John P. A. Ioannidis emphasizes the central role of prior probabilities . His conR, of 1.0 (a probability of 0.5), although his examples go as high as R = 2.0. A positive study must, by definition, increase the likelihood that the hypothesis is true. It might have been clearer had Ioannidis chosen to relate odds to odds or probability to probability; in both cases, a neutral study would produce a straight line along a 45-degree diagonal.Unfortunately, his formulation relates the post-study probability that the study's conclusion is true to the pre-study odds. The results might have been clearer had he also plotted the relation of odds to probability, a curvilinear relationship, assuming the study carried no information. Further, the various graphs are right-truncated at pre-study odds, R × LR. For a single unbiased study, LR = (1 − β)/α. When incorporating study bias, u, as defined by Ioannidis, LR = (1 − β[1 − u])/(α[1 − u] + u). For a typical study with α = 0.05 and β = 0.2 , LR = 16. When R is less than 1:16 (a probability of 0.0588), the post-study odds will be less than one—i.e., the study's hypothesis will be more likely false than true.The pre-study to post-study relation can more simply be expressed using the odds-likelihood form of Bayes rule—i.e., the post-study odds equal the pre-study odds multiplied times the likelihood ratio (LR) of the study. Then, the equations for positive predictive value (PPV) become the simple product of R = 1. Then, LR would merely need to exceed one for the study's conclusions to be more likely true than false. At the common significance levels (α) of 0.05 and 0.01, the requisite study powers would merely need to exceed 0.05 and 0.01 respectively, corresponding to maximum type II error rates (β) of 0.95 and 0.99. Such lax requirements would almost always be met for a published study. Hence, the common belief that the vast majority of studies have valid conclusions would be correct if we can assume that the pre-study odds are truly uninformative. However, as Ioannidis suggests, this is unlikely to be the case.For non-Bayesians, statistical significance testing presumes uninformative prior probability—i.e., Two more corollaries might be added. The higher the pre-study odds that the study's hypothesis is true, the lower the requisite power (study size and effect size) required to make the study's findings more likely true than false. When studies are published, the investigator should estimate the pre-study odds and report the LR implied by the observed effect.From the perspective of an epidemiologist or a statistician, the relevant question is whether the study's hypothesis is true—i.e., is the probability of the hypothesis greater than 0.5? For clinicians and their patients, the relevant question is whether a particular strategy should be followed in an individual patient or a subset of similar patients. That decision (or recommendation to the patient) will depend on the pre-study likelihood of benefit in that patient and on the relative magnitude of benefits and risks of that strategy, if the diagnosis in that patient is uncertain. For many such decisions, the “more likely true than false” criterion may not be the best decision rule. For serious diseases and treatments of only modest risk, post-study probabilities of considerably less than 0.5 may be sufficient to justify treatment .Ioannidis's provocative Essay is a timely call for careful consideration of published studies. The odds-likelihood formulation suggested herein may be helpful in providing a more intuitive model. Clinicians now need to take it to the next step.


Multicategory Support Vector Machines (MC-SVM) are powerful classification systems with excellent performance in a variety of data classification problems. Since the process of generating models in traditional multicategory support vector machines for large datasets is very computationally intensive, there is a need to improve the performance using high performance computing techniques.In this paper, Parallel Multicategory Support Vector Machines (PMC-SVM) have been developed based on the sequential minimum optimization-type decomposition method for support vector machines (SMO-SVM). It was implemented in parallel using MPI and C++ libraries and executed on both shared memory supercomputer and Linux cluster for multicategory classification of microarray data. PMC-SVM has been analyzed and evaluated using four microarray datasets with multiple diagnostic categories, such as different cancer types and normal tissue types.The experiments show that the PMC-SVM can significantly improve the performance of classification of microarray data without loss of accuracy, compared with previous work. In recent years the use of DNA microarrays has resulted in the creation of large datasets of molecular information characterizing complex biological systems. Machine learning algorithms applied to DNA microarray data based on molecular classification approach have shown to have statistical and clinical relevance for a variety of cancer types . When apOne strategy to solve classification problems is for the computer to learn the input/output functionality from training data. The learning algorithm attempts to find the underlying target function which maps from the input to the output. The estimate of the target function is known as the solution of the learning problem, which is chosen from a set of candidate functions that map from the input space to the output domain. These candidate functions are known as hypotheses. The learning algorithm takes the training data as input and selects a hypothesis from the hypotheses. The learning algorithm must have the ability to correctly classify data that are not in the training set, known as generalization .When the learning problem deals with binary outputs, it is referred to as a binary classification problem, and when the problem deals with a finite number of outputs, it is referred as multi-category classification problem. In recent years, Support Vector machines (SVMs), a supervised machine learning algorithm is being used to solve many classification problems. SVM algorithm classifies the data by finding the optimal hyperplane between the classes of data. The focus is to maximise the margin between the parallel hyperplanes . The training data which lie on the optimal hyperplane are called support vectors. This maximum-margin classifier was proposed by Vapnik Chervonenkis theory . Many imlight T, α = T and C is a constant. e is a vector of all ones. Q is the symmetric positive semi-definite matrix and entries Qi, j are defined aswhere Qi,j = yiyjK, i, j = 1,2 ..., N,     (4)K denotes a kernel function, such as linear kernel, polynomial kernel and radial basis function (RBF). RBF kernel is used in the numerical experiments of this project.where Currently, decomposition method is one of the major methods to train SVM, in which only a subset of variable is considered per iteration. The subset, denoted as B, is called working set. If B is restricted to have only two elements, this special type of decomposition method is the Sequential Minimal Optimization (SMO). There are four steps to implement SMO :1 as the initial feasible solution. Set k = 1.1. Find αkis a stationary point of (2), stop. Otherwise, find a two-element working set B = {i,j}⊂{1,..., l}. Define N ≡ {1,..., l}\ B, and  and  as sub-vector of αkcorresponding to B and N, respectively.2. If αai,j = Kii + Kjj - 2Kij > 03. If B :Solve the following sub-problem with the variableαelseSolvesubject to constra int s of (5) to be the optimal solution of (4) and  Set k ← k + 1 and go to step 2.4. Set After the PMC-SVM is trained, the decision functions are used to classify new datasets.mi, j, i, j = 1,...,k, for k categories, each denoted by Ci, i = 1,...,k. Each model mi, j is generated from categories Ci and Cj. A total of  submodels are generated from k categories in MC-SVM. For example, if we have 4 processors and 8 categories, 28 models will be generated, as shown in Figure Task decomposition is the first step for the parallel algorithm design. In multicategory classification of support vector machines, the algorithm will generate multiple binary SVM models, denoted by k0 submodels are assigned to p PEs, each PE performs the generation task of  submodels. A task scheduling algorithm is needed to determine which categories of data are used by each PE. The pseudocode of scheduling tasks is given as followsWhen Task scheduling algorithm// Task scheduling algorithm for PMC-SVM training// model_no: the index of models// p: the total number of PEs.// my_rank: processor index, 0<=rank <p// i, j: class index// k: total number of classesmodel_no = 0;for {for{model_no++;if(my_rank = = model_no % p);// processor with my_rank gets class i and class jget_data(i) // read class i dataget_data(j) // read class i datagenerate_bin_svm; // generate a submodel for category i and j}}With this scheduling algorithm, tasks are evenly distributed among multiple processors and achieve better balance.The Portable Batch System (PBS) is a flexible workload management system . It prov#PBS -S /bin/tcsh# Set up the cpus and memory#PBS -l ncpus = 4# Name this job "job_svm"#PBS -N job_svmrm job_svm.o* # Remove any PBS output files from previous runsrm job_svm.e* # Remove any PBS error files from previous runs# Do our job# make the makefilemake# run the svm-train, save the file to svm8.out/usr/local/appl/mpich-1.2.1/bin/mpirun -np 8 svm-train letter.scale >> svm.outCZ and YD initiated the project and carry out the design and analysis. PL and AR implemented the PMC-SVM and evaluated the performance. All authors contributed and approved the final manuscript.


Signal transduction pathways are usually modelled using classical quantitative methods, which are based on ordinary differential equations (ODEs). However, some difficulties are inherent in this approach. On the one hand, the kinetic parameters involved are often unknown and have to be estimated. With increasing size and complexity of signal transduction pathways, the estimation of missing kinetic data is not possible. On the other hand, ODEs based models do not support any explicit insights into possible  flows within the network. Moreover, a huge amount of qualitative data is available due to high-throughput techniques. In order to get information on the systems behaviour, qualitative analysis techniques have been developed. Applications of the known qualitative analysis methods concern mainly metabolic networks. Petri net theory provides a variety of established analysis techniques, which are also applicable to signal transduction models. In this context special properties have to be considered and new dedicated techniques have to be designed.Saccharomyces cerevisiae serves as case study.We apply Petri net theory to model and analyse signal transduction pathways first qualitatively before continuing with quantitative analyses. This paper demonstrates how to build systematically a discrete model, which reflects provably the qualitative biological behaviour without any knowledge of kinetic parameters. The mating pheromone response pathway in feasible t-invariants, which represent minimal self-contained subnets being active under a given input situation. Each of these subnets stands for a signal flow in the system. We define maximal common transition sets (MCT-sets), which can be used for t-invariant examination and net decomposition into smallest biologically meaningful functional units.We propose an approach for model validation of signal transduction pathways based on the network structure only. For this purpose, we introduce the new notion of The paper demonstrates how Petri net analysis techniques can promote a deeper understanding of signal transduction pathways. The new concepts of feasible t-invariants and MCT-sets have been proven to be useful for model validation and the interpretation of the biological system behaviour. Whereas MCT-sets provide a decomposition of the net into disjunctive subnets, feasible t-invariants describe subnets, which generally overlap. This work contributes to qualitative modelling and to the analysis of large biological networks by their fully automatic decomposition into biologically meaningful modules. In contrast to the above mentioned techniques, Petri nets provide useful unique visualisation techniques with possibilities for hierarchical modelling and for animation. Especially this aspect is of crucial importance for experimentalists and for the dialog between experimentalists and theoreticians.Carl Adam Petri introduced the basic concepts of Petri nets in his dissertation in the context of technical systems . He deveSignal transduction pathways exhibit some special properties. In contrast to metabolic networks, in signalling pathways there is no substance flow, but a signal flow, which is performed, for example, by phosphorylated and dephosphorylated protein forms. There are no stoichiometric reaction equations given, which dictate the net structure and define the arc weights. Modelling signalling pathways we have to work on another abstraction level than in modelling metabolic networks. Petri nets provide a generic description principle, applicable on any level of abstraction.In this paper we present a method, which enables the systematic deduction of the net structure from the known signalling processes in the cell, which are translated into logical terms as implication, conjunction, inclusive or exclusive disjunction, and negation. These terms can be translated unambiguously to Petri net components. The resulting model is a qualitative, discrete Petri net, which can be analysed for certain system properties to validate the model.S. cerevisiae and recall the necessary Petri net definitions. Then, we propose and discuss in detail a systematic modelling method with respect to signalling pathways. We focus on the analysis of signal flows and introduce for this purpose the new concepts of feasible t-invariants and of maximal common transition sets (MCT-sets). Feasible t-invariants stand for minimal self-contained subnets being active under a given input situation, while MCT-sets can be interpreted as smallest biologically meaningful functional units. Both concepts define a fully automatic net decomposition into biologically meaningful modules. Whereas MCT-sets provide a decomposition of the net into disjunctive subnets, feasible t-invariants describe subnets, which generally overlap. We apply our modelling technique to the analysis of the pheromone response pathway in S. cerevisiae and discuss the results. Finally, we give conclusions indicating also further projects.In the following, we give a brief introduction into the mating pheromone response pathway in S. cerevisiae is reviewed, e.g., in [S. cerevisiae there are two different mating types of haploid cells, MATα and MATa. Cells of opposite mating type can mate, i.e., fuse to a diploid cell. This process is stimulated through small peptide pheromones, the α-factor and the a-factor, respectively. A haploid cell secretes the own factor and carries receptors for detecting a cell of opposite mating type via the other factor. When a yeast cell is stimulated by pheromone, secreted by a nearby cell of the opposite mating type, it starts a complex signalling pathway to undergo a series of physiological changes in preparation for mating. These changes are controlled and regulated through the pheromone signalling pathway, which is nearly the same in MATα and MATa cells.The signalling pathway of the mating pheromone response in e.g., in ,4,30. Ina cells, so the considered cells have an α-factor-specific cell surface receptor STE2, which is coupled to a heterotrimeric G-protein. This G-protein consists of the three subunits Gα, Gβ, and Gγ, whereby the latter two can act as heterodimer Gβγ. A conformation change of the receptor catalyses an exchange of GDP to GTP in the Gα subunit leading to a dissociation of this monomer. If GTP is hydrolysed to GDP, the monomer reassociates with the dimer to the trimeric form of the G-protein [βγ subunits mediate a signal on the MAP kinase cascade by interacting with Cdc24 (Far1 transmitted), protein kinase Ste20, and scaffold protein Ste5. The Ste20, Ste11, and Ste7 kinases, which are activated serially, phosphorylate and activate itself the two MAP kinases Fus3 and Kss1. But, Fus3 attenuates the activation of Kss1 [In the following, we refer to this pathway in MAT-protein . The G-p of Kss1 . Fus3PP  of Kss1 . Pheromo of Kss1  or degra of Kss1 .P = {p1,..., pn} and transitions T = {t1,..., tm}, and directed arcs, which are weighted by natural numbers and connect only nodes of different type. Places model typically passive system elements as conditions, states, or biological species, i.e., chemical compounds as proteins. Transitions stand generally for active system elements as events, or chemical reactions as de-/activation. In graphical representations, places are depicted as circles and transitions as rectangles. Transitions without preplaces (postplaces) are called input (output) transitions and are drawn as flat rectangles. The arcs in the net describe the causal relation between active and passive elements. They are illustrated as arrows and labelled with their weight, if it is larger than one. Arcs connect an event with its preconditions, which must be fulfilled to trigger this event, and with its postconditions, which will be fulfilled, when the event takes place.Let us first recall some basic definitions of Petri nets. More formal definitions are given, e.g., in ,18,35. Pread arcs.The fulfilment of a condition is realised via tokens residing in places. Principally, a place in a discrete net may carry any integer number of tokens, indicating different degrees of fulfilment. If all preplaces of a transition are marked sufficiently (corresponding to the arc weights) with tokens, this transition may fire. If a transition fires, tokens are removed from all its preplaces and added to all its postplaces, each corresponding to the given arc weights. Thus, the tokens are the dynamic elements of the system. If a condition must be fulfilled, but the firing of an adjacent transition does not remove any tokens from it, these nodes are connected via two converse arcs. In the following, these arcs are represented by bidirectional arrows as a short-hand notation and are called From a biological viewpoint, the tokens residing in places indicate whether the corresponding chemical species is present, i.e., its concentration is above a certain concentration level (threshold) in the cell. This presence enables the chemical reactions modelled by the place's posttransitions to take place.m ∈ marking of the net. Accordingly, the initial marking m0 of a net describes the system state before any transition has fired.A current distribution of the tokens over all places, usually given as C of a given Petri net is an (n × m)-matrix . Every matrix entry cij gives the token change on the place pi by the firing of the transition tj. Thus, the incidence matrix does not reflect read arcs. A t-invariant is defined as a non-zero vector x ∈ The incidence matrix C·x = 0.     (1)realisable, if a marking is reachable, such that all transitions of the t-invariant are able to fire in a suitable partial order. Analogously, a p-invariant is defined as a non-zero vector y ∈ A t-invariant represents a multiset of transitions, which have altogether a zero effect on the marking, i.e., if all of them have fired the required number of times, a given marking is reproduced. The invariant property holds for an arbitrary initial marking. A t-invariant is called y·C = 0.     (2)y and any markings mi, mj ∈ m0 by the firing of transitions, it holdsA p-invariant characterises a token conservation rule for a set of places, over which the weighted sum of tokens is constant independently from any firing, i.e., for a p-invariant y·mi = y·mj.     (3)x are called the support of x, written as supp(x). Considering Equations (1) and (2), it is obvious that a sum of t-invariants (p-invariants) gives again a t-invariant (p-invariant). An invariant x is called minimal, if its support does not contain the support of any other invariant z, i.e.,The nodes corresponding to the non-zero entries of an invariant z : supp(z) ⊂ supp(x),     (4)∄ invariant x is one.and the greatest common divisor of all non-zero entries of A net is covered by t-invariants (p-invariants), if every transition (place) participates in a t-invariant (p-invariant). A t-invariant (p-invariant) defines a connected subnet, consisting of its support, the support's pre- and postplaces (pre- and posttransitions), and all arcs in between.A ⇒ (B ∧ C) and (D ∧ E) ⇒ F, respectively. If there is a token on place A, the precondition A is fulfilled, and the corresponding posttransition may fire. This event will fulfil the postconditions B and C getting each a token. This example illustrates that the number of tokens in a net is not generally conserved. The fulfilment of condition F in Figure D and E.First, we want to define general network components in the language of Petri nets. Considering substructures, which model logical terms as a Petri net, an ordinary implication is a trivial case. Figure G ⇒ (H ∨ I) and (J ∨ K) ⇒ L, respectively. The two posttransitions of place G are in conflict, i.e., if there is one token on place G, both its posttransitions may fire, but only one of them can actually fire. This subnet represents an exclusive disjunction. Therefore, a nondeterministic behaviour results. In contrast to that, the pretransitions of place L are concurrent and can fire independently from each other, if any. With this subnet an inclusive disjunction is represented.Figure In all implications, presented in Figure y, which holds |supp (y)| = 2 may stand for a subnet, representing a negation by two places (A and ¬A), see Figure p1 (A) and p2 (¬A) forming the p-invariant, i.e., the token depicted on place p1 circulates only between p1 and p2 by firing of t1 and t2, respectively. To preserve this invariant structure, the adjacent transitions t3 and t4, respectively, are connected via read arcs.A crucial point in signal transduction pathways are modifications of certain proteins, resulting in a functionally active or inactive form of these proteins. Many of them are activated by phosphorylation and deactivated by dephosphorylation, and act in different ways depending on their current state. A qualitative model allows by its structure to distinguish between different states of one protein. Modelling this situation, we use a subnet, which includes the different modifications (states) of one protein as places forming a p-invariant. A p-invariant Altogether, there are three cases in signal transduction models, which can be distinguished by their different use of read arcs:A does not lose its activity by interacting with a second substance B, see Figure • Case 1: A substance C triggers several events represented by transitions, which are independent of each other, see Figure • Case 2: A substance • Case 3: The above mentioned context of a p-invariant, see Figure G are in conflict. Thus this subnet represents an exclusive disjunction G ⇒ (H ∨ I). By applying read arcs in such a substructure, the posttransitions of the corresponding place are concurrent, i.e., independent from each other. Thus, the subnet in Figure C ⇒ (D ∨ E).The subnet presented in Figure Before turning to the next step, the model validation, let us summarise the main steps of the systematic model development:1. compilation of biological knowledge about the signalling pathway of interest, e.g., from the literature or by database search,2. translation of the interactions of relevant biological species into basic logical terms,3. transferring these logical terms into net components,4. assembling these net components to one Petri net,5. if necessary, include input and output transitions ensuring that there are no places without pre- or posttransitions in the net.The last but one point ensures that the developed Petri net is connected. The input and output transitions introduced in the last point make the net to a transition-bordered one. They represent the interface of the biological system to its surroundings.The input transitions of a transition-bordered Petri net cause unboundedness, i.e., there is no upper bound for the number of tokens in such a net. Therefore, the state space of the net, i.e., all reachable markings, is infinite, and the dynamic net properties cannot be decided by the computation of the reachability graph, which contains all reachable markings as nodes. But, we do get information about the model dynamics via the analysis of the net structure, represented by the incidence matrix. Thus, our approach to validate the model is mainly based on the net invariants. The definition of invariants, see Equations (1) and (2), was introduced in . The appP-invariants in signal transduction models represent also some kind of conservation relationship, however in the sense of the several modifications of a given species as introduced in Figure The next step is to consider all possible signal flows through the network represented by realisable t-invariants, and to check their biological meaning. All possible t-invariants can be computed as non-negative linear combination of the minimal ones, see Equation (1). In former case studies, which do not contain the special p-invariant construct, see Figure The calculation of t-invariants does not take into account the initial marking. Moreover, read arcs are not reflected in the incidence matrix. Therefore, the question arises, whether a minimal t-invariant is really realisable. We assume that read arcs are the only source for non-realisability, i.e., there is a sufficient token number in each of the minimal p-invariants in the initial marking.with read arcs. For example, the p-invariant-adjacent transitions t3 and t4 in Figure p1 and p2, respectively, is not reflected in the incidence matrix. If this kind of transitions is part of a t-invariant, whereby their preplaces (connected via read arcs) carry no tokens, they are actually not able to fire. Therefore, the minimal t-invariants have to be processed to get realisable t-invariants for a suitable marking.The use of read arcs corresponding to the first case described above is discussed in . For an In accordance with our objective of model validation, we are looking for minimal self-contained t-invariants, which are realisable in the initial marking, i.e., minimal multisets of transitions, which can fire in an appropriate order without the firing of the other transitions, not belonging to that t-invariant. Those t-invariants are called feasible t-invariants. To make a t-invariant feasible, all of its involved transitions have to be able to fire, i.e., for each of them it is true either that it is an input transition or that its preplaces enable it to fire. The latter case means that either the considered preplaces carry already sufficient tokens in the initial marking or these preplaces get sufficient tokens via the firing of other transitions involved in this t-invariant. We distinguish between realisable and feasible t-invariants, because the notion of realisability of a t-invariant is a general term of Petri net theory, referring to the reachability of a marking, where the t-invariant's transitions can fire. Feasible t-invariants are special realisable t-invariants, which are minimal ones concerning their realisability in the initial marking. They are introduced because of a special net structure, which causes the interruption of the t-invariant, when it leads for example to a p-invariant, which carries a special marking.z. Let us formulate this processing of the minimal t-invariants in a general form. For each read arc between a transition tj and an empty place pi of a p-invariant the following procedure has to be applied. Let t1,...,tk be all the pretransitions of pi, which are not connected via a read arc with pi, see Figure pi in the initial marking, a t-invariant is not realisable if it includes tj and excludes tl ∀ l ∈ {1,..., k}. The following combinations z of feasible t-invariants x and one non-feasible t-invariant y reestablish the given read arc between transition tj and place piTable z = x + y ∀x ∃ l ∈ {1,...,k} : xl ≠ 0xj = 0with yj ≠ 0 ∧ ∀ l ∈ {l,...,k} : yl = 0.     (5)and x providing a token on the place pi one linear combination z is constructed. Appropriate combinations have to be performed iteratively, if there occur several read arcs in the net, which have to be bridged. The combined t-invariants do not hold the criterion to be minimal, see Equation (4). But, since only inevitable combinations of minimal t-invariants are constructed, see the restrictions in Equation (5), the resulting combined t-invariants are minimal with respect to their realisability in the initial marking. Moreover, they only have non-negative entries, because they are build as a sum of two t-invariants, which are defined as vectors in Note that for each feasible minimal t-invariant Feasible t-invariants define, similarly to the t-invariants, connected subnets, characterised by its support. Generally, these subnets overlap. They represent minimal self-contained subnets being active under a given input situation. So, each of these subnets stands for a possible signal flow in the net. The examination of the feasible  t-invariants for their biological plausibility is a central step of model validation. It has to be checked, whether each feasible t-invariant has a biological meaning and whether every modelled part of the considered signalling pathway is reflected in a corresponding feasible t-invariant .Finally, it has to be checked, whether the net is covered by t-invariants. This property ensures that every transition participates in a t-invariant, i.e., every biological atomic action in the model may take place as part of the basic behaviour of the net.meet at a minimal p-invariant with a corresponding place not carrying a token.While minimal t-invariants are computed based on the net structure only, feasible t-invariants are constructed with respect to the marking of the net. Summarising the discussion above, the following situations are possible: the minimal t-invariants can reach (1) from input to output transitions, (2) from input transitions to minimal p-invariants, (3) from minimal p-invariants to minimal p-invariants, or (4) from minimal to output transitions. The transitions of a t-invariant influence a minimal p-invariant in such a way that altogether its initial state is reproduced. Not every possible combination is considered to bridge a read arc and to connect minimal t-invariants. Such a procedure would lead to t-invariants reaching from input transitions to output transitions, only. As described above, those minimal t-invariants are combined, which maximal common transition sets (MCT-sets) by their occurrence in the minimal t-invariants: ∀i, j ∈ {1,...,m} the transitions ti and tj are grouped into the same MCT-set, if and only if they participate in exactly the same minimal t-invariants, i.e., all t-invariants x holdIn order to support the check of feasible t-invariants for their biological meaning, the transitions are grouped into so-called χ{0}(xi) = χ{0}(xj),     (6)χ{0} denotes the characteristic function, binary indicating if an argument is equal to zero. This support-oriented grouping leads to maximal sets of transitions, where each set of transitions ϑ holdswhereas x ∈ X : ϑ ⊆ supp(x) ∨ ϑ ∩ supp(x) = ∅,     (7)∀X denotes the set of all minimal t-invariants x.whereas T, the set of transitions, which leads to a partition of T. The equivalence classes ϑ are the MCT-sets. Transitions, not contained in any t-invariant, form their own MCT-set. MCT-sets define subnets, but not necessarily connected ones. An example of a disconnected MCT-set will be given below in the presented model.The grouping according to Equation (6) represents an equivalence relation in The resulting MCT-sets are disjunctive and represent a possible decomposition of large biochemical networks into rather small subnets, which can be read as functional units. Because of the way, in which they are generated, each of the MCT-sets may represent a signalling unit or a building block with its own biological meaning. In the following, only those MCT-sets are considered, which contain more than one transition.S. cerevisiae. The Petri net in Figure In accordance with the elucidation above we developed a Petri net, modelling the signal transduction pathway of the mating pheromone response in α-factor (via Bar1), of the receptor Ste2 (endocytosis induced by Yck1 and Yck2), of G-protein subunits (via Sst2), of the MAPK cascade (via labelling for degradation), and of the regulation of transcription (via repression of the transcription factor Ste12).Proteins, protein complexes, and other chemical compounds are represented as places, and complex formation/cleavage, protein de-/phosphorylation, and other reactions by transitions. Tables complex2 (p17) does not lose its activity by firing of transition Ste11_phos_Ste7 (t18), i.e., in the MAPK-complex the protein Ste11 remains in its active form after activating the protein Ste7, see case 1. The place Fus3PP (p20) triggers several transitions Ste12_inhibit_phos t22), Ste12_phos (t24), Far1-phos (t31), and Sst2_phos (t34), which are independent of each other, see case 2. The place pair Kss1_phos (p38) and unphos_Kss1 (p39) forms a p-invariant as discussed in case 3.We take into account the above mentioned substructures, e.g., the place complex3 (p18), complex4 (p19), compl_without_Fus3 (p21), and the transitions Ste7_phos_Fus3 (t19), Fus3PP-release (t20), and binding_free_Fus3 (t21). By using this arc weight, it is ensured that the activation of the transition Ste7_phos_Fus3 (t19) depends on firing of transition Ste11_phos_Ste7 (t18), see Figure With one exception, there exist only arcs weighted by 1 because no substance flow with stoichiometric relations takes place, but signal flow is described by the model. This special arc is weighted by 2 to reduce the token number in the structural cycle, which is formed by the places inact_Far1 (p31) and Sst2_in_nucleus (p34), respectively, which symbolise that these proteins are already present in the cell in low concentration independently of a signal.In the initial marking there is one token residing in the place trimer_bound_to_receptor (p5) or which is dissociated in its subunits G_beta_gamma_dimer (p7) and G_alpha_GTP (p6). Therefore, both of these p-invariants include the place p5 in addition to one of the places p6 and p7, respectively. The places Kss1_phos (p38) and unphos_Kss1 (p39) indicate the modification of the involved kinase Kss1 and form the third minimal p-invariant according to case 3. Thus, all these minimal p-invariants can be interpreted in a biologically reasonable meaning.There are three minimal p-invariants in the net. Table Model validation, the places trimer_bound_to_receptor (p5) and unphos_Kss1 (p39) carry tokens ensuring that the minimal p-invariants contribute to the system behaviour.In the initial marking of the net there are some tokens already residing in special places. According to the elucidation in section MATalpha_cell(surrounding) (t1), receptor_synthesis (t3), binding_factor_to_receptor (t2), and receptor_conformation_change (t4). Therewith, this MCT-set represents the synthesis of Ste2, its binding to the present α-factor and its resulting conformation change. In Figure techn_input , compare Table For our model we got ten minimal t-invariants covering all transitions. As mentioned above, the transitions are grouped into MCT-sets, if and only if they participate in exactly the same minimal t-invariants, see Equation (6). If a huge number of t-invariants arises for a network, the concept of MCT-sets provides additional structure information. Due to their definition, MCT-sets represent sets of events, which occur only together. These sets, together with their pre- and postplaces and all arcs in between, should symbolise signalling functional units of the model with an own biological meaning, see Table G_beta_gamma_dimer (p7) remain in the net because of the minimal p-invariant 1. Therefore, the minimal t-invariants 3, 4, 5, and 6 do not fulfil the feasibility criterion, because they contain the MCT-set 3, but do not contain any transitions, which provide tokens at place p7, which is connected with the MCT-set 3 via read arcs. These non-feasible t-invariants have to be processed in the way explained above, see Equation (5). Only the minimal t-invariant 1 involves transition division  (t5), which provides tokens on place p7 and does not involve the MCT-set 3. Therefore, each of the four non-feasible t-invariants is combined with t-invariant 1. These combinations provide four feasible, non-minimal t-invariants. Table The necessarily in the net remaining read arcs endanger the feasibility, i.e., the realisability in the initial marking, of the minimal t-invariants. Following our modelling approach, minimal t-invariants are not feasible, if they involve a transition, which is connected via a read arc with an empty preplace, which does not get tokens by firing of other transitions involved in this t-invariant. To ensure the feasibility of all t-invariants, these non-feasible ones are joint with some others, which provide tokens on the critical preplaces. The read arcs adjacent to the place α cell in the surroundings, causing a receptor conformation change (MCT-set 1). In t-invariant 1 the activated receptor causes the dissociation of the G-protein (transition t5), whose subunits reassociate to a trimeric form (transition t6). In the t-invariant 2 the proteins Yck1 and Yck2 are bound through Akr1 to the plasma membrane and phosphorylate the activated receptor. This leads to a receptor endocytosis (MCT-set 7) via an ubiquitination. The t-invariants 7 to 14 include a signal transduction via the G-protein (transition t5) and the MAP kinase cascade (MCT-set 3). A release and phosphorylation of the transcription factor Ste12 (MCT-set 4) is involved in the t-invariants 7 to 10. This leads in these t-invariants to a transcription of the pheromone response genes preparing the cell for mating, including a feedback via Bar1 and Sst2 (MCT-set 5). Far1 is transported out of the nucleus (MCT-set 2) in t-invariants 8 and 10, while it initiates the cell cycle arrest (transition t32) in the t-invariants 7 and 9. By containing MCT-set 6, the t-invariants 9 and 10 include a Ste12 repression via activated Fus3. In t-invariants 7 and 8, kinase Kss1 is activated and deactivated . This p-invariant 3 contributes also to the processes described in t-invariant 14, where Kss1 is phosphorylated and dephosphorylated . Generally, the t-invariants 11 to 14 represent signalling pathways, which do not lead to a mating of the cell. All these t-invariants consist only of the receptor activation (MCT-set 1) and the MAP kinase cascade (MCT-set 3). Only t-invariant 13 involves an additional activation of Ste12 (MCT-set 4), which is repressed via Kss1 (transition t43). The t-invariants 11 and 12 include a feedback inhibition via a protein degradation (transition t38), which is initiated through a phosphorylation of these proteins .We want to discuss the biological plausibility of the feasible t-invariants. Table Recapitulating, t-invariants 7 to 10 lead to a mating of the cell, while the other t-invariants enable the cell to regulate, and modulate a pheromone induced signal by attenuated response or response desensitisation.Altogether it can be summarised that the known signalling processes are represented in the feasible t-invariants.The Petri net model can serve as basis of theoretical knockout experiments. They can be constructed by deleting the corresponding place and adjacent transitions, modelling the absence of this compound in a null mutant. Animating the resulting net shows already where some token accumulations arise under the new situation. These affected compounds can characterise a null mutant. Theoretical knockout experiments by the Petri net model could point out, which compounds indicate a mutant and nominate possible indicators for biological exploration.Ste5(scaffold) (p11) and its adjacent transitions binding_to_Ste5 (t12) and Ste5_binds_Ste11 (t13). This results in the deletion of the MCT-set 3, see Table cell_fusion (t28) , and celI_cycle_arrest_in_G1 (t32) occur only in the feasible t-invariants 7 to 10, which were deleted. Thus, the Ste5 mutant cells modelled by that mutant net are neither able to arrest the cell cycle nor to undergo a cell fusion, because the corresponding transitions do not occur in the feasible t-invariants of the net.Furthermore, a deletion of transitions leads to a deletion of feasible t-invariants, at least in a net, which is covered by t-invariants. The remaining feasible t-invariants show, which events still take place in this new cell type. For example, it is known  that SteIn this paper we describe a systematic approach to model and analyse signal transduction pathways using qualitative Petri nets. We propose a step-wise model development via the translation of the biological interactions into logical terms, which then in turn are transferred into net components.We explain the model validation step with a strong focus on the invariant analysis. In our modelling approach, we use minimal p-invariants, e.g., for the representation of the switching behaviour representing a compound in its activated and deactivated form. These p-invariants are embedded into the whole network by read arcs. Therefore, minimal t-invariants have generally to be processed to get feasible t-invariants. According to the construction principle, feasible t-invariants correspond to minimal self-contained subnets active under a given input situation. So, feasible t-invariants have to be checked for their biological meaning.Furthermore, we define maximal common transition sets, which provide a decomposition of the net into not necessarily connected subnets. These structures can be interpreted as the smallest biologically meaningful functional units or building blocks of a given network.The new concepts of feasible t-invariants and MCT-sets have been proven to be useful for model validation and the interpretation of the biological system behaviour by their fully automatic approach to decompose a given network into biologically meaningful modules. Please note that it is possible to build the MCT-sets based on the feasible t-invariants instead of the minimal ones. That makes sense in Petri nets, which contain more substructures of minimal p-invariants. MCT-sets provide a decomposition of the net into disjunctive subnets, and feasible t-invariants describe subnets, which can overlap.S. cerevisiae. The validated model denotes the known signal flows through the net, which are obtained by the invariant analysis. We are able to assign a reasonable biological meaning to all feasible t-invariants. These reflect correctly the signal transduction in such a system, i.e., the signal response behaviour. Thus, Petri net theory is also useful to model and analyse signalling pathways considering not a substance flow, but a flow of information  through the net.We illustrate this approach using the mating pheromone pathway in The Petri net model may serve as basis for theoretical knockout experiments. By deleting appointed places and adjacent transitions, null mutants can be modelled. The analysis results of the original net can be easily modified to get some information about the derived mutant cell.Having once validated the structure of the qualitative discrete Petri net, several applications and extensions are possible. The net may be refined and extended to a quantitative one by including known or estimated kinetic parameters , using continuous or hybrid Petri nets, respectively ,39.In this way, the resulting quantitative Petri net preserves the structure of the underlying net topology. Continuous Petri nets may be considered as a structured ODEs description. They can be evaluated by standard differential equation solvers. For a related case study see . Hybrid In this paper, we have chosen deliberately a rather small running example to be able to present all steps of the analysis procedure in full detail, which should help the reader to mimic the approach using his/her own case studies. It would be worth applying our approach to larger networks as presented in ,42.The development of the net has been done using the graphical Petri net editor and animator Snoopy  describeThe paper is a product of a strong co-operation between all three authors. The key concepts and case study has been elaborated by AS during the work on her diploma thesis. MH gave substantial and valuable contributions in the application of Petri net techniques. IK defined the topic and gave substantial contributions to the conceptional design of the research project. The editorial work has been done by all three authors. All authors read and approved the final manuscript.


Oc∈R((N + nm) log n), where N is the size of the output patterns and (n × m) is the size of the input with n time frames and m reaction coordinates. To date, this is the best time complexity for this problem. We next apply this to a β-hairpin folding trajectory and demonstrate that this approach extracts crucial information about protein folding intermediate states and mechanism. We make three observations about the approach: (1) The method recovers states previously obtained by visually analyzing free energy surfaces. (2) It also succeeds in extracting meaningful patterns and structures that had been overlooked in previous works, which provides a better understanding of the folding mechanism of the β-hairpin. These new patterns also interconnect various states in existing free energy surfaces versus different reaction coordinates. (3) The approach does not require calculating the free energy values, yet it offers an analysis comparable to, and sometimes better than, the methods that use free energy landscapes, thus validating the choice of reaction coordinates.  approach toward identification of global states. The approach is based on computing biclusters (or patterned clusters)—each cluster is a combination of various reaction coordinates, and its signature pattern facilitates the computation of the Z-score for the cluster. For this discovery process, we present an algorithm of time complexity pproach:  The meth β-hairpin folding trajectory and demonstrate that this approach extracts crucial information about protein folding intermediate states and mechanism.The study of protein folding mechanisms continues to be one of the most challenging problems in computational biology. Currently, the protein folding mechanism is often characterized by calculating the free energy landscape versus various reaction coordinates, such as the fraction of native contacts, the radius of gyration, RMSD from the native structure, and so on. In this paper, the authors present a combinatorial pattern discovery approach toward understanding the global state changes during the folding process. This is a first step toward an unsupervised  approach toward identification of global states. The authors apply this approach to a  Understanding protein folding is one of the most challenging problems in molecular biology –7. The iβ-strand hydrogen bonds, the number of α-helix turns, the hydrophobic core radius of gyration, and the principal components (PC) from principal component analysis th child) ignores the δ-clusters. The algorithm is efficient due to the two following factors: (1) use of a data structure (D in the pseudocode below) to store the maximal biclusters, so that searching for an arbitrary one can be done quickly, and (2) terminating the tree traversal appropriately. The data structure suggested for use is a tree so that each query takes log n time. The terminating condition (line [2.4] of the pseudocode) is such that each leaf node corresponds to either the maximal bicluster defined by the δ-clusters  j1 ≤ j2 … ≤ pj or its variants of the form q ≤ ρ.We describe a simple recursive algorithm to solve this problem. This algorithm implicitly constructs a tree in a depth-first manner where (1) each level corresponds to a distinct Generate-Set shown below, describes the algorithm. Assume a function Add-set , which inserts R, a subset of integers between one and n, in a tree data structure D, along with the accompanying set C: then a query of the form if a set R exists in D takes O(log n) time. The initial call is Generate-Set ).The pseudocode of the recursive routine R,C,j)Generate-Set (j ≤ 0) then exit(1) If ((2) For Let Let D ), add C″If ElseDAdd-set (3) Generate-Set  stored in D.The maximal biclusters are We first show that the algorithm is correct in computing all the maximal biclusters and next show that the algorithm runs in time linear with the size of the output.Correctness of the Algorithm. We first show that each computed  is a bicluster. By the construction, for each j, δ-cluster. Thus  is a bicluster. Next, we have to show that it is maximal. Assume it is not. Then there exists ijv such that j,δ-cluster. Then in the subroutine call Generate-Set  of the pseudocode , this set must have been created, leading to a contradiction. Hence, the assumption is wrong.j, δ-cluster. Then in Step 3 of the subroutine call Generate-Set, dV corresponding to j′ must have been included, leading to a contradiction. Hence, the assumption is wrong. Thus, all the computed sets are maximal biclusters. By similar arguments, it is easy to see that if there is any maximal bicluster defined on the data set, it must one of the computed R's.Next, assume there exists Complexity of the Algorithm. Assume the input elements are ij,ν 1 ≤ i ≤ n, 1 ≤ j ≤ m. Consider the first step of computing the δ-clusters for each j. The sorting of the elements iν, 1 ≤ I ≤ n takes O(n log n) time. The algorithm works by scanning the input from left to right, say i to I + s, where the set {iν , i+ν1 ,…, iνs+ } is a maximal δ-cluster. Then the input is scanned from i + 1, i + s + 1, i + s + 2,… onwards and so on. Thus, each data element is visited no more than twice. Assuming the comparison can be made in constant time, this step of the algorithm takes O(n log n + n) = O(n log n) time for each j.Generate-Set can be done in log n time. In the recursive-call tree structure (of the subroutine Generate-Set), each leaf node corresponds to a maximal bicluster. In a tree, the number of internal nodes is bounded by the number of leaf nodes and each leaf node is hit only as many times as the number of features in each pattern, thus assuming the output size is N  and the second step of the algorithm takes O(N log n) time. Thus, the time taken by the complete algorithm is O((nm + N) log n), where N is the size of the output and nm is the size of the input.Next, consider the second step of computing the maximal biclusters. Notice that the search in Step (2.4) of the subroutine


Nature Communications 10.1038/s41467-018-03311-y, published online 02 March 2018Correction to: In the original version of this Article, extraneous text not belonging to the Article was accidentally appended to the “Results” section. This error has now been corrected in both the PDF and HTML versions of the Article.


We find that for any tripartite mixed state, the unified -entropy entanglement of assistance follows a polygamy relation. This polygamy relation also holds in multi-particle systems. Furthermore, a generalized monogamy relation is provided for unified -entropy entanglement in the multi-qubit system.We investigate the entanglement distribution in multi-particle systems in terms of unified ( Monogamy property has also been discussed in the device-independent quantum information processing4, condensed matter physics5 and black-hole physics7.Quantum entanglement is an important resource in quantum information theory. Different from classical correlations, this restricted shareability of entanglement in multi-particle systems is known as monogamy property. The more entanglement shared between two parties implies the less entanglement shared with the rest of the system. Monogamy property plays a crucial role in quantum cryptography: which restricts the quantity of information captured by an eavesdropper about the secret key to be extractedet al., who considered a three-qubit system ABC8, and showed that the amount of entanglement (which is quantified by the squared concurrence) between A and B, plus the amount of entanglement between A and C, cannot be greater than the amount of entanglement between A and the pair BC. Further, Osborne et al. proved the squared concurrence follows a general monogamy inequality for the N-qubit system1. Monogamy inequalities for different entanglement measures have been noted, such as concurrence12, entanglement of formation14, negativity19, Rényi entropy entanglement21, and Tsallis entropy entanglement24. For the other physical resources, such as discord and steering, the monogamy property of them has also been discussed28.The study of monogamy property has a long history. The first monogamy relation was found by Coffman 30. Polygamy property was first provided by using the concurrence of assistance to quantify the distributed bipartite entanglement in multi-qubit systems30. Polygamy property has also considered in many entanglement measures, such as Rényi entropy20, Tsallis entropy31 and convex-roof extended negativity19.As dual to monogamy property, polygamy property in multi-particle systems has arised many interests by researchersq, s)-entropy is an important entropic measure, which can be used in many areas of quantum information theory. In this paper, we investigate the entanglement distribution in multi-particle systems in terms of unified -entropy. We find that for any tripartite mixed state, the unified -entropy entanglement of assistance follows a polygamy relation. This polygamy relation also holds in multi-particle systems. Furthermore, a generalized monogamy relation is provided for unified -entropy entanglement in the multi-qubit system.Unified -entropy and discuss the properties of unified -entropy entanglement. In the second subsection, we give our main results. We summarize our results in the third subsection.This paper is organized as follows. In the first subsection, we recall the definition of unified -entropy is defined as32q, s ≥ 0 such that Given a quantum state s tends to 1, the unified -entropy converges to Tsallis entropy Tq(ρ)33When s tends to 0, the unified -entropy converges to Rényi entropy Rq(ρ)34When q tends to 1, the unified -entropy converges to von Neumann entropy S(ρ)35When q → 1 and s → 0, we will use q = 1 and s = 1 to represent the limits in this paper. Now, let’s consider the entanglement in terms of the unified -entropy. For any pure state A and B), the unified -entropy entanglement is defined as36q, s ≥ 0.Because the limits exist in the case of ρAB, the unified -entropy entanglement can be defined via the convex-roof extensionρAB with pi ≥ 0. It is straightforward to verify that For a mixed state 31. When 20. Especially, The unified 38ρAB with pi ≥ 0.When q, s)-entropy entanglement, the unified -entropy entanglement of assistance -EOA) can be defined asρAB with pi ≥ 0. To understand -EOA better, consider a tripartite pure state 39. The entanglement supplier, Charlie, performs a measurement on his share of the tripartite state, which yields a known bipartite entangled state for Alice and Bob. Tracing over Charlie’s system yields the bipartite mixed state q, s)-EOA.As a dual quantity to the unified  can be defined asρAB, concurrence and COA are known to have analytic formula40λi being the eigenvalues, in decreasing order, of matrix For a two-qubit mixed state ρABIn ref. q ≥ 1, 0 ≤ s ≤ 1 and qs ≤ 3fq,s(x) has the formx ≤ 1.In ref. In this section, we will provide our main results. First, we have following result:ρABC in the Hilbert space q ≥ 1 and qs ≥ 1.Theorem 1. For any tripartite mixed state Proof: Let q, s)-entropy entanglement q ≥ 1 and qs ≥ 1, the unified -entropy is subadditive, that isFor any bipartite pure state Combining Eq.  with Eq.Thus, the proof is completed.q, s)-EOA in a tripartite quantum system. The upper bound of -EOA of A|BC can’t be greater than the sum of -EOA of B|AC and -EOA of C|AB. In particular, for a tripartite pure state q, s)-entropy entanglement Theorem 1. Shows a simple but interesting polygamy relation of -EOA in the multi-particle system, and gives an upper bound of -EOA of q, s)-entropy entanglement Corollary 1. Shows a constrained relationship of -EOA becomes Tsallis entropy entanglement of assistance which has discussed in ref. Corollary 2. For any multi-qubit mixed state Before our second main result, we have following lemma:q = 2 and fq,s(x) in Eq.  in Eq.  satisfieq, s)-entropy entanglement of AB and Next, the following result will provide a lower bound of unified -entropy is subadditive for any q ≥ 1 and qs ≥ 1. Thus, the following equality holdsq, s)-entropy entanglement of AB and From the definition of unified  is an increasing function for q ≥ 2, 0 ≤ s ≤ 1, and qs ≤ 336.We also note thatOn the other hand, we haveCombine Eqs  and 33)33) with Putting Eq.  into Eq.q, s)-entropy entanglement for 8.Theorem 2 shows a monogamy relation for a multi-qubit pure state q = 2 and i = 1, 2 and Example 3: Consider a pure state lity Eq. .n-qubit system, where A1A2 isExample 4: Finally, let’s consider a general W state lity Eq.  is \doculity Eq.  holds foq, s)-entropy is an important generalized entropy in quantum information theory. Many entropies such as Tsallis entropy, Rényi entropy, and von Neumann entropy can be seen as a special case for unified -entropy. In this paper, we have investigated the entanglement distribution in multi-particle systems in terms of unified -entropy. We find that for any tripartite mixed state, the -EOA follows a polygamy relation for q ≥ 1 and qs ≥ 1. This polygamy relation provides an upper bound for the bipartition A|BC, which also holds in multi-particle systems. Furthermore, for q = 2 and q, s)-entropy entanglement. This monogamy relation provides a lower bound for the bipartition Unified (Both monogamy property and polygamy property are fundamental properties of multipartite entangled states. We have studied the properties above in detail, and provided a two-parameters entropy function to study the entanglement distribution. We believe our result provides a useful methodology to understand the entanglement distribution of multi-particle entanglement.


Scientific Reports 10.1038/s41598-017-18174-4, published online 19 January 2018Correction to: The original version of this Article contained an error in the title of the paper, where the word “Reflectors” was incorrectly given as “Reectors”. This has now been corrected in the PDF and HTML versions of the Article.


Xiaoxi Li's name was omitted from the byline. In addition, the article did not correctly report the funding associated with the study. The article has since been corrected online.In the article “Predictive Value of Serum miR-10b, miR-29c, and miR-205 as Promising Biomarkers in Esophageal Squamous Cell Carcinoma Screening”,


Bogliubov-de Gennes equations are solved self-consistently to investigate the properties of bound states in chiral p-wave superconductive disks. It shows that either an s-wave or the mixed d- and s-wave state with odd-frequency and spin-triplet symmetry is induced at the vortex core, depending both on the chirality of the pairing states and on the vortex topology. It is also found that the odd-frequency triplet even parity (OTE) bound state can be manipulated with a local non-magnetic potential. Interestingly, with an appropriate potential amplitude, the zero-energy OTE bound state can be stabilized at a distance from the vortex core and from the local potential. Possible existences of the Majorana fermion modes are expected if the particle-hole symmetry property is applied to the zero-energy OTE bound state. Moreover, skyrmion modes with an integer topological charge have been found to exist. Sr2RuO4 is the first perovskite compound showing superconductivity without cupper-oxide planes1. Experimental and theoretical investigations suggest that its superconducting pairing state is most-likely a spin-triplet chiral p-wave state with the px ± ipy symmetry5. In particular, Nelson et al.6 performed the measurements to confirm an odd-parity and spin-triplet Cooper pairing symmetry in Sr2RuO4 superconductor. The px ± ipy state that has an intrinsic orbital angular momentum L = ±1 and breaks down the time-reversal symmetry has shown promising new physical properties under a applied magnetic field, e.g, the half-quantum vortices7, Majorana zero-energy modes8, Skyrmionic states10 and coreless vortices11, which is analog to the Anderson-Thouless vortices12 and Mermin-Ho vortices13 in the A phase of liquid 3He.The layered ruthenate superconductor Sr2RuO4, e.g., the odd-frequency spin-triplet s-wave pairings at a vortex core has been discussed by several research groups18. It is known that superconducting pairing amplitude has to be an odd function with respect to the permutations of electrons spin and position for equal times, as required by Pauli’s exclusion principle. In an inhomogeneous system, translational invariance was broken-down, which would lead to couplings between the even- and odd-parity pairing states. With the spin rotation symmetry invariant, the pair amplitude of opposite parity must be opposite in the Matsubara frequency that defines the frequency of the relative motion of two electrons forming a Cooper pair19. A spin-singlet even-parity and a spin-triplet odd parity pairing correlation that is even in Matsubara frequency obeys that rule, and so is the odd-frequency spin-triplet s-wave pairing state.In addition, possible existences of odd-frequency spin-triplet even parity (OTE) states in the chiral p-wave superconductor p+ state in an atomic length scale, whereas it is not the situation for the occupied states for the vortex winding parallel to the p+ chirality18. Hence, on one side, one believes that the results18 strongly support for the existence of an odd-frequency spin-triplet s-wave core state in Sr2RuO4 for an anti-winding vortex. On the other side, we should ask a question on what is the essence of the bound state at a parallel vortex core. Below a vortex with positive vorticity in relative to the chirality of p+ -state is defined as a parallel vortex, and a vortex with negative vorticity is known as an anti-parallel one.In the chiral p-wave superconductor the structure of quasiparticle around a vortex core is closely associated with the bulk topology of the pairing states and the vortex topology. Indeed, it has been showed that an odd-frequency triplet s-wave core state has an identical spatial distribution as that of the local density of states at both zero-energy and finite energy levels for a single vortex winding antiparallel to the chirality of the Sr2RuO4. Alternatively, one may ask whether the odd-frequency spin-triplet s-wave state and the Majorana zero-energy excitation can exist at a distance from the vortex core. For this end, we use the Bogoliubov-de Gennes(BdG) model20 to solve the problem of a cylindrical chiral p-wave superconductor with a two-dimensional isotropic Fermi surface in the presence of a single vortex. We show that an odd-frequency spin-triplet s-wave core state exists for an anti-parallel vortex, in agreement with those reported in previous investigations18. However, we found out that the state at the core is dominated by the odd frequency triplet d-wave for a parallel vortex. We also investigate the effect of a local non-magnetic potential on the bound states. Remarkably, a zero-energy peak in the local density of states appears at a distance from the vortex core and from the local potential well. We found that the spatial distribution of the corresponding zero-energy local density of states(LDOS) peak fits well with that of a mixed odd-frequency spin-triplet s-wave and even-frequency p-wave amplitude. To our knowledge, this is the first zero-energy OTE pairing state coexisting with the bulk px ± ipy pairing amplitudes in the chiral p-wave superconductor Sr2RuO4.Noticing also that almost all of the odd-frequency spin-triplet s-wave state and the Majorana zero-energy excitations are necessarily to be bound with an external topology defect  where the p-wave pairing amplitudes vanish or possess a local minimum. A second question arisen is if one can create an OTE bound state without involving local external topology defects in superconductor The paper is organized as follows. The mixed s- and d-wave state with odd-frequency and spin-triplet symmetry is induced at the vortex core in chiral p-wave superconductive disks and this bound state can be manipulated by introducing a local non-magnetic potential barrier are discussed in Sec. Numerical Results. Our results are summarized in Sec. Conclusion. Finally, in Sec. Theoretical Approach, we present in detail our theoretical approach and model.ν = −1 we obtain the results (not shown) that are in good agreement with those in ref. We discuss first the properties of bound states around an axial vortex. For an antiparallel vortex with ν = 1 . Therefore, states with even parity can exist around the vortex core. Importantly, the even parity triplet pairing state must be an odd function in frequency to satisfy Pauli’s exclusion principle. Indeed, as shown in our calculations, the zero energy mode is an OTE s-wave pairing state, and the first occupied state is dominated by the odd-frequency triplet d-wave state.In the presence of a vortex with vorticity ν = 1. To manipulate the odd-frequency triplet core states, we increase steadily the potential magnitude until to the gap closing occurs(The parameters are Pmax = 1.469 eV). Here, the term “gap closing” refers to all of the pairing amplitudes we concerned vanish at the core center. We choose a non-magnetic potential with Gaussian distribution to study how a local potential affect the topology of the bound state. The non-magnetic potential changes the translation invariance properties of the system. As a result, pairing states with even- and odd-parity would coexist. This is one of the mechanisms for the existence of a triplet bound state with even-parity. We find a relative wide range of parameters  can work, provided that the radius of the non-magnetic potential of ring-shaped is neither too small nor too large, e.g. in the range of Sr2Ru4 disk, which gives a ring-shaped potential with height/amplitude of about electron-volts. Another way is to irradiate the sample with a non-paraxial x-ray with a desirable spot-size.We now discuss effects of a non-magnetic potential on the spectrum and bound state properties of the system. We focus to the case of a parallel vortex with 22.Figure px ± ipy state is ν = 1 is applied to the superconductor sample. Here, Δ+(r) and Δ−(r) are both real functions, which are evaluated from the simulations. The calculated Δ+(r) and Δ−(r) are shown in Fig. ± as Δx ± iΔy. It is straightforward for us to show that Δx has, besides the one at the origin  of the polar coordinate, four zeros at , ) and at ), ) while Δy has zeroes at ), ) and at ), ). Where r1 is a radial length at which r2. Contour plots for |Δx| and |Δy| are shown in Fig. − and Δ+ are differentiable there must exists a closed loop L1 (L2) (white dashed lines) where the amplitude of Δ−(Δ+) approaches to zero-value asymptotically. The intersections in L1 and L2 are the so-called “singularities” where the relative phase θx − θy in the phase of Δx and Δy changes abruptly and discontinuously.As aforementioned, a formal expression for the pairing potential of the x and Δy lead us to think of topological defects, which opens possibilities for the induced vortex with distinctive topological index. As well-known, the relative phase θx − θy must be an integer multiple of 2π radians at an Abrikosov vortex core. For instance, as shown in Fig. θx and θy change 2π radians at the core of a single vortex with ν = 1. Meanwhile, |Δx| and |Δy| vanish simultaneously at the vortex core. We checked our data carefully and confirmed that θx(θy) could be a π winding at the half-vortex core. For instance, as shown in Fig. θy is a 2π winding while θx is a π wingding at  and at . Also θx is a 2π winding while θy is a π wingding at  and at . Therefore, the resulting θx − θy is a π winding there. In contrast to it, θx − θy changes only π radians [Fig. x and Δy amplitudes decays to zero [Fig. p+ pairing states is knownn as a r1, while the half antivortex is on a circle with a radius of r2, forming the so-called skyrmion modes  in our static skyrmion mode, which would have a resonance spectrum in the local density of states  where the amplitude of Δ+ tends to zero. It indicates that θx − θy is a constant of C(r*) [Fig. θx − θy varies from 180 degrees to 0 degrees when electron local coordinates increase crossing through the cycle C(r*) in the second and fourth quadrants. Instead, it varies from 0 degrees to 180 degrees in the first and third quadrants. It increases that phases of Δx and Δy precede p− state will be reflected on the circle to its mirror state with an opposite chirality. Superposition of the incident with its mirror particles leads to transverse resonances in the LDOS spectra, as shown in the left panel of Fig. Let us examine r*) Fig. . θx − θC(r*). Sandwiched between two superconductive domains, the cycle C(r*) where bulk p+ pairing amplitude vanishes may be regarded as an electronic subsystem of”normal state” domain wall. Electrons scattered on the wall have suffered to two processes. One is the normal reflection. Another is the Andreev retroflection. At the temperature much lower than the superconducting transition temperature the Andreev reflection dominates. In fact, it is the Andreev reflection that assists the bound state carrying the local microscopic current screening the magnetic field of the spontaneous half-quantum vortices. Consider an incident electron with an angle α measured from the polar axis. In general, the total current carried by the bound state can be expressed as n being an integer. Since each bound state with  has a mirror state of C(r*). Spatially inhomogeneous current distribution implies a time-dependent local electron density, according to the current continuality relationship. A resonant LDOS spectrum is obtained once the Fourier transformation has been performed with respect to the time variable.We discuss the major physical processes that lead to resonances in the spectra briefly by examining electrons scattered on p+-wave amplitude remains unchanged over the whole length scale. Hence, an identical r-dependence of the local electron density with that of a mixed s- and p-wave triplet amplitude  with p-wave amplitude. Very importantly, while the p-wave amplitude would be greatly suppressed with an increase of the scattering intensity of the non-magnetic impurities, the zero-energy odd-frequency triplet s-wave pairing amplitude that is insensitive to the scattering intensity remains unchanged. Then, using a TEM probe of s-wave superconductor the zero-energy odd-frequency triplet s-wave state can be tested as a zero-bias LDOS peak. When this happens the zero-energy LDOS signaled is a good indication of the Majorana excitation without an external magnetic topology defect.It is important to further note that the particle-hole symmetry of the system Hamiltonian remains hold even in the presence of an non-magnetic potential. By using this property, it is readily for us to prove that the Majorana condition We have investigated the properties of bound states in chiral p-wave superconductive disks. An odd-frequency triplet even parity (s- and/or d-wave) bound state or the zero-energy Majorana fermion is found at a single Abrikosov-type vortex core. The zero-energy OTE can exist at a distance from the external topological defect when a non-magnetic potential with appropriate amplitude has been included in the model Hamiltonian. Remarkably, half-quantum antivortex-vortex pairs and skyrmion modes are found to exist with the aid of the Andreev reflections. The topological charges have equal magnitudes but opposite signs for the 26 for the chiral p-wave superconductor Sr2RuO4 with pairing symmetry of px ± ipy:m is the electron mass, e is the electron charge, and c is the speed of light in vacuum μ is the chemical potential. A is the vector potentials. The gap equation of px ± ipy-wave is written as20± is the order parameter for the px ± ipy-wave, g is the coupling constant, We begin with the BdG equationsHere,we use the gauge R under an axial vortex of vorticity ν. In the cylindrical coordinate electron and hole quasi-particle wave functions We consider the problem of a chiral p-wave superconductor disk of radius μth Bessel function and μ and μ′ have to satisfy the following relations in order to ensure a real gauge of the BdG model. For instance, μ = μ′ when ν = 1, ν = −1, and ν = 0 etc. Quasi-particle wave functions satisfy the normalization condition:Here From the self-consistent solutions, we can calculate the LDOS as28. With the same mechanism, the OTE bound states are allowable in a chiral p-wave superconductor. The odd-frequency spin-triplet s- and d-wave pair amplitudes are defined by Since an Abrikosov vortex breaks down the translation symmetry and leads to couplings between the even-parity and odd-parity pairing states, one expects then possibilities for an odd-frequency pairing bound state around the vortex even in a conventional spin-singlet s-wave superconductor. One of the manifestations of the bound state is the enhanced LDOS at the vortex core, as observed as a zero bias conductance peak in the scanning tunneling spectroscopy26 from a standard BdG model, from which the odd-frequency spin-triplet d-wave pair amplitude is given byIn analog to Eq., one canwhere We note that the particle-hole symmetry relationship β is the standard deviation of the distribution. Different parameter values are used to adjust the location and potential intensity. The non-magnetic potential is extrmely located.This is the expression of normal distribution. nm, from which the coefficient in Eq. (R = 500 nm.In our calculation, we set the coupling strength t in Eq.  is \docu


We provide a methodology for estimating counts of single-year-of-age live-births, fetal-losses, abortions, and pregnant women from aggregated age-group counts. As a case study, we estimate counts for the 254 counties of Texas for the year 2010.We use interpolation to estimate counts of live-births, fetal-losses, and abortions by women of each single-year-of-age for all Texas counties. We then use these counts to estimate the numbers of pregnant women for each single-year-of-age, which were previously available only in aggregate. To support public health policy and planning, we provide single-year-of-age estimates of live-births, fetal-losses, abortions, and pregnant women for all Texas counties in the year 2010, as well as the estimation method source code.The online version of this article (doi:10.1186/s13104-017-2496-x) contains supplementary material, which is available to authorized users. Estimates of pregnant populations in a geographic region can be critical to assessing public health risks, such as chemical exposure , alcoholPregnant populations can be estimated from counts of live births, fetal losses and abortions. However, such data are often aggregated into 3–5 year age groups. Only a handful of studies provide single-year-of-age pregnancy estimates, including several addressing the prevalence of Down’s syndrome –10, fetaHere, we describe an interpolation method for estimating single-year-of-age pregnancy counts from readily available, aggregated year-age live births, fetal losses and abortion data. To demonstrate the method, we derive county-level estimates across the state of Texas for the year 2010.We summarize the notation for our analysis below.set of countiesset of single-year-of-age groups for which we estimate live birth/abortion/ fetal loss counts: set of aggregated year-age groups for which we have live birth/abortion/ fetal loss counts:Texas live birthsTexas abortionsTexas fetal lossesk   set of single-year age groups for which we estimate live birth/abortion/ fetal loss counts within the set of aggregated year-age group j in county icount of women in age group k in county icount of live births from women in age-group k in county icount of abortions from women in age-group k for entire countrycount of fetal-losses from women in age group j for entire statelive births from women in age-group proportion of year a woman is pregnant when she has a live birth: proportion of year a woman is pregnant when she has a fetal loss: proportion of year a woman is pregnant when she has an abortion: j in county ilive births from women in age-group j in county ifetal losses from women in age-group j in county iabortions from women in age-group i, at a single-year-of-age resolution  , and ders (DSHS) . For aboxas DSHS . For fetndations  and use a et al. . We provFurther, supplementary files for this subsection are provided in Additional files J, as a set of single-year age groups contained in the set k. For example, for We also define a subset, The National Vital Statistics System (NVSS)  providesmentclass2pt{minimble from . We descmentclass2pt{minimk, are proportional to the total live births in the entire state for that aggregated age-group, for all counties; i.e., We assume all live births in a county for aggregated age-group, Under this assumption, we do not associate live births across one aggregated age-group to another. For example, we do relate the number of live births from a mother of age [21–22) to those of age [22–23), since they both belong to the set Further, the estimates for this subsection are provided in Additional files The CDC recommends the use of national-average, as opposed to state-specific, fetal loss rates because most state reports of fetal deaths are limited to those with at least 20 weeks of gestation ; see alsk, from Ventura et al.  [k.As with live births, counts of fetal losses for single-year-of-age,  et al.  . The fet et al.   are up t et al.   does not et al.  , and divj is available from US Census Bureau [i by multiplying j. We present Next, we use a piecewise-cubic Hermite interpolating polynomial (pchip) to determine single-year-of-age counts of national fetal losses.s Bureau . The reds Bureau . We, thuFurther, the estimates for this subsection are provided in Additional files Unlike fetal losses, the CDC does not recommend the use of a national abortion rate due to wide variations in modes of data collection in geographic and demographic sub-populations . County-xas DSHS . We use Further, the estimates for this subsection are provided in Additional files j for county i can be estimated as in [Finally, the number of pregnant women of age ed as in ,1\documeFurther, the details for this subsection are provided in Additional file j as We make several comparisons between our estimates and reported data to assess the accuracy of our method. First, we compare our live birth estimates from age mentclass2pt{minimSecond, the CDC reported a national fetal loss rate for women aged 15–44 in 2008 of 17.9 fetal losses per 1000 women  reportedFinally, our estimate of 73,481 abortions among Texas residents in 2010 is close to the Texas DSHS report of 73,600 , yieldinAdditional file 1. This file contains the source information of all the input files we provide.Additional file 2. This file contains a summary of all the input and output files we provide.Additional file 3. This file contains aggregated live births for the 254 counties of Texas for the year 2010.Additional file 4. This file contains single-year-of-age live births for the entire state of Texas.Additional file 5. This file contains our estimated single-year-of-age live births for the 254 counties of Texas for the year 2010.Additional file 6. This file contains single-year-of-age national populations of women for the year 2010.Additional file 7. This file contains national aggregated fetal loss rates and corresponding populations of women for the year 2008.Additional file 8. This file contains single-year-of-age populations of women for the 254 counties of Texas for the year 2010.Additional file 9. This file contains our estimated single-year-of-age fetal losses for the 254 counties of Texas for the year 2010.Additional file 10. This file contains aggregated abortions for the 254 counties of Texas for the year 2010.Additional file 11. This file contains our estimated single-year-of-age abortions for the 254 counties of Texas for the year 2010.Additional file 12. This file contains our estimated single-year-of-age fractional pregnancy outcomes for the entire state of of Texas for the year 2010.


An alternative error bound for linear complementarity problems for  The linear complementarity problem is to find a vector mentclass2pt{minimils, see .An interesting problem for the }∥x−x∗∥∞ , that ismentclass2pt{minimils, see  and the M is a P-matrices [B-matrices. Here a matrix B-matrix [S, with In , García-matrices , which cB-matrix  if, for S-matrix  if there is set to be −∞ (∞) ifβ̃ can be very small, thus the error bound  a strictly diagonally dominant (SDD) matrix if M-matrix if its inverse is nonnegative and all its off-diagonal entries are nonpositive [First, recall some definitions and lemmas which will be used later. A matrix entclass1pt{minima\documentM is a B-matrix, then et al. in [B-matrix, the following corollary can be obtained easily by Lemma Remark here that when the matrix t al. in . Furtherentclass1pt{minimaLetbe amatrix that is not aB-matrix, and letwithsuch thatIfthenifthenwhereγsatisfies (atisfies .Consider the family of S̅): B-matrix. Then bound , we can say that bound (From Example at bound  is more an bound . TherefoP-matrix linear complementarity problem, Chen and Xiang in [P-matrix M:To measure the sensitivity of the solution of the Xiang in  introducSimilarly to the proof of Theorem 2.4 in , we can Letbe amatrix andbe the matrix given in LemmaThenwhereandifentclass1pt{minimaSimilarly, by Corollary Letbe amatrix that is not aB-matrix, and letwithsuch thatIfthenifthenwhereγsatisfies (atisfies .M is a In this paper, we give an alternative bound for mentclass2pt{minim


Parkinson’s disease (PD) is a progressive brain disorder that interferes with activities of normal life. The main pathological feature of this disease is the loss of more than 80% of dopamine-producing neurons in the substantia nigra (SN). Dopaminergic neuronal cell death occurs when intraneuronal, insoluble, aggregated proteins start to form Lewy bodies (LBs), the most important component of which is a protein called α-synuclein (α-syn). α-Syn structurally contains hexameric repeats of 11 amino acids, which are characteristic of apolipoproteins and thus α-syn can also be considered an apolipoprotein. Moreover, apolipoproteins seem to be involved in the incidence and development of PD. Some apolipoproteins such as ApoD have a neuroprotective role in the brain. In PD, apoD levels increase in glial cells surrounding dopaminergic cells. However, elevated levels of some other apolipoproteins such as ApaA1 and ApoE are reported as a vulnerability factor of PD. At present, when a clinical diagnosis of PD is made, based on symptoms such as shaking, stiff muscles and slow movement, serious damage has already been done to nerve cells of the SN. The diagnosis of PD in its earlier stages, before this irreversible damage, would be of enormous benefit for future treatment strategies designed to slow or halt the progression of PD. This review presents the roles of apolipoproteins and α-syn in PD and how some of them could potentially be used as biomarkers for PD. PD is an age-related neurodegenerative disorder characterised by dopaminergic neural cell death in the SN of the brain Schapira . PD was Dopamine loss in PD brains is the reason for motor problems and, possibly, for the cognitive deficits seen in some patients. Motor symptoms of PD include: ‘bradykinesia’, which simply means slowness of movement and is the most common clinical feature of PD; a ‘resting tremor’ that disappears during sleep and voluntary movement; ‘muscular rigidity’ referring to limb resistance to passive movements such as flexion, extension and rotation around a joint; and ‘postural instability’ occurring in the late stage of PD as a result of reduced postural reflexes  mutations, or possibly because of problems in other organelles such as the proteasome system, or mitochondria, or as a consequence of cellular stress  and some others , including APOER2, MEGF7, VLDLR, LRP1 and LRP2 . Therefore, the CNS has its own lipid metabolism pathway. ApoE is the most frequent apolipoprotein in CSF and is secreted by astrocytes. ApoJ, as a small dense apolipoprotein with very low lipid content, is also secreted by astrocytes  can lead to PD. The three isoforms of apoE may also play different roles in the pathogenesis of α synucleinopathies  that are reported in the familial form of PD. These α-syn N-terminal mutations occur in α-n tandem repeats and affect the conformation of α-syn and lead to α-syn aggregation and fibrillation Fig. . In addiMoreover, α-syn mutation can accelerate cell death by the effect on the MAPK pathway. α-Syn can interact with mitogen-activated protein kinases (MAPK) through its N-terminal domain, but its C-terminal domain is not involved in this interaction. The α-syn-MAPK interaction is important for regulation of the cell life cycle. α-Syn overexpression is accompanied by reduced active MAPK and increased α-syn-MAPK complexes and this can lead to neuronal death. These findings suggest that the inhibitory effect of α-syn on MAPK can be pathogenic in PD brains where α-syn is overexpressed Iwata . A mouseMutant α-syn can also induce mitochondria-dependent apoptosis to a higher extent than the normal protein. Mitochondria are one of the most important organelles in all cell types as they act as ATP machines that provide cells with the required energy content. In the secretory neurons, mitochondria are also involved in the maintenance of Ca2+ levels in a proper amount suitable for vesicle exocytosis and endocytosis in the presynaptic terminals. Therefore, mitochondrial dysfunction is directly linked to reduced cell viability. In dopaminergic neurons, secretion of dopamine can be reduced as a consequence of mitochondrial dysfunction, leading to oxidative damage and cell death through enhanced dopamine metabolism  may transfer to recipient cells. This is possible based on the ability of α-syn to interact with membrane lipids  aggregates much faster than full-length α-syn (α-synFL) and these kinds of aggregate are common in familial forms of PD. An in vitro experiment has proven that aggregation of α-synFL can be induced in the presence of α-synΔC. Moreover, co-expression of α-synFL with α-synΔC in rats can cause neuronal death as the consequence of elevated α-synFL aggregates while the expression of α-synFL by itself in the rat brain just leads to moderate aggregation without any neuronal death (Ulusoy et al. The formation of α-synΔC is a normal process in cells that typically leads to accumulation of α-syn, but it is not toxic in the usual circumstance. However, in familial PD, the raised levels of both soluble α-syn and truncated forms are responsible for a greater number of inclusions and amount of cell death. Therefore, the amount of α-synΔC levels in neurons is positively related to the synucleinopathy intensity. The sites of α-syn truncation at the C-terminus may be Asp119/Pro120, Asn122/Glu123 and Glu123/Ala124. It seems that α-syn cleavage is mediated by different proteases and the proteasome is believed to be responsible for the cleavage of α-synFL at the Asp119/Pro120 position, while calpain1 mediates the truncation of α-synFL at the Asn122/Glu123 position (Li et al. In conclusion, apolipoproteins that are involved in cholesterol removal from the cells and transport excess cholesterol to bile seem to have a protective role in PD. HDL particles are the major lipoproteins that remove cholesterol from the arteries and send them back to the liver. Therefore, HDL and ApoA1 as the main constituents of HDL are very important to prevent cardiovascular disease such as atherosclerosis, which is result of accumulation of extra cholesterol in the blood. ApoA1 is also has a protective role in PD. It is consistent with findings that show people who have lower heart rate variability are at a higher risk of PD (Alonso et al. AD, Alzheimer’s disease; Apo, Apolipoprotein; BBB, Blood-brain barrier; CMA, Chaperone-mediated autophagy; CNS, Central nervous system; CSF, Cerebrospinal fluid; DLB, Dementia with Lewy bodies; HDL, High-density lipoproteins; kDa, Kilodalton; LBs, Lewy bodies; LDL, Low-density lipoproteins; MAPK, Mitogen-activated protein kinases; MAPT, Microtubule-associated protein tau; MW, Molecular weight; PD, Parkinson’s disease; PEA, β-Phenylethylamine; PP2A, Protein phosphatase 2A; ROS, Reactive oxygen species; SN, Substantia nigra; TH, Tyrosine hydroxylase; VLDL, Very-low-density lipoproteins


In a recent work (Feischl et al. in Eng Anal Bound Elem 62:141–153,  The central idea of isogeometric analysis (IGA) is to use the same ansatz functions for the discretization of the partial differential equation at hand, as are used for the representation of the problem geometry. Usually, the problem geometry ented in  for finihpk-adaptive algorithm.Since CAD directly provides a parametrization of the boundary D BEM in  and for D BEM in . Unlike D BEM in , 41 and D BEM in , 38, 42 D BEM in  for fast, and to  for some, and to , 25. TheFor standard BEM with discontinuous piecewise polynomials, a posteriori error estimation and adaptive mesh-refinement are well understood. We refer to , 11, 12 While this work focusses on adaptive IGABEM, adaptive IGAFEM is considered, e.g., in , 43. A rf is given and the density p-th order NURBS. The convergence order for uniform partitions of h-refinement of the underlying partition as well as the local smoothness of the NURBS across the nodes of the adaptively refined partitions. It reflects the fact that it is a priori unknown, where the singular and smooth parts of the density We develop and analyze an adaptive algorithm for the following model problem: Let etry. In , we analired. In , we obseWe prove that the adaptive algorithm from  is rate e, e.g., , 26, 27)lyzed in  instead tive BEM .T), but also accounts for the knot multiplicity. Moreover, for Technical contributions of general interest include a novel mesh-size function shown in , 29, 30,Throughout, all results apply for piecewise smooth parametrizations The remainder of this work is organized as follows: Sect. }Γ Sect. , the dis}Γ Sect. , the mes}Γ Sect. , B-splin}Γ Sect. , and the}Γ Sect. . Section}Γ Sect.  from .Elements, partitionandLocal mesh-sizesanda, b]. This gives rise to T are equivalent, where the equivalence constants depend only on Local mesh-ratiosPatchesandmentclass2pt{minimSuppose that we are given a deterministic mesh-refinement strategy We write For the proof of our main result, we need the following assumptions on There exists a constant h-ratios  are unifFor all Each sequence For an arbitrary initial mesh h-refinement based on local bisection .The proposed refinement strategy  used in , 25 satih-refinement procedure has been analyzed in  there exists some interval p, it is sufficient to consider the following subsetI as follows: Without loss of generality, we assume that the maximum of |F| is attained at some Step 3 We show that F / w with a polynomial F of degree p and the weight function w, which is also a polynomial of degree p and which satisfies Step 4 According to [rding to , it holdis shows . \documeThe proof of linear convergence  will be Assumption p, and x nor its neighbors result from h-refinement or from multiplicity increase. Then, x or one of its neighbors result from h-refinement and/or multiplicity increase. If only multiplicity increase took place, we getFor all h-refinement, while the multiplicity of all nodes T is obtained by h-refinement.Note that the construction of p, and h-refinement is considered.For any Local projection property :  For (ii)Local The Scott–Zhang type operator  satisfieT, have support in All NURBS basis functions which are non-zero on ,p≤1 and  to see\dWe chooseLet limits ^{}\sum _{zSince we use a different mesh-refinement strategy, we cannot directly cite the following lemma from . However DefineThe next lemma reads similarly as [f :  there is a constant Then,  there holds convergence Suppose that V the weakly-singular integral operator  for fixed f. The heart of the matter are the estimates We choose operator . Moreove


We introduce new classes of generalized Orlicz-Garling sequences and Orlicz-Lorentz sequences by using a sequence of Orlicz functions and difference operator. We show that the Orlicz-Garling sequence space admits a unique 1-subsymmetric basis and a 1-dominated block basic sequence in  X and Y be infinite-dimensional Banach spaces. If X and x, which is called the support of x. When the basis Let X and Y, respectively. If there exists C-dominates C does not matter, we will simply say that C-equivalent, and we can write it as Now, suppose that C-equivalent to C-subsymmetric. Similarly, if C-equivalent to each ϱ is a permutation of C-symmetric. Note that C-symmetry implies C-subsymmetry, which in turn implies C-unconditionality. In this paper, we use standard facts and notation from Banach spaces and approximation theory. For necessary background, see [A basic sequence mentclass2pt{minimmentclass2pt{minimM is a function that is continuous, nondecreasing, and convex with M is said to satisfy An Orlicz function The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Lindenstrauss and Tzafriri  used theshown in  that eveMusielak-Orlicz function . For. For\docion (ii)(iii)Let D is minimal in X if it is minimal in X equipped with the partial order ≺, that is, if D is supported with respect to a basis We can say that X is a sequence A block basic sequence of a basic sequence Letbe a sequence of Orlicz functions, and letbe a weight. Forwe havewith norm inclusions.ψ. Then we haveψ, we get It is clear that (i)For a given sequence of signs (ii)The coordinate projection on (iii)Given (iv)we define the linear function Let us define some linear functions as follows: Given Letbe a sequence of Orlicz functions, letbe a weight, and letLetbe a sequence of signs. LetBbe a subset ofand letψbe a linear map inThen we have the following: (i)andare norm-one operators frominto(ii)andare norm-one operators frominto(iii)is an isometry frominto(iv)The standard unit vectors form a 1-subsymmetric basic sequence inφ, we get The proof of (i) is clear since it is a consequence of (ii) and Remark Letbe a sequence of Orlicz functions, letbe a weight, and letSupposeis minimal inwith respect to the canonical basis. Thenwhereis determined byφ be the function in ϕ be the inverse function of ψ restricted to H be the projection of A onto For given supremum  is attaiLetbe a sequence of Orlicz functions, letbe a weight, and letEvery normalized block basic sequence of the canonical basis ofis 1-dominated by the canonical basis ofψ, we get the following result. □Let a normalized block basic sequence be Letbe a sequence of Orlicz functions, letbe a weight, and letLetdenote the canonical basis ofSuppose thatforms a block basic sequence inThen there exists another block basic sequence formed bysuch that(i)for(ii)(iii)is a subsequence offor each(iv)(v)is 1-dominated byLet By defining the sequences Supposeis a sequence of Orlicz functions, is a weight, andLetdenote the canonical basis ofanddenote the canonical basis ofThen every symmetric normalized block sequence ofdominatesϱ of C-equivalent to all its subsequences and C-dominates Let Letbe a sequence of Orlicz functions, letbe a weight, and letLetdenote the canonical basis ofSupposeforms a normalized block basic sequence and satisfiesLetbe the sequence from Theorem corresponding toThen for eachthere exists a subsequencethat isequivalent to the canonical basissuch thatiscomplemented in(i)The coefficients of (ii)The first part of the proof follows that of Lemma 1 in . Let \doi values between Let Let us prove that a block basic sequence P extend to a linear operator on P is the projection onto p and byP is bounded by P̃ in P̃ is the projection onto To show that ma 15 in . Let us