Ideomotor theory is an influential approach to understand goal-directed behavior. In this framework, response-effect (R-E) learning is assumed as a prerequisite for voluntary action: Once associations between motor actions and their effects in the environment have been formed, the anticipation of these effects will automatically activate the associated motor pattern. R-E learning is typically investigated with (induction) experiments that comprise an acquisition phase, where R-E associations are presumably learned, and a subsequent test phase, where the previous effects serve as stimuli for a response. While most studies used stimuli in the test phase that were identical to the effects in the acquisition phase, one study reported generalization from exemplars to their superordinate category . However, studies on so-called R-E compatibility did not report such generalization. We aimed to conceptually replicate Experiment 1 of Hommel et al.  with a free-choice test phase. While we did observe effects consistent with R-E learning when the effects in the acquisition phase were identical to the stimuli in the test phase, we did not observe evidence for generalization. We discuss this with regard to recent studies suggesting that individual response biases might rather reflect rapidly inferred propositional knowledge instead of learned R-E associations. Human actions aim to attain goals and to change the environment in desired ways. The resultant changes and consequences are often referred to as action effects. Goal-driven behavior thus requires an actor to know what (motor) behavior leads to which consequence(s) in the environment. The present study investigates one particular question on learning associations between actions and their effects, namely whether the effects’ representations generalize to a higher hierarchy level.Ideomotor theory is a general framework of action control  has been addressed with an experimental approach that has sometimes been termed induction experiments  was presented. Randomly intermixing these trials aimed to prevent participants from pre-planning their response prior to the onset of each stimulus . Similar to Experiment 1 of Hommel et al. , the cathttps://aspredicted.org/g8gu4.pdf. Data are available publicly at https://osf.io/z3qc4/.This study was preregistered on AsPredicted.org before data collection. The pre-registration is available at N = 100 participants  to be included in the analysis. In total, 99 students from the University of Bremen participated for course credit and 6 other people from the Bremen area participated without any reimbursement. Five participants were excluded from further analyses due to the exclusion criteria (see below). All participants were either native German speakers or had advanced written and spoken knowledge of German and had normal or corrected-to-normal vision. Eighty-eight participants were right-handed, 12 left-handed, and 0 ambidextrous. All participants were naïve to the hypotheses of this experiment.The preregistered sequential sampling plan resulted in 10 < 1/10 was calculated for the one-sample Bayesian t test of the category group. This result would mean that participants have reacted randomly during the test phase instead of showing a response bias, in turn suggesting that no learning of associations at all took place during the acquisition phase of the experiment even in the category group.A BF10 ≥ 6 was calculated for the one-sample Bayesian t test of the category group (suggesting that learning took place for this group) and at the same time a BF10 of ≥ 6 or < 1/6 was calculated for the two-sample t test comparing the category and the exemplar group. The first result would mean that no, or at least no full, generalization occurred. The second result would mean that full generalization occurred, as the response biases would be of the same size in both groups. In the first case, a one-sample Bayesian t test would also be calculated for the exemplar group to assess whether some signs of generalization can be observed, as would be indicated by a response bias in this group (and a corresponding BF10 ≥ 1).A BFn = 50 participants per group has been reached.A maximum number of The following, pre-registered exclusion criteria were applied: Participants who did not press the left and the right key at least 80 times each (out of a maximum of 100) during the acquisition phase were excluded from further analyses  counterbalanced across participants. All participants took part in the experiment in a single session of approximately 35 min.BFs were monitored after data from an initial sample of 20 participants per group have been collected. Then, they were monitored after four additional participants . More precisely, the participants were each assigned to one of four conditions resulting from the combination of the group (category group vs. exemplar group) and R-E mapping during the acquisition phase within each group  and omissions (RT > 1000 ms) were excluded. Thus, each participant contributed 200 valid acquisition trials and 200 valid test trials. As described in our pre-registration, we used Bayesian sequential sampling  were recorded in 2.59, 4.70, and in 0.88% of all trials, respectively, and these trials were excluded from analyses. Both response keys were used about equally often . Biases during the acquisition phase were calculated by dividing the number of left responses by the number of right responses for each participant, and they ranged from 0.71 to 1.33.t tests indicate that those values are not reliably different, BF10 = 0.54, t(98) = − 1.46, p = 0.148, d = − 0.29.Mean RTs were 380 ms for the category group and 403 ms for the exemplar group. Bayesian as well as frequentist two-sample 10 = 0.85, t(98) = 1.78, p = 0.078, d = 0.36. The percentage of congruent choices was then calculated for each participant and used as the dependent variable in the following analyses. Figure Anticipations, omissions, and false alarms in no-go trials were recorded in 0.01, 0.47, and 2.35% of all trials, and these trials were excluded from analyses. Mean RTs in go trials were 433 ms in the category group and 413 ms in the exemplar group, and were not reliably different, BF10 = 48.01, t(49) = 3.69, p = 0.001, d = 0.52.For the category group, inspection of Fig. 10 = 1.95, with the frequentist t test indicating a significant difference, t(98) = 2.25, p = 0.027, d = 0.45. Indeed, the percentage of congruent choices in the exemplar group is around chance-level, and the statistical tests favor the null hypothesis of chance-level performance, BF10 = 0.32, t(49) = 1.26, p = 0.212, d = 0.18. These results suggest the absence of a response bias in the exemplar group.For the exemplar group, however, an inspection of Fig. Taken together, evidence for (a full) generalization in the exemplar group seems weak at best. If at all, the evidence seems to favor the null hypothesis of chance-level performance in the exemplar group.The present study set out to re-assess generalization (from exemplars to categories) in R-E learning. Evidence for generalization was reported by Hommel et al.  in a sert tests do not contradict the BFs, but rather are well in line with this conclusion.A limitation of the present results is the strength of evidence as indicated by the respective BFs. According to the suggestions of Jeffreys ,2 we havWhat could be the reasons for the diverging results in the literature? Of course, either our results or those reported by Hommel et al.  could reIn sum, we aimed at demonstrating evidence for generalization in the exemplar group, but rather failed to do so. In light of this, in the following, we will discuss our results against the background of a recent study. Specifically, Sun et al. , Exp. 1 That response strategies indeed play a role in free-choice tasks was also investigated by Vogel et al.  who focucrit = 0.55. Higher values point towards a bimodal distribution, whereas lower values point towards a unimodal distribution. Using the procedures of Kieslich et al. (Against this background, we also explored the distribution of the percentages of congruent choices separately for our category and exemplar groups see Fig. . Indeed,h et al. , the BCsThus, in our category group, the observed response bias seems to be driven by few participants (as in Exp. 1 of Sun et al. This interpretation would also point to a lack of generalization in the exemplar group, although it is fair to ask, what exactly is learned and expressed by participants in experiments like ours, that is, when using a free-choice test phase? According to Sun et al. , the resIn sum, the results from our experiment as well as results from R-E compatibility experiments (Földes et al.


Nature 10.1038/s41586-022-4696-z,published online 11 May 2022.Correction to: In the version of this article initially published, the surname of author Chaitanya Bandlamudi was misspelt as Bandlamundi. The error has been corrected in the HTML and PDF versions of the article


By using trapezoidal QWs, the quantum-confined Stark effect (QCSE) can be reduced, leading to an enhanced electron-hole wave function overlap, thereby increasing the recombination rate and reducing the differential carrier lifetime. In addition, the improved hole transport also creates favorable conditions for shortening the differential carrier lifetime. Furthermore, by comparing with traditional μ-LEDs with different thicknesses of QW, the modulation bandwidth of μ-LEDs with trapezoidal QWs exhibits a large advantage at lower current densities of below 2 kA/cm Light-emitting diode (LED) has undergone great developments since its birth, penetrating every corner of our lives. Recently, great opportunities for LEDs are beginning to emerge in visible light communications (VLC) ,2. HowevThe modulation bandwidth of LEDs is inversely related to the Resistance–Capacitance (RC) time constant and the carrier lifetime ,6, and gThrough the great efforts of researchers, the modulation bandwidth of μ-LEDs measured in the laboratory has been significantly improved ,27,28,29Simulation provides a device design method without relying on the epitaxial process, saving development costs. In this article, APSYS  has been used to study the modulation bandwidth of GaN-based blue μ-LEDs with different QW structures for VLC applications . In addiR) is defined as the number of carriers recombined per unit time and unit volume. Based on the ABC efficiency model [R is mainly composed of Shockley–Read–Hall (SRH) recombination, radiative recombination, and Auger recombination, which are proportional to the first, second, and third power of the carrier concentration, respectively [G) is defined as the number of carriers generated per unit time and unit volume. Under equilibrium conditions, the carrier generation rate in the active region of a μ-LED is approximately equal to the recombination rate in it, expressed by the following formula [A, B, and C represent SRH recombination coefficient, radiative recombination coefficient, and Auger recombination coefficient, respectively, n represents electron concentration, and p represents hole concentration.The carrier recombination rate G=R=AG) and the increment of the carrier concentration is (Δn) as follows:Under high current density, the excess carriers dominate, while the excess electron concentration balances with the excess hole concentration ,39,40,41τ) can be obtained from the derivative of the carrier generation rate with respect to the carrier concentration, expressed by the following formula [The differential carrier lifetime 1τ−3dBf) is defined as the corresponding frequency when the normalized power drops to half of the maximum value. Generally, the differential carrier lifetime has a relationship with the 3 dB modulation bandwidth of the LED as follows [In the frequency response of μ-LEDs, the −3 dB modulation bandwidth f−−1, and 200 ns [−31 cm−6/s [In the physical model of simulation, band offset, internal loss, and the SRH recombination lifetimes are set to 70:30, 2000 md 200 ns ,43,44, r1 cm−6/s ,45,46. O1 cm−6/s .18 cm−3 and a three-period GaN/InGaN multiple quantum well (MQW) layer. The thickness of QBs is 10 nm, where the n-type doping concentration is 3 × 1017 cm−3. The indium content in QWs is set to 20% to ensure an emission at a wavelength of around 450 nm. On the top of the active area, there is a 20 nm thick Al0.23Ga0.77N as an electron blocking layer (EBL) with a p-type doping concentration of 1.2 × 1018 cm−3 and a 50 nm thick GaN as a cladding layer with a p-type doping concentration of 1.2 × 1018 cm−3. The ohmic contact on the cladding layer is defined as the p-electrode of the μ-LED and that on the n-type GaN layer is defined as the n-electrode of the μ-LED.The structures of the μ-LEDs in this work are shown in To study the effect of QWs on the modulation bandwidth, μ-LEDs with two different QW structures have been designed, represented by μ-LED A and μ-LED B in 2. It can be seen that the electron concentration reaches a maximum near the p-side in μ-LED A, but that reaches a maximum near the n-side in μ-LED B. Besides, the hole concentration reaches a maximum at the middle in μ-LED A, but that reaches a maximum near the n-side in μ-LED B. In To understand the internal differences between the two devices, 2. The Φ is used to represent the energy gap between QB and QW, calculated by the difference between the peak energy of the next QB and the energy valley of the QW, reflecting the transport capacity of carriers. The larger the Φ, the higher the barrier of the QB, the harder the carriers escape, and the weaker the carrier transport. In In order to study the variation rules of carrier concentration, 2. In Equation (3), the concentrations of electrons and holes used to calculate 1/τ are taken from the data at the corresponding locations in Based on Equations (3) and (4), it can be concluded that the −3 dB modulation bandwidth is positive relative to 1/τ, which is positive relative to carrier concentration. 2, we calculated the carrier distribution and carrier wave function of μ-LED C  of μ-LED A, μ-LED B, and μ-LED C, as shown in The modulation bandwidths of μ-LEDs with different QW structures were compared, and the physical mechanisms involved were discussed in detail. Compared with μ-LEDs with 3 nm and 2.5 nm thick QWs, μ-LEDs with trapezoidal QWs are superior in both carrier distribution and electron-hole wave function overlap, exhibiting a higher modulation bandwidth. It is desired that the modulation bandwidth will continue to increase as the QW thickness continues to decrease, but the trapezoidal QW structure we designed is still excellent at lower current densities, where the μ-LED can operate more efficiently. Although it is natural to use thinner QWs, our research provides an alternative design for developing high-efficiency and high-speed μ-LEDs for VLC application.


This difference in island sensitivity between argument and adjunct wh’s-in-situ is known as argument–adjunct asymmetry in the theoretical literature. Recently, this long-established asymmetry is challenged by a formal judgment study. It was claimed in the study that this asymmetry is an illusion and both argument and adjunct wh’s-in-situ are subject to island constraints. The present study demonstrates that such a claim is not convincing because it is based on problematic experimental design. We designed two experiments to test the island effects on Chinese wh’s-in-situ. The results reaffirm that the argument–adjunct asymmetry in Chinese wh’s-in-situ is indeed present, contrary to the findings of previous formal judgment study, and they also corroborate our assumption that when object wh’s-in-situ like shénme ‘what’ are located inside a relative clause, they are subject to a pragmatic constraint, suggesting that the VP (formed by a verb and its wh-object) in the relative clause tends to describe the prominent/salient feature of the relativized nominal head.Unlike adjunct  The same is true of (1b).In English and many other languages of Indo-European origin, a wh-phrase generally moves overtly to a clause-initial position in wh-interrogative sentences. As is well known, such a movement cannot cross islands, namely, the structures out of which a constituent cannot be extracted. One of the famous islands is the complex NP island, where the complex NP refers to the NP modified by a clause . The comwh’s-in-situ, argument wh’s-in-situ do not seem to be subject to island constraints, as (2) shows.However, in Mandarin Chinese (hereafter ‘Chinese’) and other East Asian languages, wh-elements stay in situ. Furthermore, different from adjunct wh’s-in-situ is known as argument–adjunct asymmetry in the literature.1 This phenomenon has drawn the interest of many scholars, and many influential hypotheses have been advanced to account for it  in the relative clause tends to describe the prominent/salient feature of the relativized nominal head. Improving on the design in experimental materials, we conducted two experiments on Chinese wh’s-in-situ. The results of the experiments show that unlike adjunct wh’s-in-situ, argument wh’s-in-situ are not sensitive to island constraints, which is in line with the long-established findings on the issue at stake.Inspired by the studies of wh-in-situ, namely Let us first expound the logic of the factorial design for isolating island effects, on which the previous formal judgment study into 2 In other words, if the island effect is present, the total effect will be greater than the sum of the dependency length effect and the structure effect , and the island effect is reflected by the super-additive effect. The island/super-additive effect can be identified statistically. If the island effect is present, there will be a significant interaction effect between the two factors: Dependency Length and Structure.The island effect can be isolated with the logic of subtraction. First, the total effect is quantified by calculating the difference between (3a) and (3d) , and then the effect of dependency length can be captured by calculating the difference between (3a) and (3b) , and the effect of structure can be isolated by calculating the difference between (3a) and (3c) . The island effect can now be obtained by subtracting the effect of dependency length and that of structure from the total effect. Put differently, the island effect can be quantified with the following formula: island effect = [3a − 3d] − [3a − 3b] − [3a − 3c]. If there is no island effect, the score for the island effect will be zero . By contrast, if there is an island effect, the score for the island effect will be larger than zero.The formula for the island effect also has an equivalence like ‘island effect = [3d − 3c] − [3b − 3a].’ The formula can be interpreted as follows. [3b − 3a] can capture the effect of dependency length. If there is an island/super-additive effect, the difference between (3c) and (3d) should be greater than the difference between (3a) and (3b) though (3c) and (3d) appear to be different only by dependency length .Dependency Length (short vs. long), Structure (non-island vs. island) and Wh-Category (argument vs. adjunct). This gave the following eight conditions: (i) short + non-island + argument, (ii) long + non-island + argument, (iii) short + island + argument, (iv) long + island + argument, (v) short + non-island + adjunct, (vi) long + non-island + adjunct, (vii) short + island + adjunct, (viii) long + island + adjunct.Inspired by the experimental paradigm in studying island effects, Consider the following sentences cited from their experiment, representing one set of their stimuli.xiǎngzhīdào ‘wonder’ takes an interrogative clause as its complement. For the sake of exposition, we will name the complement of xiǎngzhīdào ‘wonder’ as CP1. In the short condition, SpecCP1/C1 and the wh-phrase form mono-clausal dependency. If the wh-phrase is to move covertly to SpecCP1, such a movement in (4), (6), (8), and (10) will be a short-distance one. By contrast, in the long condition, SpecCP1/C1 and the wh-phrase form bi-clausal dependency. If the wh-phrase is to move covertly to SpecCP1, such a movement in (5), (7), (9), and (11) will be a long-distance one. The factor Structure controls for whether the complement of xiǎngzhīdào ‘wonder’ contains an island (the island condition) or not (the non-island condition). The factor Wh-Category controls for whether the wh-phrase at hand serves the role of an argument or an adjunct.In these sentences, the verb wh’s-in-situ, there was a significant super-additive interaction effect of Dependency Length × Structure. This is an intriguing result because if proved to be true, their findings pose a challenge to the long-standing generalization on argument–adjunct asymmetry in wh’s-in-situ. That is, argument wh’s in Chinese are not subject to island constraints, whereas adjunct wh’s are . With the Complex NP Island as a test case, they argued that both argument and adjunct wh’s are subject to island constraints. This conclusion was too hastily drawn because the subject shéi ‘who’ was placed outside an island in their design ,3 and there is no compelling evidence that object wh’s-in-situ are a typical case for testing island sensitivity. Without a detailed study of island effects on subject wh’s-in-situ, it would be particularly inappropriate to jump to the conclusion that both wh-argument and wh-adjunct elements are sensitive to island constraints. Second, when the object wh-in-situ in the relative clause is shénme ‘what,’ the interpretation of the construction is susceptible to factors not bearing on island constraints. One of the factors is that when object wh’s-in-situ are in a relative clause, the construction is subject to a pragmatic constraint, suggesting that the VP (formed by a verb and its wh-object) in the relative clause tends to characterize the prominent feature of the relativized nominal head.4 To illustrate the point, consider (7). Its alleged low acceptability might be caused by pragmatic inappropriateness rather than by a violation of an island constraint. To be exact, (7) is taken to be unacceptable because—without a proper context—eating a certain thing is not the prominent feature of a girl. On the other hand, if we provide an appropriate context, (7) will become acceptable. Suppose that there are three girls in a street eating different kinds of things: One girl is eating a hamburger, another girl, an omelet, and the third girl, fried chicken. If both the speaker and the hearer share this common ground, (7) will be acceptable because, in this context, eating a certain thing is the characteristic feature of the girls at issue, and it is natural to ask what the girl that Bill met ate.There were 24 target items and 72 filler sentences for the experiment. The participants were instructed to rate the naturalness of sentences on a seven-point Likert scale, with 1 being completely unnatural and 7 being completely natural. Their findings show that for both argument and adjunct wh-in-situ is that when we replace the lexical item in (7) with other appropriate ones, the resulting sentences such as (12)–(14) will become acceptable too.The argument to the effect of displaying the role of a pragmatic factor in the interpretation of an island-internal object For example, in the context of (12), the girl’s eating habit is important because after she and Bill got married, this would affect their relationship. Perhaps it will also affect John if he knows Bill. On this condition, it is natural to ask what the girl likes eating. The same kind of construal applies to (13). The content is a prominent feature of a book because in most cases whether the book can attract a person or not is dependent on its content. Consequently, it is valid to ask what the book that Bill bought is about. Likewise, teaching is the prominent feature of a teacher as the primary duty of a teacher is to teach. Given this, it is natural to ask what he taught, as in (14). However, wh’s-in-situ, and the other is that when studying the island effects of object wh’s-in-situ, they fail to put the pragmatic confounding factor under control. These drawbacks undermine their conclusions that argument wh’s-in-situ in Chinese are subject to an island constraint, and that there is no argument–adjunct asymmetry in wh’s-in-situ. In the next section, we will introduce our two experiments, in which the weaknesses of wh’s-in-situ was tested, and in the other one, the island sensitivity of object wh’s-in-situ was tested with the pragmatic confounding factor under control.To sum up, although Ninety-six participants were recruited from a university in China, and each of them was paid 15 Yuan for taking part in the experiment.wh’s-in-situ. Just as in Dependency Length (short vs. long), Structure (non-island vs. island), and Wh-Category (argument vs. adjunct). Hence, this yielded eight conditions. The following examples are one set of the eight conditions constructed.Correcting the potential problems with In total, there were 24 target items in this experiment, and each item consisted of the eight conditions listed above. We thus had 192 target sentences. Using Latin Square, we assigned all of these sentences to eight lists. Consequently, each list had 24 test/target sentences and 72 filler sentences. With four practice sentences, each list had 100 sentences in total. Each list was pseudo-randomized, so that the sentences of the same experimental condition would not be adjacent. After that, each list was counterbalanced into four orders in order to remove the confounding factor of order.shéi ‘who’ for the argument wh’s-in-situ. In the condition of short dependency, shéi ‘who’ served as the subject of the first embedded clause, and in the condition of long dependency, shéi ‘who’ served either as the subject of the second embedded clause or as that of the relative clause/CNP. In other words, the grammatical roles of argument wh’s-in-situ were held constant across conditions, which helped to exclude the confounding factors such as thematic role and word order. In addition, we carefully selected the words to ensure that in all the conditions, wh-phrases in long dependency were placed two words/four syllables further away from xiǎngzhīdào ‘wonder’ than those in short dependency, which ensured that the effect of Dependency Length could be subtracted from the results.Just like wèishénme ‘why’ is ambiguous, the participants were told at the very beginning of the experiment that wèishénme ‘why’ in all the experimental sentences was to question the reason of an action/event, which is similar to the English counterpart why rather than for what.5 This was intended to prevent an unnecessary processing load caused by ambiguity resolution. To check whether the participants correctly understood the instructions, we asked them to provide answers to the experiment items. To be exact, they were told that xiǎngzhīdáo ‘wonder’ takes an interrogative clause as its complement. They were asked to provide an answer to the interrogative clause containing wèishénme. Their answers were supposed to be based on their interpretation of the wh-phrase in making the acceptability judgment—that is, whether under such an interpretation the relevant interrogative clause is unacceptable or not. If the participant’s answer to the question containing wèishénme is yīnwèi… ‘because…’ or yóuyú… ‘since…,’ etc, it would indicate that he/she understood the instructions correctly. By contrast, if his/her answer to wèishénme is wèile … ‘for the purpose of …,’ all his/her scores would be excluded from the data because he/she would have failed to understand our instructions.Since wèishénme can only be used in front of the future modal huì. If preceded by the future modal huì as in (23a), it cannot be interpreted as ‘why.’ However, if in this context it is interpreted as ‘for what,’ as wèile shénme is in (23b), then the sentence will be acceptable. Based on wèishénme is yīnwèi… ‘because…’ or yóuyú… ‘since….’ The reason is that giving such a high score might indicate that the participant unconsciously interpreted wèishénme as ‘for what,’ defiant of our instruction.As noted by wh’s-in-situ in terms of island effects. Put differently, there should be a super-additive interaction, or a super-additive effect between Structure and Dependency Length factors for both argument and adjunct wh’s-in-situ. If this is true, we should also expect no three-way interaction of Structure × Wh-Category × Length. On the contrary, provided that argument and adjunct wh’s-in-situ are different in terms of island effects, we should expect to find a significant three-way interaction in Structure × Wh-Category × Length.If wèishénme was correctly interpreted by all the participants. Then, following 6 The results showed a significant main effect of Structure , of Length , and of Wh-Category . Contrary to the findings of Structure × Length , Structure × Wh-Category , and Length × Wh-Category . Furthermore, we did find that there was a significant interaction in Structure × Wh-Category × Length . See also the We found that Structure × Length under the levels of Wh-Category to examine the island sensitivity effect of each wh-category. The result showed that as for the wh-adjunct wèishénme, there was a significant super-additive interaction effect between Structure and Dependency Length . As for the wh-argument shéi in subject position, however, no super-additive effect was observed . In other words, the wh-argument and the wh-adjunct exhibited a significant difference regarding island effects. This can be seen clearly in wh-in-situ and the right panel presents the interaction plot for adjunct wh-in-situ.On a par with The mean acceptability scores are presented in wh-in-situ is generally rated to be better than the adjunct wh-in-situ, contrary to the findings of wh’s-in-situ exhibit no significant difference in acceptability. Notably, in the short + non-island condition, the adjunct wh-in-situ is better than the argument wh-in-situ. However, it is in the long + island condition that the adjunct wh-in-situ is significantly worse than the argument wh-in-situ, with their mean rating scores being 2.20 and 6.14, respectively.As can be seen from wh-in-situ” that when the object wh-in-situ shénme ‘what’ is in a relative clause island, its interpretation is susceptible to factors not bearing on the Complex NP Constraint. Specifically, the object wh-in-situ in the relative clause is subject to a pragmatic constraint, such that the VP (formed by a verb and the following wh-object) in the relative clause is bound to characterize the prominent feature of the relativized nominal head. In this experiment, putting this pragmatic confounding factor under control, we intend to ascertain whether there is argument-adjunct asymmetry in island sensitivity of wh’s-in-situ.It is pointed out in Section “The logic of factorial design for isolating the island effects and the previous formal judgment study into 7 and each of them was paid 15 Yuan for taking part in Experiment 2.Another group of 64 participants were recruited from a university in China,Dependency Length (short vs. long), Structure (non-island vs. island), and Wh-Category (argument vs. adjunct). Therefore, this yielded the eight conditions. Similar to wh-in-situ in the short condition is served by shéi ‘who,’ and the one in the long condition is served by shénme ‘what.’ The following examples are one representative set of the eight conditions constructed.Just like Another set of three experimental sentences for the long + island + argument condition is shown below to demonstrate how the pragmatic confounding factor is controlled for.jǐngchá ‘policeman’ has different kinds of features. He can engage in different kinds of actions. For example, a policeman can drink water, watch TV, read books, inspect something, etc. In the context of (32), the feature like ‘inspect something’ becomes prominent because after reading ‘Yánliàng will bribe the policeman,’ one expects to know something associated with this policeman’s duty. Put differently, the use of the verb jiǎnchá ‘inspect’ can render the whole sentence coherent. By contrast, if we change this verb to chī ‘eat’ or hē ‘drink,’ the acceptability of the sentence such as (35) will decrease substantially because ‘eat something’ and ‘drink something’ are not prominent actions or features associated with jǐngchá ‘policeman.’As can be seen in (32)–(34), all the verbs in the relative clauses are related to the prominent features of the relativized nominal heads. To illustrate this point, consider (32). The relativized nominal head bàodào ‘report’ are used to describe the prominent features of the corresponding relativized nominal heads. For more discussion about the notion of prominence adopted in this study, see Consider also (33). When one wants to recruit a salesperson, he will pay attention to the person’s ability or specialty. Therefore, being good at something will be one of the prominent features of this person. The same is true of (27) and (34). The main activity associated with a hunter is to capture and kill something, and the main function of news is to report something. In other words, bǔshā ‘capture and kill’ and Other designs of this experiment are similar to those of Experiment 1. For example, just like Experiment 1, we also used four lexicalizations of the sentence type in (23a) (repeated as (36)) as part of our filler items. If a participant’s mean acceptability score for sentences like (36) was more than 5, all his/her scores would be removed. For the sake of space and simplicity, we will not reiterate the introduction of the experiment design.wh’s-in-situ in terms of island effects. In other words, there should be a super-additive interaction, or a super-additive effect between Structure and Dependency Length factors for both argument and adjunct wh’s-in-situ. Along this line, we should also expect no three-way interaction of Structure × Wh-Category × Length. Unlike wh’s-in-situ in relative clause islands, reported by wh’s-in-situ. If our assessment of Structure × Wh-Category × Length.If wèishénme as ‘for what’ rather than ‘why.’8 Then, the acceptability rating scores were analyzed with same method as that of Experiment 1. The results showed significant main effects of Structure , of Length , and of Wh-Category . Contrary to the findings of Structure × Length , Structure × Wh-Category , and Length × Wh-Category . Furthermore, there was a significant interaction in Structure × Wh-Category × Length .The rating scores of two participants were removed from subsequent analysis because both rated all the sentences like (36) as 7, which indicates that they did not follow our instruction and interpreted Structure × Length under the level of Wh-Category to examine the island sensitivity effects of each wh-category. The result showed that as for the wh-adjunct wèishénme ‘why,’ there was a significant super-additive interaction effect between Structure and Dependency Length . As for the wh-arguments, however, no super-additive effect was observed . Put differently, the wh-arguments and the wh-adjuncts exhibited a significant difference in island effects. This can be seen clearly in wh-in-situ and the right panel presents the interaction plot for adjunct wh-in-situ.On a par with The mean acceptability scores are presented in wh’s-in-situ are generally rated to be better than the adjunct wh-in-situ. Notably, in the short + non-island condition the adjunct wh-in-situ is better than the argument wh-in-situ shéi ‘who.’ Moreover, in the short + island condition the adjunct wh-in-situ is slightly better than the argument wh-in-situ shéi ‘who.’ It is worth noting that in the long + island condition the adjunct wh-in-situ is significantly worse than the argument wh-in-situ shénme ‘what,’ with their mean rating scores being 2.10 and 6.15, respectively.As can be observed, just like those of Experiment 1, the mean acceptability scores in Structure × Wh-Category × Dependency Length was observed. In this regard, our study renders reinforcing support to the long-standing generalization concerning the argument–adjunct asymmetry in wh’s-in-situ in Chinese  rather than wh-adjuncts’ sensitivity to island constraints. Our two experiments have demonstrated that the validity of their claim needs to be re-evaluated. Recall the results of Experiment 1 and Experiment 2. They reveal that the wh-arguments were not always rated to be better in acceptability than the wh-adjunct, which is different from the findings made in 10 Based on the findings in our study, we can conclude that even if wh’s-in-situ in the island condition cannot be attributed to the effects of Wh-Category.Second, wh’s-in-situ, without explicit reference to minimal pairs involving both adjunct wh’s-in-situ and argument wh’s-in-situ. It is, then, implausible to claim that the participants in these studies always implicitly construct a minimal pair, which—as In addition, it is worth noting that previous studies often focus only on island effects that arise on adjunct wh’s-in-situ and adjunct wh’s-in-situ. As can be seen from the two interaction plots of our experiments, the dotted lines for argument wh’s-in-situ in the left panels are almost flat, and the dotted ones for adjunct wh’s-in-situ in the right panels slope downward visibly. The identical result is also found in wh’s-in-situ are more sensitive to dependency length than argument wh’s-in-situ. Therefore, it is highly likely that the mean acceptability rating score for wh-adjuncts is higher than or approximately equal to that of wh-arguments  in short dependency condition, and the rating score for wh-adjuncts is lower than that of wh-arguments in long dependency condition. Given this, it will be unsurprising that in the short + non-island condition, wh-adjuncts were judged to be more acceptable than wh-arguments.Perhaps, at this moment one may wonder why the wh-adjunct in the short + non-island condition was judged to be more acceptable than the wh-argument in our experiments and why there was a difference in this regard from If we continue to investigate why there is a difference in sensitivity to dependency length, we speculate that the reason might be that wh-adjuncts and wh-arguments establish dependency in different ways: wh-adjuncts establish dependency through LF movement and wh-arguments through unselective binding . It willwh’s-in-situ are preferred over adjunct wh’s-in-situ in the short + non-island condition in Lu et al.’s experiment, a result different from ours, we speculate that one reason lies in the choice of an embedded subject. The embedded subject in the adjunct + short + non-island condition of their experiment is a proper noun, as in (8). It has been reported in the literature that a proper noun/definite phrase provokes a higher processing cost . The embedded subject may thus not have induced any additional processing cost. This may have resulted in the explicit short dependency effect of adjunct wh’s-in-situ. Consequently, the mean rating score for adjunct wh’s-in-situ is greater than that of argument wh’s-in-situ.As to the question why argument ing cost , which wwh-in-situ is in a relative clause/an island, the processing of such a sentence is susceptible to an additional pragmatic constraint. This is confirmed by Experiment 2, in which the experiment items were carefully designed so that the pragmatic confounding factor could be removed. The results of this experiment have shown that once the pragmatic confounding factor is under control, the acceptability score for the object wh-in-situ in the long + island condition is very high.Third, we suspect, as pointed above, that the low acceptability score reported by wh-in-situ is in a relative clause/an island, it is subject to an additional pragmatic constraint, and the low acceptability score reported by wh-in-situ and the object wh-in-situ behave almost in the same way. For example, in the long + island condition, both the subject wh-in-situ and the object wh-in-situ are judged to be better than the adjunct wh-in-situ, and in the short + non-island condition, both the subject wh-in-situ and the object wh-in-situ are considered to be worse than the adjunct wh-in-situ. Their parallel behaviors are well expected because both subject and object wh’s-in-situ serve as arguments. Unlike Experiment 2, wh-in-situ and the adjunct wh-in-situ in the long + island condition are rated to be very low in acceptability. If their findings are reasonable, then object wh’s-in-situ and adjunct wh’s-in-situ are distinguished in acceptability from the subject wh-in-situ in the long + island condition. Such a difference is surprising given that both subject wh-in-situ and object wh-in-situ are arguments. After fixing the experimental materials so that the pragmatic confounding effect is controlled for or eliminated, Experiment 2 has shown that the object wh-in-situ exhibits the same characteristics as the subject wh-in-situ does. This argues strongly in favor of our claim that it is the pragmatic confounding factor that prevents wh’s-in-situ. Put differently, under the impact of the pragmatic confounding factor, their result shows that just like adjunct wh’s-in-situ, there is also a significant Structure × Dependency Length interaction for argument wh’s-in-situ. Consequently, they fail to find a significant three-way interaction of Structure × Wh-Category × Length. Actually, the significant Structure × Dependency Length interaction for argument wh’s-in-situ in Moreover, a comparison of the results of Experiment 1 and Experiment 2 also suggests that we should be on the right track in assuming that when the object wh’s-in-situ in this language are not sensitive to island constraints, they usually use subject wh’s-in-situ as in (37) rather than object wh’s-in-situ as a test case.11Our claim can also explain the practice that theoretical linguists in Chinese generally rely on: when intending to show that argument wh’s-in-situ, when object wh’s-in-situ occur in a relative clause, namely an island, they are more likely subject to an additional pragmatic constraint. Therefore, it is much easier to create sentences with appropriate subject wh’s-in-situ than sentences with object wh’s-in-situ to examine the island sensitivity of argument wh’s-in-situ.According to the findings of our experiments, the reason lies in that unlike subject wh’s-in-situ in Chinese are insensitive to island constraints, unlike adjunct wh’s-in-situ. This has led to the celebrated generalization regarding the argument and adjunct asymmetry in wh’s-in-situ in this language. With the acceptability judgment experiment, wh-in-situ and the argument wh’s-in-situ are subject to island constraints. In this study, we point out that their results are not convincing because their experimental design has some drawbacks. We redesigned the experimental materials in question to examine island effects on wh-elements in situ in Chinese. The results of the two experiments in this study show that the argument versus adjunct asymmetry in wh’s-in-situ is present. Furthermore, the argument–adjunct asymmetry at issue cannot be attributed to the main effect of Wh-Category.Different from their counterparts in English, wh-elements in Chinese remain in situ in question sentences. Argument wh’s-in-situ, this study also discovers that when object wh’s-in-situ are located inside a relative clause, they are subject to a pragmatic constraint, suggesting that the verb phrase in the relative clause tends to describe the prominent/salient feature of the relativized nominal head. This contributes to the understanding of the processing of relative clauses and wh-in-situ sentences.On top of supporting the traditional theoretical generalization on Chinese Finally, it will be much fair to point out that although The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.QT conceived and designed the study, implemented the experiments, analyzed the data, drafted, and revised the manuscript. M-KP provided theoretical guidance when necessary and made revisions to the manuscript. XY participated in the conception of the study, statistical analysis, as well as the revision of part of the manuscript. All authors contributed to the article and approved the submitted version.


Vol. 219, No. 4 | 10.1084/jem.20211498 | March 10, 2022The authors regret that in the original version of their article, the symbols and lines for the “Veh; host-derived” group were inadvertently swapped with those of the “Veh; donor-derived” group in The errors appear in print and in PDFs downloaded before June 1, 2022.


Correction to: Nature Communications 10.1038/s41467-022-31202-w, published online 23 June 2022The original version of this Article contained an error in the legend for Fig. 3, in which all colour-coded findings were incorrectly attributed as ‘Direct SDG influencers’, rather than only ’Unique’, ‘Cross-sectoral’, and ‘Substitute’. This has been corrected in both the PDF and HTML versions of the Article.


Dear Editor,In a meta-analysis, Liao et al. . concludFirstly, in contrast with their stated inclusion criteria, the data from 2 trials  did not d) of −0.31  = −0.57 to −0.05) favoring n-3 PUFAs over placebo in the treatment of depression. Liao et al. report an SMD of −0.26 (95% CI = −0.47 to −0.09). Consequently, the errors do not invalidate the study conclusion, but note that the CI has widened.We carried out a random-effect meta-analysis on the corrected dataset, which yielded a Standardized Mean Difference . In the meta-analysis, the same RCT is presented as having a large and statistically significant effect (SMD = −0.82). This mismatch is due to the parameter for efficacy that Liao et al. .Notably, the results from Iranian studies put a stamp on the outcomes of the meta-analysis. When excluded from the analysis, the overall result of the meta-analysis  is a nuWe argue that the quality of the underlying data reported by Liao et al. . does no


P of soft magnetic laminations like SiFe sheets tends to be expressed through an integral over the power product H · dB/dt. Already in earlier papers, we stressed that distinctions are needed for the quantities H and B. However, they are not considered in practically consistent ways, in the so far literature. Here, we discuss these distinctions in closer ways, comparing loss determination by calculation and measurement, respectively. A physically consistent procedure is described for the determination of loss and magnetization power functions through measurement of bi-located quantities HS and BC . On the other hand, it is concluded that corresponding quantitative calculations—based on co-located quantities H and B—are impeded by the high amount of technological parameters of modern steel products. For example, they result from chemical additions, and—in particular—also from specific technologies of rolling, annealing, coating or scribing.Magnetic energy loss  However, during the last years, higher frequencies, up to an order of several kHz, gained industrial relevance. This is due to widened applications for electric drives, in particular for electro-mobility.Soft magnetic laminations like silicon iron sheets represent the standard material for cores of electric machines. Since hundred years, intensive attempts are made to reduce the magnetic energy loss of core materials, supported by standards for their experimental testing , 2. TradP that usually is given in W/kg, i.e. related to the mass of material. For example, grain-oriented (GO) steels as applied for power transformers show very low values P < 1 W/kg for a peak induction value BPEAK = 1.7 T. This is significant with respect to long-term operation. On the other hand, in the case of electric drives, loss values up to the order of 100 W/kg are accepted, however, mainly for non-oriented (NO) steel that is loaded for a rather short duration of time with f > 50 Hz.Magnetic loss is expressed by a quantity P has been strongly extended, from less than 1 W/kg, up to 100 W/kg. As a consequence, accurate loss determination gains industrial significance. This is valid for both, the experimental measurement and the mathematical calculation. In both cases, it is a convention to express loss quantities in a time-averaged way by a function of the following type:ρ is the density of sample. T = 1/f is the length of the period of sinusoidal magnetization with frequency f. B stands for induction, H for magnetic field strength.Historically, the above means that the practically relevant range of loss B and H, partly also replacing B through the—practically identical—polarization J. Different text books on magnetisms tend to use different definitions, partly lacking the corresponding physical conditions.As a reason of confusion, the literature uses different definitions for the two quantities B by the polarization J (in T), without any closer definition or comment. The quantity H is introduced as the “magnetic field strength”, as well without further specification. For the first time, in 1978, Pfützner [In a surprising way, this is even valid for international standards. In particular, IEC standards  for EpstPfützner  proved tH, the induction B, the electric field strength E and the current density S = γE (with γ the conductivity). A further interdependency is given by the vacuum permeability (magnetic constant) µo, according to B = µoH + J. All above quantities are defined for a location of the soft magnetic matter, as a macroscopic continuum. The level below is described by the Lorentz-Maxwell equations [As it is well known, expressions of type (1) are derived from Maxwell Equations (MEs). For magnetic problems, the latter offer a link between the vector field quantities magnetic field strength quations , as a liB”, we usually address the induction that results from an average over several grains and introduce this quantity into the system of MEs. Analytically or numerically, we finally can determine the “local” distribution of induction, also as a global quantity, e.g. averaged over a given region of the magnetic object.As a characteristic of a heterogeneously built-up soft magnetic steel sample, it offers itself to define five levels Table , in rathB and H correspond to each other, for a given “co-location” and a given point of time. For example, they express induction and field in the inner centre of a sample—or at any other location, i.e. volume element dV, respectively. On the other hand, comparable experimental measurements are not possible in co-local ways, due to the lack of sufficiently small sensors. Moreover, almost not any sensor types are given for interior measurements.As a specific problem for the here discussed case, calculated field quantities like B in an individual grain of GO-steel on Level L3 use smallest search coils in holes that are drilled into the tested lamination. In a—usually much smaller—grain of NO-steel, a corresponding measurement cannot be performed at all. A search coil tends to yield an average over several grains, corresponding to L5. The corresponding vectors of field H can hardly be measured at all. Smallest tangential field coils or Hall plates show dimensions of many mm for effective tests. Thus, they yield results on level L5, a priori.Apart from exceptional indirect methods, as described in , measureFor the here given task of comparing calculated loss determinations with experimental one, the above conditions cause problems of restricted comparability. Analytical or numerical calculations tend to yield local results on L4 that can be handled by Maxwell equations (MEs). Of course, they can be averaged for the determination of regional results. On the other hand, experiments are restricted to regional results, a priori. This means that their results cannot be handled through MEs in straightforward ways.V can be expressed asH is the magnetic field at the given location dV, B is the corresponding local induction. On the other hand, for the first time, it was proved in [A, total averaged losses can be expressed byHS is the magnetic field as measured at the surface, averaged over some distance that covers a sufficiently large amount of grains. BC is the induction averaged over the cross section A. p(t) is the magnetization power function.In clear ways, it follows from MEs that mere hysteresis losses at a given local volume element droved in  that forB(H) locally, for mere PH,Bozorth , p. 508 B(H) locally, for mere PE,Kneller , p. 612 B(Ha) locally, for loss portions, with Ha as the applied field,Bertotti , Ch. 12 B(H), for total P,Handley , p. 339 Ha) for cross section and surface, respectively, for measurement of total P,Fiorillo  p. 367 JB(H), for total P,Coey  (p. 442)B(H), for total P,Hilzinger  (p. 121)B(Ha) for cross section and surface, respectively, for measurement of total P.Moses  (p. 167)As already mentioned above, even standards , 2 do noNot any of the above references offers a distinction between Eqs. (2) and (3). This causes errors of interpretation in scientific studies, but even more in industrial discussions.f, this loop is very weakly affected by varying f. A well-known mechanism is the nucleation of main Bloch walls with rising f, as being typical for highly GO steel [As a resulting problem, also the term “hysteresis loop” is used in ambivalent ways. Per definition, it is applied for the quasi-static case where the loop area expresses mere hysteresis loss. But the literature uses also the term “dynamic hysteresis loop”. In fact, due to a very weak change of hysteresis with varying frequency GO steel , 15. Howf. For electric steel, the main reason for this widening is that a dynamic loop is not determined by a measurement at a single defined locality, i.e. in a co-local way, according to level L4. In all usual practice, it is measured in a bi-local way (on level L5), HS at the surface of sample and BC as an average over its cross section.On the other hand, in accordance with , we distHS at the surface (as location 1) as the driving source of progressing external and internal induction B through Bloch wall displacements. The initiated time-changes of B cause eddy currents that affect the inner magnetization (as location 2) in two ways: (i) reducing its intensity, and (ii) causing a lag of phase  would hardly be affected by increased frequency f. A thought experiment would result in a very similar loop BC(HC), from a co-local measurement that is taken as an average over the cross section. On the other hand, the practically usual, bi-local measurement of HS and BC reveals a widened dynamic loop, the area of which represents the sum of hysteresis loss and eddy current loss.We assume that performing a (fictive) co-local measurement of both P and instantaneous magnetization power functions p(t).For the practical design of loss testers, we addressed the needed distinctions in several publications like –18. HereP is measured either by the already mentioned Epstein Tester [According to international standards, loss n Tester  or by thn Tester . In the BC that is averaged over the samples cross section. Actually, it is not possible to measure the induction BS of the sample surface, approximate determinations being restricted to evaluations of surface positions of main domain Bloch walls.For loss measurements, the induction tends to be detected by a B-coil round the sample. As an alternative, it can be detected through the so-called needle method. In both cases, the result reflects the induction i. As a drawback, the result of measurement is affected by the yoke system, in particular also by the contact regions between sample ends and yoke. A more effective alternative is the arrangement of a tangential field coil (H-coil) on the sample surface . As . As i. Ace e.g. , 19). Al. Ali. AsP is performed by the well-known rise-of-temperate method. However, in cases of low P, it yields poor resolution. Thus, it tends to be replaced by the so-called electro-dynamic principle.The physically clearest measurement of loss BC and the surface field HS.Figure BC and HS allow a determination of loss P in exact ways. However, a condition is that HS is a constant along the whole surface. Later, in 2018, we confirmed this in [d, as it is illustrated by Fig. As already mentioned, in 1978, reference  proved iSPG,S [SPG,S represents the instantaneous power density that “streams” into the material, related to a surface area element. The aim of our measurement is to determine the total loss P averaged over both the duration T of the period and the sheets cross section and also related to the materials density ρ. Due to both-sides power stream, the loss results asAs a basis, we apply the surfaces Poynting vector SPG,S , 17, 18.HS(t) by means of a tangential field coil (H-coil) of just 2 mm thickness. For good averaging, its length is close to 300 mm, the width is almost 500 mm. With an effective distance of little more than 1 mm from the sample surface, the first ME promises that the detected field equals the samples surface field HS in very good approximation. It follows from the coils voltage uH prop. dHS/dt, through numerical integration.As a very effective option, we measure ES(t). Easily, it can be determined from the voltage uB of a B-coil of 300 mm length around the sample. The second ME yields the simple relationuB.The second to be detected quantity is the surfaces electric field P, related to the materials density ρ, according to (3).Finally, the above procedure yields the total loss P. However, the above described procedure is characterized by a high degree of physical consistency, thus promising good absolute accuracy. Residual sources of systematic error result from the effective distance of H-coil from the sample surface and from the problem that the B-coil windings have some distance from the sample surfaces. Both mechanisms are considered by careful manufacturing of hardware, supported by some software correction . With these measures, the accuracy can be estimated as being close to about 2%, i.e. distinctly better than that of standardized measurements on the basis of Watt-metric measurement of total power consumption that includes also loss of the magnetization yoke system.Unfortunately, so far, no golden standard exists for the measurement of HS with cross section averaged induction BC. Further, we stressed that local measurements of surface BS or field H at an interior location are not possible in practice. Assuming that we produce very small sensors for co-local measurements of induction B and field H at any single point of sample, then the experiment would yield mere hysteresis loss PH, according to the further down Eq. (9). Assuming that we also produce a very small sensor for the measurement of the current density S at this location, the equation would allow for the additional determination of the local eddy current loss PE. The sum would offer the local total loss P.At this point, we would like to add a thought experiment. According to the above, total losses are measurable in bi-local ways, i.e. linking surface field E, the dielectric polarization D, the current density S, the magnetic field strength H and the induction B. Let us now consider an electromagnetically excited volume element of solid matter. For it, the MEs yield in well-known ways the instantaneous, local electromagnetic power density asS being assumed to represent mere eddy currents, i.e. no galvanically impressed currents. The second term is the magnetization power density, the third one is the electric polarization power density that is neglected in the following.As it is well known, the Maxwell equations (MEs) describe vector interdependencies of the electric field strength Averaged over the sample thickness, we findd as sketched in Fig. H and B are parallel to each other, without other components. The crucial task is to perform an effective analytical or numerical calculation of local quantities S, H and B that are needed in order to determine the instantaneous thickness-averaged (or volume-averaged) power density.Let us now consider a soft magnetic sheet of thickness P. Per definition, it is related to the density ρ of material and averaged over the duration T of the period. With (3) and (8), this yieldsP:Theoretically, the latter promises a calculation of the total loss HS(t) and BC(t)—a task that hardly is possible in practice.(1) Straightforward analytical or numerical calculation of the two key quantities H(t), B(t) and S(t), for all locations of the sheet, followed by averaging over the cross section. This would yield eddy current loss asS resolves the 3D structure as resulting from both the classical and the excess currents round moving Bloch walls, Eq. (10) in vector expression would reflect the sum of classical and excess eddy current losses—but this is not feasible, in practice due to technological conditions as discussed further down.(2) Analytical or numerical calculation of the quantities PE, the second term represents the mere hysteresis loss PH [In analogous ways, we find hysteresis loss as loss PH , contrarmentclasspt{minimaHa as the “applied” field which however may differ from the samples surface field HS in strong ways through effects of demagnetization. Further, dynamic “hysteresis” loops are introduced as functions B(H) of strong frequency dependence. This is in contrast to the already mentioned fact that hysteresis loss PH depends on frequency in very weak ways. As already hint to in Sect. Unfortunately, the textbooks define the field quantity in demanding ways. In particular, several authors introduce B(H) that describes for a given volume element dV the very weak dependence of mere hysteresis from frequency f,“Dynamic hysteresis loop”—as the local function BC(HS) that comprises both hysteresis and eddy currents, thus depending on f in strongest ways.“Dynamic magnetization loop”—as the global function P that represent the main demand for industrial applications—irrespective of physically relevant portions.The literature reports many attempts to calculate individual portions of loss , conditions of rolling and annealing, as well as after-treatments like stress coating or laser scribing. Even best tools of calculation are not likely to consider the corresponding consequences on quantities like As a conclusion, we assume that mathematical loss predictions are not feasible, apart from rough predictions within a given class of steel, based on parameter fitting. As a benefit, loss separation favours discussions of frequency dependencies, as also stressed in , Ch. 12.P.According to (3), for the definition of instantaneous magnetization power functions, we relate to the density, according toP. Thus, we restrict the following to the experimental determination.First attempts for corresponding calculations by Petrun et al. , 23 werep(t) for grain-oriented steel, demonstrating that instantaneous values p may offer effective conclusions on the temporal “history” of loss generation. For the measurement, we applied the Low Mass SST according to Fig. uB yields dBC/dt in a direct way. The H-coil voltage uH(t) yields HS(t) after numerical integration.Recently, we reported in  our firsp(t) in [P = 2.43 W/kg for 50 Hz and BPEAK = 1.5 T. Figure p(t) for sinusoidal induction. Close to the instant t = 2 ms, p exhibits a peak value pPEAK = 3 W/kg, that indicates that peak losses may exceed averaged ones in distinct ways. Round t = 5 ms and 15 ms, we see a pair of spikes that are due to turns of atomic moments, thus being of non-dissipative nature.While the just mentioned function p(t) in  concernef = 100 Hz, where averaged loss results as P = 6.52 W/kg, corresponding to an increase of 170%. On the other hand, p(t) reveals an instantaneous peak loss, according to about pPEAK = 12 W/kg. Also, the intensity of non-dissipative spikes is strongly increased, the negative spike showing − 13 W/kg, instead of − 6 W/kg for 50 Hz.Figure As a conclusion, instantaneous power functions can be measured in easy ways. They yield both practically and theoretically relevant information about the “history” of temporal development, as arising during a period of magnetization.P uses different interpretations of terms of type o∫ H · dB(see Eq. (1)). Reference [H and B as specific local values of field and induction, respectively, for the determination of mere hysteresis loss PH. On the other hand, for total loss P, bi-locally determined values HS and BC were suggested for the surface and for an average over the materials cross section, respectively. The regional difference is characterized by phase shifts through eddy currents, as a main reason for different results.As early as in 1978, the first author of the present paper pointed to the finding that the literature on magnetic power loss eference  expresseHS and BC, a priori.So far, the above distinction has not found common use. Thus, the present paper discusses practical aspects and consequences of lacking distinction. In particular, the latter is a possible explanation for too low values of calculated losses. On the other hand, measured losses are less problematic, since they tend to be based on bi-locally quantities p(t), the suggested relation to density shows the advantage that both P and p are given in W/kg, which favours interpretations. On the other hand, the distinction between dissipative and non-dissipative portions needs deeper theoretical knowledge. Unfortunately, a restricted distinction yields errors of interpretation that may be of high practical significance.For the here also addressed instantaneous power functions


This study aimed to optimize the trade-in pricing strategy. To leverage market share, many sellers adopt trade-in strategy for advance selling, Customers can return their old products at a discount price when they buy new products. This can help increase the market share and decrease natural resource consumption.We consider a seller who sells new-generation products over two periods: advance selling and regular selling. Based on the rational expectation equilibrium, we adopt dynamic programming to construct a two-period pricing model with three different trade-in strategies–only in period 2, in both periods, and not at all–explaining the trade-in strategy as a promotion tool used by a monopolist to discriminate for advance selling between new and old customers.The results suggest that the optimal price is determined by the proportion of old customers, discount factor and product innovation level. Whether and when to give a trade-in rebate to old customers depends on these parameters. The seller’s choice of optimal trade-in strategy depends on the threshold value of the new customer demand and trade-in demand.Most existing literature focuses on advance selling strategies and trade-in strategies. To the best of our knowledge, this is a pioneering study that adopts trade-in as part of the advance selling strategy. Advance selling is a retail practice that a seller accepts customer orders before new products are launched. It has been widely adopted in various industries, such as electronics, fashion, travel and ticketing. For example, customers were encouraged to pre-order Apple smartphone XR before it was released, The advance selling period is also called the first period in this paper; after that, sales became a regular selling period where consumers could purchase in a usual manner, that is, the second period. The benefits of advance selling are threefold: period extension of product sales, more accurate demand forecasting, and improvement in cash flow because customers pay in advance .With the accelerating speed of product updates, an increasing number of customers would replace those “almost new” products with the latest generation products. According to a survey in China, it takes approximately 18 months for customers to switch to a new mobile phone. More than 80 million mobile phones are scrapped every year. Each mobile phone contains over 20 hazardous materials that threaten the environment  refers to the proportion of new high-type customers over all new customers; (d) the number of old customers is 1−λ. As explained earlier, all old customers are high-type customers; therefore, the total number of high-type customers is θλ+(1−λ). We assume λ follows the generation distribution F(⋅) and density f(⋅). The resulting distributions of the consumer segments are summarised in For the sake of simplicity, we consider the following scenario and make some necessary assumptions: (a) the market size is 1; (b) no second-hand market exists, no price competition exists between the rebates and other offers; (c) the proportion of new customers is a(0<a≤1) be an exogenously given constant that captures the innovation level, so old customers’ original valuation for the last generation product is βvH≤vL i.e. p1 and decides whether to introduce the trade-in strategy. Strategic high-type customers who observe p1 simultaneously decide to pre-order. In the second period, the seller sets the regular selling price p2, and all low type customers arrive to buy. We assume that the willingness-to-pay of all high-type customers remains constant in the second period. The seller produces a new generation product at a marginal cost c. The seller and customers are risk-neutral and forward-looking. The seller aims to optimize their total expected profit, and the customers maximize their expected net utility. It should be noted that capacity k(0<k≤1) is a limited constraint.We assume that all high-type customers are strategic. There is a risk that this product will be out of stock if strategic consumers wait to buy in the second period. Let and Carr ). Customand Carr ). The seFollowing this sequence as shown in i to denote strategy “i”, where α for future periods in different trade-in strategies and assume this to be common to both price and product valuation. Thus, the trade-in price for old customers is α is the durable parameter or discount factor for customers’ willingness to pay for old products. Then 1−α is a depreciation parameter. The old customers’ valuation for the last-generation product is αβvH in two selling periods for the new generation product. Then, the old customers’ willingness-to-pay for the new-generation product is vH−αβvH as adopting a trade-in strategy. Let the trade-in rebate be rj (rj>c) in both periods, j = 1,2 denotes the selling period, that is, r1 = (1−α)P1, r2 = (1−α)P2. The old  customers maintain the original valuation for the new-generation product and the same level of willingness-to-pay as for the last-generation one in both periods. Therefore, whenever the old customers choose a trade-in for the new-generation product, the seller optimally maintains the same trade-in price that is no more than the old high-type customers’ willingness to pay for the last-generation product  and s1>s2≥c. The unit salvage must be greater than the trade-in rebate (sj>rj), otherwise, trade-in products have no value to the monopolist.To examine the benefit of the trade-in strategy, we consider a benchmark strategy in this study. First, we consider the “base case” in which the seller offers advance selling without a trade-in, i.e. strategy “n”. Second, in addition to the advance selling strategy, we examine the case when the seller offers a trade-in strategy in the second period, i.e. strategy “s”. Finally, the seller offers a trade-in strategy in both periods, i.e. strategy “b”. Throughout this paper, we reserve superscript n and Li ). HoweveThe decision variable for the seller is pricing strategy. The monopolist sets the pre-order price, regular selling price, and trade-in price.We adopt the rational expectation (RE) framework to characterize market outcomes. In the RE equilibrium, each player decides based on individual beliefs that are rationally formed and consistent with the actual outcomes. According to the backward induction, we first calculate the optimal regular price and the available probability in the second period when new low-type customers arrive in the second period.k≤θλ, the probability of product availability in the second period k>θλ in period 1, the available probability in period 2 isThe optimal regular price is The seller’s expected revenue is For new high-type consumers, the utility function is All new high-type customers arrive to pre-order the new-generation product in the advance selling period. Because of limited capacity, the product availability probability is According to the RE equilibrium, all strategic high-type customers consistently believe in the reality of the available probabilities and prices in both periods. The utility function is Proposition 1. Under the no trade-in strategy for advance selling, there is a unique RE equilibrium in which all new high-type customers pre-order in the first period at the optimal price given by Eq  the optimal pre-order price has nothing to do with the innovation level. (ⅱ) θ.Proposition 2. The seller can set up the optimal pre-order price and trade-in price is αvL respectively, the optimal trade-in rebate is (1−α)vL. The old high-type customers would like to use the trade-in rebate to buy the new-generation one in the second period when the trade-in discount factor satisfies the condition with a. The trade-in rebate strictly decreases with the innovation level a.In Proposition 2, we find that the ceiling of the trade-in discount factor is determined by the innovation parameter. When the innovation parameter is higher, the old high-type customers would rather buy the new-generation product at a higher trade-in price than hold the old one. Then, the seller can decide a higher trade-in price.With trade-in strategy just in the second period, the seller’s expected total profit isIf the optimal release price is still k≤1−λ+θλ, the product availability probability is When the seller permits old customers to trade-in in either period, all old high-type customers would certainly choose to use the trade-in rebate to pre-order in the first period because they are strategic. When the seller’s capacity is less than the demand for new and old high-type customers, that is, λ+θλ<k<1, that is. θ satisfies the condition 0≤θ<k, which means the seller’s capacity must be so large that the trade-in strategy is worth implementing, then the product availability probability in period 2 isWhen 1−Then the expected revenue in period 2 is,For new low-type customers, the utility function in the second period isThe availability probability is The utility function is Then the trade-in price is The expected revenue is The seller’s expected total profit isvH−α⋅βvH, the old customer would trade-in. At least the seller should set up Only if the trade-in price θ<k, the ceiling of the trade-in discount factor is less than When 0≤The purchasing utility of old customers is To induce the old customers to trade-in as early as possible rather than wait until the second period, according to θ<kThen we can induce the optimal trade-in discount factor to be θ<k or k≤θ<1, the optimal trade-in discount factor is the same.Either 0≤Property 3. We find (ⅰ) there is only one optimal pre-order price as 0≤θ<k. And the optimal trade-in discount factor exists a. Recall θ as 0≤θ<k.θ<k, when all high-type customers including all old customers using the trade-in rebate pre-order in the first period, and all low-type customers buy in the second period, the seller’s expected total profit isFor 0≤λ<k) 0≤θ<k.And we assume that the new consumers demand λ is uniform distribution in the distance . Then, we can obtain the available probability in different strategies.It is obviously that Because of θ<k, we can get property 4.According to Property 4. Comparing the Eqs (θ<k. This implies that  the Eqs  with 7)Property  the Eqs , and ξ2bθ<k. This result is consistent with practice. The seller can use trade-in program in both periods to induce old and new high-type consumers to obtain a higher pre-order price.According to the Property 4, the seller can always set up highest pre-order price with the trade-in program in both periods as 0≤k≤θ≤1(ii) k≤θ≤1, it’s useless for the seller to adopt the trade-in program in both periods for advance selling strategy. Therefore, we analyze only the two strategies ahead, i = n, b. We find that k≤θ≤1.When the proportion of new high-type customers is no less than the capacity number, Theorem 2. According to Eqs  and andTheorg to Eqs , the preAccording to Theorem 2, the seller can set up highest pre-order price θ<k(i) 0≤πs−πn≥0This means that it is always beneficial for the seller only if the durable parameter or discount factor for the customers’ willingness to pay for old products is no more than its threshold.The forehead five items delegate the first profit difference, the last three items delegate the second profit difference. The second profit difference is positive.To guarantee k≤θ≤1(ii) We then analyze the different profits in the three different strategies. According to Eqs  and 8),,8), we cπs−πn≥0, asα increases with the new high-type customers’ demands. When new high-type customers’ demands are higher, the seller can control higher trade-in price, lower trade-in rebate.We find that θ.Because the unit revenue from dealing with the old product is certainly more than that of the trade-in rebate, the profit difference is more than zero. This means that it is optimal for a seller to adopt trade-in program for any Property 5. For k<θ≤1, the seller can use trade-in for advance selling in both periods, but the trade-in price must be larger than k = 0.85, vH = 3.0, vL = 2.0, c = 0.1, s1 = 0.25, s2 = 0.15, β = 0.667 (as a = 0.5), we can calculate α* = 0.5999.We set θ-axis is the demand of all new high-type customers, and the price-axis is the seller’s optimal prices under different trade-in strategies. θ under no trade-in strategy and trade-in strategy in period 2. θ<k and remains constant and the same as the customer’s high value for the new generation product for k<θ≤1.θ indicates the popularity of the new-generation products.θ-axis is the demand of all new high-type customers, and the profit-axis is the seller’s expected total profit under different trade-in strategies. The grey curve represents the seller’s expected total profit πn with just advance selling no trade-in strategy, the orange curve represents the seller’s expected total profit πs with advance selling and just trade-in strategy in second period, and the blue curve represents the seller’s expected total profit πb with advance selling and trade-in strategy in both periods. θ. The two trade-in strategies are much better than no trade-in strategy for any θ, so either trade-in strategy is always optimal for the seller because the seller can set a higher pre-order price with a higher value demand. It is also noted that there exists a threshold value of θ′ which is the intersection of the two profit curves of strategy 2 and strategy 3, and the profit is the same as θ = θ′. The profit of strategy 3 is greater than strategy 2 as θ<θ′, and the profit of strategy 2 is greater than that of strategy 3 as θ>θ′. This means more new high-type customers do not always benefit the seller under strategy 3. Because more new high-type customers buy new generation products at the optimal pre-order price in the first period, the seller can earn more profit.We now examine the impact of the high-value customer proportion and new customer number on profit in different trade-in strategies. Although the trade-in strategy and advance selling strategy have been widely studied separately in extant literature. Our paper is one of few literature combined trade-in with advance selling strategy. And we adopt trade-in strategy as part of the advance selling strategy. In practice, many sellers including retailers and manufacturers have adopted the trade-in with advance selling strategies. However the trade-in rebate can hurt the seller’s profit and the discarded old products can damage the living environment. So designing an appropriate trade-in strategy is crucial for sellers that frequently release the new generation products.θ and 03bb λ. Moreover, how the seller chooses the optimal trade-in strategy depends on the threshold value of new customer demand and trade-in demand. We helped the seller find the threshold value θ′ of new high-value customers under strategy 3. More new customers can hurt the seller when the new high-type customers are less than the threshold value. So we find that the optimal price is determined by stochastic factors such as the proportion of old customers, discount factor and product innovation level, which can help the seller make an appropriate decision of whether or not, and when to provide a trade-in rebate to old customers also depends on these parameters.In this study, based on the rational expectation equilibrium, we adopt dynamic programming to construct a two-period pricing model with three different trade-in strategies between the seller and consumers. We developed an analytical model to study how the innovation level, discount factor, and trade-in customers’ demand influence the choice of optimal trade-in strategy for advance selling. From the firm’s perspective, we identified that the seller can always make the highest pre-order price under the third strategy, i.e. offering trade-in in both periods because of old high-value strategic consumers. Given the innovation level, the seller can obtain the optimal discount factor, and both trade-in strategies are always more profitable than the no trade-in strategy for advance selling for This study only focuses on the advance selling period and the regular selling period. The next avenue is to extend the period to a third phase when the product reaches the end-of-life cycle. We mainly take innovative products into account. In practice, many large e-commerce retailers such as Jindong, Suning, Taobao have adopted trade-in strategy with advance selling for durable products. Thus future related research can also pay attention to durable products such as washing machine, refrigerator, etc. The limitation of this study is that the trade-in rebate is constant and proportional to the pre-order or regular price. In future research, it can also be considered that the seller can adjust the trade-in rebate according to the characteristics of the old product. We just combined the trade-in strategy with advance selling between the seller and consumers. So how to choose supply chain contracts can be studied with trade-in and advance selling strategy between the supplier and retailer in the future. And how to stimulate product returns by designing return incentives and optimizing pricing strategies in this type of closed-loop supply chain would be interesting to explore.


QuickXplain algorithm, which provides a divide-and-conquer computation strategy to find within a given set an irreducible subset with a particular (monotone) property. Beside its original application in the domain of constraint satisfaction problems, the algorithm has since then found widespread adoption in areas as different as model-based diagnosis, recommender systems, verification, or the Semantic Web. This popularity is due to the frequent occurrence of the problem of finding irreducible subsets on the one hand, and to QuickXplain’s general applicability and favorable computational complexity on the other hand. However, although (we regularly experience) people are having a hard time understanding QuickXplain and seeing why it works correctly, a proof of correctness of the algorithm has never been published. This is what we account for in this work, by explaining QuickXplain in a novel tried and tested way and by presenting an intelligible formal proof of it. Apart from showing the correctness of the algorithm and excluding the later detection of errors (proof and trust effect), the added value of the availability of a formal proof is, e.g., (i) that the workings of the algorithm often become completely clear only after studying, verifying and comprehending the proof (didactic effect), (ii) that the shown proof methodology can be used as a guidance for proving other recursive algorithms (transfer effect), and (iii) the possibility of providing “gapless” correctness proofs of systems that rely on (results computed by) QuickXplain, such as numerous model-based debuggers (completeness effect).In his seminal paper of 2004, Ulrich Junker proposed the  MSMP  problem  of an unsatisfiable (satisfiable) clause set is unsatisfiable (satisfiable). In fact, there are two MUSes for S, i.e., manifestation of the MSMP problem.The task of finding within a given universe an irreducible subset with a specific monotone property is referred to as the minimal unsatisfiable subsets , prime Horridge ), prime  Marquis , and mosA for a specific manifestation of the MSMP problem can be used to solve arbitrary manifestations of the MSMP problem if (i) the procedure used by A to decide the monotone predicate is used as a black-box , and (ii) no assumptions or additional techniques are used in A which are specific to one particular manifestation of the MSMP problem.Numerous algorithms to solve manifestations of the MSMP problem have been suggested in literature, e.g., Marques-Silva et al. , Rodler Not all algorithms meet these two criteria. For instance, there are algorithms that rely on additional outputs beyond the mere evaluation of the predicate  (re)used as is for suitable manifestations of MSMP  adapted in order to solve other manifestations of MSMP (Rodler (c) modified or extended, respectively, e.g., to achieve a better performance for a particular MSMP manifestation  and (ii), n Junker   explaining (2) presenting a clear and intelligible proof of Proof Effect(a) It shows (b) It creates compliance with common scientific practice. That is, every proposal of an algorithm should be accompanied with a (full and public) formal proof of correctness. This demand is even more vital for a highly influential algorithm like Didactic Effect(a) It promotes (proper and full) understanding (Hanna and Jahnke (b) It provides the basis for understanding (hundreds of) other works or algorithms that use, rely on, adapt, modify or extend Completeness Effect It is necessary to establish and prove the full correctness of other algorithms that rely on (the correctness of) Trust and Sustainability Effect It excludes the possibility of the (later) detection of flaws in the algorithm, and is thus the only basis for placing full confidence in the proper-functioning of Transfer Effect It showcases a proof template for recursive algorithms and may thus provide guidance to researchers when approaching the  proof of other recursive algorithms.The rest of this paper is organized as follows. We discuss related work in Sect. In this work, we account for this by entclass1pt{miniman Junker , i.e., wmin. As a side note in Bradley and Manna , min, although being similar to p. In this case, min would return First, d Junker , Fig. 3,mentclass2pt{minimmin given in Bradley and Manna  segment our proof into small, intuitive, and easily digestible chunks, thus putting a special focus on its clarity, elucidation, and didactic value, (2) provide visualizations of the interrelations between and of the sequence and meaning of the individual proof steps by means of diagrams  organize the proof in a way it is illustrative and amenable to a mental “tracking” in that it can be viewed as directly traversing the algorithm’s call-recursion-tree while verifying the correctness of all transitions in the tree  explicate the proof template adopted in our proof in order to promote the reader’s comprehension of the underlying general proof principle and to provide them with well-founded justifications for the individual proof steps  knowledge base  U for which we are interested in finding a minimal unsatisfiable subset (MUS) X.U be the universe (a set of elements) and p holds for p is a monotone property iff (Monotone Property) Let p is monotone iff, given that p holds for some set p also holds for any superset p does not hold for some set p does not hold for any subset So, (a) some elements of the universe must not occur in the sought minimal subset, or (b) the minimal subset of the universe should be found in the context of some reference set. Both cases (a) and (b) can be subsumed as searching for a minimal subset of the analyzed setbackgroundU  and U that are possibly faulty); in case (b), X of elements in In practical applications it is often a requirement that p-Problem-Instance) Let p be a monotone predicate. Then we call the tuple p-problem-instance (p-PI).(p-Set (given some Background)) Let p-PI. Then, we call X a p-set wrt.p-set X wrt. minimal iff there is no p-set  A p-set exists foriffis a—and the only—p-set wrt.iff(Existence of a p-PI p(X) for arbitrary p-set wrt. p-set’ is output. Note, in order to not overload the discussion, we focus in this work on QX’s property of computing a minimalp-set . That imentclass2pt{minimTrivial Cases: Before line 7 is reached, the algorithm checks if trivial cases apply, i.e., if either no p-set exists  p-set is non-empty. (Line 11): If the analyzed set mentclass2pt{minimRecursion: Subsequently, the recursion is started. The principle is to partition the analyzed set non-empty  subsets split and get functions; lines 13–15), and to analyze these subsets recursively (divide-and-conquer). In this vein, a binary call-recursion-tree is built  returns the subset of those elements  returns the subset of those elements : The first question is: Are all elements of p-set, i.e., Right subtree : Here, we can distiguish between two possible cases, i.e., either the set (i) empty or (ii) non-empty.p-set (as verified in the left subtree before returning the empty set) and every p-set is non-empty . Hence, Given (i), we know that p-set, are all elements of p-set, i.e., p-set X found by For (ii), the question is: Given the subset Finally, the union of the outcomes of left .Fig. 1CNotation: Below, we show the workings of p in line 9, the double-underlined elements are those that are already fixed elements of the returned minimal p-set, and the grayed out elements those that are definitely not in the returned minimal p-set. Finally, p-set (function p in line 9 returns 1); p-set (function p in line 9 returns 0).15HowProceeds: After verifying that there is a non-empty p-set wrt. Explanation: After splitting p-set in the left half p-set, the right subset p-set, and 8 is irrelevant. From now on, 7, as a fixed element of the p-set, takes part in all further executed predicate tests.p-set is included in p-set, i.e., the test for p-set. So, 4 is fixed. To check the relevance of 3, X, one of our minimal p-sets. In step (4), the goal is to figure out whether the left half p-set wrt. the p-PI it gets as an input, if a p-set exists, and ’no p-set’ otherwise. While reading and thinking through the proof, the reader might consider it insightful to keep track of the meaning, implications, and interrelations of the various propositions in the proof by means of Fig. In this section, we give a formal proofmentclass2pt{minimLetbe ap-PI. Thenterminates.17(Termination) p-PI First, observe that split and get functions.Now, assume an infinite sequence of nested recursive calls of entclass1pt{minimaFinally, there can only be a finite number of such nested recursive call sequences because no more than two recursive calls are made in any execution of The following proposition witnesses that returns ’nop-set’ in line 3 iff there is nop-set wrt.Ifreturnsin line 5, is a minimalp-set wrt.If the execution ofreaches line 7, holds.–(3) in turn.Proof of (1): The fact follows directly from Proposition Proof of : The facProof of (2): Because line 5 is reached, p-set wrt. p-set wrt. p-set wrt. mentclass2pt{minimProof of (3): This statement follows directly from the test in line 2 and the fact that line 7 is reached. We now characterize an invariant which applies to every call of  when it returns in line 10 or 12 p-set wrt.If the execution ofreaches line 11, thenholds.Ifreturnsin line 12, thenis a minimalp-set wrt.If the execution ofreaches line 16, whereis called, then(Invariant Causes Sound Outputs and Propagates Downwards) We prove all statements (1)–(4) in turn.Proof of (1): “p-set wrt. Proof of : “\docump-set wrt. “mentclass2pt{minimProof of (2): Proposition p-set wrt. mentclass2pt{minimProof of (3): A return in line 12 can only occur if the test in line 11 is positive, i.e., if line 11 is reached and p-set wrt. p-set wrt. p-set wrt. p-set wrt. First, mentclass2pt{minimProof of (4): Consider the call split and get functions  As per Definition p-set wrt. We first prove It remains to be shown that Overall, we have demonstrated that At this point, we know that the invariant property of Definition p-set wrt. its p-PI given as an argument, due to Proposition Due to the fact that entclass1pt{minimap-set for the p-PI given as an input argument to the call of p-set wrt. their given p-PIs—whether or not these calls directly return—the combination of these p-sets is again a minimal p-set for the respective p-PI at the call that executed lines 16 and 17.The next proposition evidences—as a special case—that the combination (set-union) of the two outputs Let the recursive callin line 16 during the execution ofreturn a minimalp-set wrt.and let the recursive callin line 17 during the execution ofreturn a minimalp-set wrt. Thenreturns a minimalp-set wrt.) The statement is a direct consequence of Lemma Letbe a partition ofIf(a) is a minimalp-set wrt.and(b) is a minimalp-set wrt.thenis a minimalp-set wrt.p-set, and then we show its minimality.We first show that p-Set Property: First, by Definition p-set wrt. p-set wrt. Minimality: To show that minimalp-set wrt. p-set wrt. X can be represented as X is a p-set wrt. p, it follows that p-set wrt. Let us first assume that X is a p-set wrt. p, and since p-set wrt. Second, assume that p-Set is Returned By Tree) Ifholds forthen it returns a minimalp-set wrt. k. Therefore, by the Induction Assumption, p-set wrt. p-set wrt. k. Consequently, p-set wrt. Induction Assumption.Because p-set wrt. their given p-PIs p-set wrt. As both recursive calls made throughout the execution of This completes the inductive proof. Letbe ap-PI. Then,terminates and returns a minimalp-set wrt.if ap-set exists forOtherwise,returns ’nop-set’.20 holds when the loop is entered , (2) remains true for the next loop iteration if it is true for the current iteration (L-maintenance), and (3) yields a property at termination of the loop that is useful to prove the algorithm’s correctness (L-termination).recursion invariant, i.e., a predicate that is true for every recursive call of a procedure, andrelies on a R-initialization),holds for the first call of the recursive procedure (remains true for any further (recursive) call of the procedureR-maintenance), and(R-termination).entails correctness of the procedure’s output . Consequence: These combine-steps need to be addressed when proving R-termination.In case the recursion implements a divide-and-conquer approach, there can be As an illustration, the following table shows how the building blocks of our proof are assigned to the three different proof phases of R-INV:24(1) The finding of a “right” invariant can be tricky and will often represent the key to success in proving a recursive algorithm by means of the R-INV template. However, once an appropriate invariant has been determined, the rest of the proof can be relatively straightforward provided that the systematic steps suggested by R-INV are followed. (2) There can be multiple  invariants that allow to prove one and the same algorithm by means of the R-INV template. Indeed, also in the case of QX, a second possible invariant exists and is given by: The foundation for our used template is provided by the proof principle for (non-recursive) algorithms based on loop invariants detailed in Cormen et al. . The ideQuickXplain , it fosters proper and full understanding of didactic effect), it is a necessary foundation for “gapless” correctness proofs of numerous algorithms, e.g., in model-based diagnosis, that rely on (results computed by) completeness effect), it makes the intuition of trust and sustainability effect), as well as it might be used as a template for devising proofs of other recursive algorithms (transfer effect). Since (i) we exemplify the workings of (ii) we put a special emphasis on the clarity and didactic value of the given proof (e.g., by segmenting the proof into small, intuitive, and easily-digestible chunks, by showing how our proof can be “directly traced” using the recursive call tree produced by