 10 times and report the average clustering error. For k-means, we repeat it five times (because the results are more stable); for IF-PCA(X), we repeat it 20 times. We use the clustering errors to rank all eight methods for each data set; in the presence of ties, we assign ranks in a way such that the total rank sum is 36 . The average rank of a method is a metric of its overall performance across multiple data sets. In addition to ranks, we also compute regrets: for each data set, the regret of a method is defined to be r = (e − emin)/(emax − emin), where e is the clustering error of this method and emax and emin are the maximum and minimum clustering error, respectively, among all the methods. The average regret also measures the overall performance of a method .W or on X), which combines k-means with non-linear dimension reduction, significantly improves k-means on some “difficult” data sets, such as BreastCancer, ColonCancer, and SuCancer. However, for those “easy” datasets, such as Leukemia and Lymphoma, the VAE significantly underperforms compared to k-means. It suggests that the non-linear dimension reduction is useful mainly on “difficult” datasets. Third, the IF-VAE (either on W or on X) improves the VAE in the majority of data sets. In some datasets, such as LungCancer (1), the error rate of the IF-VAE is much lower than that of the VAE. This observation confirms that the IF step plays a key role in reducing the clustering errors. There are several notable observations. First, somewhat surprisingly, the simple and tuning-free method, IF-PCA, has the best overall performance. It has the lowest average rank among all eight methods and achieves the smallest number of clustering errors in four out of 10 data sets. We recall that the key idea of IF-PCA is to add a tuning-free feature selection step prior to dimension reduction. The results in m, we select the m top-ranked features, perform VAE on the unnormalized data matrix X restricted to these m features, and report the average number of clustering errors over five repetitions of the VAE. m. An interesting observation is that as m increases, the clustering error first decreases and then increases . A possible explanation is as follows: when m is too small, some influential features are missed, resulting in weak signals in the VAE step; when m is too large, too many non-influential features are selected, resulting in large noise in the VAE step. There is a sweet spot between 200 and 400, and the tuning-free procedure in the IF step selects m = 251. Next, we study IF-VAE(X) more carefully with the LungCancer (1) data set. We recall that the IF step ranks all the features using KS statistics and selects the number of features by a tuning-free procedure. We use the same feature ranking but manually change the number of retained features. For each Remark 1 (comparison with other clustering methods for microarray): https://hemberg-lab.github.io/scRNA.seq.datasets). It contains scRNA-seq datasets from Human and Mouse. Among them, we selected eight datasets that have a sample size between 100 and 2,000 and can be successfully downloaded and pre-processed using the code provided by the Hemberg Group under the column ‘Scripts’. The datasets Camp1, Camp2, Darmanis, Li, and Patel come from Human, and the data sets Deng, Goolam, and Grun come from Mouse. Each data matrix contains the log-counts of the RNA-seq reads of different genes (features) in different cells (samples). The cell types are used as the true cluster labels to evaluate the performances of clustering methods. We first pre-processed all the data using the code provided by the Hemberg Group, then features (genes) with fractions of nonzero entries m is the number of selected features, N is the number of principal components in use, k0 is the number of clusters in k-nearest neighbors, and δ is a ‘resolution’ parameter. We fix  =  for all datasets  are the default ones; the default value of m is 2,000, but we found that m = 1,000 gives the same results with the eight data sets and is faster to compute). We choose a separate value of δ for each dataset in a way such that the resulting number of clusters from a modularity optimization is exactly K . We fix  fails in this case, where the error rate is approximately the same as that of random guess. The proof of Next, we study the performance of classical PCA and IF-PCA. However, before we perform that, we present a lemma on classical PCA in Suppose we have a data matrix ξ is the first singular vector of X. By basic algebra, it is the first eigenvector of the N × N matrix XX′, or equivalently, the first eigenvector of XX′ − mIN. We writeμ‖2YY′ should be much larger than that of (ZZ′ − mIN). It is important to note that ‖μ‖2YY′ is a rank-1 matrix, where the spectral norm is N‖μ‖2. In addition, by random matrix theory (ZZ′ − mIN) concentrates at In such a setting, we investigate when the PCA approach in x theory , the speWe have the following lemma.Lemma 4.1μ‖2 ≫ log(N + m). Let ξ be the first left singular vector of X. When min{N, m} → ∞, with probability 1 − o(m−3),Here, we consider Model r with ‖r‖∞ = o (1) such thatBy Therefore, if we let This says that the PCA approach can fully recover the true class labels.We now come back to Models χ2-test is feasible. As mentioned before, letχ2-testing scores and recall that approximately,We first discuss when feature selection by χ2-test for a useful feature. We have two cases.β < 1/2). In this case, the number of useful features sp is much larger than ψj for a useful feature j may be much smaller than 1, even though χ2-test is not useful. Consequently, except for a negligible probability, the IF step of IF-PCA selects all features, so IF-PCA reduces to PCA.• (Less sparse case of β > 1/2). In this case, the number of useful features sp is much smaller than ψj → ∞ if j is a useful feature. In such a case, feature selection maybe successful and IF-PCA is significantly different from PCA.•  =  and noting that in this setting,Consider the first case and suppose we apply the PCA approach in Comparing this with the definition of We have the following theorem.Theorem 4.2Region for PCA). We fix  ∈ 2 and consider the clustering problem for Models p → ∞, if.(Possibility β < 1)Consider the second case, where we may have successful feature selection, so it is desirable to use IF-PCA. We assumeS(μ) is the true support of μ andWe recall that j, the SNR isBy , for any useful feature By elementary statistics, we have that approximately,μS is the sub-vector of μ with all entries restricted to S, and XS and ZS are the sub-matrices of X and Z respectively, with columns restricted to S. Therefore, in the clustering step of IF-PCA, we are, in effect, applying the PCA approach of XS, where we recall |S|/sp ≈ 1. Applying N, m) =  and noting thatTherefore, except for a negligible probability,Combining this with Comparing this with the definition of Theorem 4.3Region for IF-PCA). We fix  ∈ 2 and consider the clustering problem for Models p → ∞, if partitions the region {: 0 < β < 1, α > 0} into two regions.Impossibility {: α > α*, 0 < β < 1}. In this region, the Hamming clustering error rate of any method with polynomial computation time is bounded away from 0.• Region of Possibility {: α < α*, 0 < β < 1}. The region further partitions into two parts: β < 1/2 (left) and β > 1/2 (right).• Region of less sparse case where the number of useful features α, β) in this region, the Hamming error rate of PCA is o(1), so PCA achieves the optimal phase transition. In addition, in this case, the signals are too weak individually and feature selection is infeasible. Therefore, in the IF step, the best we can do is to select all features, so IF-PCA reduces to PCA.• The left is the α, β) in this region, the Hamming error rate of IF-PCA is o(1), so IF-PCA achieves the optimal phase transition. In addition, in this case, the signals are strong enough individually and feature selection is desirable. Therefore, IF-PCA and PCA are significantly different.• The right is the more sparse case, where the number of useful features β < 1, (1 − θ − 2β) < α < (1 − β)/2} (shaded green region of o (1), but that of PCA is bounded away from 0. Therefore, PCA is non-optimal in this particular region.• In particular, for any fixed parameters in the region {1/2 < We fix 0 < See IF-PCA is a simple and tuning-free approach to unsupervised clustering of high-dimensional data. The main idea of IF-PCA is a proper combination of feature selection and dimension reduction by PCA. In this study, we make several contributions. First, we extend IF-PCA to IF-VAE, by replacing PCA with the VAE, a popular UDL algorithm. Second, we study the theoretical properties of IF-PCA in a simple clustering model and derive the phase transitions. Our results reveal how the feature sparsity and the feature strength affect the performance of IF-PCA and explain why IF-PCA can significantly improve the classical PCA. Third, we investigate the performances of IF-PCA and IF-VAE on two applications, the subject clustering with gene microarray data and the cell clustering with single-cell RNA-seq data, and compare them with those of some other popular methods.X instead of the normalized data matrix W. On the eight single-cell RNA-seq data sets considered in this paper, IF-PCA(X) has the second-best accuracy in almost all the data sets, showing a stable performance across multiple data sets. We think IF-PCA has significant potential for single-cell clustering, as the method is simple, transparent, and tuning-free. Although the current IF-PCA(X) still underperforms compared to the state-of-the-art methods  in some data sets, it is hopeful that a variant of IF-PCA  can outWe also find that UDL algorithms do not immediately yield improvements over classical methods with the microarray data and the single-cell data. The IF-VAE underperforms compared to IF-PCA in most data sets; there are only a few data sets in which the IF-VAE slightly outperforms IF-PCA. The reason can be either that non-linear dimension reduction has no significant advantage over linear dimension reduction in these data sets or IF-VAE is not optimally tuned. How to tune the deep learning algorithms in unsupervised settings is an interesting future research direction. Moreover, the theory on the VAE remains largely unknown . A theorThe framework of IF-PCA only assumes feature sparsity but no other particular structures on the features. It is possible that the features are grouped  or have K, is given. When K is unknown, how to estimate K is a problem of independent interest. One approach is to use the scree plot. For example, K. Another approach is based on global testing. Given a candidate K, we may first apply a clustering method with this given K and then apply the global testing methods in K such that the global null hypothesis is accepted in all estimated clusters. In general, estimating K is an independent problem from clustering. It is interesting to investigate which estimators of K work best for gene microarray data and single-cell RNA-seq data, which we leave to future work.In the real data analysis, we assume that the number of clusters,


G be a finite group, let H be a core-free subgroup and let b denote the base size for the action of G on G/H. Let H of G with G is a strongly base-two group if G has base size 2. In this paper, we study the strongly base-two finite groups with trivial Frattini subgroup.Let  H. A subset B of base for G if the pointwise stabilizer of B in G is trivial. We write b for the base size of G, which is the minimal size of a base for G. Here H is a core-free subgroup of G and b is the smallest positive integer b with the property thatG is a base-two permutation group if Let The base size of a finite permutation group is a classical invariant that has been studied for many decades, finding a wide range of applications and connections to a diverse range of problems in group theory and related areas  such that G on G/H. We call G a strongly base-two group if Let H is a proper subgroup, then  for an unspecified solvable group of order n. We adopt the standard notation for simple groups of Lie type from . As in the above analysis of case (II), we may assume that L. Since H, W], say Let G in Theorem 6.4 with A, with the Sylow 2-subgroup of L acting faithfully on L acting as scalars on G is isomorphic to Let us consider the groups Suppose L is abelian and every subgroup of F(G) is normal in G. Now apply Corollary 3.11.Since By combining Theorems 6.1, 6.2, 6.4 and 6.6, we obtain the following corollary.G is solvable and either Suppose G is a finite group with In this section, we prove Corollaries 5 and 6. As usual, we assume :L as in .Corollary 5  Let G be a finite nonsolvable group. If G has the form given in part (i) of the corollary (note that the possibilities for T are recorded in Theorem 2).c(G) and r(G) as before. Since G is nonsolvable, it follows that L and S is an abelian 2-group. In particular, G is solvable and so we may assume G is nonsolvable.Now assume Now let us turn to Corollary 6. We first establish a more general result on regular orbits.L has no regular orbit on F(G). Then P and trivially on Q. In particular, Suppose F(G) contains an involution and so L does not have a regular orbit on F(G). For the remainder, let us assume L has no regular orbit on F(G).Clearly if p be an odd prime and set R has a regular orbit on R is abelian, so let us assume R is nonabelian. This implies that R as a subgroup of O(R) is cyclic and the Sylow 2-subgroup of R is abelian. Since we are assuming R| is indivisible by p. This implies that either R acts imprimitively on R acts primitively and is contained in the normalizer of a nonsplit maximal torus.First assume (a) holds and write R-invariant and we claim that R has a regular orbit on the remaining y acts trivially on a unique line. It follows that R has no regular orbit on R contains all such involutions. However, the group generated by these involutions has a nonabelian Sylow 2-subgroup, a contradiction.Let N that act trivially on a line are involutions in R has no regular orbit, then all such involutions must be in R and once again we reach a contradiction since the Sylow 2-subgroup of the group generated by these involutions is nonabelian. This justifies the claim.A very similar argument applies in case (b). Here L acting trivially on Q. By the previous claim, p, so there exists K acts faithfully on K does not have a regular orbit. Since K is normal in L, and since K is its own automorphism group, we deduce that Now write L has no regular orbit on F(G). Then A is an abelian group of odd order.Suppose L does not have a regular orbit on F(G), it follows that G has the prescribed form, then it is easy to see that L is the only core-free subgroup H of G (up to conjugacy) with Since L has a subgroup H and L are conjugate and thus A is a complement to F(G). Therefore, A is abelian and we note that A| is even, so the hypothesis A| is odd.To complete the proof, suppose p.We will also need the following proposition, which handles a family of groups with a primitive permutation representation of degree (i)If (ii)L is abelian. Then Suppose (iii)L is nonabelian and acts imprimitively on F(G). Then r is an odd prime divisor of Suppose (iv)L is nonabelian and acts primitively on F(G). Then Suppose Let p is odd and H is a nontrivial core-free subgroup of G. If L has a regular orbit on F(G) by Proposition 7.1. In particular, if p.The result is clear if L| is indivisible by p. In particular, F(G) in G are conjugate. Let L.Since L is abelian and let D be a subgroup of order p. Then Z has composite order. On the other hand, if Z has prime order, so L/Z acts regularly on the set of order p subgroups of F(G), in which case L fixes a line, this implies that L is cyclic of order Z is the stabilizer of any line, this forces First assume L is nonabelian. We divide the analysis according to whether or not L acts imprimitively (as a linear group) on F(G).For the remainder we may assume L acts imprimitively on F(G). Then L has a normal abelian subgroup L, then clearly F(G) fixed by L. This observation, together with the fact that r and we deduce that (iii) holds. Here it is straightforward to see that the only core-free subgroups H of G with First assume L is nonabelian and primitive on F(G). Since L is contained in the normalizer of a nonsplit torus in L has an abelian subgroup Now let us assume L is a dihedral group and |L| divides L has order at most 2. Since the involutions in L are self-centralizing, we deduce that the stabilizer of a pair of conjugate lines is trivial and thus H of G. In particular, we conclude that If Z has prime order and L/Z acts regularly on the set of lines of F(G), so D : Z is a maximal core-free subgroup of G for any subgroup D of order p, so Z must be the stabilizer of every line in F(G). Since L has even order, this implies that L must be in Z and no element in L of order 4 fixes a line. Therefore, the Sylow 2-subgroup of L is cyclic of order 4 and thus Now assume Corollary 6    Let G be a finite group with socle N and a core-free maximal subgroup. Then S and positive integer d. If S is nonabelian, then Theorem 3 implies that F(G) is the unique minimal normal subgroup of G.If Finally, let us assume r(G) denotes the maximal rank of a minimal normal subgroup of G. Since G has rank 1 or 2. As usual, write In this final section, we consider the strongly base-two groups with :L as in .H of F(G) with H, which means that Recall that if We have already handled several special cases in this setting. For instance, the nonsolvable strongly base-two groups were determined in Corollary 5, while the examples with a core-free maximal subgroup are recorded in Corollary 6. The latter result shows that we can find groups with O(L) be the largest odd order normal subgroup of L. Then O(L) is abelian and S. Note that the condition G has a minimal normal subgroup G acts as a nonabelian group, then For the remainder, let us assume L has a regular orbit on F(G). As a consequence, it is straightforward to show that there are no core-free subgroups H of G such that In view of Corollary 7.2, we may assume that L has odd order.The following result gives necessary and sufficient conditions for (i)L has a regular orbit on F(G).(ii)H is a core-free subgroup of F(G), then If (iii)L does not contain a subgroup G-module.Each y acts as a nontrivial scalar on each G-module and y acts nontrivially on G.(iv)L does not contain a subgroup G-module, and J acts faithfully on U.Suppose H of G contained in F(G).The necessity of (i) is clear . Set Y be the normal closure of L. Note that each nontrivial element of Y either acts as a scalar on each G and acts trivially on Y normalizes U and we deduce that VY is the normal closure of G. Since Let us assume there exists an element J as in (iv). Let G and set VJ is the normal closure of H. Then Now suppose that there exists a subgroup H be a maximal core-free subgroup of G with X is a nontrivial proper subgroup of H by conditions (i) and (ii). Let V be the normal closure of X and write H.We have now shown that if L on L of order H acts completely reducibly on Suppose L-module) is either reducible, or irreducible and primitive, then the intersection of the stabilizers of distinct L-conjugate lines in L on L-invariant just consists of scalars). On the other hand, if L-orbit, or the intersection of the stabilizers consists of scalars. By (ii), we can choose i, which implies that h acts as a scalar on L-module and h stabilizes each of the two lines permuted by L. If we now write W is a normal subgroup of G, then there are two cases to consider.Next observe that if W, which means that W and acts as a scalar on each First assume that W. By Proposition 7.1, this implies that U . Then by arguing as in the previous paragraph, we see that V. This contradicts (iv) and completes the proof.Now suppose that J is normal in L. This is clear in case (iv) and indeed we note that y acts as a scalar on each y is central in L and thus y is not central in L. Then G-module. Let Y be the normal closure of L. If Y and so L acts on Y as an elementary abelian 2-group. In particular, Y is a completely reducible L-module with 1-dimensional summands and therefore we can choose L.Let us note that we may modify conditions (iii) and (iv) in Theorem 8.2 so that the subgroup By inspecting the proof of Theorem 8.2, together with the relevant proofs in Sections G be a finite group with H of G with N is the normal closure of H in G.Let H of G contained in F(G).The next result shows that F(G) contains a core-free subgroup H of G with Suppose H is a maximal core-free subgroup of G. In particular, K is a proper subgroup of H, then Seeking a contradiction, let us assume H in G and write L on Let L acts imprimitively on L-orbit of i, it follows that H and U are L-conjugate and so there is some element i. But this gives First assume that y acts trivially on y stabilizes y centralizes Finally, suppose i, it follows that x must normalize Let To conclude, let us briefly explain how one can use Theorem 8.2 and Proposition 8.5 to analyze the groups with H represents the unique conjugacy class of core-free subgroups of G with H must intersect F(G) nontrivially, but it cannot be contained in F(G). Furthermore, there exists a normal subgroup J of L, as described in parts (iii) or (iv) of Theorem 8.2 (see Remark 8.3). Then First observe that if C of G ].Now F(G) has a subgroup L-module, then G. Therefore, in case (iii) we may assume that y acts as a scalar on each G has no imprimitive minimal normal subgroups. We leave the task of describing the precise structure of the groups with For example, if


Quantum contextuality supports quantum computation and communication. One of its main vehicles is hypergraphs. The most elaborated are the Kochen–Specker ones, but there is also another class of contextual sets that are not of this kind. Their representation has been mostly operator-based and limited to special constructs in three- to six-dim spaces, a notable example of which is the Yu-Oh set. Previously, we showed that hypergraphs underlie all of them, and in this paper, we give general methods—whose complexity does not scale up with the dimension—for generating such non-Kochen–Specker hypergraphs in any dimension and give examples in up to 16-dim spaces. Our automated generation is probabilistic and random, but the statistics of accumulated data enable one to filter out sets with the required size and structure. Quantum contextuality, which precludes assignments of predetermined values to dense sets of states, has found applications in quantum communication ,2,3, quaThere are three classes of contextual sets elaborated on in the literature which are not of the more common kind of Kochen–Specker (KS) sets ,33,34 anThe first class consists of the operator-based state-independent contextual (SIC) sets put forward by Klyachko et al. , Yu and n vectors, where n is the dimension of space in which a hypergraph resides [The second class consists of hypergraphs built by multiples of mutually orthogonal vectors where at least one of the multiples contains less than  resides ,34,41.The third class consists of the so-called true-implies-false and true-implies-true sets ,43.All sets from these three classes as well as their hypergraph generalization that we elaborate on are contextual, and therefore, we call them non-KS contextual sets.We provide a general method for arbitrarily generating many non-KS hypergraphs in spaces of up to 16-dim. In order to achieve these goals, we make use of non-binary non-KS McKay–Megill–Pavičić hypergraphs (MMPHs) and their language. By means of our algorithms and programs, we arbitrarily obtain many MMPHs, which can be used for various applications, e.g., to generate new entropic tests of contextuality or new operator-based contextual sets.The paper is organized as follows.n-dim space correspond to vectors and their orthogonalities, i.e., m-tuples  lines passing through them.Definition 2.An n-dim non-binary MMPH (NBMMPH),  [is an MMPH 1 s and 0 s in such a way thatfor which each hyperedge contains m vertices, 1. No two vertices within any of its edges are both assigned a value of 1;2. In any of its edges, not all of the vertices are assigned a value of 0.H), n≥3,  is an MMDefinition 3.An NBMMPH in which  KS MMPH.For  theorem ,34,45,46Definition 4.An NBMMPH in which  non-KS MMPH.In this paper, we consider only those non-KS MMPHs for which Definition 5.An n-dim binary MMPH (BMMPH),  MMPH for which each hyperedge contains m vertices, 1. No two vertices within any of its edges are both assigned a value of 1;2. In any of its edges, not all of the vertices are assigned a value of 0.Definition 6.A non-KS NBMMPH to which vertices are added so as to make the number of vertices equal to n in every hyperedge is called a filled MMPH.Filled MMPHs are mostly BMMPHs.Definition 7.A critical NBMMPH is an NBMMPH that is minimal in the sense that removing any of its hyperedges turns it into a BMMPH.Definition 8.Vertex multiplicity is the number of hyperedge vertexes “i” belongs to; we denote it by Definition 9.A master is a non-critical MMPH that contains smaller critical and non-critical sub-MMPHs. A collection of sub-MMPHs of an MMPH master forms its class.parity proof ofthe contextuality of a k-l NBMMPH with odd l and where each vertex shares an even number of edges stems from its inherent contradiction: because each vertex shares an even number of hyperedges, there should be an even number of hyperedges with 1 s. At the same time, each edge can contain only one 1 by definition, and since there are an odd number of hyperedges in the MMPH, there should also be an odd number of edges with 1 s.A Definition 10.A coordinatization of a non-KS NBMMPH is a set of vectors assigned to its vertices that is a subset of n-dim vectors in  MMPH or its smallest master (they need not coincide) or any of its masters.n-tuple of orthogonal vectors, while a “coordinatization” of each hyperedge of the original non-KS NBMMPH is represented by a vector m-tuple  or any its masters. In our present approach, a coordinatization is automatically assigned to each hypergraph by the very procedure of its generation from master MMPHs, as we show below.In other words, a “coordinatization” of each hyperedge of a filled MMPH or a smallest master MMPH is represented by an k vertices and l edges is denoted as a k-l MMPH. ASCII string representation is used for computer processing. MMPH strings are handled by means of algorithms embedded in the programs SHORTD, MMPSTRIP, MMPSUBGRAPH, VECFIND, STATES01, and others [An MMPH is encoded with the help of printable ASCII characters, with the exception of “space”, “0”, “+”, “,” and “.”, organized in single strings; its hyperedges are separated by commas, and each string ends with a period. When all ASCII characters are exhausted, one reuses them prefixed by “+”, and then again by “++”, and so forth. An MMPH with d others ,49,50,51To generate non-KS NBMMPHs, we make use of the following methods.M1 consists of dropping vertices contained in single hyperedges (multiplicity ity m=1)  of eitheM2 consists of a random addition of hyperedges to MMPHs so as to obtain bigger ones, which then serve us to generate smaller non-KS NBMMPHs by stripping hyperedges randomly again;M3 consists of the random deletion of vertices in either NBMMPHs or a BMMPHs until a non-KS NBMMPH is reached.We combine all three methods to obtain an arbitrary number of non-KS NBMMPHs in an arbitrary dimension. The methods rely on the property of MMPHs where, by stripping an MMPH or NBMMPH  or BMMPH of its hyperedges, we can arrive at smaller non-KS NBMMPHs in contrast to a critical KS NBMMPH whose stripping of hyperedges can never yield another  NBMMPH.k-l MMPHs refer to hypergraphs with k vertices and l hyperedges (Definition 1), while the corresponding graphs have more than l edges. For example, in ALK corresponds to a graph clique with three edges: AL, LK, and KA.In M1. The 13-16 MMPH is not critical, and it contains four critical sub-MMPHs, the smallest of which is 10-9 [Yu-Oh’s three-dim non-KS NBMMPH, shown in  is 10-9 .Howard, Wallman, Veitech, and Emerson’s four-dim 30-108 non-KS NBMMPH, shown in Cabello, Portillo, Solís, and Svozil’s five-dim 10-9 non-KS NBMMPH, shown in Our methods can generate NBMMPHs that are critical as well as those that are not. Therefore, although none of the aforementioned examples are critical, we focus on critical ones, because they offer the simplest implementation and presentation. The rationale for adopting such an approach is that only minimal contextual sets, i.e., critical NBMMPHs, are relevant for experimental implementations, since their supersets just contain additional orthogonalities that do not change the contextuality property of their smallest critical set. Hence, while designing MMPHs for particular implementations, we should attempt to find the ones that are critical and are provided via automated generations of MMPHs.M1 and M2. Therefore, below, we give distributions and samples of just four- and five-dim critical non-KS NBMMPHs presented in n vertices (n being the dimension of the considered MMPH), none of which has the multiplicity In , we giveM1, we first generate the supermasters from the vector components. In the four-dim space, we obtain the 24-24 supermaster from the To obtain non-KS NBMMPHs via n-dim critical non-KS MMPHs under the requirement that at least one of their hyperedges must contain n vertices, of which none have a multiplicity of 1 , because We generate Cabello, Portillo, Solís, and Svozil also give a number of minimal six-dim TIFS non-KS NBMMPHs in . In dimensions greater than nine, such vertices do not appear, although even here we can avoid their generation by applying M3 to KS NBMMPHs, as shown in In the seven-dim space, masters obtained from simple vector components, such as M1. However, the MMPHs with M3, as shown in M3.The eight-dim MMPH master is big (2768-1346016), but the stripping technique can still provide us with non-KS NBMMPHs via riticals  so as toM1 applicable. Thus, after the random stripping of 12,068,200 hyperedges, we obtained submasters with 505 hyperedges. By requiring that at least one of the hyperedges contains n vertices and that some of them can have the multiplicity 4ac7efhK2 contains nine vertices.  The filled 13-16, i.e., 44-6, also shown in M2 inapplicable.The nine-dim NBMMPH master obtained from n-vertex requirement in full If we wanted to keep our pscaling ,56, as fM3. The procedure consists of removing vertices and/or hyperedges in such a way that an NBMMPH stops being critical, which enables us to generate smaller critical non-KS NBMMPHs from it via STATES01.A 10-dim or any higher-dimensional masters are too big to be generated from vector components. Therefore, to obtain the non-KS MMPH in those dimensions, we rely on minimal KS NBMMPHs obtained via dimensional upscaling  while apIn In M3.In the following sections, we stay with this approach while applying n vertices, none of which has the multiplicity It has been proven that the minimal complexity  of the dimensional upscaling of KS MMPHs does not scale up with dimension . In  5656, we giIn this paper, we first generated non-KS contextual NBMMPHs (non-binary MMP hypergraphs) with the help of master sets generated from simple vector components whose complexity exponentially scales with dimension—for dimensions four to eight—and then by means of methods whose complexity does not scale with dimension. The need for developing such methods and obtaining MMPHs in higher dimensions has emerged from recent elaborations of classes of contextual sets that are not of the KS kind, all of which have an MMP hypergraph representation. Examples of such elaborations in the literature and their correspondence with MMPHs are given in In Therefore, to ensure arbitrarily exhaustive generation of MMPHs in ever higher dimensions, we need a method whose complexity does not grow with the dimensions. For comparatively small KS MMPHs, such a method—dimensional upscaling—was recently developed in  based onhttp://puh.srce.hr/s/Qegixzz2BdjYwFL (accessed on 22 July 2023). MMPHs can be visualized via hypergraph figures consisting of dots and lines and represented as a string of ASCII characters. The latter representation enables the processing of billions of MMPHs simultaneously via supercomputers and clusters. For the latter elaboration, we developed other dynamical programs specifically to handle and parallelize jobs with arbitrary numbers of MMP hypergraph vertices and edges.The methods used to handle quantum contextual sets rely on algorithms and programs within the MMP language: VECFIND, STATES01, MMPSTRIP, MMPSHUFFLE, SUBGRAPH, LOOP, and SHORTD developed in ,51,57,58To summarize, based on elaborations of non-KS sets that recently appeared in the literature and of which we provided several examples in


Correction: Mol Cancer 18, 82 (2019)https://doi.org/10.1186/s12943-019-1016-0https://doi.org/10.18632/oncotarget.18204) are indeed the same and this image belongs to the ‘si-NC of NOZ/Dox’. During that time, we were working on several projects simultaneously and the error should be occurred in the data storage. And we had contacted the editorial office of Oncotarget to explain and correct this error.After receiving the comment, we have carefully re-checked the raw data and confirmed the error in Fig. 4A. The two images of ‘si-NC of NOZ/Dox’ in Fig. 4A in this article and the ‘si-NC of NOZ’ in Fig. 6C in our other article in Oncotarget (Besides, in order to make sure no further problems will be pointed out after the correction of this time, we have self-examined all the raw data of all the projects conducted simultaneously during that time for article . And we


Retraction Note: BMC Cancer 19, 224 (2019)10.1186/s12885-019-5419-5The Editors have retracted this article because of significant concerns with a number of figures, specifically:Figure 7d, CD133 panel, appears to be the same as Fig. 2D, Oct 4 panel, in [Figure 4b, FAP-a panel, appears to be the same as Fig. 3D, p53 panel, in Figure 3a, b-actin panel, appears to be the same as Fig. 3K, b-actin panel, in [The authors were contacted and asked for an explanation but did not provide one. The Editors therefore no longer have confidence in the integrity of the data in this article.Kai-qun Ren agrees with the retraction. None of the other authors have responded to any correspondence from the Editors about this retraction.


In the published article, there was an error in the article title. Instead of “SARS-CoV-2 antibody response after mRNA vaccination in healthcare workers with and without previous COVID-19, a follow-up study from a university hospital in Poland during 6 months 2021”, it should be “SARS-CoV-2 antibody response after mRNA vaccination in healthcare workers with and without previous COVID-19, a follow-up study from the University Hospital in Krakow, Poland”.The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.


Nature Communications 10.1038/s41467-023-36884-4, published online 01 March 2023Correction to: The original version of this Article contained an error in Fig. 1, in which panel b was missing, and panels c–e were incorrectly labelled as b–d respectively. The correct version of Fig. 1 is:which replaces the previous incorrect version:This has been corrected in both the PDF and HTML versions of the Article.


Nature Ecology& Evolution 10.1038/s41559-023-02058-0. Published online 18 May 2023.Correction to: x-axis label order: “Phloem” and “Blood,” in the second and third positions, respectively, were accidentally reversed. The labels have been corrected in the HTML and PDF versions of the article.In the version of this article originally published, there was an error in the Fig. 1c