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https://www.nature.com/articles/s41467-021-26338-0	Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. # Mid-lateral cerebellar complex spikes encode multiple independent reward-related signals during reinforcement learning ## Abstract Although the cerebellum has been implicated in simple reward-based learning recently, the role of complex spikes (CS) and simple spikes (SS), their interaction and their relationship to complex reinforcement learning and decision making is still unclear. Here we show that in a context where a non-human primate learned to make novel visuomotor associations, classifying CS responses based on their SS properties revealed distinct cell-type specific encoding of the probability of failure after the stimulus onset and the non-human primate’s decision. In a different context, CS from the same cerebellar area also responded in a cell-type and learning independent manner to the stimulus that signaled the beginning of the trial. Both types of CS signals were independent of changes in any motor kinematics and were unlikely to instruct the concurrent SS activity through an error based mechanism, suggesting the presence of context dependent, flexible, multiple independent channels of neural encoding by CS and SS. This diversity in neural information encoding in the mid-lateral cerebellum, depending on the context and learning state, is well suited to promote exploration and acquisition of wide range of cognitive behaviors that entail flexible stimulus-action-reward relationships but not necessarily motor learning. ## Introduction The cerebellum has been classically considered to be a center for supervised motor learning in the brain, where the predicted results of movement are compared with the animal’s actual performance, in order to correct the errors in the action that led to the mismatch1,2,3,4. The cerebellar cortex has been posited to achieve this via its two distinct types of inputs to its principle output cells, the Purkinje cells (P-cells). First, the mossy fibers, relayed through the parallel fibers of the granule cells, contain a number of sensory and efference copy signals, which are read out as high frequency simple spikes (SS)5. Second, the climbing fibers arising from the inferior olive (IO), which evoke complex spikes (CS), signal unexpected events or errors to facilitate learning6. The precisely timed relationship between the coincidence of CS and SS causes synaptic plasticity at the granule cell->P-cell synapse, thereby effecting learning. One such mechanism is long-term depression (LTD)1,2,4,7. This flow of information and circuitry explains many simple motor learning behaviors: connections that led to erroneous and undesirable behavior could be carefully pruned by the instructions provided by the CS. However, motor learning and optimization do not always entail CS activity providing a teaching signal for SS responses8,9,10. Furthermore, recent evidence suggest that cerebellar activity is correlated with aspects of behavior that do not involve correcting the kinematics of movement: for example classical conditioning11, stimulus prediction12,13, and the magnitude of predicted reward14,15. The cerebellum’s role in these aspects of reward-related learning behavior cannot be readily explained by the present classical error-based learning models, nor do they necessarily entail CS activity affecting SS responses14 by an LTD mechanism. This is because, in reward-based learning, rather than pruning connections that led to erroneous behavior, the brain must strengthen connections that would lead to the preferred behavior16. When the non-human primates learn to associate arbitrary visual symbols with hand movement choices, the SS encode a reinforcement error signal during learning, which gradually diminishes through learning, and disappears once the learning is completed17. This error signal, which could contribute significantly to reinforcement learning18, is encoded as the difference in SS activity between recent correct and wrong outcomes of P-cells17. However, (a) the role of concurrent CS activity, (b) the interaction between SS and CS, and (c) their relationship to complex reinforcement learning and decision making are all still unknown. Here, we show that while the SS carry a reinforcement learning signal, which has information about the outcome of the animal’s most recent decision, the concurrent CS do not carry such information nor do they instruct a change in SS’s activity. Instead, the CS encoded two different signals: first, a response to the beginning of the trial that may have predicted the possibility of reward given successful performance of the task, independent of both the state of reinforcement learning and the cell type. Second, a cell type and learning-state-specific learning response that occurred after two specific events: the symbol onset and the animal’s decision, describing the general probability of failure but not the actual outcome of the prior or current trial. Neither of these types of signals correlated with any changes in the motor kinematics. These results show that although the mid-lateral cerebellum contributes to reinforcement learning18, the mechanism by which this learning occurs does not require CS-induced changes at the parallel fiber-P-cell synapse through an error-based mechanism. Rather, CS and SS form two independent channels of information, both encoding different aspects of reward-based learning depending on the context. Such differences in neural information encoding in the mid-lateral cerebellum and their complex interplay depending on the context and learning state may promote exploration and acquisition of wide range of cognitive behaviors that entail flexible stimulus–action–reward relationships. ## Results Two non-human primates performed a two-alternative forced-choice discrimination task where, in each session, they associated one of two visual symbols with a left-hand movement and the other visual symbol with a right-hand movement17. They grabbed the two bars, each with one hand to initiate the trial. A small square (cue1) appeared on the top-left corner of the screen briefly (see Methods). After a fixed duration (523 ms), cue1 reappeared in the same position, along with another cue (cue2) at the center of the screen. Again, after a fixed duration (800 ms), one of the two symbols briefly appeared on the screen and they released the hand associated with that symbol, as soon as possible, with a well-learned stereotypic hand movement to earn a liquid reward (delivered 1 ms after correct movement onset) (Fig. 1a). The kinematics or the dynamics of hand movement were task irrelevant and only the choice of hands used to release the bars (associated with the symbols) merited reward. The animals usually performed ~30 trials of an overtrained (OT) association at the beginning of each session. Then, we presented them with two novel symbols that they learned to associate with specific choices (hand releases), through trial and error. They typically achieved criterion for learning (see Methods) in ~50–70 trials on an average through an adaptive learning mechanism (Fig. 1b). Their reaction time was high during early learning and decreased significantly through learning (Fig. 1b). The animals were free to move their eyes and thus occasionally made task-irrelevant eye movements. Here we analyzed the CS activity P-cells recorded in Crus I and II of the non-human primate cerebellum whose SS activity we previously reported17 (see Methods). We identified P-cells by the presence of CS online (Fig. 1c), and offline by the (i) spike waveforms (Fig. 1d), (ii) the SS and CS interspike interval distribution (Fig. 1d), and (iii) a pause in SS after a CS (Fig. 1d, e)19. The CS fired at a very low firing rate, and in a minority of trials, consistent with prior reports20,21 (Fig. 1f), although they varied in number of spikelets and duration21 (Fig. 1g). We only analyzed activity from those cells (n = 25) with reliably detected CS that were stable throughout the entire recording (see Methods; Fig. 1h, i). ### P-cell response characteristics during the overtrained task During the OT condition, the SS activity significantly changed from the baseline only during the hand movement (Fig. 1j). In contrast, there were significant changes in CS responses in three epochs: after the cue1 onset (cue1 epoch), after the symbol onset (symbol epoch), and after the animal’s decision (reward epoch) (Fig. 1k; see Methods). The majority of the cells responded in more than one epoch (Supplementary Table S1). The CS responses in any of the three epochs could not be explained by any obvious changes in motor kinematics, such as movement of the responding hand (Fig. 1l), the non-responding hand (Fig. 1m), licking (Fig. 1n), or eye movements (Fig. 1o). The CS responded only in about 20% of trials in the cue1 epoch, 21% of trials in the symbol epoch, and in 19% of trials in the reward epoch (Supplementary Fig. S1a). Furthermore, we did not see any modulation in CS waveform duration among these three epochs (Supplementary Fig. S1b). The CS responses in the symbol epoch and during hand movement were not selective for the symbol (Supplementary Fig. S2a) or the choice of hand respectively (Supplementary Fig. S2b). ### CS activity after symbol onset was cell type specific and learning dependent The mid-lateral cerebellar P-cell SS encode a reinforcement error signal when animals learn a new visuomotor association, by reporting the outcome of the most recent decision in short epochs called “delta epochs” in a manner entirely independent of the kinematics of the movement with which the animal made the response, or the various sensory events associated with reward delivery17. During learning, roughly half of the P-cells were selective for the wrong outcome (wP-cells; Supplementary Fig. S3a) and the remaining were selective for the correct outcome (cP-cells; Supplementary Fig. S3b) during these delta epochs17. The difference between the SS activities of the cP and wP-cells provides the error signal, which approaches zero as the animals learn the new association17. We studied the learning-related changes in the CS activity after the symbol presentation in cP-cells (n = 14 cells) and wP-cells (n = 11 cells) separately. We analyzed the CS responses in 100 ms epoch (50–150 ms after symbol onset) in four different learning states (illustrated in Fig. 1b): last 20 trials of OT condition, the beginning of learning (Lbeg; the first 20 trials after the symbol switch), the middle of learning (Lmid; the first 40–60 trials after the symbol switch), and at the end of learning (Lend; 20 trials after the animal reached the criterion for learned; see Methods). The CS peak firing rate of the wP-cells changed with learning: CS increased their firing rate during early learning from OT (OT-Lbeg: P < 0.01; two-tailed Wilcoxon signed rank test) and after learning, returned to an activity that was not different from OT (OT-Lend: P = 0.24; two-tailed Wilcoxon signed rank test; Fig. 2a, b). However, the CS peak firing rate of cP-cells did not show any learning-related changes (P = 0.10, two-way Friedman test, 55 d.f. across all learning conditions that is, OT, Lbeg, Lmid, and Lend; Fig. 2d, e). Instead, the CS activity of cP-cells was more sustained or temporally dispersed (estimated as the full width at half maximum firing rate, fwhm) during learning, compared to the OT condition (OT-Lbeg: P < 0.001; two-tailed Wilcoxon signed rank test; Fig. 2d, f). After the animals learned the association between the symbols and the movements, the CS activity became temporally less dispersed (i.e., more temporally precise) as the symbols predicted a future reward more accurately (Lbeg -Lend: P < 0.001; two-tailed Wilcoxon signed rank test; Fig. 2d, f) and was no longer different from the OT condition (OT-Lend: P = 0.22; two-tailed Wilcoxon signed rank test; Fig. 2d, f). The duration of the CS waveform also differed during learning in a cell type-dependent way. Although the wP-cells did not show any learning-related changes in their CS waveform durations (P = 0.44, two-way Friedman test, 57 d.f. across all learning conditions; Fig. 2g), the CS waveform for cP-cells was longer at the beginning of learning compared to the OT condition (OT-Lbeg: P < 0.01, two-tailed Wilcoxon signed rank test; Fig. 2g) and decreased after learning, resembling the waveform in the OT condition (OT-Lend: P = 0.06, two-tailed Wilcoxon signed rank test; Fig. 2g). During learning, after the symbol onset, there were no changes in motor kinematics of the non-human primate (hand movement of the responding hand, the non-responding hand, licking, or the eye movement) and the motor behavior did not differ between correct and wrong trials (Fig. 2h). After the symbol onset, neither type of P-cells predicted the impeding decision’s outcome (Fig. 2i; wP-cell: P = 0.85, two-tailed Wilcoxon signed rank test, and cP-cell: P = 0.88, two-tailed Wilcoxon signed rank test). ### CS activity after the non-human primate’s decision was also cell type specific and learning dependent CS activity in the reward epoch was also cell type specific. Here, the firing rate of wP-cells significantly increased at the beginning of learning from OT (OT-Lbeg: P < 0.01, two-tailed Wilcoxon signed rank, Fig. 3a, b) and decreased to a lower activity in the mid learning and finally decreasing even further, comparable to the activity in the OT condition after the animals learned the task (OT-Lend: P = 0.90, two-tailed Wilcoxon signed rank test, Fig. 3a, b). There were no learning-related changes in the temporal dispersion (P = 0.61 two-way Friedman test, 42 d.f. across all learning conditions, Fig. 3a, c). However, across learning, the cP-cells did not show any significant learning-related changes either in their peak firing rate (P = 0.33, two-way Friedman test, 54 d.f. across all learning conditions, Fig. 3d, e) or the temporal dispersion of activity (P = 0.36, two-way Friedman test, 52 d.f. across all learning conditions; Fig. 3d, f). Consistent with learning-related changes in peak firing rate for wP-cells, the duration of CS was longer during the beginning of learning compared to the OT condition (OT-Lbeg: P < 0.05 two-tailed Wilcoxon signed rank test, Fig. 3g) and the duration decreased after learning and was comparable to OT (OT-Lend; P = 0.16, two-tailed Wilcoxon signed rank test; Fig. 3g). The CS waveform duration for cP-cells did not change in this epoch during learning (P = 0.21, two-way Friedman test, 55 d.f. across all learning conditions; Fig. 3g). Finally, during learning, after the non-human primate’s decision, there were no changes in motor kinematics of the non-human primate (hand movement of the responding hand, the eye movement, non-responding hand, licking) between correct and wrong trials (Fig. 3h). Neither type of P-cells reported the recent decision’s outcome (Fig. 3i; cP-cell: P = 0.31 and wP-cell: P = 0.88 two-tailed Wilcoxon signed rank test), contrary to prior reports13. They did not predict the next trial’s outcome either (Supplementary Fig. S4). ### CS responded to the stimulus that signaled the beginning of the trial On every trial, before we presented the symbols that instructed the hand movements, we presented two additional cues: cue1 and cue2 with a fixed interval of 523 ms between them (see Methods; Fig. 4a). Both types of P-cells only fired for cue1 but not for cue2. That is, for both types of P-cells, CS activity in response to cue1 was significantly higher than the baseline (cP-cells: P < 0.001; wP-cells: P < 0.001 two-tailed Wilcoxon signed rank test; Fig. 4b) and was significantly higher than that for cue2 (cP-cells: P < 0.001, wP-cells: P < 0.001, two-tailed Wilcoxon signed rank test; Fig. 4b), which was not different from the baseline value (cP-cells: P = 0.36; wP-cells: P = 0.54 two-tailed Wilcoxon signed rank test; Fig. 4b). For both types of P-cells, there was no learning-related modulation in either the peak activity (wP-cells: P = 0.49, two-way Friedman test, 43 d.f.; Fig. 4c, d; cP-cells: P = 0.44, two-way Friedman test, 54 d.f. across all learning conditions; Fig. 4f, g) or temporal dispersion of activity (wP-cells: P = 0.18, two-way Friedman test, 40 d.f. across all learning conditions; Fig. 4c, e; cP-cells: P = 0.62, two-way Friedman test, 55 d.f. across all learning conditions; Fig. 4f, h). There were no changes in CS waveform duration between the two groups or through learning (wP-cells: P = 0.96, two-way Friedman test, 60 d.f., cP-cells: P = 0.07, two-way Friedman test, 124 d.f. across all learning conditions, Fig. 4i). During this epoch, there were no changes in motor kinematics of the non-human primate (hand movement of the responding hand, the eye movement, non-responding hand, or licking). And, the motor behavior did not differ between correct and wrong trials for any of the effectors (Fig. 4j). The CS activity in this epoch did not encode prior decision’s outcome (wP-cell: P = 0.36, two-tailed Wilcoxon signed rank test, cP-cell: P = 0.92, two-tailed Wilcoxon signed rank test, Fig. 4k). ### CS activity was unrelated to SS activity or behavior during learning of novel visuomotor associations Finally, we investigated whether the CS activity related to the SS activity and the behavior during learning. In motor learning, CS acts as a teaching signal, instructing the SS output and the motor behavior through an error-based supervised learning framework3. However, we have several lines of evidence suggesting CS activity does not affect SS activity during learning of novel visuomotor associations First, the time of delta epoch was not temporally related to the time of CS activity in cue, symbol, or reward epoch for either type of P-cells (Fig. 5a; wP-cells: cue: P = 0.72, symbol: P = 0.42, reward: P = 0.59; cP-cells: cue: P = 0.43, symbol: P = 0.79, reward: P = 0.13 circular Rayleigh z test, see Supplementary Fig. S5a–c for single cell examples). Furthermore, 2/25 P-cells with delta epochs did not show any significant modulation in CS during any of the three times at which we found significant responses in the majority of P-cells (Supplementary Fig. S5d). This indicates that the time of delta epoch is unrelated to the time of CS responses during learning17 suggesting a causal dissociation between the two. That is, the CS activity did not cause the delta epoch during learning. Second, during certain types of motor learning, for instance, smooth pursuit learning, CS activity has a profound effect on SS activity in on the next trial20. When non-human primates learn to predict a smooth pursuit direction change, the presence of a CS in the prior trial is associated with a decrease of SS activity in the current trial, which occurs 175–50 ms before the time at which the CS occurred in the prior trial, as if the presence of the CS depressed the response of the P-cell to the parallel fiber activity that had occurred during learning20. However, in our reinforcement learning task, if CS were present in the previous trial during learning, the SS activity in the next trial 175–50 ms before the CS was not different from the SS activity in the same epoch for which CS was absent on the previous trial. This was true both across trial type and cell type (Fig. 5b; correct trials: cP-cells: P = 0.89, wP-cells: P = 0.80, two-tailed Wilcoxon ranksum test, Pearson r = 0.91, P < 0.001; wrong trials: cP-cells: P = 0.51, wP-cells: P = 0.65, two-tailed Wilcoxon ranksum test, Pearson r = 0.69, P < 0.001). Third, also in smooth pursuit learning, the duration of CS is longer during the instruction epoch compared to the fixation epoch (a task irrelevant epoch)21. In contrast, in our task, we found no changes in CS waveform duration (Fig. 5c) at the beginning, during, or end of delta epoch for either type of cells during learning. Although CS activity is frequently correlated with some aspect of the non-human primate’s behavior, we have two lines of evidence that this is not the case in reward-based visuomotor association learning. First, the CS activity in the prior trial could affect the behavioral performance of the non-human primate in the next trial during motor learning. For example, during smooth pursuit learning, the presence of a CS in a given trial was associated with a change of pursuit velocity in the next trial20,21. Similarly, during a saccade adaptation task, the CS encoded the error in saccade amplitude and direction that allowed for correction of that error in the text trial, improving the behavioral performance. However, in our reinforcement learning task, if CS were present in the previous trial during learning, the probability that the next trial would be correct was not significantly higher than chance level (cP-cells: P = 0.42, wP-cells: P = 0.33, one sample t-test; Fig. 5d). This means that CS responses did not affect behavior through an error-based learning mechanism. Second, the CS had no information about the outcome of the prior trial during learning, even at a time in the trial when the SS reported the outcome of the prior trial17. The CS activity at the beginning, during, or end of delta epoch during learning did not carry information about the prior trial outcome (Fig. 5e start of the delta epoch: cP-cells: P = 0.29, wP-cells: P = 0.81, two-tailed Wilcoxon signed rank test; middle of delta epoch: cP-cells: P = 0.75, wP-cells: P = 0.80, two-tailed Wilcoxon signed rank test; end of delta epoch cP-cells: P = 0.23, wP-cells: P = 0.75, two-tailed Wilcoxon signed rank test). All these provide strong converging evidence that CS were unlikely to instruct a change in SS activity through the classical error-based learning framework14,22. This furthermore suggests that the CS neural activity is entirely unrelated to SS activity. ## Discussion A comprehensive role for the cerebellum in reinforcement learning is not well understood. Several recent studies show cerebellar activity correlated with reward-based paradigms12,13,14,15,17. However, all these reinforcement learning-based studies have focused primarily on only one aspect of neural encoding in the cerebellum (either SS or CS). In this study, we show that when a non-human primate learns a new visuomotor association (Fig. 1), classifying CS responses based on their SS properties (depending on whether the SS preferentially encoded success on the prior trial, cP-cells, or failure, wP-cells)17 revealed distinct cell type-specific encoding of the probability of failure after the symbol onset (Fig. 2) and the animal’s decision (Fig. 3), but not the decision’s outcome (which is encoded by SS). CS from both cell types, from the same cerebellar area also responded to the symbol that signaled the beginning of the trial (Fig. 4). Importantly, all these CS signals were independent of changes in any motor kinematics (Figs. 24). The CS did not instruct changes in concurrent SS activity during reinforcement learning (Fig. 5), nor was CS activity related to the outcome of the prior or current trial. ### Multiple channels of information encoding in the cerebellum during reinforcement learning Unlike studies of motor learning20,23 and in contrast to the classic Marr–Albus model of the cerebellum, we did not find any relationship between the learning properties of CS activity and that of SS activity. One might have expected that a CS signal could have served as a teaching signal for the delta epoch of SS during learning if the classical error correcting framework were to apply to non-motor learning3. This was not at all the case (Fig. 5). There are several reasons why CS signals are unlikely to play the role of a teaching signal in our experiment. First, at the symbol switch between the OT and learning conditions, the SS suddenly express large differences in activity in the delta epoch (~30 sp/s). It is unlikely that this difference in the SS rate could have been caused solely by synaptic depression elicited by CS that has only been shown to cause a maximum of 8–10 sp/s changes in SS activity (with the longest CS waveforms)20,21. In addition, if the CS were causing the delta epochs, we should have seen a tight temporal relationship between the two, but we did not. It may be that CS only provide error signals during certain types of motor learning, and not for other types of learning. For example, the CS in the flocculus signal both the expected amount of reward and the motor properties14. During our reinforcement learning task, SS encode the magnitude of the reinforcement learning error, reporting the result of the most recent decision, while CS encode the probability of failure without having information about the result of the most recent decision. Both these signals disappear with learning (Figs. 2 and 3). This is in contrast with the recent reports in mice where the CS activity persists after learning, either reporting the trial outcome13,15 or predicting the reward12. The role of concurrent SS in these studies is unclear. Furthermore, in our task, the SS and CS signals form two distinct channels of neural information encoding during reinforcement learning as they do not seem to interact at the level of the cerebellar cortex (Fig. 5). However, they could impact downstream processing at the level of deep cerebellar nuclear (DCN) neurons (Supplementary Fig. S6). Apart from the reward-based, learning-dependent, and cell type-dependent signals encoded by CS after symbol and the animal’s decision, the CS also encoded a learning- and cell type-invariant response to the cue1 that signaled the beginning of the trial that was also the first of a series of temporally paired stimuli (Fig. 4). Cue1 occurred at the beginning of the trial. After its presentation, the animal’s prediction that it would get a chance to earn a reward would change. However, after the presentation of cue2, the animal does not update its prediction since cue2 occurs after a fixed interval after cue1. Keeping with this, both types of P-cells only fired for cue1 but not for cue2. Because cue1 occurred at different times after correct or wrong trials (due to an additional timeout of 2200 ms after wrong trials, see Methods), it could have not been a late response to the termination of the hand movement in the prior trial24. The response was unlikely to be just a visual response to cue1: the same stimulus (as cue1) reappeared along with cue2, but the P-cells did not respond to it. Every cell that responded to cue1 also responded to the symbol and/or after the animal’s decision. The stimulus evoking the cue1 response appeared after the symbol but not after the decision, which shows that the stimulus per se was not necessary for the response. Because of the fixed timing between cue1 and the symbol appearance, it is possible that this was a learned response to a stimulus, which was similar to the conditioned stimulus of a classical Pavlovian association, in this case the appearance of the symbols. This is consistent with a temporal difference error signal11, although the signal was not linked to the presence of reward, but rather to the possibility of performing a task to earn a reward. Since we performed electrophysiological recordings months after training both non-human primates with repeated presentation of temporally paired stimuli, we could not confirm if both the cues originally evoked a CS response that migrated eventually to cue1. Nevertheless, since the appearance of cue1 always preceded the symbols (that instructed the hand movement), it could also serve as an alerting response preparing the animal for the trial. Together, these results show that individual CS in the same cerebellar area are flexible in that they can encode very different non-motor signals, depending on the context—a reinforcement learning-dependent and cell type-dependent signal when the animal learns to make a decision, and a reinforcement learning-independent and cell type-independent response to the stimulus that signaled the beginning of the trial, consistent with a temporal difference error during classical conditioning. This mixed selectivity suggests new and general roles for CS signals that are disparate from classical error-based supervised learning. ### A cerebellar circuit that contributes to reinforcement learning The reinforcement learning signal encoded by the SS could be a transformation of the reward signals provided by the granule cells25, which in turn receive convergent reward and sensory input from diverse brain areas. However, if the CS also carry reward-related information, where could this information come from? One such key source of input to the IO is the meso-diencephalic junction (MDJ)26, a midbrain region composed of multiple nuclei, some of which integrate DCN output and project to either downstream neurons in the IO27. The MDJ also integrates descending input from cortical pyramidal tract neurons28, thus allowing the IO to represent higher order cortical computations. This is a good candidate to transmit the types of reward-related information. While CS activity in cP-cells showed both activity related to the probability of failure after both the symbol onset and decision, CS activity in wP-cells only showed the latter (Figs. 2 and 3). The waveform duration of CS also mirrored these changes. If different types of P-cells (cP-cells and wP-cells) projected to different types of DCN cells, and this segregation were maintained in the projection from the DCN to the IO, the IO neurons could maintain this functional difference as well. Therefore, just like there are cP-cells and wP-cells, we suggest that there may be cIO-cells and wIO-cells that project to these respective P-cell populations (Supplementary Fig. S6 shows schematics of the circuits by which P-cells with the two different types of CS could contribute to visuomotor association learning). However, unlike SS, the climbing fiber activity did not carry information about the most recent decision during learning. Extracellular recording in the non-human primates cannot provide information about functional or molecular segregation of P-cells. This is unlike the mouse, where functional differences in P-cells could be reflected in molecular expression of different proteins (Adolase or antigen, Zebrin)29 or differences in anatomical location (microzones)13. However, the neural basis of this functional differences in IO cells is yet unknown. Interestingly, although both these cell types responded to the stimulus that signaled the beginning of the trial in the same way in both the OT and learning contexts, they encoded different information during learning, suggesting that the information about the trial-beginning stimulus could be projected onto both cell types from an upstream to the IO. Both the climbing fiber and ~50 P-cells30 project to a single DCN neuron. The two information channels (SS and CS) carrying different information (as discussed above) could sculpt the information encoded in the DCN (Supplementary Fig. S6). The DCN is connected to the striatum31 and the PFC32 through the thalamus and is monosynaptically connected to the ventral tegmental area (VTA)33. Optogenetic stimulation of the DCN reliably evokes postsynaptic responses in both GABAergic as well as dopaminergic VTA neurons, contributing to reward-related behavior and social behavior34. Suppressing this connection is sufficient to abolish social behavior in mice34. VTA dopaminergic neurons have two key downstream targets: the ventral striatum35 and the prefrontal cortex36 both of which have been shown to be critically involved in reward processing37,38. Although it is clear from our results that the CS do not inform the SS about the results of the prior trial, other cerebellar structures might. The SS synapse, affected by the CS in motor learning, is not the only modifiable synapse in the cerebellum39. For example, the calcium responses of molecular layer interneurons become selective for the rewarded odorant as mice learn which of a pair of odorants is associated with a reward, and which with a punishment (a brief timeout) and an optogenetic inactivation of these cells slows the learning process40. The question then arises whether the different signals encoded by the CS, at the beginning of the trial and the probability of failure, which have no relationship to trial-by-trial error or reward, could also contribute to the process of visuomotor association learning. One mechanism by which they could is to provide a parallel motivational signal through the cerebellar projections to the dopaminergic system via the DCN. The DCN neurons project to several dopaminergic areas, including the VTA34 and the substantia nigra pars compacta41. Dopamine neurons are not exclusively related to reward. Different dopamine neurons respond to alerting and motivating signals as well as reward42. The CS responses that we have discovered could, via the direct projection of the climbing fibers to the DCN, excite the midbrain dopamine system, providing a cerebellar contribution to behavior entirely independent from associative learning. A lack of this signal to the basal ganglia could contribute to the learning deficit caused by mid-lateral cerebellar inactivation. Our results suggest that the SS and CS in the cerebellum have signals that could be useful for two different networks in the brain, a traditional error signal in the SS that project to the sensorimotor network, and, possibly, a motivational or arousing signal from the CS, which projects to the dopaminergic system. This synergy between the sensorimotor and motivational contributions of cerebellar processes may provide the flexibility necessary for sophisticated cognitive functions. ## Methods ### Experimental model and subject details We performed all experiments on two adult male non-human primates (Macaca mulatta), B (age: 14 years) and S (age: 7 years), weighing 10–11 kg each, for the experiments. All experimental protocols were approved by the Animal Care and Use Committees at Columbia University and the New York State Psychiatric Institute, and complied with the guidelines established by the Public Health Service Guide for the Care and Use of Laboratory Animals. ### Method details We used the NIH REX-VEX system for behavioral control. The non-human primates sat inside a dimly lit recording booth, with its head firmly fixed, in front of a back-projection screen upon which visual images were projected. The two-alternative forced-choice discrimination task began with the non-human primates grasping two bar-manipulanda, one with each hand, after which two cues (white square) appeared sequentially. The first one was briefly flashed on the top-left corner of the screen to signal a photocell that there was a programming change in the VEX display system. This square appeared at every subsequent change in the video display. The computer began to monitor whether the non-human primates had pressed the bars 20 ms after this cue. On 97% of the trials, the non-human primates had pressed both bars during the inter-trial interval (ITI) and on those after a fixed interval of 525 ms, the second one was flashed at the center of the screen for 800 ms. On the remaining 3% of the trials, the non-human primates waited until after cue1 to press the bar, so there was a variable time between the two cues. Then one of a pair of fractal symbols, that the non-human primate had never seen before, appeared briefly for 100 ms, at the center of gaze. There was no jitter in time between the time of cue onset and the time of symbol onset. One symbol signaled the non-human primate to release the left bar and the other to release the right bar. We rewarded the non-human primates with a drop of liquid juice reward for releasing the hand associated with that symbol as soon as possible. From the initiation of the hand movement, there was an 800 ms delay (ITI) until the next trial started. On wrong trials, we increased this ITI from 800 ms to (800 ms ITI + 2200 ms timeout) 3000 ms, to increase the non-human primates’ motivation to perform the task. The non-human primates were free to move their eyes and make any hand movement as long as they released the correct bar associated with the presented symbol. Although this was the case, non-human primates made very stereotypic hand movements that did not change across trials. In the OT condition, the non-human primates were repeatedly presented with the same familiar pair of symbols for which the non-human primates have learned the associations over 4–6 months. In the novel condition, the non-human primates were presented with a different pair of novel symbols that they have never seen before. They learned the association between these novel symbols and left- or right-hand release through trial and error. On every recording session, we started with the OT condition and after ~30 trials, switched to the learning condition. A correct trial was defined as the trial in which the non-human primate released only the one correct hand associated with the symbol. The non-human primates received reward only for correct trials. We defined a wrong trial as the trial in which the non-human primate released the hand not associated with the symbol. Trials where the non-human primates released both hands anytime during the trial, or released the hand(s) before the symbol onset or released the hand(s) after 2800 ms from symbol onset were considered abort trials and were neither rewarded nor analyzed. We constructed the learning curve for every session by calculating the percent correct trials in a sliding window of 10 trials shifted by 5 trials. If the non-human primates reached >90% correct through the above method and remained above 80% for at least the next 20 trials, the associations were considered “learned.” #### Single unit recording Here, we analyzed CS and SS activity from a previous study17. Briefly, we used two recording cylinders, on the left hemisphere of each non-human primate. We introduced glass-coated tungsten electrodes with an impedance of 0.8-1.2 MOhms (FHC) into the left mid-lateral cerebellum of non-human primates every day that we recorded using a Hitachi microdrive. We passed the raw electrode signal through a FHC Neurocraft head stage, and amplifier, and filtered through a Krohn-Hite filter (bandpass: lowpass 300 Hz to highpass 10 kHz Butterworth), then through a Micro 1401 system, CED electronics. We used the NEI REX-VEX system coupled with Spike2 (CED electronics) for event and neural data acquisition. We verified all recordings offline to ensure that we had isolated P-cells and that the spike waveforms had not changed throughout the course of each experiment. To do this, we correlated the spikes from the beginning and the end of a recording session and used only those sessions that had at least a correlation of 0.85 (Fig. 1h, i). The CS of 25 cells satisfied this criterion. #### Hand tracking We painted a spot on the non-human primates’ right hand with a UV-blacklight reactive paint (Neon Glow Blacklight Body Paint) prior to every session. We used a 5 W DC converted UV blacklight illuminator to shine light on the spot. Then we used a high speed (250 fps) camera (Edmund Optics), mechanically fixed to the primate chair, to capture a video sequence of the hand movement while the non-human primates performed the tasks. We used the track mate Image J43,44 and custom written software in MATLAB to semi-manually track the fluorescent paint spot painted on the non-human primate’s hand. #### Licking We recorded licking at a sampling rate of 1000 Hz using a capacitive touch sensor coupled to the metal water spout that delivered liquid water reward near the non-human primate’s mouth. Raw binary lick traces were used to generate instantaneous lick rate by trial averaging and smoothing it with a Gaussian kernel of sigma = 20. #### Eye movements We tracked the non-human primate’s left eye positions at 240 Hz sampling rate with an infrared pupil tracker (ISCAN, Woburn, MA USA) interfaced with Spike2 (CED electronics) where it was upsampled to 1000 Hz and synced with the event markers from NEI REX-VEX system. ### Quantification and statistical analysis #### Quantitation of CS activity To study the event related CS activity, for each cell, we first aligned the CS responses to cue1, cue2, symbol, and reward onset. Then, for each condition, we binned the CS responses in 1 ms bins and convolved the resulting function with a Gaussian kernel of sigma = 20 ms to obtain spike density functions, for each cell. Then, we quantified the firing rate and the temporal dispersion (estimated as the full width at half maximum firing rate, fwhm) in a 100 ms window (50–150 ms after respective event onset) for each condition, and averaged across single cell results to provide our final estimates. We confirmed the independence of these two measures through a lack of significant correlation. #### Epochs of significant CS activity We estimated the epochs where the CS had significant activity by performing a two-tailed t-test between the population CS activity (across all cells and all trials) in every 100 ms bins and a baseline activity (–100 to 0 ms aligned to cue2 onset). Then, we corrected for multiple comparisons using the Benjamini and Hochberg/Yekutieli false discovery rate method. Through this method, we found three epochs with significant CS activity: after cue1, symbol, and reward epochs (Supplementary Fig. S1). However, to be consistent in our analysis, we only analyzed data in 100 ms bins in all three epochs. Therefore, we analyzed the CS responses 50–150 ms after symbol onset, reward onset, and cue1 onset. Furthermore, we analyzed the data from a condition of a cell only if it had at least one CS across trials in that condition’s interval. #### Measurements of CS morphology The validity of the data presented in Figs. 2g, 3g, 4g, and 5c depends on the accuracy of our CS duration measurements. One of the authors manually made all these measurements while being blind to the type of cell or the epoch in which the CS was present. We measured each CS duration from the beginning of the first deflection of the extracellular potential to the time of the return to baseline potential (as indicated above panel Supplementary Fig. S1b). To reduce the bias in measurements, another author randomly verified the measurements and made independent measurements of randomly selected CS spikes, to crosscheck the results, while also being blind to the type of cell or the epoch in which the CS was present. Furthermore, random errors in measurements should not be prominent in a population study. #### CS tuning to symbol and choice of hand The CS responses in the symbol epoch and during movement were not selective for symbol or choice of hand respectively. To show this, we first calculated the contrast function (A – B) / (A + B) in the symbol epoch (50–250 ms after symbol onset) for preferences between the two symbols and in the movement epoch (50 ms before to 250 ms after the movement onset) for preferences between the hand movements and the symbols. To verify if this tuning were meaningful and not just due to extreme differences in sampling number and noise (due to sparseness in firing rate and low trial number), we generated a null distribution of spike times through a gamma distribution45 that was matched with the parameters of the experimental data (we obtained the shape parameter, $$k$$, the ISI distribution fit and took the scale parameter, $$\theta$$, as the inverse of firing rate) and calculated a similar tuning function on this null distribution. We found that the CS responses during the symbol (Supplementary Fig. S2a) or the movement epochs (Supplementary Fig. S2b) were not statistically different from a null distribution (symbol selectivity: P = 0.51; t-test; choice selectivity: P = 0.48; t-test). ### Statistics and reproducibility All the experimental analyses were performed on CS from 25 P-cells, collected from two non-human primates. ### Reporting summary Further information on research design is available in the Nature Research Reporting Summary linked to this article. ## Data availability All the relevant data that support the findings of this study are available at https://github.com/naveen-7/Cerebellum_reward. A reporting summary for this article is available as a Supplementary Information file. Source data are provided with this paper. ## Code availability The codes used for the analyses that support the findings of this study are available from https://github.com/naveen-7/Cerebellum_reward. ## References 1. 1. Marr, D. A theory of cerebellar cortex. J. Physiol. 202, 437–470 (1969). 2. 2. Ito, M. The Cerebellum and Neural Control. (Raven Pr, 1984). 3. 3. Raymond, J. L. & Medina, J. F. Computational principles of supervised learning in the cerebellum. Annu. Rev. Neurosci. 41, 233–253 (2018). 4. 4. Albus, J. S. A theory of cerebellar function. Math. Biosci. 10, 25–61 (1971). 5. 5. Lisberger, S. & Fuchs, A. Role of primate flocculus during rapid behavioral modification of vestibuloocular reflex. II. Mossy fiber firing patterns during horizontal head rotation and eye movement. J. Neurophysiol. 41, 764–777 (1978). 6. 6. Stone, L. & Lisberger, S. Visual responses of Purkinje cells in the cerebellar flocculus during smooth-pursuit eye movements in monkeys. II. Complex spikes. J. Neurophysiol. 63, 1262–1275 (1990). 7. 7. Suvrathan, A., Payne, H. L. & Raymond, J. L. Timing rules for synaptic plasticity matched to behavioral function. Neuron 92, 959–967 (2016). 8. 8. Avila, E. et al. Purkinje cell activity during suppression of voluntary eye movements in rhesus macaques. Preprint at bioRxiv (2021). 9. 9. Streng, M. L., Popa, L. S. & Ebner, T. J. Complex spike wars: a new hope. Cerebellum 17, 735–746 (2018). 10. 10. Ke, M. C., Guo, C. C. & Raymond, J. L. Elimination of climbing fiber instructive signals during motor learning. Nat. Neurosci. 12, 1171–1179 (2009). 11. 11. Ohmae, S. & Medina, J. F. Climbing fibers encode a temporal-difference prediction error during cerebellar learning in mice. Nat. Neurosci. 18, 1798–803 (2015). 12. 12. Heffley, W. et al. Coordinated cerebellar climbing fiber activity signals learned sensorimotor predictions. Nat. Neurosci. 21, 1431–1441 (2018). 13. 13. Kostadinov, D., Beau, M., Pozo, M. & Häusser, M. Predictive and reactive reward signals conveyed by climbing fiber inputs to cerebellar Purkinje cells. Nat. Neurosci. 22, 950–962 (2019). 14. 14. Larry, N., Yarkoni, M., Lixenberg, A. & Joshua, M. Cerebellar climbing fibers encode expected reward size. Elife 8, e46870 (2019). 15. 15. Heffley, W. & Hull, C. Classical conditioning drives learned reward prediction signals in climbing fibers across the lateral cerebellum. eLife 8, e46764 https://doi.org/10.7554/eLife.46764 (2019). 16. 16. Catz, N., Dicke, P. W. & Thier, P. Cerebellar complex spike firing is suitable to induce as well as to stabilize motor learning. Curr. Biol. 15, 2179–2189 (2005). 17. 17. Sendhilnathan, N., Ipata, A. E. & Goldberg, M. E. Neural correlates of reinforcement learning in midlateral cerebellum. Neuron 106, 188–195.e5 (2020). 18. 18. Sendhilnathan, N. & Goldberg, M. E. The mid-lateral cerebellum is necessary for reinforcement learning. Preprint at biorXiv https://doi.org/10.1101/2020.03.20.000190 (2020). 19. 19. Dijck, G. et al. Probabilistic identification of cerebellar cortical neurones across species. PloS One 8, e57669 https://doi.org/10.1371/journal.pone.0057669 (2013). 20. 20. Medina, J. F. & Lisberger, S. G. Links from complex spikes to local plasticity and motor learning in the cerebellum of awake-behaving monkeys. Nat. Neurosci. 11, 1185–1192 (2008). 21. 21. Yang, Y. & Lisberger, S. G. Purkinje-cell plasticity and cerebellar motor learning are graded by complex-spike duration. Nature 510, 529–532 (2014). 22. 22. Sendhilnathan, N., Ipata, A. E. & Goldberg, M. E. Mixed selectivity in the cerebellar Purkinje-cell response during visuomotor association learning. Preprint at bioRxiv (2021). 23. 23. Herzfeld, D. J., Kojima, Y., Soetedjo, R. & Shadmehr, R. Encoding of error and learning to correct that error by the Purkinje cells of the cerebellum. Nat. Neurosci. 21, 736–743 https://doi.org/10.1038/s41593-018-0136-y (2018). 24. 24. Khilkevich, A., Zambrano, J., Richards, M.-M. & Mauk, M. D. Cerebellar implementation of movement sequences through feedback. Elife 7, e37443 (2018). 25. 25. Wagner, M. J., Kim, T., Savall, J., Schnitzer, M. J. & Luo, L. Cerebellar granule cells encode the expectation of reward. Nature 544, 96–100 (2017). 26. 26. De Zeeuw, C. I. et al. Microcircuitry and function of the inferior olive. Trends Neurosci. 21, 391–400 (1998). 27. 27. Onodera, S. Olivary projections from the mesodiencephalic structures in the cat studied by means of axonal transport of horseradish peroxidase and tritiated amino acids. J. Comp. Neurol. 227, 37–49 (1984). 28. 28. Veazey, R. B. & Severin, C. M. Afferent projections to the deep mesencephalic nucleus in the rat. J. Comp. Neurol. 204, 134–150 (1982). 29. 29. Hawkes, R. & Herrup, K. Aldolase C/zebrin II and the regionalization of the cerebellum. J. Mol. Neurosci. 6, 147–158 (1995). 30. 30. Person, A. L. & Raman, I. M. Purkinje neuron synchrony elicits time-locked spiking in the cerebellar nuclei. Nature 481, 502–505 (2011). 31. 31. Hoshi, E., Tremblay, L., Féger, J., Carras, P. L. & Strick, P. L. The cerebellum communicates with the basal ganglia. Nat. Neurosci. 8, 1491–1493 (2005). 32. 32. Middleton, F. A. & Strick, P. L. Cerebellar projections to the prefrontal cortex of the primate. J. Neurosci. 21, 700–712 (2001). 33. 33. Beier, K. T. et al. Circuit architecture of VTA dopamine neurons revealed by systematic input-output mapping. Cell 162, 622–634 (2015). 34. 34. Carta, I., Chen, C. H., Schott, A. L., Dorizan, S. & Khodakhah, K. Cerebellar modulation of the reward circuitry and social behavior. Science 363, eaav0581 (2019). 35. 35. Kelley, A. E. Ventral striatal control of appetitive motivation: role in ingestive behavior and reward-related learning. Neurosci. Biobehav. Rev. 27, 765–776 (2004). 36. 36. Tzschentke, T. The medial prefrontal cortex as a part of the brain reward system. Amino Acids 19, 211–219 (2000). 37. 37. Histed, M. H., Pasupathy, A. & Miller, E. K. Learning substrates in the primate prefrontal cortex and striatum: sustained activity related to successful actions. Neuron 63, 244–253 (2009). 38. 38. Pasupathy, A. & Miller, E. K. Different time courses of learning-related activity in the prefrontal cortex and striatum. Nature 433, 873–876 (2005). 39. 39. De Zeeuw, C. I., Lisberger, S. G. & Raymond, J. L. Diversity and dynamism in the cerebellum. Nat. Neurosci. 24, 160–167 (2021). 40. 40. Ma, M. et al. Molecular layer interneurons in the cerebellum encode for valence in associative learning. Nat. Commun. 11, 1–16 (2020). 41. 41. Watabe-Uchida, M., Zhu, L., Ogawa, S. K., Vamanrao, A. & Uchida, N. Whole-brain mapping of direct inputs to midbrain dopamine neurons. Neuron 74, 858–873 (2012). 42. 42. Bromberg-Martin, E. S., Matsumoto, M. & Hikosaka, O. Dopamine in motivational control: rewarding, aversive, and alerting. Neuron 68, 815–834 (2010). 43. 43. Tinevez, J.-Y. Y. et al. TrackMate: an open and extensible platform for single-particle tracking. Methods 115, 80–90 (2017). 44. 44. Schindelin, J. et al. Fiji: an open-source platform for biological-image analysis. Nat. Methods 9, 676–682 (2012). 45. 45. Sendhilnathan, N., Basu, D. & Murthy, A. Assessing within-trial and across-trial neural variability in macaque frontal eye fields and their relation to behaviour. Eur. J. Neurosci. 52, 4267–4282 (2020). ## Acknowledgements We thank Glen Duncan for electronic assistance, John Caban and Matthew Hasday for machining, Dr Girma Asfaw, Dr Moshe Shalev, Dr Christina Winnicker, and the Columbia Institute for Comparative Medicine for animal care, and Lisa Kennelly, Whitney Thomas, and Holly Cline for facilitating everything. This work was supported by the Keck, Zegar Family, and Dana Foundations and the National Eye Institute (R24 EY-015634, R21 EY-017938, R21 EY-020631, R01 EY-017039, and P30 EY-019007 to M. E. G., PI). ## Author information Authors ### Contributions N.S. conceptualized the study; N.S. and A.I. collected the data; N.S. analyzed the data and made all the figures; N.S. and M.E.G. wrote, revised, and edited the manuscript. ### Corresponding authors Correspondence to Naveen Sendhilnathan or Michael E. Goldberg. ## Ethics declarations ### Competing interests The authors declare no competing interests. Peer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ## Rights and permissions Reprints and Permissions Sendhilnathan, N., Ipata, A. & Goldberg, M.E. Mid-lateral cerebellar complex spikes encode multiple independent reward-related signals during reinforcement learning. Nat Commun 12, 6475 (2021). https://doi.org/10.1038/s41467-021-26338-0 • Accepted: • Published: • DOI: https://doi.org/10.1038/s41467-021-26338-0	2022-01-28 09:02:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5483046174049377, "perplexity": 5022.9977324452375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305423.58/warc/CC-MAIN-20220128074016-20220128104016-00341.warc.gz"}
https://answerbun.com/mathematics/putting-an-a-otimes-amathrmop-module-structure-on-operatornamehomvv/	# Putting an $A otimes A^{mathrm{op}}$-module structure on $operatorname{Hom}(V,V')$ Observe that given an algebra $$A$$, we can put a $$A otimes A^{mathrm{op}}$$ structure on $$operatorname{Hom}(V,V’)$$ by $$((a otimes a’)f)(v)=af(a’v)$$ Indeed, we have begin{align} {}& ((a otimes a’)(b otimes b’))f(v) \ ={}& (ab otimes b’a’))f(v) \ ={}& ab(f(b’a’v)) \ ={}& a((b otimes b’)f(a’v)) \ ={}& (a otimes a’)((b otimes b’)f(v)) end{align} I’m just a bit confused why we need the $$A^{mathrm{op}}$$ rather than just $$A$$… Can we put an $$A otimes A$$ structure on $$operatorname{Hom}(V,V’)$$ by the same formula? begin{align} {}& ((a otimes a’)(b otimes b’))f(v) \ ={}& (ab otimes a’b’))f(v) \ ={}& ab(f(a’b’v)) \ ={}& (a otimes a’)bf(b’v)) \ ={}& (a otimes a’)((b otimes b’)f(v)) end{align} Maybe they both work… Thanks in advance!! Mathematics Asked by A Dragon on December 28, 2020 Letting $$(af)(v) = f(av)$$ does not define an $$A$$-action, because it breaks associativity. $$((ab)f)(v)$$ should equal $$f(abv)$$, but $$(a(bf))(v) = (bf)(av) = f(bav)$$, reversing the multiplication of $$a$$ and $$b$$. This is why this is an $$A^{op}$$-structure. Correct answer by Jacob FG on December 28, 2020 ## Related Questions ### Total expectation under measure theory 1 Asked on December 6, 2020 by jiexiong687691 ### complex norm inequality 1 Asked on December 6, 2020 by stranger ### How do you define “independence” in combinatorics? 3 Asked on December 6, 2020 by trivial-math-is-difficult ### Show that the function $f(z)=frac{1-cos z}{z^n}$ has no anti-derivative on $mathbb Csetminus{0}$ if $n$ is an odd integer. 1 Asked on December 6, 2020 by riyasudheen-t-k ### Checking a ring is not Cohen-Macaulay 2 Asked on December 6, 2020 ### Unique representation: $a!cdot b!cdot c!=m!cdot n!cdot p!$ 4 Asked on December 6, 2020 by felipeuni 0 Asked on December 6, 2020 by twosigma ### Is $x^2$ analytic in $mathbb{R}$ 1 Asked on December 5, 2020 by joey ### The best strategy to maximize the last number on a die. 2 Asked on December 5, 2020 by student ### How to find bounds when doing a double integral? 1 Asked on December 5, 2020 by dr-suess-official ### A simple looking second-order DE is giving me tough time 1 Asked on December 5, 2020 by curious-2-learn ### Convergence of a series of integrals: $S=sum_{ngeq 1}(-1)^{n}int_{n}^{n+1}frac{1}{t e^t},dt$ 3 Asked on December 5, 2020 by ramana ### Understanding Seifert Van Kampen 0 Asked on December 5, 2020 by moooose ### Trying to evaluate a complex integral? 4 Asked on December 5, 2020 by gray ### Random variable $X$ has uniform distribution on section $[0,2]$. What’s the expected value of variable $Y=frac{X^{4}}{2}$ 2 Asked on December 5, 2020 by fakeraker-p ### Show that this iterative Richardson iteration may diverge 1 Asked on December 5, 2020 ### Estimating number of infected people and getting bounds of its probability based on a few samples. 1 Asked on December 5, 2020 by user_9	2022-01-20 11:22:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 18, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4077412784099579, "perplexity": 4387.208037772703}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301737.47/warc/CC-MAIN-20220120100127-20220120130127-00245.warc.gz"}
https://sdenisov.wordpress.com/	The celebrated Carleson theorem for trigonometric Fourier series solved positively the Lusin’s conjecture. It says, in particular, that for every $f\in L^2(\mathbb{T})$, the Fourier series $\sum_{n}\widehat f_ne^{in\theta}$ converges for a.e. $\theta\in \mathbb{T}$. This results follows from the estimate on the Carleson’s maximal function. In the preprint with R. Bessonov, we studied the analog of Lusin’s conjecture for polynomials orthogonal on the unit circle. Suppose $\mu$ is probability measure on $\mathbb{T}$. It belongs to Szego’s class if $\log \mu'\in L^1(\mathbb{T})$. This class plays crucial role in many branches of classical analysis and probability (see other posts). If $\mu$ is in Szego’s class, we can define Szego’s function by the formula $D(z)=\exp\left( \frac{1}{2\pi } \int_{\mathbb{T}} \frac{1+\bar\xi z}{1-\bar\xi z}\log \sqrt{ \mu'(\theta)}d\theta\right),\, \xi=e^{i\theta}\,,z\in \mathbb{D}.$ Denote the polynomials orthonormal with respect to $\mu$ by $\{\phi_n(z,\mu)\}$. The analog of Lusin’s conjecture says: For $\mu$ in Szego’s class, prove that $\sup_{n}\|\phi_n(z,\mu)\|<\infty$ for a.e. $z\in \mathbb{T}$. We were not able to solve this problem but we proved a few results that reformulate pointwise convergence of polynomials in different terms. Each measure $\mu$ generates the sequence of Schur functions (analytic contractions on $\mathbb{D}$), that we denote by $\{f_n(z)\}$. Given a parameter $\rho\in (0,1)$ and a point $\xi\in \mathbb{T}$, define the Stolz angle $S_\rho^(\xi)$ to be the convex hull of $\rho\mathbb{D}$ and $\xi$. Our central result is the following theorem Theorem. Let $\mu$ be Szego measure and $Z(\phi_n) = \{z \in \mathbb{D}: \; \phi_n(z) = 0\}$. Take any $a>0$ and denote $r_{a,n}= 1-a/n$. Then, for almost every $\xi\in \mathbb{T}$, the following assertions are equivalent: (a) $\lim_{n \to \infty} \|\phi_{n}^{}(\xi)\|^2= \|D_{\mu}^{-1}(\xi)\|^2$, (b) $\lim_{n \to \infty} {\rm dist}(Z(\phi_n), \xi)\,n = +\infty$, (c) $\lim_{n \to \infty} f_n(r_{a,n}\xi) = 0$, (d) $\lim_{n\to \infty}\sup_{z\in S_\rho^(\xi)}\|f_n(z)\|=0$ for every $\rho\in (0,1)$. The idea of the proof is based on the analysis of new entropy function which generalizes the standard Szego entropy. This quantity is well-behaved for a.e. boundary point and this gives uniform in $n$ control over the oscillation of $f_n(z)$ in $z$ on the circle. Roman Bessonov and I uploaded a new paper titled “de Branges canonical systems with finite logarithmic integral” to arxiv today. This is a continuation of an earlier manuscript, see also the previous topic on “Szego theorem on the real line and Krein strings”. In the current version, we completed the project of describing the measures on the line with finite logarithmic integral. Assume $\sigma$ is Poisson-finite measure, i.e., that $\displaystyle \int_{-\infty}^\infty \frac{d\sigma}{1+x^2}<\infty.$ We define its logarithmic integral as $\displaystyle \int_{-\infty}^\infty \frac{\log \sigma'}{1+x^2}dx.$ Existence of logarithmic integral, i.e., condition $\displaystyle \int_{-\infty}^\infty \frac{\log \sigma'}{1+x^2}dx>-\infty$ plays a role in the theory of Gaussian stationary stochastic processes: it holds if and only if the future of the process with spectral measure $\sigma$ can not be predicted by its past. Every Poisson-finite measure $\sigma$ gives rise to a function $m$ in Herglotz-Nevanlinna class, i.e., the class of functions analytic in $\mathbb{C}^+$ with non-negative imaginary part, by a Herglotz formula $\displaystyle m(z)=az+b+\frac{1}{\pi}\int_{-\infty}^\infty \left(\frac{1}{x-z}-\frac{x}{1+x^2}\right)d\sigma,$ where $a\ge 0, b\in \mathbb{R}$. The de Branges theory of Hilbert spaces of functions of exponential type provides a bijection between Herglotz-Nevanlinna class and the class of all canonical Hamiltonian systems. Canonical Hamiltonian system can be written as the following Cauchy problem $JM'=zHM, \quad M(0,z)=I_{2\times 2}, \quad J=\left( \begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right), z\in \mathbb{C}$ where $2\times 2$ Hamiltonian $H(t)$ is nonnegative locally summable matrix-function on $[0,\infty)$. In short, the Weyl-Titchmarsh theory for canonical systems provides $m$ for each $H$ and the converse is true by de Branges theory. In the paper, we characterize all $H$ for which the logarithmic integral of $\sigma$ exists. Consider $H$ for which $\sqrt{\det H}\notin L^1(\mathbb{R}^+)$. Then, define the grid of points $\{\eta_n\}$ by the formula $\displaystyle \eta_n=\min\{t: \int_0^t \sqrt{\det H(\tau)}d\tau=n\}$ and consider the quantity $\displaystyle K(H)=\sum_{n=0}^\infty \left(\det\left( \int_{\eta_n}^{\eta_{n+2}} H(\tau)d\tau\right)-4\right).$ Our main theorem, which is stated below, is a natural generalization of the Szego theorem in the theory of polynomials orthogonal on the unit circle. #### THEOREM. The measure $\sigma$ has finite logarithmic integral if and only if $K(H)<\infty$ for $H$ generated by $\sigma$. We quantify it by the sharp two-sided estimate. The sum in the definition of $K(H)$ can be written in the form reminiscent of the matrix $A_2$ Muckenhoupt condition but we do not understand yet how the problem is connected to Muckenhoupt classes. There are multiple applications of our theory to scattering problems for Dirac and wave equations. Great news! The next ICM will be in Saint-Petersburg, Russia. We will be monitoring the progress here August 18, 2020. Two preprints were submitted lately. One is “Jacobi matrices on trees generated by Angelesco system: asymptotics of coefficients and essential spectrum” with A. Aptekarev and M. Yatselev. For Angelesco system with two real-analytic weights, we performed the Riemann-Hilbert analysis to obtain asymptotics of polynomials of the first and second type when the multi-index $n=(n_1,n_2)$ goes to infinity in any direction, including the marginal ones. That was used to characterize the right limits and the essential spectrum of the corresponding Jacobi matrix on the binary tree. The second preprint Spectral theory of Jacobi matrices on trees whose coefficients are generated by multiple orthogonality with M. Yattselev finishes the program initiated in the original paper A. Aptekarev, S. Denisov, and M. Yattselev, Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials, Trans. Amer. Math. Soc., Vol. 373, N2, 2020, 875-917 now published in Transactions of AMS. When two measures of orthogonality are involved, we study Jacobi matrixes generated by MOPs of the first and of the second types. Those of the second type give rise to a Jacobi matrix on the finite binary tree, which is self-adjoint in indefinite metric (it is sign-definite for Angelesco system and is non-sign-definite for Nikishin system). We find the spectrum and construct the basis of eigenvectors each of which can be written in terms of these MOPs. The analysis for MOPs of the first type is more involved since the associated Jacobi matrix is defined on the infinite binary tree. For Angesleco system, this Jacobi matrix is self-adjoint in the standard metric and we find the convenient decomposition of the Hilbert space into the orthogonal sum of cyclic subspaces. Each subspace is described by a generalized eigenvector which is written via the MOPs of the first type. The spectral measures of generators are found explicitly thus making the complete analysis of the spectral type possible. In short, this paper puts the connection between MOP and the operator theory on the solid ground. Unlike for one-dimensional theory, the feature of multi-dimensional case is that often times the theory of self-adjoint operators in indefinite metric (in Krein spaces) starts to play the role. ********************************************************************** June 18, 2018. In the recent preprint with A. Aptekarev and M. Yattselev, we found the missing link between the spectral theory of self-adjoint operators and the theory of multiple orthogonal polynomials. Recall that given a measure on the real line with compact support, we can construct a sequence of orthonormal polynomials which satisfy three-term recurrence that defines one-sided Jacobi matrix. This matrix is actually a self-adjoint operator in $\ell^2(Z^+)$ for which many standard quantities of operator theory (e.g., Green’s function, spectral measures, etc) can be computed through the measure we started with and the associated orthogonal polynomials. Conversely, given a bounded self-adjoint Jacobi matrix, we can uniquely find the measure of orthogonality that generates it. This correspondence proved to be very useful both in spectral theory and approximation theory (however, to my knowledge never played a crucial role in proving deeper analytical results). The multiple orthogonal polynomials (MOPs) can be of two types and they are defined given $d$ measures through some orthogonality conditions (see the paper for details). These polynomials depend on multi-index $\vec{n}$ of dimension $d$ and satisfy recurrence relations on the integer lattice. It turns out that, to define the corresponding operator, one has to untwine these recurrences to the rooted $d+1$ homogeneous tree. The resulting operator is a self-adjoint Jacobi matrix which is defined on that tree. We obtained, among other things, the formulas connecting the Green’s function to MOPs and found its spatial asymptotics for analytic weights using matrix Riemann-Hilbert analysis. The usefulness of this connection is illustrated by, e.g., reproving some known results. In the recent preprint (to appear in Comm. Math. Phys.) “On the growth of the support of positive vorticity for 2D Euler equation in an infinite cylinder”, Kyudong Choi and I obtained an upper bound for the diameter of the support of positive vorticity in the 2D Euler dynamics $\theta_t+u\cdot \nabla \theta=0, \quad \theta(x,y,0)=\theta_0(x,y).$ We considered the problem on the infinite cylinder which is equivalent to $2\pi$-periodic initial data in one direction. If one takes $\theta_0$ as nonnegative bounded function with compact support, then the weak solution exists globally in time. If $d(t)$ denotes the diameter of its support, then the trivial bound reads $d(t)\leq C(1+t).$ In the paper, we improve it to $d(t)\leq C(1+t)^{\frac 13} \log^2(2+t).$ Our argument is based on controlling the sequence of specially chosen moments on the dyadic spatial scale. The crucial part of the argument was to exploit the uniform in time estimate on the first moment of vorticity. For the problem on the whole plain, similar results were obtained previously in the paper by Iftimie, Sideris, and Gamblin “On the evolution of compactly supported planar vorticity”. Our method is different from the one used previously in that it uses another conserved quantity and different set of moments. The 2D Euler evolution in infinite cylinder is remarkable model because the kernel in the Biot-Savart law that expresses velocity $u$ in terms of $\theta$ has the exponentially decaying first component. That means two distant parts of vorticity are essentially decoupled. One would hope that this, along with conservation of horizontal center of mass, should yield much stronger bound on $d(t)$, for example, $d(t)\leq C_\epsilon(1+t)^\epsilon$. This, however, seems difficult to achieve due to possible “diffusive dynamics” of $\theta$. One possible interesting direction is to study the evolution of patch of vorticity in the active scalar equation when no conserved quantities are known but the kernel in the Biot-Savart law is short-range and has some basic symmetries. One would think that this should be enough to prove strong confinement results. Roman Bessonov and I just posted the paper “A spectral Szego theorem on the real line” on arxiv. You can also read it here. Given a probability measure $\sigma$ on the unit circle, one can ask when the analytic polynomials are NOT dense in $L^2(\sigma)$. The theorem of Szego claims that this is so iff any of the following conditions holds: • $\int_{T}\log \sigma'd\theta>-\infty$ • The sequence of recurrence parameters $\{a_n\}$ (or Schur parameters) of polynomials orthogonal with respect to $\sigma$ belongs to $\ell^2$. Given a measure on the real line $\mu$ that satisfies normalization $\displaystyle \int_{R}\frac{d\mu(t)}{1+t^2}<\infty$, we can ask the question when the set of functions $\displaystyle \int_0^\infty e^{itx}f(x)dx, f\in C^\infty_c(0,\infty)$ is NOT dense in $L^2(\mu)$. The answer is given by the theorem of Kolmogorov-Krein-Wiener: it is iff $\displaystyle \int_{R} \frac{\log \mu'}{1+t^2}dt>-\infty$ However, the spectral characterization of this condition has been missing. In the paper, we consider the Krein string, – the “mother of all non-negative self-adjoint operators with simple spectrum”. It is given by the formal differential operator $\displaystyle S=-\frac{\partial}{\partial M} \left( \frac{\partial}{\partial t}\right)$ where $M$ is any non-decreasing function on $R^+$. The corresponding self-adjoint operator can be defined and its spectral measure $\mu$ along with one additional real parameter determines $M$ completely. In the paper, we characterize all strings $M$ for which the logarithmic integral of $\mu$ converges. This is done by proving analogous statement for diagonal De Branges canonical systems. The existence of the entropy is important for the prediction theory of stationary Gaussian processes with continuous time. It is likely that the obtained characterization will allow one to quantify some statements in this theory. In the recent preprint, I study the wave equation for the elliptic operator in divergence form. In $\mathbb{R}^3$, define $H=-{\rm div }(1+V)\nabla,$ where $V$ oscillates and decays at infinity. More precisely, $V={\rm div} \,Q$ where $\\|V\\|<\infty, \\|Q\\|<\infty$ and the norm $\\|f\\|$ is defined as $\\|f\\|=\left(\sum_{n=0}\max_{\|x\|\in [n,n+1]}\|f\|^2\right)^{1/2}.$ I also assume $\\|V\\|_\infty<1$ to make sure that $H$ is non-negative operator. The wave equation for $H$ is $u_{tt}+Hu=0, \, u(x,0)=f_1,\, u_t(x,0)=f_2.$ The main result of the paper states that the following wave operators $\lim_{t\to\pm \infty} e^{it\sqrt{H}}e^{-it\sqrt{-\Delta}}f=W^{\pm}f$ exist for every $f\in L^2(\mathbb{R}^3)$ and the limit is understood in $L^2(\mathbb{R}^3)$ norm. The condition on $V$ is optimal in some sense, i.e., the rate of decay is sharp and the oscillation is necessary if the potential is not short-range. The proof is based on the analysis of the asymptotical behavior of the Green’s function $G(x,y,k^2)$ where $k\in \mathbb{C}^+$, $y$ is fixed, and $\|x\|\to\infty$. The result about asymptotics is similar to that for the orthogonal polynomials on the circle in the Szego case. The main difference with the one-dimensional situation is that the resulting “Szego” function belongs to the vector-valued Hardy space. The operator $e^{it\sqrt{H}}$ can be written through the resolvent $(H-z)^{-1}$ by the contour integral and this is how the Green’s function enters the proof. The method is quite general and can be adapted to wave equations for the Schrodinger equation and other problems. In the recent preprint with Jen Beichman we considered the 2D Euler evolution on the tube $S=\mathbb{R}\times \mathbb{T}$. Each rectangle $\Omega_L=[-L,L]\times\mathbb{T}$ is a steady state. We proved that if $L$ is sufficiently large, then these steady states are stable for all time. For example, if one takes a patch $\Omega$ such that $\|\Omega\Delta\Omega_L\|$ is small, then the Euler evolution $\Omega(t)$ of this patch will have $\|\Omega(t)\Delta \Omega_L\|$ small for arbitrary $t$. This result generalizes analogous statement for the stability of the disc on the plane (proved by Sideris-Vega). As Sideris and Vega, we used the method of V. Arnold. The idea of this method is to study the variational problem associated to the conserved quantities. In our case, these are $I_0=\|\Omega(t)\|,\,\, I_1=\int_{\Omega(t)} xdxdy,\,\, I_3=\int_{\Omega(t)}\psi dxdy$ where $\psi$ is a stream function given by $\psi=\Delta^{-1}\chi_{\Omega(t)}.$ Then, we set up a variational problem with constraint $I_3\to\min,\, I_2=0,\, I_1=4\pi L$ We proved that the global minimizer is $\Omega_L$ and if $I_3(E)$ is close to the minimum value for some patch $E$, then $E$ is close to $\Omega_L$ in a weak topology. This essentially gives the required stability. March 3, 2014. If one considers the 2d Euler equation of incompressible inviscid fluids on the plane in the vorticity form and takes the initial data as the characteristic function of a certain domain, then the Yudovich theory guarantees that the solution will exist globally and will be equal to the characteristic function of a time-dependent domain which is homeomorphic to the original one for all times. The numerical experiments dating back to the works of P. Saffman and Zabusky et al. indicate the existence of the centrally-symmetric V-states, i.e. a symmetric pair of patches that rotates with constant angular velocity around the origin without changing shape. If the distance between the patches in the pair equals to $\lambda$ and $\lambda>0$, then the boundary of the V-state seems to be smooth. However, when $\lambda=0$, the both patches form a 90 degrees angle at the point of contact. The analytical proof for the existence of these V-states has never been obtained and this is an interesting problem. In the recent preprint, I considered the analogous equation with the cut-off. Loosely speaking, this corresponds to looking at the window around the origin where the contact of the patches is supposed to happen. Mathematically, the model with cut-off is important as it possesses the explicit singular solution: $y_0(x)=\|x\|$. Then, I addressed the problem of existence of the curve of smooth solutions that converge to $y_0$ in the uniform metric when the parameter $\lambda\to 0$. Technically, this boils down to application of the implicit function theorem and is somewhat tedious. This technique might be important to better understand the mechanism of the merging and the sharp corner formation in the Euler dynamics. Another important problem is to prove that the merging in finite time is possible for the $\alpha$-model when $\alpha<1$ and is close to $1$. May 2011. Consider $H_0=-\Delta, x\in \mathbb{R}^d$. Going on the Fourier side one can see that $H_0$ is equivalent to multiplication by $\|\omega\|^2$ and so the spectrum of $H_0$ is purely absolutely continuous (a.c.). From the physics perspective, the presence of a.c. spectrum is an indication that the wave propagation governed by $\displaystyle i\psi_t=H_0\psi$ does have a transport effect (though without much specifics). Perturb the Laplacian as follows $H=H_0+V(x)$ where $V(x)$ is potential and ask the question what is a minimal assumptions on $V$ to guarantee that the a.c. spectrum is preserved. Make it a perturbation theory question, assume that $V$ is in some weighted Lebesgue space $L^p_w$. Then, what are the critical $p$ and $w$? In one-dimensional case, one answer is $L^2(\mathbb{R})$. This is a critical space. This result was proved for Schrodinger in a great paper by Deift and Killip but for Dirac it was known for at least half a century and dates back to the works by Mark Krein. Krein’s result on Dirac, however, is only a continuous analog of the classical results for polynomials orthogonal on the unit circle (Szego case). For $d>1$, the conjecture by Barry Simon is that $\displaystyle \int_{\mathbb{R}^d} \frac{V^2(x)}{\|x\|^{d-1}+1}dx <\infty$ is sufficient for the preservation of a.c. spectrum. Very little is known so far. Only the case of Schrodinger on the Cayley tree is well-understood. Take a rooted Cayley tree $\mathbb{B}$ with the origin at $O$ and assume that each vertex has exactly three neighbors while $O$ has only two. Consider the Laplacian on $\mathbb{B}$ defined at each point as the sum over the neighbors and then a simple calculation shows that the spectrum of $H_0$ is $[-2\sqrt 2, 2\sqrt 2]$ and it is purely a.c. Then, perturb by $V$. The multidimensional $L^2$ result reads as follows. Consider all paths that go from $O$ to infinity without self-intersections (rays). Put the probability measure on them by tossing a Bernoulli coin at each vertex. Then, the claim is that the a.c. spectrum contains the a.c. spectrum of unperturbed operator if with positive probability the potential $V(X_n)\in \ell^2$ where $X_n$ denotes the path from the origin. There is more quantitative version, of course, which implies Simon’s conjecture if the Jensen inequality is applied. The condition we have here is more general and more physically appealing: it says that we only need enough directions were the potential is small for the particle to propagate. For $d>1$, sparse or slowly decaying and oscillating potentials can be handled. If the potential does not oscillate, then the scattering process is quite complicated, it is governed by very intricate evolution equation (that captures semiclassical WKB correction as very special case). This evolution equation is poorly studied and much work is needed in this direction. Soft one-dimensional methods seem to be of little help. In Euclidean case, what would be the analog of the probability space on the set of paths that escape to infinity? This question was addressed here. It turns out that there is a natural Ito’s stochastic equation that describes these paths. The statement we have is somewhat weak though. It says that the a.c. spectrum contains the positive half-line if $V(X_t)\in L^1(\mathbb{R}^+)$ with positive probability and we can not yet replace summability by the square summability over the path $X_t$. Nevertheless, even this result gives rise to interesting questions like how one computes probabilities given by this Ito’s calculus? That conventionally can be reduced to the analysis of the corresponding potential theory and the modified harmonic measure. The potential theory one encounters in this case is somewhat in between elliptic and the parabolic one: on the large scale it is parabolic and on the small scale it is elliptic. The estimates on the harmonic measure in terms of the geometric properties of support can be found in my paper with Kupin. ___________________________________________________________________ May 2012. I recently finished writing a survey for the Nikolskii conference volume, it contains more details.	2021-04-11 01:27:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 174, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9076388478279114, "perplexity": 242.86200401690377}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038060603.10/warc/CC-MAIN-20210411000036-20210411030036-00214.warc.gz"}
https://itprospt.com/num/8139628/1-0l-irout-welghts-1-u-405-5-grams-special-food-1-pue-t0lnbox	5 # 1 0l irout welghts 1 U 405.5 grams? special food 1 pue T0lnbox trouliinacnings in 0 a standard deviallon of 10,8 1 1 Based = 1 onuiarge 1 H inesd Qulu 1... ## Question ###### 1 0l irout welghts 1 U 405.5 grams? special food 1 pue T0lnbox trouliinacnings in 0 a standard deviallon of 10,8 1 1 Based = 1 onuiarge 1 H inesd Qulu 1 1 0l irout welghts 1 U 405.5 grams? special food 1 pue T0lnbox trouliinacnings in 0 a standard deviallon of 10,8 1 1 Based = 1 onuiarge 1 H inesd Qulu 1 #### Similar Solved Questions ##### Slab slab Written Question #2: Uniformly Charged is oricnted that its insulating material has thickness 2d and parallel to the yz-plane and given by the planes I faces are and z-dimensions of the slab are very large and % The be treated essentially infinitc. The slab compared t0 and IaY has uniform negative charge densityfield due: to the slab (a.) 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Find the interest rate: Round to the nearest percent:The interest rate is approximately% Use the model A=Pert or A=P 1+3) where A is the future value of P dollars invested at interest rate r compounded continuously or n times per year for years. A S25,000 investment grows to S52,680 in 15 years compounded monthly. Find the interest rate: Round to the nearest percent: The interest rate i... ##### 1. (8 points) Describe explicitly the set of all 2 X 2 matrices such that both e1 eigenvectors of A[ej and e2 12\| are 1. (8 points) Describe explicitly the set of all 2 X 2 matrices such that both e1 eigenvectors of A [ej and e2 12\| are... ##### $7-14$ . Graph the function, not by plotting points, but by starting from the graph of $y=e^{x}$ in Figure $1 .$ State the domain, range, and asymptote. $$y=e^{x-3}+4$$ $7-14$ . Graph the function, not by plotting points, but by starting from the graph of $y=e^{x}$ in Figure $1 .$ State the domain, range, and asymptote. $$y=e^{x-3}+4$$...	2022-08-11 17:58:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5462831258773804, "perplexity": 14029.374353819207}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571483.70/warc/CC-MAIN-20220811164257-20220811194257-00596.warc.gz"}
https://wormax.ioground.com/proxy/https:/mathoverflow.net/questions/425727/fixed-point-free-diffeomorphisms-of-surfaces-fixing-no-homology-classes	# Fixed-point free diffeomorphisms of surfaces fixing no homology classes One of my graduate students asked me the following question, and I can't seem to answer it. Let $$\Sigma_g$$ denote a compact oriented genus $$g$$ surface. For which $$g$$ does there exist an orientation-preserving diffeomorphism $$f\colon \Sigma_g \rightarrow \Sigma_g$$ with the following two properties: 1. $$f$$ has no fixed points. 2. The action of $$f$$ on $$H_1(\Sigma_g)$$ fixes no nonzero elements. Since $$f$$ has no fixed points, you can use the Lefschetz fixed point theorem to deduce that the trace of the action of $$f$$ on $$H_1(\Sigma_g)$$ must be $$2$$. From this, you can easily see that no such $$f$$ can occur for $$g=0$$ and $$g=1$$. However, I can't figure out what is going on here for $$g \geq 2$$. Goodwillie's construction (in genus two) generalises to all higher genus as follows. Let $$P_n$$ be the regular $$n$$-gon in the plane with vertices at roots of unity. When $$n$$ is even, we can glue opposite (and thus parallel) sides to obtain an oriented surface $$F_n$$. Suppose that $$n = 4g + 2$$. In this case $$F_n$$ has genus $$g$$; also the rotation by $$2\pi / (4g + 2)$$ induces a homeomorphism $$f_n$$ of $$F_n$$ with exactly one fixed point, at the origin. Now we take copies of $$F_{4g + 2}$$ and $$F_{4h + 2}$$, remove small disks about the origin of each, and glue along the so created boundaries. The resulting connect sum $$F$$ has genus $$g + h$$. In a neighbourhood of the gluing we interpolate between the homeomorphisms $$f_{4g + 2}$$ and $$f_{4h + 2}$$ (this is called a "fractional Dehn twist" in some places). The resulting homeomorphism $$f \colon F \to F$$ has the desired properties. • This is great, thanks!!! Jul 3 at 1:14 Here's an example with $$g=2$$. Let $$T$$ be the torus $$\mathbb C/L$$, where $$L$$ is the lattice spanned by $$1$$ and $$\zeta=e^{2\pi i/6}$$. Let $$f:T\to T$$ be induced by multiplication by $$\zeta$$. This is a diffeomorphism fixing one point $$0\in T$$ and fixing no non-zero elements of $$H_1(T)$$. Now remove a little disk centered at $$0$$ and stick together two copies of this punctured torus, and let the map act like $$f$$ on both copies. But I don't immediately see how to learn anything about $$g>2$$ from this example. • Your construction generalises to all genera greater than one, as follows. Let $P_n$ be the regular $n$-gon in the plane with vertices at roots of unity. When $n$ is even, we can glue opposite (and thus parallel) sides to obtain an oriented surface $F_n$. Suppose that $n = 4g + 2$. In this case $F_n$ has genus $g$; also the rotation by $2\pi / (4g + 2)$ induces a homeomorphism $f_n$ of $F_n$ with exactly one fixed point, at the origin. Now we... Jun 30 at 8:02 • take copies of $F_{4g + 2}$ and $F_{4h + 2}$, remove small disks about the origin of each and glue. This gives a surface of genus $g + h$. Also, in a neighbourhood of the gluing we interpolate between the homeomorphsims $f_{4g + 2}$ and $f_{4h + 2}$ (this is called a "fractional Dehn twist" in some places). Jun 30 at 8:03 • Sam Nead: I hadn't thought of interpolating like that. You should make this an answer. Jun 30 at 10:48 • This is great, but Sam Nead's answer sounds even better! So I'm going to hold off on accepting an answer until he has a chance to write one. Jun 30 at 21:29	2022-08-16 19:35:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 45, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9246446490287781, "perplexity": 158.8995366195676}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572515.15/warc/CC-MAIN-20220816181215-20220816211215-00237.warc.gz"}
http://mathhelpforum.com/business-math/203167-checking-my-answer-calculating-expenses.html	# Thread: Checking my answer: Calculating expenses 1. ## Checking my answer: Calculating expenses What are the total amount of expenses shown in the picture? I get: 186,200. Because there is a service revenue of 541,100, there is a net income of 354,900. Is that correct? Thank You	2017-04-30 00:46:51	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8194360733032227, "perplexity": 4569.273675645968}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123632.58/warc/CC-MAIN-20170423031203-00232-ip-10-145-167-34.ec2.internal.warc.gz"}
https://brilliant.org/problems/a-calculus-problem-by-avineil-jain/	Deducing From The Integral Calculus Level 5 A differentiable function $$f$$ is defined on the positive real numbers such that $\int_{1}^{xy} f(t) \, dt = y\int_{1}^{x} f(t) \, dt + x\int_{1}^{y} f(t) \, dt.$ If $$f(1) = 3$$, what is $$f(e)$$? × Problem Loading... Note Loading... Set Loading...	2017-07-27 08:54:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9499044418334961, "perplexity": 2748.447998643541}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549427750.52/warc/CC-MAIN-20170727082427-20170727102427-00106.warc.gz"}
https://arts.units.it/handle/11368/3040360	By using variational quantum Monte Carlo techniques, we investigate the instauration of stripes (i.e., charge and spin inhomogeneities) in the Hubbard model on the square lattice at hole doping delta = 1/8, with both nearest- (t) and next-nearest-neighbor hopping (t'). Stripes with different wavelengths lambda (denoting the periodicity of the charge inhomogeneity) and character (bond- or site-centered) are stabilized for sufficiently large values of the electron-electron interaction U/t. The general trend is that lambda increases going from negative to positive values of t'/t and decreases by increasing U/t. In particular, the lambda = 8 stripe obtained for t' = 0 and U/t = 8 [L.F. Tocchio, A. Montorsi, and F. Becca, SciPost Phys. 7, 21 (2019)] shrinks to lambda = 6 for U/t greater than or similar to 10. For t'/t < 0, the stripe with lambda = 5 is found to be remarkably stable, while for t'/t > 0, stripes with wavelength lambda = 12 and lambda = 16 are also obtained. In all these cases, pair-pair correlations are highly suppressed with respect to the uniform state (obtained for large values of vertical bar t'/t vertical bar), suggesting that striped states are not superconducting at delta = 1/8. ### Stripes in the extended $t-t^\prime$ Hubbard model: A Variational Monte Carlo analysis #### Abstract By using variational quantum Monte Carlo techniques, we investigate the instauration of stripes (i.e., charge and spin inhomogeneities) in the Hubbard model on the square lattice at hole doping delta = 1/8, with both nearest- (t) and next-nearest-neighbor hopping (t'). Stripes with different wavelengths lambda (denoting the periodicity of the charge inhomogeneity) and character (bond- or site-centered) are stabilized for sufficiently large values of the electron-electron interaction U/t. The general trend is that lambda increases going from negative to positive values of t'/t and decreases by increasing U/t. In particular, the lambda = 8 stripe obtained for t' = 0 and U/t = 8 [L.F. Tocchio, A. Montorsi, and F. Becca, SciPost Phys. 7, 21 (2019)] shrinks to lambda = 6 for U/t greater than or similar to 10. For t'/t < 0, the stripe with lambda = 5 is found to be remarkably stable, while for t'/t > 0, stripes with wavelength lambda = 12 and lambda = 16 are also obtained. In all these cases, pair-pair correlations are highly suppressed with respect to the uniform state (obtained for large values of vertical bar t'/t vertical bar), suggesting that striped states are not superconducting at delta = 1/8. ##### Scheda breve Scheda completa 2022 Pubblicato https://scipost.org/10.21468/SciPostPhys.12.6.180 File in questo prodotto: File SciPostPhys_12_6_180.pdf accesso aperto Tipologia: Documento in Versione Editoriale Licenza: Creative commons Dimensione 504.59 kB Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/3040360	2023-03-26 12:41:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6444799900054932, "perplexity": 3132.4119550132336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945472.93/warc/CC-MAIN-20230326111045-20230326141045-00362.warc.gz"}
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-8-section-8-1-matrix-solutions-to-linear-systems-exercise-set-page-895/67	## Precalculus (6th Edition) Blitzer $60$ units, $7700$ dollars. Step 1. Using the given function $y=ax^2+bx+c$ and the data from the table, we can set up the system of equations as $\begin{cases} a(30)^2+b(30)+c=5900 \\ a(50)^2+b(50)+c=7500 \\ a(100)^2+b(100)+c=4500 \end{cases}$ or $\begin{cases} 900a+30b+c=5900 \\ 2500a+50b+c=7500 \\ 10000a+100b+c=4500 \end{cases}$ Step 2. Taking the difference between the first and second, then the second and the third equations, we have $\begin{cases} 1600a+20b=1600 \\ 7500a+50b=-3000 \end{cases}$ or $\begin{cases} 80a+b=80 \\ 150a+b=-60 \end{cases}$ Step 3. Taking the difference between the two equations, we have $70a=-140$ and $a=-2$ Step 4. Back-substitute to get $b=80-80(-2)=240$ and $c=5900-900(-2)-30(240)=500$ and the function becomes $y=-2x^2+240x+500$ Step 5. The maximum of the function can be found at $x=-\frac{b}{2a}=-\frac{240}{2(-2)}=60$ with a maximum profit of $y=-2(60)^2+240(60)+500=7700$ dollars.	2021-10-27 06:59:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8872303366661072, "perplexity": 102.80079185518491}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588102.27/warc/CC-MAIN-20211027053727-20211027083727-00107.warc.gz"}
http://mathoverflow.net/feeds/question/41123	Cliques of hyperedges - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:38:34Z http://mathoverflow.net/feeds/question/41123 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41123/cliques-of-hyperedges Cliques of hyperedges Dave Pritchard 2010-10-05T10:27:22Z 2010-10-08T10:40:34Z <p>Suppose we have a graph, with multiple edges allowed. An edge-clique is a set $C$ of edges so that every two edges in $C$ share at least one endpoint. Note that any edge-clique falls into one of two categories:</p> <ul> <li>A star: there is a vertex such that every edge of $C$ contains it</li> <li>A triangle: there are three vertices such that every edge of $C$ goes between two of them</li> </ul> <p>What kind of patterns persist for hypergraphs of rank $\le r$? (It has a vertex set, and "hyper"edges which are arbitrary sets of size at most $r$. Again an edge-clique is a family of pairwise intersecting edges.)</p> <p>I believe one can show the following structure theorem: for every clique $C$, there is a set $S$ of at most $f(r)$ vertices, such that every pair of edges $C$ meet at at least one vertex in $S$. But the bound I have on $f(r)$ is something like doubly exponential. What is the best possible?</p> <p>Note that $f(r) \ge \Omega(r^2)$ by considering projective planes.</p> <p>Cross-post asking for a good algorithm to find a max-clique: <a href="http://cstheory.stackexchange.com/questions/1846/max-clique-in-line-graph-of-hypergraph" rel="nofollow">http://cstheory.stackexchange.com/questions/1846/max-clique-in-line-graph-of-hypergraph</a></p> <p>Answer: I have found there was a series of results on this function $f$, and that this set $S$ is often called a <i>kernel</i>. The best current bounds on $f$ are due to Tuza (Tuza, Z. (1985) Critical hypergraphs and intersecting set-pair systems. J. Combin. Theory Ser. B 39 134–145.); in short $f(r) = \Theta(4^r/\sqrt{r})$.</p>	2013-05-19 06:38:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7861147522926331, "perplexity": 835.73721690005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368696384181/warc/CC-MAIN-20130516092624-00055-ip-10-60-113-184.ec2.internal.warc.gz"}
https://vietlod.com/93-cau-trac-nghiem-kinh-te-luong-phan-1	Bài sắp đăng Home \| Trắc nghiệm \| Kinh tế lượng \| 93 câu trắc nghiệm Kinh tế lượng – Phần 1 Tổng hợp 93 câu trắc nghiệm kinh tế lượng (có đáp án) bằng tiếng anh. # 93 câu trắc nghiệm Kinh tế lượng – Phần 1 Tổng hợp 93 câu trắc nghiệm Kinh tế lượng cơ bản trong tài chính bằng tiếng anh (có đáp án kèm theo). Nội dung được phân thành 9 chương, được chia làm 2 phần. Các câu hỏi trắc nghiệm phần 1 bao gồm: KTL_002_C1_1: The numerical score assigned to the credit rating of a bond is best described as what type of number? ○ Continuous ○ Cardinal ● Ordinal ○ Nominal KTL_002_C1_2: Suppose that we wanted to sum the 2007 returns on ten shares to calculate the return on a portfolio over that year. What method of calculating the individual stock returns would enable us to do this? ● Simple ○ Continuously compounded ○ Neither approach would allow us to do this validly ○ Either approach could be used and they would both give the same portfolio return KTL_002_C1_3: Consider a bivariate regression model with coefficient standard errors calculated using the usual formulae. Which of the following statements is/are correct regarding the standard error estimator for the slope coefficient? (i) It varies positively with the square root of the residual variance (s) (ii) It varies positively with the spread of X about its mean value (iv) It varies positively with the sample size T ● (i) only ○ (i) and (iv) only ○ (i), (ii) and (iv) only ○ (i), (ii), (iii) and (iv). KTL_002_C1_4: In a time series regression of the excess return of a mutual fund on a constant and the excess return on a market index, which of the following statements should be true for the fund manager to be considered to have “beaten the market” in a statistical sense? ● The estimate for $$\alpha$$ should be positive and statistically significant ○ The estimate for $$\alpha$$ should be positive and statistically significantly greater than the risk-free rate of return ○ The estimate for $$\alpha$$ should be positive and statistically significant ○ The estimate for $$\alpha$$ should be negative and statistically significant. KTL_002_C1_5: What result is proved by the Gauss-Markov theorem? ○ That OLS gives unbiased coefficient estimates ○ That OLS gives minimum variance coefficient estimates ● That OLS gives minimum variance coefficient estimates only among the class of linear unbiased estimators ○ That OLS ensures that the errors are distributed normally KTL_002_C1_6: The type I error associated with testing a hypothesis is equal to ○ One minus the type II error ○ The confidence level ● The size of the test ○ The size of the sample KTL_002_C1_7: Which of the following is a correct interpretation of a “95% confidence interval” for a regression parameter? ● We are 95% sure that the interval contains the true value of the parameter ○ We are 95% sure that our estimate of the coefficient is correct ○ We are 95% sure that the interval contains our estimate of the coefficient ○ In repeated samples, we would derive the same estimate for the coefficient 95% of the time KTL_002_C1_8: Which of the following statements is correct concerning the conditions required for OLS to be a usable estimation technique? ● The model must be linear in the parameters ○ The model must be linear in the variables ○ The model must be linear in the variables and the parameters ○ The model must be linear in the residuals. KTL_002_C1_9: Which of the following is NOT a good reason for including a disturbance term in a regression equation? ○ It captures omitted determinants of the dependent variable ● To allow for the non-zero mean of the dependent variable ○ To allow for errors in the measurement of the dependent variable ○ To allow for random influences on the dependent variable KTL_002_C1_10: Which of the following is NOT correct with regard to the p-value attached to a test statistic? ● p-values can only be used for two-sided tests ○ It is the marginal significance level where we would be indifferent between rejecting and not rejecting the null hypothesis ○ It is the exact significance level for the test ○ Given the p-value, we can make inferences without referring to statistical tables KTL_002_C1_11: Which one of the following is NOT an assumption of the classical linear regression model? ○ The explanatory variables are uncorrelated with the error terms. ○ The disturbance terms have zero mean ● The dependent variable is not correlated with the disturbance terms ○ The disturbance terms are independent of one another. KTL_002_C1_12: Which of the following is the most accurate definition of the term “the OLS estimator”? ○ It comprises the numerical values obtained from OLS estimation ● It is a formula that, when applied to the data, will yield the parameter estimates ○ It is equivalent to the term “the OLS estimate” ○ It is a collection of all of the data used to estimate a linear regression model. KTL_002_C1_13: Two researchers have identical models, data, coefficients and standard error estimates. They test the same hypothesis using a two-sided alternative, but researcher 1 uses a 5% size of test while researcher 2 uses a 10% test. Which one of the following statements is correct? ○ Researcher 2 will use a larger critical value from the t-tables ● Researcher 2 will have a higher probability of type I error ○ Researcher 1 will be more likely to reject the null hypothesis ○ Both researchers will always reach the same conclusion. KTL_002_C1_14: Consider an increase in the size of the test used to examine a hypothesis from 5% to 10%. Which one of the following would be an implication? ● The probability of a Type I error is increased ○ The probability of a Type II error is increased ○ The rejection criterion has become more strict ○ The null hypothesis will be rejected less often. KTL_002_C1_15: What is the relationship, if any, between the normal and t-distributions? ○ A t-distribution with zero degrees of freedom is a normal ○ A t-distribution with one degree of freedom is a normal ● A t-distribution with infinite degrees of freedom is a normal ○ There is no relationship between the two distributions.	2019-01-17 08:41:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7222802639007568, "perplexity": 1223.891694465468}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583658901.41/warc/CC-MAIN-20190117082302-20190117104302-00312.warc.gz"}
https://solvedlib.com/n/pointssercpt1-3-2-0p-009-panther-jumps-mjaheight-when-ienving,6983568	# PointsSERCPT1 3.2.0P.009.panther jumps mjaheight =when Ienving the growndancle4Jo. Whalinitial speedets)Wvcsth dround?Need Help?AeaalHv Nole eIn1 pointsPreviou: AnersSERCPI1 3.2.0P.013.todt HnantEa Fn to9ted ###### Question: points SERCPT1 3.2.0P.009. panther jumps mja height = when Ienving the grownd ancle 4Jo. Whal initial speed ets) Wvcsth dround? Need Help? Aeaal Hv Nole e In1 points Previou: Aners SERCPI1 3.2.0P.013. todt HnantEa Fn to9ted tne buildino buvlalnd anale Aeoyn Won TORanos Orcuno udrefAE Hocd Hoip ' Mv Noici -71pl4nt- Sercp1j],2,0p.015 Wott 3 MntA0fe lu #### Similar Solved Questions ##### 19 arter Quarter Quarter Quarter 10.100 1.100 10,000 its produced 00 19 arter Quarter Quarter Quarter 10.100 1.100 10,000 its produced 00... ##### Milliliter What is the cost of 6 onions in Ayarceeville if 3 onions weigh 15 lb... milliliter What is the cost of 6 onions in Ayarceeville if 3 onions weigh 15 lb and the price of Ces equals 150 US dollars one pound of onions is 35 cents in the US? US x 10 tun second of bonors given son si costo 50 Bonion 1.Si 11. 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Write the vector as a linear combination of the standard unit vectors $\mathbf{i}$ and $\mathbf{j}.$ $\begin{array}{cc}\text{Initial Point} && \text{Terminal Point} \\ (-1,-5) && (2,3) \end{array}$ The initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors $\mathbf{i}$ and $\mathbf{j}.$ $\begin{array}{cc}\text{Initial Point} && \text{Terminal Point} \\ (-1,-5) && (2,3) \end{array}$... ##### Fallen company Exercise 22-5 Fallon Company uses flexible budgets to control its selling expenses. Monthly sales... fallen company Exercise 22-5 Fallon Company uses flexible budgets to control its selling expenses. Monthly sales are expected to range from $173,000 to$210,200. Variable costs and their percentage relationship to sales are sales commissions 7%, advertising 4%, traveling 4%, and delivery 24. F... ##### The North bond of an interstate freeway is in need of a beam structure to support... 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You must support your conclusion to receive credit: Let X be a random variable with mean and standard deviation 14 and 1.2,respectively. Let Y = 4X and z=X+Xz +X3 +X4 + Xs where each Xi is an independent observation on the variable X. (a) Which has the greater mean, Y or Z? You must support your conclusion to receive credit: (b) Which has the greater... ##### 8 Q 0 E(X?) =8?{17I00%Exercise 10.5. Let X be random variable with density function f(r) = 1 for <I < 0 f(r) =1-I for 0 < r < 1, and f(r) 0 elsewhere: (a) Justily gcometrically that E(X) = 0 (without calculating it) (6) Find Var(.) .AdoCont or ExcExercise 10.6 (Bonus) . Suppose 1,, 1, are independent with E(X;) and Var ( ) for all = 1 Find E[(1; 1n)?]. (Hint: Use tnt IE(} 2 Var( ) ) (E()) 1o.7 Crnduate). Show" that E((T c12) is Minimize [ br taking Exercise U3 Cakculus to study 8 Q 0 E(X?) =8? {17 I00% Exercise 10.5. Let X be random variable with density function f(r) = 1 for <I < 0 f(r) =1-I for 0 < r < 1, and f(r) 0 elsewhere: (a) Justily gcometrically that E(X) = 0 (without calculating it) (6) Find Var(.) . Ado Cont or Exc Exercise 10.6 (Bonus) . Suppose 1,... On February 1, 2021, Cromley Motor Products issued 6% bonds, dated February 1, with a face amount of $85 million. The bonds mature on January 31, 2025 (4 years). The market yield for bonds of similar risk and maturity was 8%. Interest is paid semiannually on July 31 and January 31. Barnwell Industri... 5 answers ##### Find the points at which the following plane intersects the coordinate axes and find equations of the lines where the plane intersects the coordinate planes_ Sketch graph of the plane_2y+2=6Find the points at which the plane intersects the coordinate axes Select the correct choice below and fill in the answer boxes to complete your choice_ (Simplify your answers Type integers or decimals ) OA The plane intersects the X-axis at does not intersect the y-axis, and intersects the z-axis atThe plane Find the points at which the following plane intersects the coordinate axes and find equations of the lines where the plane intersects the coordinate planes_ Sketch graph of the plane_ 2y+2=6 Find the points at which the plane intersects the coordinate axes Select the correct choice below and fill i... 5 answers ##### Conductiny bar of length 0.40 m rotates counterclockwise direction with constant ongular speed of +2.0 radls about pivot 0t one end shown: uniform magnetic field (magnitude 8.6 T) is directedl out of the puper. What is the potentiol difference Vp-Va?-138-0.34 V4103 V03 V0.34v conductiny bar of length 0.40 m rotates counterclockwise direction with constant ongular speed of +2.0 radls about pivot 0t one end shown: uniform magnetic field (magnitude 8.6 T) is directedl out of the puper. 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T4 true. 21 false 32 False . 13 false. T4 false_but it should output false in these Cascs;T1false 92tre j true, r4 true 22 False, ,3 true, r4false ,T1false ,Prove that YOUI" AHSWCT is correc (6 marks) Design circuit that takes four inputs cl 12. - 24 and outputs true if (and only if) all of the false inputs are consecutive For instance your circuit should output true in the following four cases: 11 truc_ 22 truc_ 23 truc_ 14 truc. 11 false, v2 true, 1 true, 4 true. 11 = true_ 22 false;... 5 answers ##### Use linear approximation (or differentials) to estimate the given number (Round your answer to five decimal placesV12600800 Use linear approximation (or differentials) to estimate the given number (Round your answer to five decimal places V126 00800... 1 answer ##### 4. At the arteriole end of the capillary, pressure exceeds pressures at the verñule end, pressure... 4. At the arteriole end of the capillary, pressure exceeds pressures at the verñule end, pressure exceeds out of the capillary bed at the arteriole end, whereas fluids into the capillary bed at the venule end. -Pressure pushes water pressure draws pressure. e. _ T Activity 2: Tracing Blood Fl... 5 answers ##### 12) Determine whether the series is absolutely convergent; conditionally convergent; or divergent Clearly indicate which of the following test(s) You used: Alternating Series Test; Ratio Test, Root Test; Test for Divergence, Integral Test; Comparison Test; or Limit Comparison Test2 ( 12) Determine whether the series is absolutely convergent; conditionally convergent; or divergent Clearly indicate which of the following test(s) You used: Alternating Series Test; Ratio Test, Root Test; Test for Divergence, Integral Test; Comparison Test; or Limit Comparison Test 2 (... 1 answer ##### Control engineering subject Hz(s) Y(s) H (s) H3(s) Figure 1: Block diagram Simplify the block diagram... 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True False... 5 answers ##### The number of hours per week that the televisioni qurned onis determined for each family in median Is 34.2 hours: Twenty- sample fourpfithe farnilies The mean of the data is 38 hours and the data hours; tne sample turned on the television for 23 hours or less for the week The 146 percentile of theStep 2 of 5: Approximately how mary (amilies are the 5ample Raund yourlanswer to the nearest integer.Answer (How Enter}PointsTables The number of hours per week that the televisioni qurned onis determined for each family in median Is 34.2 hours: Twenty- sample fourpfithe farnilies The mean of the data is 38 hours and the data hours; tne sample turned on the television for 23 hours or less for the week The 146 percentile of the S... 1 answer ##### Describe the opportunities and threats right now for a firm entering the ___________ (Choose one from... Describe the opportunities and threats right now for a firm entering the ___________ (Choose one from below) market: Bangladesh India China South Korea Minimum 400 words... 1 answer #####$17,000 180,000 Multiple-Step and Single-Step Income Statements, and statement of Comprehensive Income On December 31, 2019,... $17,000 180,000 Multiple-Step and Single-Step Income Statements, and statement of Comprehensive Income On December 31, 2019, Opgenorth Company listed the following items in its adjusted trial balance: Loss from fire (pretax)$8,000 General and administrative expenses Interest revenue 3,000 Sales Sel... ##### Question 11 ptsSelect the three most important general questions we think about in Linear Algebra.A Are there free variables?B. Is the system consistent?C What is the Reduced Row Echelon Form of the corresponding augmented matrix?D: If consistent; how many solutions?E: Can we describe the solutions,either analytically or geometrically? Question 1 1 pts Select the three most important general questions we think about in Linear Algebra. A Are there free variables? B. Is the system consistent? C What is the Reduced Row Echelon Form of the corresponding augmented matrix? D: If consistent; how many solutions? E: Can we describe the sol... ##### MR Demand 10 20 30 40 50 60 70 80 Duantity Refer to Figure 15-20. The... MR Demand 10 20 30 40 50 60 70 80 Duantity Refer to Figure 15-20. The deadweight loss caused by a profit-maximizing monopoly amounts to a. $900. b.$225. c. $1,350. d.$450 Price MC 4+ F + 1 + 2 + 4 Demand 10 11 12 3 5 6 7 8 9 Quantity Refer to Figure 15-11. Which area represents the deadweight loss... ##### The following table shows summary data on mercury concentration in mackerel (in ppm) from two different areas of Atlantic Ocean:Area 1:m = 12,x 0.0740,51 0.0030 Area 2:n = 13,y = 0.0825,52 0.0027Do the data suggest that the mercury concentration is higher in mackerel from Area 2? Compute the value of the test statistic (round off to second decimal place). Assume normal populations with equal variance: The following table shows summary data on mercury concentration in mackerel (in ppm) from two different areas of Atlantic Ocean: Area 1:m = 12,x 0.0740,51 0.0030 Area 2:n = 13,y = 0.0825,52 0.0027 Do the data suggest that the mercury concentration is higher in mackerel from Area 2? Compute the value... ##### (6 points) Find the general solution for the foloning Euler-Cowchy equeva 22v' ~Iv+v=0, 1>0. (0) u(z) =e/(C,Vi+6+ (6) v(z) = 2M(C, M1z+01) (c) vlr) =Cie+Cel:,(d) u(z) =Cr+CzV.Answer; (6 points) Find the general solution for the foloning Euler-Cowchy equeva 22v' ~Iv+v=0, 1>0. (0) u(z) =e/(C,Vi+6+ (6) v(z) = 2M(C, M1z+01) (c) vlr) =Cie+Cel:, (d) u(z) =Cr+CzV. Answer;... ##### If n=16, ū=35, and s=13, construct a confidence interval at a 90% confidence level. Assume the... If n=16, ū=35, and s=13, construct a confidence interval at a 90% confidence level. Assume the data came from a normally distributed population. Give your answers to one decimal place. *<u<... ##### 12: A: Determine the equivalent resistance of the circuit. O2 201 B: Determine the current being... 12: A: Determine the equivalent resistance of the circuit. O2 201 B: Determine the current being delivered by the battery and the total power delivered by the battery. deivesed to C: Determine the power disipatedby each resistor... ##### Find the average rate of change On the interval(g(z) 2r" on [1, 3]PreviewTIPEnter youf Iblef a: number (like 5; -3,2.2172) Of a3 calculation (like 573,2 3,5+4) Enter DNE for Does Not Exist; 00 for Infinity Find the average rate of change On the interval (g(z) 2r" on [1, 3] Preview TIP Enter youf Iblef a: number (like 5; -3,2.2172) Of a3 calculation (like 573, 2 3,5+4) Enter DNE for Does Not Exist; 00 for Infinity... 4. Management has come to you, as they are concerned with the drop in estimated sales from $50,000 in May to only$20,000 in June. In your opinion, is there a reason for them to be concerned? Yes or No and WHY (OR WHY NOT)? VOLUNTARY COMPUTATION AREA... ##### Reaction of $2.0 mathrm{~L}$ of hydrogen gas with $1.0 mathrm{~L}$ of oxygen gas yields $2.0 mathrm{~L}$ of water vapor. All gases are at the same temperature and pressure. Show how these data support the idea that oxygen gas is a diatomic molecule. Must we consider hydrogen to be a diatomic molecule to explain these results? Reaction of $2.0 mathrm{~L}$ of hydrogen gas with $1.0 mathrm{~L}$ of oxygen gas yields $2.0 mathrm{~L}$ of water vapor. All gases are at the same temperature and pressure. Show how these data support the idea that oxygen gas is a diatomic molecule. Must we consider hydrogen to be a diatomic molecul...	2022-10-05 14:52:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4403858780860901, "perplexity": 8327.842680996982}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337631.84/warc/CC-MAIN-20221005140739-20221005170739-00522.warc.gz"}
http://serverfault.com/questions/280483/smb-share-through-the-web	# SMB Share Through the Web I'm looking to setup an SMB/windows share that you can access through the internet on either a windows 2003 or windows 2008 server. This would be for a server behind a firewall, I have a handle on NAT. I'm mainly looking for a tool that can setup which share that needs to be shared publicly and authentication. I got the idea from the \live.sysinternals.com\tools site (read here). I've seen lots of posts here and on google about setting up a web based SMB share, but I'm looking more to setup a full \sharename.domain.com access back to the server (read only of course), so I can map to it from a remote location. Can this be done with native tools, or would I need a 3rd party app? - you could use webdav and map that if you want. – tony roth Jun 14 '11 at 22:23 As @tony roth pointed out. webdav did the trick for me. They key here was needing to install the update for it found here to get it working on my windows 7 machine. After that I was actually able to map web sites to drive letters net use z: https://share.domain.com , wasn't exactly what I was asking for, but this did exactly what I needed it for.	2014-08-28 11:16:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22640706598758698, "perplexity": 1278.7578503618658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830746.39/warc/CC-MAIN-20140820021350-00267-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.degruyter.com/view/j/htmp.2015.34.issue-8/htmp-2014-0164/htmp-2014-0164.xml	Jump to ContentJump to Main Navigation Show Summary Details More options … # High Temperature Materials and Processes Editor-in-Chief: Fukuyama, Hiroyuki Editorial Board: Waseda, Yoshio / Fecht, Hans-Jörg / Reddy, Ramana G. / Manna, Indranil / Nakajima, Hideo / Nakamura, Takashi / Okabe, Toru / Ostrovski, Oleg / Pericleous, Koulis / Seetharaman, Seshadri / Straumal, Boris / Suzuki, Shigeru / Tanaka, Toshihiro / Terzieff, Peter / Uda, Satoshi / Urban, Knut / Baron, Michel / Besterci, Michael / Byakova, Alexandra V. / Gao, Wei / Glaeser, Andreas / Gzesik, Z. / Hosson, Jeff / Masanori, Iwase / Jacob, Kallarackel Thomas / Kipouros, Georges / Kuznezov, Fedor IMPACT FACTOR 2018: 0.427 5-year IMPACT FACTOR: 0.471 CiteScore 2018: 0.58 SCImago Journal Rank (SJR) 2018: 0.231 Source Normalized Impact per Paper (SNIP) 2018: 0.377 Open Access Online ISSN 2191-0324 See all formats and pricing More options … Volume 34, Issue 8 # Metal Loss of Steam-Oxidized Alloys after Exposures at 675°C and 725°C for 500 Hours T. Dudziak • Corresponding author • Foundry Research Institute, Centre for High Temperature Studies, Zakopianska 73, 30-418, Krakow, Poland • Email • Other articles by this author: • De Gruyter OnlineGoogle Scholar / S. Grobauer / N. Simms / U. Krupp / M. Łukaszewicz Published Online: 2015-01-14 \| DOI: https://doi.org/10.1515/htmp-2014-0164 ## Abstract The paper reports metal loss data and general overview of steam oxidation behaviour of the selected alloys in water steam under atmospheric pressure in temperature range of 675–725°C. In this research, T22, T23, TP347HFG, HR3C and 718+ alloys were used. In this study, kinetics and metal loss data were obtained for better understanding corrosion degradation of the exposed materials. All the samples after 500-h exposure were characterized using environmental scanning electron microscopy (ESEM) in backscattered mode (BSE) with energy-dispersive x-ray spectroscopy (EDX). The results indicate that the highest corrosion degradation and metal loss were observed in T22, T23 alloys, moderate degradation was found in TP347HFG, whereas HR3C and 718+ nickel-based alloy showed good corrosion resistance. Keywords: steam oxidation; metal loss; high temperature; alloy PACS.: 81 ## Introduction In order to secure the energy needs for fast civilization development, the efficiency of coal-fired power plants has to be improved significantly. Steam oxidation of the materials used for boiler components such as superheater and reheater (SH/RH) tubing has become an important research subject due to the increasing demand for a higher steam temperature and pressure, needed to achieve a higher efficiency of pulverized fuel power plants. In this regard, it is crucial to recognize that the energy efficiency of convectional pulverized coal power plant is a strong function of steam conditions; higher steam temperature increases power plant efficiency. However, due to the severity of operational conditions at elevated temperatures, the materials used for the ultra-supercritical (USC) applications have to exhibit better mechanical and physical properties [1]. The existing power plants can be classified as subcritical power plants (560–580°C, steam pressure 140 bar, efficiency around 37%); advanced power plants, with higher operating temperature and pressure (600°C, steam pressure of 280 bar, efficiency 40–45%); USC power plants where steam operating temperatures could reach 760°C with steam pressures of 350 bar and efficiencies of around 60% [2]. Currently in the UK coal-fired power plant boilers operate with ~37% efficiency and with steam temperatures around 560–580°C, where mainly ferritic materials (T22, T23 alloys) with 2–3 wt% of Cr or ferritic-martensitic steels (e.g. T91, T92) with ~9 wt% of Cr are used. Unfortunately, these alloys (especially T22 and T23) show poor high-temperature steam oxidation protection at temperatures above 550°C. In these alloys, thick and non-protective scales are forming [3]. Thus, in order to increase operating temperature of steam in a power plant, it is necessary to use materials with higher Cr contents. Such materials have to form an adherent and a thin protective oxide scale. Alternatively, it is possible to use cheaper materials and accept shorter components life. In this study degradation of materials is shown not only by standard kinetic measurement but additionally through the change in wall thickness (metal loss). Metal loss is a crucial factor highlighting lifetime of the structural materials, especially in the energy sector where reliable materials are fundamental for constant energy supply. ## The materials In this work five different materials were studied: two ferritic steels T22, T23; two austenitic stainless steels HR3C, TP347HFG; and one nickel-based alloy 718+. The chemical compositions of the alloys are shown in Table 1. Prior to steam oxidation tests at high temperatures, the samples were accurately measured using a digital micrometer (Multico with ± 0.01 µm error). The shapes of the samples used in this study are shown in Figure 1. All surfaces of these tube segments were finished with 600 grit paper (Ra<0.4 μm). Prior to the exposure the samples were cleaned in an ultrasonic bath for 20 min in Volasil followed by rinsing in isopropyl alcohol (IPA), mass gain of the exposed samples was recorded each 100 h. Figure 1 The samples geometry used in high-temperature tests Table 1 Chemical compositions of the alloys used in this study in wt% ## Oxidation test Steam oxidation tests at 675°C and 725°C were carried out in the test rig presented in Figure 2. In this type of steam oxidation test facility [4], steam is generated by pumping water from a reservoir into the furnace. Then the steam passes over the test samples and flows into a condenser before the water returns to the reservoir. The water used in the reservoir is double de-ionized. The whole system is sealed and thoroughly purged using oxygen free nitrogen (OFN). This purge continues through the water reservoir throughout the sample exposure period to minimize the level of oxygen in the system [4]. In this study, five cycles, 100 h each at 675°C and 725°C were performed in order to check the performance of the selected alloys. After each 100-h cycle the furnace was cooled down in natural rate due to the power switch off, and the weight of the exposed materials was measured using a digital balance (SATORIUS CP225D) with a resolution of ± 0.01 mg for masses <80 g. In the test five samples from each alloy were exposed in order to provide better traceability of the corrosion progress. Each individual alloy was removed after each 100 h. The samples were placed on ceramic plates within hot zone, calibrated accurately prior to the experiments at both temperatures. Figure 2 Steam oxidation rig used in this study The balance was calibrated frequently using its internal calibration function and periodically with test weights. Prior to the exposure, furnace calibration was required in order to detect hot zone. The calibration ensured where the best place for the samples was to get the desired test temperature ±5°C. The second test at 725°C was conducted with the same procedure as the 675°C test. The exposed specimens were metallographically prepared, using cold mounting process. The specimens were placed in moulds that were filled with epoxy resin (Buehler Epoxicure Epoxy, mix ratio: five parts of resin and one part of hardener). The resin needed around 24 h to harden, and then the samples were sliced close to the middle of the sample in a cutting machine (ISOMet 5000). Finally, the samples were ground (on paper 240, 600 and 1200 μm) and polished (Motopol) (using 6 and 1 μm diamond suspension) to enable further investigations, i.e. cross sections. ## Metal loss data The technique for metal loss assessment of the corroded samples was used as a digital image analyser. By comparing sample dimensions before and after the exposure, the apparent change in metal and the change in sound metal (change in metal + internal damage) can be calculated. These data sets can then be re-ordered (from greatest to least metal loss) and corrected for calibration differences (using data from reference samples). Next, the processed data can be plotted as a change in metal vs cumulative probability; effectively, this type of plot indicates the probability (e.g. 4%) of a certain degree of damage being observed, for the data acquisition the software Axio-vision was used. The image analyser is connected with an optical microscope. Figure 3 shows a schematic on the x–y stage for the analysis. 1, Origin; 2, polished cross section; 3, mount; 4, rectangular sample; 5, motorized calibrated XY stage. Figure 3 Schematic of a rectangular samples cross section on the digital image analyser stage First of all, a sample is placed on the motorized X–Y stage (cross-sectioned, grinded and polished). An important step is to ensure that the long side of specimen is parallel to the x-motion of the stage. After this, the main cross-sectional locations (top, bottom, right and left) on the sample will be fixed. Using these, the machine calculated the x–y co-ordinates around the sample. For the best results ~55 or more points around the sample is required. The images are recorded during measurement, the software automatically made nine individual pictures of each point. Pictures were stitched together and the obvious metal losses in each of these images were pinpointed. Figure 4 demonstrates the function of image analyser (e.g. at point B the x-value=b2 and the y value=a2). To summarize, in this work the polished cross sections will be all measured using an image analyser to generate accurate measurements of the amount of metal remaining after the corrosion tests; these measurements will be compared to the pre-exposure metal thickness data to produce distributions of the change in metal resulting from the exposures. Figure 4 Illustration of function of image analyser. (A) Stage and sample, (B) determining metal loss from the images recorded ## Mass change data (kinetics) of the exposed alloys at 675°C and 725°C Figures below present comparisons of the mass change between the same alloys exposed at two different temperatures. Figure 5 shows mass change data of the samples oxidized in steam: (A) T22, (B) T23, (C) TP347HFG, (D) HR3C and (E) 718+ alloy exposed for up to 500 h at 675°C (solid line) and 725°C (dashed line) in steam oxidation environment T22 alloy exposed at 675°C and 725°C. Figure 5 Steam oxidation mass change data: (A) T22, (B) T23, (C) TP347HFG, (D) HR3C and (E) 718+ alloy exposed for up to 500 h at 675°C (solid line) and 725°C (dashed line) in steam oxidation environment The results achieved after steam oxidation test shows that the poorest resistance were shown by T22 and T23 alloys with the lowest Cr content. Both alloys developed non-protective thick oxide scales with good adherence. In comparison to ferritic alloys (T22, T23) mass gain of the austenitic alloys with medium and high Cr content showed much better performance as expected. It has been found that mass gain of TP347HFG samples in both temperatures showed high degree of instability, where flaky oxide scale spalled off from the material. The observed behaviour probably originates from mismatch of coefficient of thermal expansion (CET) between the formed oxide scale and the substrate. The HR3C alloy showed the formation of protective scale with lack of spallation and good adherence to the bulk material. Nickel-based alloy 718+ showed the smallest mass gain from all of the exposed alloys due to the formation of protective oxide scale; lack of spallation was observed. The experimental data points from every test temperature can be reasonably fitted only for two ferritic steels T22 and T23 by a parabolic curve represented by the equation: ${d}^{2}={k}_{\mathrm{p}}t$(1)where d donates the total thickness of the oxide scale and t steam oxidation time. The values of kp obtained from curve fitting at every temperature are placed in Table 2. Expressing the relationship between the parabolic rate constant and temperature in an Arrhenius type of equation, kp is given by the following equation [5]: ${k}_{\mathrm{p}}={k}_{\mathrm{o}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{Q}{RT}\right)$(2)where Q is the activation energy of steam oxidation, R the gas constant and T the absolute temperature (K). Using the values from the experiment it was possible to determine the specific activation energy Q (kJ/mol) for steam oxidation process. Calculated values of activation energies for T22 and T23 alloys shown the values of –386.1 and –342.9 kJ/mol respectively. The values of activation energy are in good agreement with the other values achieved by other researchers and are shown in Table 3. Aríztegui et al. [6] found that the activation energy for T22 alloy in temperature range subjected to isothermal and non-isothermal oxidation treatments in water steam at several temperatures ranging from 550°C to 700°C for over 1000 h reached –324 kJ/mol. Table 2 kp values (mg2/cm4/s) for the exposed alloys at 657°C and 725°C for 500 h Table 3 Activation energies (kJ/mol) for T22 and T23 alloys exposed at 657°C and 725°C for 500 h ## Surface analysis This section presents the microstructural analysis of the exposed materials at 675°C and 725°C; it is divided into the surface analysis of ferritic steels (T22 and T23), of austenitic (TP347HFG and HR3C) and Ni-based 718+ alloys. The environmental scanning electron microscopy (ESEM) with energy-dispersive x-ray spectroscopy (EDX) analysis data were obtained after 500 h only. ## Exposure at 675°C and 725°C Figure 6 shows the surface microstructures of the alloys exposed at 675°C and 725°C for 500 h. During the exposure in both temperatures, rough and non-protective oxide was observed in T22 and T23 alloys, the external part of the oxide based on EDX analyses consisted of Fe2O3 phase (hematite). Figure 6 A comparison of the scale morphologies formed on the exposed materials after 500 h at 675°C and 725°C: (A) T22, (B) T23, (C) TP347HFG, (D) HR3C and (E) 718+ High alloyed steels TP347HFG (~20 wt% Cr) and HR3C (~25 wt% Cr) showed much better corrosion resistance due to the formation of the mixture consisted of Cr–Ni–Fe oxide scales [7]. At 725°C TP347HFG alloy showed a different morphology than that formed after exposure at 675°C; the surface of the material was covered by crystals consisting of Fe–Mn oxides with composition: ~27, ~18 and ~9 wt% of Mn, Fe and Cr, respectively. HR3C alloy at 675°C formed oxide scale with rich content of Cr (~38 wt %), medium content of Fe (~19 wt %), relatively high Mn concentration (6 wt %) and low Ni content was found with concentration of ~2.2 wt %. At higher temperature, HR3C showed very similar behaviour as was shown at 675°C; the developed oxide contained mainly Cr (~35 wt%), Fe (~19 wt%), Mn (~6 wt%) and Ni (~6.5 wt%). The surface of the exposed 718+ sample was covered after 500 h by the layer of the oxide containing ~19 wt% of Cr and ~34 wt% of Ni, in some places Nb-rich crystal formed with following composition: ~55 wt% of Nb, 6.0 wt% of Ni, 3.6 wt% of Co, 12.4 wt% of Cr and 4 wt% of Ti. ## Cross section of exposed materials The cross-sectioned ESEM images of the exposed materials at 675°C and 725°C in steam environment for 500 h are shown in Figure 7. It needs to be pointed out that EDS concentration profiles are not shown here due to high number of profiles, only overall summary of the outcome of analyses are presented. Figure 7 A comparison of the cross-sectioned samples: after 500-h exposure at 675°C and 725°C: (A) T22, (B) T23, (C) TP347HFG, (D) HR3C and (E) 718+ T22 alloy showed thicker scale than that observed in T23 alloy after 500 h at 675°C; similar observations were found at higher temperatures. EDX analyses performed on T22 and T23 alloys after exposure showed that the top layer consisted of the phase with 70 wt% of Fe suggesting the formation of Fe2O3 (hematite), the middle layer was occupied by the phase with 73 wt% of Fe suggesting development of Fe3O4 (magnetite), the phase between the Fe3O4 and the substrate consisted of the highest level of Fe suggesting the formation of FeO (wustite) phase. In addition during the exposure of T22 and T23 alloys, in the interface of the oxide scale, enriched region containing up to 5 wt% of Cr and W in T23 was found. Figure 5 shows as well that hematite layer detached from the magnetite layer due to the difference in CTE between both oxides. In contrast, top layer in T23 showed lack of detachment. Both ferritic alloys showed the formation of voids within the developed oxide scale. The voids result from different fluxes of ions with the oxide scale, and different diffusion coefficient with the oxide layers. The voids formation with the oxide scale was discovered and theoretically described by Kirkendall [8]; the effect is enhanced where Fe–Al coating is deposited on Fe-based alloy and heat treated at high temperature [9]. Austenitic steels TP347HFG and HR3C alloys formed much thinner oxides as ferritic steels. The formed oxide in TP347HFG alloy at 675°C consisted 51 wt% of Fe, 20 wt% of Cr, 2.5 wt% of Mn and 7.6 wt% of Ni. At higher temperature, according to EDS concentration profiles, Cr reached only 17 wt%, Mn as well showed lower concentration than that found at 675°C. At 725°C higher concentration of Ni was detected in the oxide scale. The thickness of the oxide scale in both temperatures was relatively low; the thickness was assessed for ~ 2 μm (thickness measurement performed in higher magnification, not shown here). Nevertheless, despite the formation of thin oxide scale, TP347HFG alloy developed as well nodules rich in Fe3O4 with Cr addition (not shown here). Such behaviour was not observed in the alloy with higher content of Cr (HR3C), where thin oxide scale with lack of nodules was formed rich in Cr (20 wt%) and Ni (18 wt%) at 675°C. Further analyses at higher temperature show that the oxide scale formed on HR3C alloy possesses over 26 wt% of Cr and 21 wt% of Ni. Concentration of both Mn and Fe within the oxide scale at both temperatures showed lowered values. Based on these findings, it can be suggested that HR3C alloy showed better corrosion resistance and formation of more protective oxide scale enriched in Cr can be developed at higher temperature. The exposure of 718+ Ni-based alloy developed thin (~1μm) mixture of Ni-Cr oxides with high additions of the other elements (Co, Al and Ti). The alloy showed slightly higher concentration of Cr and Ni at 725 than at 675°C. However, Cr concentration within the oxide scale was much lower than that found in HR3C alloy at both temperatures. ## Dimensional metrology of T22 and T23 alloys exposed at 675°C and 725°C The polished cross sections were all measured using an image analyser to generate accurate assessments of the amount of metal remaining after steam oxidation tests; these measurements were compared to the pre-exposure sample thickness data in order to analyse the distribution of the change in metal (metal loss) resulting from the exposures. By comparing sample dimensions before and after exposure, the apparent change in metal and the change in sound metal (change in metal + internal damage) can be calculated. These data sets can then be re-ordered (from greatest to least metal loss) and corrected for calibration differences (using data from reference samples). The processed data can then be plotted as a change in metal vs cumulative-probability; effectively, this type of plot indicates the probability (e.g. 4%) of a certain degree of damage being observed. The data can be summarized more readily on cumulative probability plots (for which the data have been ordered from most to least damage). This type of plot highlights the variability of corrosion damage around samples, particularly for the lower Cr content steels. Figure 8 illustrates the change in metal as a function of cumulative probability of T22 and T23 alloys exposed at 675°C and 725°C. In this section only two alloys are presented, T22 and T23, which showed the highest metal loss, significantly exceeding the tolerance of the analyser (5 μm). The other materials (TP347HFG, HR3C and 718+ alloys) showed low metal losses which are very difficult to measure. The results show that the damage levels of the T22 samples was much higher than that observed in T23 material, the metal loss increased with increasing temperature. Median values of metal loss are presented in Figure 9(A) and (B). Figure 8 Change in metal vs cumulative probability showing metal loss data of T22 alloy (A and B) and T23 alloy (C and D) after exposure at 675°C and 725°C Figure 9 Median values of metal loss of T22 and T23 materials for 500 h at 650°C (A) and 725°C (B) ## Discussion The aim of this study was to calculate and show the results of metal loss of the currently useable ferritic steels at high temperature. Additionally, corrosion behaviour of the ferritic steels in steam oxidation was compared with better alloys containing higher Cr content within the matrix. In this study five alloys were taken into consideration: T22, T23, TP347HFG, HR3C and 718+. The two ferritic alloys (T22 and T23) were used in this study to show metal loss of low Cr content materials. Other alloys such as TP347HFG, HR3C and 718+ alloys contained ~19, ~25 and ~18 wt% of Cr, respectively. Due to the different mechanism of the degradation of the exposed alloys in steam environment, the structure of this discussion part is divided into following sections. ## Steam oxidation behaviour of the ferritic alloys (T22 and T23) The thickness of the oxide scales formed on T22 and T23 alloys during steam oxidation are not acceptable for components used for boiler components such as SH/RH. The upper limit for the use of these materials is ~580°C, in subcritical power plants with steam pressures of 140 bar [2, 10]. The mechanism of the degradation of T22 and T23 alloys can be demonstrated. Thus when the T22 and T23 alloys are exposed to steam environments, the partial pressure of the oxygen results from the dissociation of steam via following reaction: ${\mathrm{H}}_{2}\mathrm{O}\left(\mathrm{g}\right)↔{\mathrm{H}}_{2}\left(\mathrm{g}\right)+\frac{1}{2}{\mathrm{O}}_{2}\left(\mathrm{g}\right)$(3) The calculated dissociation oxygen partial pressures of steam and the oxides of Fe and Cr as a function of temperature show that the oxygen partial pressure of steam after dissociation is much higher than that needed for the formation of the Fe2O3 phase, where Fe2O3 is the least stable iron oxide phase. The formation of Fe2O3 in steam environment is often disputed, due to the fact that partial pressure of oxygen after water dissociation is not high enough. However in this work OFN was used that contained 99.998% nitrogen – i.e. an oxygen partial pressure of ~2 × 10–5 atm. This is more than sufficient to oxidize low-alloyed steel and form Fe2O3 layer. Moreover, in a well-sealed container, OFN will reduce the dissolved oxygen level in water (and hence steam) to around 20 ppb. Again this is quite sufficient to oxidize reactive metals such as iron. Thus in steam environment it is likely to observe mainly formation of FeO, Fe3O4 and Fe2O3 oxides. Grabke et al. [11, 12] described, where five pathways of the oxide growth, as follows: • (1) Outward diffusion of Fe ions, which react with oxygen at the interface (oxide steam), oxygen origin from the dissociation of H2O at high temperature. • (2) Dissociation of steam at the oxide–steam interface and diffusion of oxygen inward via defects in the oxygen lattice or hydrogen defects. Norby et al. [13] proposed that hydrogen ions entering the oxide lattice could associate with oxygen ions (on their normal lattice sites) to form “hydrogen defects”, each of which have an effective charge that would encourage increased cation transport, so increasing the rate of oxide growth and presumably delaying the formation of a more protective layer. • (3) Dissociation of steam on the steam–oxide interface and diffusion of oxygen ions inward to the oxide–alloy interface, reaction with chromium in the alloy, to form internal chromia precipitates. • (4) Steam is transported through the scale; there dissociation of steam occurs (on the oxide–alloy interface) to form iron-chromium oxides. This scenario is unlikely to be possible, unless the formed oxide scale has significant porosity [14]. • (5) Dissociation of Fe3O4 phase at the interface with the inner and outer layers the released iron ions may diffuse to the oxide–gas interface to react and form new oxide [15]. According to Gala and Grabke et al. [10] the inner layer (Fe3O4) dissociates according to the reaction shown below: $\mathrm{F}{\mathrm{e}}_{3}{\mathrm{O}}_{4}\left(\mathrm{s}\right)↔2\mathrm{F}{\mathrm{e}}^{3+}+\mathrm{F}{\mathrm{e}}^{2+}+4{\mathrm{O}}^{2-}$(4)The dissociation of magnetite-released iron ions, which may diffuse outward, and oxygen ions which may diffuse inward in order to form layer of FeO which is more stable at higher temperatures than Fe3O4 (magnetite). The voids in the oxide scale were observed after 500 h of exposure at 675°C and 725°C. Quadakkers et al. [16] showed that the voids formation can be observed on the ferritic oxide scale after long-term exposures. The voids can be unevenly distributed or can coalesce to form a crack or a gap at the interface between the inner and outer layer. ## The role of W in low-alloyed steels Slightly better corrosion resistance of T23 in comparison to T22 can be related to W addition, when added, W through the chemical reaction with C forms WC carbide: $\mathrm{W}+\mathrm{C}\to \mathrm{W}\mathrm{C}$(5)Thus, due to the higher activity of Cr in T23, more Cr diffuses to the interface where Fe2CrO4 spinel can form. In contrast, in low-alloyed steel such as T22, Cr directly reacts with C to form carbides such as Cr3C2, Cr7C3 and Cr23C6 [17], and significantly decreases Cr activity and concentration, thus much lower amount of Cr diffuses to the interface, as a result thicker non-protective scale based on Fe-oxide can be formed. ## Steam oxidation behaviour of TP347HFG alloy The exposed TP347HFG alloy formed two regions with different degree of the degradation at both temperatures; region with higher degree of resistance formed Fe–Cr spinel with small Ni content. The other region showed a higher degree of the degradation, where small nodules rich in Fe3O4 phase with a small amount of Cr formed on the surface. The bottom part of the nodules consisted of Fe3O4 with a small amount of Mn. These findings are in good agreement with the results obtained by the other researchers [7, 19]. The authors of these studies found that the austenitic steels, especially with Cr content lower than 20 wt%, in steam environment are covered by two layers; an outer magnetite layer (Fe3O4, sometimes with Fe2O3 patches) and an inner layer (Fe–Cr spinel). In this work such structures, as mentioned by the authors, were not observed, however, such morphology can be developed after long exposure time (2,000 h). Thus it can be considered that this model can be fitted to TP347HFG alloy. Otsuka et al. [18] found that after exposure at 700°C for 2,000 h in 1 bar steam pressure, alloys having a lower content of Cr than 20 wt% may develop a relatively uniform layer, with non-formation of healing Cr-rich layer or penetration along alloy grain boundaries. The exposures with longer time cause the formation of irregular thickness developed due to penetration of Cr-rich oxide along alloy grain boundaries. The behaviour of TP347HFG alloy in steam environment where nodules formed suggests the limited corrosion resistance for a longer period of exposure (10,000 h). According to Quan et al. [19], two mechanisms in nodules formation can be involved: • Disruption or mechanical failure in an initially formed passive layer with subsequent short-circuit diffusion • The co-development of two different scale phases from the onset of exposure [20, 21] The formation of volatile CrO2(OH)2 phase could as well contribute to the higher rate of degradation of TP347HFG alloy. The volatile chromium oxy-hydroxide CrO2(OH)2 is formed via [22]: $\frac{1}{2}\mathrm{C}{\mathrm{r}}_{2}{\mathrm{O}}_{3}\left(\mathrm{s}\right)+{\mathrm{H}}_{2}\mathrm{O}\left(\mathrm{g}\right)+\frac{3}{4}{\mathrm{O}}_{2}\left(\mathrm{g}\right)\to \mathrm{C}\mathrm{r}{\mathrm{O}}_{2}{\left(\mathrm{O}\mathrm{H}\right)}_{2}\left(\mathrm{g}\right)$(6)It is believed that oxygen in eq. (3) comes from dissociation of steam: ${\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{H}}_{2}+\frac{1}{2}{\mathrm{O}}_{2}$(7)Finally, it can be concluded that better steam oxidation resistance of the TP347HFG alloy than T22 and T23 alloys is mainly associated with its higher chromium content (~19 wt%) in the matrix and ability to form protective scale [23]. During these tests at 675°C and 725°C thin layers of Fe–Cr spinel and Fe–Mn spinel have been formed. The development of these phases is a function of temperature, a faster development was observed at higher temperature (725°C) than at lower (675°C). The formation was associated with a faster diffusion at higher temperature [20]. This statement is in agreement with the findings of Trindade et al. [24] that a low concentration of Cr (17.5 wt%) is required to develop a Cr2O3 oxide scale when Fe–Cr alloys exhibit a fine-grained microstructure. In the case of samples with coarse grains (~65 μm), a higher amount of Cr (>17.5 wt%) is required to develop a slow-growing Cr2O3 scale as the influence of the grain boundary diffusion is minimized in these materials. ## Steam oxidation behaviour of the HR3C alloy The results obtained here suggest that in order to receive the protective oxide scale, Cr content in the alloy has to be higher than ~17 wt%. Thus in this study HR3C alloy was also exposed. The alloy with ~25 wt% of Cr showed much better corrosion resistance than TP347HFG; the formation of nodules was not observed at 675°C and 725°C after 500 h of exposure. In contrast to TP347HFG alloy, thin protective scale formed, containing mainly Fe-Cr spinel with high amount of Ni (~18 wt%). Thus, the results obtained in this study are in contradiction with the results presented by Otsuka et al. [15], where the alloys with Cr content higher than 20 wt% generally form continuous Cr2O3 layers. It is generally accepted that steels containing high levels of Cr have better corrosion resistance, as was mentioned in this work previously. It was observed by Otoguro et al. [25] that the improvements of the alloys with high Cr content in steam oxidation is also caused by high Ni content. The benefit of Ni additions was also reported by Croll et al. [26], who demonstrated the benefit on the oxidation resistance in Fe–Cr–Ni alloys. Caplan et al. [27] also showed that Ni increased oxidation resistance of stainless steels in wet air. It is suggested that this is due to the formation of a Ni-enriched layer. ## Steam oxidation behaviour of Ni-based alloy 718+ It was observed that Ni-based alloy 718+ which was exposed at 675°C and 725°C showed the best corrosion resistance in steam oxidation environment from the all exposed alloys. The formation of NiO + Cr2O3 or NiCr2O4 spinel scale was observed. Essuman et al. [28] found that the oxidation of Ni–20Cr in both dry air and wet air resulted in the formation of three oxides. These were Cr2O3, NiO and the spinel oxide NiCr2O4. Whilst in the dry air the NiO phase gradually disappeared with time by reaction with chromia to generate the spinel, in the wet atmosphere, the NiO persisted. Significant cracking and spallation is reported to have occurred after longer exposures at higher temperatures; in this study, the exposed Ni-based alloy showed lack of spallation of the oxide scale. Similar to Fe-based alloy with medium content of Cr, volatilization of compounds rich in chromium would exert an over-pressure, which would cause the oxide layer to burst. It was found, when chromia layer created continuous layer at the alloy/oxide interface, compressive stresses would be induced causing breakdown of the NiO and NiCr2O4 layers. The chromia layer is then in contact with the atmosphere and would volatilize as CrO2(OH)2. Thus from now the exposed alloy depletes in chromium would re-oxidize and a new layer of NiO, and eventually spinel, would develop. It can be suggested that higher content of water vapour in the atmosphere, faster volatilization of Cr can be observed. It was found in this work, that in some places on the surface during steam oxidation at 675°C and 725°C Nb-rich oxide crystals formed, (~55 wt% of Nb). It can be assumed that the formation of these crystals was due to Nb outward diffusion through the grain boundaries. ## Conclusions The aim of this work was to investigate metal loss degree and steam oxidation behaviour of selected alloys (T22, T23, TP347HFG, HR3C and 718+) in steam environment at 675°C and 725°C for up to 500 h of exposure. The results obtained in this study clearly show that T22 and T23 ferritic steels showed a similar behaviour, where thick and non-protective Fe oxide scales are formed. The metal loss data from the ferritic steel T22 showed a higher degree of degradation compared with T23 alloy. It is believed that slightly better steam oxidation resistance is driven by the addition of W to T23. The alloys with higher Cr content (austenitic stainless steels TP347HFG and HR3C and Ni-based alloy 718+) developed a thin and protective oxide layer. The alloy TP347HFG with ~19 wt% of Cr showed Fe3O4 nodules formation; on the other hand, the same alloy formed more protective FeCr2O4 spinel. The alloy HR3C with 25 wt% of Cr showed better oxidation resistance than that observed in TP347HFG, where lack of nodules formation was observed. The alloy developed thin oxide layer of Fe–Cr with high Ni content (18 wt%). Ni-based alloy 718+ due to the formation of a thin oxide scale containing NiO and Cr2O3 or NiCr2O4 spinel showed good steam oxidation resistance at 675°C and 725°C. However, the surface of the alloy was covered by the oxide crystals rich in Nb (~55 wt %). ## Acknowledgement Authors like to acknowledge the support of The Energy Programme, which is a Research Councils UK cross council initiative led by EPSRC and contributed by ESRC, NERC, BBSRC and STFC, and specifically the Supergen initiative (Grants GR/S86334/01 and EP/F029748) and the following companies; Alstom Power Ltd., Doosan Babcock, E.ON, National Physical Laboratory, Praxair Surface Technologies Ltd, QinetiQ, Rolls-Royce plc, RWE npower, Siemens Industrial Turbomachinery Ltd. and Tata Steel, for their valuable contributions to the project. ## References • 1. Fry TA, Wright IG, Simms NJ, McGhee B, Holcomb GR. Steam oxidation of fossil power plant materials: collaborative research to enable advanced steam power cycles. Mater High Temp 2013;30:261–70. • 2. Holcomb RG. Steam turbine materials and corrosion. Albany NY: National Energy Technology Laboratory, 2008. Google Scholar • 3. Fry A, Osgerby S, Wright M. Oxidation of alloys in steam environments-a review. London: NPL Materials Centre, 2002. Google Scholar • 4. Łukaszewicz M, Simms NJ, Dudziak T, Nicholls JR. Effect of Steam Flow Rate and Sample Orientation on Steam Oxidation of Ferritic and Austenitic Steels at 650 and 700°C. Oxid Met 2013;79:473–83. • 5. Laverde D, Gomez-Acebo T, Castro F. Continuous and cyclic oxidation of T91 ferritic steel under steam. Corros Sci 2004;46:613–31. • 6. Aríztegui A, Gómez-Acebo T, Castro F. Steam oxidation of ferritic steels: kinetics and microestructure. Bol Soc Esp Cerám Vidrio 2000;39:305–11. • 7. Neroth G, Vollenschaar D, der Metalle DK. Wendehorst Baustoffkunde 2011:735–49. http://www.springerprofessional.de/978-3-8348-9919-4---wendehorst baustoffkunde/1808484.html Crossref • 8. Smigelskas AD, Kirkendall EO. Zinc Diffusion in Alpha Brass. Trans AIME 1947;171:130–42. Google Scholar • 9. Fan DW, Kim HS, Oh JK, Chin KG, De Cooman BC. Influence of isothermal deformation conditions on the mechanical properties of 22MnB5 HPF steel. ISIJ Int 2010;50:561–8. • 10. Dooley RB. Program on technology innovation: oxide growth and exfoliation on alloys exposed to steam. Palo Alto, CA: EPRI, 2007. Google Scholar • 11. Gala A, Grabke HJ. Arch Eisenhüttenwes 1972;43:463–9. Google Scholar • 12. Grabke HJ, Viefhaus H. Phys Chem 1980;84:152–9. Google Scholar • 13. Norby T. Hydrogen defects in inorganic solids. O. Johannessen, AG Andersen, editors. Amsterdam: Elsevier, 1989. Google Scholar • 14. Mayer P, Manolescu AV. Proceedings of international corrosion conference series NACE-6, Houston, 1983:368–379. Google Scholar • 15. Wright IG, Wood GC. The isothermal oxidation of Co-Cr alloys in 760 Torr oxygen at 1000°C. Oxid Met 1977;11:163–91. • 16. Quadakkers W. Mechanisms of Steam Oxidation in High Strength Martensitic Steels. Int J Pres Ves Pip 2007;84:75–81. • 17. Ellis J, Haw M, Carbides C. Wear Resistant Coatings. Mater World 1997;5:1–36. Google Scholar • 18. Otsuka N, Fujikawa H. Scaling of Austenitic Stainless Steels and Nickel-Base Alloys in High-Temperature Steam at 973 K. Corrosion 1991;47:240–8. • 19. Quan NS, Young DJ. Sulfidation behavior of an aluminum-manganese steel. Oxid Met 1986;25:107–19. • 20. Sakiyama M, Tomasazewicz P, Wallwork GR. Oxidation of iron-nickel aluminum alloys in oxygen at 600–800°C. Oxid Met 1979;13:311–30. • 21. Smith PJ, Beauprie RM, Smeltzer WW, Stevanovic DV. The effect of aluminum ion implantation on the oxidation of an Fe-6 at.% Al alloy at 1173 K. Oxid Met 1987;28:259–76. • 22. Wright I, Dooley R. A review of the oxidation behaviour of structural alloys in steam. Philadelphia, PA: Maney for the Institute and ASM International, 2010. Google Scholar • 23. Wright I, Pint B. Proceedings of the NACE conference, August 8–11, 2002, Denver, 2002. Google Scholar • 24. Trindade VB, Krupp U, Wagenhuber E-G, Christ H-J. Oxidation Mechanisms of Cr-Containing Steels and Ni-Base Alloys at High Temperatures – Part I: The Different Role of Alloys Grain Boundaries. Mater Corros 2005;56:785–90. • 25. Otoguro Y, Sakakibara M, Saito T, Ito H, Inoue Y. Corrosion behaviour of austenitic heatresisting steels in a high temperature and high pressure steam environment. Trans ISIJ 1988;28:761–8. • 26. Croll JE, Wallwork GR. The high-temperature oxidation of iron-chromium-nickel alloys containing 0–30% chromium. Oxid Met 1972;4:121–40. • 27. Caplan D, Cohen M. Influence of Impurities on the Oxidation of Fe-26Cr Alloys. Nature 1965;205:690–702. • 28. Essuman E, Walker LR, Maziasz PJ, Pint BA. Oxidation behaviour of cast Ni–Cr alloys in steam at 800°C. Mater Sci Tech 2013;29:822–7. Work carried out at Centre for Energy and Resource Technology, Cranfield University, Bedfordshire MK43 0AL, UK. ## About the article #### T. Dudziak T. Dudziak Currently employed at Foundry Research Institute, Centre for High Temperature Studies, Zakopiańska 73, 30–418 Krakow, Poland. #### M. Łukaszewicz M. Łukaszewicz Currently employed at National Physical Laboratory, Hampton Rd, Teddington, Middlesex TW11 0LW, UK. Received: 2014-09-18 Accepted: 2014-11-11 Published Online: 2015-01-14 Published in Print: 2015-12-01 Citation Information: High Temperature Materials and Processes, Volume 34, Issue 8, Pages 783–798, ISSN (Online) 2191-0324, ISSN (Print) 0334-6455, Export Citation ©2015 by De Gruyter. ## Citing Articles Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page. ## Comments (0) Please log in or register to comment. Log in	2019-08-22 19:47:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6383549571037292, "perplexity": 7152.424005708035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317359.75/warc/CC-MAIN-20190822194105-20190822220105-00378.warc.gz"}
https://learn.microsoft.com/en-us/dax/poisson-dist-function-dax	# POISSON.DIST Returns the Poisson distribution. A common application of the Poisson distribution is predicting the number of events over a specific time, such as the number of cars arriving at a toll plaza in 1 minute. ## Syntax POISSON.DIST(x,mean,cumulative) ### Parameters Term Definition x Required. The number of events. mean Required. The expected numeric value. cumulative Required. A logical value that determines the form of the probability distribution returned. If cumulative is TRUE, POISSON.DIST returns the cumulative Poisson probability that the number of random events occurring will be between zero and x inclusive; if FALSE, it returns the Poisson probability mass function that the number of events occurring will be exactly x. ## Return value Returns the Poisson distribution. ## Remarks • If x is not an integer, it is rounded. • If x or mean is nonnumeric, POISSON.DIST returns the #VALUE! error value. • If x < 0, POISSON.DIST returns the #NUM! error value. • If mean < 0, POISSON.DIST returns the #NUM! error value. • POISSON.DIST is calculated as follows. • For cumulative = FALSE: $$\text{POISSON} = \frac{e^{-\lambda} \lambda^{x}}{x!}$$ • For cumulative = TRUE: $$\text{CUMPOISSON} = \sum^{x}_{k=0} \frac{e^{-\lambda} \lambda^{x}}{k!}$$ • This function is not supported for use in DirectQuery mode when used in calculated columns or row-level security (RLS) rules.	2023-03-27 21:38:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49507561326026917, "perplexity": 2496.6329810480165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948684.19/warc/CC-MAIN-20230327185741-20230327215741-00576.warc.gz"}
https://stacks.math.columbia.edu/tag/03JJ	Lemma 64.12.4. Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $Z \subset X$ be a closed subspace. Assume $Y$ is reduced. A morphism $f : Y \to X$ factors through $Z$ if and only if $f(\|Y\|) \subset \|Z\|$. Proof. Assume $f(\|Y\|) \subset \|Z\|$. Choose a diagram $\xymatrix{ V \ar[d]_ b \ar[r]_ h & U \ar[d]^ a \\ Y \ar[r]^ f & X }$ where $U$, $V$ are schemes, and the vertical arrows are surjective and étale. The scheme $V$ is reduced, see Lemma 64.7.1. Hence $h$ factors through $a^{-1}(Z)$ by Schemes, Lemma 26.12.7. So $a \circ h$ factors through $Z$. As $Z \subset X$ is a subsheaf, and $V \to Y$ is a surjection of sheaves on $(\mathit{Sch}/S)_{fppf}$ we conclude that $X \to Y$ factors through $Z$. $\square$ There are also: • 2 comment(s) on Section 64.12: Reduced spaces In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	2021-04-21 20:11:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9934411644935608, "perplexity": 245.09087555458973}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039550330.88/warc/CC-MAIN-20210421191857-20210421221857-00202.warc.gz"}
http://www.algebra.com/algebra/homework/Graphs/Graphs.faq	Questions on Algebra: Graphs, graphing equations and inequalities answered by real tutors! Algebra -> Graphs -> Questions on Algebra: Graphs, graphing equations and inequalities answered by real tutors! Log On Question 994548: For the equation 9 x ^3 - 9 y ^2 = -6 answer the following questions. Is the equation symmetric with respect to the y-axis? ( yes or no) Is the equation symmetric with respect to the x-axis? ( yes or no) Is the equation symmetric with respect to the origin? ( yes or no) I have substitute both axis and the origin with the corresponding transformation of -x and -y. However, I couldn't find the right answer and I would like to know what I did wrong. For example, x axis: 9x^3-9(-y)^2=-6 9x^3-9y^2=-6 <-- same equation as original therefore Yes Y axis: 9(-x)^3-9y^2=-6 -9x^3-9y^2=-6 <--- not the same equation as original therefore No Origin: 9(-x)^3-9(-y)^2=-6 -9x^3-9y^2=-6 <--- not the same equation as original therefore No You can put this solution on YOUR website! Looks symmetric to the horizontal axis to me. And it certainly is not symmetric to either the vertical axis or the origin. So I don't know what you mean by "couldn't find the right answer" because the right answer is right there in your post. John My calculator said it, I believe it, that settles it Question 994416: 3x + 23 = 23 You can put this solution on YOUR website! 3x + 23 = 23 3x = 0 x = 0 Question 994433: The sum of two numbers is 48, and one number is twice the other. Find the numbers You can put this solution on YOUR website! Hi there, The sum of two numbers is 48, and one number is twice the other x + 2x = 48 3x = 48 x = 48/3 x = 16 First number = 16 Second number = 32 Hope this helps :-) Question 994381: system of linear equations using elimination for -x-6y=11 and 5x-6y=-19 You can put this solution on YOUR website! -x-6y= 11 5x-6y=-19 --------------- Subtract Question 994374: system of linear equations using substitution y=5x-11, y=-3x+5 You can put this solution on YOUR website! system of linear equations using substitution y=5x-11, y=-3x+5 ---- y=5x-11 y=-3x+5 --- sub for y 5x-11 = -3x+5 8x = 16 etc Question 994298: Determine whether the function is an odd or even function: You can put this solution on YOUR website! . This function is odd. Change x for -x, and you will see that the function will change the sign. Question 994299: Determine whether the function is an odd or even function: You can put this solution on YOUR website! . Neither odd nor even. See the plot in the Figure: Plot of = Question 994306: If 4 < a ≤ 8 and 1/4 < b < 1/2, then what are the possible values of ab? A. 1 ≤ ab ≤ 4 B. None of these C. 1 ≤ ab < 4 D. 1 < ab ≤ 4 E. 1 < ab < 4 You can put this solution on YOUR website! . Correct answer is E. 1 < ab < 4 Question 994086: Write equations for the vertical and horizontal lines passing through the point (-1,-5) You can put this solution on YOUR website! Write equations for the vertical and horizontal lines passing through the point (-1,-5) ----- Vertical:: x = -1 Horizontal:: y = -5 ----- Cheers, Stan H. ----------- Question 994085: How do you do h-(-2) is greater than or equal to 10? You can put this solution on YOUR website! h-(-2) > or = 10 - time - is a + h + 2 > = 10 subtract 2 from both sides of inequality h > = 8 Question 994024: Find the equation of the line that contains the given point and has the given slope. Express the equation in the form Ax + By = C, where A, B and C are integers. (5, 6), m = 5/6 ____=-11 You can put this solution on YOUR website! Find the equation of the line that contains the given point and has the given slope. Express the equation in the form Ax + By = C, where A, B and C are integers. (5, 6), m = 5/6 ____=-11 m=∆y/∆x=(y-6)/(x-5)=5/6 6y-36=5x-25 5x-6y=-9 (equation of the line) Question 994036: Which equation represent the line with slope -2 that passes through the point (4,-1) Y= -2x-9 Y= -2x-7 Y= -2x+2 Y= -2x+7 Found 2 solutions by vleith, josgarithmetic: You can put this solution on YOUR website! they all have a slope of -2. You need to see which equation has the point (4,-1) on it. Just substitute 4 for x and -1 for y in each equation. Then find the equation that ends up being a "true" result. It's D since which is true You can put this solution on YOUR website! Two ways to solve. Plug-in the coordinate values and see which equation is true. OR Plug slope and given point into point-slope formula and simplify into general form. Question 993987: Find the slope if possible of the line passing through the points, (1,2) and (1,-1) and describe the line. You can put this solution on YOUR website! Find the slope if possible of the line passing through the points, (1,2) and (1,-1) and describe the line. slope of given line is undefined. It can be described as a vertical line passing thru x=1 Question 993287: For what values of "a" is the distance between P(a, 3) and Q(6, 2a) greater than 29? You can put this solution on YOUR website! Distance between P(a,3) and Q(6,2a) is = = For this distance to be greater than 29; => => Solving the quadratic we get roots as -10.44 and 15.24 Since coefficient of a^2 is positive, the graph is opening upwards So the value of this quadratic is negative between these two roots. So the required value of a for the quadratic to be positive, is All values of a except [-10.44,15.24] range. Question 993988: The equation, in general form, of the line that passes through the point (3,5) and is parallel to the line 6x-9y+2=0 is Ax+By+C=0 where A,B and C are what? Found 2 solutions by anand429, josgarithmetic: You can put this solution on YOUR website! For parallel lines: A1/A2 = B1/B2 So, => -3A=2B ----------------------------(i) Now it passes through (3,5) so, A3 +B5 +C=0 => 3A+5B + C=0 => 3B = -C (using (i))-----------(ii) Simplfying given equation in terms of single variable we get, Ax+By+C=0 => => You can put this solution on YOUR website! A study about that STANDARD form equation format would help you to understand or be able to determine that the slope is . Your given line accordingly has a slope . You find the value for C using the point (3,5) which must be on the line parallel to the given line. Be careful. The original given line is 6x-9y+2=0, and using this, the slope is also , so you might want to stay with the less simplified form just to avoid complicating mistakes if you want. Parallel lines in the plane has equal slopes, so the line you are trying to find will also have slope . Staying in STANDARD FORM, the equation you want will be . . Values for A and B were already explained and shown. Your description mentioned "general form" at the beginning. If you want that, then solve for y from . Notice that if you simplify this standard form equation it is . Using this simplified form of the standard form equation, you have . Question 993873: whats the slope of (5,8) (-5,8) You can put this solution on YOUR website! Slope is Points: (x1, y1) --> (5, 8) and (x2, y2) --> (-5,8) Question 993863: what is y=3/4x-2 when you graph it You can put this solution on YOUR website! what is y=3/4x-2 when you graph it -------------- Depends. Is it (3/4)x - 2 ? or (3/4x) - 2 ? or 3/(4x - 2) ? Question 993828: how do you transform y=3x+18 into a standard form Found 2 solutions by MathTherapy, josgarithmetic: You can put this solution on YOUR website! how do you transform y=3x+18 into a standard form 3x + 18 = y ----------- Reversing sides 3x - y + 18 = 0 ------- Subtracting y to get variables on left-hand side --------- Subtracting 18 to get the constant on the right-hand side DONE!! You can put this solution on YOUR website! Done, now shown fully in the previous question. If you have any slope intercept line equation , in which the slope is , converting into standard form is a matter of very simple algebra. This result matches the form . Carefully note how the coefficients A and B are used in forming the slope. Slope is. Question 993823: how do you transform y=3x+18 into a slope intercept form You can put this solution on YOUR website! Converting into STANDARD FORM? You want a form like . The equation as you have is almost like what you need, only requiring ONE SIMPLE STEP. , I am using more than just one step; or you can use the equivalent equation, . Do you understand how? Question 993714: plase help me solve this. Rearrange this equation into that form that can be plotted as a straight line graph y=10ex. the x is a power.Thank you. You can put this solution on YOUR website! http://slc.umd.umich.edu/slconline/TRANSF/ generally speaking, you are transforming y = 10 * e^x into the equation ln(y) = x + ln(10) that's done in the following manner: take natural log of both sides to get: ln(y) = ln(10 * e^x) this becomes: ln(y) = ln(10) + x * ln(e) which becomes: ln(y) = ln(10) + x rearrange the terms to get: ln(y) = x + ln(10) which is in the form of a linear equation. i don't pretend to understand what they doing or why they're doing it, but the reference i found on the web should be helpful. Question 993775: If there is an improper fraction such as -17 over 3 how would u change it into a mixed fraction to graph it? You can put this solution on YOUR website! If there is an improper fraction such as -17 over 3 how would u change it into a mixed number to graph it? ---- Divide -17 by 3 to get: Ans: -(5 2/3) ------------ Cheers, Stan H. ------- Question 993679: What is the slope of the line through the points (2,-3) and (6,5)? You can put this solution on YOUR website! Slope is found by m = (y2 - y1)/(x2 - x1) m = (5-(-3)) / (6 - 2) m = 8 / 4 = 2 Question 993563: If I have -(x+3)^2+5 how do I make a vertical compression that restricts to 20% of the original? I'm thinking a would need to equal 20, but I'm not sure. You can put this solution on YOUR website! 20% is 1/5. For vertical compression, you need a "less than 1" lead coefficient. John My calculator said it, I believe it, that settles it Question 993532: How do you solve 4x+5y=25 and 8x-5y=35 when using substitution You can put this solution on YOUR website! . How do you solve 4x+5y=25 and 8x-5y=35 when using substitution. ---------------------------------------------------------------- . You can express 4x from the first equation, 4x = 25-5y, and then substitute it into the second equation. You will get 2(25-5y) - 5y = 35. Simplify and solve: 50 - 10y - 5y = 35, 15y = 30 - 35, 15y = 15, y = 1. Now substitute the found value of y=1 into the first equation to find x. You will get 4x + 51 = 25, 4x = 20, x = 5. Answer. x = 5, y = 1. Question 993524: m:3/2,(4,6) You can put this solution on YOUR website! Slope and included point of a line? Learn to understand and use Point-Slope formula. Line with slope m and point on the line, (h,k) has an equation . Question 993522: How to find the slope thru a pair of points You can put this solution on YOUR website! Imagine one point is (a,b), and another point is (c,d) on a graph (these letters represent x and y values). Slope: rise/run= (d-b) divided by (c-a) Question 993481: Given the function y = ( x+3 ) ( x^2+3 x−3 ) find the coordinates of the points of intersection with the x-axis and y-axis. Please enter your answer as a set of coordinate pairs, where a single coordinate pair (a, b) constants should all be given exactly, using the sqrt function and fractions as necessary THANK YOU You can put this solution on YOUR website! intersection with the x-axis occurs where and intersection with the y-axis occurs where so set and find intersection with the x-axis ...will be true for or , or both if => solutions: exact solutions or approximate solutions: or so, ordered pairs are: (,) (,) (,) or (,) (,) (,) now find y-intercept: the y-axis occurs where , so set ordered pair is: (,) see it on the graph: Question 993482: Given the function y = ( x - 4 ) ( x^2 + 3x + 2 ) find the coordinates of the two stationary points and the point of inflection. Note. A stationary point is a critical point at which the derivative is 0. Please enter your answer as a list of coordinate pairs, where a single coordinate pair (a, b) THANK YOU You can put this solution on YOUR website! In order to find the stationary points, you need to find the zeros of the first derivative. In order to find the point of inflection you will need to find the zero of the second derivative. None of this is very rigorous, but will work just fine for a polynomial function. You have two choices for finding the first derivative. Since you have the product of two functions, you can use the product rule: Or you can just perform the indicated multiplication and take the derivative by repeated applications of the power rule. Either way, after you have the first derivative, set the quadratic equal to zero and solve. You will get the values of the two stationary points. The second derivative is just the derivative of the first derivative quadratic polynomial Set the second derivative equal to zero and solve for the value of the coordinate of the inflection point. John My calculator said it, I believe it, that settles it Question 993404: The perimeter of a rectangle is 32in. If the length of the rectangle is 1 more than twice the width. Find the dimensions of the rectangle . You can put this solution on YOUR website! Hi there, Make the width = 'x' Length = 1 + 2x Perimeter = 2 x Width + 2 x Length 32 = 2(x) + 2(1 + 2x) 32 = 2x + 2 + 4x Collect like terms. 2x + 4x = 32 - 2 6x = 30 x = 5 Width = 5 in Length = 11 in Hope this helps :-) Question 993369: NEEDING HELP WITH THIS QUESTION PLEASE. f(x) = 3x2 − 12x + 8 INCREASING=? DECREASING=? HOW DO I FIND/WHAT FORMULA DO I USE TO FIND THIS? THANK YOU! You can put this solution on YOUR website! you look at what the highest order exponent term is doing. it's 3x^2 when x is negative, x^2 is positive, so this function should be increasing on the left and on the right. here's what the graph looks like: here's a good reference on figuring out how to analyze these. Question 993358: I know how to find increasing and decreasing but I am having trouble finding this one: f(x) = −2(x + 1)2 + 4 Increasing:? Decreasing:? Found 2 solutions by stanbon, Fombitz: You can put this solution on YOUR website! f(x) = −2(x + 1)^2 + 4 f(x) = -2(x^2+2x+1) + 4 f(x) = -2x^2-4x+2 ------------------- ---------------------- Take the derivative to get: f'(x) = -4x-4 --- Increasing:? Solve: -4x-4 > 0 -4x > 4 x < -1 Increasing on (-oo,-1) ----------------------------- Decreasing:? Solve: -4x-4 < 0 -4x < 4 x > -1 Decreasing on (-1,+oo) =========== Cheers, Stan H. You can put this solution on YOUR website! This equation is in vertex form. The vertex of the parabola is (,). Since the multiplier for the quadratic term is negative ,, the parabola opens downwards and the maximum value occurs at the vertex. So for x values before the vertex the y values are increasing and for x values after the vertex, the y values are decreasing. Increasing : Decreasing : . . . . Question 993204: A line has a slope of - 3/4. What is the slope of the line parallel to it? What is the line perpendicular to it? You can put this solution on YOUR website! A line has a slope of - 3/4. What is the slope of the line parallel to it? Ans: -3/4 -------------- What is the slope of the line perpendicular to it? Ans: 4/3 ------------- Cheers, Stan H. ========= Question 993150: I am doing homework and part of it deals with graphs and I just don't understand how to come up with the solution. The problem is -2x+y=6 Thank you! :) Found 2 solutions by ikleyn, Alan3354: You can put this solution on YOUR website! The plot is here Plot y = 2x + 6. You can put this solution on YOUR website! I am doing homework and part of it deals with graphs and I just don't understand how to come up with the solution. The problem is -2x+y=6 =========================== There is no unique solution to the equation. --- If you mean how to graph it: It's a straight line so you need 2 points. You can use the 2 intercepts. At the y-intercept x = 0 --> y = 6 --> point (0,6) At the x-intercept y = 0 --> x = -3 --> point (-3,0) ---------- Plot the 2 points, draw a line thru them. Question 993132: How would I graph a slope 1/3 and (-4,-1) You can put this solution on YOUR website! First plot the point (-4,-1). Slope is change in y over change in x. So in this case a change in x of 3 units gives you a change in y of 1 unit. So another point on the line would be, (-4,-1)+(3,1)=(-1,0) Plot that point. Then draw the line connecting the points. . . . . Question 993060: x + y = 10 is this a function or not You can put this solution on YOUR website! yes it is. it is a non-vertical line.So it passes the vertical line test for being a function Question 992932: How could I use the point-slope form to find the equation of the line? The answer has to be in slope-intercept form. One point on the line is (−2, −6). The slope is 2. Thank you! You can put this solution on YOUR website! Just plug your givens into point-slope format and do the algebra steps you need. Question 992858: all reals greater than or equal to -3 and less than 6 what is the interval representation You can put this solution on YOUR website! interval notation: [,) Question 992304: Find the point of intersection of the two lines to 2x+y=5 and y=x-1 You can put this solution on YOUR website! . You need to solve this system of two equations with two unknowns: . Substitute y from the second equation to the first one. You will get an equation for the single unknown x only. Then solve it. Then substitute the found value of x into either equation of the original system to find y. Is it clear? Good luck! Question 992378: Suppose admission to the fair is $3. a) Write an equation slope-intercept form for the total cost of admission and any number of tickets at the rate of 6 tickets for$5. b) Write an equation in slope-intercept form for the total cost of admission and any number of tickets at the rate of 12 tickets for $8. Answer by solver91311(20879) (Show Source): You can put this solution on YOUR website!$3 is a fixed cost and is therefore the -coordinate of the -intercept. For a) you get 6 tickets for \$5, therefore you get 1 ticket for . The rate per independent variable unit is the slope. John My calculator said it, I believe it, that settles it Question 992349: Graph f(x)=x^2 and show f(1), f(2), f(3)	2015-10-07 04:17:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6987060308456421, "perplexity": 1109.0797689753333}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736682102.57/warc/CC-MAIN-20151001215802-00007-ip-10-137-6-227.ec2.internal.warc.gz"}
https://quantumcomputing.stackexchange.com/questions/9224/how-to-plot-histogram-or-bloch-sphere-for-multiple-circuits	# How to plot histogram or Bloch sphere for multiple circuits? I have tried to plot a histogram for the multiple circuits for the code given below. I think I have done it correctly but don't know why it's not working. Please help me out. If possible please solve it for the Bloch sphere also # quantum_phase_bloch.py import numpy as np from qiskit.visualization import plot_histogram from qiskit.tools.monitor import job_monitor from matplotlib import style from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, execute, Aer from qiskit.tools.visualization import plot_bloch_vector # Define the Quantum and Classical Registers q = QuantumRegister(1) c = ClassicalRegister(1) # Build the circuits pre = QuantumCircuit(q, c) pre.h(q) pre.barrier() meas_x = QuantumCircuit(q, c) meas_x.barrier() meas_x.h(q) meas_x.measure(q, c) meas_y = QuantumCircuit(q, c) meas_y.barrier() meas_y.s(q).inverse() meas_y.h(q) meas_y.measure(q, c) meas_z = QuantumCircuit(q, c) meas_z.barrier() meas_z.measure(q, c) bloch_vector = ['x', 'y', 'z'] exp_vector = range(0, 2) circuits = [] print(len(exp_vector)) for exp_index in exp_vector: middle = QuantumCircuit(q, c) phase = 2np.piexp_index/(len(exp_vector)-1) middle.u1(phase, q) circuits.append(pre + middle + meas_x) circuits.append(pre + middle + meas_y) circuits.append(pre + middle + meas_z) print(len(circuits)) # Execute the circuit job = execute(circuits, backend = Aer.get_backend('qasm_simulator'), shots=1024) result = job.result() # trying to plot histogram counts = [] # count = np.zeros([0]) for x in range(len(circuits)): # count[x] = result.get_counts(circuits[x]) count = result.get_counts(circuits[x]) counts.append(count) plot_histogram(counts , legend=['1','2','3','4','5','6']) print(counts) No output shows for your code as you have a line underneath the call to plot_histogram(). This should be the last line of the section in the Jupyter notebook if you would like the image to be displayed. I was able to run your code by removing the final line (print(counts)) and it displayed the histogram below.	2020-04-02 01:00:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21340395510196686, "perplexity": 7171.751318795661}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370506477.26/warc/CC-MAIN-20200401223807-20200402013807-00196.warc.gz"}
http://tex.stackexchange.com/questions/35319/a-boxed-alternative-with-minimal-spacing	# A \boxed alternative with minimal spacing? The following code comes from this post. The problem is that I would like a way to automate the size of the box so as to be very small with a few of spacing, but the following solution doesn't work with fractions for example. \documentclass[border=5pt]{standalone} \usepackage{tikz} \usetikzlibrary{calc,shapes} \newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};} \newcommand{\DrawBox}[1][red]{% \tikz[overlay,remember picture]{ \draw[#1] ($(bl)+(-0.2em,0.9em)$) rectangle ($(br)+(0.2em,-0.3em)$);} } \newcommand{\MyBox}[2][red]{\tikzmark{bl}#2\tikzmark{br}\DrawBox[#1]} \begin{document} Lorem ipsum dolor sit amet. \MyBox{Lorem ipsum} dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. \MyBox{$\frac{\frac{2}{5} - 7}{4 + x^3}$} Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum \MyBox[blue]{$3.29 \times 10^{29}$} dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem \MyBox[draw=red,fill=yellow!20,,opacity=0.3]{$3.29 \times 10^{29}$} ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. \end{document} - Do you need the answer to be TikZ-based or could a different approach be acceptable? – Gonzalo Medina Nov 18 '11 at 18:39 Any kind of flexible solution is accepted. – projetmbc Nov 18 '11 at 20:56 Related Question: drawing boxes around words – Peter Grill Nov 28 '11 at 17:20 Related Question: mdframed: size frame to content. – Peter Grill Dec 30 '12 at 18:44 With the help of the xparse package, and using a \fboxcolor, I defined a \MyBox command which has 5 optional arguments and a mandatory one; the syntax is \MyBox[<frame color>][<fill color>]{<contents>}[\fboxsep value][<box depth>][<box height>] The command is very flexible; you can change the colors (for both the frame and the fill), the value for \fboxsep used; and gives you additional independent control over the box height and depth (this command is inspired in Herbert's answer to the question you linkd to). The code and an example (I used some exaggerated values in some cases just to show the flexibility of the command): \documentclass{article} \usepackage{xparse} \usepackage{xcolor} \newbox\FBox \NewDocumentCommand\MyBox{O{black}O{white}mO{0.5pt}O{0pt}O{0pt}}{% \setlength\fboxsep{#4}\sbox\FBox{\fcolorbox{#1}{#2}{#3\rule[-#5]{0pt}{#6}}}\usebox\FBox} \begin{document} Lorem ipsum dolor sit amet. \MyBox{Lorem ipsum} dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. \MyBox[red]{$\frac{\frac{2}{5} - 7}{4 + x^3}$} Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem \MyBox[red!30][yellow!10]{$3.29 \times 10^{29}$} ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem \MyBox[olive][red!40]{$3.29 \times 10^{29}$}[8pt] ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem \MyBox[olive][red!40]{$3.29 \times 10^{29}$}[4pt][2pt][33pt] ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem \MyBox[olive][red!40]{$3.29 \times 10^{29}$}[4pt][23pt][2pt] ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. \end{document} - That's great even if I don't understand all of that small code. – projetmbc Nov 19 '11 at 10:12 \newcommand\MyBox[2][red]{\mbox{\tikzmark{bl}#2\tikzmark{br}\DrawBox[#1]}} but the vertical spacing is still wrong. However, tikz is not the right tool for this problem. You already got other solutions which also can have colored frames. - I completely agree; using TikZ here is not the best approach. – Gonzalo Medina Nov 18 '11 at 18:52 @GonzaloMedina/Herbert: Am curious as to what is wrong with a tikz solution? Besides the package overhead, is there some other reason you are thinking about? If you are already using tikz, then there is no additional package overhead. – Peter Grill Nov 19 '11 at 0:24 Here is a enhanced version of the earlier solution provided at this earlier question about a boxed alternative with nicer spacing. This version measures the height and depth of the text that the box encloses. This two parameters \@DrawBoxHeightSep specify the separation to be applied to the height and depth. Here is the relevant section zoomed in: ## Notes: • This does require two runs: the first to compute the positions of the box, and the second to draw it in the correct spot. ## Known Issues: • This won't work if the text crosses line boundaries. If this is an issue and you are willing to consider highlighting instead of a box you should refer to Cool Text Highlighting in LaTeX. ## Code: \documentclass[border=1pt]{standalone} \usepackage{tikz} \usetikzlibrary{calc,shapes} \makeatletter \newcommand{\@DrawBoxHeightSep}{0.030em}% \newcommand{\@DrawBoxDepthSep}{0.025em}% \newcommand{\@DrawBox}[3][red]{%#1= style, #2=height, #3=depth \tikz[overlay,remember picture]{ \draw[#1] ($(bl)+(-0.2em,#2+\@DrawBoxHeightSep)$) rectangle ($(br)+(0.2em,-#3-+\@DrawBoxDepthSep)$);} } \newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};} \newdimen\@myBoxHeight% \newdimen\@myBoxDepth% \newcommand{\MyBox}[2][red]{% \settoheight{\@myBoxHeight}{#2}% Record height of box \settodepth{\@myBoxDepth}{#2}% Record height of box \tikzmark{bl}#2\tikzmark{br}\@DrawBox[#1]{\@myBoxHeight}{\@myBoxDepth}% Draw the box } \makeatother \begin{document} Lorem ipsum dolor sit amet. \MyBox{Lorem ipsum} dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. \MyBox{$\frac{\frac{2}{5} - 7}{4 + x^3}$} Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum \MyBox[blue]{$3.29 \times 10^{29}$} dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem \MyBox[draw=red,fill=yellow!20,,opacity=0.3]{$3.29 \times 10^{29}$} ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet. \end{document} - Thanks for this solution using TiKz. – projetmbc Nov 19 '11 at 10:12	2015-01-27 19:11:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7885912656784058, "perplexity": 3185.054374713088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115856041.43/warc/CC-MAIN-20150124161056-00202-ip-10-180-212-252.ec2.internal.warc.gz"}
http://www.unofficialgoogledatascience.com/2018/03/quicker-decisions-in-imperfect-mobile.html	Compliance bias in mobile experiments by DANIEL PERCIVAL Randomized experiments are invaluable in making product decisions, including on mobile apps. But what if users don't immediately uptake the new experimental version? What if their uptake rate is not uniform? We'd like to be able to make decisions without having to wait for the long tail of users to experience the treatment to which they have been assigned. This blog post provides details for how we can make inferences without waiting for complete uptake. Background At Google, experimentation is an invaluable tool for making decisions and inference about new products and features. An experimenter, once their candidate product change is ready for testing, often needs only to write a few lines of configuration code to begin an experiment. Ready-made systems then perform standardized analyses on their work, giving a common and repeatable method of decision making. This process operates well under ideal conditions; in those applications where this process makes optimistic or unrealistic assumptions, data scientists must creep from the shadows and provide new approaches. In the Google Play app store, challenges such as these regularly occur, as the basic framework of mobile technology introduces many wrinkles in experimentation and measurement. The Play Data Science team works to develop appropriate approaches to these cases and develops reusable methodologies to broaden the capabilities of Google Play experimenters at large. When we set a treatment and a control group, we typically assume that, instantaneously, each unit within the treatment experiences the treatment condition, and each unit within the control instead gets some manner of baseline condition. This assumption of total compliance allows us to make strong inferences about the impact of the treatment condition. However, there are common cases where this assumption could be broken, most notably in mobile. Naturally, this issue is of particular concern to us in the Play Data Science team. Suppose a developer of a game wants to release a new major version of the game. They have a lot of users already playing their game, and the developer is eager to see what effect the update will have on them. They have a clever thought to run an experiment, offer some users the new version with a popup notification within the game, and don’t tell the rest about it. Then, by comparing the two groups, they can see exactly what benefits (and problems!) they get with the new version. Once they proceed with this, immediately they run into problems. First, as soon as they offer the version, not everybody agrees to upgrade. Some users immediately rush to upgrade, others are slower because they rarely play the game, or have some technical trouble installing the update until days or weeks after the offer is sent. Some users even refuse to update, being happy with the current version and adverse to change. Even worse, some users in the control group manage to get ahold of the new version. Clearly, drawing conclusions from this messy experiment will require more sophisticated analysis. Abstracting this problem a bit, the total compliance assumption breaks down here --- some units assigned the treatment do not receive it immediately. Instead, a non-random subset of units receives the treatment, where the membership of a unit within this set is a function of covariates and time. In our example, users who rarely interact with the game will likely adopt the treatment more slowly, causing them to be underrepresented in the set of treated users in comparison to the population of users. Further, units in the control group may manage to receive the treatment, despite the assignment setup. In our example, this corresponds to sideloading, when a user obtains the new version of the game despite it not being offered. Given this situation, a natural response is this: why not just wait until all the users in the treatment have upgraded? Even if we ignore the sideloaders (users in the control who take the treatment), we can’t always take this path. Though sometimes we can wait until most users have completed the update to draw conclusions, we usually want to make inference quickly. For example, we might want to stop the process if we measure harmful effects early. We also might want to use early data to make a decision whether or not to release the treatment to more users, giving us more measurement power later on. In these cases, we must address the bias on the early data, when only a fraction of users who can eventually update their game have done so, and those that have are a non-random subset of the population. For example, perhaps users who have been historically more engaged with the game will update first. To summarize the rest of this post, we first characterize the main issues with these sorts of experiments. We then build our methods from a simple baseline method: intent-to-treat analysis. Finally, we expand our methods to include notions of which users are treated, and using the control to match users with similar upgrade probabilities. Characterizing the main issues The analysis here focuses on an experiment where a set of users is randomly assigned to one of two experiment arms:treatment or control. We observe for each user a set of units with covariates or user features $X_i$ and an associated response of interest $Y_i$. In our introductory example, this response might be the number of times a user opens the game each day. User features might be the country, device quality, and a bucketed measure of how often the user has played the game in a predefined period prior to the experiment. We are interested then in measuring the effect of the treatment on our response $Y_i$. We make the simplifying assumption that the impact of the treatment is instantaneous and does not further evolve over time. From here, the situation is complicated in a few ways, which we now explore in detail. Issue 1: the empirical mismatch between treatment assignment and experience Following our running example, after the game developer begins their experimental release they immediately observe that not all users assigned the treatment (the new game version) actually experience the treatment. Further, some users in the control group manage to experience the treatment. That is, users may have an actual experience that differs from their experiment arm assignment. To index these situations, we introduce two binary variables as follows: • $Q$ indexes the user’s experiment arm assignment • $Q=0$: the treatment is not assigned to the user; that is, they are the control group • $Q=1$: the treatment is assigned to the user • $U$ indexes the treatment experience: • $U=0$: user does not experience the treatment condition • $U=1$: user experiences the treatment We can track the users in the treatment group $(Q=1)$ to see how many are actually experiencing the treatment ($U=1$) at any given time. We can then produce graphs of this upgrade percentage over time similar to the following: Fig 1: Only a fraction of users in the treatment group adhere to the treatment. Some take a significant time to adhere. Issue 2: the users experiencing treatment are not a simple random subsample of the population A natural question that follows from the graph of user treatment adoption: what users are actually receiving the treatment here? Typical factors that could influence update speed might be system properties like connectivity (better means faster upgrade), hardware quality (higher means faster), or core operating system software. More user-based covariates could be factors like country or current frequency of use of the product. From the point of view of making valid statistical inference, we can assess if the mix of units who have actually received the treatment reflect the overall population. Loosely put, the degree to which the treated units represent a simple random sample of the population has implications for our ability to draw generalizable conclusions from our experiment. One way to assess this is compare the set of treated units within the treatment group ($Q=1$; $U=1$) to all units assigned to the treatment ($Q=1$). Since units satisfying $Q=1$ are a random subset of the overall population, this comparison will give us hints as to how effectively we can generalize our conclusions. To take our example, suppose we are able to measure how engaged each user has been with the game over the past month. We then can bucket these data into six groups, giving a spectrum of engagement. We can then compare the distribution over these buckets for users with the new version to the entire treatment group: Fig 2: Treatment units experiencing treatment are rather different from the population as a whole. This is shown here for one particular dimension, usage. It is clear from this plot that the two distributions do not match. Further, we see some expected discrepancies: users who have been more highly engaged in the game are more likely to actually experience the treatment. Perhaps they are more enthusiastic to upgrade, or perhaps this covariate is correlated with other more impactful covariates such as the quality of the user’s hardware. In any case, it is clear that we cannot make inference without some caution in this case. Issue 3: the need to make a timely decision Following our running example, after the game developer begins their experimental release they want draw conclusions about its impact as soon as possible. As mentioned previously, we could wait to do the analysis until all the users who may eventually upgrade actually, but this is typically impractical. However the strategy of waiting until all (who will comply) have received the treatment is useful as a ground truth for evaluating our methods in the following. For each method, we can compare the conclusions we can draw at the beginning of the process to those we would get at the end. We can then choose the methods where these two conclusions match, or at least where the first is a more useful nuanced view. We adopt the following notation and assumptions to make the time component of the problem clear: • We build our models at two time points: $t=T_{\mathrm{measure}}$, and $t=T_{\mathrm{final}}$. • $T_{\mathrm{measure}}$ represents a time point at an early stage, where we would do our experiment analysis in a ‘real’ situation. • $T_{\mathrm{final}}$ represents a later time point, where virtually all of the users who eventually may upgrade have done so, and is used to benchmark the performance of the models and estimates produced at $T_{\mathrm{measure}}$ • We assume that the effect of the treatment does not evolve over time. This allows us to compare the two results directly. This is often an unrealistic assumption in applications. For example, users of a game will probably behave differently across days of the week, and their behavior will evolve as they learn the game through experience. We leave it aside in this post so we can clearly explain and explore the remaining issues. We can visualize some typical values of $T_{\mathrm{measure}}$ and $T_{\mathrm{final}}$ by annotating the adoption figure given above as follows: Fig 3: The above figure illustrates a typical case: we can afford to wait a short amount of time for a reasonable percentage of users to adopt before doing analysis ($T_{\mathrm{measure}}$). In order to get more comprehensive adoption ($T_{\mathrm{final}}$), we would have to wait a significantly longer time. Intent to Treat (ITT) and Treatment on the Treated (TOT) analysis With the main issues characterized, we now turn to analysis methods for the application. Before we begin, we should state our overall goal estimand, which is the expected effect of the treatment on the average single unit: $$\theta = E(Y\|U=1) - E(Y\|U=0)$$ In the language of counterfactuals, this gives the difference between the outcomes under the two different treatment conditions. In a simple experiment, we assume that the random assignment to experiment arms is enough to give us a reasonable estimate of this quantity from standard methods, which rely on each group being a simple random sample from the population over which we wish to make inference. A simple baseline to analyze this kind of experiment is an Intent to Treat (ITT) approach. The intent to treat estimand measures the effect of assigning the treatment to a user:$$\theta_{\mathrm{ITT}} = E(Y \| Q=1) - E(Y\|Q=0)$$ That is, we simply compare the two experiment groups on the basis of their treatment status assignment, rather than their actual experience of the treatment. To estimate this quantity, we could compare the mean value of a metric Y between treatment and control in the standard way:$$\hat{\theta}_{\mathrm{ITT}} = \frac{1}{N_{Q=1}} \sum_i Y_i[Q_i = 1] - \frac{1}{N_{Q=0}} \sum_j Y_j[Q_j = 0]$$ where $Y_i$ refers to the measured outcome for a single unit, $N_{Q=1}$ and $N_{Q=0}$ are the number of units measured in the treatment and control group, respectively. $[ \cdot]$ evaluates to $0$ or $1$ depending on whether the boolean expression within evaluates to false or true (Iversonian notation). This method has a few weaknesses we can anticipate from the previous general assessment of the treatment group. A basic issue is that since many units in the treatment do not actually experience the treatment, we would expect that for such units, there is no impact of the treatment. This would effectively shrink our estimates of the effect towards a null point. To refine the analysis, we could focus on estimating the effect of the Treatment on the Treated (TOT). That is, we consider only the units in the treatment group that actually received the treatment to the entire control group. We could adjust for this in a simple way by scaling our estimated effect by the fraction of impacted units as follows:$$\hat{\theta}^*_{\mathrm{TOT}} = \left( \frac{1}{N_{Q=1}} \sum_i Y_i [Q_i=1] - \frac{1}{N_{Q=0}} \sum_j Y_j [Q_j=0] \right) \frac{N_{Q=1}}{N_{Q=1, U=1}}$$ Here, we introduce the additional notation $N_{Q=1, U=1}$, which represents the number of units that received the treatment within the treatment group. This estimator adjusts for the gross fraction of users who actually receive the treatment. A more direct estimator of the TOT effect slices the treatment group:$$\hat{\theta}_{\mathrm{TOT}} = \frac{1}{N_{Q=1,U=1}} \sum_i Y_i [Q_i=1 \cap U_i=1] - \frac{1}{N_{Q=0}} \sum_j Y_j [Q_j=0]$$ We can now do our first evaluation of these methods, by comparing ITT and TOT estimates computed during the beginning ($T_{\mathrm{measure}}$) and end ($T_{\mathrm{final}}$) of the observation period. The following figure displays these results: Fig 4: How the ITT and TOT estimates evolve over time. Here, we see that the ITT method performs poorly; it estimates quite a different effect at the beginning ($t=T_{\mathrm{measure}}$) than at the end ($t=T_{\mathrm{final}}$) of the period. This is likely because the number of users experiencing the treatment is increasing over time, and so the earlier estimate is shrunk more strongly towards zero. The TOT method performs somewhat better in terms of stability, but the estimate declines between the two time points. The differences between the distribution of users experiencing the treatment and the population are likely to be a key factor here. Indeed, both of these estimators do not well estimate the treatment effect for all users if this set of treated users is not a random subset of the population, and if the covariates that differ between this subset and the population are also correlated with our outcome and the treatment effect. From our earlier analysis, we can see that this is not the case by comparing the distribution over one categorical covariate. To proceed, we fully characterize the types of units that are more likely to adopt the treatment at an earlier stage in our experiment. Compliance Bias A central issue in this application is that users assigned treatment sometimes do not actually experience the treatment at $T_{\mathrm{measure}}$, and furthermore this set of users is not random. Here, we can draw a direct analogy to Compliance Bias, which is primarily described in literature on the analysis of medical studies. This type of bias can occur when users do not adhere to their assignment to an intervention plan, for example when patients with less acute disease symptoms more often refuse to take a drug they were given. To make this issue precise, we expand our language of potential outcomes, and introduce a set of four potential outcomes for each unit accounting for both availability of the treatment and the actual application thereof. To index these situations, we combine our two binary variables $Q$ for assignment and $U$ for treatment experience to give the following table of potential outcomes: Potential outcome index (Q, U) Description $(0, 0)$ Control group user not experiencing treatment $(0, 1)$ Control group user experiencing the treatment $(1, 0)$ Treatment user not experiencing treatment $(1, 1)$ Treatment user experiencing treatment In the context of this application, these rows have already so far been roughly explored. Users in the control are expected to behave as $(0, 0)$ indicates (not offered the upgrade, don’t take the upgrade), and users in the treatment group may be in state $(1, 0)$ or $(1, 1)$. Users realized in the case $(0, 1)$ may seem impossible or surprising, as they represent a sort of leakage of the treatment condition into the control group. These correspond to sideloading behavior, where a user obtains the update without an offer, possibly through internet backchannels, which was discussed above. It further helps to map out the full potential outcomes for each unit. That is, for a fixed unit, what are the pair of potential outcomes we might see if we vary the group assignment Q? This gives the following table (see [1]): Potential outcomes for unit over groups (Q, U) ; (Q’, U’) User Type Rough Description $(0, 0); (1, 0)$ Never-taker User that will never upgrade $(0, 0); (1, 1)$ Complier User that will upgrade if offered $(0, 1); (1, 0)$ Defier User that will avoid upgrades if offered, seek them if not offered $(0, 1); (1, 1)$ Always-taker User that will seek out the upgrade in all conditions (sideloading) While they may exist in theory, do we have units of each of these types? In our application, never-takers (units who can never execute an update) and compliers (units who will upgrade if given the chance) seem reasonably common. Note here that we evaluate $U$ at $T_{\mathrm{measure}}$, so never-takers here are those who would not experience the treatment at $T_{\mathrm{measure}}$, regardless of assignment $Q$. We will ignore always-takers and defiers, since in our application the number of users with realized $(0, 1)$ outcomes are extremely rare ($\ll 1\%$). This implies our population consists overwhelmingly of compliers and never-takers. This greatly simplifies our situation, and makes full conditional observable or conditional compliance modeling approaches unnecessary [1]. Conditional compliance models estimate causal effects for each of the four types of users in the table. Conditional observable models try to estimate relationships between all four counterfactual quantities for each user. Propensity scoring within the treatment We now explore statistical strategies for estimation that account for the difference in users who experience the treatment. A starting idea is to analyze the treatment alone as an observational study (an analysis on $Q=1$ only). Here, we ignore any control group, and analyze the treatment group as a self-contained observational study for units where $Q=1$. We attack this via propensity modeling, using $U$ as the new ‘treatment’ variable and reducing our set of potential outcomes to $\{Y(1, 1), Y(1, 0)\}$. In this case, we fit the following logistic regression:$$\mathrm{logit}( \Pr(U \| X, Q=1) ) = X \beta$$We then estimate the effect using reweighting, matching or stratification methods. Before proceeding to effect estimates, it is useful to examine the output of this model. At both the start and end of the study, we can produce a histogram of the propensity scores for $Q=1$, sliced by $U$, that is, the probability that a user within the treatment group will actually compete the upgrade (receive the treatment) Fig 5: Estimated probability of experiencing the treatment in the treatment group. Observe the subset unlikely to uptake. We see that a collection of never-takers immediately stands out with very low estimated scores, a clear conclusion even from the start of the study. These users can be safely discarded from our effect estimation analysis. Further, the existence of never-taker users calls into the question of the ITT analysis, even at $T_{\mathrm{final}}$. If there are users who will never experience the counterfactual treatment state, then ITT will never estimate the difference between these states if those users are included in the estimation. Otherwise, the model produces a decent range of $\Pr(U=1 \| X)$ predictions, and we can see from the plot that the number of users with $U=1$ are more common at higher probabilities. As a classifier, the model gives merely decent performance, which is actually advantageous for propensity methods. If in this case we instead had a clean separation of the two classes for all users, this would imply that certain user factors completely determine adoption. Therefore, within the treatment group, many users with $U=0$ would be without a peer user with $U=1$, so we could not reasonably estimate our target effect. We then would not be able to generalize our conclusions to the entire user population. This problem would be somewhat mitigated if only a subset of users have estimated probability $0$ or $1$, and we were therefore able to understand clearly for which users we cannot estimate the effect / cannot find matching users. For estimation, we consider stratification or bucketing methods. That is, we take our range of estimated propensity scores, and partition them into buckets. We then perform analysis within each bucket, and collect the within-bucket estimates to form an overall estimate. Since the scores are estimated from a model depending on many covariates, this bucketing has the effect of partitioning users based on their covariates, reduced to a single univariate measure: $\Pr(U=1 \| X)$. We adopt a simple form of stratification where we partition the range $0$ to $1$ into twenty $0.05$-width strata $S_k$ for $1\leq k \leq 20$. Let $\hat{p}_i$ be the estimated propensity score for unit $i$. We can compute a propensity weighted comparison by users whose scores fall into the $k$th stratum $S_k$ as follows: $$\hat{\theta}_{\mathrm{p-weight}, k} = \frac{\sum_i \frac{Y_i}{\hat{p}_i} [\hat{p}_i \in S_k \cap Q_i=1 \cap U_i=1] }{\sum_i \frac{1}{\hat{p}_i}[\hat{p}_i \in S_k \cap Q_i=1 \cap U_i=1]} - \frac{\sum_i \frac{Y_i}{1-\hat{p}_i} [\hat{p}_i \in S_k \cap Q_i=1 \cap U_i=0] }{\sum_i \frac{1}{1-\hat{p}_i}[\hat{p}_i \in S_k \cap Q_i=1 \cap U_i=0]}$$ We can then collect these into a single estimate by taking a weighted combination of these strata comparisons, where $N_k$ is the number of users falling into the $k$th stratum:$$\hat{\theta}_{\mathrm{p-weight}} = \frac{ \sum_k N_k \hat{\theta}_{\mathrm{p-weight}, k}} {\sum_k N_k}$$ It is useful to examine the results per strata, as the following plot does. Here, we compare the results on users at both $T_\mathrm{measure}$ and $T_\mathrm{final}$, where the strata and $\hat{p}$ are defined by the model fit at $T_\mathrm{measure}$ only. Fig 6: Propensity scores estimated for each stratum. For the most part, the estimates are in good agreement, but differ significantly in the high propensity strata. We first see that the model performs well for the lower strata (e.g. $(0.45, 0.5]$), in that the estimate at $T_\mathrm{final}$ is close to that made at $T_\mathrm{measure}$. The approach performs worse for higher buckets, which is expected as there are fewer users with $Q=1, U=0$ here. A more striking overall point is that the effect is not uniform across the buckets. In fact, the impact of the update increases as our estimate of $\Pr(U=1 \| X)$ increases. This means that for different users, the new game has a different effect. This is a valuable insight that the ITT and TOT approaches do not provide, as their estimand assumes that the treatment effect is a universal mean shift across all users. Propensity score matching to the control Another approach is to leverage the control group along with our propensity scores. With a control group, we have access to many users for whom we (mostly) observe $Y(0, 0)$ for the entire range of estimated propensity scores. In contrast, in our within treatment approach, there are fewer users with realized outcome $Y(1, 0)$ as the estimated propensity score increases. After fitting a propensity model to the treatment group, we can estimate the probability of each member of the control group experiencing treatment by assuming that $\Pr( U =1 \| X, Q = 0) = \Pr( U =1\| X, Q = 1)$ and $Y(0, 0) = Y(1, 0)$. With a $\Pr(U=1 \| X)$ available in for each unit, we can now perform some form of matching, either exact or stratified, between units and take paired differences between the resulting groups. To obtain an estimate of $E(Y(1, 1) - Y(1, 0))$, we select only groups containing units receiving the treatment in the treatment group and compare them as follows:$$\hat{\theta}_{\mathrm{p-match}, k} =\frac{\sum_i \frac{Y_i}{\hat{p}_i} [\hat{p}_i \in S_k \cap Q_i=1 \cap U_i=1] }{\sum_i \frac{1}{\hat{p}_i}[\hat{p}_i \in S_k \cap Q_i=1 \cap U_i=1]} - \frac{\sum_i \frac{Y_i}{1-\hat{p}_i} [\hat{p}_i \in S_k \cap Q_i=0 \cap U_i=0] }{\sum_i \frac{1}{1-\hat{p}_i}[\hat{p}_i \in S_k \cap Q_i=0 \cap U_i=0]}$$ This approach has the distinct advantage in comparison to our ‘within treatment’ stratification analysis pool of users without the treatment at all time points for all strata. Again, we can plot the performance at both the start and end of the study: Fig 7: Analogous to Fig 6, but with better agreement in the high propensity strata The results here are similar to the previous approach in the lower propensity score buckets. The main improvement comes at the higher buckets, where the estimates at $T_\mathrm{final}$ and $T_\mathrm{measure}$ are now as close as in the other buckets. The core reason for this improvement is that in this approach, we have many users available in the control ($Q=0$) with similar user features as those in the treatment ($Q=1$) that would produce higher propensity scores. In contrast, the $Q=0$ users have primarily $U=1$ users with high propensity scores, leading to poor estimates in these strata. Fig 8: As expected, propensity matching is more consistent over time than propensity weighting Conclusion Experiment analysis often cannot rely on the assumption of faithful adoption of the treatment condition. Here, we’ve explored a case where many users assigned the treatment do not actually experience the treatment for a long time period after the beginning of the experiment. Moreover, waiting until a steady state of treatment adoption to draw inference is often impractical, so we have to make do with a biased early subset of users. As we’ve shown, adjustments are possible, but a litany of assumptions and concerns must be dealt with. Several complexities are left unaddressed here, such as effects that evolve over time or large volumes of users falling into ‘defier’ or ‘always-taker’ categories that require further refinements of approaches. Nonetheless, propensity based models often provide insightful refinements to the basic ITT or TOT approaches, and would form the basis for methods that would address these complexities. References [1] Have, Thomas R. Ten, et al. “Causal Models for Randomized Physician Encouragement Trials in Treating Primary Care Depression.” Journal of the American Statistical Association, vol. 99, no. 465, 2004, pp. 16–25. JSTOR, JSTOR, www.jstor.org/stable/27590349.	2020-07-04 21:34:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.541681706905365, "perplexity": 648.2856736933302}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886706.29/warc/CC-MAIN-20200704201650-20200704231650-00581.warc.gz"}
https://stacks.math.columbia.edu/tag/0EWN	Lemma 54.97.1. With notation as above, if $K$ is in the essential image of $R\epsilon _*$, then the maps $c^ K_{X, Z, X', E}$ of Cohomology on Sites, Lemma 21.26.1 are quasi-isomorphisms. Proof. Denote ${}^\#$ sheafification in the h topology. We have seen in More on Flatness, Lemma 37.35.7 that $h_ X^\# = h_ Z^\# \amalg _{h_ E^\# } h_{X'}^\#$. On the other hand, the map $h_ E^\# \to h_{X'}^\#$ is injective as $E \to X'$ is a monomorphism. Thus this lemma is a special case of Cohomology on Sites, Lemma 21.29.3 (which itself is a formal consequence of Cohomology on Sites, Lemma 21.26.3). $\square$ In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	2018-12-18 20:52:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9937995076179504, "perplexity": 2155.6259591312682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829812.88/warc/CC-MAIN-20181218204638-20181218230638-00627.warc.gz"}
http://grouper.ieee.org/groups/802/3/10G_study/email/save/msg05391.html	# Re: [802.3ae] 10GBASE-X PCS; status register definition? "THALER,PAT (A-Roseville,ex1)" wrote: > > Ed, > > Comment 126 requested that such a mapping be added to Clause 48 and my > recollection is that the comment was accepted. Therefore, there should be no > need for a recirculation comment. > > For Clause 51 and for the PMA functions in Clause 48, there are no state > machines and the ability to detect synchronization is an implementation > dependent function which is why there is not a mapping. Possibly one could > add a statement that if the optional sync_err signal is implemented, the > state of the management bit should be the dependent on the state of > sync_err, though it is not clear to me that it is necessary to do so. sync_err is not even optional for clause 48/53; it just doesn't exist (except in implementor's heads :)). Even if no reference is necessary, the net effect appears to be that of a mandatory management bit that doesn't have to go anywhere; the ability to detect synchronisation is not mentioned at all in C48. PMA_SIGNAL.indicate is only mentioned in C49. What if I get sync on lanes 0-2 but not lane 3? What should the bit value be? > > Regards, > Pat > Gareth > -----Original Message----- > From: Ed Turner [mailto:[email protected]] > Sent: Wednesday, January 23, 2002 5:30 AM > To: IEEE HSSG > Subject: Re: [802.3ae] 10GBASE-X PCS; status register definition? > > Gareth, > > You are correct to highlight this and are not failing to spot a reference, > the definition of receive link status has not been mapped explicitly to any > primitives (or variables). > Management is pervasive throughout the PHY and the MDIO register bits do not > necessarily have to map directly to any primitives or variables. > In earlier versions of the draft, there was an additional register with > lane-by-lane bits for synchronization and a global bit when all lanes were > synchronized. The receive link status bit was defined as a latching > reflection of this global sync bit. This lane-by-lane register was > (correctly) removed since the synchronization function is part of the PCS > for 10GBASE-X rather than the PMA. > There would be less ambiguity if we were to map this bit directly to some > primitive or variable and reference out to Clauses 51 and 48. The question > is how we do it. As Pat said in her e-mail yesterday, this would have to be > a re-circ comment, but there's no change against which to comment. It may be > stretching the definition of an editorial comment to make this change to > Clauses 45 and 48. > I would also be interested in hearing the views of the Clause 51 and 48 > people. > > Regards > Ed > (Clause 45 editor) > > Gareth Edwards wrote: > > > All, > > > > I'm looking for clarification on how the PMA/PMD management register > > 1.1.2, "Receive Link Status" should behave when the PHY instance is a > > 10GBASE-X PCS/PMA. The specification describes it thus: > > > > \begin{quote} > > When read as a one, bit 1.1.2 indicates that the PMA is locked to the > > received signal. When read as a zero, bit 1.1.2 indicates that the PMA > > be implemented with latching low behavior as defined in the introductory > > text of 45.2. > > \end{quote} > > > > which I guess is aimed at the optional sync_err signal on the XSBI for > > the clause 49 PCS and clause 51 PMA. Thing is, it's not explicitly > > mapped to any similar signal (or should I say primitive) on the > > 10GBASE-X PCS/PMA boundary, nor is it stated how it should relate to the > > state of PMA lock of each and any of the 4 PMA lanes. > > > > Does the draft need to be refined at this point? Or am I just failing to > > spot the reference? > > > > Cheers > > Gareth > > > > -- > > / /\/\ Gareth Edwards mailto:[email protected] > > \ \ / Design Engineer > > / / \ System Logic & Networking Phone: +44 131 666 2600 x234 > > \_\/\/ Xilinx Scotland Fax: +44 131 666 0222	2019-01-21 08:33:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5204257369041443, "perplexity": 6051.954897842303}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583763839.28/warc/CC-MAIN-20190121070334-20190121092334-00276.warc.gz"}
https://codereview.meta.stackexchange.com/questions/6538/how-do-we-feel-about-canonical-post-for-repeat-review-concepts?noredirect=1	# How do we feel about canonical post for repeat review concepts? [duplicate] I had started to write a question where I felt I was just going to repeat myself in reviews. Well I've been in the city since I was two and I certainly wouldn't say that I was stuck in a rut... stuck in a rut ... stuck in a rut... stuck in a rut... There are things that I have been seeing in several questions where my suggestion would be the same. I am tempted then to make, and save, some custom info blocks that I can use. I don't however want my answers to just be copy pastes of myself (code biting so to speak). I then saw in the similar post sections My reviews feel like a broken record which then led me to Justify your review!. The accepted answer put forward the idea for make a canonical for such concepts that might be repeated. The suggested conditions of which I will re-post here with a minor change: • Find a good looking question which uses using <>; • Post an answer to this question which only addresses this specific issue. The answer should be very detailed, in-depth, and specific. When new questions are posted with a bad use of using <>, answers can comment: Don't use using <> and have a link pointing back to this canonical, in-depth, detailed explanation on why using <> is bad. Where <> could be a poor coding style, unnecessary use of tools / resources etc. If it matters I was specifically thinking about the use of backticks as line continuation characters. While they are fine and have their place they do not always need to be used and have much cleaner and friendly alternatives. So getting to the point. What does the community think about canonical posts like this. Assuming that they can stand alone and meet the sites standards? I have no strong desire either way but since I am just getting started I would just like to get a feel of where to go.	2019-06-18 00:55:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3840453326702118, "perplexity": 974.1566991465425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998600.48/warc/CC-MAIN-20190618003227-20190618025227-00015.warc.gz"}
https://www.orbiter-forum.com/threads/external-mfds-for-orbiter.26055/	# External MFDs for Orbiter #### kamaz ##### Unicorn hunter 2 cheap Android tablets: 100 EUR 2 Thrustmaster Cougar MFDs: 80 EUR 2 weeks programming the add-on: 0 EUR 2 working Orbiter MFDs: priceless (Details will be posted during the weekend. I must get some sleep now.) #### Attachments • cougar.jpg 127.6 KB · Views: 1,163 • cougar2.jpg 111.2 KB · Views: 1,243 Last edited: #### yagni01 Donator Look forward to seeing your approach. :thumbup: #### Mojave ##### Doug Uses Light Theme... Moderator whoa, those looks absolutely neat. Good luck on your venture. #### kamaz ##### Unicorn hunter I will post the display code first, and discuss intefacing the Cougars in the next part in a few days. This is because the Cougar-related code requires a major cleanup. (Hey, I hacked it in one evening on the day I got the Cougars!) But, first of all, I will document how I came to the solution I currently use. So, let's discuss the grand question of simpit building: how to make a dedicated MFD display? One step in the right direction is using ExtMFD. ExtMFD allows you to open your MFD in an external window. Unfortunately, ExtMFD suffers from a focus problem, meaning that if you click mouse on the MFD button, the window focus will switch from the main Orbiter window, so your keyboard input will go to the MFD window, instead of the main window. So you loose control of the ship until you click your mouse on the main window. Fortunately there are some ways to solve that and so you can use a second monitor to display MFDs like that (and even glue the Cougars to the screen for added realism). But, that approach requires fitting a second extra monitor in your setup -- and, a PC monitor is too large for me. So, not good. At that point, I must mention ExternalMFD/Orb::Connect. ExternalMFD is a program running on a separate computer, which queries the Orb::Connect server running in Orbiter, and displays the parameters it reads. A very cool idea, but it requires reimplementing each and every MFD on the client side. So, no TransX or IMFD for you. So, not good. So, I went back to the idea of adding an extra small screen to my PC to just display the MFDs. I looked at using the MIMO USB touch screens together with ExtMFD. That would work, but MIMOs are a bit expensive (140 EUR + 19% VAT each at minipc.de). So, I went looking for a cheaper 7" or 8" display... ...and I didn't find any. I mean, yes, you can buy 7-8" automotive VGA display, but these babies cost around 300 EUR. Each. (Unless you can get them in China yourself.) So not only this is more expensive than MIMOs, you also have two add extra VGA outputs to your PC, which is a major headache. So, still no good. Then I realized that my local MediaMarkt is well-stocked in cheap 7" displays -- in the form of low-end Android tablets (Lark FreeMe 70.0 at 50 EUR each)! So, I thought that I will export the MFD over network and use a cheap tablet as a display device. The first idea was to use WebMFD. However, as aptly documented in the WebMFD thread WebMFD has its own issues -- notably, it's very picky about which browser is used. Some work, others don't. And of course, the browser included in my Lark would not work. Oh well. But, WebMFD served as important inspiration. I realized that a similar solution exists: instead of using HTTP, export the MFD screen using the VNC remote desktop protocol. Then, you can use any client device which can run a VNC client -- and there is a multitude of these (including mobile phones). Next, since the tablet has a touchscreen, and VNC can obviously send mouse clicks, then I have the button input problem solved as well! Simply draw buttons around the actual MFD screen area, export the whole thing over VNC and check where the user clicks. In other words, ExtMFD over the network. And so, VNCMFD was born. #### Hielor ##### Defender of Truth Donator Beta Tester Very cool. What do you end up doing with the extra buttons on the Cougar units, and the fact that they only have 5 buttons on each side? I agree that the RemoteMFD requiring re-implementing all MFDs was kind of what killed the project in the end. There's definitely tons of room for improvement in some of the default MFDs, but rewriting things like IMFD is kind of infeasible. One thing I considered was some kind of host process that can load MFD dlls on a second computer and provide the right functions via Orb::Connect, but that's also kind of difficult and basically forces you into using Windows computers, which aren't very cheap. The presence of modern super-cheap tablets makes those tempting, and it's good that you've found a way to do that. Is there a list of browsers supported by WebMFD somewhere? Seems like the best solution might be to find a cheap tablet that supports that... #### kamaz ##### Unicorn hunter VNCMFD - export MFD over the network using the VNC protocol VNC is a widely used remote desktop protocol. This addon creates starts a separate VNC server for each MFD. The first server is created on port 5900, the second on 5901, and so on. Additional servers can be started by invoking the add-on via Ctrl-F4 (or setting MAX_DISPLAYS in VNCMFD.cpp). Once the server is started, you can use any VNC client to display the MFD panel (screen and surrounding, clickable buttons). The add-on is covered by the GPL license. The VNC server implementation uses GPL'd LibVNCserver library. The code has actually been sourced from vnccast, which included a version of linvncserver hacked to compile on windows and conveniently packaged as a Visual Studio project. A logfile named VNCMFD.log is created in Orbiter's main directory. The panel size and layout is currently hard-coded (see VNCMFD.cpp and MFDPanel.cpp). The panel size is 480x480, with a 400x400 MFD display area to match an 800x480 tablet screen. An extra button is provided which sets the refresh rate of the screen. This allows you to balance each MFD's responsivity vs. Orbiter framerate. The MFD image is sent using progressive scan. Meaning, at 1Hz refresh rate, 1/10th of the screen is sent every 100ms. (If progressive scan is not used, and the MFD screen is copied all at once, main window animation is not smooth. This is because doing BitBlt() on the MFD screen globally locks the renderer until the image copy (BitBlt()) is done. The problem is particularly visible with the inline graphic client. This is the same issue that ExtMFD and WebMFD run into.) Developed and tested on Orbiter 2010P1 (clean install). Tested graphic clients: both inline and D3C9Client work. Tested VNC clients: UltraVNC on PC and android-vnc-client on Android. In principle, any VNC client should work. Important notice: the add-on creates a network server with no security of any kind. There is no authentication or encryption of VNC session. And old VNC code is used, which can have remotely exploitable bugs. Please use a suitably configured firewall! Both source and compiled DLL are included. Built with VC++ 2010 Express. #### Attachments • ultravnc.jpg 302.5 KB · Views: 379 • nocougar.jpg 160.8 KB · Views: 279 • VNCMFD-rev15-noCougar.zip 2.1 MB · Views: 116 #### kamaz ##### Unicorn hunter Very cool. What do you end up doing with the extra buttons on the Cougar units, and the fact that they only have 5 buttons on each side? Remember that the basic code is VNCMFD. Input from Cougars is handled by an extra layer which is hacked atop VNCMFD. (Actually, the real reason I have Cougars is that the tablets I used have abysmal touchscreen... ) Since VNCMFD draws the buttons by itself (and decodes button presses), there is no problem with rearranging them so they match the button layout of the Cougar. So I have buttons 1-5 on the left, buttons 7-11 on the right, and the bottom row is 6, Hz, SEL, MNU, 12 (I do not need the PWR button ) The standard layout is on screenshots in post #6, whereas the Cougar layout is in post #1. One reason I want to rewrite the Cougar code before releasing it is that all that is hardcoded now, and I'd rather put that in a config file. So the panel layout can be user-configurable. This approach works when the MFD screen is displayed. It runs into a problem when selection screen or help screen is displayed (i.e. when you press "SEL" or "MNU") because then the button location does not match the descriptions. From the documentation, the button layout should be configurable via the MFDSPEC structure. So I plan to investigate that in detail. I currently have 5 unused top buttons on each MFD (plus 4 rockers). When I get to reworking the Cougar code, I think I will make them configurable. One idea is to use them to send Orb::Connect commands, so they could be used to control autopilots and such... Ideas on what to do with these unused buttons are welcome. Is there a list of browsers supported by WebMFD somewhere? Seems like the best solution might be to find a cheap tablet that supports that... Also, to make this clear. I actually did two projects: (1) make VNCMFD which converts a cheap Android tablet into a touchscreen MFD and (2) modify VNCMFD to use Cougar buttons instead of touchsreen. So, if touchscreen MFD is enough for you and you can get a tablet with a half-decent touchscreen, then you don't need Cougars -- just install VNCMFD from post #6 and you're set. #### csanders An extra button is provided which sets the refresh rate of the screen. This allows you to balance each MFD's responsivity vs. Orbiter framerate. The MFD image is sent using progressive scan. Meaning, at 1Hz refresh rate, 1/10th of the screen is sent every 100ms. (If progressive scan is not used, and the MFD screen is copied all at once, main window animation is not smooth. This is because doing BitBlt() on the MFD screen globally locks the renderer until the image copy (BitBlt()) is done. The problem is particularly visible with the inline graphic client. This is the same issue that ExtMFD and WebMFD run into.) Did you try that code I posted in the WebMFD thread? Just curious if it blew up or anything... #### kamaz ##### Unicorn hunter Did you try that code I posted in the WebMFD thread? Just curious if it blew up or anything... Not yet I was going to look into that, but then the shipment with the Cougars arrived and my attention shifted #### yagni01 Donator Tested VNC clients: UltraVNC on PC and android-vnc-client on Android. In principle, any VNC client should work. I noticed TightVNC has a Java-based client, which could help for any difficult non-windows client devices. #### Quick_Nick ##### Passed the Turing Test Donator I noticed TightVNC has a Java-based client, which could help for any difficult non-windows client devices. TightVNC has been a bit buggy for me trying to use android-vnc-client. I use RealVNC reliably. Generally I'm over wifi for speed and full quality. However, I don't believe Orbiter shows up this way. (don't want to give the wrong impression) I noticed this has little to do with this situation, since we're only talking about clients. Last edited: #### yagni01 Donator Very cool. What do you end up doing with the extra buttons on the Cougar units, and the fact that they only have 5 buttons on each side? I agree that the RemoteMFD requiring re-implementing all MFDs was kind of what killed the project in the end. There's definitely tons of room for improvement in some of the default MFDs, but rewriting things like IMFD is kind of infeasible. One thing I considered was some kind of host process that can load MFD dlls on a second computer and provide the right functions via Orb::Connect, but that's also kind of difficult and basically forces you into using Windows computers, which aren't very cheap. The presence of modern super-cheap tablets makes those tempting, and it's good that you've found a way to do that. Is there a list of browsers supported by WebMFD somewhere? Seems like the best solution might be to find a cheap tablet that supports that... We have the same concerns about driving a remote Orbiter facade to host MFDs, either with O:C or something new and more efficient (OMP-ish, perhaps). As much as our work on RemoteMFD was limited, it fulfilled a goal of my own simpit, which was customize the look of displays to more closely resemble modern avionics displays as much as possible and eliminate the 2D panels. Even the STS MEDS are so 80's looking. So I have a couple questions about the VNC version. Since it appears the buttons can be independently drawn, can multiple MFDs be 'stacked' so that the sel and mnu buttons of the top one can be place below alongside the bottom ones (I also don't need the PWR buttons)? Could this stacking accomodate combinations in different sizes/resolutions? MYgoal has changed from airliner (e.g. B737/A320) style to more like a regional carrier/bizjet (Bombardier Challenger 305) style. Picture the PFD/MFD from here http://wallpaper.goodfon.com/image/177658-3600x2400.jpg with edge buttons. Can the VNC client stitch together 'standard' mfds in this type of format? ---------- Post added at 09:49 PM ---------- Previous post was at 09:40 PM ---------- I currently have 5 unused top buttons on each MFD (plus 4 rockers). When I get to reworking the Cougar code, I think I will make them configurable. One idea is to use them to send Orb::Connect commands, so they could be used to control autopilots and such... Ideas on what to do with these unused buttons are welcome. I'm working on a version 2 of Orb:Connect for O2010p1; Is that something that would help you? Hadn't heard anyone was really using it, so posting it hasn't been on my list ---------- Post added 07-29-12 at 01:22 PM ---------- Previous post was 07-28-12 at 09:49 PM ---------- Copied the folders into my O2010p1 dir and when I tried to enable modules I now get popup with "The procedure entry point GetThreadId could not be located in the dynamic link library KERNEL32.dll The other modules it contains is DX9 client, Orb:Connect and XR2. Same problem in a clean instal and I see "Error loading module Modules\Plugin\|VNCMFD.dll (code 127)" in the orbiter.log Last edited: #### kamaz ##### Unicorn hunter Copied the folders into my O2010p1 dir and when I tried to enable modules I now get popup with "The procedure entry point GetThreadId could not be located in the dynamic link library KERNEL32.dll The other modules it contains is DX9 client, Orb:Connect and XR2. Same problem in a clean instal and I see "Error loading module Modules\Plugin\|VNCMFD.dll (code 127)" in the orbiter.log Ooops... GetThreadId() was introduced in Vista. You're on XP I guess? Okay, will try to hack around that tomorrow... ETA: Well, turned out to be easier than I thought. Please try the attached version. #### Attachments • VNCMFD-rev16-noCougar.zip 2.1 MB · Views: 44 Last edited: #### yagni01 Donator Ooops... GetThreadId() was introduced in Vista. You're on XP I guess? Indeed. #### kamaz ##### Unicorn hunter So I have a couple questions about the VNC version. Since it appears the buttons can be independently drawn, can multiple MFDs be 'stacked' so that the sel and mnu buttons of the top one can be place below alongside the bottom ones (I also don't need the PWR buttons)? Uh... can you draw the panel layout you want? Could this stacking accomodate combinations in different sizes/resolutions? In principle, yes. However, at the moment, panel/button geometry is hardcoded. Which is why I want to redo the panel layout logic. Can the VNC client stitch together 'standard' mfds in this type of format? The VNC client does not stitch anything. Actually, the VNC client is very dumb: it just displays the image sent from server. What I'm actually doing is that I make a bitmap, which has an MFD screen in the center surrounded by buttons and send all that to the client. When the user clicks mouse (touches) the panel, the client sends me back xy coordinates of the click. Then I look up which button these coordinates correspond to, and send the appropriate event to the Orbiter. The actual MFD code does not care where the buttons are; it just reads the events. So as far as MFD operation is concerned, button placement is completely arbitrary. I.e. you can make a layout which has 3 buttons above the screen and 12 buttons below the screen or whatever. The only problem is that Orbiter makes certain assumptions about button placement when in the "SEL" or "MNU" mode... #### yagni01 Donator You can look at this image of my flight deck using 2 instances of RemoteMFD each holding 2 displays http://imageshack.us/photo/my-images/291/remobemfdops2.jpg/, but essentially it means sending 2 mfds in one vnc window to make it "appear" to be a single display. Picture a surfaceMFD above and orbitMFD aligned vertically below, with the orbitMFD moved up to cover the upper PWR/SEL/MNU buttons, and without the second window bar. The "hidden" SEL button(s) for the surfaceMFD would be painted alongside the ones for the lower orbitMFD. Hope this was clearer than mud. #### kamaz ##### Unicorn hunter This can be done client-side. 1. Configure the server to send just the MFD screen image without rendering buttons. This functionality will be provided soon. 2. Write a client rendering two VNC framebuffers in one window producing the geometry you want (i.e. a client which instantiates a VNC viewer class twice in one window). There is a multitude of embeddable VNC client classes: - Java applet for web pages; - Java class; - SDL: http://www.ferzkopp.net/Software/SDL_vnc/ - MFC: http://www.pjtec.com/Products/GovVNC/index.htm - .NET: http://dotnetvnc.sourceforge.net/ 3. Then we just need some socket interface to transmit button presses and labels between the server and client. In principle, the server could render two or more MFDs into the same VNC server, but the code handling button presses is currently built on assumption that each MFD has its own VNC server. Last edited: #### kamaz ##### Unicorn hunter New version with Cougar support Attached is a new version of the plug-in (rev 25), with Cougar support included. Changes: - VNC servers keep running between simulations. No need to reconnect the client. - Panel layout and buttons are fully configurable via the VNCMFD.ini file. This file must be placed in Orbiter's root directory. See comments in the file for details. - Number of MFDs is set using the config file. There is no hard limit. - Thrustmaster Cougar MFD support is added. If you don't have Cougars, please use VNCMFD-noCougars.ini file instead (i.e. rename it to VNCMFD.ini). Have fun #### Attachments • VNCMFD-rev25.zip 2.1 MB · Views: 47 • transx_1280.jpg 123.1 KB · Views: 100 • iss_docking_1280.jpg 137.6 KB · Views: 104 #### SolarLiner ##### It's necessary, TARS. There is a way to put the input box into the VNC client ? Because, for example I'm on IMFD, setting target, and I must go on my computer to enter the target. In the case of a "Mission Central Command" addon witch takes your VNC support, this couldn't be a realistic way. #### kamaz ##### Unicorn hunter Note on API access and Orb::Connect support MFDs running under VNCMFD can be accessed using Orbiter's API calls, e.g. oapiOpenMFD() and remotely via Orb::Connect. Orb::Connect access to MFDs running under VNCMFD has been tested by me and it works, subject to one caveat below. Orbiter API requires MFD identifier to be passed to the function call. In case of MFDs embedded on spacecraft panels, these identifiers are known (0 for the left MFD, 1 for the right MFD). In case of MFD displays derived from ExternMFD class (including VNCMFD), these MFD identifiers are unpredictable 32-bit numbers (it is simply an object pointer cast to UINT, if you are really interested in such details). To my knowledge, there is no API function which can be used to obtain the list of identifiers of all MFDs currently running. For debugging and testing purposes, the MFD identifier can be obtained from the VNCMFD.log file, where a line like the following is written for each MFD instance: Code: 2012-07-31 23:56:14.025 Thread 4724 MFDPanel.cpp: 26:MFDPanel(1001)::MFDPanel Creating instance Id()=85388328 where 85388328 is the MFD identifier. Of course, this is no solution for programming. For this reason, VNCMFD.dll provides two extra DLL entry points, which allow other modules to obtain the identifiers in question. The export definition is: Code: extern "C" __declspec(dllexport) UINT GetMfdIds(void); extern "C" __declspec(dllexport) LPSTR GetMfdNames(void); Calling GetMfdIds() returns a NULL-terminated array of MFD identifiers. Calling GetMfdNames() returns a NULL-terminated array of corresponding MFD names. The following drop-in function can be used in another module to obtain the identifiers (no need to mess with linker settings): Code: UINT GetVncMfdIds(void) { HMODULE hMod = GetModuleHandle("VNCMFD.dll"); if ( ! hMod ) return NULL; if ( hMod ) { if ( ! lpfnGetMfdIds ) return NULL; return (UINT) lpfnGetMfdIds(); } } Here is usage example: Code: UINT ids = GetVncMfdIds(); UINT id = ids; while (id) { Log("Panel id=%d", id); id++; } ---------- Post added at 11:17 PM ---------- Previous post was at 10:37 PM ---------- There is a way to put the input box into the VNC client ? No. This is a limitation of the Orbiter core. MFDs call oapiOpenInputBox() to read user data, and oapiOpenInputBox() always renders the dialog box into the main window.	2021-10-23 14:46:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22711306810379028, "perplexity": 5206.879746782077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585696.21/warc/CC-MAIN-20211023130922-20211023160922-00687.warc.gz"}
https://www.queryhome.com/puzzle/10175/square-rhombus-stand-base-what-would-ratio-area-square-rhombus	If a square and a rhombus stand on the same base, then what would be the ratio of the area of square and rhombus? 457 views If a square and a rhombus stand on the same base, then what would be the ratio of the area of square and rhombus? posted Sep 29, 2015 Let the base of square and rhombus be a units . Square all sides equal ;;; Area of square=a^2;; let the height of rhombus , i.e perpendicular distance from the top edge to the base be h units and included angle be x then area of rhombus =ah=a^2sinx ratio = (a^2)/(a^2sinx) === 1/sinx and since 0<x<90 ;; 0<sinx<1;;; the ratio will always be greater than 1 ,,,, answer Sep 29, 2015 Similar Puzzles	2023-01-28 20:59:24	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8306747078895569, "perplexity": 1129.4258887448157}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499654.54/warc/CC-MAIN-20230128184907-20230128214907-00001.warc.gz"}
https://github.com/fgnass/node-dev/pull/50	# fgnass/node-dev ### Subversion checkout URL You can clone with or . Closed wants to merge 1 commit into from +20 −3 ### 2 participants • add livescript support(use nearly the same code as the code for coffeescript) • disable the notifications of growl.. not a good Idea, though. The problem I came across is that there are always too many icons stay in GNOME3, and here's my screenshot. I tested growl({sticky: false}) before but didn't work as expetcted. So I'm trying to diable it. But that's also a bad solution... jiyinyiyong add livescript support and use sticky:false in growl d47b11a Owner I added --hint=int:transient:1 to the notify-send call so that the notifications should no longer pile up: fgnass/node-growl@f7d592c Could try node-dev with my node-growl fork on your machine? If this indeed fixes the issue I'll send TJ a pull-request. Thanks. It don't pile up now. To be honest, I don't think it's that useful to pop up the notifications. Personaly I just keep my terminal in the corner on the screen. The notifications poped is not as useful as the restarting info in the terminal. So I think it's not the best idea to use poped up notifications. @fgnass removed the commit about growl.. now only to add support for LiveScript. This was referenced Closed Closed closed this	2015-05-28 10:33:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2665565013885498, "perplexity": 7459.497834013016}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929272.95/warc/CC-MAIN-20150521113209-00180-ip-10-180-206-219.ec2.internal.warc.gz"}
https://solvedlib.com/3-predict-the-products-of-each-of-the-following,385526	# 3. Predict the products of each of the following reactions show stereochemistry where necessary. If both... ###### Question: 3. Predict the products of each of the following reactions show stereochemistry where necessary. If both a major and a minor product are possible, draw only the major product (2 pts each) OH 1. TsCl, pyr. 2. t-BuOK ton conc. H2SO4 6 #### Similar Solved Questions ##### 1. A homogeneously doped piece of silicon at T = 300 K has impurity doping concentrations of NA = 2* 1016( cm and Np = 4 X 1016cm-3 everywhere inside it. Assume all dopants are ionized, and that Si has an intrinsic carrier concentration of ni = 15* 1010( cm at T= 300 KPlease find: ia} majority carrier type and density; {bj niinorjty carrier type Jd dersit; nner 1. A homogeneously doped piece of silicon at T = 300 K has impurity doping concentrations of NA = 2* 1016( cm and Np = 4 X 1016cm-3 everywhere inside it. 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Flocks of these pigeons pose a threat to crops, livestock, and human health as their large numbers can decimate crops and act as disease vectors for pathogens t... ##### 3-86 The demand for electric power is usually much higher during the day than it is... 3-86 The demand for electric power is usually much higher during the day than it is at night, and utility companies often sell power at night at much lower prices to encourage con- sumers to use the available power generation capacity and to avoid building new expensive power plants that will be use... ##### Find Ihe derivalive Of Ihe funcilon ((XY) =Ine: poIni (0,duectnIne function ducteases Iloet (unl0 A 0 8.0 D Find Ihe derivalive Of Ihe funcilon ((XY) = Ine: poIni (0, duectn Ine function ducteases Iloet (unl 0 A 0 8. 0 D... ##### Determine which of the following series converge and which diverge_ For those that converge find their RuI 2()' (ii) 276, (iii) 50 (iv) '2"() rent series 5 d, and >bn with bn 0for all n, but E an/bn diverges Determine which of the following series converge and which diverge_ For those that converge find their RuI 2()' (ii) 276, (iii) 50 (iv) '2"() rent series 5 d, and >bn with bn 0for all n, but E an/bn diverges... ##### Use Euler's method with step size 0.5 to compute the approximate Y-values Y1, Yz, Y3 and Y4 of the solution of the initial- value problem Y' =Y - 3X, Y(3) = 1- Y1 Y2 Y3 Y4Submit AnswerViewing Saved Work Revert to Last Response Use Euler's method with step size 0.5 to compute the approximate Y-values Y1, Yz, Y3 and Y4 of the solution of the initial- value problem Y' =Y - 3X, Y(3) = 1- Y1 Y2 Y3 Y4 Submit Answer Viewing Saved Work Revert to Last Response... ##### Froblem Considet tle' solution to the Laplace equation Uvn (T,0) â‚¬ D = {(I,9) â‚¬ R' : I<r+v" < 9 wilh Ditiehlet hxuundluy eonditions %l") 6 \| 0vr7 OH1 Uhe eueleo nclius ( = Ute' cuele o Hulius Assume Ilut Ulee solution Ccmmu MaX iut and thc: minimum 0l u ."l FM tc Froblem Considet tle' solution to the Laplace equation Uvn (T,0) â‚¬ D = {(I,9) â‚¬ R' : I<r+v" < 9 wilh Ditiehlet hxuundluy eonditions %l") 6 \| 0vr7 OH1 Uhe eueleo nclius ( = Ute' cuele o Hulius Assume Ilut Ulee solution Ccmmu MaX iut and thc: minimum 0l u . ... ##### Exaluate the following points) integrals Sre Fr? ~using dr proper ~limit notationdpoints) Exaluate the following points) integrals Sre Fr? ~using dr proper ~limit notation d points)... ##### Point) Consider the function f(x)2 In(x),Sx<6.The absolute maximum value isand this occurs at x equalsThe absolute minimum value isand this occurs at x equalsNote: You can earn partial credit on this problem: point) Consider the function f(x) 2 In(x), Sx<6. The absolute maximum value is and this occurs at x equals The absolute minimum value is and this occurs at x equals Note: You can earn partial credit on this problem:... ##### Amanda Corporation makes two kinds of boat hulls: regular and deluxe. The following data are available.... Amanda Corporation makes two kinds of boat hulls: regular and deluxe. The following data are available. Regular DeluxeOpening WIP inventory$280,000$147,500RM used690,000545,000Direct labor consumed985,0001,342,600Cost of goods manufactured 2,250,0... ##### Suppose that f: [-1,1] ~ R is an even function (f(-x) f(r)) Show that f',f(xJdx = 2f6 f()dx b. give an example t0 illustrate your answer: Suppose that f: [-1,1] ~ R is an even function (f(-x) f(r)) Show that f',f(xJdx = 2f6 f()dx b. give an example t0 illustrate your answer:... ##### And the market prce for the glven pair of supply and demand equations:9p 1160D = 2760 T1p And the market prce for the glven pair of supply and demand equations: 9p 1160 D = 2760 T1p... ##### Find the general solution of the given second-order differential equation2y" Sy' 6yy(x) Find the general solution of the given second-order differential equation 2y" Sy' 6y y(x)... ##### Wnal two nonnegative real numbers with Sum of 48 have the largest possible product? Let x be one of the numbers and Iet 5 be the product of Ine two numbers Write Ihe objeciive lunction lens Ol x P-L (Type-an oxpression ) Wnal two nonnegative real numbers with Sum of 48 have the largest possible product? Let x be one of the numbers and Iet 5 be the product of Ine two numbers Write Ihe objeciive lunction lens Ol x P-L (Type-an oxpression )... ##### Consider the following R expressions: string1 = c("A", "B", "C") and string2 = str_c("A", "B","C") What... Consider the following R expressions: string1 = c("A", "B", "C") and string2 = str_c("A", "B","C") What is the value of the expression below? length(string1)*length(string2) + str_length(string2) answer: 6 But I don't know why the answer is 6. I do... ##### PLEASE WATCH THIS Video THROUGH KANOPY OR ANY OTHER WAY AND ANSWER THE FOLLOWING QUESTION (WE... PLEASE WATCH THIS Video THROUGH KANOPY OR ANY OTHER WAY AND ANSWER THE FOLLOWING QUESTION (WE WERE CHILDREN FILM) asap Film Questions “We Were Children” If someone was to ask you “What is an Indian Residential Schools”. What would you tell them? What are some of t... ##### What is the purpose of using sodium sulfite or sodiummetabisulfite in the preparation of benzoic acid? Showreaction.HW for organic chemistry 2 course at a university. What is the purpose of using sodium sulfite or sodium metabisulfite in the preparation of benzoic acid? Show reaction. HW for organic chemistry 2 course at a university.... ##### A solution of potassium sodium oxalate NazooccoO pecmanganate KMnOa is standardized using under acidic conditions solution of represented by the following unbalanced The titration half-reactions. can be MnOa (aq) Mn2+ (pal OOcCOO? (aq) COzle) 1f13.9 mL of acidified NazOOccoo, the CeMho; is used to titrate 50.0 mL of 0.100 concentration of the KMnoa Mol/l answer to 3 decimal places solution is Mol/L Round your Your Answer:Answer: A solution of potassium sodium oxalate NazooccoO pecmanganate KMnOa is standardized using under acidic conditions solution of represented by the following unbalanced The titration half-reactions. can be MnOa (aq) Mn2+ (pal OOcCOO? (aq) COzle) 1f13.9 mL of acidified NazOOccoo, the CeMho; is used to t... ##### Decscribe the use of B2B and B2C in health care Decscribe the use of B2B and B2C in health care... ##### The number of milligrams D (h) of certain drug that is in patient's bloodstream h hours after the drug is injected is given by the following function_D(h) = 20e OshWhen the number of milligrams reaches 3_ the drug is to be injected again. How much time is needed between injections?Round your answer to the nearest tenth; and do not round any intermediate computations_hours The number of milligrams D (h) of certain drug that is in patient's bloodstream h hours after the drug is injected is given by the following function_ D(h) = 20e Osh When the number of milligrams reaches 3_ the drug is to be injected again. How much time is needed between injections? Round your... ##### Problem 2.22 Making use of Eq 2.42 of the text in the form detA = â‚¬ijkA,i Azj Azk write Eq 2.71 asItij ASi;l = tijk (t1i Sui) (t2j 82j (t3k S3;) = 0 and show by expansion of this equation that13 ~tiiA? + 2 (tittjj tijtji) ^ Eijktlitzityk = 0to verify Eq 2.72 of the text_det A {ijk AilAjAk3 Eijk AliAzj Ajk(2.42)Itij ASijl = 0(2.71)1> ItA2 +ItA Mt =0(2.72) Problem 2.22 Making use of Eq 2.42 of the text in the form detA = â‚¬ijkA,i Azj Azk write Eq 2.71 as Itij ASi;l = tijk (t1i Sui) (t2j 82j (t3k S3;) = 0 and show by expansion of this equation that 13 ~tiiA? + 2 (tittjj tijtji) ^ Eijktlitzityk = 0 to verify Eq 2.72 of the text_ det A {ijk AilAjAk... -- 0.022667--	2022-07-06 15:23:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6651403307914734, "perplexity": 9597.72772049041}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104675818.94/warc/CC-MAIN-20220706151618-20220706181618-00685.warc.gz"}
https://math.stackexchange.com/questions/2651165/let-n-be-a-6-digit-number-perfect-square-and-perfect-cube-if-n-6-is-not-ev/2651170	# Let $n$ be a 6-digit number, perfect square and perfect cube. If $n-6$ is not even or a multiple of 3, find $n$ Let $n$ be a 6-digit number, perfect square and perfect cube. If $n-6$ is not even or a multiple of 3, find $n$. My try Playing with the first ten perfect squares and cubes I ended with: The last digit of $n \in (1,5,9)$ If $n$ last digit is $9$, then the cube ends in $9$, Ex: if $n$ was $729$, the cube is $9^3$ (ends in $9$) and the square ends in $3$ or $7$ If $n$ last digit is 5, then the cube ends in 5 and the square ends in 5 If $n$ last digit is 1, then the cube ends in 1 and the square ends in 1 By brute force I saw that from $47^3$ onwards, the cubes are 6-digit, so I tried some cubes (luckily for me not for long) and $49^3 = 343^2 = 117649$ worked. So I found $n=117649$ but I want to know what is the elegant or without brute force method to find this number because my method isn't very good, just pure luck maybe. Note that the required number is both a square and a cube, so it must be a sixth power. Already $10^6=1000000$ has seven digits and $5^6=15625$ has only five digits, so that leaves us with $6^6,7^6,8^6,9^6$ to test. Furthermore, we are given that $n-6$ is not even and not a multiple of 3, which implies that $n$ itself is also not even and not a multiple of 3. This eliminates $6^6,8^6$ and $9^6$ immediately, leaving $7^6$ as the only possible answer. • Pretty clever, i don't know how i didn't think about that. Thanks – Rodrigo Pizarro Feb 15 '18 at 2:29 If $n$ is both a perfect square and perfect cube then. $n = a^6$ If $n-6$ is neither even nor divisible by $3$, then $n$ is not even nor divisible by $3$ and $a$ is not even or divisible by 3. $a^6$ is a $6$ digit number $6<a<10$ $7$ is the only integer in that interval that is not divisible by $2$ or by $3.$ $n = 7^6$	2021-03-04 19:42:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6623613238334656, "perplexity": 77.52112165097368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178369512.68/warc/CC-MAIN-20210304174506-20210304204506-00005.warc.gz"}
https://www.physicsforums.com/threads/modulus-of-rigidity.78129/	# Modulus of rigidity (1 Viewer) ### Users Who Are Viewing This Thread (Users: 0, Guests: 1) #### chandran what is modulus of rigidity and shear modulus? What do they define? #### Dr.Brain There are three moduli of rigidity: 1. Young's Modulus 2.Bulk Modulus 3.Shear Modulus Modulus is generally defined as Stress/Strain 1.Young's Modulus is generally used for solid materials( In problems, for wires..) $Y= \frac{Longitudinal Stress}{Longitudinal Strain}$ 2. Bulk Modulus is generally used for Liquids and Gases $B= \frac{Volumetric Stress}{Volumetric Strain}$ 3. Shear Modulus is used where tangential stress is applied and the object bends or tangentially bends making some angle with vertical. I assume you know what stress and strain is. #### Pyrrhus Homework Helper Rigidity is the required force to produce a unit incrementum of length. In prismatic beams, the product of EA is known as axial rigidity. $$\delta = \frac{PL}{EA}$$ where $\delta$ is the change in length, P is the force applied at the centroid, L is the original length, E is the modulus of elasticity (assuming the material is at the elastic-linear region) and A is the cross sectional area. Of course this is for Homogenous materials. In general the rigidity will be a measure of a structural member "opposing the change in length", with rigidity it's often used flexibility, which is inverse to the rigidity. #### Pyrrhus Homework Helper Maybe you are refering to the modulus of elasticity in shear stress, also know as modulus of rigidity. According to Hooke's Law in shear (elastic-linear region) $$\tau = G \gamma$$ where $\tau$ is the shear stress, G is the modulus of rigidity or elasticity in shear and $\gamma$ is the angle of distorsion or the unit deformation. The rigidity here is about measuring the structural element resistance to the "change of its shape". Last edited: #### sindhujahere what does the product of rigidity modulus and moment of inertia of a beam mean?? #### rotin089 can some one please tell me the derivation of modulus of rigidity or shear modulus i stuck i need to finish with this equation: G=E/2(1+U) please help out if you can thanks #### FredGarvin Science Advisor Last edited by a moderator: #### engineers can someone tell me in brief about modulas of elasticiy along with pictures ### The Physics Forums Way We Value Quality • Topics based on mainstream science • Proper English grammar and spelling We Value Civility • Positive and compassionate attitudes • Patience while debating We Value Productivity • Disciplined to remain on-topic • Recognition of own weaknesses • Solo and co-op problem solving	2019-03-24 10:43:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6413246393203735, "perplexity": 3035.2108241921437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203438.69/warc/CC-MAIN-20190324103739-20190324125739-00195.warc.gz"}
https://www.physicsforums.com/threads/show-the-critical-flow-occurs.902867/	# Show the critical flow occurs 1. Feb 6, 2017 ### fonseh 1. The problem statement, all variables and given/known data I am not sure how to get the Mmin . Why δA will become B (refer to the pink circled part) 2. Relevant equations 3. The attempt at a solution Is there anything wrong with the author's steps ? I know the author's intention is to show that the Mmin occur when the Fr = (Q^2)B/ g(A^3) = 1 , but we have (Q^2)δA / g(A^3) ... How to make the B=δA ??? File size: 30.9 KB Views: 18 2. Feb 9, 2017 ### haruspex I'm afraid I cannot make sense of the development over those two pages. On the left page, the parameter appears to be distance along the stream. At any such distance, there is an A, a $\bar y$ etc. The height, y, is not a parameter here. On the right page we are concerned with the minimum M has a function of height, and yet the equations appear to be derived from those on the left page. The step "dividing by ∂y" is not adequately justified by that description. I see no explanation of what B represents.	2017-10-23 17:56:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5597477555274963, "perplexity": 1233.6260681549631}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187826210.56/warc/CC-MAIN-20171023164236-20171023184236-00123.warc.gz"}
http://openstudy.com/updates/504c13bae4b0985a7a58d1b7	## mostfinnest 3 years ago In 1980, median family income was about $19,000, and in 2000 it was about$40,000. Find the slope of the line passing through points (1980, 19000) and (2000,40000) m= $slope=m=\frac{ y _{2} - y _{1}}{ x _{2} - x _{1}}$ If you treat (1980, 19000) as $(x _{1}, y _{1})$ and (2000,40000) as $(x _{2}, y _{2})$	2016-05-01 23:17:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.509418249130249, "perplexity": 3672.017165121596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860117244.29/warc/CC-MAIN-20160428161517-00117-ip-10-239-7-51.ec2.internal.warc.gz"}
http://wiki.socr.umich.edu/index.php/SOCR_EduMaterials_Activities_BoxPlot	# SOCR EduMaterials Activities BoxPlot ## Summary This activity describes the construction of the box-and-whisker plot (or simply box plot) in SOCR. The applets can be accessed at SOCR Charts under the Miscellaneous folder. ## Goals The aims of this activity are to: • show the importance of the box plot in exploratory data analysis (EDA) • illustrate how to use SOCR to construct a box plot • present some unusual pathologies of a box plot ## Background & Motivation The box plot (or box-and-whisker-plot), invented by John Tukey in 1977, is an efficient way for presenting data, especially for comparing multiple groups of data. In the box plot, we can mark-off the five-number summary of a data set (minimum, 25th percentile, median, 75th percentile, maximum). The box contains the $$50 %$$ of the data. The upper edge of the box represents the 75th percentile, while the lower edge is the 25th percentile. The median is represented by a line drawn in the middle of the box. If the median is not in the middle of the box then the data are skewed. The ends of the lines (called whiskers) represent the minimum and maximum values of the data set, unless there are outliers. Outliers are observations below $$Q_1 -1.5 (IQR)$$ or above $$Q_3 + 1.5(IQR)$$, where $$Q_1$$ is the 25th percentile, $$Q_3$$ is the 75th percentile, and $$IQR=Q_3-Q_1$$ (called the interquartile range). The advantage of a box plot is that it provides graphically the location and the spread of the data set, it provides an idea about the skewness of the data set, and can provide a comparison between variables by constructing a side-by-side box plots. ## Examples & Exercises • Example 1: Go to the SOCR Charts and first, click on the Miscellaneous folder and then on BoxAndWhiskerChartDemo1. In the Demo1 box plot, we can see side-by-side box plots of two categories for each of three series. These demonstration data can be viewed by clicking on DATA. Clicking on MAPPING you can choose the variables. Clicking on SHOW ALL the applet will present the graph, the data, and the mapping environment. Let’s clear this data set (click on CLEAR) so that we can enter our own data. After you click on CLEAR button, click on the DATA tab to enter data into the spreadsheet. The following data will be entered (don’t forget to separate the data by commas!): C1 C2 C3 Series 1 1,2,3,4,5,6 2,4,6,8,10,12 Series 2 3,4,5,6,7,8 6,8,10,12,14,16,18 Series 3 5,6,7,8,9 10,16,18,20,22 When you finish entering your data, click on MAPPING to select the series and categories, and finally click on UPDATE_CHART to view the box plots. The following snapshot shows how the above data entered into SOCR: The following snapshot shows the mapping of the data: The following snapshot shows the side-by-side box plots: The following snapshot shows the data, the mapping, and the box plots in one screen: • Example 2: If we are working with a single variable, we can use the BoxAndWhiskerChartDemo2. Double click this link to see the demonstration of the construction of the box plot with one variable. As we did in example 1, we will enter our own data. Click on CLEAR to enter your data in the spreadsheet. The data we want to enter are the following: 60, 95, 72, 87, 88, 75, 76, 91, 100, 58, 78, 81, 73, 94, 65. When you finish entering your data, click on MAPPING to select the category (here only C1), and finally click on UPDATE_CHART to view the box plot. The following snapshot shows how the above data entered into SOCR: The following snapshot shows the mapping of the data: The following snapshot shows the box plot: The following snapshot shows the data, the mapping, and the box plots in one screen: ### Box Plot Pathologies Box plots can show unusual pathologies. For the following box plots, enter the data in the SOCR Charts spreadsheet that created them. • Example 1: • Example 2: • Example 3: • Example 4: • Example 5: ## Other Forms of Data Alternatively, the user can import data by clicking on FILE OPEN. Note here that the data must be saved previously as a comma delimited (CSV) in order to be accessed in SOCR.	2020-01-28 16:55:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4702455699443817, "perplexity": 1173.489558145007}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251779833.86/warc/CC-MAIN-20200128153713-20200128183713-00361.warc.gz"}
https://people.maths.bris.ac.uk/~matyd/GroupNames/480i1/C2xD10.D6.html	Copied to clipboard ## G = C2×D10.D6order 480 = 25·3·5 ### Direct product of C2 and D10.D6 Series: Derived Chief Lower central Upper central Derived series C1 — C30 — C2×D10.D6 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C2×C3⋊F5 — C22×C3⋊F5 — C2×D10.D6 Lower central C15 — C30 — C2×D10.D6 Upper central C1 — C22 — C23 Generators and relations for C2×D10.D6 G = < a,b,c,d,e \| a2=b10=c2=d6=1, e2=b4c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b7, cd=dc, ece-1=b6c, ede-1=b5d-1 > Subgroups: 1324 in 264 conjugacy classes, 81 normal (27 characteristic) C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C23, D5, D5, C10, C10, C10, Dic3, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, F5, D10, D10, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C22×C6, C22×C6, C3×D5, C3×D5, C30, C30, C30, C2×C22⋊C4, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C6.D4, C22×Dic3, C23×C6, C3⋊F5, C6×D5, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C22⋊F5, C22×F5, C23×D5, C2×C6.D4, C2×C3⋊F5, C2×C3⋊F5, D5×C2×C6, D5×C2×C6, D5×C2×C6, C22×C30, C2×C22⋊F5, D10.D6, C22×C3⋊F5, D5×C22×C6, C2×D10.D6 Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, F5, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C2×F5, C6.D4, C22×Dic3, C2×C3⋊D4, C3⋊F5, C22⋊F5, C22×F5, C2×C6.D4, C2×C3⋊F5, C2×C22⋊F5, D10.D6, C22×C3⋊F5, C2×D10.D6 Smallest permutation representation of C2×D10.D6 On 120 points Generators in S120 (1 87)(2 88)(3 89)(4 90)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 77)(22 78)(23 79)(24 80)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120) (1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120) (1 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 26)(22 25)(23 24)(27 30)(28 29)(31 38)(32 37)(33 36)(34 35)(39 40)(41 48)(42 47)(43 46)(44 45)(49 50)(51 52)(53 60)(54 59)(55 58)(56 57)(61 68)(62 67)(63 66)(64 65)(69 70)(71 78)(72 77)(73 76)(74 75)(79 80)(81 82)(83 90)(84 89)(85 88)(86 87)(91 98)(92 97)(93 96)(94 95)(99 100)(101 108)(102 107)(103 106)(104 105)(109 110)(111 112)(113 120)(114 119)(115 118)(116 117) (1 16 29 117 40 105)(2 17 30 118 31 106)(3 18 21 119 32 107)(4 19 22 120 33 108)(5 20 23 111 34 109)(6 11 24 112 35 110)(7 12 25 113 36 101)(8 13 26 114 37 102)(9 14 27 115 38 103)(10 15 28 116 39 104)(41 83 66 71 53 96)(42 84 67 72 54 97)(43 85 68 73 55 98)(44 86 69 74 56 99)(45 87 70 75 57 100)(46 88 61 76 58 91)(47 89 62 77 59 92)(48 90 63 78 60 93)(49 81 64 79 51 94)(50 82 65 80 52 95) (1 50 6 45)(2 43 5 42)(3 46 4 49)(7 48 10 47)(8 41 9 44)(11 95 16 100)(12 98 15 97)(13 91 14 94)(17 93 20 92)(18 96 19 99)(21 58 22 51)(23 54 30 55)(24 57 29 52)(25 60 28 59)(26 53 27 56)(31 68 34 67)(32 61 33 64)(35 70 40 65)(36 63 39 62)(37 66 38 69)(71 120 74 119)(72 113 73 116)(75 112 80 117)(76 115 79 114)(77 118 78 111)(81 102 88 103)(82 105 87 110)(83 108 86 107)(84 101 85 104)(89 106 90 109) G:=sub<Sym(120)\| (1,87)(2,88)(3,89)(4,90)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,77)(22,78)(23,79)(24,80)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,26)(22,25)(23,24)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,52)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,70)(71,78)(72,77)(73,76)(74,75)(79,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,98)(92,97)(93,96)(94,95)(99,100)(101,108)(102,107)(103,106)(104,105)(109,110)(111,112)(113,120)(114,119)(115,118)(116,117), (1,16,29,117,40,105)(2,17,30,118,31,106)(3,18,21,119,32,107)(4,19,22,120,33,108)(5,20,23,111,34,109)(6,11,24,112,35,110)(7,12,25,113,36,101)(8,13,26,114,37,102)(9,14,27,115,38,103)(10,15,28,116,39,104)(41,83,66,71,53,96)(42,84,67,72,54,97)(43,85,68,73,55,98)(44,86,69,74,56,99)(45,87,70,75,57,100)(46,88,61,76,58,91)(47,89,62,77,59,92)(48,90,63,78,60,93)(49,81,64,79,51,94)(50,82,65,80,52,95), (1,50,6,45)(2,43,5,42)(3,46,4,49)(7,48,10,47)(8,41,9,44)(11,95,16,100)(12,98,15,97)(13,91,14,94)(17,93,20,92)(18,96,19,99)(21,58,22,51)(23,54,30,55)(24,57,29,52)(25,60,28,59)(26,53,27,56)(31,68,34,67)(32,61,33,64)(35,70,40,65)(36,63,39,62)(37,66,38,69)(71,120,74,119)(72,113,73,116)(75,112,80,117)(76,115,79,114)(77,118,78,111)(81,102,88,103)(82,105,87,110)(83,108,86,107)(84,101,85,104)(89,106,90,109)>; G:=Group( (1,87)(2,88)(3,89)(4,90)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,77)(22,78)(23,79)(24,80)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,26)(22,25)(23,24)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,52)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,70)(71,78)(72,77)(73,76)(74,75)(79,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,98)(92,97)(93,96)(94,95)(99,100)(101,108)(102,107)(103,106)(104,105)(109,110)(111,112)(113,120)(114,119)(115,118)(116,117), (1,16,29,117,40,105)(2,17,30,118,31,106)(3,18,21,119,32,107)(4,19,22,120,33,108)(5,20,23,111,34,109)(6,11,24,112,35,110)(7,12,25,113,36,101)(8,13,26,114,37,102)(9,14,27,115,38,103)(10,15,28,116,39,104)(41,83,66,71,53,96)(42,84,67,72,54,97)(43,85,68,73,55,98)(44,86,69,74,56,99)(45,87,70,75,57,100)(46,88,61,76,58,91)(47,89,62,77,59,92)(48,90,63,78,60,93)(49,81,64,79,51,94)(50,82,65,80,52,95), (1,50,6,45)(2,43,5,42)(3,46,4,49)(7,48,10,47)(8,41,9,44)(11,95,16,100)(12,98,15,97)(13,91,14,94)(17,93,20,92)(18,96,19,99)(21,58,22,51)(23,54,30,55)(24,57,29,52)(25,60,28,59)(26,53,27,56)(31,68,34,67)(32,61,33,64)(35,70,40,65)(36,63,39,62)(37,66,38,69)(71,120,74,119)(72,113,73,116)(75,112,80,117)(76,115,79,114)(77,118,78,111)(81,102,88,103)(82,105,87,110)(83,108,86,107)(84,101,85,104)(89,106,90,109) ); G=PermutationGroup([[(1,87),(2,88),(3,89),(4,90),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,61),(18,62),(19,63),(20,64),(21,77),(22,78),(23,79),(24,80),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,26),(22,25),(23,24),(27,30),(28,29),(31,38),(32,37),(33,36),(34,35),(39,40),(41,48),(42,47),(43,46),(44,45),(49,50),(51,52),(53,60),(54,59),(55,58),(56,57),(61,68),(62,67),(63,66),(64,65),(69,70),(71,78),(72,77),(73,76),(74,75),(79,80),(81,82),(83,90),(84,89),(85,88),(86,87),(91,98),(92,97),(93,96),(94,95),(99,100),(101,108),(102,107),(103,106),(104,105),(109,110),(111,112),(113,120),(114,119),(115,118),(116,117)], [(1,16,29,117,40,105),(2,17,30,118,31,106),(3,18,21,119,32,107),(4,19,22,120,33,108),(5,20,23,111,34,109),(6,11,24,112,35,110),(7,12,25,113,36,101),(8,13,26,114,37,102),(9,14,27,115,38,103),(10,15,28,116,39,104),(41,83,66,71,53,96),(42,84,67,72,54,97),(43,85,68,73,55,98),(44,86,69,74,56,99),(45,87,70,75,57,100),(46,88,61,76,58,91),(47,89,62,77,59,92),(48,90,63,78,60,93),(49,81,64,79,51,94),(50,82,65,80,52,95)], [(1,50,6,45),(2,43,5,42),(3,46,4,49),(7,48,10,47),(8,41,9,44),(11,95,16,100),(12,98,15,97),(13,91,14,94),(17,93,20,92),(18,96,19,99),(21,58,22,51),(23,54,30,55),(24,57,29,52),(25,60,28,59),(26,53,27,56),(31,68,34,67),(32,61,33,64),(35,70,40,65),(36,63,39,62),(37,66,38,69),(71,120,74,119),(72,113,73,116),(75,112,80,117),(76,115,79,114),(77,118,78,111),(81,102,88,103),(82,105,87,110),(83,108,86,107),(84,101,85,104),(89,106,90,109)]]) 60 conjugacy classes class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A ··· 4H 5 6A ··· 6G 6H ··· 6O 10A ··· 10G 15A 15B 30A ··· 30N order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 ··· 4 5 6 ··· 6 6 ··· 6 10 ··· 10 15 15 30 ··· 30 size 1 1 1 1 2 2 5 5 5 5 10 10 2 30 ··· 30 4 2 ··· 2 10 ··· 10 4 ··· 4 4 4 4 ··· 4 60 irreducible representations dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + - + + + image C1 C2 C2 C2 C4 C4 S3 D4 Dic3 D6 Dic3 C3⋊D4 F5 C2×F5 C3⋊F5 C22⋊F5 C2×C3⋊F5 D10.D6 kernel C2×D10.D6 D10.D6 C22×C3⋊F5 D5×C22×C6 D5×C2×C6 C22×C30 C23×D5 C6×D5 C22×D5 C22×D5 C22×C10 D10 C22×C6 C2×C6 C23 C6 C22 C2 # reps 1 4 2 1 6 2 1 4 3 3 1 8 1 3 2 4 6 8 Matrix representation of C2×D10.D6 in GL6(𝔽61) 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 , 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 60 0 0 0 0 1 0 60 0 0 0 1 0 0 0 0 60 1 0 0 , 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 1 0 0 0 60 0 1 0 0 60 0 0 1 0 0 0 0 0 1 , 0 1 0 0 0 0 1 0 0 0 0 0 0 0 17 54 0 7 0 0 0 10 54 7 0 0 7 54 10 0 0 0 7 0 54 17 , 0 50 0 0 0 0 11 0 0 0 0 0 0 0 7 14 47 54 0 0 0 14 54 7 0 0 14 0 47 7 0 0 14 7 0 54 G:=sub<GL(6,GF(61))\| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,1,1,1,1,0,0,60,0,0,0,0,0,0,60,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,1,1,1,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,0,7,7,0,0,54,10,54,0,0,0,0,54,10,54,0,0,7,7,0,17],[0,11,0,0,0,0,50,0,0,0,0,0,0,0,7,0,14,14,0,0,14,14,0,7,0,0,47,54,47,0,0,0,54,7,7,54] >; C2×D10.D6 in GAP, Magma, Sage, TeX C_2\times D_{10}.D_6 % in TeX G:=Group("C2xD10.D6"); // GroupNames label G:=SmallGroup(480,1072); // by ID G=gap.SmallGroup(480,1072); # by ID G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,2693,14118,2379]); // Polycyclic G:=Group<a,b,c,d,e\|a^2=b^10=c^2=d^6=1,e^2=b^4c,ab=ba,ac=ca,ad=da,ae=ea,cbc=b^-1,bd=db,ebe^-1=b^7,cd=dc,ece^-1=b^6c,ede^-1=b^5*d^-1>; // generators/relations ׿ × 𝔽	2021-06-16 17:03:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 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https://www.cs.toronto.edu/~frossard/topics/gradient-descent/	Introduction Linear regression is a method used to find a relationship between a dependent variable and a set of independent variables. In its simplest form it consist of fitting a function $\boldsymbol{y} = w.\boldsymbol{x}+b$ to observed data, where $\boldsymbol{y}$ is the dependent variable, $\boldsymbol{x}$ the independent, $w$ the weight matrix and $b$ the bias. Illustratively, performing linear regression is the same as fitting a scatter plot to a line.	2019-12-12 16:54:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35073670744895935, "perplexity": 172.0903388674977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540544696.93/warc/CC-MAIN-20191212153724-20191212181724-00533.warc.gz"}
http://mathoverflow.net/revisions/41986/list	MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4). 3 added 27 characters in body Let $M$ be a closed orientable $n$-manifold containing the compact set $X$. Given an $n-q-1$-cocyle on $X$ (I am choosing this degree just to match with the notation of the Wikipedia article to which you linked), we extend it to some small open neighbourhood $U$ of $X$. By Lefschetz--Poincare duality on the open manifold $U$, we can convert this $n-q-1$-cocylce into a Borel--Moore cycle (i.e. a locally-finite cycle made up of infinitely many simplices) on $U$ of degree $q+1$. Throwing away those simplices lying in $U \setminus X$, we obtain a usual (i.e. finitely supported) cycle giving a class in $H_{q+1}(U,U\setminus X) = H_{q+1}(M,M\setminus X)$ (the isomorphism holding via excision). Alexander duality for an arbitrary manifold then states that the map $H^{n-q-1}(X) \to H_{q+1}(M,M \setminus X)$ is an isomorphism. (If $X$ is very pathological, then we should be careful in how define the left-hand side, to be sure that every cochain actually extends to some neighbourhood of $X$.) Now if $M = S^{n+1}$, then $H^i(S^n)$ H^i(S^{n+1})$is almost always zero, and so we may use the boundary map for the long exact sequence of a pair to identify$H_{q+1}(M,M\setminus H_{q+1}(S^{n+1}, S^{n+1}\setminus X)$with$H_{q}(M\setminus H_{q}(S^{n+1}\setminus X)$modulo worrying about reduced vs. usual homology/cohomology (to deal with the fact that$H^i(S^n)$H^i(S^{n+1})$ is non-zero at the extremal points $i = 0$ or $n$). So, in short: we take a cocycle on $X$, expand it slightly to a cocyle on $U$, represent this by a Borel--Moore cycle of the appropriate degree, throw away those simplices lying entirely outside $X$, so that it is now a chain with boundary lying outside $X$, and finally take this boundary, which is now a cycle in $S^{n+1} \setminus X$. (I found these notes of Jesper Moller helpful in understanding the general structure of Alexander duality.) One last thing: it might help to think this through in the case of a circle embedded in $S^2$. We should thicken the circle up slightly to an embedded strip. If we then take our cohomology class to be the generator of $H^1(S^1)$, the corresponding Borel--Moore cycle is just a longitudinal ray of the strip (i.e. if the strip is $S^1 \times I$, where $I$ is an open interval, then the Borel--More cycle is just $\{\text{point}\} \times I$). If we cut $I$ down to a closed subinterval $I'$ and then take its boundary, we get a pair of points, which you can see intuitively will lie one in each of the components of the complement of the $S^1$ in $S^2$. More rigorously, Alexander duality will show that these two points generate the reduced $H^0$ of the complement of the $S^1$, and this is how Alexander duality proves the Jordan curve theorem. Hopefully the above sketch supplies some geometric intuition to this argument. 2 added 946 characters in body One last thing: it might help to think this through in the case of a circle embedded in $S^2$. We should thicken the circle up slightly to an embedded strip. If we then take our cohomology class to be the generator of $H^1(S^1)$, the corresponding Borel--Moore cycle is just a longitudinal ray of the strip (i.e. if the strip is $S^1 \times I$, where $I$ is an openinterval, then the Borel--More cycle is just $\{\text{point}\} \times I$). If we cut $I$ down to a closed subinterval $I'$ and then take its boundary, we get a pairof points, which you can see intuitively will lie one in each of the componentsof the complement of the $S^1$ in $S^2$. More rigorously, Alexander duality will show that these two points generate the reduced $H^0$ of the complement of the $S^1$, and this is how Alexander duality proves the Jordan curve theorem. Hopefully the above sketch supplies some geometric intuition to this argument. 1 Let $M$ be a closed orientable $n$-manifold containing the compact set $X$. Given an $n-q-1$-cocyle on $X$ (I am choosing this degree just to match with the notation of the Wikipedia article to which you linked), we extend it to some small open neighbourhood $U$ of $X$. By Lefschetz--Poincare duality on the open manifold $U$, we can convert this $n-q-1$-cocylce into a Borel--Moore cycle (i.e. a locally-finite cycle made up of infinitely many simplices) on $U$ of degree $q+1$. Throwing away those simplices lying in $U \setminus X$, we obtain a usual (i.e. finitely supported) cycle giving a class in $H_{q+1}(U,U\setminus X) = H_{q+1}(M,M\setminus X)$ (the isomorphism holding via excision). Alexander duality for an arbitrary manifold then states that the map $H^{n-q-1}(X) \to H_{q+1}(M,M \setminus X)$ is an isomorphism. (If $X$ is very pathological, then we should be careful in how define the left-hand side, to be sure that every cochain actually extends to some neighbourhood of $X$.) Now if $M = S^{n+1}$, then $H^i(S^n)$ is almost always zero, and so we may use the boundary map for the long exact sequence of a pair to identify $H_{q+1}(M,M\setminus X)$ with $H_{q}(M\setminus X)$ modulo worrying about reduced vs. usual homology/cohomology (to deal with the fact that $H^i(S^n)$ is non-zero at the extremal points $i = 0$ or $n$). So, in short: we take a cocycle on $X$, expand it slightly to a cocyle on $U$, represent this by a Borel--Moore cycle of the appropriate degree, throw away those simplices lying entirely outside $X$, so that it is now a chain with boundary lying outside $X$, and finally take this boundary, which is now a cycle in $S^{n+1} \setminus X$. (I found these notes of Jesper Moller helpful in understanding the general structure of Alexander duality.)	2013-06-19 14:33:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9305490255355835, "perplexity": 341.23863296581567}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368708142388/warc/CC-MAIN-20130516124222-00099-ip-10-60-113-184.ec2.internal.warc.gz"}
http://stats.stackexchange.com/tags/logarithm/hot	# Tag Info 36 I always hesitate to jump into a thread with as many excellent responses as this, but it strikes me that few of the answers provide any reason to prefer the logarithm to some other transformation that "squashes" the data, such as a root or reciprocal. Before getting to that, let's recapitulate the wisdom in the existing answers in a more general way. Some ... 21 I always tell students there are three reasons to transform a variable by taking the natural logarithm. The reason for logging the variable will determine whether you want to log the independent variable(s), dependent or both. To be clear throughout I'm talking about taking the natural logarithm. Firstly, to improve model fit as other posters have noted. ... 12 I am very wary of using logarithmic axes on bar graphs. The problem is that you have to choose a starting point of the axis, and this is almost always arbitrary. You can choose to make two bars have very different heights, or almost the same height, merely by changing the minimum value on the axis. These three graphs all plot the same data: An alternative ... 11 Charlie provides a nice, correct explanation. The Statistical Computing site at UCLA has some further examples: http://www.ats.ucla.edu/stat/sas/faq/sas_interpret_log.htm , and http://www.ats.ucla.edu/stat/mult_pkg/faq/general/log_transformed_regression.htm Just to compliment Charlie's answer, below are specific interpretations of your examples. As always, ... 9 Some additional ideas: (1) You needn't confine yourself to a logarithmic transformation. Search this site for the "data-transformation" tag, for example. Some data lend themselves well to certain transformations like a root or a logit. (Such transformations--even logs--are usually to be avoided when publishing graphics for a non-technical audience. On ... 9 Yes. Quantiles can be transformed under any monotonically increasing transformation. To see this, suppose $Y$ is the random variable and $q_{0.16}$ is the 16% quantile. Then $$\text{Pr}(Y\le q_{0.16}) = \text{Pr}(\log(Y)\le\log(q_{0.16})) = 0.16.$$ Generally, if $f$ is monotonic and increasing then $$\text{Pr}(Y\le q_{\alpha}) = \text{Pr}(f(Y)\le ... 8 In the log-log- model, see that$$\begin{equation}\beta_1 = \frac{\partial \log(y)}{\partial \log(x)}.\end{equation}$$Recall that$$\begin{equation} \frac{\partial \log(y)}{\partial y} = \frac{1}{y} \end{equation}$$or$$\begin{equation} \partial \log(y) = \frac{\partial y}{y}. \end{equation}$$Multiplying this latter formulation by 100 gives the ... 8 You could also use the open source plotting package Gnuplot for this task. The first section of a very readable tutorial here shows how to plot with a log scale. 8 R is good and can be freely downloaded from http://www.r-project.org/ R takes some time to get used to, but here's a simple example ("#" indicates that a comment follows): x <- rnorm(20) # generate a sample of size 20 from N(0,1) y <- 10^x # define y_i = 10^(x_i) for each i=1,...,20 plot(x, y) # plot y vs x plot(x, y, log="y") # plot y vs x with ... 8 Reasons to use logged variables fall into two categories: Statistical and substantive. Statistically, if your variables are right-skew (that is, they have a long tail at the high end) then a measure such as correlation or regression can be influenced a lot by one or a few cases at the high end on one or both variables (outliers, leverage points, influential ... 6 I agree with onestop. You may also find this blog post from Econometrics Beat useful in learning how to interpret the coefficients on dummy variables when the dependent variable is logged: http://davegiles.blogspot.com/2011/03/dummies-for-dummies.html The Cliffs Notes version is that for a model like \ln(Y) = a + b \cdot \ln(X) + c \cdot ... 6 You tend to take logs of the data when there is a problem with the residuals. For example, if you plot the residuals against a particular covariate and observe an increasing/decreasing pattern (a funnel shape), then a transformation may be appropriate. Non-random residuals usually indicate that your model assumptions are wrong, i.e. non-normal data. Some ... 6 One typically takes the log of an input variable to scale it and change the distribution (e.g. to make it normally distributed). It cannot be done blindly however; you need to be careful when making any scaling to ensure that the results are still interpretable. This is discussed in most introductory statistics texts. You can also read Andrew Gelman's ... 5 For more on whuber's excellent point about reasons to prefer the logarithm to some other transformations such as a root or reciprocal, but focussing on the unique interpretability of the regression coefficients resulting from log-transformation compared to other transformations, see: Oliver N. Keene. The log transformation is special. Statistics in Medicine ... 4 In case you are using LaTeX for your report writing, the package pgfplots can read in data files and plot single or double logarithmic axis. In case you need to do calculations you can escape to gnuplot. It just looks this tiny bit better if your text font matches your axis labels font. 3 I think the more important point is suggested in @whuber's comment. Your whole approach is misfounded because by taking logarithms you effectively are throwing out of the dataset any students with zero missing days in either 2010 or 2011. It sounds like there are enough of these people to be a problem, and I am sure your results will be wrong based on the ... 3 Correlation(pearson) measures a linear relationship between two continuous variables. There is no such choice for (X,Y) or (log X, log Y). Scatter plot of the variables can be used for understanding of the relationship. The following link may answer regarding normality issue. link 3 Isn't the derivative that you want actually \frac{dy}{dx_{1}}=\frac{300}{x_{1}}-\frac{30}{x_{1}}\ln x_{1}? Your derivative is the change in y for a small change in \ln x_{1}. I think it's probably easier to think about about changing x_{1} on the original (non-logged) scale. That shows that the marginal effect of x_{1} starts out positive for small ... 3 A few quick points about logs The following R code is a reminder that the log of a negative number is not a number and that the log of zero is negative infinity. Thus, if you are going to take a log of a z-score, you first need to make all values obtained greater than zero. > values <- c(-2, -1, 0, .001, .1, 1, 10) > data.frame(values=values, ... 3 If we consider "approximation" in a fairly general sense we can get somewhere. Let's say that a is "approximately normal" (and concentrated near the mean) in a sense that we can handwave away the concerns about a coming near 0 (and its subsequent impact on the moments of \log(a), because a doesn't 'get down near 0'), but with the same low order ... 2 Can you just use a scale comprised of powers of (1+r) for some small r, and round to the nearest integer? For example, in R, with r = 0.25: > x <- unique(round(1.25^(0:50))) > x [1] 1 2 3 4 5 6 7 9 12 15 18 23 28 36 44 56 69 87 108 136 169 212 265 331 414 [26] 517 646 ... 2 As I understand it, you've generally discretized to create a set of n points, x_1, \dots, x_n, with probability p_1, \dots, p_n, and you then calculate the cumulative probabilities, say c_i = \sum_{j=1}^i p_j. So you can draw U \sim Uniform(0,1) and then take X = x_{i^} where i^* = \min_i \{i:c_i \ge U\}, or something like that. But your ... 2 On normal scale the extreme points would be far from the other point and to fit them on the same curve would either require putting a break on the curve or making the difference in consecutive tick marks large enough to fit everything in. But that would make it very difficult to discern differences between point in the center of the distribution. Using a ... 2 You have forgotten a crucial term in the PDFs: the measure dx. Really,$$p(x; \mu, \sigma) = \frac{log_{10}(e)}{x \sigma \sqrt{2 \pi}} e^{-\frac{(log_{10}(x) - \mu)^2}{2 \sigma^2}}\ dx,$$whence$$\eqalign{ g(y=log_{10}(x); \mu, \sigma) &= \frac{log_{10}(e)}{10^y \sigma \sqrt{2 \pi}} e^{-\frac{(y - \mu)^2}{2 \sigma^2}}\ d(10^y) \\ ... 2 I agree with other respondents, especially with respect to the form of the model. If I understand the motivation of your question, however, you are addressing general audiences and want to convey the substantive (theoretical) meaning of your analysis. For this purpose I compare predicted values (e.g. estimated days missed) under various "scenarios." Based on ... 2 Log-scale informs on relative changes (multiplicative), while linear-scale informs on absolute changes (additive). When do you use each? When you care about relative changes, use the log-scale; when you care about absolute changes, use linear-scale. This is true for distributions, but also for any quantity or changes in quantities. Note, I use the word ... 2 If you assume a model form that is non-linear but can be transformed to a linear model such as $\log Y = \beta_0 + \beta_1t$ then one would be justified in taking logarithms of $Y$ to meet the specified model form. In general whether or not you have causal series , the only time you would be justified or correct in taking the Log of $Y$ is when it can be ... 2 Your model is not identifiable. Consider that if you double $a_1$ and halve $a_2$, or double both $a_0$ and $a_1$ - or many other combinations of increases and decreases, the $Y$ value is unchanged - what could possibly tell you which of an infinite number of possible combinations of $a$'s goes with some particular value of $\frac{Y X_1 X_2}{X_0}$? If you ... 1 If we have the model $Y = bX$, then we might expect that a 1 unit increase of $X$ yields a b unit increase in Y. Instead, if we have $Y = b \log(X)$, then we expect a 1 percent increase in $X$ to yield $b\log(1.01)$ unit increase in Y. Edit: whoops, didn't realize that your dependent variable was also log transformed. Here's a link with a good example ... 1 You cannot assign arbitrary Mean and SD to covert z-score data into Raw data (x). However, you can check a shape of the distribution of z-scores by calculating skewness or kurtosis. Log-transform only useful if you're data is positively skewed. Moreover, it would be good if you explain that what is your objective? as @Karl asked. It might be helpful to visit ... Only top voted, non community-wiki answers of a minimum length are eligible	2013-05-26 07:51:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8020455241203308, "perplexity": 744.4101874493467}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706637439/warc/CC-MAIN-20130516121717-00062-ip-10-60-113-184.ec2.internal.warc.gz"}
https://phoenixcarpetcleaners.us/l9ti8x/o3-electron-geometry-6d8be0	Why don't libraries smell like bookstores? This will result in the end O groups being pushed down giving the O3 molecule a bent molecular geometry or V shape. VSEPR theory is a model used in chemistry to determine the geometry of individual molecules. What are wildlife sanctuaries national parks biosphere reserves? ... Electron geometry describes the arrangement of the electron groups in a molecule, whereas the molecular geometry describes the arrangement of the atoms. SiCl4 is tetrahedral because Si has 4 Cl bound to it. Based on the VSEPR (valance shell electron pair repulsion) theory, electrons will repel the electron cloud of the two oxygen atoms on each end. Molecular Geometry: ? Is the molecule polar: Here's what I get. Thus, the electron-pair geometry is tetrahedral and the molecular structure is bent with an angle slightly less than 109.5°. the electron pair geometry would be trigonal planar because there is a lone pair on the oxygen atom. Based on the VSEPR (valance shell electron pair repulsion) theory, electrons will repel the electron cloud of the two oxygen atoms on each end. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. Draw Lewis structures for Ozone, O3 Number of electron sets (groups): ? Identify the electron-pair geometry based on the number of regions of electron density: linear, trigonal planar, tetrahedral, trigonal bipyramidal, or octahedral (Figure $$\PageIndex{2}$$). trigonal planer: What is the shape of H2O? Figure 5. Is evaporated milk the same thing as condensed milk? Predict the central atom of the molecule. To determine the molecular geometry: • Draw the Lewis structure • Count the number of electron pairs (bond pairs and lone pairs but count multiple bonds as one pair) • Arrange electron pairs to minimise repulsion • Position the atoms to minimise the lone pair - lone pair repulsion if > 1 lone pair Electronic Geometry: ? See the answer. linear: What is the shape of BeBr2? What is the electron pair geometry for O3? Get a free answer to a quick problem. From an electron-group-geometry perspective, GeF 2 has a trigonal planar shape, but its real shape is dictated by the positions of the atoms. (b) The trigonal pyramidal molecular structure is determined from the electron-pair geometry. Based on the VSEPR (valance shell electron pair repulsion) theory, electrons will repel the electron cloud of the two oxygen atoms on each end. Number of non-bonding electron sets (groups) or lone pairs: ? for sulfur, oxygen, and TWO chlorine atoms, I make it $6+6+2×7=26•\text{electrons}$, i.e. 0 0. Does pumpkin pie need to be refrigerated? Wherewhen and how do you apply for a job at Winco foods in indio ca.? When and how lovebirds will enter into the nest box? The A represents the central atom, the X represents the number of atoms bonded to A and E represents the number of lone electron pairs surrounding the central atom. A link to the app was sent to your phone. The following are the steps used in this determination. so that's 4 electron domains about Br ... 3 bonding and 1 non-bonding . Is the molecule polar: Source(s): https://shrink.im/bafAS. Use the number of lone pairs to assign an AX m E n designation and determine the molecular geometry. The answer to this question is simple; this structure helps in understanding the basic structure, electrons that take part in bond formation along with the charges on a given atom.Lewis structure is based on the octet rule. Number of bonding electron sets (groups): ? No packages or subscriptions, pay only for the time you need. the electron pair geometry would be trigonal planar because there is a lone pair on the oxygen atom. Show transcribed image text. In applying the VSEPR model to the molecule, we focus on the arrangement of electrons around the central O. When applying VESPR theory, the AXE method of electron counting is often used. > a) "BrF"_5 The Lewis structure is The central "Br" atom has six electron domains, so the electron geometry is octahedral. NO, PO3 3−, SO3. The molecular pair geometry would be bent. The molecular geometry is thus trigonal pyramidal: (b) For ozone the central atom and the outer atoms are oxygen atoms. Well, how many valence electrons do we consider …. Use the following Lewis structure of H2O to identify the electron-pair geometry and molecular structure of H2O. Expert Answer . Sulfur trioxide has a trigonal planar electron geometry, according to David Roth of Tutoring & Homework Help. It should be the most electronegative atom. Molecular Geometry: ? Here's what I get. Here is a chart that describes the usual geometry for molecules based on their bonding behavior. Who is the longest reigning WWE Champion of all time? All the "Br-F" bonds are polar, The two opposing pairs in the horizontal plane cancel each other. If present, do the dipole cancel each other: ? The molecular pair geometry would be bent. Kathleen M. The most stable geometry for the electron pairs, bonding and non-bonding is a tetrahedron, a prediction of #"VSEPR"#. 3 0. mulock. From an electron-group-geometry perspective, GeF 2 has a trigonal planar shape, but its real shape is dictated by the positions of the atoms. Start here or give us a call: (312) 646-6365, © 2005 - 2020 Wyzant, Inc. - All Rights Reserved, a Question trigonal bipyramidal. Most questions answered within 4 hours. That makes three domains with electrons. Lv 4. answered 11/12/19. A. bent B. linear C. tetrahedral D. trigonal pyramidal E. not enough information When did organ music become associated with baseball? Or if you need more Molecular vs Electron Geometry practice, you can also practice Molecular vs Electron Geometry practice problems. You can view video lessons to learn Molecular vs Electron Geometry. An example of bent molecular geometry that results from tetrahedral electron pair geometry is H 2 O. Figure $$\PageIndex{9}$$: (a) H 2 O has four regions of electron density around the central atom, so it has a tetrahedral electron-pair geometry. Where can i find the fuse relay layout for a 1990 vw vanagon or any vw vanagon for the matter? Thus, the electron-pair geometry is tetrahedral with three of the corners occupied by the bonding pairs of electrons. How many bonds have a dipole: ? The molecular geometry is square pyramidal. In fact, the bond angle is 104.5°. The oxygen has 6 valence electrons and thus needs 2 more electrons from 2 hydrogen atoms to complete its octet. As one electron goes to s orbital, three occupy the p orbital, and the last one enters the d orbitals of the central atom, the hybridization of Pbr5 is sp3d. The molecular geometry is the shape of the molecule. Therefore, the electron pair geometry is tetrahedral and the molecular geometry is tetrahedral. Number of non-bonding electron sets (groups) or lone pairs: ? linear: What is the shape of BF3? Why did cyclone Tracy occur in 1974 at Darwin? bent: What is the shape of NH3? Solution for Electron Pair Molecular Molecule Lewis Dot Structure Arrangement Geometry Polarity H2S .. H- S-H CH,CI H-E-: O3 Electronic Geometry, Molecular Shape, and Hybridization Page 1 The Valence Shell Electron Pair Repulsion Model (VSEPR Model) The guiding principle: Bonded atoms and unshared pairs of electrons about a central atom are as far from one another as possible. Electronic Geometry: ? Choose an expert and meet online. CO3 2−, NO3 −, and O3. please answer #19 #20. Thus, the electron-pair geometry is tetrahedral and the molecular structure is bent with an angle slightly less than 109.5°. Now, a lot of people ask why it is necessary to know the Lewis structure of any given molecule or compound. How to Determine Electron Geometry. SO3 has a central sulfur atom and three surrounding oxygens, with a total of 24 valence electrons. Electronic Geometry, Molecular Shape, and Hybridization Page 1 The Valence Shell Electron Pair Repulsion Model (VSEPR Model) The guiding principle: Bonded atoms and unshared pairs of electrons about a central atom are as far from one another as possible. Because the oxygen-centred lone pairs are close to the oxygen (and not bound to a neighbouring atom), these tend to compress the #/_C-O-C# bond angle to give a value of #105^@# rather than #109.5^@#. This will result in the end O groups being pushed down giving the O3 molecule a bent molecular geometry or V shape. Water: Chime in new window. This problem has been solved! All Rights Reserved. If present, do the dipole cancel each other: ? Predicting Molecular Geometry . Draw Lewis structures for Ozone, O3 Number of electron sets (groups): ? Molecular Geometry of PBr5 In many cases, the lewis structure of the compound helps in understanding the molecular geometry of the compound. Is Series 4 of LOST being repeated on SKY? This will result in the end O groups being pushed down giving the, SN1SN2 - Nucleophilic Substitution Reactions, Science Study Tip - Especially for Organic Chemistry. Without drawing the Lewis structures, predict which of the species below will be free radicals. The electron-pair geometry provides a guide to the bond angles of between a terminal-central-terminal atom in a compound. The electron geometry can be obtained using VSEPR theory. Two oxygens form single bonds with sulfur, while one forms a double bond. How do you put grass into a personification? The electron geometry gives the spatial arrangement of all the bonds and lone pairs of a molecule. H2CO Lewis Structure Molecular Geometry Valence Electrons Electron Geometry Lewis Structure 20. (a) The electron-pair geometry for the ammonia molecule is tetrahedral with one lone pair and three single bonds. 120 sp2 BF3, NO3- 3 2 1 AB2N Bent <120 sp2 SnCl2, O3 # Electron Groups on Central Atom # Atoms bound to Central atom # Lone Pairs Type of Geometry Shape Molecular Depiction Ideal Bond Angle Hybridization on Central Atom Example 4 4 0 AB4 Tetrahedral 109.5 sp3 CH4, CCl2F2 4 3 1 AB3N Pyramidal <109.5 sp3 NH3 4 2 2 AB2N2 Bent <109.5 sp3 H2O Nitrate ion == The Lewis structure for NO3^- must accommodate 24 valence electrons. NO. CO2, CO3 2−, NO3 −, O3. 4 years ago. The electron-pair geometry provides a guide to the bond angles of between a terminal-central-terminal atom in a compound. For Free, Based on the VSEPR (valance shell electron pair repulsion) theory, electrons will repel the electron cloud of the two oxygen atoms on each end. Save teachers time and engage students with a new, simpler interface! So when asked to describe the shape of a molecule we must respond with a molecular geometry. The water molecule is so common that it is wise to just memorize that water is a BENT molecule. A B; What is the shape of H2? A quick explanation of the molecular geometry of O2 including a description of the O2 bond angles. Is there a way to search all eBay sites for different countries at once? Electron Geometry Single Bond Lone Molecular Geometry Pairs Real Molecule Formula Model Bond Angle Real Bond Model what is the molecular geometry following item? In fact, the bond angle is 104.5°. What is the electron pair geometry of SF4? 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Smarties Candy Ingredients, Average Calories In Korean Bbq, The Girl With The Dragon Tattoo Order, Lane College Acceptance Rate, How To Get An Afro With Straight Hair For Guys, Is Apple Good For Acid Reflux, Whole Foods Broccoli Salad Calories, Fellowship In Pediatrics In Delhi, Samsung Galaxy S7 Edge Price In Usa, Metallurgy Class 10 Important Questions Cbse,	2022-07-01 04:04:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25950154662132263, "perplexity": 3199.9372950499405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103920118.49/warc/CC-MAIN-20220701034437-20220701064437-00665.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/introductory-algebra-for-college-students-7th-edition/chapter-1-review-exercises-page-110/100	# Chapter 1 - Review Exercises: 100 10 #### Work Step by Step Plug in -2 for x and simplify the expression. -x$^2$-7x -(-2)$^2$-7(-2) -(4)-7(-2) -4-7(-2) -4-(-14) -4+14 10 After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2018-06-18 21:14:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5685453414916992, "perplexity": 7552.402970931077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267861163.5/warc/CC-MAIN-20180618203134-20180618223134-00534.warc.gz"}
https://idomus.com.au/blogs/news/cold-plasma-in-medicine-and-healthcare-the-new-frontier-in-low-temperature-plasma-applications	Cold Plasma in Medicine and Healthcare: The New Frontier in Low Temperature Plasma Applications Airtemp Bipolar Ultra plasma Ion anti-virus indoor enviromernts purifiers made this technology available for everyone. Please read the excerpt or if you like read the entire paper: click the link • Electrical and Computer Engineering Department, Old Dominion University, Norfolk, VA, United States Low temperature plasmas that can be generated at atmospheric pressure and at temperatures below 40°C have in the past couple of decades opened up a new frontier in plasma applications: biomedical applications. These plasma sources produce agents, such as reactive species (radicals and non-radicals), charged particles, photons, and electric fields, which have impactful biological effects. Investigators have been busy elucidating the physical and biochemical mechanisms whereby low temperature plasma affects biological cells on macroscopic and microscopic scales. A thorough understanding of these mechanisms is bound to lead to the development of novel plasma-based medical therapies. This mini review introduces the reader to this exciting multidisciplinary field of research. Introduction Plasma medicine is about using low temperature atmospheric pressure plasmas to generate controllable amounts of specific chemically reactive species that are transported to react with biological targets including cells and tissues. The remarkable achievement of this plasma application is that it took only about 25 years to take it from initial discovery, to fundamental scientific investigation stage, and finally to applications on actual patients. How did this happen in a relatively short time? A brief answer to this question is that although the field started in a rather modest and unexpected way, it did not take long for the plasma physics community to realize its great potential and its revolutionary promise. This was accentuated by the recruitment of health science experts (biochemists, microbiologists, etc.) who joined the various research endeavors and greatly advanced the ongoing research. Up until the present the mechanisms of action of plasma on cells and tissues are still not fully understood but the body of knowledge has been steadily growing and our understanding has expanded significantly to include a relatively good grasp on the physical and biochemical pathways whereby plasma impacts biological matter. The field started in the mid-1990s by few proof of principle experiments which showed that low temperature plasma (LTP) possesses efficient bactericidal property [15]. It was realized from the very beginning that the reactive species generated by LTP, which include reactive oxygen species (ROS) and reactive nitrogen species (RNS) played a pivotal role in the observed biological outcomes [1, 6]. It also became quickly apparent that LTP can not only be used to inactivate pathogens, such as bacteria, on abiotic surfaces but it can also be used to disinfect biological tissues and therefore can be employed for wound healing. In due time these early bold ideas, backed by some preliminary experimental data, resonated strongly within the LTP research community, which by then (around 2005) realized what these new, promising but not fully explored applications meant and joined this emerging research field in droves. Consequently, advances and new milestones were reached at relatively “break neck” speed, and by the beginning of the second decade of the 2000s clinical trials on chronic wounds were conducted with some success [7]. In addition small doses of LTP were found to selectively kill cancer cells without harming healthy ones. This opened up another research avenue sometimes referred to as “plasma oncology.” Investigators from research labs around the world reported promising in vitro and in vivo results on the killing of various cancer cell lines (see review [8] and references therein). The cell lines included those associated with leukemia, carcinoma, breast cancer, brain cancer, prostate cancer, colorectal cancer, etc. [8]. In addition, more recently, cold plasma was used in Germany in limited preliminary trials as a palliative therapy for head and neck cancer patients [9]. The above described various efforts finally culminated in the US Food & Drug Administration (FDA) approval of the first clinical trials in the USA in 2019. This constitutes yet another major milestone for the efforts to develop novel LTP-based cancer therapies. In this mini review, descriptions of LTP sources used in plasma medicine is first given, then some major medical applications are briefly described. Cold Atmospheric Pressure Plasma Sources Two types of plasma discharges have been used extensively in biomedical applications: The dielectric barrier discharge (DBD) and the non-equilibrium atmospheric pressure plasma jet (N-APPJ). Figure 1 illustrates two photographs showing a DBD ignited in argon gas (left photo) and the plasma plume emanating from a N-APPJ operated with helium (right photo). FIGURE 1 Figure 1. Two sources of low temperature atmospheric pressure plasma: Dielectric Barrier Discharge in argon driven by repetitive short duration (ns - μs) high voltage pulses (a); A micro-jet using helium as operating gas, generating a cold plasma plume about 2.5 cm in length (b). Dielectric Barrier Discharge (DBD) Dielectric Barrier Discharges are ideal for the generation of large volume non-equilibrium atmospheric pressure diffuse plasma. Extensive investigations allowed for a good understanding and improvement of their operation [1023]. DBDs use a dielectric material, such as glass or alumina, to cover at least one of the electrodes. The electrodes are driven by high AC voltages in the kV range and at frequencies in the kHz. Plasmas generated by DBDs have been used for ozone generation, for material surface modification, as flow control actuators, etc. DBDs most recent domain of application has been in biomedicine after their successful early use in the mid-1990s to inactivate bacteria [1]. Today they are used in various biomedical applications including wound healing and the destruction of cancer cells and tumors [2427]. Sinusoidal voltages with amplitudes in the kV range and frequencies of few the kHz were originally used to power DBDs. However, since the early 2000s it was found that repetitive high voltage short pulses (ns - μs) offered a more efficient way to enhance the chemistry of such discharges [2830]. DBDs are able to maintain the non-equilibrium state of the plasma due surface charge accumulation on the dielectric surface as soon as a discharge is ignited. This creates an electrical potential that counteracts the externally applied voltage and results in a self-limited pulsed current waveform. The electron energy distribution function (EEDF) defines/controls the chemistry in the plasma. Short repetitive high voltage pulses allow for preferential heating of the electrons population and therefore an increase of ionization and excitation [29]. Pulses with widths less than the characteristic time of the onset of the glow-to-arc transition maintain stable non-equilibrium low temperature plasma [29, 30]. To extend the operating frequency range below the kHz few methods were proposed. For example, Okazaki and co-workers used a dielectric wire mesh electrode to generate a discharge at a frequency of 50 Hz [16]. Laroussi and co-workers used a high resistivity layer/film to cover one of the electrodes in a device they referred to as the Resistive Barrier Discharge (RBD) [31]. The RBD can be operated with low frequencies extending all the way to DC. The film barrier usually has a resistivity of few MΩ.cm. The high resistivity film plays the role of a distributed resistive ballast which inhibits the discharge from localizing and the current from reaching high values. Non-equilibrium Atmospheric Pressure Plasma Jets (N-APPJ) Although plasma jets were previously employed for material processing applications [32, 33] biotolerant plasma jets developed specifically for plasma medicine have been in use only since the mid-2000s [34, 35]. These jets can emit low temperature plasma plumes in the surrounding air. Because they can maintain temperatures below 40°C, they can come in touch with soft matter, including biological tissues, without causing thermal damage. These plasma sources proved to be very useful for various applications including biomedical applications [26, 34, 35]. Because the plasma propagates away from the high voltage electrodes and into a region free from high voltage the plasma does not cause electrical shock/damage to the target cells or tissues. However, the plasma plume does exhibit a very high instantaneous and local electric field at its tip. This field plays a role in the propagation of the plasma plume and can also affect the treated target. Investigators discovered that the plasma plumes generated by N-APPJs are not continuous volumes of plasma but discrete plasma packets/bullets propagating at high velocities, up to 105 m/s [36, 37]. The mechanisms governing the generation and propagation of these plasma bullets were reported by both experimental and modeling investigations [3849]. A photoionization model was proposed by Lu and Laroussi who first investigated the dynamics of the plasma bullet [37]. Further investigations also showed that the high electrical field at the head of the plume plays a role in the propagation process. The average strength of this electric field was experimentally measured to be in the 10–30 kV/cm range [5052]. The low temperature plasma sources described above produce chemically reactive species including reactive oxygen species (ROS) and reactive nitrogen species (RNS), which are known from redox biology to play important biological roles [53]. Other agents generated by these plasma sources are also suspected to play active roles in biological applications. These include charges particles (electrons and ions), UV and VUV radiation, and electric fields. For example the electric field can cause electroporation of cell membranes, allowing molecules (including ROS and RNS) to enter the cells and cause damage to the cell's internal organelles (including mitochondria) and macromolecules such as lipids, proteins, and DNA. To learn more about the physics and design of LTP sources the reader is referred to the following references [23, 27, 5457]. Applications of Cold Plasma in Biology and Medicine The early groundbreaking experiments using low temperature atmospheric pressure plasma for biomedical applications were conducted in a decade spanning from 1995 to 2004 [16, 5860]. The earliest experiments involved the use of dielectric barrier discharge to inactivate bacteria on surfaces and in liquids [1, 58] and to generate pulsed plasma in saline solutions for surgical applications [61, 62]. Works on using cold plasma for the disinfection of wounds, enhancement of proliferation of fibroblasts, and cell detachment soon followed [25, 59, 60]. Eventually these seminal works attracted the interest of the low temperature plasma research community and the field witnessed a substantial growth in the years following 2005 and until the present. Applications in wound healing, dentistry, cancer treatment, etc. have since then been pursued in various laboratories and research centers around the world leading to a remarkable increase in the number of journal manuscripts on the topic and to the publication of several books [6366]. The ability of cold atmospheric plasma to inactivate bacteria recently gained more relevance because modern society has been facing several serious healthcare challenges. Amongst these are: (1) Antibiotic resistant strains of bacteria such as Methicillin Resistant Staphylococcus aureus (MRSA) and Clostridium difficile (C-diff) are sources of hospital acquired infection (HAI), which can be fatal to patients with a compromised immune system; (2) Chronic wounds, such as diabetic ulcers, do not heal easily or at all, and one of the problems is the high level of infection caused by a spectrum of bacteria. The inability of conventional methods to satisfactorily deal with these problems necessitated the need for novel approaches based on new technologies. Cold atmospheric plasma has been shown to effectively inactivate bacteria such as MRSA and to greatly reduce the bioburden in infected chronic wounds, making it a very attractive technology that can be used to help overcome the challenges listed above. In 2010, the first clinical trials on the treatment of chronic wounds with cold atmospheric plasma took place and yielded encouraging results [7]. Today there are several plasma devices on the market which have been licensed as medical instruments and which can be used in medicine, including the treatment of various dermatological diseases. LTP can be applied in two different ways. The first is what is referred to as “direct” exposure. In this mode of application the plasma comes in direct contact with the biological target and therefore all plasma-produced agents act on the cells/tissues. The second mode is what is referred to as “indirect” exposure. In this case only the afterglow of the plasma is used or the plasma is first used to activate a liquid medium then the plasma-activated liquid is applied on top of cells/tissues. One of the advantages of the latter is that the plasma activated liquid (PAL) can be stored and used at a later time, giving a degree of flexibility that direct exposure does not offer. Direct Exposure As mentioned earlier under direct exposure the biological target is subjected to all plasma agents including charged particles, photons, electric field, and reactive species. These agents act alone and/or in synergy to produce certain biological outcomes. In the case of bacteria inactivation, all the above agents were reported to play a role. Lysing of vegetative cells as well spores were reported after direct exposure to LTP, but cell death without lysis was reported as well for gram-positive bacteria [67, 68]. The inactivation of bacteria by LTP has several applications ranging from sterilization of heat sensitive medical tools, to the destruction of biofilms, to disinfection of wounds, to decontamination of liquids, food, and agricultural products. Direct exposure has also been used in a non-lethal way to affect eukaryotic cell functions, by modulating cell signaling pathways [69], and in a lethal way for the destruction of cancer cells and tumors [7074]. Experiments using various cell lines have been reported which showed that under a certain exposure dose LTP can kill cancer cells in a selective manner [7074]. Investigators reported that LTP exposure leads to an increase in intracellular ROS concentrations. Since cancer cells are under high oxidative stress, the increase in ROS leads to severe redox imbalance, which can lead to one or more of the following: DNA damage, mitochondrial dysfunction, caspase activation, advanced state of oxidation of proteins, etc. Such acute stress ultimately leads to cancer cells death. Indirect Exposure In this section we limit the discussion to the case of plasma activated liquids (PAL). In this mode of exposure only long lived chemical species that diffuse and solvate into the aqueous state play a role. This eliminates the effects of photons, electric field, short lived species, and heat. Liquids that have been used include water to make plasma activated water (PAW) and biological culture media to make plasma activated media (PAM). The following discussion focuses on the use of PAM to destroy cancer cells. Over the past few years investigators have reported encouraging results on the use of PAM in vitro and in vivo to kill cancer cells and reduce tumors [7581]. The anticancer characteristic of PAM has been attributed to the long lived species produced in the liquid phase after LTP exposure. These species include hydrogen peroxide, H2O2, nitrite, ${\text{NO}}_{2}^{-}$, nitrate, ${\text{NO}}_{3}^{-}$, peroxynitrite, ONOO, and organic radicals. The making of PAM involves the exposure of a liquid medium to an LTP source, most frequently the plasma plume of a plasma jet, for a certain length of time. Media used include Eagle's Minimum Essential Medium (EMEM), Dulbecco's Modified Eagle Medium (DMEM), Ringer's Lactate solution (RL), Roswell Park Memorial Institute medium (RPMI), with additives such as serum (e.g., bovine serum), glutamine, and antibiotics (e.g., mix of Penicillin/Streptomycin). As an example of producing PAM, a 24-well plate can be used where a few ml of fresh cell culture media is added to each well. Each well can be treated by LTP for a certain length of time, this way producing different PAMs with different “strengths.” To illustrate the effects of PAM on cancer cells the following work done at the author's laboratory is summarized [81]. In this experiment, to make PAM, 1 ml of fresh cell culture media (MEM) was added to each well of a 24 well plate. Each well was exposed to the plume of the plasma pencil (a pulsed plasma jet) for a designated time. After exposure, the media on top of cells grown in a 96-well plate was replaced by 100 μl of PAM. After PAM application, cells were stored at 37°C in a humidified incubator with 5% CO2. Media not exposed by LTP was used for the control sample. The cancer cell line used was SCaBER (ATCC® HTB3™) cell line from a urinary bladder tissue with squamous cell carcinoma. Cell viability was quantified at different times of incubations using The CellTiter 96® AQueous One Solution Cell Proliferation Assay (MTS) (Promega, Madison, MI, USA). To quantify the MTS assay results trypan blue exclusion assay was used [81]. Figure 2 shows the results. FIGURE 2 Figure 2. Viability of SCaBER cells after PAM treatment, using MTS assay. Exposure time indicates the time the liquid media was exposed to plasma to make PAM. Measurements were made after 12, 24, and 48 h of PAM application. Data is based on three independent experiments using two replications each. This figure is plotted based on data previously published in Mohades et al. [81]. As can be seen in Figure 2, PAM that was created by longer LTP treatment times causes a greater cell kill. PAM created with an exposure time greater than 3 min induces more than 90% cell reduction. However, for the 2 min case, over time the proliferation of live cells overtakes the destruction of cells and therefore an increase in viability at 24 and 48 h was observed. To investigate the role of the reactive species in the killing of SCaBER cells measurements of hydrogen peroxide, H2O2, produced in PAM were made. It was found that the concentration of H2O2 in PAM increased with exposure time and correlated well with the reduction in cell viability of SCaBER [81]. This was in agreement with works of various investigators which have shown the key role H2O2 plays in the anticancer efficacy of PAM. Recently Bauer proposed the hypothesis that H2O2 and nitrite lead to the generation of singlet oxygen (1O2) which causes inactivation of catalase [82]. Catalase, which is normally expressed on the membrane of cancer cells, protects them from intercellular ROS/ RNS signaling. With enough inactivation of catalase an influx of H2O2 via aquaporin occurs. Therefore the inactivation of the protective catalase causes ROS mediated signals that lead to apoptosis of malignant cells. Since healthy cells do not express catalase on their surface they are subject to an influx of ROS such as H2O2 or peroxynitrite. So if they are exposed to very high ROS concentrations they also can be damaged. Therefore, the applied dose of ROS/RNS has to be below a certain threshold to achieve selective killing of cancer cells. Conclusion The application of low temperature atmospheric pressure plasma in biomedicine opened up new frontiers in science and technology. On a scientific level, new fundamental knowledge (albeit still incomplete) regarding the interaction of plasma with soft matter has been created. Before the mid-1990s basic scientific understanding of the physical and biochemical effects of plasma on cells and tissues was simply missing. Today, 25 years later and after extensive and hectic scientific investigations, our knowledge has greatly grown and many of the mechanisms involved have been elucidated on the cellular and sub-cellular levels. This allowed remarkable advances in the quest of developing novel plasma-based therapies to overcome various healthcare challenges. The recent approval of the US Food & Drug Administration of clinical trials using plasma for cancer treatment is a critical milestone and a sign that low temperature plasma may be on its way to be accepted as a promising and exciting healthcare technology. To learn more about the future directions of the field the reader is referred to references [83, 84]. In addition, low temperature plasma has obvious merits as a viable technology for space medicine. As deep-space long-duration space travel becomes a reality it is crucial to have available adequate methods to meet medical emergencies in space. In this context plasma offers a practical “energy-based” and “dry” technology that can replace perishable drugs. Author Contributions The author confirms being the sole contributor of this work and has approved it for publication. Conflict of Interest The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. References 1. Laroussi M. Sterilization of contaminated matter with an atmospheric pressure plasma. IEEE Trans Plasma Sci. (1996) 24:1188–91. doi: 10.1109/27.533129 2. Kelly-Wintenberg K, Montie TC, Brickman C, Roth JR, Carr AK, Sorge K, et al. Room temperature sterilization of surfaces and fabrics with a one atmosphere uniform glow discharge plasma. J Indust Microbiol Biotechnol. (1998) 20:69–74. doi: 10.1038/sj.jim.2900482 3. Laroussi M, Saylor G, Glascock B, McCurdy B, Pearce ME, Bright NG, et al. Images of biological samples undergoing sterilization by a glow discharge at atmospheric pressure. IEEE Trans Plasma Sci. (1999) 27:34−5. doi: 10.1109/27.763016 4. Herrmann HW, Henins I, Park J, Selwyn GS. Decontamination of Chemical and Biological Warfare (CBW) agents using an atmospheric pressure plasma jet. Phys Plasmas. (1999) 6:2284–9. doi: 10.1063/1.873480 5. Laroussi M, Alexeff I, Kang W. Biological decontamination by non-thermal plasmas. IEEE Trans Plasma Sci. (2000) 28:184–8. doi: 10.1109/27.842899 6. Laroussi M. Non-Thermal decontamination of biological media by atmospheric pressure plasmas: review, analysis, and prospects. IEEE Trans Plasma Sci. (2002) 30:1409–15. doi: 10.1109/TPS.2002.804220 7. Isbary G, Morfill G, Schmidt HU, Georgi M, Ramrath K, Heinlin J, et al. A first prospective randomized controlled trial to decrease bacterial load using cold atmospheric argon plasma on chronic wounds in patients. Br J Dermatol. (2010) 163:78. doi: 10.1111/j.1365-2133.2010.09744.x 8. Laroussi M. From killing bacteria to destroying cancer cells: twenty years of plasma medicine. Plasma Process Polym. (2014) 11:1138–41. doi: 10.1002/ppap.201400152 9. Metelmann HR, Nedrelow DS, Seebauer C, Schuster M, von Woedtke T, Weltmann K-D, et al. Head and neck cancer treatment and physical plasma. Clin Plasma Med. (2015) 3:17–23. doi: 10.1016/j.cpme.2015.02.001 10. Bartnikas R. Note on discharges in helium under AC conditions. J Appl Phys D Appl Phys. (1968) 1:659. doi: 10.1088/0022-3727/1/5/417 11. Donohoe KG. The development and characterization of an atmospheric pressure nonequilibrium plasma chemical reactor. (Ph.D Thesis). California Institute of Technology, Pasadena, CA, United States (1976). Google Scholar 12. Kanazawa S, Kogoma M, Moriwaki T, Okazaki S. Stable glow at atmospheric pressure. J Phys D Appl Phys. (1988) 21:838. doi: 10.1088/0022-3727/21/5/028 13. Yokoyama T, Kogoma M, Moriwaki T, Okazaki S. The mechanism of the stabilized glow plasma at atmospheric pressure. J Phys D Appl Phys. (1990) 23:1128. doi: 10.1088/0022-3727/23/8/021 14. Massines F, Mayoux C, Messaoudi R, Rabehi A, Ségur P. Experimental study of an atmospheric pressure glow discharge application to polymers surface treatment. In: Proceeding of GD-92, Vol. 2. Swansea (1992). p. 730–3. Google Scholar 15. Roth JR, Laroussi M, Liu C. Experimental generation of a steady-state glow discharge at atmospheric pressure. In: Proceeding of IEEE International Confrence Plasma Science. Tampa, FL (1992). p. 170−1. Google Scholar 16. Okazaki S, Kogoma M, Uehara M, Kimura Y. Appearance of a stable glow discharge in air, argon, oxygen and nitrogen at atmospheric pressure using a 50 Hz source. J Phys D Appl Phys. (1993) 26:889. doi: 10.1088/0022-3727/26/5/025 17. Massines F, Rabehi A, Decomps P, Gadri RB, Ségur P, Mayoux C. Experimental and theoretical study of a glow discharge at atmospheric pressure controled by a dielectric barrier. J Appl Phys. (1998) 8:2950. doi: 10.1063/1.367051 18. Gherardi N, Gouda G, Gat E, Ricard A, Massines F. Transition from glow silent discharge to micro-discharges in nitrogen gas. Plasma Sources Sci Technol. (2000) 9:340. doi: 10.1088/0963-0252/9/3/312 19. Gheradi N, Massines F. Mechanisms controlling the transition from glow silent discharge to streamer discharge in nitrogen. IEEE Trans Plasma Sci. (2001) 29:536. doi: 10.1109/27.928953 20. Shi JJ, Deng XT, Hall R, Punnett JD, Kong M. Three modes in a radio frequency atmospheric pressure glow discharge. J Appl Phys. (2003) 94:6303. doi: 10.1063/1.1622110 21. Massines F, Gherardi N, Naude N, Segur P. Glow and townsend dielectric barrier discharge in various atmosphere. Plasma Phys Contrl Fusion. (2005) 47:B557. doi: 10.1088/0741-3335/47/12B/S42 22. Kogelschatz U. Silent discharges for the generation of ultraviolet and vacuum ultraviolet excimer radiation. Pure Appl Chem. (1990) 62:1667. doi: 10.1351/pac199062091667 23. Kogelschatz U, Eliasson B, Egli W. Dielectric-Barrier discharges: principle and applications. J Physique IV. (1997) 7:47. doi: 10.1051/jp4:1997405 24. Fridman G, Friedman G, Gutsol A, Shekhter AB, Vasilets VN, Fridman A. Applied plasma medicine. Plasma Process Polym. (2008) 5:503. doi: 10.1002/ppap.200700154 25. Laroussi M. Low temperature plasmas for medicine? IEEE Trans Plasma Sci. (2009) 37:714. doi: 10.1109/TPS.2009.2017267 26. Weltmann KD, Kindel E, von Woedtke T, Hähnel M, Stieber M, Brandenburg R. Atmospheric-pressure plasma sources: prospective tools for plasma medicine. Pure Appl Chem. (2010) 82:1223. doi: 10.1351/PAC-CON-09-10-35 27. Laroussi M, Lu XP, Keidar M. Perspective: the physics, diagnostics, and applications of atmospheric pressure low temperature plasma sources used in plasma medicine. J. Appl Phys. (2017) 122:020901. doi: 10.1063/1.4993710 28. Mildren RP, Carman RJ. Enhanced performance of a dielectric barrier discharge lamp using short-pulsed excitation. J Phys D Appl Phys. (2001) 34:3378. doi: 10.1088/0022-3727/34/1/101 29. Stark R, Schoenback KH. Electron heating in atmospheric pressure glow discharges. J Appl Phys. (2001) 89:3568. doi: 10.1063/1.1351546 30. Laroussi M, Lu XP, Kolobov V, Arslanbekov R. Power consideration in the pulsed dbd at atmospheric pressure. J Applied Phys. (2004) 6:3028. doi: 10.1063/1.1777392 31. Laroussi M, Alexeff I, Richardson JP, Dyer FF. The resistive barrier discharge. IEEE Trans Plasma Sci. (2002) 30:158. doi: 10.1109/TPS.2002.1003972 32. Koinuma H, Ohkubo H, Hashimoto T. Development and application of a microbeam plasma generator. Appl Phys Lett. (1992) 60:816. doi: 10.1063/1.106527 33. Janca J, Klima M, Slavicek P, Zajickova L. HF plasma pencil – new source for plasma surface processing. Surf Coating Technol. (1999) 116119:547–51. doi: 10.1016/S0257-8972(99)00256-X 34. Laroussi M, Lu XP. Room temperature atmospheric pressure plasma plume for biomedical applications. Appl Phys Letts. (2005) 87:113902. doi: 10.1063/1.2045549 35. Laroussi M, Akan T. Arc-free atmospheric pressure cold plasma jets: a review. Plasma Process Polym. (2007) 4:777–88. doi: 10.1002/ppap.200700066 36. Teschke M, Kedzierski J, Finantu-Dinu EG, Korzec D, Engemann J. High-Speed photographs of a dielectric barrier atmospheric pressure plasma jet. IEEE Trans Plasma Sci. (2005) 33:310. doi: 10.1109/TPS.2005.845377 37. Lu XP, Laroussi M. Dynamics of an atmospheric pressure plasma plume generated by submicrosecond voltage pulses. J Appl Phys. (2006) 100:063302. doi: 10.1063/1.2349475 38. Mericam-Bourdet N, Laroussi M, Begum A, Karakas E. Experimental investigations of plasma bullets. J Phys D Appl Phys. (2009) 42:055207. doi: 10.1088/0022-3727/42/5/055207 39. Sands BL, Ganguly BN, Tachibana K. A streamer-like atmospheric pressure plasma jet. Appl Phys Lett. (2008) 92:151503. doi: 10.1063/1.2909084 40. Walsh JL, Kong MG. Contrasting characteristics of linear-field and cross-field atmospheric plasma jets. Appl Phys Lett. (2008) 93:111501. doi: 10.1063/1.2982497 41. Xiong Q, Lu XP, Liu J, Xian Y, Xiong Z, Zou F, et al. Temporal and spatial resolved optical emission behaviors of a cold aatmospheric pressure plasma jet. J Appl Phys. (2009) 106:083302. doi: 10.1063/1.3239512 42. Karakas E, Koklu M, Laroussi M. Correlation between helium mole fraction and plasma bullet propagation in low temperature plasma jets. J Phys D Appl Phys. (2010) 43:155202. doi: 10.1088/0022-3727/43/15/155202 43. Naidis GV. Modeling of plasma bullet propagation along a helium jet in ambient air. J Phys D Appl Phys. (2011) 44:215203. doi: 10.1088/0022-3727/44/21/215203 44. Naidis GV. Modeling of streamer propagation in atmospheric pressure helium plasma jets. J Phys D Appl Phys. (2010) 43:402001. doi: 10.1088/0022-3727/43/40/402001 45. Yousfi M, Eichwald O, Merbahi N, Jomma N. Analysis of ionization wave dynamics in low-temperature plasma jets from fluid modeling supported by experimental investigations. Plasma Sources Sci Technol. (2012) 21:045003. doi: 10.1088/0963-0252/21/4/045003 46. Boeuf JP, Yang L, Pitchford L. Dynamics of guided streamer (plasma bullet) in a helium jet in air at atmospheric pressure. J Phys D Appl Phys. (2013) 46:015201. doi: 10.1088/0022-3727/46/1/015201 47. Karakas E, Laroussi M. Experimental studies on the plasma bullet propagation and its inhibiton. J Appl Phys. (2010) 108:063305. doi: 10.1063/1.3483935 48. Jarrige J, Laroussi M, Karakas E. Formation and dynamics of the plasma bullets in a non-thermal plasma jet: influence of the high voltage parameters on the plume characteristics. Plasma Sources Sci Technol. (2010) 19:065005. doi: 10.1088/0963-0252/19/6/065005 49. Sakiyama Y, Graves DB, Jarrige J, Laroussi M. Finite element analysis of ring-shaped emission profile in plasma bullets. Appl Phys Lett. (2010) 96:041501. doi: 10.1063/1.3298639 50. Begum A, Laroussi M, Pervez MR. Atmospheric pressure helium/air plasma jet: breakdown processes and propagation phenomenon. AIP Adv. (2013) 3:062117. doi: 10.1063/1.4811464 51. Stretenovic GB, Krstic IB, Kovacevic VV, Obradovic AM, Kuraica MM. Spatio-temporally resolved electric field measurements in helium plasma jet. J Phys D Appl Phys. (2014) 47:102001. doi: 10.1088/0022-3727/47/10/102001 52. Sobota A, Guaitella O, Garcia-Caurel E. Experimentally obtained values of electric field of an atmospheric pressure plasma jet impinging on a dielectric surface. J Phys D Appl Phys. (2013) 46:372001. doi: 10.1088/0022-3727/46/37/372001 53. Lu X, Naidis GV, Laroussi M, Reuter S, Graves DB, Ostrikov K. Reactive species in non-equilibrium atmospheric pressure plasma: generation, transport, and biological effects. Phys Rep. (2016) 630:1–84. doi: 10.1016/j.physrep.2016.03.003 54. Lietz AM, Kushner MJ. Molecular admixtures and impurities in atmospheric pressure plasma jets. J Appl Phys. (2018) 124:153303. doi: 10.1063/1.5049430 55. Winter J, Brandenburg R, Weltmann KD. Atmospheric pressure plasma jets: an overview of devices and new directions. Plasma Sources Sci Technol. (2015) 24:054001. doi: 10.1088/0963-0252/24/6/064001 56. Becker KH, Shoenbach KH, Eden JG. Microplasmas and applications. J Phys D Appl Phys. (2006) 39:R55. doi: 10.1088/0022-3727/39/3/R01 57. Lietz AM, Kushner MJ. Electrode configurations in atmospheric pressure plasma jets: production of reactive species. Plasma Sources Sci Technol. (2018) 27:105020. doi: 10.1088/1361-6595/aadf5b 58. Laroussi M. Sterilization of liquids using a plasma glow discharge. US Patent # 5,876,663 (1999). Google Scholar 59. Shekhter AB, Kabisov RK, Pekshev AV, Kozlov NP, Perov Yu L. Experimental and clinical validation of plasmadynamic therapy of wounds with nitric oxide. Bull Exp Biol Med. (1998) 126:829–34. doi: 10.1007/BF02446923 60. Stoffels E, Flikweert AJ, Stoffels WW, Kroesen GMW. Plasma needle: a non-destructive atmospheric plasma source for fine surface treatment of biomaterials. Plasma Sources Sci Technol. (2002) 11:383–88. doi: 10.1088/0963-0252/11/4/304 61. Stalder KR, Woloszko K, Brown IG, Smith CD. Repetitive plasma discharges in saline solutions. Appl Phys Lett. (2001) 79:4503. doi: 10.1063/1.1429752 62. Woloszko K, Stalder KR, Brown IG. Plasma characteristics of repetitively-pulsed electrical discharges in saline solutions used for surgical procedures. IEEE Trans Plasma Sci. (2002) 30:1376–83. doi: 10.1109/TPS.2002.801612 63. Laroussi M, Kong M, Morfill G, Stolz W. Plasma Medicine: Applications of Low-Temperature Gas Plasmas in Medicine and Biology. Cambridge: Cambridge Univ. Press. (2012). Google Scholar 64. Fridman A, Friedman G. Plasma Medicine. New York, NY: Wiley (2013). Google Scholar 65. Metelmann HR, von Woedtke T, Weltmann KD. Comprehensive Clinical Plasma Medicine. Berlin: Springer (2018). Google Scholar 66. Toyokuni S, Ikehara Y, Kikkawa F, Hori M. Plasma Medical Science. Cambridge, MA: Academic Press (2018). Google Scholar 67. Laroussi M, Richardson JP, Dobbs FC. Effects of non-equilibrium atmospheric pressure plasmas on the heterotrophic pathways of bacteria and on their cell morphology. Appl Phys Lett. (2002) 81:772–4. doi: 10.1063/1.1494863 68. Laroussi M, Mendis DA, Rosenberg M. Plasma interaction with microbes. N J Phys. (2003) 5:41.1–10. doi: 10.1088/1367-2630/5/1/341 69. Barekzi N, Laroussi M. Fibropblasts cell morphology altered by low temperature atmospheric pressure plasma. IEEE Trans Plasma Sci. (2014) 42:2738. doi: 10.1109/TPS.2014.2315787 70. Keidar M, Walk R, Shashurin A, Srinivasan P, Sandler A, Dasgupta S, et al. Cold plasma selectivity and the possibility of paradigm shift in cancer therapy. Br J Cancer. (2011) 105:1295. doi: 10.1038/bjc.2011.386 71. Barekzi N, Laroussi M. Effects of low temperature plasmas on cancer cells. Plasma Process Polym. (2013) 10:1039. doi: 10.1002/ppap.201300083 72. Keidar M, Shashurin A, Volotskova O, Stepp MA, Srinivasan P, Sandler A, et al. Cold atmospheric plasma in cancer therapy. Phys Plasmas. (2013) 20:057101. doi: 10.1063/1.4801516 73. Köritzer J, Boxhammer V, Schäfer A, Shimizu T, Klämpfl TG, Li YF, et al. Restoration of sensitivity in chemo-resistant glioma cells by cold atmospheric plasma. PLoS ONE. (2013) 8:e64498. doi: 10.1371/journal.pone.0064498 74. Vandamme M, Robert E, Pesnele S, Barbosa E, Dozias S, Sobilo J, et al. Antitumor effects of plasma treatment on U87 glioma xenografts: preliminary results. Plasma Process Polym. (2010) 7:264–73. doi: 10.1002/ppap.200900080 75. Tanaka H, Mizuno M, Ishikawa K, Nakamura K, Kajiyama H, Kano H, et al. Plasma-activated medium selectively kills glioblastoma brain tumor cells by down-regulating a survival signaling molecule, AKT kinase. Plasma Med. (2011) 1:265. doi: 10.1615/PlasmaMed.2012006275 76. Utsumi F, Kjiyama H, Nakamura K, Tanaka H, Mizuno M, Ishikawa K, et al. Effect of indirect nonequilibrium atmospheric pressure plasma on anti-proliferative activity against chronic chemo-resistant ovarian cancer cells in vitro and in vivo. PLoS ONE. (2013) 8:e81576. doi: 10.1371/journal.pone.0081576 77. Tanaka H, Mizuno M, Ishikawa K, Takeda K, Nakamura K, Utsumi F, et al. Plasma medical science for cancer therapy: toward cancer therapy using nonthermal atmospheric pressure plasma. IEEE Trans Plasma Sci. (2014) 42:3760. doi: 10.1109/TPS.2014.2353659 78. Mohades S, Barekzi N, Laroussi M. Efficacy of low temperature plasma against scaber cancer cells. Plasma Process Polym. (2014) 11:1150–55. doi: 10.1002/ppap.201400108 79. Laroussi M, Mohades S, Barekzi N. Killing of adherent and non-adherent cancer cells by the plasma pencil. Biointerphases. (2015) 10:029410. doi: 10.1116/1.4905666 80. Mohades S, Laroussi M, Sears J, Barekzi N, Razavi H. Evaluation of the effects of a plasma activated medium on cancer cells. Phys Plasmas. (2015) 22:122001. doi: 10.1063/1.4933367 81. Mohades S, Barekzi N, Razavi H, Maramuthu V, Laroussi M. Temporal evaluation of antitumor efficiency of plasma activated media. Plasma Process Polym. (2016) 13:1206. doi: 10.1002/ppap.201600118 82. Bauer G. Cold atmospheric plasma and plasma activated medium: antitumor cell effects with inherent synergistic potential. Plasma Med. (2019) 9:57–88. doi: 10.1615/PlasmaMed.2019029462 83. Weltmann KD, Kolb JF, Holub M, Uhrlandt D, Šimek M, Ostrikov K, et al. The future of plasma science and technology. Plasma Process Polym. (2019) 16:e1800118. doi: 10.1002/ppap.201800118 84. Bekeschus S, Favia P, Robert E, von Woedtke T. White paper on plasma for medicine and hygiene: future in plasma health sciences. Plasma Process Polym. 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https://itprospt.com/qa/14144/how-do-you-balance-albr	1 # How do you balance AlBr_3 + K_2SO_4 -> KBr + Al_2(SO_4)_3? ## Question ###### How do you balance AlBr_3 + K_2SO_4 -> KBr + Al_2(SO_4)_3? How do you balance AlBr_3 + K_2SO_4 -> KBr + Al_2(SO_4)_3? #### Similar Solved Questions ##### Use rational zeros theorem to find all the real zeros in the polynomial function use rational zeros theorem to find all the real zeros in the polynomial function. use zeros to factor f over real numbers. f(x)=x^3+10x^2-13x-22 find real zeros of f? use real zeros to factor f?... ##### Distillation experiment In vacuumdistillation, why is it important that heating of the flask isbegun only after the system is evacuated to the desiredpressure?... ##### Intangible Assets Use the note below from DirecTV’s 2012 annual report to answer the questions. Why... Intangible Assets Use the note below from DirecTV’s 2012 annual report to answer the questions. Why does DirecTV not amortize its orbital slots? 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In either case,$31,360... Taco Time Corporation is evaluating an extra dividend versus a share repurchase. In either case, $31,360 would be spent. Current earnings are$3.00 per share, and the stock currently sells for \$82 per share. There are 4,900 shares outstanding. Ignore taxes and other imperfections. What will the comp...	2022-05-26 13:31:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34869152307510376, "perplexity": 8505.620843954648}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662606992.69/warc/CC-MAIN-20220526131456-20220526161456-00662.warc.gz"}
https://blog.csdn.net/qq_41129306/article/details/102770647	# 1058 A+B in Hogwarts (20 分) 部分正确已解决 If you are a fan of Harry Potter, you would know the world of magic has its own currency system -- as Hagrid explained it to Harry, "Seventeen silver Sickles to a Galleon and twenty-nine Knuts to a Sickle, it's easy enough." Your job is to write a program to compute A+B where A and B are given in the standard form of Galleon.Sickle.Knut (Galleon is an integer in [0,107], Sickle is an integer in [0, 17), and Knut is an integer in [0, 29)). ### Input Specification: Each input file contains one test case which occupies a line with A and B in the standard form, separated by one space. ### Output Specification: For each test case you should output the sum of A and B in one line, with the same format as the input. ### Sample Input: 3.2.1 10.16.27 ### Sample Output: 14.1.28 #include<cstdio> const int Sickle = 29; const int Galleon = 1729; int main() { long long g1,s1,k1; long long g2,s2,k2; long long a1,a2,sum; long long g,s,k; scanf("%lld.%lld.%lld %lld.%lld.%lld",&g1,&s1,&k1,&g2,&s2,&k2); a1=Galleong1 + Sickles1 + k1; a2=Galleong2 + Sickles2 + k2; sum=a1+a2; k = sum%29; sum=(sum - k)/29; s = sum%17; sum=(sum-s)/17; g = sum; printf("%lld.%lld.%lld",g,s,k); return 0; } #include<cstdio> const int Sickle = 29; const int Galleon = 1729; int main() { int g1,s1,k1; int g2,s2,k2; int a1,a2,sum; int g,s,k; scanf("%d.%d.%d %d.%d.%d",&g1,&s1,&k1,&g2,&s2,&k2); a1=Galleong1 + Sickles1 + k1; a2=Galleong2 + Sickles2 + k2; sum=a1+a2; k = sum%29; sum=(sum - k)/29; s = sum%17; sum=(sum-s)/17; g = sum; printf("%d.%d.%d",g,s,k); return 0; } 03-10 42 07-17 590 01-02 169 09-08 322	2021-01-18 15:06:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27647408843040466, "perplexity": 11795.884264566896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703514796.13/warc/CC-MAIN-20210118123320-20210118153320-00314.warc.gz"}
https://questions.examside.com/past-years/jee/question/the-minimum-distance-of-a-point-on-the-curve-y-x24-from-jee-main-mathematics-trigonometric-functions-and-equations-yktu1cpal0jipgrf	1 ### JEE Main 2016 (Online) 9th April Morning Slot The minimum distance of a point on the curve y = x2−4 from the origin is : A ${{\sqrt {19} } \over 2}$ B $\sqrt {{{15} \over 2}}$ C ${{\sqrt {15} } \over 2}$ D $\sqrt {{{19} \over 2}}$ ## Explanation Let point on the curve y = x2 $-$ 4 is ($\alpha$2, $\alpha$2 $-$ 4) $\therefore$ Distance of the point ($\alpha$2, $\alpha$2 $-$ 4) from origin, D = $\sqrt {{\alpha ^2} + {{\left( {{\alpha ^2} - 4} \right)}^2}}$ $\Rightarrow$ D2 = $\alpha$2 + $\alpha$4 + 16 $-$ 8$\alpha$2 $=$ $\alpha$4 $-$ 7$\alpha$2 + 16 $\therefore$ ${{d{D^2}} \over {d\alpha }}$ = 4$\alpha$3 $-$ 14$\alpha$ Now, ${{d{D^2}} \over {d\alpha }}$ = 0 $\Rightarrow$ 4$\alpha$3 $-$ 14$\alpha$ = 0 $\Rightarrow$ 2$\alpha$ (2$\alpha$2 $-$ 7) = 0 $\alpha$ = 0 or $\alpha$2 = ${7 \over 2}$ ${{{d^2}{D^2}} \over {d{\alpha ^2}}} = 12{\alpha ^2} - 14$ $\therefore$ ${\left( {{{{d^2}{D^2}} \over {d{\alpha ^2}}}} \right)_{at\,\,\alpha = 0}} = - 14 < 0$ ${\left( {{{{d^2}{D^2}} \over {d{\alpha ^2}}}} \right)_{at\,\,{\alpha ^2} = {7 \over 2}}} = 28 > 0$ $\therefore\,\,\,$ Distance is minimum at $\alpha$2 = ${7 \over 2}$ $\therefore$ Minimum distance D = $\sqrt {{{49} \over 4} - {{49} \over 4} + 16}$ = ${{\sqrt {15} } \over 2}$ 2 ### JEE Main 2016 (Online) 10th April Morning Slot Let C be a curve given by y(x) = 1 + $\sqrt {4x - 3} ,x > {3 \over 4}.$ If P is a point on C, such that the tangent at P has slope ${2 \over 3}$, then a point through which the normal at P passes, is : A (2, 3) B (4, $-$3) C (1, 7) D (3, $-$ 4), ## Explanation Given, y = 1 + $\sqrt {4x - 3}$ $\therefore$ ${{dy} \over {dx}}$ = ${1 \over {2\sqrt {4x - 3} }} \times 4 = {2 \over 3}$ $\Rightarrow$ 4x $-$ 3 = 9 $\Rightarrow$ x = 3 $\therefore$ y = 1 + $\sqrt {12 - 3}$ = 4 $\therefore$ Equation of normal at point P(3,4) y $-$ 4 = $-$ ${3 \over 2}$ (x $-$ 3) $\Rightarrow$ 2y $-$ 8 = $-$ 3x + 9 $\Rightarrow$ 3x + 2y $-$ 17 = 0 3 ### JEE Main 2017 (Online) 8th April Morning Slot The tangent at the point (2, $-$2) to the curve, x2y2 $-$ 2x = 4(1 $-$ y) does not pass through the point : A $\left( {4,{1 \over 3}} \right)$ B (8, 5) C ($-$4, $-$9) D ($-$2, $-$7) ## Explanation As, ${{dy} \over {dx}}$ = $-$ $\left[ {{{{{\delta f} \over {\delta x}}} \over {{{\delta f} \over {\delta y}}}}} \right]$ ${{{\delta f} \over {\delta x}}}$ = y2 $\times$2x $-$ 2 ${{{\delta f} \over {\delta y}}}$ = x2 $\times$ 2y + 4 $\therefore\,\,\,$ ${{dy} \over {dx}}$ = $-$ $\left( {{{2x{y^2} - 2} \over {2{x^2}y + 4}}} \right)$ ${\left[ {{{dy} \over {dx}}} \right]_{(2, - 2)}}$ = $-$ $\left( {{{2 \times 2 \times 4 - 2} \over {2 \times 4 \times ( - 2) + 4}}} \right)$ = $-$ $\left( {{{14} \over { - 12}}} \right)$ = ${7 \over 6}$ $\therefore\,\,\,$ Slope of tangent to the curve = ${7 \over 6}$ Equation of tangent passes through (2, $-$ 2) is y + 2 = ${7 \over 6}$ (x $-$ 2) $\Rightarrow$$\,\,\,$ 7x $-$ 6y = 26 . . . . .(1) Now put each option in equation (1) and see which one does not satisfy the equation. By verifying each points you can see ($-$ 2, $-$ 7) does not satisfy the equation. 4 ### JEE Main 2017 (Online) 9th April Morning Slot A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point : A $\left( {{1 \over 3},24} \right)$ B $\left( {{1 \over 2},4} \right)$ C $\left( {2,{1 \over 8}} \right)$ D $\left( {3,{1 \over 28}} \right)$	2021-10-23 02:00:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8055638074874878, "perplexity": 2574.4576366518254}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585537.28/warc/CC-MAIN-20211023002852-20211023032852-00456.warc.gz"}
https://mathsgee.com/36392/subset-finite-dimensional-linear-spaces-rightarrow-linear	0 like 0 dislike 272 views Let $U \subset V$ and $W$ be finite dimensional linear spaces and $L: V \rightarrow W$ a linear map. Show that $\operatorname{dim}\left(\left.\operatorname{ker} L\right\|_{U}\right) \leq \operatorname{dim} \operatorname{ker} L=\operatorname{dim} V-\operatorname{dim} \operatorname{Im}(L)$ \| 272 views 1 like 0 dislike 1 like 0 dislike 2 like 0 dislike 1 like 0 dislike 0 like 0 dislike 2 like 0 dislike 1 like 0 dislike 1 like 0 dislike 0 like 0 dislike	2023-02-02 01:56:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7644431591033936, "perplexity": 7767.020106384212}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499954.21/warc/CC-MAIN-20230202003408-20230202033408-00424.warc.gz"}
http://www.opuscula.agh.edu.pl/om-vol34iss4art9	Opuscula Math. 34, no. 4 (2014), 777-788 http://dx.doi.org/10.7494/OpMath.2014.34.4.777 Opuscula Mathematica On some subclasses of the family of Darboux Baire 1 functions Gertruda Ivanova Elżbieta Wagner-Bojakowska Abstract. We introduce a subclass of the family of Darboux Baire 1 functions $$f:\mathbb{R}\rightarrow\mathbb{R}$$ modifying the Darboux property analogously as it was done by Z. Grande in [On a subclass of the family of Darboux functions, Colloq. Math. 17 (2009), 95-104], and replacing approximate continuity with $$\mathcal{I}$$-approximate continuity, i.e. continuity with respect to the $$\mathcal{I}$$-density topology. We prove that the family of all Darboux quasi-continuous functions from the first Baire class is a strongly porous set in the space $$\mathcal{DB}_1$$ of Darboux Baire 1 functions, equipped with the supremum metric. Keywords: Darboux property, strong Świątkowski property, Baire property, $$\mathcal{I}$$-approximate continuity, quasi-continuity. Mathematics Subject Classification: 26A15, 54C08. Full text (pdf)	2018-04-23 05:47:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8152466416358948, "perplexity": 1115.011372254674}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945793.18/warc/CC-MAIN-20180423050940-20180423070940-00629.warc.gz"}
http://www.senacmoda.info/museo-nacional-ucs/yje4f.php?1dfdfd=unit-matrix-of-order-2%C3%972	2. Note a that an m n matrix has mn elements. 2. Question: 9. Answer. A square matrix A with 1s on the main diagonal (upper left to lower right) and 0s everywhere else is called a unit matrix. Problems about idempotent matrices. v i. e. i (or simply . The unit group of the matrix ring Mn(R) is the general linear group GL(n;R) of n n invertible matrices over R. 3. This square of matrix calculator is designed to calculate the squared value of both 2x2 and 3x3 matrix. If the 2 × 2 matrix A whose rows are (2, 3) and (4, 5) is multiplied by itself, then the product, usually written A 2, has rows (16, 21) and (28, 37). We â¦ Identity Matrix is also called Unit Matrix or Elementary Matrix.Identity Matrix is denoted with the letter â I n×n â, where n×n represents the order of the matrix. 2 2 0 1 1 1 1 A: REMARK: The corresponding U and L in UL decomposition are typically dif-ferent from the ones obtained in the LU decomposition. : 4 3 1 1! Square matrix. Learn what an identity matrix is and about its role in matrix multiplication. A matrix O with all its elements 0 is called a zero matrix. To ï¬nd a Jordan chain of length 2, we pick a vector v1 that lies in the latter null space, but not in the former. In linear algebra, square matrix is a matrix which contains same number of rows and columns. 4. Prove that det A = u 11 u 11 â¦ u nn. Square Matrix Calculator. Number of rows and columns are equal therefore this matrix is a square matrix. 2 â2 2 â2 ââ 1 â1 0 0 , so the null space of A â 3I is one-dimensional. Type of Matrices 1. 2 2 -3 2 If A = 10 0, Then Show That A - 9A + 1013 = 0, Where 13 And 3 -1 -3 Are Unit Matrix And Null Matrix Of Order 3 Respectively. It is denoted by I n, or simply by I if the size is immaterial or can be trivially determined by the context. Consider the $2\times 2$ zero matrix. Eigenvalues and Eigenvectors of a 3 by 3 matrix Just as 2 by 2 matrices can represent transformations of the plane, 3 by 3 matrices can represent transformations of 3D space. For example, I3 = 1 0 0 0 1 0 0 0 1 . Preliminaryexample Suppose we calculate the product of the two matrices 4 3 1 1! But how to solve it in O(1) space? Counterexample We give a counterexample. A zero matrix or a null matrix is a matrix that has all its elements zero. Each entry in the matrix is called an element. Is A[3 1 -1 2] find A2-5A+7I where I is the unit matrix of order 2×2 - 25774322 Step-by-step explanation: 9No. If I is the unit matrix of order 2 × 2 and M â 2 I = 3 [â 1 4 0 1 ], then find the matrix M. December 26, 2019 Deboshree Mirza. and 1 â3 â1 4! The identity matrix of order m, written Im (or simply I, when no confusion arises) is a square m-by-m matrix with ones along the diagonal and zeros elsewhere. The zero matrix is a diagonal matrix, and thus it is diagonalizable. d) order: 2 × 2. Let matrix be A where A = [ 8(ð11&ð12@ð21&ð22)] Now it is given that ail = ð/ð Ex 3.1, 4 Construct a 2 × 2 matrix, A = [aij], whose elements are given by: (iii) aij =(ð +2ð)2/2 Since it is a 2 × 2 matrix it has 2 rows & 2 column. Solve related Questions. For eigen values of a matrix first of all we must know what is matric polynomials, characteristic polynomials, characteristic equation of a matrix. If AB=A, BA=B, then A is idempotent. An n x n matrix â¦ A matrix with one row is called a row matrix (or a row vector). 4 3 1 1! On the other hand, to multiply A on the left by the identity, you have to use I 2, the 2×2 identity, in order to have the right number of columns: That is, if you are dealing with a non-square matrix (such as A in the above example), the identity matrix you use will depend upon the side that you're multiplying on. Deï¬nition. v. i) can be used to denote a vector. U(Z) = f 1;1g is a cyclic group of order 2. For example matrices with dimensions of 2x2, 3x3, 4x4, 5x5 etc., are referred to as square matrix. Propertiesof the3× 3 rotationmatrix A rotation in the xây plane by an angle Î¸ measured counterclockwise from the positive x-axis is represented by the real 2×2 special orthogonal matrix,2 cosÎ¸ âsinÎ¸ sinÎ¸ cosÎ¸ . De nition 1.3.4 A ring with identity is â¦ In this post, we explain how to diagonalize a matrix if it is diagonalizable. A matrix is a rectangular array of numbers (or other mathematical objects) for which operations such as addition and multiplication are defined. Another notation is the . A inverse exists. Matrices are classified by the ... the product matrix AB exists, and has order 2×2. December 26, 2019 Toppr. Problem 5: (a)Write down a permutation matrix P that reverses the order of the rows of a 3 3 matrix. Matrices are represented in the Wolfram Language with lists. Let C be invertible such that C-1 AC = U is an n´ n upper triangular matrix. e) order: 1 × 1. If A is an idempotent matrix, then so is I-A. à¤¸à¥ à¤¨à¤à¤¦ à¤ªà¥à¤°à¤¾à¤ªà¥à¤¤ (Received Cash from N. Co.) an inverse matrix and how the inverse of a 2× 2 matrix is calculated. User can select either 2x2 matrix or 3x3 matrix for which the squared matrix to be calculated. Structural Analysis IV Chapter 4 â Matrix Stiffness Method 9 Dr. C. Caprani LinPro LinPro is very useful as a study aid for this topic: for example, right click on a member and select âStiffness Matrixâ to see the stiffness matrix for any member. For example, A = is a 2 2 matrix and B = is a 2 3 order matrix. However, the zero matrix is not [â¦] How to Diagonalize a Matrix. They can be entered directly with the { } notation, constructed from a formula, or imported from a data file. Since the matrix n x n then it has n rows and n columns and obviously n diagonal elements. The symbolic notation . Given 4 2 -1 1 M = 6i , Where M is a Matrix and I is Unit Matrix of Order 2×2. So we. Number of rows and columns are not equal therefore not a square matrix. Determine k such that I-kA is idempotent. Let A be a 2x2 matrix â¦ For a 2´ 2 matrix A over a field F, show the equation AX = I, where X is a 2´ 2 matrix, has a solution and that it is unique iff det A ¹ 0; and thereby obtain a formula for A-1. = 1 0 0 1! Number of rows and columns are equal therefore this is square matrix. You should verify this: 1 â3 â1 4! We denote the unit matrix having n rows (and n columns) by In. 1 â3 â1 4! The latest version (2.7.3) has a very useful âStudy Modeâ, which exposes the structure general form for the matrix representation of a three-dimensional (proper) rotations, and examine some of its properties. If you're seeing this message, it means we're having trouble loading external resources on our website. In linear algebra, the identity matrix (sometimes ambiguously called a unit matrix) of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. 7.1.2 Matrix Notation . 1. Definition. A diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero . A square matrix is of order m × m. A square matrix is symmetric if For example: 1 2 2 2 8 9 5 9 4; A square matrix is skew-symmetric if For example 0 2 5-2 0 9-5-9 0; Diagonal Matrices. The unit group of Q is denoted Q and consists of all non-zero rational numbers. Example: O is a zero matrix of order 2 × 3 A square matrix is a matrix with an equal number of rows and columns. c) order: 1 × 4. If M =(1, 2),(2, 1) and I is a Unit Matrix of the Same Order as that of M Show that M^2 = 2m + 3i Concept: Multiplication of Matrix. Check that P2 = I. Note that a unit matrix is a scalar matrix with is on the main diagonal. The Wolfram Language also has commands for creating diagonal matrices, constant matrices, and other special matrix types. v and index notation . We allocate memory for n x n matrix and for every element starting from nn to 1, we start filling out matrix in spiral order. Similar results can be obtained for Hermitian matrices of order In other words, a square matrix A is Hermitian if and only if the following two conditions are met. so that they are unit vectors. ... View Answer. Identity Matrix (Unit Matrix) 2 ×2 matrix, and interpret their significance in relation to an associated plane transformation. 9.0 Introduction A matrix is a rectangular array of numbers. (b)Given a lower-triangular matrix L, show how you can multiply (possibly mul- 1. We work through two methods of finding the characteristic equation for Î», then use this to find two eigenvalues. Such a set of orthogonal unit vectors is called an ... 2 2 3 23 2 22 1 21 1 1 3 13 2 12 1 11 b a ... One free index, as here, indicates three separate equations. I. Solution. The answer is No. Online calculator to perform matrix operations on one or two matrices, including addition, subtraction, multiplication, and taking the power, determinant, inverse, or transpose of a matrix. In many areas such as electronic circuits, optics, quantum mechanics, computer graphics, probability and statistics etc, matrix is used to study. We can add or multiply any two square matrices that are of the same order. On the other hand, (A â 3I)2 is the zero matrix, so its null space is two-dimensional. OK. Let us first analyse condition given Det(A) not equal to zero which implies that the matrix A is not non zero matrix. 5. A square matrix in which all the main diagonal elements are 1âs and all the remaining elements are 0âs is called an Identity Matrix. To maintain the spiral order four loops are used, each for top, right, bottom and left corner of the matrix. Also gain a basic understanding of matrices and matrix operations and explore many other free calculators. Step by Step Explanation. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. The vectors and can also be shown to be unit vectors. (I) State the Order of Matrix M. (Ii) Find the Matrix M. Concept: Matrices Examples. It is important to note that for any m-by-m matrix B, BIm = Im B â¦ If we re-order the matrices and recalculate we will obtain the same result. = 1 0 0 1! Can use first conditions that det(A) not equal to zero For any. Is idempotent be shown to be unit vectors all its elements 0 is called an Identity matrix behind web! Being zero if you 're seeing this message, it means we 're having loading. Row is called a zero matrix a null matrix is a rectangular array of numbers of all non-zero numbers., are referred to as square matrix that has all its elements is... Can also be shown to be unit vectors will obtain the same order triangular matrix, BIm = Im â¦! Can use first conditions that det a = u 11 u 11 u u. Domains .kastatic.org and .kasandbox.org are unblocked constructed from a formula, or imported a. ( Ii ) Find the matrix representation of a â 3I ) 2 is the zero matrix are used each. If you 're behind a web filter, please make sure that the domains .kastatic.org ... That an M n matrix has mn elements ) Write down a permutation matrix P that reverses order! 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Comentários	2022-01-27 06:00:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8396808505058289, "perplexity": 610.3027416549529}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305141.20/warc/CC-MAIN-20220127042833-20220127072833-00457.warc.gz"}
https://www.physicsforums.com/threads/moment-of-inertia-of-disk.651538/	# Moment of Inertia of disk ## Homework Statement I am trying to find the moment of inertia of a disc (let the mass be m and radius R) inclined at an angle θ to the vertical axis. (See attachment 1) ## The Attempt at a Solution I started by taking a small element of area dA. (see attachment 2) The mass of this small element is dA*Mass density of disc. dA=xd$\phi$dx and mass density=m/($\pi$R^2) Now, moment of inertia is defined as I=∫dmr^2 (Here r=xsinθ) $$I=\int_{0}^{2\pi} \int_{0}^{R} \frac{m}{\pi R^2}xd \phi dx(x\sinθ)$$ Solving this, i get $$I=\frac{4}{3}mRsinθ$$ which i think is completely wrong. #### Attachments • 6.9 KB Views: 410 • 5.2 KB Views: 391 ## Answers and Replies ehild Homework Helper Hi Pranav, It is easier to imagine the problem if you keep that disk horizontal, in the xy plane and the axis in the yz plane is inclined to it with angle β. The distance d of a point P of the disk from the axis is r sin(θ) where θ is the angle the axis and the position vector r enclose: It is not the same as β. ehild #### Attachments • 10 KB Views: 424 Hi Pranav, It is easier to imagine the problem if you keep that disk horizontal, in the xy plane and the axis in the yz plane is inclined to it with angle β. The distance d of a point P of the disk from the axis is r sin(θ) where θ is the angle the axis and the position vector r enclose: It is not the same as β. ehild Yes, its now easier to understand the problem. In my method, i did not realize that θ too varies. But now how am i going to form the integral? ehild Homework Helper At given β, d depends both on φ and r. Find the expression first. ehild At given β, d depends both on φ and r. Find the expression first. ehild I have tried it for more than half hour but still have no clue, i am unable to express d in terms of φ and r. ehild Homework Helper "d" is the distance of a point from the axis. Can you find the distance of a point from a straight line? A point on the axis can be written as the vector b=bt where t is the unit vector along the axis. t=cosβj+sinβk. A point P of the disk is represented by the vector r=rcosφi+rsinφj. The distance between a point of the axis and P is equal to the magnitude of the difference r-b, and the distance of P from the axis is the shortest distance. Find the minimum of (r-b)2. ehild Last edited: Can you find the distance of a point from a straight line? Yes. ehild said: The distance between a point of the axis and P is equal to the magnitude of the difference r-b, and the distance of P from the axis is the shortest distance. Find the minimum of (r-b)2. ehild I get r2(1-cos2φcos2β) as the minimum value of (r-b)2. Is this correct? ehild Homework Helper Yes. I get r2(1-cos2φcos2β) as the minimum value of (r-b)2. Is this correct? I got the same. You can integrate now for the disk ehild I got the same. You can integrate now for the disk ehild Thanks for the help ehild! Here's my attempt: $$dI=\frac{m}{\pi R^2}r(d\varphi)(dr)\cdot r^2(1-\cos^2\varphi \cos^2\beta)$$ Integrating the above expression under the appropriate limits, i get $$I=\frac{mR^2}{2}\left(1-\frac{cos^2\beta}{2}\right)$$ Is this correct? Is it valid to use the parallel axis theorem here if i want the MI about an axis parallel to the given axis in the question? haruspex Homework Helper Gold Member 2020 Award $$I=\frac{mR^2}{2}\left(1-\frac{cos^2\beta}{2}\right)$$ Looks right. Is it valid to use the parallel axis theorem here if i want the MI about an axis parallel to the given axis in the question? Sure. Why not? ehild Homework Helper Checking my calculation again, I got d2=r2(1-cos2β sin2φ) with φ the angle with respect to the x axis, as shown in my post #2. But that does not influence the final result. The parallel axis theorem is valid for any shape and any parallel axes. ehild	2021-01-19 12:51:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.765599250793457, "perplexity": 545.2316476801435}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703518240.40/warc/CC-MAIN-20210119103923-20210119133923-00192.warc.gz"}
https://gsebsolutions.com/gseb-solutions-class-12-maths-chapter-6-ex-6-5/	# GSEB Solutions Class 12 Maths Chapter 6 Application of Derivatives Ex 6.5 Gujarat Board GSEB Textbook Solutions Class 12 Maths Chapter 6 Application of Derivatives Ex 6.5 Textbook Questions and Answers. ## Gujarat Board Textbook Solutions Class 12 Maths Chapter 6 Application of Derivatives Ex 6.5 Question 1. Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = (2x – 1)² + 3 (ii) f(x) = 9x² + 12x + 2 (iii) f(x) = – (x -1)² + 10 (iv) g(x) = x³ + 1 Solution: (i) Minimum value of (2x – 1)² is zero. ∴ Minimum value of (2x – 1)² + 3 is 3. There is no finite maximum value. (ii) f(x) = 9x² + 12x + 2 = (3x + 2)² – 2 Minimum value of (3x + 2)² is zero. ∴ Minimum value of (3x + 2)² – 2 = 9x² + 12x + 2 is – 2. f(x) does not have any finite maximum value. (iii) f(x) = – (x- 1)² + 10 Maximum value of – (x – 1)² is zero ∴ Maximum value of f(x) = – (x – 1)² + 10 is 10. f(x) does not have any finite minimum value. (iv) f(x) = x³ + 1 As x → ∞, f(x) → ∞. As x → – ∞, f(x) → – ∞. Moreover, f(x) = 3x² + 1 = +ve for all x → R. ∴ f(x) is an increasing function throughout its domain. Hence, there is no maximum and minimum values. Question 2. Find the maximum and minimum values, if any, of the following functions given, by (i) f(x) = \|x + 2\| – 1 (ii) g(x) = -\|x + 1\| + 3 (iii) h(x) = sin (2x) + 5 (iv) f(x) = \| sin 4x + 3 \| (v) h(x) = x + 1, x ∈ (-1,1) Solution: (i) \| x + 2 \| > 0, x ∈ R, f(x) = [x + 2] – 1 > – 1 Minimum value of f(x) is – 1. There is no maximum value. (ii) Let g(x) = – \| x + 1\| + 3. Now, [x + 1] ≥ 0 for all x ∈ R. ⇒ – [x + 1] ≤ 0 for all x ∈ R. – [x + 1] + 3 ≤ 3 ⇒ g(x) < 3. Thus, the maximum value of f(x) is 3, there is no minimum value. (iii) Let f(x) = sin 2x + 5. Maximum value of sin 2x is 1. Maximum value of sin 2x + 5 = 1 + 5 = 6. Minimum value of sin 2x is – 1. ∴ Minimum value of ski 2x + 5 = – 1 + 5 = 4. (iv) Let fix) = [sin 4x + 3]. Maximum value of sin 4x is 1. ∴ Maximum value of [sin 4x + 3] is [1 + 3] = 4. Minimum value of sin 4x is – 1. ∴ Minimum value of fix) is [-1 + 3] = [2] = 2. (v) Let h(x) = x + 1. As x → ∞, h(x) → ∞. Also, x → – ∞, h(x) → – ∞. Thus, there is no maximum or minimum value. However, at x = 1, h(x) = 1 + 1 = 2 and at x = -1, h(x) = -1 + 1 = 0. ∴ Greatest value of h(x) is 2 and least value is 0. Question 3. Find the local maxima or local minima, if any, of the following functions. Find also the local maximum and local minimum values, as the case may be : (i) f(x) = x² (ii) g(x) = x³ – 3x (iii) h(x) = sin x + cos x, 0 < x < $$\frac { π }{ 2 }$$ (iv) f(x) = sin x – cos x, 0 < x < 2π (v) f(x) = x³ – 6x² + 9x + 15 (vi) g(x)= $$\frac { x }{ 2 }$$ + $$\frac { 2 }{ x }$$, x> 0 (vii) g(x) = $$\frac{1}{x^{2}+2}$$ (viii) f(x) = x$$\sqrt{1-x}$$, x > 0 Solution: (i) Let f(x) = x² ⇒ f'(x) = 2x. Now, f'(x) = 0 ⇒ 2x = 0 i.e., x = 0. At x = 0 : When x is slightly < 0, f'(x) is – ve. When x is slightly > 0,f'(x) is + ve. ∴ f'(x) changes sign from – ve to + ve as x increases through 0. ⇒ f(x) has a local minima at x = 0. Local minimum valus = f(0) = 0. (ii) Let g(x) = x³ – 3x ∴ g'(x) = 3x² – 3 = 3(x²-1) = 3(x-1)(x+1) Now, g’ (x) = 0 ⇒ 3(x-1)(x+1) = 0 ⇒ Either x = 1 or x = – 1. At x = 1: When x is slightly < 1, g’(x) is (+)(-)(+), i..e, – ve. When x is slightly > 1, g'(x) is (+)(+)(+), i.e.,+ve. Thus g'(x) changes sign from negative to positive as x increases through 1 and hence x = 1 is a point of local minima. At x = – 1: When x is slightly < – 1, g'(x) is (+)(-)(-), i.e., +ve. When x is slightly > – 1, g'(x) is (+)(-)(+), i.e. -ve. Thus g'(x) changes sign from positive to negative as x increases through -1 and hence x = -1 is a point of local maxima. Local minimum value = f(1) = 1 – 3 = – 2. Local maximum value = f(- 1) = – 1 + 3 = 2. (iii) h(x) = sin x + cos x ∴ h’(x) =cos x – sin x = cos x(1 – tan x) h’(x) = 0 ⇒ cos x – sin x = 0. or tan x = 1 ⇒ x = $$\frac { π }{ 4 }$$. x = $$\frac { π }{ 4 }$$ ∈ (0, $$\frac { π }{ 2 }$$). At x = $$\frac { π }{ 4 }$$, h’(x) = cos x (1 – tan x) = 0 When x is slightly < $$\frac { π }{ 4 }$$, cos x = + ve and tan x = 1 – h, where h is small. So, 1 – tan x = 1 – (1 – h) = + ve. ⇒ h’(x) = cos x (1 – tan x) = (+)(+) = +ve. When x is slightly > $$\frac { π }{ 4 }$$, cos x = + ve and tan x = 1 + h, where h is small ∴ 1 – tan x = 1 – (1 + h) = – h = – ve. ∴ h’(x) = cos x(1 – tan x) = (+)(-) = – ve. Hence, there is a local maxima at x = $$\frac { π }{ 4 }$$ Local maximum value = h$$\frac { π }{ 4 }$$ = sin $$\frac { π }{ 4 }$$ + cos $$\frac { π }{ 4 }$$ = $$\frac{1}{\sqrt{2}}$$ + $$\frac{1}{\sqrt{2}}$$ = $$\frac{2}{\sqrt{2}}$$ = $$\sqrt{2}$$. (iv) f(x) = sin x – cos x, 0 < x < 2π f’(x) = cos x + sin x = cos x(1 + tan x) f’(x) = 0 ⇒ cos x + sin x = 0 = tan x = – 1. ∴ x = $$\frac { 3π }{ 4 }$$, $$\frac { 7π }{ 4 }$$ At x = $$\frac { 3π }{ 4 }$$ (a) When x is slightly < $$\frac { 3π }{ 4 }$$, cos x = – ve, tan x = – (1+h) = – 1 – h ∴ 1 + tan x = 1 + (- 1 – h) = – h = + ve (b) When x is slightly < $$\frac { 3π }{ 4 }$$, cos x = – ve, tan x = – (1+h) = – 1 – h ∴ 1 + tan x = 1 + (- 1 – h) = – h = + ve ∴ f'(x) = cos x (1 + tan x) = (-)(+) = – ve. ∴ f'(x) changes sign from +ve to – ve ∴ At x = $$\frac { 3π }{ 4 }$$, there is point pf local maxima. Local maximum value = f ($$\frac { 3π }{ 4 }$$) = sin $$\frac { 3π }{ 4 }$$ – cos $$\frac { 3π }{ 4 }$$ = $$\frac{1}{\sqrt{2}}$$ + $$\frac{1}{\sqrt{2}}$$ = $$\frac{2}{\sqrt{2}}$$ = $$\sqrt{2}$$. At x = $$\frac { 7π }{ 4 }$$, (a) When x is slightly < $$\frac { 7π }{ 4 }$$, cos x = – ve, tan x = – (1+h) ⇒ 1 + tan x = 1 – (1+h) = – h ∴ f'(x) = cos x (1 + tan x) = (-)(+) = – ve. (b) When x is slightly > $$\frac { 7π }{ 4 }$$, cos x = – ve, tan x = – (1+h) ⇒ 1 + tan x = 1 – (1 + h) = – h ∴ f'(x) = cos x (1 + tan x) = (-)(+) = – ve. ∴ f'(x) changes sign from – ve to + ve. Hence, there is a local minima at x = $$\frac { 7π }{ 4 }$$. (v) Let f(x) = x³- 6x² + 9x + 15 f(x) = 3x² -12x + 9 = 3(x² – 4x + 3) = 3(x – 1)(x – 3). Now, f(x) = 0 ⇒ x = 1, 3. At x = 1: When x is slightly < 1, f(x) = (-)(-) = +ve. When x is slightly > 1, f(x) = (+)(-) = -ve. ∴ f'(x) changes sign from +ve to -ve as x increases through 1. ⇒ f(x) has a local maxima at x = 1. Local maximum value = f( 1) = 1 – 6 + 9 + 15 = 19. At x = 3 : When x is slightly < 3, fix) = (+)(-) = -ve. When x is slightly > 3, f\x) = (+)(+) = +ve. f'(x) changes sign from -ve to +ve as x increases through 3. ⇒ f(x) has a local minima at x = 3. ∴ Local minimum value = f(3) = (3)³ – 6(3)² + 9(3) + 15 = 27-54 + 27 + 15 = 15. [Note : How to determine the change of sign of f(x) as x increases through a particular point? Let f’ (x) = (x – a)(x – b)(x – c) At x = a, x – a = a – a = 0, x – b = a – b, x – c = a – c. Sign corresponding to x – a changes from -ve to +ve, sign responding to x – b is that of a – b and remains the same. Sign corresponding to x – c is that of a – c and remains the same. Let f'(x) = (x – 1)(x – 3) At x = 1, x – 1 = 1 – 1 = 0, x – 3 = 1 – 3 = – 2 = – ve Factor (x – 1)(x – 3) Signs when x is slightly < 1 are (-)(-) = +ve. Signs when x is slightly > 1 are : (+)(-) = -ve ∴ f'(x) changes from +ve to -ve At x = 3, x – 1 = 3 – 1 = 2 = +ve, x – 3 = 3 – 3 = 0 Factor (x-1)(x-3) Sign when x is slightly < 3, (+)(-) = -ve Sign when x is slightly > 3, (+)(+) = +ve ∴ f(x) changes sign from -ve to +ve (vi) Since it is given that x > 0, hence x = – 2 is rejected. For x slightly < 2, g'(x)= $$\frac{(-)(+)}{(+)}$$ = – ve. For x slightly > 2, g ‘(x) = $$\frac{(+)(+)}{(+)}$$ = +ve. Thus, g'(x) changes sign from -ve to +ve. Hence, g(x) has a local minima at x = 2. Local minimum value = g(x) = $$\frac { 2 }{ 2 }$$ + $$\frac { 2 }{ 2 }$$ = 1 + 1 = 2. (vii) Let g(x) = $$\frac{1}{x^{2}+2}$$ ∴ g'(x) = – $$\frac{2 x}{\left(x^{2}+2\right)^{2}}$$ Now, g'(x) = 0 ⇒ x = 0 At x = 0 : When x is slightly < 0, g'(x) = $$\frac{(-)(-)}{(+)}$$ = + ve. When x is slightly > 0, g'(x) = $$\frac{(-)(+)}{(+)}$$ = – ve. ∴ g'(x) changes sign from – ve to + ve as x increases through 0. ⇒ f(x) has a local minima at x = 0. Local minimum valus = g(0) = $$\frac { 1 }{ 2 }$$. (viii) When x is slightly < $$\frac { 2 }{ 3 }$$ f(x) = $$\frac { (-)(-) }{ (+) }$$ = + ve. When x is slightly > $$\frac { 2 }{ 3 }$$ f'(x) = $$\frac { (-)(+) }{ (+) }$$ = – ve. ∴ f'(x) changes sign from +ve to -ve as x increases through x = $$\frac { 2 }{ 3 }$$ ∴ f(x) has a local maxima at x = $$\frac { 2 }{ 3 }$$. Local maximum value = f($$\frac { 2 }{ 3 }$$) = $$\frac { 2 }{ 3 }$$$$\sqrt{1-\frac{2}{3}}$$ = $$\frac { 2 }{ 3 }$$ . $$\frac{1}{\sqrt{3}}$$ = $$\frac{2 \sqrt{3}}{9}$$ Question 4. Prove that the following functions do not have maxima or minima : (i) f(x) = ex (ii) g(x) = log x (iii) h(x) = x³ + x² + x + 1 Solution: (i) f(x) = ex ⇒ f'(x) = ex. For maxima and minima, f'(x) = 0. ⇒ ex = 0, which is not defined for any finite value. Hence f'(x) = ex does not have maxima and minima. (ii) We have : g(x) = log x. ∴ g'(x) = $$\frac { 1 }{ x }$$. Now, g'(x) = 0 ⇒ $$\frac { 1 }{ x }$$ = 0, which is not defined for any real x. Hence, g(x) = log x does not have maxima and minima. (iii) We have: h(x) = x³ + x² + x + 1. h'(x) = 3x² + 2x + 1 Now, h'(x) = 0 ⇒ 3x² + 2x + 1 =0 ⇒ x = $$\frac{-2 \pm \sqrt{4-12}}{6}$$ = $$\frac{-1 \pm \sqrt{-2}}{3}$$ i.e., h'(x) = 0 at imaginary points. i.e., h'(x) ≠ 0 for any real value of x. Hence, there is neither maximum nor minimum. Question 5. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) f(x) = x³[-2, 2] (ii) f(x) = sin x + cos x, x ∈ [0, π] (iii) f(x) = 4x – x², x ∈ [-2, $$\frac { 9 }{ 2 }$$] (iv) f(x) = (x – 1)² + 3, x ∈ [-3, 1] Solution: (i) We have: f(x) = x³ in [- 2, 2]. ∴ f'(x) = 3x² Now, f'(x) = 0 at x = 0, f(0) = 0. Now, f(- 2) = (-2)³ = – 8; f(0) = (0)³ = 0 and f(2) = (2)³ = 8 Hence, the absolute maximum value of f(x) is 8, which is attained at x = 2 and absolute minimum value of f(x) = – 8, which is attained at x = – 2. (ii) We have : f(x) = sin x + cos x in [0, π]. ∴ f'(x) = cos x – sin x For extreme values, f'(x) = 0. ⇒ cos x – sin x = 0 ⇒ 1 – tan x = 0 ⇒ tan x = 1 ⇒ x = $$\frac { π }{ 4 }$$. Now, we find f(x) at x = 0, $$\frac { π }{ 4 }$$ and π. ∴ Absolute maximum value = $$\sqrt{2}$$ at x = $$\frac { π }{ 4 }$$ and absolute minimum value = 3 at x = π. (iii) At x = 4, absolute maximum value = 8. At x = – 2, absolute minimum value = – 10. (iv) We have : f(x) = (x – 1)² + 3 in [- 3,1]. ∴ f'(x)= 2(x-1). For extreme values, f'(x) = 0. ⇒ 2(x-1) = 0 ⇒ x – 1 = 0 ⇒ x = 1. Now, we find f(x) at x = 1 and – 3. f(1) = (1-1)² + 3 = 0 + 3 = 3. f(-3) = (- 3 – 1)² + 3 = 16 + 3 = 19. Question 6. Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 – 24x -18². Solution: Profit function is p(x) = 41 – 24x – 18x². ∴ p'(x) = – 24 – 36x = – 12(2 + 3x) For maxima and minima, p'(x) = 0. Now, p'(x) = 0 ⇒ – 12(2 + 3x) = 0. ⇒ 2 + 3x = 0 ⇒ x = – $$\frac { 2 }{ 3 }$$ p’(x) changes sign from +ve to -ve as it passes through – $$\frac { 2 }{ 3 }$$ ⇒ p(x) has a maximum value at x = – $$\frac { 2 }{ 3 }$$. ∴ Maximum profit = p$$\frac { 2 }{ 3 }$$ = 41 – 24(-$$\frac { 2 }{ 3 }$$) – 18(-$$\frac { 2 }{ 3 }$$)² = 41 + 16-8 = 49. Question 7. Find both the maximum and minimum values of 3x4 – 8x³ + 12x² – 48x + 25 on the interval [0, 3]. Solution: Let f(x) = 3x4 – 8x³ + 12x² – 48x + 25. ∴ f'(x) = 12x³ – 24x² + 24x – 48 = 12(x³ – 2x² + 2x – 4) = 12[x²(x – 2) + 2(x – 2)] = 12(x² + 2)(x – 2) For maxima and minima, f'(x) = 0. ⇒ 12(x² + 2)(x – 2) = 0 ⇒ x = 2. Now, we find f(x) at x = 0, 2 and 3. f(0) = 25, f(2) = 3(2)4 – 8(2)³ + 12(2)² – 48(2) + 25 = 48 – 64 + 48 – 96 + 25 = – 39. and f(3) = 3(3)4 – 8(3)³ + 12(3)2 – 48(3) + 25 = 243 – 216 + 108 -144 + 25 = 16. Hence, at x = 0, maximum value = 25, and at x = 2, minimum value = – 39. Question 8. At what points in the interval [0, 2π], does the function sin 2x attains its maximum value? Solution: We have : f(x) = sin 2x in [0, 2π]. ∴ f’ (x) = 2 cos 2x. For maxima and minima f'(x) = 0. ⇒ 2 cos 2x = 0 ⇒ cos 2x = 0 ⇒ 2x = $$\frac { π }{ 2 }$$, $$\frac { 3π }{ 2 }$$, $$\frac { 5π }{ 2 }$$, $$\frac { 7π }{ 2 }$$ x = $$\frac { π }{ 4 }$$, $$\frac { 3π }{ 4 }$$, $$\frac { 5π }{ 4 }$$, $$\frac { 7π }{ 4 }$$. Now, we find f(x) at x = 0, $$\frac { π }{ 4 }$$, $$\frac { 3π }{ 4 }$$, $$\frac { 5π }{ 4 }$$, $$\frac { 7π }{ 4 }$$ and 2π. Hence, maximum value of f(x) = 1 is at x = $$\frac { π }{ 4 }$$ and x = $$\frac { 5π }{ 4 }$$. Question 9. What is the maximum value of the function sin x + cos x? Solution: Consider the interval [0, 2π]. Let f(x) = sin x + cos x. ⇒ f'(x) – cos x – sin x. , For maxima and minima, f'(x) = 0 ⇒ cos x- sin x = 0 ⇒ tan x = 1. ⇒ x = $$\frac { π }{ 4 }$$ and $$\frac { 5π }{ 4 }$$ Now, we find f(x) at x = 0, $$\frac { π }{ 4 }$$, $$\frac { 5π }{ 4 }$$ and 2π. Hence, maximum value of f(x) = $$\sqrt{2}$$ Question 10. Find the maximum value of 2x³ – 24x +107 in the interval [1, 3]. Find the maximum value of the same function in [- 3, -1]. Solution: We have : f(x) = 2x³ – 24x + 107 in [1, 3]. f'(x) = 6x² – 24. For maxima and minima, f'(x) = 0. ⇒ 6x² – 24 = 0 ⇒ x = ± 2. For the interval [1, 3], we find the values of f(x) at x = 1, 2, 3 f(1) = 2(1)³ – 24 x 1 + 107 = 85, f(2) = 2(2)³ – 24 x 2 + 107 = 75 and f(3) = 2(3)³ – 24 x 3 + 107 = 89. Hence, maximum value of f(x) = 89 at x = 3. For the interval [- 3,-1], we find the values off(x) at x = – 3, – 2 and – 1. f(- 3) = 2(- 3)³ – 24 (- 3) + 107 = 125, f(- 2) = 2(- 2)³ – 24 (- 2) + 107 = 139 and f(-1) = 2(-1)³ – 24 (-1) + 107 = 129. Hence, maximum value of f(x) = 139 at x = – 2. Question 11. It is given that at x – 1, the function x4 – 62x² + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a. Solution: We have: f(x) = x4 – 62x² + ax + 9. ∴ f'(x) = 4x³ – 124x + a. Now, f'(x) = 0 at x = 1 [Given] ∴ f'(1) = 4 – 124 + a = 0 ⇒ a = 120. Now, f”(x) = 12x² – 124 At x = 1 : f”(1) = 12 – 124 = – 112 < 0. ⇒ f(x) has a maximum at x = 1, when a = 120. Hence, a = 120. Question 12. Find the maximum and minimum values of x + sin 2x on [0,2 π]. Solution: We have: f(x) = x + sin 2x on [0, π]. f'(x) = 1 + 2 cos 2x. For maxima and minima, Hence, maximum value of f(x) = 2π and minimum value of f(x) = 0. Question 13. Find two numbers whose sum is 24 and whose product is as large as possible. Solution: Let the required numbers be x and (24 – x). Their product P = x(24 -x) = 24x – x². Now, $$\frac { dp }{ dx }$$ = 0 ⇒ 24 – 2x = 0 ⇒ x = 12. Also, $$\frac{d^{2} P}{d x^{2}}$$ = – 2 < 0. ⇒ P is maximum at x = 12. Hence, the required numbers are 12 and (24 -12), i.e., 12. Question 14. Find two positive numbers x and y such that x + y = 60 and xy³ is maximum. Solution: The value x = 60 is rejected as it makes y = 0. At x = 15 : When x is slightly < 15, $$\frac { dp }{ dx }$$ = (+)(+) = +ve. When x is slightly > 15, $$\frac { dp }{ dx }$$ = (+)(-) = -ve. ⇒ $$\frac { dp }{ dx }$$ changes sign from +ve to -ve as x increases through 15. ∴ P is maximum at x = 15. Hence, the required numbers are 15 and (60 – 15), i.e., 45. Question 15. Find two positive numbers x and y such that their sum is 35 and the product x²y5 is a maximum. Solution: We have : x + y = 35 ⇒ y = 35 – x … (1) So, product: P = x²5 – x²(35 – x)5 [Using (1)] ∴ $$\frac { dp }{ dx }$$ = x².5(35 – x)4(-1) + (35 – x)5.2x = x(35-x)4[-5x + 2(35-x)] = x(35 – x)4 (70 – 7x) Now, $$\frac { dp }{ dx }$$ = 0. ⇒ x(35 – x)4(70 – 7x) = 0. ⇒ x = 0, 10, 35. Only admissible values is x = 10 as x = 0 and 35 are rejected. At x = 10: When x is slightly < 10, $$\frac { dp }{ dx }$$ = (+)(+)(+) = +ve. When x is slightly > 10, $$\frac { dp }{ dx }$$ = (+)(+)(-)= -ve. ⇒ $$\frac { dp }{ dx }$$ changes sign from + veto-ve as x increases through 10. ⇒ P is maximum at x = 10. From (1), y = 35 -10 = 25. Hence, the required numbers are 10 and 25. Question 16. Find two positive numbers whose sum is 16 and sum of whose cubes is minimum. Solution: Let two numbers be x and 16 – x. Hence, the required numbers are 8 and (16-8), i.e., 8. Question 17. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box ¡s the maximum possible? Solution: Let each side of the square lo be cut off be x cm. ∴ For the box: Volume is maximum, when x = 3, i.e., square of side = 3 cm is cut from each comer. Question 18. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cuttting off square from each comer and folding up flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible? Solution: Let each side of the square cut off from each comer be x cm. ∴ Sides of the rectangular box are (45 – 2x), (24 – 2x) and x cm. Then, volume V of the box is given by ∴ V is maximum at x = 5, i.e., square of side 5 cm is cut off from each comer. Question 19. Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius a is a square side $$\sqrt{2}$$a. Solution: Let the length and breadth of the rectangle inscribed in a circle of radius a be x and y respectively. ∴ x² + y² = (2a)² ⇒ x² + y² = 4a² … (1) ∴ Perimeter = 2(x + y). For P(x) to be maximum, P'(x) = 0 and P”(x) < 0. From (2), P'(x) = 0. From (1), y² = 4a² – x² = 4a² – 2a² = 2a² ⇒ y = $$\sqrt{2}$$ a Thus, x = y. Hence, the rectangle is a square of side $$\sqrt{2}$$ a. Question 20. Show that the right circular cylinder of the given surface area and maximum volume is such that its height is equal to the diameter of the base. Solution: Let S be the given surface area of the circular cylinder whose radius is r and height h. Let V be the its volume. Then, Surface area S = 2πr² + 2πrh ∴ V is maximum. Thus, volume is maximum, when h = 2r, i.e., when height of the cylinder = diameter of the base. Question 21. Of all the closed cylindrical cans (right circular cylinders) of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area. Solution: Let r be the radius and h be the height of cylindrical can. ∴ Volume = πr²h = 100 cm³. ∴ h = $$\frac{100}{\pi r^{2}}$$ Now, total surface area of the can ⇒ S is minimum or least, when r = $$\left(\frac{50}{\pi}\right)^{\frac{1}{3}}$$. Hence, the total surface area is least when radius of base is $$\left(\frac{50}{\pi}\right)^{\frac{1}{3}}$$ can and h = $$\frac{100}{\pi r^{2}}$$ = $$\frac { 100 }{ π }$$$$\left(\frac{\pi}{50}\right)^{\frac{2}{3}}$$ cm. Question 22. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum? Solution: Let one part be of length x, then the other part will be 28-x. Let the part of the length x be convetred into a circle of radius r. ∴ 2πr = x ⇒ r = $$\frac { x }{ 2π }$$ ∴ Area of circle = πr² = π = $$\frac{x^{2}}{4 \pi^{2}}$$ = $$\frac{x^{2}}{4 \pi}$$ Now second part of length 28 – x is converted into a square. Differentiating (1) w.r.t. x, we get $$\frac{d^{2} A}{d x^{2}}$$ = $$\frac { 1 }{ 2π }$$ + $$\frac { 1 }{ 8 }$$ ⇒ A is minimum. So, the two parts are x = $$\frac { 28π }{ 4+π }$$ and 28 – x = $$\frac { 112 }{ 4+π }$$ Question 23. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is $$\frac { 8 }{ 27 }$$ of the volume of the sphere. Solution: Let a cone VAB of greatest volume be inscribed in the sphere. Let ∠AOC = θ. ∴ Radius AC of the base of the cone v = R sin θ and VC = VO + OC = R + R cos θ = height of the cone. ∴ V, the volume of the cone For maximum and minimum, we have : $$\frac { dv }{ dθ }$$ = 0. But cos θ ≠ – 1 as cos θ = – 1 ⇒ θ = π, which is not possible. When θ is slightly < cos-1$$\frac { 1 }{ 3}$$ sin θ = +ve 3 cos θ – 1 = +ve [as θ decreases, cos θ increases) and cos θ + 1 = +ve ∴ $$\frac { dv }{ dθ }$$ = (+)(+)(+) = +ve. When θ is slightly > cos-1$$\frac { 1 }{ 3}$$ sin θ = +ve, 3 cos θ – 1 = – ve and cosθ+ 1 = +ve. [Since as θ increases, the value of cos θ decreases] ∴ $$\frac { dv }{ dθ }$$ = (+)(-)(+) = – ve. Thus, changes sign from +ve to —ve, when passes thorough cos-1$$\frac { 1 }{ 3}$$. Hence, V is maximum at θ = cos-1$$\frac { 1 }{ 3}$$. Now, maximum volume of cone Question 24. Show that the right circular cone of least curved surface area and given volume has an altitude equal to $$\sqrt{2}$$ times the radius of the base. Solution: Let r and h be the radius and height of the cone respectively. Also, $$\frac { dS }{dr}$$. changes sign from -ve to +ve as r increases through the point k² = 2r6. ⇒ S is least at this point. From (1), k² = h²4 ⇒ h²4= 2r6 or h² = 2r². ∴ h = $$\sqrt{2}$$r. Question 25. Show that the semi-vertical angle of the cone of maximum volume and of given slant height is tan-1$$\sqrt{2}$$. Solution: Let V be the volume, l be the slant height and 0 be the semi-vertical angle of the cone. Now, volume V of the cone = $$\frac { 1 }{ 3}$$πr²h, vertical height h = l cos θ and radius r = l sin θ. When θ is slightly < tan-1$$\sqrt{2}$$ sin θ cos²θ = + ve, tanθ – $$\sqrt{2}$$ = – ve, tan θ + $$\sqrt{2}$$ = + ve ∴ $$\frac { dv }{ dθ }$$ = (-)(+)(-)(+) = +ve. When θ is slightly > tan-1$$\sqrt{2}$$ sin θ cos² θ = +ve, tan θ – $$\sqrt{2}$$ = +ve, tan θ + $$\sqrt{2}$$ = +ve ∴ $$\frac { dv }{ dθ }$$ = (-)(+)(+)(+) = – ve. ∴ $$\frac { dv }{ dθ }$$ changes sign from +ve to – ve, when θ increases through tan-1$$\sqrt{2}$$ ∴ V is maximum at θ = tan-1$$\sqrt{2}$$. Question 26. Show that semi-vertical angle of a right circular cone of given surface area and maximum volume is sin-1$$\frac { 1 }{ 3}$$. Solution: Let r be the radius, l be the slant height and h be the vertical height of a cone of semi- vertical angle CL. ∴ Surface area S = πrl + πr² … (1) or l = $$\frac{S-\pi r^{2}}{\pi r}$$. The volume of the cone Now $$\frac{d V^{2}}{d r}$$ = 0 for maximum or minimum of V. ⇒ $$\frac { S }{ 9 }$$(2Sr – 8πr³) = 0 or S – 4π³ = S = 4π². Putting S = 4πr² in (2), we get $$\frac{d^{2} V^{2}}{d r^{2}}$$ = $$\frac { S }{ 9 }$$ [8πr² – 24πr²] = – ve. ⇒ V is maximum, when S = 4π². Putting this value of S in (1), we get 4π² = πrl + πr². or 3πr² = πrl. or $$\frac { r }{ l }$$ = sin α = $$\frac { 1 }{ 3 }$$ ∴ l = sin-1$$\frac { 1 }{ 3}$$. Thus, for a given surface area, V is maxi*ium, when α = sin-1$$\frac { 1 }{ 3}$$. Question 27. The point on the curve x² = 2y which is nearest to the point (0, 5) is ______. (A) (2$$\sqrt{2}$$, 4) (B) (2$$\sqrt{2}$$, 0) (C) (0, 0) (D) (2, 2) Solution: Let P(x, y) be a point on the curve x² = 2y. The other point is A(0,5) ⇒ Z is minimum at (2$$\sqrt{2}$$, 4). ⇒ $$\sqrt{Z}$$ is minimum at (2$$\sqrt{2}$$, 4). ∴ Part (A) is the correct answer. Question 28. For all real values of x, the minimum value of $$\frac{1-x+x^{2}}{1+x+x^{2}}$$ is ______. (A) 0 (B) 1 (C) 3 (D) $$\frac { 1 }{ 3 }$$ Solution: Question 29. The maximum value of [x(x-1)+1]$$\frac { 1 }{ 3 }$$, 0 ≤ x ≤ 1 is (A) ($$\frac { 1 }{ 3 }$$)$$\frac { 1 }{ 3 }$$ (B) $$\frac { 1 }{ 2 }$$ (C) 1 (D) $$\frac { 1 }{ 3 }$$ Solution: Also, $$\frac { dy }{ dx }$$ changes sign from – ve to + ve at x = $$\frac { 1 }{ 2 }$$, ∴ y is miniinum at x = $$\frac { 1 }{ 2 }$$ Value of y at x= 0, (0 + 1)$$\frac { 1 }{ 3 }$$ = 1$$\frac { 1 }{ 3 }$$ = 1, Value of y at x= 1, (0 + 1)$$\frac { 1 }{ 3 }$$ = 1$$\frac { 1 }{ 3 }$$ = 1, ∴ The maximum value of y is 1. ∴ Part (C) is the correct answer.	2022-05-24 15:57:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.693785548210144, "perplexity": 1263.671042791431}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662573053.67/warc/CC-MAIN-20220524142617-20220524172617-00727.warc.gz"}
http://math.stackexchange.com/questions/555165/what-is-the-equivalence-class-of-a-relations-element	# What is the equivalence class of a relation's element? I'm studying about equivalence relations. My book has the following definition for an equivalence class: If $R=(G,A,A)$ is a relation of equivalence over the set $A$, the equivalence class of $a$ is denoted as $[a]$ is the set $$[a] = \{b \in A : a\mathbin{R}b\}$$ Recently, I found a document online about the topic: http://www2.uca.es/matematicas/Docencia/ESI/1710003/Apuntes/Leccion8.pdf The document is in Spanish, but here is its definition for an equivalence class: If $R$ is a relation of equivalence over a set $A$, for each $a \in A$, we'll call the equivalence of $a$ to the set formed by all elements in $A$ that are related to it. It will be denoted $[a]$, that is: $$[a] = \{x \in A : x\mathbin{R}a\}$$ I am a bit confused now. It seems to me that both documents' definitions don't quite match. This is just an idea, but maybe it doesn't matter, since the relation is supposed to be symmetric? - To get the curly braces you have to use \{ and \}; plain { and } vanish. – Brian M. Scott Nov 7 '13 at 13:13 Your idea in the last line is correct: we talk about equivalence classes only when we have an equivalence relation, which by definition is symmetric. Thus, the two definitions are equivalent: for any $a,x\in A$, $a\mathbin{R}x$ if and only if $x\mathbin{R}a$, and therefore $$\{x\in A:a\mathbin{R}x\}=\{x\in A:x\mathbin{R}a\}\;.$$	2015-10-10 14:28:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9259881377220154, "perplexity": 171.93719746383312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737956371.97/warc/CC-MAIN-20151001221916-00014-ip-10-137-6-227.ec2.internal.warc.gz"}
http://openwetware.org/index.php?title=Physics307L:People/Gibson/Formal_Lab_Report_2&oldid=174151	# Physics307L:People/Gibson/Formal Lab Report 2 (diff) ←Older revision \| Current revision (diff) \| Newer revision→ (diff) Jump to: navigation, search MEASURING THE SPEED OF LIGHT BY A TIME DELAYED SIGNAL BETWEEN A LIGHT EMITTING DIODE AND PHOTOMULTIPLIER TUBE Author: Zane Gibson Experimentalists: Zane Gibson Matthew Gooden, Department of Physics and Astronomy University of New Mexico Albuquerque, NM ## Abstract This experiment was done to determine the value of c or the value of the speed of light. To complete this experiment, a time amplitude converter (TAC) and an oscilloscope were used to measure the time between a light signal from an light emitting diode (LED) and a photo multiplier tube (PMT). By changing the distance between the PMT and the LED, we were able to record various voltages for various distances, then we converted the voltage to a time and constructed a trend line of our distances and times using a least squares fit analysis. The slope of this trend line was the constant we were originally looking for, the speed of light. For this experiment we found our constant to be $c=\left(2.892 \pm .077\right)\times10^8 m/s$ which is very close to the theoretical approximation of $2.99\times10^8 {m/s}$. ## Introduction The speed of light has been used in many branches of physics in many different ways. The speed of light was first seen as constant by Albert Einstein and as such is a basis for the branch of physics known as special realitivity [1] As such, this value has been disputed and many experiments have been performed to find this value. The first accurate way to find the value was done around 1960 where lasers first started to become used widely in optical physics. It wasn't until 1972, when the National Institute of Standards and Technology employed the developing laser technology to measure the speed of light at 299,792,458 meters per second [2]. Since then, the speed of light has been used in a variety of applications from optical physics to biomedical physics, more specifically using light to shrink cancer cells [3]. In this experiment we consider how close we can approach the value of c, while minimizing our error and uncertainty in our measurements. ## Methods and Materials Equipment List 1.Time-Amplitude Converter (TAC) - EG&G Ortec Model 567 TAC/SCA 2.Light Emitting Diode 3.Photomultiplier Tube (PMT) - Magnetic Shield Co., 22P50 4.DC Power supply (LED) - Harrison Laboratories, Model 6207A 0-160V,0-.2A 5.DC Power supply (PMT)- Bertan Associates,Inc. Model 315, DC Power Supply 0-5000V,0-5mA 6.Oscilloscope - Tektronix TDS 1002 2 Channel Digital Storage Oscill., 60MHz 1GS/s 7.BNC Connector Cables - 6 8.Delay box - Canberra NSEC Delay Box, Model 2058 9.Several Meter Sticks - 4 10.Long Tube (cardboard or other nontransparent material) - 4 meters long Procedure Set up - To begin the experiment we first collected all our equipment, then proceeded to first get power to the LED. We then did the same for the PMT (NOTE: Do not expose the PMT to massive quantities of light while operating, this can destroy the PMT) then placed them in the long cardboard tube. We then bolted the LED to one of the ends of a meter stick (so we could vary the distance) and taped the sticks end to end to allow us plenty of variable distances. Using a connector cable, we connected the PMT signal output into channel 1 of the oscilloscope and plugged the response from the LED (when it sent a signal) to the TAC. We then ran a cable from the TAC to the time delay box, and then to channel 2 of the oscilloscope. The distance of how far in or out the LED was is arbitrary. Taking whatever place as the zero works fine and recording the differences in the distances is what is important. Collection- To begin we left the delay box set to zero delay for the experiment, as none was ever needed. We then adjusted the voltage knob on the PMT power source so that the PMT was receiving a voltage of approximately 1900 volts (as per recommendation from last years lab manual [4]. Then we adjusted the LED power supply to approximately 200 volts. Then turning to the oscilloscope we saw two signals present on the screen, one from the PMT and the other from the TAC. The PMT signal was characterized by a sharp downward voltage drop, while the TAC signal was characterized by a rapidly oscillating square wave. Using the acquire button on the oscilloscope we chose to average over the values that the oscilloscope is receiving to allow measurements to be taken must easier, rather than constantly adjust the equipment to find a steady value. We chose the 128 setting under the average command, to greatly stabilize the signal we were viewing. At this point we began our data collection, having one of us at the other end of the tube to move the LED, and the other at the end with the PMT and other equipment. We picked a value of the voltage on Ch 1 and attempted to maintain that value for the duration of the measurements made for each data set, this is need because as the LED is moved toward or away from the PMT the intensity of the signal changes and by aligning or unaligning the polarizers we were able to keep the signal constant. This was accomplished by turning the PMT left/right to align or unalign its polarizer with that of the LED. Taking care to note our chosen zero mark, we then moved the LED away from the PMT a short distance, recording this change in distance, while holding the PMT steady. For this experiment, we are not concerned with the actual distance between the PMT and LED. Rather at any particular distance between the PMT and LED we make a measurement and then we are only concerned with the change in that distance for the next measurement, since we are measuring the time delay of the signal from the LED pulse and the PMT signal. Once the LED was moved the PMT was adjusted to regain the chosen voltage, and then using the measure function to allow us to see the values outputted to channels 1 and 2, we recorded the voltage output from the TAC. To complete the particular data set we made several more measurements in the same fashion always recording the change in distance from the original zero mark. Every data set has different settings for voltages we were aiming for from the PMT, as listed below in the results and discussions. Analyzing- In our analysis we used two programs to construct graphs and do averages/ standard deviations for our results. These programs were Microsoft Excel and Mathworks Matlab. ## Results and Discussion Below are the 4 data sets we took and a discussion of each set. In discovering the times for each data set, we were required (since we didn't measure time) to use the equation V=G*T where V is voltage; T is the time we want, and G is a constant set before we began taking measurements = 1/10 volt/nanosecond. Here we took down measurements as listed in the procedure and placed them below: Table 1 Data Set 1 Measurments PMT Ch1 Voltage TAC Time delay voltage Distance (cm) Time (ns) 1 $600 \pm 8 {mV}$ $3.24\pm.04$ 60 32.4 2 $600 \pm 8 {mV}$ $3.2 \pm.04$ 80 32 3 $600 \pm 8 {mV}$ $3.08 \pm.04$ 100 30.8 4 $600 \pm 8 {mV}$ $3.00 \pm.04$ 120 30 5 $600 \pm 8 {mV}$ $2.96\pm.04$ 140 29.6 6 $600 \pm 8 {mV}$ $3.00 \pm.04$ 130 30 7 $600 \pm 8 {mV}$ $3.12 \pm.04$ 90 31.2 8 $600 \pm 8 {mV}$ $3.2 \pm.04$ 70 32 Table 1 shows a constant for the speed of light being around $c=\left(2.68 \pm 0.18\right)\times10^{8} m/s$ given all the settings labeled above. Table 2 Data Set 2 Measurments PMT Ch1 Voltage TAC Time delay voltage Distance (cm) Time (ns) 1 $800 \pm 8 {mV}$ $2.44\pm.04$ 70 24.4 2 $800 \pm 8 {mV}$ $2.4 \pm.04$ 80 24 3 $800 \pm 8 {mV}$ $2.36 \pm.04$ 90 23.6 4 $800 \pm 8 {mV}$ $2.32 \pm.04$ 100 23.2 5 $800 \pm 8 {mV}$ $2.32\pm.04$ 110 23.2 Table 2 shows a speed of light being $c=\left(2.94 \pm 0.42\right)\times10^{8} m/s$ given all the settings labeled in the table above. Table 3 Data Set 3 Measurments PMT Ch1 Voltage TAC Time delay voltage Distance (cm) Time (ns) 1 $720 \pm 8 {mV}$ $2.8\pm.04$ 27.5 28 2 $720 \pm 8 {mV}$ $2.64 \pm.04$ 70 26.4 3 $720 \pm 8 {mV}$ $2.52 \pm.04$ 110 25.2 4 $720 \pm 8 {mV}$ $2.44 \pm.04$ 130 24.4 5 $720 \pm 8 {mV}$ $2.72\pm.04$ 50 27.2 Table 3 shows a speed of light being $c=\left(2.89 \pm 0.08\right)\times10^{8} m/s$ given all the settings labeled above. • NOTE: Table 4 was analyzed separately from the other data sets. This was because the delay box was used to help improve our accuracy of getting to the theoretical value of the speed of light, as discussed in the conclusions. Table 4 Data Set 2 Measurments PMT Ch1 Voltage TAC Time delay voltage Distance (cm) Time (ns) 1 $640 \pm 8 {mV}$ $3.16\pm.02$ 15 31.6 2 $640 \pm 8 {mV}$ $3.26 \pm.02$ 30 32.6 3 $640 \pm 8 {mV}$ $3.26 \pm.02$ 45 32.6 4 $640 \pm 8 {mV}$ $3.32 \pm.02$ 60 33.2 5 $640 \pm 8 {mV}$ $3.36\pm.02$ 75 33.6 6 $640 \pm 8 {mV}$ $3.44\pm.02$ 90 34.4 Table 4 shows a value of $c=\left(2.85 \pm 0.03\right)\times10^{8} m/s$ given a delay of 2 nanosecs and all labeled settings above. • NOTE: See acknowledgments section Once we completed calculating our times for each of the trials, we then proceeded to construct 3 least squares plots to determine what our value of the speed of light would be. To do this we simply took the distance recorded vs time which then gave us a very linear graph... the slope of this line is the constant we were looking for. These are plot of data sets 1-3 along with least-squares fit lines This is a plot of the data from data set 4 given above • The slopes for each data set are listed below: Data Set 1 $c=\left(2.68 \pm 0.18\right)\times10^{8} m/s$ Data Set 2 $c=\left(2.94 \pm 0.42\right)\times10^{8} m/s$ Data Set 3 $c=\left(2.89 \pm 0.08\right)\times10^{8} m/s$ Average of all three data sets: $c=\left(2.83 \pm .23\right)\times10^8 m/s$ From here we calculated our relative error to determine the amount off of the theoretical value: $Relative Error=\frac{\left\|2.83\times10^{8}-2.99\times^{8}\right\|}{\left\|2.99\times10^{8}\right\|}=0.0535=5.3%$ Possible sources of uncertainties have to do with the equipment. We ran the same experiment several times over the same values to see how close our values would be to one another after several repeated measurements. We found quite a discrepancy and we've concluded it has to do with some sort of systematic error which we are unable to account for at this time. ## Conclusions Thus our closest approach to the theoretical value of the speed of light is $c=\left(2.892 \pm .077\right)\times10^8 m/s$. This is a good value to receive especially with our unfamiliarity with most of the equipment. Additional error analysis shows our reported value we find having a relative error given by $RE=\frac{\left\|2.99\times10^{8}-c\right\|}{\left\|2.99\times10^{8}\right\|}=0.0328=3.3%$ which is again very good considering the time to take measurements was somewhat constrained. With data set 4 we attempted to improve on this error by taking more measurements and possibly shrinking this value. The ideas used are explained next to the table above but as can be seen from the data set none of these things were able to improve the results, and actually left us with a value of c that is farther off than either of data sets 2 or 3. The data given above for set 4 is the first and more accurate attempt made for that set in comparison to the other trials. It must be concluded that there is some form of systematic error in the experiment that is not being accounted for and at this time with our little knowledge of the inner workings of the electronics we cannot specifically pin point the problem. Our most accurate and reported value is $c=\left(2.892 \pm .077\right)\times10^8 m/s$. ## Acknowledgments/References 1. [Special Relativity] 2. [Measuring the Speed of Light] as done by NIST. 3. [Cancer] article and using light to shrink cells 4. [Previous Course lab manual] from Dr. Gold I would like to thank Dr. Steve Koch for help with theory, instrumentation and experiment design. I would also like to thank my lab partner Matt Gooden for his help in presentation of the data.	2014-09-02 16:06:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 62, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6126524806022644, "perplexity": 937.6871462542281}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535922087.15/warc/CC-MAIN-20140909051142-00290-ip-10-180-136-8.ec2.internal.warc.gz"}
https://xianblog.wordpress.com/tag/artificial-intelligence/	## Data science for social good fellowships [DSSGx UK 2023] Posted in Kids, Statistics, Travel, University life with tags , , , , , , , , , , , , , on November 9, 2022 by xi'an Warwick is (again) running a 12-week summer programme bringing together some of the top student talents from data science, machine learning and artificial intelligence, all over the World, to work on real-world data science challenges and deliver positive social impact. Applications for DSSG 2023 are now OPEN! Click here for the application form (please read the information carefully) and click here for the FAQs for 2023. (The application also works for a similar programme in Kaiserslauten, Germany. DSSG helps not-for-profit organisations and government bodies to achieve more with their data by enhancing their services, interventions and outreach, helping fulfil their mission of improving the world and people’s lives. The programme gives not-for-profit organisations and government bodies unprecedented access to inspiring, top-tier data science talent. This helps build their capacity to use cutting-edge quantitative methods to address societal challenges in areas such as education, health, energy, public safety, transportation and economic development. At the same time, it provides intensive case-based and supported training to students to create industry-standard data science products in collaboration with government agencies and NGOs, to deliver positive social impact. And it builds a world-wide community of data scientists who care about the social good. In 2019, the University of Warwick together with the Alan Turing Institute brought DSSG to the UK. The University of Warwick has run it each year since and now preparation is well underway for DSSGx UK 2023, which will be held at the University of Warwick, UK, from 5 June to 25 August. ## Ada L. at the ATI [6 October 2022] Posted in Statistics with tags , , , , , , , , , , on September 29, 2022 by xi'an ## Klara and the Sun [book review] Posted in Books, Kids with tags , , , , , , , , , , , , , on April 22, 2022 by xi'an Klara and the Sun is the latest book of Kazuo Ishiguro. I am a big admirer of Ishiguro’s books and always moved by their bittersweet exploration of humanity (or humanness?!). The remains of the day is one of my favourite books, competing with Graham Greene’s The end of the affair, and I deeply enjoyed When we were orphans, Never let me go, and The buried giant. While this latest book exhibits the same craftsmanship in depicting human feelings and incomplete (in the sense of unsatisfactory) relations, I feel like I missed some component of the book, too many hints, the overall message… Not that I rushed through it, contrary to my habit, reading a few chapters at a time during lunch breaks. But I cannot set the separation between the subjective perception of Klara [the robotic friend], which is very clearly limited, both by her robotic sensors [lacking a sense of smell for instance] and her learning algorithm, furthermore aggravated by her wasting (?) some material to sabotage a machine, and the real world [within the novel, a vague two-tiered USA]. Because the perspective is always Klara’s. This confusion may be completely intentional and is in that sense brilliant. But I remained perplexed by the Sun central episode in the novel, which I fear reveals a side of the story I did not get. Like Джозі в якийсь момент перетворилася на робота? [Using Ukrainian to avoid spoilers for most readers!] (In a way, Klara and the Sun is a variation on Never let me go, both dealing with a future where copies of humans could be available, for those who could afford it.) ## Prairie/MIAI Artificial Intelligence summer school [5-9 July 2021] Posted in Books, Statistics, University life with tags , , , , , on June 16, 2021 by xi'an ## the Ramanujan machine Posted in Books, Kids, pictures, University life with tags , , , , , , , , , , , on February 18, 2021 by xi'an Nature of 4 Feb. 2021 offers a rather long (Nature-like) paper on creating Ramanujan-like expressions using an automated process. Associated with a cover in the first pages. The purpose of the AI is to generate conjectures of Ramanujan-like formulas linking famous constants like π or e and algebraic formulas like the novel polynomial continued fraction of 8/π²: $\frac{8}{{{\rm{\pi }}}^{2}}=1-\frac{2\times {1}^{4}-{1}^{3}}{7-\frac{2\times {2}^{4}-{2}^{3}}{19-\frac{2\times {3}^{4}-{3}^{3}}{37-\frac{2\times {4}^{4}-{4}^{3}}{\ldots }}}}$ which currently remains unproven. The authors of the “machine” provide Python code that one can run to try uncover new conjectures, possibly named after the discoverer! The article is spending a large proportion of its contents to justify the appeal of generating such conjectures, with several unsuspected formulas later proven for real, but I remain unconvinced of the deeper appeal of the machine (as well as unhappy about the association of Ramanujan and machine, since S. Ramanujan had a mystical and unexplained relation to numbers, defeating Hardy’s logic, “a mathematician of the highest quality, a man of altogether exceptional originality and power”). The difficulty is in separating worthwhile from anecdotal (true) conjectures, not to mention wrng conjectures. This is certainly of much deeper interest than separating chihuahua faces from blueberry muffins, but does it really “help to create mathematical knowledge”?	2022-12-01 16:11:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4288070499897003, "perplexity": 2717.310764991413}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710829.5/warc/CC-MAIN-20221201153700-20221201183700-00451.warc.gz"}
https://deepai.org/publication/testing-tensor-products	# Testing tensor products A function f:[n]^d→F_2 is a direct sum if it is of the form f((a_1,...,a_d)) = f_1(a_1)+... + f_d (a_d), for some d functions f_1,...,f_d:[n]→F_2. We present a 4-query test which distinguishes between direct sums and functions that are far from them. The test relies on the BLR linearity test and on the direct product test constructed by Dinur and Steurer. We also present a different test, which queries the function (d+1) times, but is easier to analyze. In multiplicative ± 1 notation, the above reads as follows. A d-dimensional tensor with ± 1 entries is called a tensor product if it is a tensor product of d vectors with ± 1 entries. In other words, it is a tensor product if it is of rank 1. The presented tests check whether a given tensor is close to a tensor product. ## Authors • 9 publications • 1 publication • ### Towards a General Direct Product Testing Theorem The Direct Product encoding of a string a∈{0,1}^n on an underlying domai... 01/18/2019 ∙ by Elazar Goldenberg, et al. ∙ 0 • ### Simple, Fast Semantic Parsing with a Tensor Kernel We describe a simple approach to semantic parsing based on a tensor prod... 07/02/2015 ∙ by Daoud Clarke, et al. ∙ 0 • ### Faster Johnson-Lindenstrauss Transforms via Kronecker Products The Kronecker product is an important matrix operation with a wide range... 09/11/2019 ∙ by Ruhui Jin, et al. ∙ 0 • ### Symmetric Tensor Completion from Multilinear Entries and Learning Product Mixtures over the Hypercube We give an algorithm for completing an order-m symmetric low-rank tensor... 06/09/2015 ∙ by Tselil Schramm, et al. ∙ 0 • ### Tensor Matched Subspace Detection The problem of testing whether an incomplete tensor lies in a given tens... 10/23/2017 ∙ by Cuiping Li, et al. ∙ 0 • ### Tensor GMRES and Golub-Kahan Bidiagonalization methods via the Einstein product with applications to image and video processing In the present paper, we are interested in developing iterative Krylov s... 05/15/2020 ∙ by M. El Guide, et al. ∙ 0 • ### Addressing Computational Bottlenecks in Higher-Order Graph Matching with Tensor Kronecker Product Structure Graph matching, also known as network alignment, is the problem of findi... 11/17/2020 ∙ by Charles Colley, et al. ∙ 0 ##### This week in AI Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday. ## 1 Introduction Given functions , their direct sum is the function given by where addition is in the field . We denote . We study the testability question: given a function test if it is a direct sum, namely if it belongs to the set DirectSumn,d={f1⊕⋯⊕fd\|fi:[n]→F2}. We suggest and analyze a four-query test which we call the “square in a cube” test, and show that it is a strong absolute local test for being a direct sum. By this we mean that neither the number of queries nor the testability constant depend on the parameters and . We also describe a simpler -query test, whose easy analysis we defer to section 4. Our square in a cube test is as follows 1. Choose uniformly at random 2. Choose two subsets uniformly at random and let be their symmetric difference. 3. Accept iff f(a)+f(aSb)+f(aTb)+f(aUb)=0, where is the string whose -th coordinate equals if and otherwise. We prove ###### Theorem 1 (Main). There exists an absolute constant s.t. for all and given , dist(f,DirectSumn,d)≤c⋅Pra,b,S,T[f(a)+f(aSb)+f(aTb)+f(aS△Tb)≠0] where are chosen independently and uniformly from the domain of , and are random subsets of , and refers to relative Hamming distance, namely . ###### Remark 2. The above theorem is true in a greater generality. Namely, the same proof can be adapted to the case of a function , where the corresponding subspace of direct sums is DirectSumn1,…,nd={f1⊕⋯⊕fd\|fi:[ni]→F2}. ##### Testing if a tensor has rank 1. An equivalent way to formulate our question is as a test for whether a -dimensional tensor with entries has rank . Indeed moving to multiplicative notation and writing and , we are asking whether there are such that h=h1⊗⋯⊗hd. Denoting TensorProductn,d={h1⊗⋯⊗hd\|hi:[n]→{−1,1}} we have ###### Corollary 3. There exists an absolute constant s.t. for all and , for every , dist(h,TensorProduct)≤c⋅Pra,b,S,T[h(a)⋅h(aSb)⋅h(aTb)⋅h(aS△Tb)≠1]. ##### Background. Direct sum is a natural construction that is often used in complexity for hardness amplification. It is related to the direct product construction: a function is the direct product of as above if for all . The testability of direct products has received attention [GS97, DR06, DG08, IKW12, DS14] as abstraction of certain PCP tests and it was not surprising to find [DDG17] that there is a connection between testing direct products to testing direct sum. However, somewhat unsatisfyingly this connection was confined to testing a certain type of symmetric direct sum. A symmetric direct sum is a function that is a direct product with all components equal; namely such that there is a single such that In [DDG17] a test was shown for testing if a given is a symmetric direct sum, and the analysis carried out relying on the direct product test. It was left as an open question to devise and analyze a test for the property of being a (not necessarily symmetric) direct sum. ##### Method. Our proof, similarly to [DDG17], relies on a combination of the BLR linearity testing theorem [BLR93] and the direct product test of [DS14]. The trick is to find the right combination. We first observe that once we fix , the test is confined to a set of at most points in the domain, and can be viewed as performing a BLR (affinity rather than linearity) test on this piece of the domain. From the BLR theorem we deduce an affine linear function on this piece. The next step is to combine the different affine linear functions, one from each piece, into one global direct sum, and this is done by reducing to direct product. ## 2 Tensor Product We refer to as a -dimensional binary tensor. ###### Definition 4. A -dimensional binary tensor is a tensor product, if there exist one-dimensional tensors, i.e., vectors, such that . ###### Definition 5. A -dimensional binary tensor is -close to tensor product, if there exists tensor product such that Prx∼[n]d(g(x)=g′(x))≥1−ε, where is chosen uniformly at random. In the next two sections we present two different approaches for testing whether a -dimensional binary tensor is close to a tensor product. ## 3 Square in a Cube Test We start by introducing some notation. Given two vectors , define • ; • the induced subcube is the binary cube ; • the projection map defined for as ρa,b(x)(i)=⎧⎨⎩a(i)=b(i),i∉Δ(a,b);b(i),i∈Δ(a,b) and x(i)=1;a(i),i∈Δ(a,b) and x(i)=0; • the function as . The following test is the same as the on preceeding the formulation of Theorem 1. ###### Test 6. Square in a Cube test. Given query access to a function : • Choose uniformly at random. • Choose uniformly at random. • Query at and . • Accept iff . ###### Theorem 7. Suppose a function passes Test 6 with probabilty , then is -close to a tensor product. ### 3.1 The BLR affinity test The Blum-Luby-Rubinfeld linearity test was introduced in [BLR93], where its remarkable properties were proven. Later a simpler proof via Fourier analysis was presented, e.g. see [BCH95]. Below we give a variation of this test for affine functions, see [O’D14, Chapter 1]. ###### Definition 8. A function is called affine, if there exists a set and a constant such that for every vector g(x)=c+∑i∈Sxi. ###### Definition 9. A function is said to be -close to being affine, if there exists an affine function such that Prx∼Fd2(g(x)=g′(x))≥1−ε, where is chosen uniformly at random. Note that (see [O’D14, Exercise 1.26]) a function is affine iff for any two vectors it satisfies g(0)+g(x)+g(y)+g(x+y)=0. (1) The BLR test implies that if a function satisfies (1 ) with high probability, then it is close to an affine function. ###### Test 10. • Choose and independently and uniformly at random. • Query at and . • Accept if . ###### Theorem 11 ([Blr93]). Suppose passes the affinity test with probability . In other words, satisfies Prx,y∼Fd2(g(0)+g(x)+g(y)+g(x+y)=0)=1−ε. Then is -close to being affine. ∎ ### 3.2 Direct Product Test ###### Definition 12. A function , where , is called a direct product if it is of the form . Given functions , their direct product is a function denoted and defined as . In [DS14], Dinur and Steurer presented a -query test which distinguishes between direct products and functions that are far from direct products with constant probability. ###### Test 13. – Two-query test with intersection . Given query access to a function : • Choose a set of size uniformly at random. • Choose uniformly at random, conditioned . • Query at and . • Accept iff . ###### Theorem 14. [DS14, Theorem 1.1] Let be positive integers, let , where , and let . Let be given such that PrA,x,y(g(x)A=g(y)A)≥1−ε, where are chosen w.r.t. the test distribution . Then there exists a direct product function such that , where stand for the Hamming distance. In particular, when this implies Prx(f(x)=g(x))≥1−O(ε), as k→∞% and N→∞. ###### Remark 15. Note that Theorem 14 is true for for all , and not just . More precisely, the following statement holds: If a function passes Test 13 with probability at least for wtih , then passes Test 13 with probability at least for , where is a positive integer. This reduction shows that Theorem 14 is true as it is stated for for all , as the reduction affects only the constant in the notation. For a more detailed explanation, see Appendix (Section 6). ### 3.3 Proof of Theorem 7 For a positive integer , we denote by the distribution on , where each coordinate is equal to with probability and to with probability . We use the following proposition in the course of the proof. ###### Proposition 16. Let be a set and be the corresponding linear function, i.e., . Suppose Prx∼μ\nicefrac23(FD2)(χS(x)=0)>23, then . ###### Proof. Consider . Then Prx∼μ\nicefrac23(FD2)(χS(x)=0)=Prx∼μ\nicefrac23(FD2)((−1)χS(x)=1). Also the following holds 13<∣∣ ∣∣2Prx∼μ\nicefrac23(FD2)((−1)χS(x)=1)−1∣∣ ∣∣=∣∣Ex∼μ\nicefrac23(FD2)(−1)χS(x)∣∣= ∣∣ ∣∣∏i∈[D]Exi∼μ\nicefrac23(F2)(−1)xi∣∣ ∣∣=∣∣ ∣∣(−13)\|S\|∣∣ ∣∣=(13)\|S\|, and the statement follows. ∎ ###### Proof. (of Theorem 7.) Assume Test 6 fails on a function with probability less than , i.e., Pra,b∼[n]dx,y∼Ca,b(fa,b(0)+fa,b(x)+fa,b(y)+fa,b(x+y)=0)>1−ε, where all distributions are uniform. Recall that is a shorthand for . Then there exists such that Prb∼[n]dx,y∼Ca,b(fa,b(0)+fa,b(x)+fa,b(y)+fa,b(x+y)=0)>1−ε. W.l.o.g. we assume that and that . We can assume this, since if needed we can re-index the tensor, and flip it, i.e., add the constant one tensor element-wise. We write for and for . Then for every , Prx,y∼Cb(fb(0)+fb(x)+fb(y)+fb(x+y)=0)=1−εb. The BLR theorem (Theorem 11) implies that there exists a subset , such that Prx∼Cb(fb(x)=χS(b)(x))=1−εb. ###### Remark. By the BLR theorem, there should be the “greater or equal to” sign instead of the equality. We assume equality to ease of the proof. Let be a function defined as follows. For each , the set can be viewed as a subset of , since . Then is defined as the element of corresponding to the set . We now show that passes Test 13 with high probability and hence is close to a direct product. Let be chosen uniformly at random, and let be chosen with respect to the following distribution . For each , b′i={bi,w.p. \nicefrac34;chosen uniformly at random from [n]∖{bi},w.p. \nicefrac14. Note that the distribution on pairs , where is chosen uniformly from and w.r.t. , is equaivalent to the following: for each , {bi=b′i chosen uniformly from [n],w.p.% \nicefrac34;bi≠b′i both chosen uniformly from [n]w.p. \nicefrac14. (2) In particular, it is symmetric in the sense that choosing uniformly at random first, and then , leads to the same distribution on pairs as the one described above. For such a pair define distribution on as follows. For a vector , xi=⎧⎪⎨⎪⎩0,if i∈Δ(b,b′);0,w.p. \nicefrac13;bi=b′iw.p. \nicefrac23.if i∉Δ(b,b′). Note that the distribution is supported on a binary cube of dimension inside . Denote εb,b′=Prx∼Db,b′(f(x)≠χF(b)(x)). We claim that the following holds εb=Prx∼Cb(f(x)≠χF(b)(x))=Eb′∼D(b)εb,b′. (3) To see (3) note that since is chosen uniformly, is chosen w.r.t. , and , the resulting distribution for is xi={0,w.p. \nicefrac12;biw.p. \nicefrac12, which is exactly the uniform distribution on . We now show that Prb∼[n]db′∼D(b)(εb,b′+εb′,b>13)<6ε (4) First note that it follows from the definitions that Eb∼[n]dEb′∼D(b)εb,b′=Eb∼[n]dεb=ε. And by the symmetry of the distribution on pairs , Eb∼[n]dEb′∼D(b)εb′,b=Eb′∼D(b)Eb∼[n]dεb′,b=ε. Combined together, the previous two equations imply that Eb∼[n]dEb′∼D(b)(εb,b′+εb′,b)=2ε, and by the Markov inequality, Inequality 4 follows. By the definition of , Prx∼Db,b′(χF(b)(x)=χF(b′)(x))>1−(εb,b′+εb′,b). which is equivalent to Prx∼Db,b′(χF(b)ΔF(b′)(x)=1)>1−(εb,b′+εb′,b). Proposition 16 implies that if , then F(b)Cb∩Cb′=F(b′)Cb∩Cb′. By Theorem 14, the function is close to a direct product, i.e., there exist functions such that Prb∼[n]d(F(b)=(F1(b1),…,Fd(bd)))≥1−O(ε). Therefore, Prb∼[n]d(f(b)=d∑i=1Fi(bi))≥1−O(ε). ## 4 The Shapka Test In [KL14] Kaufman and Lubotzky showed that -coboundary expansion of a -dimensional complete simplicial complex implies testability of whether a symmetric -matrix is a tensor square of a vector. The following test is inspired by their work and in a way generalizes it. Given two vectors , for denote by the vector which coincides with in every coordinate except for the -th one, where it coincides with , i.e., (aib)j={aj,if j≠i;bi,if j=i. ###### Test 17. • Choose uniformly at random. • Define the query set to consist of , for all , and iff is even. • Query at the elements of . • Accept iff . ###### Remark 18. Shapka is the Russian word for a winter hat (derived from Old French chape for a cap). The name the Shapka test comes from the fact that the set consists of the two top layers of the induced binary cube (and also the bottom layer if is even). ###### Theorem 19. Suppose a function passes Test 17 with probabilty , then is -close to a tensor product. ### 4.1 Proof of Theorem 19 ###### Proof. Denote by the normalized distance from to the subspace of tensor products, i.e., there exists a tensor product such that Prx∼[n]d(f(x)≠g(x))=δ. For a vector , for , define a function as follows. For , fka(x)=f(akx). For , the defintion of depends on the parity of and reads as follows. For , Given a collection of vectors, , we denote their tensor product by . In other words, for a vector , the following holds (T(g1,…,gd))(x1,…,xd)=∑i∈[d]g1(x1). (5) In these notations, the following holds for any , (f−T(f1a,…,fda))(b1,…,bd)=∑q∈Qa,bf(q). As is a tensor product, it is at least -far from for any vectors , and hence for any , (6) Assume now that fails Test 17 with probabilty , i.e., ε=Pra,b∼[n]d⎛⎜⎝∑q∈Qa,bf(q)=1⎞⎟⎠. Combining this equality with (5) and (6), we get the following ε=Ea∼[n]dPrb∼[n]d((f−δ0(f1a,…,fda))(b1,…,bd)=1)≥(Ea∼[n]dδ)=δ, which completes the proof. ∎ ## 5 Further Directions 1. Can the original function be reconstructed by a voting scheme using the Shapka test 17? 2. It is plausible that the Square in the Cube test 6 be analyzed by the Fourier transofrm approach similarly to the analysis of the BLR test. 3. Another test in the spirit of the presented above is the following. ###### Test 20. • Choose uniformly at random. • Choose uniformly at random. • Query at and . • Accept iff . We conjecture that this test is also good, i.e., if a function passes the test with high probability then it is close to a tensor product. ## 6 Appendix: Proof of Remark 15 In [DS14], Dinur and Steurer proved Theorem 14 for . The following reduction shows that the theorem is true for all by a reduction from to some . Recall that Test 13 makes two queries according to the distribution , which is the following distribution: (1) Choose a set of size uniformly at random. (2) Choose uniformly at random, conditioned . ###### Proposition 21. Let denote the probability that a function passes Test 13 with respect to distribution . If for some , then for , where and . In addition, is , then also . ###### Proof. Fix a function , and suppose for some , i.e., PrA,x,y∼T(αk)(g(x)A=g(y)A)≥1−ε. We will show that where and . Note that satisfies . Given a pair of random vectors and a set distributed according to , we construct a sequence of vectors such that for all , the pair is distributed according to . The complement of has size . Partition it randomly into parts of equal size , . Denote for all . For each , construct such that it agrees with on the coordinates in and with on the rest of the coordinates . Then for each , agrees with on the set of the size . Therefore, Pr(g(xi−1)Ai=g(xi)Ai)≥1−ε. Hence, 1−r⋅ε≤Pr(∀1≤i≤r:g(xi−1)Ai=g(xi)Ai)≤PrAr,x,y∼T(α′k)(g(x0)Ar=g(xr)Ar). The case of has to be treated separately as it is not covered by Theorem 14. In this case there is a reduction to as follows. Given two vectors distributed w.r.t. construct an intermidiate random vector which agrees on exactly half of the coordinates with both and . ∎ ###### Corollary 22. Let be positive integers, let , where , and let . Let be given such that PrA,x,y(g(x)A=g(y)A)≥1−ε, where are chosen w.r.t. the test distribution . Then there exists a direct product function such that , where stand for the Hamming distance. In particular, cwhen this implies Prx(f(x)=g(x))≥1−O(ε), as k→∞% and N→∞. #### Funding The first author is supported by ERC-CoG grant number 772839. A substantial part of the work was done while the second author held a joint postdoctoral position at The Weizmann Institute and Bar-Ilan University funded by the ERC grant number 33628. Currently, the second author is supported by the SNF grant number 200020_169106. ## References • [BCH95] Mihir Bellare, Don Coppersmith, Johan Håstad, Marcos A. Kiwi, and Madhu Sudan. Linearity testing in characteristic two. In 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, USA, 23-25 October 1995, pages 432–441, 1995. • [BLR93] Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of computer and system sciences, 47(3):549–595, 1993. • [DDG17] Roee David, Irit Dinur, Elazar Goldenberg, Guy Kindler, and Igor Shinkar. Direct sum testing. SIAM J. Comput., 46(4):1336–1369, 2017. • [DG08] Irit Dinur and Elazar Goldenberg. Locally testing direct products in the low error range. In Proc. 49th IEEE Symp. on Foundations of Computer Science, 2008. • [DR06] Irit Dinur and Omer Reingold. Assignment testers: Towards combinatorial proofs of the PCP theorem. SIAM Journal on Computing, 36(4):975–1024, 2006. Special issue on Randomness and Computation. • [DS14] Irit Dinur and David Steurer. Direct product testing. In 2014 IEEE 29th Conference on Computational Complexity (CCC), pages 188–196, 2014. • [GS97] Oded Goldreich and Shmuel Safra. A combinatorial consistency lemma with application to proving the PCP theorem. In RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science. LNCS, 1997. • [IKW12] Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. New direct-product testers and 2-query PCPs. SIAM J. Comput., 41(6):1722–1768, 2012. • [KL14] Tali Kaufman and Alexander Lubotzky. High dimensional expanders and property testing. In Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, ITCS ’14, pages 501–506, New York, NY, USA, 2014. ACM. • [O’D14] Ryan O’Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.	2020-11-28 20:09:54	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9660224914550781, "perplexity": 1123.1447526467591}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141195745.90/warc/CC-MAIN-20201128184858-20201128214858-00003.warc.gz"}
http://learning.maxtech4u.com/category/uncategorized/big-data/page/2/	Information Retrieval System and Applications / December 24, 2017 Information retrieval (IR) is the field of computer science that deals with the processing of documents containing free text, so that they can be rapidly retrieved based on keywords specified in a user’s query. The effectiveness of IR systems is measured by comparing performance on a common set of queries and documents. The meaning of the term IR can be very broad. Just getting a credit card out of your wallet so that you can type in the card number is a form of IR. However, as an academic field of study, information retrieval might be defined thus: What is information retrieval ? Information retrieval is generally considered as a subfield of computer science that deals with the representation, storage, and access of information. Information retrieval is concerned with the organization and retrieval of information from large database collections Information Retrieval (IR) is the science of searching for information within relational databases, documents, text, multimedia files, and the World Wide Web. Information retrieval is accomplished by means of an information retrieval system and is performed manually or with the use of mechanization or automation. Human beings are indispensable in information retrieval. Depending on the character of the information contained in the… Introduction of HopField Neural Network / December 22, 2017 Human beings are constantly thinking since ages about the reasons for human capabilities and incapabilities. Successful attempts have been made to design and develop systems that emulate human capabilities or help overcome human incapabilities. The human brain, which has taken millions of years to evolve to its present architecture, excels at tasks such as vision, speech, information retrieval, complex pattern recognition, all of which are extremely difficult tasks for conventional computers. A number of mechanisms have been which seems to enable human brain to handle various problems. These mechanisms include association; generalization and self-organization. A brain similar computational technique namely HopField Neural Network is explained here. Working of Hop Field Neural Network A neural network (or more formally artificial neural network) is a mathematical model or computational model inspired by the structure and functional aspects of biological neural networks. It consists of an interconnected group of artificial neurons. The original inspiration for the term Artificial Neural Network came from examination of central nervous systems and their neurons, axons, dendrites and synapses which constitute the processing elements of biological neural networks. One of the milestones for the current renaissance in the field of neural networks was the associative model proposed by Hopfield at… An Introduction of Data Visualization / December 14, 2017 A picture is worth a thousand words – especially when we are trying to understand and discover insights from data. Visuals are especially helpful when we’re trying to find relationships among hundreds or thousands of variables to determine their relative importance – or if they are important at all. Regardless of how much data we have, one of the best ways to discern important relationships is through advanced analysis and high-performance data visualization. If sophisticated analyses can be performed quickly, even immediately, and results presented in ways that showcase patterns and allow querying and exploration, people across all levels in our organization can make faster, more effective decisions. Data Visualization : Definition Data visualizations are surprisingly common in our everyday life, but they often appear in the form of well-known charts and graphs. A combination of multiple visualizations and bits of information are often referred to as infographics. Data visualizations can be used to discover unknown facts and trends. You may see visualizations in the form of line charts to display change over time. Bar and column charts are useful when observing relationships and making comparisons. Pie charts are a great way to show parts-of-a-whole. And maps are the best way… Introduction of Bioinformatics / December 13, 2017 Quantitation and quantitative tools are indispensable in modern biology. Most biological research involves application of some type of mathematical, statistical, or computational tools to help synthesize recorded data and integrate various types of information in the process of answering a particular biological question. Bioinformatics involves the use of computers to collect, organize and use biological information to answer questions in fields like evolutionary biology. Definition of Bioinformatics Bioinformatics is an interdisciplinary research area at the interface between computer science and biological science. A variety of definitions exist in the literature and on the World Wide Web; some are more inclusive than others. Bioinformatics involves the technology that uses computers for storage, information retrieval, manipulation, and distribution of information related to biological macromolecules such as DNA, RNA, and proteins. The emphasis here is on the use of computers because most of the tasks in genomic data analysis are highly repetitive or mathematically complex. The use of computers is absolutely indispensable in mining genomes for information gathering and knowledge building. Bioinformatics is the science of developing computer databases and algorithms for the purpose of speeding up and enhancing biological research. It can be defined more specifically, “Bioinformatics combines the latest technology with biological… What is Knowledge Graph / December 9, 2017 Knowledge graphs are large networks of entities and their semantic relationships. They are a powerful tool that changes the way we do data integration, search, analytics, and context-sensitive recommendations. Knowledge graphs have been successfully utilized by the large Internet tech companies, with prominent examples such as the Google Knowledge Graph. Open knowledge graphs such as Wikidata make community-created knowledge freely accessible. Overview of Knowledge graphs The World Wide Web is a vast repository of knowledge, with data present in multiple modalities such as text, videos, images, structured tables, etc. However, most of the data is present in unstructured format and extracting information in structured machine-readable format is still a very difficult task. Knowledge graphs aim at constructing large repositories of structured knowledge which can be understood by machines. Such knowledge graphs are being used to improve the relevance and the quality of search in case of search engines like Google and Bing. Knowledge graphs are also being used by applications like Google now, Microsoft Cortana and Apple Siri which are capable of understanding natural language queries and answer questions, making recommendations, etc. to the user. The construction of knowledge graphs is thus a major step towards making intelligent personalized machines. Web… What is Deep Learning / December 6, 2017 Machine-learning technology powers many aspects of modern society: from web searches to content filtering on social networks to recommendations on e-commerce websites, and it is increasingly present in consumer products such as cameras and smartphones. Machine-learning systems are used to identify objects in images, transcribe speech into text, match news items, posts or products with users’ interests, and select relevant results of search. Increasingly, these applications make use of a class of techniques called deep learning. Deep learning is a machine learning technique that teaches computers to do what comes naturally to humans: learn by example. Deep Learning Overview Deep Learning is a subfield of machine learning concerned with algorithms inspired by structure and function of brain called artificial neural networks. In deep learning, a computer model learns to perform classification tasks directly from images, text, or sound. Deep learning models can achieve state-of-the-art accuracy, sometimes exceeding human-level performance. Models are trained by using a large set of labeled data and neural network architectures that contain many layers. Deep learning is a subset of machine learning. Usually, when people use the term deep learning, they are referring to deep artificial neural networks, and somewhat less frequently to deep reinforcement learning…. Social Network Analysis / November 22, 2017 Network analysis is still a growing field with a great deal of opportunity for new and transformative contributions. The term social network refers to the articulation of a social relationship, official or achieved, among individuals, families, households, villages, communities, regions, and so on. Each of them can play dual roles, acting both as a unit or node of a social network as well as a social actor Social Network Analysis : Definition Social network theory views a network as a group of actors who are connected by a set of relationships. Social networks develop when actors meet and form some kind of relation between each other. These can be of an informal as well as of a formal nature. Hereby actors are often people, but can also be nations, organizations, objects etc. Social Network Analysis (SNA) focuses on patterns of relations between these actors. It seeks to describe networks of relations as fully as possible. This includes teasing out the prominent patterns in such networks, tracing the flow of information through them, and discovering what effects these relations and networks have on people and organizations. It can therefore be used to study network patterns of organizations, ideas, and people that connected… What is Ensemble Learning / November 19, 2017 Ensemble learning typically refers to methods that generate several models which are combined to make a prediction, either in classification or regression problems. This approach has been the object of a significant amount of research in recent years and good results have been reported. This section introduced basic of the ensemble learning of classification. Ensemble Learning : Overview Ensemble learning is a machine learning paradigm where multiple learners are trained to solve the same problem. In contrast to ordinary machine learning approaches which try to learn one hypothesis from training data, ensemble methods try to construct a set of hypotheses and combine them to use. An ensemble contains a number of learners which are usually called base learners. The generalization ability of an ensemble is usually much stronger than that of base learners. Actually, ensemble learning is appealing because that it is able to boost weak learners which are slightly better than random guess to strong learners which can make very accurate predictions. So, “base learners” are also referred as “weak learners”. It is noteworthy, however, that although most theoretical analyses work on weak learners, base learners used in practice are not necessarily weak since using not-so-weak base learners often… Community Detection : Unsupervised Learning / November 9, 2017 Advances in technology and computation have provided the possibility of collecting and mining a massive amount of real-world data. Mining such “big data” allows us to understand the structure and the function of real systems and to find unknown and interesting patterns. This section provides the brief overview of the community structure. Introduction of Community Detection In the actual interconnected world, and the rising of online social networks the graph mining and the community detection become completely up-to-date. Understanding the formation and evolution of communities is a long-standing research topic in sociology in part because of its fundamental connections with the studies of urban development, criminology, social marketing, and several other areas. With increasing popularity of online social network services like Facebook, the study of community structures assumes more significance. Identifying and detecting communities are not only of particular importance but have immediate applications. For instance, for effective online marketing, such as placing online ads or deploying viral marketing strategies [10], identifying communities in social network could often lead to more accurate targeting and better marketing results. Albeit online user profiles or other semantic information is helpful to discover user segments this kind of information is often at a coarse-grained level… Ariori Algorithm: Example and Algorithm Description / November 2, 2017 With the quick growth in e-commerce applications, there is an accumulation vast quantity of data in months not in years. Data Mining, also known as Knowledge Discovery in Databases (KDD), to find anomalies, correlations, patterns, and trends to predict outcomes. Apriori algorithm is a classical algorithm in data mining. It is used for mining frequent itemsets and relevant association rules. It is devised to operate on a database containing a lot of transactions, for instance, items brought by customers in a store. It is very important for effective Market Basket Analysis and it helps the customers in purchasing their items with more ease which increases the sales of the markets. It has also been used in the field of healthcare for the detection of adverse drug reactions. It produces association rules that indicate what all combinations of medications and patient. Figure 1 Apriori algorithm example application Ariori Algorithm : Overview One of the first algorithms to evolve for frequent itemset and Association rule mining was Apriori. Two major steps of the Apriori algorithm are the join and prune steps. The join step is used to construct new candidate sets. A candidate itemset is basically an item set that could be either Frequent or… Insert math as $${}$$	2018-04-22 16:37:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3250890076160431, "perplexity": 1147.8191900757295}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945624.76/warc/CC-MAIN-20180422154522-20180422174522-00403.warc.gz"}
http://en.wikipedia.org/wiki/Toilet_paper_orientation	# Toilet paper orientation Toilet paper orientation The over orientation The under orientation Toilet paper when used with a toilet roll holder with a horizontal axle parallel to the wall has two possible orientations: the toilet paper may hang over (in front of) or under (behind) the roll. The choice is largely a matter of personal preference, dictated by habit. In surveys of American consumers and of bath and kitchen specialists, 60–70% of respondents prefer over.[1] While many people consider this topic unimportant, some hold strong opinions on the matter. Advice columnist Ann Landers said that the subject was the most controversial issue in her column's history. Defenders of either position cite advantages ranging from aesthetics, hospitality, and cleanliness to paper conservation, the ease of detaching individual squares, and compatibility with a recreational vehicle or a cat. Celebrities are found on both sides. Some writers have proposed connections to age, sex, or political philosophy; and survey evidence has shown a correlation with socioeconomic status.[2] Solutions range from compromise, to using separate dispensers or separate bathrooms entirely, or simply ignoring the issue altogether. One man advocates a plan under which his country will standardize on a single forced orientation, and at least one inventor hopes to popularize a new kind of toilet roll holder which swivels from one orientation to the other.[3] ## Context and relevance In the article "Bathroom Politics: Introducing Students to Sociological Thinking from the Bottom Up",[4] Eastern Institute of Technology sociology professor Edgar Alan Burns describes some reasons why toilet paper politics is worthy of examination. On the first day of Burns' introductory course in sociology, he asks his students, "Which way do you think a roll of toilet paper should hang?"[5] In the following fifty minutes, the students examine why they picked their answers, exploring the social construction of "rules and practices which they have never consciously thought about before".[6] They make connections to larger themes of sociology, including gender roles, the public and private spheres, race and ethnicity, social class, and age. Moreover, Burns argues that there is an additional lesson: Sociologists are often concerned that their discipline is seen merely as an elaboration of the trivial or the obvious. Therefore, the theoretical point illustrated through the paper-hanging exercise is not that small-scale realities are the opposite of big picture sociology, but rather that the big picture does not exist separately "out there." Minor details and "taken for granted" rules and beliefs are the built-in meta-narratives of society, and this is what makes them so powerful.[6] Burns' activity has been adopted by a social psychology course at the University of Notre Dame, where it is used to illustrate the principles of Berger and Luckmann's 1966 classic The Social Construction of Reality.[7] Similar everyday topics that have been used to awaken the sociological imagination include games of tic-tac-toe, violations of personal space, the rules of walking, and the etiquette by which men choose urinals in public restrooms.[8] Christopher Peterson, a professor of psychology at the University of Michigan, classifies the choice of toilet paper orientation under "tastes, preferences, and interests" as opposed to either values or "attitudes, traits, norms, and needs". Other personal interests include one's favorite cola or baseball team. Interests are an important part of identity; one expects and prefers that different people have different interests, which serves one's "sense of uniqueness". Differences in interests usually lead at most to teasing and gentle chiding. For most people, interests don't cause the serious divisions caused by conflicts of values; a possible exception is what Peterson calls "the 'get a life' folks among us" who elevate interests into moral issues.[9] Morton Ann Gernsbacher, a professor of psychology at the University of Wisconsin–Madison, compares the orientation of toilet paper to the orientation of cutlery in a dishwasher, the choice of which drawer in a chest of drawers to place one's socks, and the order of shampooing one's hair and lathering one's body in the shower. In each choice, there is a prototypical solution chosen by the majority, and it is tempting to offer simplistic explanations of how the minority must be different. She warns that neuroimaging experiments—which as of 2007 were beginning to probe behaviors from mental rotation and facial expressions to grocery shopping and tickling—must strive to avoid such cultural bias and stereotypes.[10] In his book Conversational Capital, Bertrand Cesvet gives toilet paper placement as an example of ritualized behavior—one of the ways designers and marketers can create a memorable experience around a product that leads to word-of-mouth momentum. Cesvet's other examples include shaking a box of Tic Tacs and dissecting Oreo cookies.[11] Sometimes toilet paper is simply entertaining. In between songs at a concert, John Hiatt will sometimes tell the tale of his wife switching her preference.[12] Broadcaster Jim Bohannon, who once spent an hour on toilet paper orientation, explains that such issues are good for talk radio: "It is an interactive medium, a certain kind of clash, it doesn't have to be a violent clash, but at least a disagreement would certainly be at the top of the list. It has to be something that's of general interest."[13] There is a difficulty in the medium of television: on the major American networks NBC and CBS, as of 1987, toilet paper was not allowed to be shown hanging next to the toilet.[14] The 1970s sitcom All in the Family was the first show to include a discussion of toilet paper, when Archie yelled at Meathead for hanging the paper under.[15] In a 1995 episode of The Simpsons, "Home Sweet Homediddly-Dum-Doodily", the children are confiscated by Child Protective Services, who hand Marge a note citing her home as a "squalid hellhole" where the toilet paper is "hung in improper overhand fashion".[16] ## Preliminaries In their 2006 book Why Not?, Barry Nalebuff and Ian Ayres write that the debate over toilet paper is a debate about symmetry. (They also write that too much paper has been wasted on the issue, and that they prefer over.) By taking an approximately symmetric situation and flipping it around, one can sometimes arrive at a new solution to a problem with its own surprising advantages. Other physical examples include peeling a banana from the apex rather than the pedicel, or steering a car from the rear rather than the front.[17] There is a reflection symmetry between the left and right sides of the roll, so whether it rotates clockwise or counterclockwise is ambiguous; it depends on one's point of view.[18] The up/down and front/back symmetries are broken by the force of gravity and the locations of the wall and the user, so one can distinguish between two orientations: • Over: the end hangs away from the wall and dispenses over the top of the roll when pulled. • Under: the end hangs next to the wall and dispenses under the bottom of the roll. This nomenclature can also be read ambiguously. In 1991, a customer wrote to Herb Kelleher, chairman of Southwest Airlines, with an unusual complaint: "Dear Herb: ... Last week in my journey to SFO someone put the toilet paper in wrong. Any damn fool knows the papers come out the bottom of the roll and not over the top. I couldn't figure out how to correct the error ..."[19] Kelleher replied, copying his senior management committee, general counsel, and customer relations manager: "Dear Jim: What the hell were you doing upside down in our lavatory?"[20] Kevin and Jackie Freiberg cite this episode in their book Nuts! as an example of Southwest's unconventional approach to customer service.[21] There are other everyday objects that dispense a sheet of material from a roll: fax machines, cash registers, plastic wrap, aluminum foil, and parchment paper. One columnist who believes in the importance of toilet paper orientation writes, "all have to exit in the correct direction or it doesn't work, or you cut yourself, or both."[22] ## Arguments for over or under Folded and sealed toilet paper with cover, Hotel Monasterio 2009 Paper mounted under with upside-down images and text The main reasons given by people to explain why they hang their toilet paper a given way are ease of grabbing and habit.[23] Some particular advantages cited for each orientation include: • Over reduces the risk of accidentally brushing the wall or cabinet with one's knuckles, potentially transferring grime and germs.[24] • Over makes it easier to visually locate and to grasp the loose end.[25] • Over gives hotels, cruise ships, office buildings, public places and homeowners with guest bathrooms the option to fold over the last sheet to show that the room has been cleaned.[26] • Over is generally the intended direction of viewing for the manufacturer's branding, so patterned toilet paper looks better this way.[27] • Under provides a tidier appearance, in that the loose end can be more hidden from view.[28][29] • Under reduces the risk that a toddler or a house pet, such as a dog or cat, will completely unroll the toilet paper when batting at the roll.[30] • Under in a recreational vehicle may reduce unrolling during driving.[31] Partisans have claimed that each method makes it easier to tear the toilet paper on a perforated sheet boundary, depending on the direction of pulling and the use of a second hand to stabilize the roll.[32] (A traveller from the U.S. to China in 1991 noted a different setup: non-perforated paper with a metal cutter above the roll, which obliges the over direction.)[33] It is unclear if one orientation is more economical than the other. The Centralian Advocate attributes a claim that over saves on paper usage to Planet Green.[34] A reader of The Orange County Register found a "six-month study" by a "university in the U.S." that came to the same conclusion.[35] But a reader of the Cape Argus wrote that a "British loo paper manufacturer" came to the opposite conclusion.[36] In his humor compilation How Hemlines Predict the Economy, Peter FitzSimons writes that placing the hanging flap against the wall "is generally twice as economical".[37] In the academic field of evaluation, Michael Scriven writes that the question of the correct way to insert toilet paper is a "one-item aptitude test" for measuring one's evaluation skills. These skills include the evaluative attitude, practical logical analysis, empathy, teaching, and being a quick study. To prove one's competence, one may either derive the "one right answer" or prove that the test is or is not culturally biased.[38] ## Survey results The question "Do you prefer that your toilet tissue unwinds over or under the spool?" is featured on the cover of Barry Sinrod and Mel Poretz's 1989 book The First Really Important Survey of American Habits. The overall result: 68% chose over.[39] Sinrod explained, "To me, the essence of the book is the toilet paper question ... Either people don't care, or they care so much that they practically cause bodily injury to one another."[40] Poretz observed, "The toilet-paper question galvanizes people almost like the Miller Lite tastes-great/less-filling commercial."[41] In Bernice Kanner's 1995 book Are You Normal?, 53% of survey respondents prefer over, while "a fourth" prefer under and 8% don't know or care.[42] Sitting Pretty: The History of the Toilet, a travelling exhibition that tours Canadian museums, asks visitors to register their preferred roll direction. When the exhibition reached Huntsville, Ontario, in June 2001, 13,000 visitors had taken the survey, with 67% preferring over.[43] At the Saint Boniface Museum in Winnipeg in February 2005, a voting machine registered 5,831 over versus 5,679 under, or 51% over. Saint Boniface's director noted, "I think there's been some cheating, though."[44] Georgia-Pacific commissioned a survey of Americans' bathroom habits in 1993 to launch its new Quilted Northern brand, and more surveys followed:[45] • 1993 Practices and Preferences of Toilet Paper Users: 73% over out of 1,200 respondents. The press release claims, "A first-of-its-kind survey has settled, once and for all, the great toilet paper debate."[46] • 1994 Toilet Paper Report: 59% over,[47] out of 1,000 respondents; conducted by KRC Research and Consulting[48] • 1995 Bathroom Tissue Report: 59% over versus 29% under,[49] out of 1,000 respondents; conducted by KRC Research and Consulting[47] • 2001 Bathroom Confidential: 63% over out of 1,001 respondents; conducted by Impulse Research[50] • 2004 Bathroom Confidential: 72% over[51] In 1993, American Standard Brands conducted a poll of "designers, contractors, dealers, distributors and other bath and kitchen reps"[52] at the Kitchen/Bath Industry Show & Conference in Atlanta, Georgia. The question: "What is the correct and only way to hang the toilet paper – under or over?"[53] Over won 59% of the vote, 1,826 to 1,256.[52] American Standard spokeswoman Nora Monroe observed, "The bathroom is a territorial place. You'd be surprised how many people have definite opinions on this issue."[54] In 2008, American Standard commissioned the 2008 Bathroom Habits Survey, a more traditional format conducted by Opinion Research Corporation with 1,001 respondents. This time, "three-quarters" answered over.[55] In 1995, a survey by Scott Paper Company's "Cottonelle College of Freshness Knowledge" had "most Americans over 50" preferring over.[56] In another Cottonelle survey in 1999, 68% of respondents preferred over to 25% under. Columnist Bonnie Henry hypothesizes of the others: "Meanwhile, 7 percent – no doubt bored beyond belief at this point by the inane questioning – had slipped into a deep, irreversible coma."[57] On January 27, 2010, the 100th anniversary of Thomas Crapper's death,[23] Cottonelle launched a "Great Debate" advertising campaign, inviting American consumers to vote their preference at a Kimberly-Clark website.[58] The result was announced during the 82nd Academy Awards: 72% had voted over.[59] In a more traditional preliminary survey of 1,000 Americans, Cottonelle found that "overs" are more likely than "unders" to notice a roll's direction (74%), to be annoyed when the direction is incorrect (24%), and to have flipped the direction at a friend's home (27%).[60] Besides orientation, toilet paper manufacturers and survey authors have studied other private practices around toilet paper: how much is used; whether it is torn off with one hand or two; whether it is torn off right-to-left or left-to-right; and whether it is crumpled or folded before use.[46] ## Themes ### Sex and age Poretz and Sinrod break down the results of their 1989 survey by sex and age. These are the percentages of respondents who roll their paper over:[61] 21–34 35–44 45–54 55 + Total Grand total Male 71% 81% 60% 63% 69% 68% Female 81% 65% 62% 83% 67% The book does not note the number of respondents in each segment, so it is difficult to say whether any of the deviations are statistically significant, but there does not seem to be a difference between men's and women's preferences. Nonetheless, such a difference has been claimed by other authors, in both directions. The American Standard conference poll concluded: "Many men voted for over, saying it made the paper easier to reach."[54] Inventor Curtis Batts arrives at a different conclusion from his personal experience: "Women like it over, and men like it under. I think it bugs women when it touches the wall."[62] Advice columnist Ms Maud of The Press asserts that women prefer over because they are "logical thinkers".[63] A Cottonelle survey indicated that men were more likely than women to notice, and become annoyed with, a toilet roll hung against their preference.[64] A popular-culture occurrence of a gender theory is found in the Weekly World News, a supermarket tabloid that runs outlandish stories for comedic effect. In the 2003 story North Korea Shocker!, the WWN claimed that North Korean leader Kim Jong-il was secretly female. As supporting evidence, Kim supposedly watched the Home Shopping Network, is a member of Oprah's Book Club, and "Yells at staffers who leave the toilet seat up and hang toilet paper rolls outward instead of inward."[65] According to W. C. Privy's Original Bathroom Companion, Number 2, "By more than 4 to 1, older folks prefer to have their toilet paper dispense over the front."[66] The same claim is made by James Buckley's The Bathroom Companion for people older than 50.[67] ### Class and politics Sinrod observed of his survey, "60 percent of those who earn $50,000 or more prefer it to be over and 73 percent of those who earn less than$20,000 prefer under".[40] On what that proves: "I don't know, but it's sure interesting."[39] In one local election in Saskatoon, Saskatchewan, new voting machines were given a trial run by asking the question, "Are you in favor of toilet paper in all public washrooms being installed with the loose end coming up and over the front of the roll?" The answer was yes: 768 to 196, or 80% over. It was thought to be a question "which carried no political association".[68] Yet one teenager's science project at the Southern Appalachian Science and Engineering Fair, and a favorite of the fair's coordinator, was a survey concluding that liberals roll over while conservatives roll under.[69] ### Character In his 2003 book 10 Steps to Sales Success, Tim Breithaupt proposes a set of four personality types evolving from Carl Jung's work: Socializer, Director, Thinker, and Relater. Breithaupt writes that toilet paper management is an important detail for Thinkers, while Directors don't care so long as the paper is available.[70] In her 2001 book Three Keys to Self-Understanding, Pat Wyman locates having an opinion on toilet paper hanging on the Enneagram of Personality, which classifies people as Ones, Twos, Threes, and so on: "Ones know the answer to such dilemmas."[71] Gilda Carle, a therapist and Cottonelle consultant, offers her theories on character traits: If you roll over, you like taking charge, crave organization and are likely to over-achieve. If you roll under, you're laid-back, dependable and seek relationships with strong foundations. If you don't care as long as it's there, you aim to minimize conflict, value flexibility and like putting yourself in new situations.[64] David Grimes, a columnist, takes a more sarcastic attitude towards bathroom-informed personality tests: If you are the kind of person who prefers the paper to roll over the top, then you are an outgoing, free-spending type who gets his kicks trying to sneak 11 items through the 10-items-or-less line at the grocery store; if you are the kind of person who prefers the paper to roll from the bottom, then you are a naturally suspicious sort who vacuums his house three times a day and thinks Jerry Springer is god. Or perhaps the other way around.[72] A reporter for the trade journal Fund Action relays a story of a mutual fund firm that profiled job candidates with questions that would be analyzed by a psychologist. One of the questions was "Which way do you hang toilet paper? So it unrolls from the front or the back?". The story does not reveal the name of the firm or its preferred answer.[73] ## Consequences Toilet paper orientation is often mentioned as a hurdle for married couples.[74] The issue may also arise in businesses and public places.[75] Even at the Amundsen–Scott Research Station at the South Pole, complaints have been raised over which way to install toilet paper. During the six-month-long polar night, a few dozen residents are stuck living together, and while many of the headaches of modern life are far away, food and hygiene are not. Despite the challenges posed by the hostile Antarctic climate, "It is in the more mundane trials of everyday life that personality clashes are revealed."[76] ## Similar controversies Domestic strife can arise from many other situations where a household item, such as a tube of toothpaste, is left in the wrong state.[77] Some closely related examples: • Which way should a paper towel hang in the kitchen? When Ann Landers was asked this question in 1997, she replied, "I'm still trying to recover from the flak ... I'm not giving any more advice on how to hang anything."[78] • Should a toilet seat be left up or down? This debate involves a stronger asymmetry between the sexes, as women rarely want the seat up.[48] • Should a twist tie be tightened clockwise or counter-clockwise?[79] Since some store-bought products are pre-tied by machine, this question also pits consumer against engineer.[80] • For a public restroom stall with a dispenser holding two rolls of paper, Donald Knuth proposes classifying users into big-choosers (those who take paper from the roll that is currently larger) and little-choosers (those who do the opposite). Letting $p$ denote the probability that a random user is a big-chooser and $q$ that of a little-chooser, Knuth uses contour integration and generating functions to find the expected number of sheets left on the larger roll when the smaller one runs out. He shows (Theorem 1) that if $\|p-1/2\|$ is of order at least $1/\sqrt n$, then $M_n(p)=\begin{cases}p/(p-q) + O(r^n), & q p \end{cases}$ where $r$ is an arbitrary parameter larger than 4pq and n is the number of sheets in a roll. He also separately analyzes the case where p=q.[81] ## Solutions Some of the proposed solutions to this problem involve more or better technology, while others concentrate on human behavior. ### Mechanical The Tilt-A-Roll is a swiveling toilet paper dispenser invented by Curtis Batts, a Dallas-native industrial engineer.[62] His patents on the invention published in 1996 and 1997, US 5588615 and US 5690302 , summarize its design: An adjustable angle coupling secures the yoke to the mounting assembly and permits rotation of the yoke about an axis directed orthogonally through the spindle such that the paper roll can be oriented to unroll paper either from over or from under the roll as desired. Batts explains that his parents argued over toilet paper placement "all the time", as do he and his wife; the device's motto is "Let Tilt-A-Roll save your marriage!"[62] The Tilt-A-Roll has been featured on a variety of newspapers, magazines, radio interviews, and TV shows, including The Tonight Show with Jay Leno.[82] Batts entered the Tilt-A-Roll at the 1999 INPEX invention show, the world's largest invention show with 800 inventions,[83] and it won third place "for its appeal and simplicity".[84] Batts started out constructing the devices in a workshop in his garage. He explains, "I've had a few sleepless nights where I went 24 hours making these things. I've got to find a manufacturer for it. I can't keep up with the volume."[62] As of 2000, Batts was still "in the process of selecting a distributor";[85] he has tried The Home Depot[82] and QVC.[86] As of 2008, the Tilt-A-Roll is on sale at Batt's website.[87] An inventor named Rocky Hutson demonstrated a similar device he called the T.P. Swivel to the producers of the television program PitchMen in late 2009. Of 173 entrants gathered at Ybor City, Tampa, Florida, Hutson was one of the 20 chosen to pitch their products to Anthony Sullivan.[88] For his part, Hutson pulls his paper from the top of the roll.[89] Twelve rolls in a mix of states Another solution: install two toilet paper dispensers, as is more common in public restrooms and hotels.[90] A reader of the Annie's Mailbox column recommends using a holder large enough to fit two rolls, noting that the roll mounted over is more popular. Another reader sidesteps the issue by foregoing the holder, instead piling five or six rolls in a big wicker basket.[91] Even using separate bathrooms can help.[92] Other solutions include vertical holders, or simply not using a toilet roll holder at all. ### Behavioral A Grand Rapids, Michigan, toilet paper enthusiast named Bill Jarrett argues that previous polls have been too small. He wants a national referendum with at least one million votes, with the result to decide a "national toilet paper hanging way" to be enforced by "the toilet paper police".[93] Jarrett refuses to reveal his own preference; he even removed the toilet paper from his house's bathrooms before inviting in an AP reporter for an interview. "I'm not saying because I don't want to influence the vote."[94] Voting requires the purchase of a $5 debate kit. His value proposition to the nation: assuming that one can spend half an hour per year searching for the end of the toilet paper, the United States should save 90 million hours at home per year—and$300 million at the workplace.[95] Toilet paper orientation has been used rhetorically as the ultimate issue that government has no business dictating, in letters to the editor protesting the regulation of noise pollution[96] and stricter requirements to get a divorce.[97] In 2006, protesting New Hampshire's ban on smoking in restaurants and bars, representative Ralph Boehm (R-Litchfield) asked "Will we soon be told which direction the toilet paper must hang from the roll?"[98] In a column in the Houston Chronicle, Jack Brewer observes that it only takes five seconds to turn the roll "the right way" around (over), which is much less than the time it takes to "start a fuss" with his wife.[99] In a column in The Grand Rapids Press, Karin Orr relates her chance discovery that her husband and sister both turn the toilet paper around in others' houses—and in opposite directions. Orr writes, "You just can never really know another person."[100] David O'Connor's 2005 book Henderson's House Rules: The Official Guide to Replacing the Toilet Paper and Other Domestic Topics of Great Dispute aims to solve disagreements with a minimum of debate or compromise by offering authoritative, reasonable rules.[101] The "House Rule" for toilet paper is over and out, and a full page is dedicated to a diagram of this orientation. But O'Connor writes that "if a female household member has a strong preference for the toilet paper to hang over and in, against the wall, that preference prevails. It is admittedly an odd preference, but women use toilet paper far more often than men—hence the rule."[102] ## Noted preferences Multi-orientable toilet paper holder Advice columnist Ann Landers (Eppie Lederer) was once asked which way toilet paper should hang. She answered under, prompting thousands of letters in protest; she then recommended over, prompting thousands more.[103] She reflected that the 15,000 letters made toilet paper the most controversial issue in her column's 31-year history,[104] wondering, "With so many problems in the world, why were thousands of people making an issue of tissue?"[103] In November 1986, Landers told the Canadian Commercial Travellers Association that "Fine-quality toilet paper has designs that are right side up" in the over position.[104] In 1996, she explained the issue on The Oprah Winfrey Show, where 68% of the studio audience favored over; Oprah suggested that under uses more paper.[105] In 1998, she wrote that the issue "seems destined to go on forever", insisting, "In spite of the fact that an overwhelming number of people prefer the roll hung so that the paper comes over the top, I still prefer to have the paper hanging close to the wall."[68] On the day of her last column in 2002, Landers wrote, "P.S. The toilet paper hangs over the top."[106] Her commentary on the issue has even continued after her death. 2005 saw the premiere of a one-woman play written by David Rambo: a character study of Ann Landers titled The Lady with All the Answers. Toilet paper comes up once again, and the actress surveys the audience for their opinions.[107] In his article in Teaching Sociology, Burns writes that the toilet paper hanging exercise is valuable in part because "[the] subject matter is familiar to everybody; everyone is an expert, and everyone has an opinion."[108] The media have published the opinions of entertainers, advice columnists, and businesspeople, including the following: Over Indifferent • Annie's Mailbox, advice column by Kathy Mitchell and Marcy Sugar: "If the toilet paper has a pattern, it should roll from the top over. Otherwise, it doesn't matter. Really. However you like it is just fine with us."[123] • Heloise, advice columnist: "Both ways. I try not to get obsessed about these things."[124] • Erik Seidel, brand director for Kimberly-Clark's Scott portfolio: "That's a key debate in my house. The 'under' looks cleaner and neater because you don't see it. The 'over' is more convenient."[125] Under • Ann Landers, advice columnist: "I'm very compulsive about it. The toilet paper needs to be hung down along the wall. I'll actually rearrange it myself if I'm over at someone's home and I see it hung over the top."[126] • Dean McDermott, actor: "Under just says Zen to me. When it rolls over, the toilet paper seems so aggressive. Calmly rolled under, it's symmetrical and orderly and there when I need it."[64] • Mel Poretz, author and marketing executive: "I'm an 'under' and thought we were in the majority by far. I never thought there were civilized people who put toilet paper over the top."[41] "... I married an under-the-roll girl. If not, we'd probably be divorced."[127] • Gerhard Richter's 1965 oil painting Klorolle depicts a roll hung under, as a comment on Marcel Duchamp's Fountain. Unusually, the source material is a photograph he took himself.[128] The third version of the work (CR 75-3) has been exhibited in six museums, and in 1995, it sold at Sotheby's for £120,000.[129] • Barry Sinrod, author and marketing executive: "I'm an under person, I don't know why."[40] • Gene Weingarten, journalist: "The main reason is aesthetic. It looks better. I will warrant that 80 to 85 percent of artists, architects and interior decorators have it spool out from below, with the overage hanging against the wall, not flappying down from the top into the middle of the room. I am so right. I am inarguably correct. I cannot even believe we are having this discussion."[130] ## Notes 1. ^ This paragraph summarizes material in the body; details and citations are found below. For definitions of the choices, see Preliminaries. Habit is discussed in Arguments. See Survey results for statistics. 2. ^ For pros and cons, including RVs and cats, see Arguments; for celebrities and experts, including Ann Landers, see Noted preferences; for theories, see Themes. 3. ^ The enthusiast, Bill Jarrett, and the inventor, Curtis Batts, are described in Solutions. 4. ^ 5. ^ Burns 2003, p. 111. 6. ^ a b Burns 2003, p. 113. 7. ^ 8. ^ Paul 2006. The previous topics are discussed in the section "Finding Sociology in everyday places: a review". 9. ^ Peterson 2006, pp. 173–175. 10. ^ 11. ^ Cesvet, Babinski & Alper 2008, p. 68. 12. ^ 13. ^ 14. ^ 15. ^ This is described as a "first" by Magill (1993, p. 2236). The substance of the argument is mentioned in Landers (1992). 16. ^ Cantor 2003, p. 76. 17. ^ Nalebuff & Ayres 2006, pp. 115–118. 18. ^ 19. ^ 20. ^ Grant 1991a. A slightly different quotation is in Dayton Daily News (1996). 21. ^ Freiberg & Freiberg 1998, p. 270. 22. ^ 23. ^ a b 24. ^ Ode 2010: "The Kimberly-Clark company cites three advantages for rolling over: perforation control, viewing advantage and wall avoidance."; Garton 2005; Jarski & Jarski 2007. 25. ^ 26. ^ Lind 1992; "The Grand Princess cruise ship replaces its toilet paper with the leading edge over the front, so that it can be folded as is done in five-star hotels. (Yes, someone really did ask this question.)" (Carpenter 1999); Rosencrans 1998; Garton 2005. 27. ^ 28. ^ Jarski & Jarski 2007 29. ^ 30. ^ Darbo 2007; Garton 2005; O'Connor 2005, p. 63. 31. ^ 32. ^ 33. ^ 34. ^ 35. ^ 36. ^ 37. ^ FitzSimons 2009, p. 99. 38. ^ Scriven 1991, "EVALUATION SKILLS", pp. 151–153, especially p. 153 for the quotations. 39. ^ a b 40. ^ a b c 41. ^ a b 42. ^ Kanner 1995, pp. 56, 120. 43. ^ 44. ^ 45. ^ The 1996 report, which may not have contained this question, was the fourth annual report: (McCarthey 1996) 46. ^ a b 47. ^ a b 48. ^ a b 49. ^ 50. ^ 51. ^ 52. ^ a b 53. ^ 54. ^ a b 55. ^ 56. ^ 57. ^ 58. ^ CottonelleRollPoll.com 59. ^ 60. ^ 61. ^ Poretz & Sinrod 1989, p. 34. 62. ^ a b c d 63. ^ 64. ^ a b c d 65. ^ 66. ^ Barrett & Mingo 2003, p. 400. 67. ^ Buckley 2005, p. 106. 68. ^ a b 69. ^ 70. ^ Breithaupt 2003, pp. 126, 135. 71. ^ Wyman 2001, p. 61. 72. ^ 73. ^ 74. ^ Wolf 1999, pp. 74–75; Hogan & Hogan 2000, p. 200. 75. ^ 76. ^ Daily Express 1999, p. 39. 77. ^ 78. ^ 79. ^ 80. ^ 81. ^ Knuth, Donald E. (October 1984). "The Toilet Paper Problem". American Mathematical Monthly 91 (8): 465–470. JSTOR 2322567. 82. ^ a b 83. ^ 84. ^ a b Greenberg 2007, p. 149. 85. ^ 86. ^ 87. ^ Greenberg 2007, p. 149. See The Tilt-A-Roll homepage at Curtis Batts Online 88. ^ 89. ^ 90. ^ 91. ^ 92. ^ 93. ^ Godfrey 2006, p. 103. 94. ^ 95. ^ 96. ^ 97. ^ 98. ^ 99. ^ 100. ^ 101. ^ O'Connor 2005, pp. 2–3. 102. ^ O'Connor 2005, pp. 63–64; Davis 2006. 103. ^ a b 104. ^ a b 105. ^ 106. ^ a b 107. ^ 108. ^ Burns 2003, p. 116. 109. ^ 110. ^ 111. ^ 112. ^ 113. ^ a b c 114. ^ 115. ^ 116. ^ 117. ^ 118. ^ 119. ^ 120. ^ 121. ^ 122. ^ "How Do You Roll? ", MarthaStewart.com. 123. ^ 124. ^ 125. ^ 126. ^ 127. ^ 128. ^ Elger & Solaro 2010, pp. 104–105, 119. 129. ^ 130. ^ ## References • "For Your Information", The News & Observer, 4 October 1993: C1, Factiva rnob000020011101dpa400p2p References "a Reader's Digest poll". Primary source unclear. • Brody, Ed (2002), Spinning Tales, Weaving Hope: Stories, Storytelling, and Activities for Peace, Justice and the Environment, New Society Publishers, p. 158 • Cameron, W. Bruce (2004), How to Remodel a Man: Tips and Techniques on Accomplishing Something You Know Is Impossible But Want to Try Anyway, Macmillan, p. 185 • Carpenter, Richard P. (28 March 1999), "It's swimming vs. snoozing", The Boston Globe: M4, Factiva bstngb0020010825dv3s00asy • Freeman, Kim (8 February 1986), "Vox Jox", Billboard: 16 References a poll by Ric Hanson featured in USA Today. • Grossvogel, David I. (1987), Dear Ann Landers: our intimate and changing dialogue with America's best-loved confidante, Contemporary Books, p. 257 • Kelly, William Jude (1988), Models in process: a rhetoric and reader, Macmillan, p. 154 • Kogan, Rick (2004), America's Mom: The Life, Lessons, and Legacy of Ann Landers, Thorndike Press, p. 224 • Praeger, Dave (2007), Poop Culture: How America Is Shaped by Its Grossest National Product, Feral House, p. 72 • Selby, David (1995), Earthkind: a teachers' handbook on humane education, Trentham Books, p. 367 • Singular, Stephen (1987), Talked to death: the life and murder of Alan Berg, p. 305 Mentions Bob Palmer of Denver's KCNC-TV doing a show on this topic. • Society of Automotive Engineers (2004), Reliability and robust design in automotive engineering, p. 412 Presents a statistical test to determine gender differences in toilet paper orientation. • Trachtenberg, Robert (2005), When I Knew, HarperCollins, p. 69	2014-09-02 02:35:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 6, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2815556228160858, "perplexity": 6882.41525470547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535921318.10/warc/CC-MAIN-20140909054252-00444-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/science/chemistry/introductory-chemistry-5th-edition/chapter-13-solutions-exercises-problems-page-479/34	## Introductory Chemistry (5th Edition) According to the given data, A $KCl$ solution containing 42 g of $KCl$ per 100 g of water is cooled from 60 °C to O °C. The solubility of $KNO_3$ at 60 °C = 45 g / 100 g of water. So, the given solution acts as an unsaturated solution at this temperature. The solubility of $KNO_3$ at 0 °C = 27 g / 100 g of water. At this temperature, the given solution acts as a supersaturated solution. So, cooling down from 60 °C to 0 °C results in the recrystallization of $KCl$.	2019-06-16 12:50:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.783599317073822, "perplexity": 591.361965375319}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998238.28/warc/CC-MAIN-20190616122738-20190616144738-00203.warc.gz"}
https://math.stackexchange.com/questions/895889/function-with-continuous-inverse-is-continuous	# Function with continuous inverse is continuous? If function $\textbf{F}^{-1}(x)$ is an inverse of function $\textbf{F}$ and $\textbf{F}^{-1}(x)$ is continuous. Is it true that $\textbf{F}(x)$ is continuous too? Take $f^{-1}(x) = x$ on $[0,1)$ and $f^{-1}(x) = x-1$ on $[2,3]$. Then $f(x) = x$ on $[0,1)$ and $f(x) = 1+x$ on $[1,2]$. Here $f^{-1}:[0,1) \cup [2,3] \rightarrow [0,2]$ is continuous, and $f$ is discontinuous at $x=1$. I think the map $f:[0,1) \rightarrow S^1$ given by $x \rightarrow (\sin 2\pi x, \cos 2\pi x)$ is a counter. For a somewhat-trivial example , take $id: (X,\text{discrete} )\rightarrow (X, \text{indiscrete})$ , for any set $X$ . As a general result, if $f A \rightarrow B$ is a continuous bijection; $A$ is compact and $B$ is Hausdorff, then $f^{-1}$ is necessarily continuous. For counters from a space to itself with the same topology, see: https://mathoverflow.net/questions/30661/non-homeomorphic-spaces-that-have-continuous-bijections-between-them The answer is no: take $f^{-1}(x) = e^{ix}$, defined from $[0,2\pi)$ to $\mathbb{S}^1$ (the unit sphere in the plane) This function is clearly continuous. Unfortunately, its inverse cannot be continuous since otherwise $[0,2\pi)$ would be compact being the image of the compact set $\mathbb{S}^1$ under a continuous function. I hope it helps, let me know if the details are clear enough :) EDIT: something easier to check: the identity map from the reals with the lower limit topology ($\mathbb{R}_l$) to the real with the standard topology ($\mathbb{R}$) is another counterexample! This is immediate to check since the inverse of the identity map is of course the identity map itself which now goes from $\mathbb{R}$ to $\mathbb{R}_l$. Clearly the pre image of the open set $[a,b)$ is not open! • Don't you mean [0, 2\pi) would be compact? – user99680 Aug 13 '14 at 6:47 • yeah, of course I do! :) thank you for pointing it out! – user67133 Aug 13 '14 at 6:48	2019-06-19 02:06:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9593901634216309, "perplexity": 159.25454241431012}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998879.63/warc/CC-MAIN-20190619003600-20190619025600-00426.warc.gz"}
https://www.ideals.illinois.edu/handle/2142/16340/browse?value=Feuerman%2C+Kenneth+Edward&type=author	# Browse Dissertations and Theses - Mathematics by Author "Feuerman, Kenneth Edward" • (1991) The Hanna Neumann Conjecture states that if two subgroups of a finitely generated free group have finite ranks m and n, then their intersection has rank N which satisfies $N$ $-$ 1 $\leq$ ($m$ $-$ 1)($n$ $-$ 1). The current ... application/pdf PDF (4MB)	2017-07-22 09:00:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4870796203613281, "perplexity": 1426.9147757696587}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423927.54/warc/CC-MAIN-20170722082709-20170722102709-00201.warc.gz"}
http://hackage.haskell.org/package/polysemy-1.2.1.0/docs/Polysemy-Internal-Strategy.html	polysemy-1.2.1.0: Higher-order, low-boilerplate, zero-cost free monads. Polysemy.Internal.Strategy Synopsis # Documentation data Strategy m f n z a where Source # Constructors GetInitialState :: Strategy m f n z (f ()) HoistInterpretation :: (a -> n b) -> Strategy m f n z (f a -> m (f b)) GetInspector :: Strategy m f n z (Inspector f) type Strategic m n a = forall f. Functor f => Sem (WithStrategy m f n) (m (f a)) Source # Strategic is an environment in which you're capable of explicitly threading higher-order effect states to the final monad. This is a variant of Tactics (see Tactical), and usage is extremely similar. Since: 1.2.0.0 type WithStrategy m f n = '[Strategy m f n] Source # Since: 1.2.0.0 runStrategy :: Functor f => Sem '[Strategy m f n] a -> f () -> (forall x. f (n x) -> m (f x)) -> (forall x. f x -> Maybe x) -> a Source # Internal function to process Strategies in terms of withWeavingToFinal. Since: 1.2.0.0 getInspectorS :: forall m f n. Sem (WithStrategy m f n) (Inspector f) Source # Get a natural transformation capable of potentially inspecting values inside of f. Binding the result of getInspectorS produces a function that can sometimes peek inside values returned by bindS. This is often useful for running callback functions that are not managed by polysemy code. See also getInspectorT Since: 1.2.0.0 getInitialStateS :: forall m f n. Sem (WithStrategy m f n) (f ()) Source # Get the stateful environment of the world at the moment the Strategy is to be run. Prefer pureS, liftS, runS, or bindS instead of using this function directly. Since: 1.2.0.0 pureS :: Applicative m => a -> Strategic m n a Source # Embed a value into Strategic. Since: 1.2.0.0 liftS :: Functor m => m a -> Strategic m n a Source # Lifts an action of the final monad into Strategic. Note: you don't need to use this function if you already have a monadic action with the functorial state threaded into it, by the use of runS or bindS. In these cases, you need only use pure to embed the action into the Strategic environment. Since: 1.2.0.0 runS :: n a -> Sem (WithStrategy m f n) (m (f a)) Source # Lifts a monadic action into the stateful environment, in terms of the final monad. The stateful environment will be the same as the one that the Strategy is initially run in. Use bindS if you'd prefer to explicitly manage your stateful environment. Since: 1.2.0.0 bindS :: (a -> n b) -> Sem (WithStrategy m f n) (f a -> m (f b)) Source # Embed a kleisli action into the stateful environment, in terms of the final monad. You can use bindS to get an effect parameter of the form a -> n b into something that can be used after calling runS on an effect parameter n a. Since: 1.2.0.0	2020-02-18 08:52:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.657670795917511, "perplexity": 4691.984104733845}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143635.54/warc/CC-MAIN-20200218055414-20200218085414-00078.warc.gz"}
https://abcpy.readthedocs.io/en/v0.6.3/postanalysis.html	# 5. Post Analysis¶ The output of an inference scheme is a Journal (abcpy.output.Journal) which holds all the necessary results and convenient methods to do the post analysis. ## Basis Analysis¶ One can easily access the sampled parameters and corresponding weights using: print(journal.get_parameters()) print(journal.get_weights()) The output of get_parameters() is a Python dictionary. The keys for this dictionary are the names you specified for the parameters. The corresponding values are the marginal posterior samples of that parameter. Here is a short example of what you would specify, and what would be the output in the end: a = Normal([[1],[0.1]], name='parameter_1') b = MultivariateNormal([[1,1],[[0.1,0],[0,0.1]]], name='parameter_2') If one defined a model with these two parameters as inputs and n_sample=2, the following would be the output of journal.get_parameters(): {'parameter_1' : [[0.95],[0.97]], 'parameter_2': [[0.98,1.03],[1.06,0.92]]} These are samples at the final step of ABC algorithm. If you want samples from the earlier steps of a sequential algorithm you can get a Python dictionary for that step by using: journal.get_parameters(step_number) Since this is a dictionary, you can also access the values for each step as: journal.get_parameters(step_number)["name"] For the post analysis basic functions are provided: # do post analysis print(journal.posterior_mean()) print(journal.posterior_cov()) Also, to ensure reproducibility, every journal stores the parameters of the algorithm that created it: print(journal.configuration) And certainly, a journal can easily be saved to and loaded from disk: journal.save("experiments.jnl") new_journal = Journal.fromFile('experiments.jnl') ## Posterior plots and diagnostics¶ You can plot the inferred posterior distribution of the parameters in the following way: journal.plot_posterior_distr(path_to_save="posterior.png") The above line plots the posterior distribution for all the parameters and stores it in posterior.png; if you instead want to plot it for some parameters only, you can use the parameters_to_show argument; in addition, the ranges_parameters argument can be used to provide a dictionary specifying the limits for the axis in the plots: journal.plot_posterior_distr(parameters_to_show='parameter_1', ranges_parameters={'parameter_1': [0,2]}) For journals generated with sequential algorithms, we provide a way to check the convergence by plotting the estimated Effective Sample Size (ESS) at each iteration, as well as an estimate of the Wasserstein distance between the empirical distributions defined by the samples and weights at subsequent iterations: journal.plot_ESS() journal.Wass_convergence_plot() Instead, for journals generated by MCMC, we provide way to plot the traceplot for the required parameters: journal.traceplot() ## Posterior resampling and predictive check¶ In some cases, you may want to resample (for instance, bootstrapping or subsampling) the posterior samples stored in a Journal, by tacking into account the posterior weights. This can be done using the resample() method. Behind the scenes, this uses the numpy.random.choice method, and it inherits arguments from it. It allows to do different things, for instance: • if the set of posterior samples (weighted or unweighted) is too large, you can obtained a subsampled (without replacement) set by doing: new_journal = journal.resample(n_samples=100, replace=False) • Alternatively, if the used algorithm returns weighted posterior samples, you may want instead an unweighted set of samples obtained by sampling with replacement (commonly called bootstrapping); this can be done with the following line (where the number of required bootstrapped samples in the new journal is unspecified and therefore corresponding to the number of samples in the old journal): new_journal = journal.resample() Finally, in some cases you may want to generate simulations from the model for parameter values sampled from the posterior, for instance in order to check similarity with the original observation (predictive check). ABCpy provides the output.GenerateFromJournal to do that. This class needs to be instanstiated by providing to it the model and the backend which you want to use for the simulation; then, you can pass a Journal as argument to the generate() method in order to generate simulations from the posterior samples contained there: generate_from_journal = GenerateFromJournal([model], backend=backend) parameters, simulations, normalized_weights = generate_from_journal.generate(journal)	2021-10-23 02:31:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5842053294181824, "perplexity": 1079.6620374296022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585537.28/warc/CC-MAIN-20211023002852-20211023032852-00397.warc.gz"}
https://www.vedantu.com/question-answer/solve-the-following-equation-log-16x-+-log-4x-+-class-11-maths-jee-main-5efbb685954ada76881ab4c1	QUESTION # Solve the following equation:${\log _{16}}x + {\log _4}x + {\log _2}x = 7$ Hint: We have to use the necessary logarithmic properties to find the value of x. It is given to us that ${\log _{16}}x + {\log _4}x + {\log _2}x = 7$ We know that $\left[ {\because {{\log }_a}b = \dfrac{1}{{{{\log }_b}a}}} \right]$ So, we will get $\Rightarrow \dfrac{1}{{{{\log }_x}16}} + \dfrac{1}{{{{\log }_x}4}} + \dfrac{1}{{{{\log }_x}2}} = 7$ On simplification, we get $\Rightarrow \dfrac{1}{{{{\log }_x}{2^4}}} + \dfrac{1}{{{{\log }_x}{2^2}}} + \dfrac{1}{{{{\log }_x}2}} = 7$ Now since we know that $\left[ {\because n{{\log }_a}M = {{\log }_a}{M^n}} \right]$ And hence on following the above formula we have, $\Rightarrow \dfrac{1}{{4{{\log }_x}2}} + \dfrac{1}{{2{{\log }_x}2}} + \dfrac{1}{{{{\log }_x}2}} = 7$ And hence on taking $\dfrac{1}{{{{\log }_x}2}}$ common, we get, $\left[ {\dfrac{1}{4} + \dfrac{1}{2} + 1} \right]\dfrac{1}{{{{\log }_x}2}} = 7$ And hence on doing the simplification, we have $\left[ {\dfrac{7}{4}} \right]\dfrac{1}{{{{\log }_x}2}} = 7$ $\Rightarrow \dfrac{1}{{{{\log }_x}2}} = 7 \times \left[ {\dfrac{4}{7}} \right]$ Here 7 will get cancelled out $\Rightarrow {\log _2}x = 4$ $\left[ {\because {{\log }_a}b = \dfrac{1}{{{{\log }_b}a}}} \right]$ $\Rightarrow {2^4} = x$ $\Rightarrow x = 16$ Note: This question consists of equations comprising logarithmic functions. So we just need to use the appropriate logarithmic properties. Mistakes should be avoided in application of these properties.	2020-07-11 11:24:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8853399157524109, "perplexity": 602.9300705201422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655929376.49/warc/CC-MAIN-20200711095334-20200711125334-00261.warc.gz"}
https://socratic.org/questions/5487b383581e2a1c00615a87	# Question #15a87 Dec 13, 2014 There are a few rules that we can apply to the ground state electron configuration of an element to find what ions it can form. First it is important to know that the transition elements do not obey the octet rule. So predicting their ions is a little more difficult, but not much more. Rules for obtaining the electron configuration of Cations: 1. The first electrons lost by an atom or ion are those from the shell with the largest value n ( the outer s and p electrons). 2. Inside a given shell, the electrons are removed from the highest energy subshell before the lower energy subshell is emptied (p emptied before s.) 3. Electrons in the d orbitals are emptied after both s and p are empty since they are the lowest in energy. 4. When the d orbital is present, the s and p orbitals lose all their electrons at once, the d orbitals lose their electrons one at a time, if at all . 5. For transition elements the first electrons lost are from the s. If additional electrons are lost, they come from the lower energy d subshell one at a time. Example when d orbital is present: Tin (Sn) Ground state electron configuration: $S n \left[K r\right] 5 {s}^{2} 4 {d}^{10} 5 {p}^{2}$ First rewrite in the order of how they will lose their electrons. $S n \left[K r\right] 4 {d}^{10} 5 {s}^{2} 5 {p}^{2}$ Now we follow our rules to understand the different possible tin ions. A) $S {n}^{2 +} \left[K r\right] 4 {d}^{10} 5 {s}^{2}$ Notice we followed rule 2 and 4. Since electrons are negatively charged, losing 2 negative charges leaves a 2+ positive charge (since we didn't remove positively charged protons-but that's another lesson entirely.) B) $S {n}^{4 +} \left[K r\right] 4 {d}^{10}$ Since the full d orbital is pretty stable, it would take alot of energy to keep removing electrons from this point. Example transition element: notice: I wrote the electron configuration so that we can just begin removing electrons--skipping the first step shown in the last example Iron (Fe) Ground state electron configuration: $F e \left[A r\right] 3 {d}^{6} 4 {s}^{2}$ A) $F {e}^{2 +} \left[A r\right] 3 {d}^{6}$ Iron loses it's s electrons fairly easily. B) $F {e}^{3 +} \left[A r\right] 3 {d}^{5}$ Rule 5 Now that the 3d subshell is half filled it is in a decently stable state and would be hard pressed to lose more electrons.	2022-05-22 16:21:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5038528442382812, "perplexity": 1266.669479078937}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662545875.39/warc/CC-MAIN-20220522160113-20220522190113-00315.warc.gz"}
https://physics.stackexchange.com/questions/333962/why-in-su5-we-do-not-consider-bar-nu-l	# Why in $SU(5)$ we do not consider $\bar{\nu}_L$? In GUT, why in representation $\bar{5}+10$ of $SU(5)$ we do not consider $\bar{\nu}_L$? One says that there are 15 particles-antiparticles per generation but, for me, there are 16 particles-antiparticles. • why do you think that $\nu_L \in 5$ describes a different physical state than $\nu_L \in \bar{5}$? – jak May 18 '17 at 8:39 • Of course but I meant why in $\bar{5}+10$ (or $5+\bar{10}$) do we have both $e_L^+$ and $e_L^-$ and not both $\bar{\nu}_L$ and $\nu_L$ ? – ketherok May 19 '17 at 22:10 When Georgi and Glashow discovered the first GUT model, they noticed that all standard model particles fit perfectly in the $\bar{5}\oplus 10$ representation of $SU(5)$. That's why they proposed that it is a good idea to study this kind of model with $SU(5)$ as a GUT group. However, if you like you can consider an $SU(5)$ model with fermions in the $\bar{5}\oplus 10 \oplus 1$ representation, i.e. simply add the right-chiral fermion by hand to the model. This is, in fact, exactly what you get when you consider $SO(10)$ as a GUT group. The $15$ standard model fermions plus the right-chiral neutrino fit perfectly in the $16$ of $SO(10)$. When you break $SO(10)$ to $SU(5)$ you get $$16 \to \bar{5}\oplus 10 \oplus 1$$ • @AccidentalFourierTransform I think the singlet makes no difference for the anomaly cancellation. The anomalies within $\bar{5}\oplus 10$ cancel and equally withing $\bar{5}\oplus 10 \oplus 1$. Otherwise we would have a strong argument for or against right-chiral neutrinos – jak Jun 2 '17 at 6:25	2019-11-17 16:09:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8746541738510132, "perplexity": 221.79912661721468}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669183.52/warc/CC-MAIN-20191117142350-20191117170350-00509.warc.gz"}
https://maker.pro/forums/threads/can-we-perform-pause-operation-while-replaying-wav-file-in-ltspice.295495/	# Can we perform pause operation while replaying .wav file in ltspice? #### santlal123 Jul 19, 2021 1 I have recorded a node voltage value in a .wav file using Ltspice. While replaying the recorded signal values, I want a pause of 5 millisecond after every 10millisecond. Please, suggest the way. I stucked at this point. Thank you so much. Santlal Prajapati #### Alec_t Jul 7, 2015 3,285 Welcome to EP! You could use the .measure directive to sample the node voltage at chosen time points. Replies 13 Views 4K Replies 3 Views 2K B Replies 11 Views 1K mleuck M M Replies 28 Views 2K Paul Hovnanian P.E. P E Replies 0 Views 1K E	2023-02-03 20:42:44	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8027287721633911, "perplexity": 11763.761213372767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500074.73/warc/CC-MAIN-20230203185547-20230203215547-00299.warc.gz"}
http://mathreports.ru/en/articles/splines-for-three-point-rational-interpolants-with-autonomous-poles/	# Splines for three-point rational interpolants with autonomous poles ### DOI: 10.31029/demr.7.2 For arbitrary grids of nodes $\Delta: a=x_0<x_1<\dots<x_N=b$ $(N\geq 2)$ smooth splines for three--point rational interpolants are constructed, the poles of interpolants depend on nodes and the free parameter $\lambda$. Sequences of such splines and their derivatives for all functions $f(x)$ respectively of the classes of $C_{[a,b]}^{(i)}$ $(i=0,1,2)$ under the condition $\\|\Delta\\| \to 0$ uniformly in $[a,b]$ converge respectively to $f^{(i)}(x)$ $(i=0,1,2)$ (depending on the parameter $\lambda$). Bonds for the convergence rate are found in terms of the distance between the nodes. Keywords: splines, interpolation splines, rational splines.	2021-09-19 10:35:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9724478125572205, "perplexity": 1186.2583329494212}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056856.4/warc/CC-MAIN-20210919095911-20210919125911-00342.warc.gz"}
https://labs.tib.eu/arxiv/?author=J.M.C.%20Rawlings	• ### Unified Models of Molecular Emission from Class 0 Protostellar Outflow Sources(1308.5111) Aug. 23, 2013 astro-ph.GA Low mass star-forming regions are more complex than the simple spherically symmetric approximation that is often assumed. We apply a more realistic infall/outflow physical model to molecular/continuum observations of three late Class 0 protostellar sources with the aims of (a) proving the applicability of a single physical model for all three sources, and (b) deriving physical parameters for the molecular gas component in each of the sources. We have observed several molecular species in multiple rotational transitions. The observed line profiles were modelled in the context of a dynamical model which incorporates infall and bipolar outflows, using a three dimensional radiative transfer code. This results in constraints on the physical parameters and chemical abundances in each source. Self-consistent fits to each source are obtained. We constrain the characteristics of the molecular gas in the envelopes as well as in the molecular outflows. We find that the molecular gas abundances in the infalling envelope are reduced, presumably due to freeze-out, whilst the abundances in the molecular outflows are enhanced, presumably due to dynamical activity. Despite the fact that the line profiles show significant source-to-source variation, which primarily derives from variations in the outflow viewing angle, the physical parameters of the gas are found to be similar in each core. • ### Episodic Explosions in Interstellar Ices(1301.2982) Jan. 14, 2013 astro-ph.GA We present a model for the formation of large organic molecules in dark clouds. The molecules are produced in the high density gas-phase that exists immediately after ice mantles are explosively sublimated. The explosions are initiated by the catastrophic recombination of trapped atomic hydrogen. We propose that, in molecular clouds, the processes of freeze-out onto ice mantles, accumulation of radicals, explosion and then rapid (three-body) gas-phase chemistry occurs in a cyclic fashion. This can lead to a cumulative molecular enrichment of the interstellar medium. A model of the time-dependent chemistries, based on this hypothesis, shows that significant abundances of large molecular species can be formed, although the complexity of the species is limited by the short expansion timescale in the gas, immediately following mantle explosion. We find that this mechanism may be an important source of smaller organic species, such as methanol and formaldehyde, as well as precursors to bio-molecule formation. Most significantly, we predict the gas-phase presence of these larger molecular species in quiescent molecular clouds and not just dynamically active regions, such as hot cores. As such the mechanism that we propose complements alternative methods of large molecule formation, such as those that invoke solid-state chemistry within activated ice mantles. • ### The JCMT Legacy Survey of the Gould Belt: mapping 13CO and C18O in Orion A(1201.5483) March 15, 2012 astro-ph.GA The Gould Belt Legacy Survey will map star-forming regions within 500 pc, using HARP (Heterodyne Array Receiver Programme), SCUBA-2 (Submillimetre Common-User Bolometer Array 2) and POL-2 (Polarimeter 2) on the James Clerk Maxwell Telescope (JCMT). This paper describes HARP observations of the J = 3-2 transitions of 13CO and C18O towards Orion A. The 1500-resolution observations cover 5 pc of the Orion filament, including OMC1 (inc. BN-KL and Orion Bar), OMC 2/3 and OMC 4, and allow a comparative study of the molecular gas properties throughout the star-forming cloud. The filament shows a velocity gradient of ~1 km/s /pc between OMC 1, 2 and 3, and high velocity emission is detected in both isotopologues. The Orion Nebula and Bar have the largest masses and line widths, and dominate the mass and energetics of the high velocity material. Compact, spatially resolved emission from CH3CN, 13CH3OH, SO, HCOOCH3, C2H5OH, CH3CHO and CH3OCHO is detected towards the Orion Hot Core. The cloud is warm, with a median excitation temperature of ~24 K; the Orion Bar has the highest excitation temperature gas, at >80 K. The C18O excitation temperature correlates well with the dust temperature (to within 40%). The C18O emission is optically thin, and the 13CO emission is marginally optically thick; despite its high mass, OMC 1 shows the lowest opacities. A virial analysis indicates that Orion A is too massive for thermal or turbulent support, but is consistent with a model of a filamentary cloud that is threaded by helical magnetic fields. The variation of physical conditions across the cloud is reflected in the physical characteristics of the dust cores....continued • ### The JCMT Legacy Survey of the Gould Belt: a first look at Taurus with HARP(1002.2020) Feb. 10, 2010 astro-ph.GA As part of a JCMT Legacy Survey of star formation in the Gould Belt, we present early science results for Taurus. CO J=3-2 maps have been secured along the north-west ridge and bowl, collectively known as L 1495, along with deep 13CO and C18O J=3-2 maps in two sub-regions. With these data we search for molecular outflows, and use the distribution of flows, HH objects and shocked H2 line emission features, together with the population of young stars, protostellar cores and starless condensations to map star formation across this extensive region. In total 21 outflows are identified. It is clear that the bowl is more evolved than the ridge, harbouring a greater population of T Tauri stars and a more diffuse, more turbulent ambient medium. By comparison, the ridge contains a much younger, less widely distributed population of protostars which, in turn, is associated with a greater number of molecular outflows. We estimate the ratio of the numbers of prestellar to protostellar cores in L 1495 to be ~ 1.3-2.3, and of gravitationally unbound starless cores to (gravitationally bound) prestellar cores to be ~ 1. If we take previous estimates of the protostellar lifetime of ~ 5 x 10^5 yrs, this indicates a prestellar lifetime of 9(+/-3) x 10^5 yrs. From the number of outflows we also crudely estimate the star formation efficiency in L 1495, finding it to be compatible with a canonical value of 10-15 %. We note that molecular outflow-driving sources have redder near-IR colours than their HH jet-driving counterparts. We also find that the smaller, denser cores are associated with the more massive outflows, as one might expect if mass build-up in the flow increases with the collapse and contraction of the protostellar envelope. • ### The JCMT Legacy Survey of the Gould Belt: a first look at Orion B with HARP(0908.4162) Oct. 16, 2009 astro-ph.GA The Gould Belt Legacy Survey will survey nearby star-forming regions (within 500 pc), using HARP (Heterodyne Array Receiver Programme), SCUBA-2 (Submillimetre Common- User Bolometer Array 2) and POL-2 (Polarimeter 2) on the James Clerk Maxwell Telescope (JCMT). This paper describes the initial data obtained using HARP to observe 12CO, 13CO and C18O J = 3 - 2 towards two regions in Orion B, NGC 2024 and NGC 2071. We describe the physical characteristics of the two clouds, calculating temperatures and opacities utilizing all three isotopologues. We find good agreement between temperatures calculated from CO and from dust emission in the dense, energetic regions. We determine the mass and energetics of the clouds, and of the high-velocity material seen in 12CO emission, and compare the relative energetics of the high- and low-velocity material in the two clouds. We present a CLUMPFIND analysis of the 13CO condensations. The slope of the condensation mass functions, at the high-mass ends, is similar to the slope of the initial mass function. • ### Molecular tracers of PDR-dominated galaxies(0903.3191) March 18, 2009 astro-ph.GA Photon-dominated regions (PDRs) are powerful molecular line emitters in external galaxies. They are expected in galaxies with high rates of massive star formation due to either starburst (SB) events or starburst coupled with active galactic nuclei (AGN) events. We have explored the PDR chemistry for a range of physical conditions representing a variety of galaxy types. Our main result is a demonstration of the sensitivity of the chemistry to changes in the physical conditions. We adopt crude estimates of relevant physical parameters for several galaxy types and use our models to predict suitable molecular tracers of those conditions. The set of recommended molecular tracers differs from that which we recommended for use in galaxies with embedded massive stars. Thus, molecular observations can in principle be used to distinguish between excitation by starburst and by SB+AGN in distant galaxies. Our recommendations are intended to be useful in preparing Herschel and ALMA proposals to identify sources of excitation in galaxies. • ### Tracing high density gas in M 82 and NGC 4038(0808.2815) Aug. 20, 2008 astro-ph We present the first detection of CS in the Antennae galaxies towards the NGC 4038 nucleus, as well as the first detections of two high-J (5-4 and 7-6) CS lines in the center of M 82. The CS(7-6) line in M 82 shows a profile that is surprisingly different to those of other low-J CS transitions we observed. This implies the presence of a separate, denser and warmer molecular gas component. The derived physical properties and the likely location of the CS(7-6) emission suggests an association with the supershell in the centre of M 82. • ### Molecular tracers of high mass star-formation in external galaxies(0712.2776) Dec. 17, 2007 astro-ph Hot core molecules should be detectable in external active galaxies out to high redshift. We present here a detailed study of the chemistry of star-forming regions under physical conditions that differ significantly from those likely to be appropriate in the Milky Way Galaxy. We examine, in particular, the trends in molecular abundances as a function of time with respect to changes in the relevant physical parameters. These parameters include metallicity, dust:gas mass ratio, the H$_{2}$ formation rate, relative initial elemental abundances, the cosmic ray ionization rate, and the temperature of hot cores. These trends indicate how different tracers provide information on the physical conditions and on evolutionary age. We identify hot core tracers for several observed galaxies that are considered to represent spirals, active galaxies, low-metallicity galaxies, and high-redshift galaxies. Even in low-metallicity examples, many potential molecular tracers should be present at levels high enough to allow unresolved detection of active galaxies at high redshift containing large numbers of hot cores. • ### CO abundances in a protostellar cloud: freeze-out and desorption in the envelope and outflow of L483(0710.2814) Nov. 27, 2007 astro-ph CO isotopes are able to probe the different components in protostellar clouds. These components, core, envelope and outflow have distinct physical conditions and sometimes more than one component contributes to the observed line profile. In this study we determine how CO isotope abundances are altered by the physical conditions in the different components. We use a 3D molecular line transport code to simulate the emission of four CO isotopomers, 12CO J=2-1, 13CO J=2-1, C18O J=2-1 and C17O J=2-1 from the Class 0/1 object L483, which contains a cold quiescent core, an infalling envelope and a clear outflow. Our models replicate JCMT (James Clerk Maxwell Telescope) line observations with the inclusion of freeze-out, a density profile and infall. Our model profiles of 12CO and 13CO have a large linewidth due to a high velocity jet. These profiles replicate the process of more abundant material being susceptible to a jet. C18O and C17O do not display such a large linewidth as they trace denser quiescent material deep in the cloud. • ### Determining the cosmic ray ionization rate in dynamically evolving clouds(astro-ph/0511064) Nov. 2, 2005 astro-ph The ionization fraction is an important factor in determining the chemical and physical evolution of star forming regions. In the dense, dark starless cores of such objects, the ionization rate is dominated by cosmic rays; it is therefore possible to use simple analytic estimators, based on the relative abundances of different molecular tracers, to determine the cosmic ray ionization rate. This paper uses a simple model to investigate the accuracy of two well-known estimators in dynamically evolving molecular clouds. It is found that, although the analytical formulae based on the abundances of H3+,H2,CO,O,H2O and HCO+ give a reasonably accurate measure of the cosmic ray ionization rate in static, quiescent clouds, significant discrepancies occur in rapidly evolving (collapsing) clouds. As recent evidence suggests that molecular clouds may consist of complex, dynamically evolving sub-structure, we conclude that simple abundance ratios do not provide reliable estimates of the cosmic ray ionization rate in dynamically active regions.	2021-02-25 22:55:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6157002449035645, "perplexity": 3013.961903120438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178355937.26/warc/CC-MAIN-20210225211435-20210226001435-00340.warc.gz"}
https://www.physicsforums.com/threads/cabibbo-theory.236431/	# Cabibbo Theory Hi, I'm trying to cover the basics of Cabibbo theory, yet the materials Ive been presented with give a very jumbled description, and I'd just like to ask here to obtain some clarity... Is it fair to say that the Cabibbo angle is a means of quantifying the different coupling strengths between different generations of quarks? Or is this too much a simplification or just plain ol wrong? Regards Luke	2021-03-03 18:35:57	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8198843002319336, "perplexity": 1102.0608707329943}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178367183.21/warc/CC-MAIN-20210303165500-20210303195500-00636.warc.gz"}
https://tex.stackexchange.com/questions/380688/circle-and-line-diagram-tex	# Circle and line diagram tex [closed] I saw this diagram I wonder how we type this up in Latex. I have seen that TikZ package produces similar outcome(?) ## closed as too broad by TeXnician, Zarko, Stefan Pinnow, Paul Gaborit, JesseJul 15 '17 at 2:48 Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. • So far I only understood how to draw circle and lines. Could not add labels nor connect them. – CL. Jul 14 '17 at 15:13 • Please add that as code example (MWE). – TeXnician Jul 14 '17 at 15:13 • Your observation is correct! You can draw this as three (growing right), as automaton or as simple image of three circle connected bylines. Maybe for star you look in texample.net. There is a lot of examples of TikZ pictures. – Zarko Jul 14 '17 at 15:13 • I have found this one, as almost the thing I wanted. Now I would like to add circles around text... which I have no idea how. – CL. Jul 14 '17 at 15:21 • Then you should simply post some code... – TeXnician Jul 14 '17 at 15:23 Here is a simple solution with pst-tree (can be compiled with pdflatex , with switch --enable-write18 under MiKTeX,-shell-escape` under TeX Live and MacTeX): \documentclass[border=3pt]{standalone} \usepackage[utf8]{inputenc} \usepackage{pst-tree} \usepackage{auto-pst-pdf} \begin{document} \everypsbox{\scriptstyle} \psset{shortput=tab, labelsep=0pt} \MakeShortTnput{\tnput} $\begin{psmatrix} \pstree[treemode=R, arrows=->, arrowinset=0]{% \Tcircle{\mathsf{S₁}}\tnput[labelsep=24pt]{\mathsf{t = 0}}}% {% \Tcircle{\mathsf{S₃}}^{\mathsf{p}} \Tcircle{\mathsf{S₂}}\tnput[labelsep=5pt, tnpos=b]{\mathsf{t = dt}}_{\mathsf{1-p}}}% \end{psmatrix}$ \end{document}	2019-06-25 11:32:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9100275635719299, "perplexity": 1838.4630990827598}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999838.23/warc/CC-MAIN-20190625112522-20190625134522-00109.warc.gz"}
https://github.com/IngoScholtes/kdd2018-tutorial/blob/master/code/1_8_exploration.py	IngoScholtes/kdd2018-tutorial Fetching contributors… Cannot retrieve contributors at this time 59 lines (43 sloc) 2.87 KB #%% import markdown from IPython.core.display import display, HTML def md(str): display(HTML(markdown.markdown(str + " "))) #%% md(""" # 1.8 Exploration: Multi-order analysis of [paths and time-stamped social networks](https://github.com/IngoScholtes/kdd2018-tutorial/tree/master/data) Ingo Scholtes Data Analytics Group Department of Informatics (IfI) University of Zurich August 22 2018 In the last (open-ended) exploration, you get the chance to apply multi-order representation in the analysis of real data. In addition to the pathway data from session 1, we will consider data that we provide in the SQLite database temporal_networks.db. You can check which tables it contains by checking the metadata table: """) #%% In [2] import pathpy as pp import sqlite3 con = sqlite3.connect('data/temporal_networks.db',) con.row_factory = sqlite3.Row for row in con.execute('SELECT * from metadata'): print('{0} \t\t {1}'.format(row['tag'], row['name'])) #%% md(""" Details on the origin of these data can be found [here](https://github.com/IngoScholtes/kdd2018-tutorial/tree/master/data). Below, we include boilerplate code to load these data sets into the TemporalNetwork class in pathpy: """) #%% In [3] table = 'manufacturing_email' # Check whether network is directed or not directed_network = bool(con.execute("SELECT directed FROM metadata WHERE tag='{0}'".format(table)).fetchone()['directed']) t = pp.TemporalNetwork.from_sqlite(con.execute('SELECT source, target, time FROM ' + table), directed=directed_network) print(t) #%% md(""" Using these data and the methods introduced in our tutorial, we suggest to study the following problems (in ascending order of difficulty): - Generate higher-order visualisations of the US Flight and London Tube data and visually compare the graph layouts calculated for the first and optimal-order models. - Use the MultiOrderModel class to learn the optimal order of a temporal network. How does the detected optimal order change with the time scale $\delta$ that you use in the extraction of causal paths? - Use the MultiOrderModel class to learn the optimal order of the London Tube data set. How does the detected optimal order compare to the prediction performance studied in exploration 1.4? - Study the change in the algebraic connectivity between the second-order model and the second-order null model for (i) a temporal network data set and (ii) the US Flights data. - Perform a spectral clustering of a dynamic social network based on the Laplacian of higher-order networks at different orders. How does the clustering differ from a first-order clustering? Again, these are only suggestions and you are welcome to use the time to study other data sets or questions that come to your mind. We'll be happy to help you with the analysis. """)	2019-01-22 17:45:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24768854677677155, "perplexity": 3939.8481191741757}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583857993.67/warc/CC-MAIN-20190122161345-20190122183345-00008.warc.gz"}
https://www.physicsforums.com/threads/phase-constant-in-wave-propagations-what-are-its-effects.367515/	# Phase constant in wave propagations: what are its effects? 1. Jan 6, 2010 ### Ionito In the electromagnetism theory, the phase factor or constant (usually BETA) in wave propagation for lossy medium has the unit rad/m. I understood that it must be interpreted as the amount of phase shift that occurs as the wave travels one meter. However, differently of the attenuation factor (usually ALFA), I cannot see examples relating the phase factor to the distance. In other words, we can see the signal attenuation as the form of 8.69ALFAd, where d is the distance between the sender and the receiver. However, this distance "d" is not used in conjunction with the phase factor BETA. Is it right? Is there any correlation between BETA and the signal attenuation? If not, what are the effects of having a HIGH and LOW BETA? Can anyone provide me a complete example of the total attenuation (in dBs), given ALFA, BETA, frequency, and distance d, for a plane wave propagating in a lossy medium? Thanks 2. Jan 6, 2010 ### Born2bwire The wave number/vector is a complex quantity. The real part dictates the phase progression with regard to spatial displacement. This is your \beta. The imaginary part gives rise to an attenuating behavior with regards to space, this is your \alpha loss. They are both related with the position in space/distance travelled since they both come out of the phase term: $$e^{i\mathbf{k}\cdot\mathbf{r}}$$ So the real part will dictate the phase progression and the effective wavelength of the wave while the imaginary part dictates the attenuation of the wave. Both of these are dependent on not only the physical properties of the medium of propagation but also the frequency of the wave as well since the wave number is defined as: $$k = \|\mathbf{k}\|^{\frac{1}{2}} = \omega\sqrt{\epsilon\mu}$$ 3. Jan 6, 2010 ### Ionito Thank you for the answer. Besides the velocity, wavelength, and phase shift effects caused by different BETAs of lossy media, is it possible (and how) that these BETAs are also related to the attenuation behavior (a secondary factor in addition to the ALFA factor)? 4. Jan 6, 2010 ### Born2bwire If you are talking about \beta and \alpha in a strictly mathematical sense, then no, there is no relation. However, they are derived from the same material properties and frequency of the wave by the fact that they come from the wave number, defined above. So the two are linked physicaly by the fact that they are directly determined by the material properties of your surrounding medium. If we had a dielectric of a given permittivity and permeability, the introduction of loss, without modifying its dielectric constant (real part of the permittivity), results in the shortening of the wavelength as the square root of the permittivity increases when we increase its magnitude. So increasing the loss of the material, via the manipulation of the material's conductivity, will decrease the wavelength (increase \beta) and increase the attenuation factor (increase \alpha).	2018-03-25 01:56:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.87443608045578, "perplexity": 552.6083544711796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257651481.98/warc/CC-MAIN-20180325005509-20180325025509-00618.warc.gz"}
https://www.oecd-ilibrary.org/sites/05bd280b-en/index.html?itemId=/content/component/05bd280b-en	1. Developments in Agricultural Policy and Support In 2020, agricultural policies and support to the sector were significantly affected by the outbreak of the coronavirus SARS-CoV-2, the subsequent spread of the COVID-19 virus, and substantial restrictions to populations and enterprises aimed at containing the virus. These factors caused economic growth to slow significantly or even turn strongly negative in all economies, while in many countries unemployment rose as companies were forced to lay off employees. Commodity markets were affected as well, but the implications for global agricultural markets remained comparatively limited as, despite some significant stresses, food systems and supply chains proved relatively robust. The pandemic caused major dislocations to food markets, in particular with the closure of restaurants, and the shift in consumption away from food outside the home. But the overall demand for food was fairly stable, as food supply was generally recognised to be essential and thus exempt from lockdowns, while consumers prioritised food among their expenditures. However, several advanced economies saw increased recourse to food banks among low income consumers who had seen a drastic fall in their incomes. Labour intensive sectors, such as meat processing and sectors requiring seasonal labour for planting or harvesting, were also deeply affected by the virus and measures to contain it. As a land based activity, the production of most commodities was generally able to withstand the pandemic, although those products requiring more labour input – principally fruits and vegetables – or where supply is destined primarily for the restaurant trade, were more affected. In general, developments on agricultural markets were driven as much by non-COVID factors as by the impacts of the pandemic. Overall, the agricultural sector proved remarkably resilient, with farm incomes increasing in 2020 for a majority of countries covered in this report. To help people and companies to cope with the economic consequences of both the virus and containment strategies, governments introduced a wide set of policies as of early 2020. In looking at changes made to agricultural policies and support, this report therefore begins by discussing policy responses to the COVID-19 pandemic that focus on, or strongly affect, agricultural producers, other actors along the food supply chain, and food consumers. The report then analyses the level and structure of agricultural support, in particular in terms of the extent to which they help or hinders the performance of food systems, gauged in terms of their contribution to the “triple challenge” of: 1. 1. Achieving food security and nutrition for a growing world population. 2. 2. Providing livelihoods to farmers and others connected to the sector, either vertically along the value chain or spatially across rural economies. 3. 3. Reducing the environmental footprint of the sector and contributing to lower greenhouse gas (GHG) emissions. Drawing on insights from the OECD Framework for Productivity, Sustainability and Resilience, this part of the report also explores how current policies perform across the three dimensions of productivity, sustainability and resilience, which are identified as key channels through which agriculture can contribute to the challenges facing food systems. Lastly, this part of the report concludes with an assessment of the developments in policies and support, and with recommendations for concrete actions to improve the performance of agricultural policies in meeting the challenges facing global food systems. Conditions in agricultural markets are strongly influenced by macro-economic factors, such as economic growth (measured by gross domestic product, GDP), which generates the income supporting demand for agricultural and food products, as well as prices for crude oil and other energy sources which affect the prices of numerous production inputs in agriculture, such as fuel, chemicals and fertiliser. Energy prices also affect the demand for cereals, sugar crops and oilseeds through the market for biofuels produced from these feedstocks. Global economic growth, which slowed to below 3% in 2019, came to an abrupt halt in the wake of the COVID-19 pandemic. Global output in 2020 is estimated to have been more than 4% below that in 2019, reflecting policy responses to the pandemic, which included substantial restrictions in both personal and economic activities (OECD, 2020[1]).1 GDP growth in all OECD economies turned negative. The contraction was particularly significant in the Euro area, where economic output declined by 7.5% in 2020, after low growth of 1.3% the year before. Japan was significantly hit as well, with GDP shrinking by 5.3% in 2020, after some first signs of rebounding growth in 2019 at +0.7%. The contraction was less pronounced in the United States, where economic output, which grew by more than 2% in 2019, declined by 3.7% in 2020. The downturn in OECD economies was associated with a decreased demand for labour. Across the OECD area, unemployment, which had fallen slightly to 5.4% in 2019, increased to 7.2% in 2020. In many countries, the negative impact on employment was mitigated by substantial public interventions, including notably the widespread application of publicly supported short-time work.2 Average inflation declined further to 1.5%, driven in particular by falling energy prices (see below). Growth in emerging economies also fell substantially, although the extent of the downturn varied strongly. Argentina’s GDP, which had seen negative growth for the last two years already, shrank by 12.9%, the first double-digit economic contraction since the currency and debt crisis of 2001-02. India’s GDP contracted by 9.9%, more than 14 percentage points below 2019 growth, while South Africa’s GDP fell by 8.1%, following stagnation in 2019. On the other hand, the People’s Republic of China (hereafter, “China”) is the only country covered in this report that maintained positive growth in 2020, at 1.8% compared with 6.1% the year before. The Indonesian economy also fared comparatively well, with a slight contraction of 2.4%, following 5% growth in 2019. The consequences of the COVID-19 pandemic and of related restrictions are strongly visible in international trade. In real terms, global trade declined by more than 10% year-on-year, following already slow growth in 2019. Lower economic growth and restrictions on personal and economic mobility put significant pressure on prices for energy and other non-food commodities (IMF, 2021[2]). On average, energy prices in 2020 were 30% lower than in 2019, and more than 40% below their 2018 levels. Crude oil prices, which had fallen to levels close to (and on certain markets even below) zero in April 2020, averaged 33% lower over the full year compared to 2019. Lower energy prices also pulled down fertiliser prices, which on average were 9% lower year-on-year. In comparison, food prices remained robust. After dropping by 7% in the second quarter of 2020, average international food prices increased towards the end of the year, and annual averages ended 3% higher than in 2019, with contrasting movements between crop and livestock markets, as explained below. Global food markets saw prices for crops and livestock products moving in opposite directions. World meat markets had seen production decline in 2019 primarily due to the impact of African Swine Fever (ASF) on China’s pig meat sector. While the disease continued to limit production in China and other countries such as Viet Nam during 2020, herds began to rebuild. In spite of the lower Chinese output, however, global meat prices were under significant downward pressure in 2020 due to logistical difficulties and reduced demand following the COVID-19 pandemic, which together dampened meat import demand from several key importing countries. On average, meat prices in 2020 fell by 4.5% year-on-year. The pandemic also had significant, though varied, impacts on dairy markets. While away-from-home consumption in many countries suffered as a result of widespread confinement measures, larger retail sales for at-home consumption partly offset these losses. Fresh dairy products were particularly vulnerable to disruptions in supply chains, but many countries were able to adjust their production chains relatively quickly. As a consequence, while the effects of the pandemic varied across regions, global dairy prices changed only little year-on-year, with lower prices in the second quarter balanced by rising prices towards the end of the year. In contrast to livestock markets, world prices for crop commodities mostly rose in 2020. Following short-term disruptions due to the COVID-19 pandemic, oilseeds markets were driven by strong demand notably for imported soybeans into China as the country began to rebuild pig herds. At the same time, lower supply growth of palm oil resulted in relatively short supplies on international markets. As a consequence, international prices rose significantly in 2020, with prices for soybeans and vegetable oils averaging 7% and almost 20% higher than in 2019. Increased feed demand from the rebuilding pork sector in China, logistical difficulties in some major producing countries, and some temporary export restrictions following the COVID-19 pandemic, drove prices upwards in cereal markets. Pushed by increases notably towards the end of the year, average cereal prices were almost 7% higher in 2020 than in the preceding year. Continued shortfalls in sugar production due to unfavourable weather conditions in some of the major producing countries offset lower import demand for sugar and notably reduced biofuel demand in light of reduced mobility due to the pandemic, resulting in average sugar prices increasing slightly year-on-year, but remaining well below levels seen in 2016. Overall, food supply chains were recognised as essential services in most countries implementing COVID-19 related restrictions on economic activities, as a result of which the sector was affected by those restrictions more indirectly than directly. Often, both domestic and international trade in food products were facilitated through green corridors and other measures notwithstanding disruptions affecting trade overall. Labour shortages due to restrictions on people’s movement were alleviated through exceptions for agricultural and food chain workers, and through schemes encouraging workers laid off in other sectors or students to temporarily work in agriculture and the food industry. However, income losses and economic uncertainties, together with restrictions for restaurants and other away-from-home food suppliers, generated changes in food demand which the industry needed to cope with. But the impact of economic contractions on food expenditure was mitigated through public support partly compensating for income losses, and reductions in disposable incomes seem to have led to higher shares of income being spent on food. Partly with the help of government policy responses, food systems have therefore proven remarkably resilient. Indeed, after short-term disruptions in international food markets in the early phase of the pandemic, these markets appear to have been impacted more by other factors such as livestock diseases and climatic conditions than by the pandemic itself. As governments started implementing containment measures to slow the spread of the COVID-19 virus early in 2020, they also began introducing measures to limit impacts of the virus and associated containment measures on the agriculture and agro-food supply chains.3 Most government responses in the sector were introduced in the first few months of the pandemic, largely in response to the shock to specific subsectors. Still, as the year went by, as new waves and strands of infection developed, governments in many countries shifted their attention towards medium-term issues by bolstering early relief measures and introducing economic recovery packages. This section presents an overview of government measures introduced in 2020 in the 54 countries covered in this report, using different categorisations, focusing mainly on the number and type of measures, and associated budget figures. The dataset used for analysis was compiled based on the information on domestic and international trade related COVID-19 policy developments provided in country chapters in this report.4 While the reported set of measures is comprehensive, and covers all the most important policy responses, it does not claim to capture all measures in place in all countries covered in the study. Governments of the covered countries and the European Union introduced 776 unique policy measures to respond to the COVID-19 related crisis during 2020, of which 496 were introduced in the first four months of 2020 (OECD, 2020[3]; Gruère and Brooks, 2021[4]). The overall number of unique measures for the year 2020 increases to 1 086 applied policy measures if EU-wide measures, applicable to all member states, are added to unique measures for each of the EU Member States (including for the period covered, the United Kingdom). The nature of the government responses varied widely. OECD (2020[3]) distinguished seven categories of measures: 1) Sector-wide and institutional measures; 2) Information and co-ordination measures; 3) Measures on trade and product flows (enhancing trade or restricting trade); 4) Labour measures (biosecurity and workforce related measures); 5) Agriculture and food support (or support for agriculture and food companies); 6) General support (including packages that apply to the sector); and 7) Food assistance and consumer support (demand side interventions).5 Unique government measures were distributed across those categories, with 37% of the 776 measures focusing on agriculture and food support, 5% on institutional measures, and 8% on food assistance measures, with the remaining four categories covering between 11% and 14% of measures (Figure 1.2). These proportions changed since the four first months of 2020, from a focus on information and co-ordination to agriculture and food support measures. The share of agriculture and food support measures increased by 14 percentage points over the year, while the share of measures on information and co-ordination and general support declined by 7 and 4 percentage points, respectively. This evolution might reflect the need for information and communication in the early period, followed by the increased importance that some governments attached to providing support to agriculture and food companies to cushion the impact of the first wave of the virus. Shares for other categories of measures remained stable, indicating a moderate increase in the use of these measures across countries. A wide range of measures adopted is also observed among the 54 covered countries, underscoring the comprehensiveness of government responses. Thirty-eight of the covered countries applied measures in all seven categories, while ten countries applied measures in six of the seven categories. Fifty or more countries applied trade and product flow measures, information measures or agriculture and food support measures, while the other categories of measures were each applied by at least 46 countries (Figure 1.3). At the same time, differences in the number of measures by category can be seen among regions and countries. In particular, 54% of measures undertaken by governments in OECD countries focused on the three categories of support (agriculture and food support, general support and food assistance and consumer support measures), including the largest proportion on agriculture and food support (35%), while 58% of measures undertaken by emerging economies were in the non-support categories of measures (sector wide and institutional, information and co-ordination, trade and product flows and labour measures), including the largest proportion of measures (26%) in the trade and product flow category. This difference may reflect the existing policies covering the sector in the respective groups of countries, but may also be due to differences in structures of the sector as well as the type of shocks associated with the COVID-19 pandemic and associated containment measures. A further factor may be differences in budgetary and fiscal scope to provide additional support. Among OECD countries, Asian and European countries favoured agriculture and food support measures, South American countries focused on information and co-ordination measures, Oceanian countries prioritised labour measures, and North American countries displayed no clear dominance across categories of measures. Only 11% of the unique measures recorded explicitly built on existing policy measures already in place, almost all in the agriculture and food support category in the form of flexibility or changes in existing policy programmes. This suggests that governments often introduced new programmes, funding or approaches to respond to the crisis, or that they relied on existing policies without making notable changes. Innovative approaches were used for instance to re-channel food unused by closed schools towards families, to hire temporarily unemployed workers from cities in fields, or via the use of digital tools to ease market transactions and custom controls. Government responses also differed in their timing and scope, from the initial imposition of lockdown measures, to policies aiming to temper the impacts of the crisis on specific supply chains or consumers or in the medium term. At the same time, several measures taken to facilitate the functioning of production or supply chains could usefully have been taken before of the COVID-19 crisis. To highlight these distinctions and better understand the implication of government responses, measures were organised in three groups:6 • Urgent measures to ensure supply: these emergency measures were taken at the onset of the crisis to ensure supply and keep the sector functioning. Examples include biosafety measures; declaring agriculture and food as an essential sector; measures to ensure the functioning of government agencies; co-ordination of responses with the private sector; and national and international logistic and transport measures, including setting up green lanes to ensure the continuation of trade. These measures are intrinsically linked to the pandemic, and would either be lifted or no longer relevant after the COVID-19 crisis. This group includes 150 unique measures (19% of the total). • No regrets measures: these measures improve market functioning and thereby contribute to improved resilience. They could have been taken before, and should be maintained or even scaled up after the COVID-19 crisis. This group includes measures supporting digital innovations that facilitate e-commerce; exchange of information; agriculture job-matching information centres; and training or trade facilitation measures. This group includes 75 unique measures (10% of the total). • Temporary relief measures: these measures seek to contain the impact of the crisis on agriculture and food sector actors, from producers to consumers. Governments considered them necessary but they should include sunset clauses to avoid outliving their original rationale. These measures comprise largely temporary trade and markets measures to relieve domestic economic pressure, agricultural support measures, including those that compensate producers and agro-food chain actors for damages incurred; consumer and food assistance7 measures and measures that lifted or limited regulatory requirements for farmers. This group is the largest, with 537 unique measures (69% of the total). The remaining 14 measures (2%) could not be attributed to any of the groups. As expected, measures in the three support categories (5, 6 and 7) are overwhelmingly temporary relief measures, but measures in other categories, belong to different groups (Figure 1.4). Urgent measures to ensure supply include institutional and informational measures, but also labour measures and trade and product flow measures (categories 1 to 4). No regrets measures were mostly information and co-ordination measures and product and trade flow measures that enhance the functioning of markets (categories 2 and 3). A large majority of countries implemented measures that belong to each of these groups, even if some differences are observed among countries. All but two countries applied one or more urgent measures to ensure supply, and the same number of countries applied temporary relief measures; fewer countries (46) applied at least one no regrets measure. OECD countries applied relatively more temporary relief measures than emerging economies, who applied relatively more of measures in the other two groups. An additional distinction was made to identify measures that could at least temporarily be potentially market and trade distorting or environmentally harmful.8 These mostly temporary relief measures include trade bans or export restrictions that were temporarily put in place by several countries, but also market price controls, relaxed environmental regulations, and specific agricultural support measures for different agricultural commodities. Eighty-five unique measures (11% of the total) introduced by 47 countries were identified to have potential impact on markets or the environment, belonging to the agriculture and food support category, the trade and product flow, and the food assistance categories. One of the key ways in which governments have addressed the economic impact of the COVID-19 pandemic and associated lockdowns is by offering liquidity, credits, and funding for relief measures. Governments in many countries have adopted comprehensive economic recovery packages, with measures that included new lines of credits, subsidised loans, flexibilities in taxes, or subsidies and which included firms in the agriculture and food sector. At the same time, governments in many countries created specific financial support measures to the agriculture and food sector. This section provides a preliminary assessment of budgetary allocation in response to the COVID-19 impact based on collected information. It therefore only focuses on the subset of measures for which financial information was available (in total 119 unique measures in 41 countries). A review of the reported budgetary figures associated with the collected COVID-19 responses comes with several important caveats. First, it is impossible to track how much of the general recovery packages were used on the agricultural sector, so these are largely excluded from the assessment. Second, while these numbers include some expenditures incurred in 2020, a larger set of programmes that were announced in 2020 has not yet been delivered to the sector. As such, a majority of the numbers presented are not reflected in the 2020 data in this year’s agriculture support estimate database. Third, funding for sector-wide and institutional measures (category 1) and information and co-ordination measures (category 2) was not available. Fourth, some of the measures provide support for targeted or affected individuals on the basis of unit costs, but there is no estimate of the number of individuals or firms that benefitted from the support, so these support measures are excluded from the assessment. Fourth, governments may have used existing policies and measures, potentially with budget adjustments or changes in implementation, without reporting those as related to COVID-19. All these caveats suggest the reported figures are likely to represent minimum estimates of financial support measures in the 54 countries. In total, governments dedicated USD 157 billion in response to impacts to the sector (Table 1.2). Of this total, USD 116 billion was earmarked in the form of grants, payments or other funding, while USD 41 billion was offered in in the form of subsidised rates loans, new credit lines, and other mechanisms. At the same time, USD 5.6 trillion was provisionally identified in general recovery packages that included the food and agriculture sector (category 6 - general support). This support was not specific to the sector. Sector specific earmarked funding primarily focused on relief measures for agriculture and food actors, and food assistance measures (83% as shown in Figure 1.5). Twelve per cent of financial support focused on general services, such as infrastructure development, e-commerce development and measures easing trade, which are listed under the category of measures on product and trade flows. The remaining 5% of support was directed towards addressing labour shortfall, via compensation mechanisms for migrant or new farm workers, and implementing bio-sanitary measures, including compensation to the culling of minks potentially infected by the COVID-19 virus as well as equipment support. There are significant differences in the reported financial support between OECD countries and emerging economies (Figure 1.6). OECD countries’ financial support amounted to USD 75 billion, almost entirely dedicated to relief measures expressed in terms of agriculture and food support (USD 32 billion) and food assistance (USD 41 billion), with the remaining funding going towards labour and biosafety measures. New and expanding food assistance programmes were observed in OECD countries (Box 1.1). In contrast, emerging economies reported USD 82 billion of financial support, with USD 34 billion going to agriculture and food support and USD 24 billion to food assistance, implying lower shares of overall support in these categories, with a higher share (23%) dedicated to general services enhancing market and trade. With regard to agriculture and food support measures, OECD countries favoured funding mechanisms, such as direct payments, grants or increased allocation to existing support programmes (83%), while emerging economies supported the sector via preferential loans and credit mechanisms (99%). Large countries on both sides drive this pattern, with the United States accounting for 69% of total agriculture and food support via earmarked funding, and India accounting for 90% of loans and credits to be granted to the sector in response to the COVID-19 crisis. Sixteen of the 119 measures displaying financial support were identified as potentially market and trade distorting or environmentally harmful. These agriculture and support measures amounted to USD 731 million, which is significant but remains marginal compared to the total earmarked funds dedicated to the agriculture and food support (USD 35 billion) or to the global agriculture support estimates conveyed in this report. While policies for agriculture and food have been strongly influenced by the COVID-19 pandemic, other changes were also made in 2020. Specific information on the developments is summarised below, with details on adjustments made to policies and programmes within countries available in the country chapters within this report. Several countries have revised their agricultural policy frameworks. Colombia introduced the “Together for the Countryside” (Juntos por el campo) initiative, including a range of new policy programmes and subsidies for transportation, machinery and equipment, and variable inputs. Indonesia introduced specific programmes to increase production capacity on about 165 000 hectares of swampy land in Central Kalimantan, and to expand rice planting areas with 250 000 hectares of rice, maize, shallots and chilies in deficit areas. Japan revised its “Basic Plan for Food, Agriculture and Rural Areas”, which sets out policy directions, food self-sufficiency goals and commodity production targets for the next ten years. Mexico published the Sectoral Programme for Agriculture and Rural Development 2019-2024, focusing on improving agricultural productivity for food self-sufficiency, reducing poverty rates in rural areas, and increasing small-scale agricultural producers’ incomes. Viet Nam approved a series of strategies, plans and programmes to promote agricultural and rural development, including a new Livestock Development Strategy for 2021-30; a plan to promote investment in the agricultural and rural sector for 2021-25; a Master Programme on Sustainable Agricultural Development and Adaptation to Climate Change in the Mekong River Delta for 2030; a Scheme for Developing Organic Agriculture for 2020-30; and an irrigation strategy for 2030. The European Union also released a number of major policy initiatives: the European Parliament and the Council agreed on transitional rules for the Common Agricultural Policy (CAP) for 2021-22, while negotiations continue on CAP reform. In May 2020, the European Commission released more details on proposed Green Deal initiatives most relevant to the agricultural sector – specifically, the Farm to Fork and the Biodiversity strategies, which seek to halt biodiversity loss in Europe, transform EU food systems into global standards for competitive sustainability, protect human and planetary health and safeguard the livelihoods of all actors in the food value chain. New support measures and reforms to existing policies were introduced. Argentina shifted to more active export restrictions, reintroducing taxes that were reduced or eliminated between 2015 and 2018. Brazil created financial mechanisms to attract funds for rural credit, reducing preferential annual interest rates provided by Pronaf, the main credit programme for small farmers. Korea established a new direct payment system, combining the direct payments for rice, upland crops and less favoured areas into a single scheme. The income compensation scheme for rice, which has been the main payment scheme in Korea, was converted into a decoupled payment programme and accompanied by environmental cross compliance regulations. Norway eliminated its last export subsidies on cheese and processed agricultural products as of the end of 2020. The Philippines established a Rice Competitiveness Enhancement Fund to support investments in machinery and equipment, breeding and distribution of high quality rice seeds, credit and expansion. The Russian Federation (hereafter “Russia”) expanded its railroad tariff subsidies to cover the transportation of soybean meal, vegetables and mineral fertilisers. Viet Nam extended a land tax exemption to the end of 2025, allowing farm households and organisations to avoid paying an agricultural land use tax or continue benefiting from a land tax reduction. A number of countries developed new climate-related policies and strategies. Canada has established a new Natural Climate Solutions for Agriculture Fund, which will support carbon sequestration and beneficial management practices, such as cover crops or shelterbelts, through development, testing, peer-to-peer learning and solution sharing with farmers. Furthermore, under the “A Healthy Environment and A Healthy Economy” plan, the government of Canada plans to invest USD 123 million over seven years to support the agricultural sector in developing transformative clean technologies, reducing emissions from fertilisers to 30% below 2020 levels, boosting climate-smart agriculture, and supporting the production and use of low-carbon fuels. Japan published a national Green Growth Strategy in December 2020, outlining a comprehensive plan to achieve net-zero GHG emissions across the economy by 2050. The Ministry of Agriculture, Forestry and Fisheries has also announced a strategy for sustainable food systems, named “Measures for Achievement of Decarbonisation and Resilience with Innovation”, which aims to achieve zero CO2 emissions from agriculture, reduce the use of chemical pesticides and fertilisers, and make all subsidies carbon neutral by 2040. Korea released the 2050 Carbon Neutral Strategy, a long-term plan for GHG emissions mitigation. The strategy sets out four tasks for the agricultural sector: transition to smart farming; develop and deploy low-carbon agricultural practices; promote participatory policies for farmers and consumers; and scale up the deployment of eco-friendly energy. New Zealand has developed a ten-year roadmap for boosting primary sector export earnings while reducing biogenic methane emissions in accordance with the 2019 Zero Carbon Act. In addition, the “He Waka Eke Noa – Primary Sector Climate Action Partnership” seeks to reduce agricultural GHG emissions and enhance the sector’s resilience to climate change. Ukraine introduced new legislation to outline its strategy on environmental policies, along with a framework to monitor, report and verify the country’s GHG emissions. Chile, Iceland, Israel and Viet Nam also outlined new strategies and objectives in 2020 to reduce their GHG emissions from agriculture. In addition, several countries took steps to improve the sustainable management of their water resources. This group includes Chile (currently developing a Ministerial water plan), New Zealand (through the 2020 National Environment Standards for Freshwater), and Viet Nam (via the Irrigation Strategy to 2030). This follows a more general trend in OECD countries, where governments changed their agriculture and water policies, in the last decade, broadly in line with the OECD Council Recommendation on Water (Gruère, Shigemitsu and Crawford, 2020[11]; OECD, 2021[12]).9 Several countries strengthened their promotion of organic farming. Notably, the European Union’s Farm to Fork Strategy includes several agriculture-specific targets, one of which is to increase the share of farmland under organic farming to at least 25%. Furthermore, increasing the area of organic farming is also a key policy objective of Japan’s Ministry of Agriculture, Forestry and Fisheries. Russia introduced a new law providing requirements for the production and labelling of organic products. The creation of a system of certification for organic products is ongoing, with 64 producers currently certified. Viet Nam approved a Scheme for Developing Organic Agriculture in 2020-30, setting out specific goals to increase the share of organic production in agricultural land use and for improving the value per hectare of organic production by 2030. Some countries developed new solutions to tackle food loss and waste. Canada is investing USD 15 million to establish the Food Waste Reduction Challenge, encouraging innovative business models to develop solutions to prevent or divert food waste along the food supply chain. Turkey published a national strategy document and action plan on Prevention, Reduction and Monitoring of Food Loss and Waste, setting four strategic goals and 13 targets. Risk management and disaster assistance policies were strengthened. Australia introduced drought resilience response programmes through the Future Drought Fund, and provided support to farm clean up and emergency response activities through the National Bushfire Recovery Fund. China’s Ministry of Agriculture and Rural Development and the Ministry of Finance jointly allocated USD 47 million to a new disaster relief fund assisting agricultural producers in flood-hit southern provinces. Kazakhstan’s mandatory crop insurance system was transformed into a voluntary insurance scheme with a view towards expanding crop and livestock insurance markets in the country. In New Zealand, a flooding event and significant drought affecting large parts of the country triggered public support for recovery and relief, as well as to individual farmers in hardship through Rural Assistance Payments. Turkey provided additional coverage through the state-supported agricultural insurance scheme, issuing 2.1 million insurance policies and USD 250 million of state insurance premium support. The United States provided an additional USD 1.5 billion for the continuation of disaster assistance programme delivery, adding several new qualifying disaster events and eligible participants under the Wildfire and Hurricane Indemnity Program Plus (WHIP+). The USDA’s Risk Management Agency also introduced a new policy to help farmers recover from hurricanes, covering 70 different crops. New laws and regulations on animal and plant health were introduced. Chile’s animal and plant health agency promoted electronic certification, now established for exports to 34 countries and covering around 70% of all phytosanitary certificates. Costa Rica’s animal and plant health institutions established a single export window to deal with sanitary and phytosanitary procedures, and created an online system for consulting phytosanitary certificates for agricultural exports in real time. Switzerland introduced new plant health legislation, requiring stricter regulations and stronger preventive measures to protect plants from harmful pests. In the United States, the USDA’s Animal and Plant Health Inspection Service (APHIS) published the Sustainable, Ecological, Consistent, Uniform, Responsible, Efficient (SECURE) rule, the first comprehensive revision of the Agency’s biotechnology regulations in over 30 years. The new rule puts in place a more efficient process to identify plants that would be subject to regulation, focusing on the properties of the plant rather than on its method of production. Concerning land reform and investment, Russia increased support for investments in agriculture, including purchases of agricultural machinery, goods and processing equipment. The company Rosagroleasing aims to supply 9 000 units of equipment in one year, which represents a 40% increase on last year’s numbers. South Africa established the Agriculture Development Agency to support the development of sustainable land reform programmes and reduce barriers to the commercialisation of small-scale farmers. Ukraine passed new legislation ending the ban on the sale of agricultural land. As of July 2021, individual citizens of Ukraine will be permitted to purchase up to 100 hectares of land, while from January 2024 purchases of up to 10 000 hectares will be made available to legal entities whose founders or final beneficiaries are Ukrainians, and which do not have business abroad or in offshore companies. Viet Nam approved a plan to promote investment in the agricultural and rural sector in 2021-25, including the following priorities: evaluating market potentials, trends and investment partners; building a database on investment promotion activities; establishing a list of projects calling for investment; and providing support to enterprises and investors. Some countries provided new support to agricultural innovation and the development of digital technologies. Japan published the Smart Agriculture Comprehensive Policy Package, identifying key measures to advance data-driven agriculture over the next five years. The Ministry of Agriculture, Forestry and Fisheries also established the Conception and Projects for DX of Agriculture Initiative, which provides a roadmap for the development of artificial intelligence, big data, and the digitalisation of administrative procedures. Korea established the Smart Agriculture Project, which aims to promote the application of new technologies and attract young and innovative farmers to the agricultural sector. Young farmers can benefit from concessional leasing of agricultural facilities and farmlands in smart farm complexes, and cross-sectoral R&D projects will be conducted to support the development of new technologies. Turkey introduced the Digital Agriculture Market (DITAP), a digital platform to help develop supplier linkages between smallholders and large-scale food processing and retail firms. DITAP also helps small farmers to access markets for inputs such as seeds and fertilisers, and provides a platform for farmers to lease their land. Numerous countries have concluded bilateral and regional trade agreements. On 15 November 2020, the Regional Comprehensive Economic Partnership (RCEP) was concluded by fifteen countries in the Asia-Pacific region, including Australia, China, Indonesia, Japan, New Zealand, the Philippines, Korea and Viet Nam. The Agreement will reduce tariffs on goods among the 15 participating economies by 90% over two decades from entry into force, and provides a framework for strengthening co-operation in the areas of standards, technical regulations, and conformity assessment procedures, as well as for streamlining rules of origin and border processes for perishable goods. The Canada-United States-Mexico Agreement (CUSMA) entered into force on 1 July 2020, preserving the existing agricultural commitments under the North American Free Trade Agreement (NAFTA). The European Union and Mexico finished negotiations on a new EU-Mexico trade agreement, which will further liberalise more than 85% of the agricultural tariff lines that were left out of the original EU-Mexico Global Agreement that has been in force since 2000. On 31 January 2020, the United Kingdom left the EU Single Market and Customs Union, ending the free movement of people, goods and services with the European Union. The rules governing trade and movement between the two are laid down in the draft EU-UK Trade and Cooperation Agreement, which was agreed on 24 December 2020 and ratified by the European Parliament on 27 April 2021. Of particular relevance to agriculture, the trade component of the agreement includes duty- and quota-free imports on all goods that comply with rules-of-origin provisions. Several additional bilateral free trade agreements (FTAs) were negotiated or came into effect in 2020 and 2021, helping to facilitate bilateral trade in agricultural products. These include: the Canada–United Kingdom Trade Continuity Agreement; Colombia-Israel FTA; European Union-Viet Nam FTA; Indonesia-Australia Comprehensive Economic Partnership Agreement (CEPA); Indonesia-Korea CEPA; Japan-US FTA; Korea-Israel FTA; Ukraine-Israel FTA; United Kingdom-Israel FTA (and related protocol for the mutual recognition of organic produce); United Kingdom-Japan CEPA; United Kingdom-Korea FTA; United Kingdom-Mexico Trade Continuity Agreement; United Kingdom-Ukraine political, free trade and strategic partnership agreement; United Kingdom-Viet Nam FTA; United States-China Phase One Trade Agreement. Numerous other FTA negotiations are ongoing. Trade promotion and market development policies were introduced by a number of countries. India initiated reforms to remove limits on private stocking, trading or buying of commodities, allow farmers to sell their agricultural products outside of government-regulated markets, and promote barrier-free inter and intra-state trade of agricultural commodities. The government also established a new Agriculture Infrastructure Fund to support farmers, producer organisations and agribusinesses through subsidised loans for post-harvest infrastructure such as cold storage, collection centres and processing units. To facilitate the exports of processed food products, the Ministry of Trade of Indonesia adopted measures to simplify the certificate of origin service and introduce automatic authentication procedures in licensing processes. Japan introduced the Act on Facilitating the Export of Agricultural, Forestry and Fishery Products and Food, which streamlines export policies for these products. The Strategy to Realize Export Expansion of Agricultural, Forestry, Fishery Products and Food designates products to prioritise resources and actions for agricultural export expansion. Russia introduced a programme to support exports of agricultural products, including additional financing for export infrastructure, simplification of border procedures, veterinary and phytosanitary services, information support, and support to promotion and market access. Food systems face a daunting “triple challenge”. First and foremost, they are expected to achieve food security and nutrition for a growing world population. Second, they have an essential role to play in providing incomes and livelihoods for hundreds of millions of people involved in farming and other segments of the food chain. And third, they must do so in a sustainable manner, without depleting land, water and biodiversity resources, while contributing to reductions in greenhouse gas (GHG) emissions. The urgency of these challenges is reflected in the international political timetable, with food and agriculture at the heart of foreseen discussions in 2021 at the COP-26 UN Climate Change Conference, the COP-15 meeting of the Conference of the Parties to the Convention on Biological Diversity, and the UN Food Systems Summit. Agricultural support policies have played a major role in shaping today’s food systems. Historically, the provision of support to agriculture has been motivated by a variety of policy objectives, which have included ensuring food security, supporting farmers’ incomes and livelihoods and improving environmental outcomes – key components of the “triple challenge”. The instruments chosen to pursue these objectives have varied widely. Some countries have relied on trade and open access to markets to ensure food security, while others have stressed domestic production and high rates of self-sufficiency, maintained via subsidies and trade protection. Countries have similarly varied in the extent to which they see income support as a goal for agricultural policy (as opposed to being covered by wider social protection programmes), and in the instruments they have chosen to deliver it. Most countries also have specific agri-environmental programmes, but many of the environmental impacts of agricultural policies stem from the choice of policies to address the first two objectives. This section begins with an overview of the level and composition of agricultural support policies across countries. This is followed by an assessment of the implications of agricultural support for the performance of food systems, reflected in the extent to which they may be helping or hindering progress in meeting the triple challenge. Finally, the section considers the effectiveness of agricultural support policies in strengthening the overall productivity, sustainability and resilience of the agricultural sector – key channels for improving the performance of food systems. The OECD has been monitoring developments in agricultural support in OECD countries on an annual basis since 1988, with an increasing number of economies outside the OECD area included since then. This exercise quantifies different forms of policy intervention according to their implementation criteria, and forms the basis for an assessment of policy performance against stated objectives. The current assessment covers 54 countries across six continents, including all OECD member countries, 5 non-OECD EU Member States, and 12 emerging and developing economies.10 Together, these countries represent three-quarters of global agricultural value-added. The assessment also discusses aggregate results for OECD member countries, the emerging economies, and all countries combined. In these aggregates, however, Costa Rica, which became the 38th Member of the OECD in May 2021, is included as one of the 12 Emerging Economies. The European Union is presented as one economic region, and includes the United Kingdom, which has left the European Union in early 2020 but remained part of the single market and continued to implement the Common Agricultural Policy through to the end of 2020 (a separate set of support indicators is presented in this report for the United Kingdom for 2017-20). Figure 1.7 provides an overview of the structure of agricultural support indicators. The Total Support Estimate (TSE) is the OECD’s broadest indicator of support. It comprises policy expenditures in general services for primary agriculture that benefit the sector as a whole (General Services Support Estimate or GSSE); policy transfers to individual producers (Producer Support Estimate or PSE); and budgetary support to consumers included in the Consumer Support Estimate (CSE). Annex 1.A provides definitions of the OECD indicators of agricultural policy support. In 2018-20, agricultural support policies across the 54 countries covered in this report generated USD 720 billion per year in transfers to agriculture. This was counter-acted by more than USD 104 billion per year in implicit taxation of farmers. Individual producers received USD 540 billion in support per year (about 75% of all positive transfers to agriculture) through various support measures, including higher prices paid by consumers. Governments employ a variety of different policy measures to deliver agricultural support (Figure 1.8). An important share of support is delivered through measures that modify domestic prices relative to world market prices. These policies do not result in government expenditures per se, but rather represent market transfers from consumers to producers, or vice-versa: • Market price support (MPS) arises from policies that create a price gap between domestic market prices and border prices for specific agricultural commodities (Box 1.2). Import licences, tariffs, tariff rate quotas and minimum prices are examples of measures that would result in higher prices paid by consumers. Total positive MPS amounted to USD 272 billion per year in 2018-20. • Some emerging and developing countries (Argentina, India, Viet Nam, Kazakhstan, Russia and Indonesia) implicitly tax producers on some or all agricultural commodities through measures that depress the domestic prices of these products, such as export taxes and export restrictions (resulting in negative market price support). Overall, negative MPS amounts to more than USD 104 billion per year. The remaining support measures amounted to USD 447 billion per year, and are delivered in the form of budgetary payments and expenditures targeted to the agricultural sector (i.e. they represent transfers from taxpayers to producers, consumers, or to the sector as a whole): • Other most distorting support refers to subsidies linked to output or the unconstrained use of variable inputs (USD 66 billion per year), which have similar propensity to create market distortions to those generated by MPS. • Other producer support (USD 202 billion per year) includes payments based on land area, animal numbers, receipts or income, or payments not linked to the production of agricultural commodities, such as payments based on historical entitlements. These subsidies are considered to be “less coupled” to production and therefore more efficient in transferring income to the owners of land and other production factors. Payments can also be conditional on specific production practices and input uses designed to support environmental objectives. This category also includes specific payments designed to encourage farmers to adopt environmentally friendly technologies and practices. • Policies that benefit the agricultural sector as a whole include investments in R&D and innovation, infrastructure (including off-farm irrigation systems, transportation and the provision of information and communication technologies), biosecurity, marketing and public stockholding. These policies are measured by the General Services Support Estimate (GSSE), which amounted to USD 102 billion per year, or 14% of all positive transfers to agriculture. • Subsidies for consumers (such as food assistance programmes) amounted to USD 78 billion per year, or 11% of all positive transfers to agriculture. Total support to agriculture has grown considerably in nominal terms over the past two decades, largely driven by increasing support in large emerging economies (Figure 1.9). The nominal value of the total support estimate (TSE) for OECD countries has remained relatively stable, reaching USD 329 billion in 2018-20, with reforms stalling over the last decade following some previous reforms. At the same time, the share of total support in GDP has declined steadily from 1.0% in 2000-02 to 0.6% in 2018-20, reflecting the declining importance of the sector. In the 12 emerging economies, the TSE grew from USD 44 billion in 2000-02 to USD 280 billion in 2018-20, driven by increasing rates of producer support in the largest emerging economies – in particular, China, India and Indonesia. The TSE for emerging economies averaged 1.2% of GDP in 2018-20, reflecting the importance of support in the largest emerging economies, which are home to large agricultural sectors with sizeable agricultural populations. Additionally, emerging economies subjected their producers to more than USD 104 billion in negative market price support (i.e. implicit taxation) in 2018-20. Aggregate figures mask the diversity in levels of support across countries (Panel A in Figure 1.10). The share of TSE in GDP (%TSE) indicates the cost of support to the sector for the overall economy. It was highest in Indonesia (2.5%), the Philippines (2.5%), and China (1.6%), partly reflecting the fact that agriculture has a comparatively high weight in the economies of these countries. The largest reductions in the %TSE since 2000-02 (in percentage points) have occurred in Turkey, Colombia and Korea – countries where the burden of support was initially high, but nonetheless still remained above 1.2% in 2018-20. The level of total support in OECD countries continues to be high when measured relative to agricultural value added, amounting to 42% in 2018-20 (Panel B in Figure 1.10). Total support relative to the size of the sector varies widely across OECD countries, from 146% in Switzerland, 81% in Korea, and 78% in Japan, to less than 10% in just three countries: Australia, Chile and New Zealand. In comparison, total support in the 12 emerging economies represented just 15% of agricultural value added in 2018-20. The importance of support to the sector is highest in the Philippines (27%), China (22%), and Kazakhstan (21%). Total support is low relative to agricultural value added in India (4%) and Brazil (7%), and negative in Argentina and Viet Nam. The total effective tax on agriculture relative to the size of the sector was 54% in Argentina and 8% in Viet Nam. The Producer Support Estimate (PSE) measures policy transfers to individual agricultural producers. Transfers to producers in the PSE comprise market price support (MPS) provided through domestic market prices that are higher (or lower if support is negative) than world prices, and budgetary payments from the government to farmers (Figure 1.7). The price gaps generated by trade policies and domestic market interventions are typically calculated as a differential between domestic and reference prices, but in some cases alternative methods are used for these calculations (Box 1.2). The average %PSE (producer support as a share of gross farm receipts) for all 54 countries has been declining over the past two decades, from 18% in 2000-02 to 11% in 2018-20 (Figure 1.11). Within this average is a clear pattern of a decreasing rate of producer support in OECD countries and increasing rate of producer support in emerging and developing economies from the beginning of the century until 2015. In OECD countries, the %PSE fell from 28% in 2000-02 to 18% in 2018-20. Most of this decline was driven by reforms initiated prior to 2008; the pace of decline has been markedly slower since and reversed to a slight increase after 2014. In contrast, the %PSE in emerging economies almost doubled from 3.8% in 2000-02 to 7.4% in 2018-20. The %PSE in emerging economies reached a peak of 10.8% in 2015 and subsequently declined to 7.4% in 2020. This is in part due to higher levels of negative market price support, which depressed the domestic prices of certain commodities in some of these countries. Indeed, the %PSE represents the balance of positive and negative MPS elements, and tends to underestimate the extent of price distortions when both positive and negative price support are present. Support remains highly concentrated. In 2000-02 the overall value of producer support was concentrated in OECD countries, in particular the European Union, the United States and Japan. Since then, support in some large emerging economies (China, India and Indonesia) has become increasingly important. Four countries accounted for the vast majority of the aggregate net Producer Support Estimate in 2018-20: China (44%), the European Union (24%), the United States (10%) and Japan (9%). Negative market price support was predominantly provided by India (78%). The size of the agricultural sectors in these countries means that any policy will automatically result in large absolute numbers. For this reason, it is often useful to express the producer support estimate relative to gross farm receipts, as is done in Figure 1.12 below. Countries differ widely in their tendency to support (or tax) their farmers. The countries with the highest levels of producer support when measured as a share of gross farm receipts are all in the OECD area. In Norway, Iceland, Switzerland, Korea and Japan, agricultural policy transfers arising from tariffs and other support measures generate between 40% and 60% of the revenues received by farmers. Producer support is above the OECD average of 18% in the Philippines, Indonesia, the United Kingdom, the European Union, and Israel. Seven countries have low levels of support, below 5%: Kazakhstan, South Africa, Chile, Australia, Ukraine, Brazil and New Zealand. Finally, three countries have negative levels of producer support, as a consequence of farmers facing implicit taxation through suppressed producer prices: Argentina, Viet Nam and India. The level of producer support as a share of gross farm receipts has declined across OECD countries relative to the levels observed in 2000-02. Support has also declined in a number of emerging economies, notably Brazil, South Africa, Kazakhstan and Costa Rica. As mentioned previously, some of the larger emerging economies increased their level of support as measured by the %PSE, including Ukraine, Indonesia, China, the Philippines and Russia. Support to producers became more negative in Argentina and India, while Viet Nam’s %PSE turned from positive in 2000-02 to negative in 2018-20. According to the FAO, “a person is food insecure when they lack regular access to enough safe and nutritious food for normal growth and development and an active and healthy life.” The severity of food insecurity can vary by time and degree, ranging from mild (uncertainty regarding one’s ability to obtain food) to moderate (compromising on food quality and variety, reducing food quantity, skipping meals) to severe food insecurity (no access to food for more than a day) (FAO, 2020[17]). The world as a whole is not on target to achieve the United Nations Sustainable Development Goals target 2.1, of “ensuring access to safe, nutritious and sufficient food for all people all year round”, nor target 2.2, of “eradicating all forms of malnutrition”. While the proportion of people who are undernourished declined significantly over the past few decades, this trend has reversed in recent years. The prevalence of undernourishment increased from 8.6% in 2014 to 8.9% in 2019, and the absolute number of people affected by hunger increased by 60 million over the same period. Nearly 750 million people, or 10% of the world’s population, were considered to be severely food insecure in 2019, while an estimated 2 billion people (26% of the global population) experienced moderate or severe food insecurity, meaning that they did not have regular access to safe, nutritious and sufficient food. Africa and Asia currently account for 92% of the world’s undernourished, or 631 million out of 688 million people. If current trends persist, the number of people affected by hunger is projected to exceed 840 million in 2030, of which 762 million (91%) will be in Africa and Asia. The COVID-19 pandemic has also led to a significant worsening of the situation, potentially resulting in an additional 83-132 million undernourished people in the world in 2020 (FAO, IFAD, UNICEF, WFP and WHO, 2020[18]). Food security is linked to multiple areas of government policy, including macroeconomic policies that raise incomes and thereby improve access to food, trade policies that influence food availability, and public health and sanitation policies that improve food safety and nutritional outcomes. Tackling this complex and multi-faceted problem requires ensuring that sufficient food is available, that people have access11 to food, and that food leads to good nutritional outcomes. A fourth requirement is the stability of these three dimensions over time, which implies effective risk management (OECD, 2013[19]). This section assesses the specific impact of agricultural support policies on the four dimensions of food security: availability, access, nutrition, and stability. A global lack of food has not been a fundamental cause of continued food insecurity around the world. Global agricultural production has increased four-fold since 1960, with the amount of food available per person growing by 56%. This remarkable growth in supply can be largely attributed to productivity growth and yield improvements, as agricultural production has rapidly outpaced population growth and the expansion of agricultural land (Figure 1.13). The OECD-FAO Agricultural Outlook 2020-2029 projects that the pace of demand growth for agricultural commodities will slow over the coming decade, and will continue to be outpaced by efficiency gains in crop and livestock production (OECD/FAO, 2020[20]). Even so, some countries have suffered from a lack of food availability due to prolonged conflicts and extreme fragility. More commonly, however, food insecurity in these countries is driven by poverty and a lack of access to food. Across 15 countries with a protracted crisis for which food price data are available, the cost of a healthy diet (USD 3.80) is roughly in line with the global average (USD 3.75), yet healthy diets are unaffordable for 86% of the population (compared with the global average of 38%) (FAO, IFAD, UNICEF, WFP and WHO, 2020[18]). Thus, the notions of food availability and access to food are closely linked. Governments can improve the availability of food by stimulating the domestic supply of food with non-distorting policies (e.g. through productivity improvements, reduced post-harvest losses, or reduced diversion of food crops to biofuels), and by limiting excess food demand (e.g. through reductions in over-consumption and consumer waste). International trade also plays a vital role in increasing the availability of food by balancing the deficits of net food importers with the surpluses of net food exporters, and permitting an allocation of production across countries that reflects relative differences in resource abundance. Trade is particularly important for the food security of regions experiencing growing food demand, which often do not correspond to the areas in which supply can be increased in an efficient and sustainable manner. Agricultural support policies have adverse implications for global food availability by encouraging a sub-optimal allocation of resources, altering the relative mix of products grown, and displacing production to less efficient locations (OECD, 2016[21]). Many countries provide support to their agricultural sectors through measures that artificially stimulate domestic production and distort trade, with potentially significant consequences for global food availability. The most distorting measures – market price support, payments based on output and payments based on variable inputs without constraints – represent more than half of all transfers to and from producers in many countries, although some countries have implemented reforms that have decoupled support from production levels (Figure 1.14). Agricultural support policies are therefore concentrated on measures that seek to increase domestic food availability, but often do so in an inefficient way (e.g. by raising prices), rather than through productivity-enhancing investments in R&D, innovation and infrastructure. These policies may contribute to domestic supply increases, but also encourage crops to be diverted away from human food consumption and towards the production of animal feed, biofuels, and the expansion of stocks (Pingali, 2015[22]). Policies to reduce the overconsumption of food and reduce food waste have so far had limited success, but can also play an important role in increasing domestic food availability. The most distorting support policies reduce the global availability of food by impeding international trade (Brooks and Matthews, 2015[23]). Market price support policies such as import tariffs, quotas, and minimum prices may boost domestic production but also raise domestic prices, thus reducing domestic demand and food imports. These policies also reduce access to food for low income consumers (discussed further in the section on “Access to food”) Export taxes and restrictions (discussed further in the section on “Stability”) lead to higher prices and lower exports, effectively amounting to an implicit tax on farmers (negative market price support). Such measures discourage production and long-term investments in productive capacity. Collectively, these policies also influence the pattern of specialisation across countries, causing production to shift from more efficient to less efficient locations. Farmers in countries with export potential and low levels of government assistance face lower returns, due to restrictions in market access and reduced opportunities to sell into protected markets (OECD, 2013[19]; Anderson and Valenzuela, 2021[24]). The trade-distorting effects of agricultural support policies in OECD countries have declined considerably compared with earlier decades. Export subsidies were banned under WTO rules in 2015, and many countries have replaced market price supports for individual products with less distorting measures that are decoupled from current production. For example, Switzerland provides significant direct payments to farms, almost all of which are subject to environmental cross-compliance. These have increased over time, from around 20% of support to farmers in the 1980s to almost 50% in recent years. Successive reforms to the European Union’s Common Agricultural Policy (CAP) since the early 2000s have decoupled nearly half of budgetary support from production, by reducing distortive price supports and increasing direct payments to producers (of which nearly 60% are contingent on mandatory environmental constraints). Area-based payments and direct income payments have a weaker influence on production decisions, as they are not directly tied to output. The importance of market price support is reflected in the fact that higher tariffs continue to be applied to trade in agricultural and food products, in spite of extensive tariff reductions since the 1994 Uruguay Round Agreement on Agriculture. The average applied tariff on agricultural products globally in 2018 was 7.8% (compared with 4.6% for industrial goods). At the same time, the gap between tariff rates bound under WTO rules and applied rates means that countries can raise tariffs on agricultural products to an average of 48.9% (compared with 27.1% for industrial goods). This significant water in the tariffs for agriculture adds to policy risks. Furthermore, average tariff rates mask distortions along specific product lines; while many tariff lines are at zero, some are considerably higher and may even exceed 100%, and there are many instances where tariff rates increase with higher levels of processing (OECD, 2020[5]). To further illustrate this point, Figure 1.15 shows that levels of market price support (as a share of gross farm receipts) vary widely across countries and commodities. Only Australia, Chile, Brazil and Kazakhstan have low average levels of market price support, at or below 6% for all commodities. All other countries have at least one commodity with price support above 20%.12 Six countries (Korea, Japan, Iceland, the Philippines, Norway and Switzerland) have high average levels of market price support in excess of 20% of gross farm receipts, while average market price support is negative in Kazakhstan, Viet Nam, India and Argentina. Figure 1.15 also demonstrates that there is significant dispersion of market price support within countries, albeit with varying distributions across commodities. In several countries, some commodities are supported whilst others are taxed, creating significant additional distortions to prices and market signals. Broad based multilateral reform of trade and domestic support policies is likely to generate large and widespread benefits for food availability, by facilitating shifts in production to regions that are best able to meet the growing global demand for food and agricultural raw materials. OECD (2016[21]) found that the removal of all trade-related and domestic support to agriculture would increase trade in both intermediate and final agro-food commodities (the largest effect was observed for final food products, due to higher applied tariffs on processed products and the fact that products may face tariffs on multiple occasions as intermediate goods travel across borders). Removing barriers to market access therefore has the potential to boost trade (including in intermediate agricultural products) and strengthen participation in agro-food global value chains (GVCs) (Greenville et al., 2019[25]). Intra-regional trade can improve food availability in countries that face difficulties accessing world markets and integrating in global supply chains. Bilateral and more extensive trade agreements have become increasingly prevalent in the global agricultural trading environment since the early 1990s, in part due to the slow progress of multilateral negotiations. These agreements are often viewed as a vehicle for economic and political integration amongst members, and have resulted in substantial improvements in market access, delivering reduced tariffs across a broad range of agricultural commodities (Thompson-Lipponen and Greenville, 2019[26]). In some cases, however, preferential trade agreements may cause rents to shift to participating countries, rather than creating new market opportunities. Reforming trade-distorting support can strengthen global food availability by allowing countries to benefit from improved market access and providing an important springboard for export-led growth. Trade openness can also improve access to food and contribute to faster economic growth, by raising the incomes of exporters (allowing them to profit from higher prices than would be received in the absence of trade) and importers (who benefit from lower prices than would otherwise be paid) (Brooks and Matthews, 2015[23]). However, it is important to recognise that reforms to the most distorting forms of support are likely to impose short-term costs on some stakeholders. In particular, producers that formerly benefited from protection, exporters that benefited from preferential market access, and consumers that benefited from former policy arrangements may face difficulties adapting to a more competitive trading environment. In such cases, it may be necessary to provide transitional assistance. Social safety nets can facilitate structural adjustment, by ensuring adequate incomes for those with few viable economic alternatives (Brooks and Matthews, 2015[23]; OECD, 2002[27]). It is particularly important to reform the most distorting policies that stifle innovation and hamper the agricultural sector’s long-term productivity and sustainability. In recent decades, agricultural productivity growth has played an essential role in increasing the global supply of food and contributing to widespread improvements in food availability. Productivity growth has also put significant downward pressure on food prices, resulting in improved access to food for poor consumers worldwide. The growth in agricultural productivity owes much to efforts by governments to facilitate the provision of public goods and services and create enabling conditions to strengthen the competitiveness of agriculture. Continued policy attention in these areas will be fundamental to achieving sustained improvements in food security. The General Services Support Estimate (GSSE) includes expenditures on R&D and innovation, inspection services, infrastructure development and maintenance, marketing and promotion, and public stockholding. Despite its potential to contribute to sustainable productivity growth and strengthen food security, the GSSE tends to be much lower than support provided directly to producers: in 2018-20, it represented 13% of the Total Support Estimate (TSE) in OECD countries, and 20% of the TSE across the 12 emerging economies. When measured as a share of agricultural value added, the GSSE stood at just 5.6% in OECD countries and 3.0% in the 12 emerging economies in 2018-20 (Figure 1.16). Expenditures on general services were highest in Switzerland (16% of agricultural value added), Japan (16%) and Korea (12%). In the remaining countries, the GSSE ranged between 1.0% of agricultural value added in Iceland and 6.1% in the United States. The composition of expenditure also varies widely across countries: agricultural knowledge and innovation systems accounted for just 5% of GSSE expenditures in Indonesia, and 92% in Brazil13. Spending on infrastructure development and maintenance ranged from 3% of the GSSE in Ukraine to 86% in Japan. R&D plays a vital role in strengthening productivity in agricultural production, food processing and delivery to consumers. There is ample evidence that public investments in agricultural R&D generate large rates of return (Alston et al., 2010[28]; Piesse and Thirtle, 2010[29]), and can have positive implications for food security (Kristkova, van Dijk and van Meijl, 2017[30]). Public funding is crucial in areas where private investors are missing, and can help to stimulate private investment, including through public-private partnerships (PPPs). Governments should also work to create an enabling environment for private investments, provide stable funding for knowledge infrastructure, and strengthen linkages within the agricultural innovation system between R&D and technical assistance. Making innovation systems more collaborative and demand-driven can improve the impact of public expenditure. Efforts to improve the governance of the agricultural innovation system may include the development of long-term strategies for agricultural innovation, involving stakeholders more formally and earlier in the process, and strengthening evaluation frameworks (OECD, 2019[31]). Agricultural R&D remains dominated by the public sector in many countries, while private research tends to focus on specific areas (e.g. genetic improvements, fertilisers and chemicals, machinery, food processing). However, growth in public agricultural R&D investment has been slowing over the past decade in high-income countries (Heisey and Fuglie, 2018[32]). In addition to maintaining strong levels of investment in agricultural R&D, investments in productivity-enhancing infrastructure can also strengthen food availability. Well-developed transportation infrastructure, including rural road networks and access to port facilities, can help to connect farmers with markets and allow them to take advantage of export opportunities. Ensuring affordable access to ICTs in rural areas can provide farmers with real-time information on food prices and weather conditions, improve the reach of early-warning systems, and facilitate the adoption of new digital technologies and innovations. At the same time, some investments to expand irrigation infrastructure may slow structural change and hamper the development of diversified farming systems, with potential negative consequences for environmental sustainability. Access to food is fundamentally driven by two related factors: the price of food, and real incomes. High agricultural prices can impede access to food for low-income consumers, who typically spend a large proportion of their household budgets on food. Food prices have been declining since the mid-1970s and are low by historical standards (Figure 1.17). With no major structural shifts in agricultural commodity demand on the horizon, the OECD-FAO Agricultural Outlook 2020-2029 projects flat to declining real agricultural prices over the next ten years (OECD/FAO, 2020[20]). It is important to recognise that farmers are affected by food prices as both buyers and sellers. Whilst higher prices can improve incomes and access to food for some farmers, the majority of the rural poor are net buyers of food staples (OECD, 2013[19]). Sharp increases in the prices of food staples – as was witnessed during the 2007-08 food price crisis – can therefore lead to lower real incomes and weaken the purchasing power of poor farmers as well as consumers, undermining food security objectives. Several studies have found that higher food prices have a negative impact on poverty and welfare outcomes, particularly for poor households who tend to spend a greater share of their incomes on food (Filipski and Covarrubias, 2012[33]; Ivanic and Martin, 2008[34]). The prospect of continued low prices for food staples bodes well for the overall accessibility of food. However, there are concerns that healthy and nutritious foods remain unaffordable for much of the world’s population, leading to rising rates of hunger, food insecurity and malnutrition. According to the State of Food Security and Nutrition in the World 2020, healthy diets14 cost 60% more than diets that only meet the requirements for essential nutrients, and are nearly five times more expensive than diets that only meet the basic dietary energy needs through a starchy staple. More than 1.5 billion people cannot afford a diet that meets the required levels of essential nutrients, and over 3 billion people cannot afford the cheapest healthy diet (FAO, IFAD, UNICEF, WFP and WHO, 2020[18]). Agricultural support policies are often implemented by raising domestic prices above world market prices, leading to higher costs for the consumers of agricultural commodities. The percentage Consumer Support Estimate (%CSE) expresses the monetary value of the transfers to consumers (both through prices and through food assistance programmes) as a percentage of consumption expenditure (measured at farm gate). When domestic prices are higher than world market prices, consumers are effectively subjected to implicit taxation. In most countries, consumers are harmed by market price support policies, resulting in negative values for the %CSE (Figure 1.18). The level of this implicit tax ranges from zero in Australia to more than 35% in Iceland, Korea, Japan and Norway. Some emerging economies (India, Argentina, Kazakhstan and Viet Nam) have a positive %CSE, meaning that they implicitly tax producers and support consumers by artificially lowering the prices for agricultural commodities. The United States is the only OECD country with a positive %CSE, due to the high level of budgetary transfers for food assistance programmes. Market price support policies generally result in lower real incomes and reduced access to food. Poor consumers are disproportionately burdened by higher agricultural prices, as food accounts for a greater share of their household budgets. In addition, small farmers in emerging and developing economies are often net buyers of agricultural commodities, and therefore bear a part of these costs. Market price support also has a negative influence on the competitiveness of downstream segments of the food chain: livestock producers face higher costs for animal feed, and food processing industries face higher prices for their inputs. Furthermore, if support measures are sufficient to cause countries to have an export surplus, they can curtail export opportunities for farmers in countries with low levels of government assistance (such as Australia, Brazil and New Zealand) (Anderson and Valenzuela, 2021[24]). While prices clearly matter and have a strong influence on the affordability of food, real incomes and poverty levels also play an essential role in determining access to food. If incomes are extremely low, even cheap food can be out of reach for the poor (OECD, 2021[35]). In many emerging and developing countries, increases in food prices such as those experienced during the 2007-08 food price crisis were largely compensated for by robust growth in incomes. Countries therefore have much better prospects of strengthening access to food by raising incomes and tackling poverty than by attempting to lower domestic prices below world levels (OECD, 2013[19]). Governments have a range of policy tools at their disposal to support the incomes of rural households and improve access to food (discussed further in the section on “Incomes and livelihoods. Conditional cash transfers have been a popular and effective tool deployed by many developing countries in recent years. Such programmes provide cash to poor households on the condition that they make pre-determined investments (e.g. in schooling for their children). Emergency food reserves can also be used to protect the most vulnerable, provided they supply food to specific groups without disrupting private markets (OECD, 2013[19]). In addition, many countries have introduced social safety nets and food assistance programmes to provide low-income households with better access to food. Examples include the USDA’s Supplemental Nutrition Assistance Programme and National School Lunch Programme, Korea’s Food Voucher Assistance Programme, and the United Kingdom’s Healthy Start scheme (Placzek, 2021[10]). The COVID-19 pandemic has also had a measurable impact on access to food, mainly through declines in income and increases in global poverty (Laborde et al., 2020[36]). In response to the crisis, for example, India’s food subsidy allocation increased from USD 13 billion in the 2020-21 budget estimate to USD 48 billion in the revised budget estimates, reflecting the additional cost of free food grain distribution in the wake of the COVID-19 pandemic. Poor nutrition is a significant threat to the health and well-being of the world’s population. According to estimates from the State of Food Security and Nutrition in the World 2020, 144 million children (21%) under the age of five were stunted, 47 million (6.9%) were affected by wasting, and 38 million (5.6%) were overweight in 2019. At least 340 million children suffer from micronutrient deficiencies (FAO, IFAD, UNICEF, WFP and WHO, 2020[18]). Countries are also facing a growing public health burden linked to poor quality diets: more than two billion people (about 40% of the world’s adult population in 2016) are overweight or obese, and adult obesity is rising in all regions across the globe (Figure 1.19). Across the OECD, almost 60% of the population is overweight or obese, and nearly 25% of people are obese (OECD, 2019[37]). Malnutrition and obesity have significant negative consequences for health, quality of life, productivity and economic outcomes. Poor diets have been associated with increased rates of type II diabetes, cancer, cardiovascular diseases and other non-communicable diseases, as well as shorter lifespans. According to the Global Burden of Diseases, Injuries, and Risk Factors Study, dietary risks15 such as a high intake of salt, sugar and red or processed meat, and a low intake of whole grains, fruits and vegetables, were responsible for 7.9 million deaths among adults aged 25 years and older in 2019 (GBD 2019 Risk Factors Collaborators, 2020[38]). In OECD countries, overweight and obesity will claim an estimated 92 million lives by 2050, reducing life expectancy by nearly three years (OECD, 2019[37]). Poor diets and unhealthy food choices impose considerable economic costs on society, including reduced school performance for children, higher rates of workplace absenteeism, and lower labour productivity. The combined economic impact of overweight on life expectancy, health expenditure and labour market productivity will reduce GDP by an estimated 3.3% per annum in OECD countries between 2020 and 2050 (OECD, 2019[37]). The causes of poor nutrition in developed countries are complex and highly context-dependent, and include urbanisation, changes in lifestyles, socio-economic factors, as well as the low cost and widespread availability of processed and fast food (Placzek, 2021[10]). In addition, there are concerns that agricultural support policies may have contributed to worsening health and nutritional outcomes. Since the late 1960s, many countries have pursued national food security goals through an overarching focus on achieving self-sufficiency in the production of cereal crops such as wheat, maize and rice. Agricultural R&D was heavily biased towards staple grains, through large-scale public investments in the development of new crop varieties and advances in plant breeding. Policies such as price supports, preferential credit, input subsidies, and grain procurement for public stocks, as well as infrastructure investment (e.g. in irrigation networks), strongly encouraged farmers to specialise in the production of staple crops. As a result, global grain production increased substantially, and developing countries experienced rapid increases in yields per hectare during the Green Revolution: between 1960 and 2000, yields rose by 208% for wheat, 109% for rice, 157% for maize, 78% for potatoes, and 36% for cassava (Pingali, 2012[39]). Over the past few decades, agricultural productivity growth has been a fundamental driver of poverty reduction and widespread improvements in global food security (Alston et al., 2010[28]; Kristkova, van Dijk and van Meijl, 2017[30]; Piesse and Thirtle, 2010[29]). In particular, productivity-led declines in food prices have substantially improved access to food for poor consumers, resulting in increased calorie availability per capita and a significant fall in the prevalence of undernourishment globally. However, an excessive policy focus on staple crops may have reduced dietary diversity by promoting the production of energy-dense cereals at the expense of micronutrient-rich non-staple foods, such as fruits, vegetables and pulses (Pingali, 2015[22]). As land and resources were increasingly allocated towards staple crops, important sources of critical micronutrients were displaced and became relatively less affordable (Bouis, 2000[40]; Kataki, 2002[41]). For example, during the 1970s and 1980s, farmers in India diverted land away from pulses to produce wheat and rice, leading to sharp increases in the price of pulses and a drop in their per capita consumption (Hazell, 2009[42]). More recently, work by the OECD has demonstrated that agricultural policies promote staple products such as rice and wheat at the expense of other production activities (OECD, 2016[21]). Today, diets across many societies are characterised by an over-consumption of processed foods, sugar and fat, and insufficient consumption of fruits and vegetables (Giner and Brooks, 2019[43]). With the exception of Asia and some upper-middle income countries, most countries do not have enough fruits and vegetables available to meet the FAO/WHO recommendation of consuming a minimum of 400 g per person per day (FAO, IFAD, UNICEF, WFP and WHO, 2020[18]). The current structure of agricultural support policies may have significant consequences for nutritional outcomes. Figure 1.20 shows the transfers to specific commodities (expressed as a share of commodity gross farm receipts), which collectively represented more than 47% of producer support in 2018-20. Sugar has the highest reliance on government support, with transfers amounting to 28% of commodity gross farm receipts. Milk is highly supported in many OECD countries, although the aggregate %SCT hides significant variation in milk policies across countries (including -33% of implicit taxation in India). Energy-dense foods such as vegetable oils (rapeseed), staple crops (maize and rice) and meat also feature prominently, while relatively limited support is provided for fruits and vegetables. These measures ossify production and increase the supply of these commodities. To the extent that support measures encourage the production of nutrient-poor commodities, this may hamper incentives for farmers to diversify their production towards foods that are potentially richer in micronutrients. At the same time, it is worth noting that most commodity-specific transfers come from increased domestic prices through policies such as import tariffs, quotas and minimum prices. Their immediate effect would therefore be to reduce the domestic consumption of these products. However, this effect may be small if consumers are not very responsive to higher prices (e.g. if demand is inelastic, or if the value of agricultural commodities accounts for a small share of overall food expenditures), and may be overwhelmed by the price-depressing effects of other support policies, such as taxpayer-financed subsidies and investments in R&D (Beghin and Jensen, 2008[44]; Pingali, 2015[22]). Reducing trade-distorting support could therefore facilitate a transition towards more diverse agricultural production systems, providing consumers with access to a broader range of nutritious foods necessary for a healthy diet (Brooks and Matthews, 2015[23]). Decoupled payments allow farmers to follow market signals in their production decisions, without biasing choices on what to produce, or whether to remain in the sector at all. Furthermore, there may be scope to rebalance support measures that directly encourage the production of staple crops towards the provision of a greater diversity of nutrient-rich perishable foods (Global Panel on Agriculture and Food Systems for Nutrition, 2020[45]). Additional public and private investments may be needed to strengthen market infrastructure and information systems for nutrient-rich perishable foods (Pingali, 2015[22]). Investments in transport and storage infrastructure (including cold chains) can help to retain the nutritional value of fresh produce and high-value food products (FAO, IFAD, UNICEF, WFP and WHO, 2020[18]). Public funding for R&D and innovation focused on micronutrient-rich foods and food fortification, along with efforts to strengthen farmers’ knowledge and capacities, can provide further incentives for the production of nutrient-rich foods and the development of diversified farming systems (Bowman and Zilberman, 2013[46]; Global Panel on Agriculture and Food Systems for Nutrition, 2020[45]). In countries where per capita meat consumption exceeds healthy levels, a shift towards more plant-based diets, with lower levels of ruminant meat consumption, would have the twin potential of benefiting public health while lowering GHG emissions (Giner and Brooks, 2019[43]). While there may be a need to rebalance agricultural investments across sub-sectors and towards more nutrition-sensitive investment, agriculture and trade policies are not always the best instrument to address the complex and multifaceted challenges of global malnutrition. Work by the OECD suggests that governments should favour demand side strategies for encouraging healthier food choices, with a parallel need to work with industry at the supply-demand interface, and in some cases impose stricter regulations on retailers, for example in the marketing of specific food products, in particular to children (Giner and Brooks, 2019[43]). Given alarming trends in public health, some governments are also giving increased consideration to fiscal measures. In particular more than 40 countries have imposed consumption taxes on sugar and sweetened beverages, a product category where consumption levels often exceed by a large margin those recommended by health guidelines (Hattersley et al., 2020[47]). The announcement in the United Kingdom of a soft drinks levy resulted in several major companies reformulating their products ahead of the introduction of the tax, suggesting that the credible threat of policy action can play an important role in prompting change and may be as important as the action itself. Building stability in food systems is fundamental to achieving food security over the long term. Farmers and consumers are increasingly confronted with risks relating to climate change, natural disasters, price volatility and external shocks, such as the COVID-19 pandemic. Stability can also be influenced by agricultural support policies, including through sudden and unanticipated changes in the policy landscape. Trade plays an essential role in maintaining stability in the global food system. By allowing produce to flow from food surplus areas to food deficit areas, trade helps to absorb the impacts of local and regional supply shocks. This generally results in lower price volatility, reduced uncertainty of supply, and greater integration of global and regional markets (OECD, 2013[19]). Where production variability is weakly correlated among countries, trade can help to mitigate supply volatility and manage domestic food shortages driven by poor harvests, droughts, floods and other catastrophic events (Brooks and Matthews, 2015[23]). The stabilising role of trade is only likely to increase in importance, as domestic production shocks become more frequent due to climate change. Policy distortions that impede trade’s role in maintaining stability in food systems can be measured by comparing the prices received by producers with world market prices (Box 1.3). Many countries have attempted to pursue self-sufficiency in staple crop production through border interventions such as import tariffs, quotas, and export restrictions. These measures ostensibly attempt to protect domestic constituents and prevent the transmission of international food price volatility onto domestic markets. The viability of such strategies is questionable however, as reducing a country’s integration with world markets will only increase its exposure to volatility in domestic output and prices. Domestic shocks tend to be more frequent and severe than international shocks, with output in individual countries varying to a much greater degree than the world output of individual food commodities (Brooks, 2012[48]). Trade policy interventions such as export taxes and restrictions are often introduced with the stated intention of stabilising domestic markets, but have the perverse effect of withdrawing products from world markets, reducing food availability and contributing to higher and more volatile world prices. During the 2007-08 food price crisis, several countries placed temporary export restrictions on staple crops as a means to protect their domestic consumers from rising food prices. A number of grain-exporting emerging and developing economies adopted export bans, whilst several major grain-importing nations reacted by reducing pre-existing import tariffs and relaxing tariff-rate quotas. These measures exacerbated the increases in world prices and ultimately undermined the reputations of exporting countries as reliable suppliers on world markets, resulting in reduced long-term demand from traditional trading partners (Deuss, 2017[49]). The reallocation of trade caused by export restrictions may encourage importing countries to lose confidence in international markets, and pursue less efficient policy objectives such as self-sufficiency and the expansion of public stocks. Public stockholding policies are almost always implemented using other policy instruments such as administered prices, trade policy measures, and import and export monopolies. These policies are often ineffective in reducing domestic price volatility, and may lead to negative spill-overs in international markets. In comparison to private stockholding, public stocks are arguably less responsive to market developments, and may therefore exacerbate rather than mitigate volatility if stock changes are misaligned with market needs. In particular, the acquisition of large amounts of grain to build or replenish public stocks can decrease the available supply on international markets, potentially putting upward pressure on world market prices. Conversely, the sudden release of large amounts of grains from public stocks can depress world market prices (Deuss, 2015[50]). Trade interventions have had limited success in stabilising domestic market prices, and can result in significant welfare losses for poorer food-deficit countries (Anderson and Nelgen, 2012[51]). Whilst price stabilisation policies have on occasion been successful in containing the impact of large international price movements, they can transfer instability onto world markets and often prove to be fiscally unsustainable. Moreover, heavy trade distortions on some agricultural products make them susceptible to trade retaliation, thus adding to instability and uncertainty. Removing trade restrictions and market distortions could further strengthen the capacity of trade to stabilise markets and reduce price volatility, by allowing regions with better harvests to supply output to regions with worse harvests. If trade measures cannot be avoided, governments should design rules to limit their negative spill-over effects on other countries (OECD, 2013[19]). Trade’s role in promoting stability can be further strengthened through investments in transport and storage infrastructure, as well as efforts to improve the transparency of information on supply, demand, stocks and prices – including through international initiatives such as the G20-led Agricultural Market Information System (AMIS). However, trade openness may not be sufficient to contain rare but severe international shocks, such as simultaneous harvest failures, price spikes on world markets, and supply chain disruptions such as those witnessed during the onset of the COVID-19 pandemic (OECD, 2021[35]). It may be necessary to gather more information on market concentration at various stages of food supply chains, and where appropriate, to actively support the geographic diversification of food and feed supplies in order to limit the risks of bottlenecks. Beyond agricultural support policies, a range of other measures can be introduced to strengthen stability in the food system. Market-based mechanisms such as weather-indexed insurance can help to finance food imports during weather-related shortfalls in domestic production, without requiring costly monitoring of individual farms. Care should be taken to avoid subsidised insurance products that do not accurately reflect producers’ risk profiles, as such programmes can hamper incentives for on-farm risk management and crowd out private insurance options (OECD, 2020[52]). Well-functioning futures markets for agricultural commodities can play a significant role in reducing price fluctuations, through option contracts that lock in future import purchases at pre-determined prices. Furthermore, targeted social programmes (including cash transfers) can be an effective tool to mitigate the impacts of international price volatility on low-income households (OECD, 2013[19]). Food systems are a major source of incomes and livelihoods around the world. Primary agriculture accounted for 27% of total employment in 2019, and recent estimates suggest that there are at least 570 million farms worldwide, most of which are small (less than 2 hectares) and family-operated (World Bank, 2021[53]; Lowder, Skoet and Raney, 2016[54]). Food systems jobs represent the majority of self and wage employment in developing countries, with farming generating about 68% of rural income in Africa and about half of rural income in South Asia (Townsend et al., 2017[55]). Beyond farm production, food systems support job creation in a range of upstream and downstream industries, such as input supply, food processing, transport and logistics, supermarket chains and restaurants. The structural transformation of economies has important influences on the development of agriculture and food systems. As countries develop, productivity improvements lead to rising agricultural output yet a decline in agriculture’s share in GDP, releasing labour out of agriculture and into faster growing non-agricultural sectors. With growing rural to urban migration and a consolidation of farm structures, agriculture’s share in total employment tends to decline as per capita incomes rise (Figure 1.22). Structural change is also accompanied by transformations in the food system, with greater job opportunities offered by other segments of the value chain such as food processing, retail and other food services. Urbanisation and higher per capita incomes lead to changes in consumer preferences and new demands for fresh, processed and convenience foods. In low income countries (e.g. in eastern and southern Africa) agriculture accounts for about 90% of food-related employment, while in high income countries such as the United States, food services account for about two-thirds of all jobs in the food system (Townsend et al., 2017[55]). Food and beverage manufacturing now ranks among the top three manufacturing sub-sectors by value added in 27 OECD countries (OECD, 2021[35]). At the same time, the agricultural sector is becoming increasingly integrated into global value chains (GVCs), providing new sources of employment and opportunities for farmers to grow their incomes. Foreign direct investment (FDI) and trade have facilitated greater participation in GVCs, spurred by the liberalisation of investment, falling tariffs, and reductions in trade-distorting support for agricultural producers (Punthakey, 2020[56]). Trade and GVC participation account for an estimated 20-26% of total agricultural labour income globally, with significant employment spill-overs in other supporting sectors such as industry and services (Greenville, Kawasaki and Jouanjean, 2019[57]). Agricultural development can play an essential role in improving livelihoods and reducing rural poverty. However, it is important to recognise that rural regions are diverse and complex socio-economic systems that extend beyond agriculture, and encompass a broad range of manufacturing and service sector activities (e.g. mining, renewable energies, tourism). Indeed, many farm households derive a substantial share of their income from non-agricultural sources (OECD, 2003[58]). This implies that policies and investments to strengthen incomes and livelihoods should aim to offer multiple development pathways for farm households: improving competitiveness and productivity within agriculture, diversifying income sources among household members, and facilitating the transition of labour into non-agricultural sectors (Brooks, 2012[48]). In 2018-20, the governments of 54 countries provided USD 540 billion per year in support to farm incomes, either through higher prices paid by consumers or direct payments to farmers. This represents 75% of the USD 720 billion in positive transfers to agriculture. In contrast, a relatively low proportion of total support (14%: USD 102 billion) is provided in the form of general services, a category that includes public goods and services such as R&D and innovation, inspection services, infrastructure development and maintenance, and public stockholding (discussed previously, in the section on “Food Availability”). Consequently, the current structure of agricultural support does not encourage farmers to diversify their income sources and provides disincentives for them to leave the sector, thereby limiting adjustment pathways beyond agriculture. Government intervention in agriculture is often justified by the need to improve the incomes of farmers. While support policies may have some success in raising farm household incomes, they often do so at considerable cost to consumers and taxpayers. Policies tend to be poorly targeted: official policy statements are seldom clear about which farm households should qualify for support, and policies often fail to establish explicit eligibility criteria and discriminate between high income and low income farm households (OECD, 2002[27]; de Frahan, Dong and De Blander, 2017[59]). They are also inequitably distributed, with support based on output or factors of production resulting in a greater share of the benefits accruing to large-scale farms. Finally, they result in significant leakages, meaning that a substantial share of support accrues to other unintended beneficiaries (e.g. input suppliers, downstream industries, landowners, programme administration costs). Evidence suggests that there is a clear inverse relationship between the tendency of a policy to distort markets and its efficiency in transferring income benefits to farmers (Dewbre, Antón and Thompton, 2001[60]). In other words, policies that pay farmers without affecting their production decisions generally result in a greater share of support being retained by the household (while also minimising impacts on production and trade). This result is confirmed by estimates of the income transfer efficiency of support policies for OECD countries, which show that the share of monetary transfers accruing to farmers are just 17% for input subsidies, 23% for market price support, 26% for deficiency payments, and 47% for area payments (OECD, 2003[58]). This is because market price support and other distorting policies stimulate output, and much of the value of the support is paid out to input suppliers or capitalised into land values (especially for area payments, where over 90% of the benefits are absorbed in increased land values). Such policies raise costs for farmers who want to buy or lease land, and slow structural change. In contrast, direct income payments have a far higher income transfer efficiency, as they can be decoupled from agricultural activity and targeted to households that are in need of assistance (e.g. through the imposition of limitations on payment levels) (OECD, 2003[58]). The vast majority of the world’s farmers are small-scale producers with less than 2 hectares of land, who collectively produce an estimated 30-34% of the global food supply (Ricciardi et al., 2018[61]). Policies to strengthen incomes in the food system will therefore need to focus on improving productivity and connecting small farmers with markets. Increasing investments in public goods such as rural infrastructure, agricultural R&D, technology transfer, extension and advisory services, can help farmers to increase their competitiveness (Brooks, 2012[48]). New technologies can reduce transaction costs and increase efficiencies: digitalisation is facilitating greater financial inclusion, and e-commerce platforms are increasingly linking entrepreneurial producers with national and foreign markets. Standards, labelling and certification schemes aim to create more differentiated products and can sometimes be explicitly designed with the intention of improving farmers’ livelihoods (e.g. Fairtrade certification). Digital technologies also have significant potential to create efficiencies in Sanitary and Phytosanitary systems (SPS), and can enhance trade in agricultural and food products (OECD, 2021[62]). While policies need to enable farmers to take advantage of the rising opportunities offered by agricultural development, they also need to protect those who are unable to adjust to competitive pressures. Productivity growth puts pressure on the incomes of less competitive farmers, due to declining real prices which are not fully offset by a decline in production costs. Improving agricultural productivity therefore inevitably implies that some less productive farmers that are unable to adjust will need to leave the sector. If less productive farmers have access to viable economic alternatives in non-agricultural sectors, income support may not be necessary and may hamper the transition out of agriculture. If viable alternatives do not exist, then transitional assistance to another economic activity may be more effective than income support (OECD, 2002[27]). Ultimately, many of the policies required to improve farmers’ incomes are non-agricultural. They include investments in education and healthcare, peace and political stability, sound macroeconomic management, developed institutions, property rights, and governance (Brooks, 2012[48]). Labour market and regional development policies can facilitate the absorption of labour into other sectors, including downstream processing sectors. Social safety nets (e.g. conditional cash transfer programmes) can be an effective means for providing income support whilst ensuring equal treatment between agricultural and non-agricultural households. Income objectives and appropriate indicators should be clearly defined, with comprehensive information on the economic situation of farm households collected to allow for a more accurate assessment and monitoring of income deficiencies (OECD, 2003[58]). The food systems underpinning the world’s current food consumption patterns are a major driver of climate change and a significant source of environmental pressures worldwide. Agriculture, forestry and other land use activities contribute an estimated 16-27% of total anthropogenic greenhouse gas (GHG) emissions, including 13% of carbon dioxide (CO2), 44% of methane (CH4), and 81% of nitrous oxide (N2O). Other pre- and post-production segments of global food systems (e.g. energy, transport and industry) account for approximately 5-10% of emissions from human activity (IPCC, 2019[63]). Direct GHG emissions from agriculture vary across regions and emanate from a variety of sources (Figure 1.23). Two-thirds of direct emissions from agriculture come from livestock, with enteric fermentation16 alone accounting for 40%. Emissions from manure contribute another 26% to direct emissions. Synthetic fertilisers are responsible for 13% of direct emissions from agriculture, and rice cultivation accounts for 10%. In recent decades, growth in agricultural production has put increasing pressure on natural resources. Agriculture currently uses approximately half of the world’s habitable land (IPCC, 2019[63]). Livestock occupies about 78% (40 million km2) of all agricultural land; this includes 35% of global crop production which is devoted to the production of animal feed (Dasgupta, 2021[64]). Irrigated agriculture accounts for an estimated 70% of global freshwater usage (equivalent to 2 797 km3 per year in withdrawals from surface and groundwater resources), and an even higher share of consumptive water use (i.e. water that is not returned to the environment) due to the evapotranspiration of crops (United Nations, 2021[65]). Empirical studies have shown that agricultural expansion is a major cause of deforestation (Busch and Ferretti-Gallon, 2017[66]) Recent estimates suggest that large-scale commercial agriculture (i.e. cattle ranching, soy production and palm oil plantations) accounts for about 40% of tropical and sub-tropical deforestation, while local subsistence agriculture is responsible for a further 33% (Hosonuma et al., 2012[67]; FAO and UNEP, 2020[68]). Food production is also the world’s most significant driver of terrestrial and marine biodiversity loss. Around 80% of all threatened terrestrial bird and mammal species are in danger of habitat loss due to agricultural expansion (Tilman et al., 2017[69]). The conversion of natural ecosystems for crop production or pasture has been the biggest cause of habitat loss globally, driving an 82% decline in the collective weight of wild mammals since 1970. Farmed animals such as cows and pigs now account for 60% of the global biomass of all mammal species (compared with just 4% for wild mammals), while farmed chickens, ducks and turkeys account for 71% of the global biomass of all bird species (wild birds make up 29%) (Benton et al., 2021[70]). In many regions, soil and pollinator biodiversity have deteriorated considerably due to the over application of chemical fertilisers and pesticides, along with farm practices such as tilling and ploughing (Dasgupta, 2021[64]). Agricultural intensification has also been identified as a leading cause of widespread declines in insect biodiversity, together with climate change (Raven and Wagner, 2021[71]). Beyond their effects on production and trade, agricultural support policies have significant consequences for the environment and resource use. Support policies can induce negative environmental impacts on the intensive margin (e.g. increased input use, livestock numbers, water use), on the extensive margin (e.g. reallocating land and other inputs between different outputs), or on the entry-exit margin (e.g. expansion or contraction of agricultural land relative to other land uses) (Henderson and Lankoski, 2019[72]). Market price support, payments based on commodity output and payments based on unconstrained variable input use are among the potentially most environmentally harmful support policies (Henderson and Lankoski, 2019[72]; Henderson and Lankoski, 2020[73]; OECD, 2020[74]; DeBoe, 2020[75]). Such policies are coupled to farmers’ production decisions and cannot be easily targeted, thus providing incentives for the intensification of input use, the allocation of land for supported crops, and the entry of land to the agricultural sector. Studies have shown their negative impacts on water quality and direct agricultural GHG emissions, and they may negatively influence biodiversity by promoting less diverse agricultural systems (DeBoe, 2020[75]; Lankoski and Thiem, 2020[76]). At the global level, however, the widespread adoption of these policies may constrain emissions by lowering production as a result of resource inefficiencies (Laborde et al., 2021[77]). Payments based on variable inputs without appropriate constraints can encourage the excessive use of fertilisers, herbicides and pesticides. Over application of fertilisers and animal manure leads to substantial nutrient surpluses and nitrogen and phosphorus run-off. Nitrogen pollution causes severe damage to freshwater ecosystems, harming invertebrates and fish, causing acidification and eutrophication, stimulating the growth of toxic algae and lowering oxygen levels in water (hypoxia). Excessive or inadequate pesticide use has been associated with declines in populations of birds, insects, amphibians and aquatic and soil communities, as well as negative impacts on human health (Guerrero, 2018[78]; Sud, 2020[79]). In most countries, support based on input use is provided without constraints to protect against the over application of variable inputs. India has the largest rate of support based on inputs, primarily allocated to electricity price subsidies for groundwater pumping and irrigation, and fertiliser subsidies. These measures were worth 7.2% of gross farm receipts in 2018-20 (Figure 1.24). Kazakhstan and Iceland provide support based on inputs amounting to 6.4% and 6.1% (respectively) of gross farm receipts, although in Iceland most support based on input use is directed to fixed capital formation (i.e. on-farm investments), which are potentially less environmentally damaging than general fertiliser subsidies. The optimal policy mix for support that encourages the use of environmentally harmful inputs would be to impose a tax to account for the damage they cause to waterways and natural ecosystems (Anderson and Valenzuela, 2021[24]). Well-designed environmental policies and regulations can play an essential role in containing some of the adverse environmental impacts of input use. Policy makers have a range of instruments at their disposal, including regulatory procedures for pesticide use, targets for reducing nitrogen and phosphorus discharges, fertiliser accounting systems, nitrogen quota systems, bans on manure application on bare fields, fertiliser taxes for non-agricultural uses, taxes on phosphorus content in feed, as well as agri-environmental schemes and advisory services (OECD, 2021[35]). Water pricing or market mechanisms related to the scarcity of water can help to encourage more efficient water use and prevent the depletion of surface and groundwater resources. However, irrigation prices typically do not reflect the full cost of water use, and many countries only partially recover the operational, maintenance and capital costs associated with water use (Gruère, Shigemitsu and Crawford, 2020[11]). Governments of OECD countries have undertaken a number of policy changes related to water in agriculture since 2009, increasing their alignment with OECD recommendations in this area (Box 1.4). Payments based on current land area create incentives to expand cropping areas and maintain marginal lands in production. Non-uniform crop area payments may have mixed environmental impacts, depending on whether less or more emission intensive crops are favoured with non-uniform payment rates. If crop area payments favour arable farming over livestock production, they may induce a shift away from livestock and a reduction in agricultural GHG emissions and nutrient surpluses. Conversely, area payments may increase GHG emissions in countries where crops account for the dominant share of agricultural GHG emissions (Henderson and Lankoski, 2019[72]). Payments based on animal numbers without constraints will generally result in increased livestock numbers, which can be achieved either through increased stocking densities or increased area, and in either case are likely to cause negative environmental effects (DeBoe, 2020[75]). Fully decoupled payments based on non-current crop area (e.g. payments based on historical entitlements or overall farming income) are among the least environmentally harmful support policies (Henderson and Lankoski, 2019[72]). These measures allow farmers to follow market signals in their production decisions, and in some cases, production is not required for farmers to receive support payments. If historical acreage is fixed for payments, then there is no incentive to bring additional land into the sector (Lankoski and Thiem, 2020[76]). However, payments based on historical entitlements could still affect incentives, if farmers expect their current decisions to influence future payments (DeBoe et al., 2020[80]). Moreover, by supplementing farmer incomes and making agriculture more profitable relative to other land uses, decoupled payments could still stifle structural change and hinder the conversion of agricultural land to more sustainable land uses. Ultimately, the environmental impact of decoupled payments depends on the type and effectiveness of mandatory environmental conditions and requirements (cross compliance) that accompany payments (DeBoe, 2020[75]). Reorienting agricultural support towards decoupled payments and away from the most production distorting forms of support could reduce environmental pressures and substantially strengthen the sustainability of production. At the same time, it is important to recognise that agricultural policies can shape the structure and intensity of production over the long term. Decoupling is therefore unlikely to be sufficient on its own, particularly in countries with a high livestock density and intensive production systems (OECD, 2020[74]; Lankoski and Thiem, 2020[76]). In such cases, additional measures may be needed to ensure that policies and market prices reflect the negative environmental externalities associated with agricultural production. Agricultural policies can also be specifically designed to generate positive environmental outcomes, by encouraging farmers to provide environmental goods and services such as carbon sequestration, preservation of rural landscapes, resilience to natural disasters, pollination, habitat provision, and control of invasive species. Agri-environmental payments that encourage the use of environmentally friendly inputs or practices (e.g. compliance with fertiliser use restrictions) are potentially among the most environmentally beneficial types of support measures (DeBoe, 2020[75]). However, just USD 1.5 billion of the USD 268 billion per year of budgetary payments to producers in 2018-20 was linked clearly to the provision of environmental public goods (i.e. payments based on specific non-commodity outputs). Some policies, such as support based on non-commodity output, can occasionally have positive environmental effects. For example, land retirement policies can create incentives for farmers to switch from crop production to permanent pasture or forests, encouraging a contraction of agricultural land and reducing environmental pressures. However, if not managed well, a contraction of agricultural land resulting from land abandonment can in some instances lead to negative environmental outcomes such as biodiversity loss, increases in invasive species, or a greater risk of wildfire (DeBoe et al., 2020[80]). While reductions in agricultural land use often have beneficial environmental effects, they can also be accompanied by the intensification of production on remaining land areas, potentially resulting in unintended negative environmental impacts. This underscores the importance of carefully managing the reform process to account for potential unintended environmental consequences. For example, reductions in market price support can also result in land abandonment and further intensification of production, with potential negative consequences for biodiversity and landscape ecology. Agri-environmental payments can create adverse environmental impacts in mixed dairy and crop production systems, particularly if they favour crop production and encourage land use changes from pasture to cereals (Henderson and Lankoski, 2019[72]). Policy makers should therefore take a proactive approach to managing the process of policy reform and subsequent land use transitions. Furthermore, agri-environmental schemes could benefit from improvements in their design and in the design of mandatory constraints to better deliver environmental improvements (DeBoe, 2020[75]). OECD work with national-level collaborators seeks to exploit such potential benefits. The world faces a daunting “triple challenge” of providing safe and nutritious food for all, improving incomes and livelihoods along the food supply chain, and contributing to environmental sustainability. Meeting this challenge will require effective responses and co-ordination across many areas of public policy. With respect to the agro-food sector, simultaneous progress in achieving sustainable productivity growth and improved resilience will be essential for the sector to contribute effectively to each dimension of the triple challenge. The OECD Agro-Food Productivity-Sustainability-Resilience Policy Framework provides a structured tool for identifying policy priorities that strengthen long-term productivity, enhance environmental performance, and increase resilience. The Framework highlights the importance of developing coherent and integrated policy approaches that encompass the wider enabling policy environment for food systems. Governments should seek to establish synergies across the objectives of productivity, sustainability and resilience, while managing trade-offs and avoiding contradictory policy signals. With the global population projected to reach 10 billion by 2050, food systems are facing growing pressure to use resources sustainably, protect ecosystems, preserve biodiversity, and reduce greenhouse gas emissions. Strengthening productivity and sustainability is therefore fundamental to enable food systems to produce more with the use of less inputs and natural resources. At the same time, vulnerabilities to climate change highlight the need to build resilience to natural disasters and strengthen capacities to respond to an evolving risk environment. Figure 1.26 shows that the drivers of agricultural output growth have shifted dramatically over time, with important consequences for resource use and environmental sustainability. Historically, most of the growth in food production came from increases in the total area of agricultural land used for crop and animal production. After 1960, however, more intensive use of inputs (e.g. synthetic fertilisers, pesticides, labour and machinery) became the most important driver of output growth. This trend persisted until the 1990s, when improvements in total factor productivity (i.e. efficiency improvements such as better farm management practices, improvements in crop varieties and breeds) took over as the most important factor contributing to global agricultural production. Total factor productivity growth has driven a “decoupling” of food production and land use, enabling global food production to increase four-fold since 1960, while agricultural land use has grown by just 10% (see the section on Food availability). Land use changes from agriculture are still a major concern, driving deforestation, declines in biodiversity, GHG emissions, and the depletion of soil organic carbon (IPCC, 2019[63]). Nonetheless, productivity growth has been indispensable in enabling agriculture to feed the world, while preventing worse and potentially catastrophic outcomes for environmental sustainability. There are important synergies to be realised in policies to promote productivity, sustainability and resilience. For example, improvements in technology and farm management practices have facilitated a decline in the emissions intensity of agriculture (i.e. emissions per unit of output) across most regions. Direct emissions from agriculture grew by approximately 0.5% per year between 1990 and 2016, while crop production grew by an estimated 2.5% per year and livestock production grew by about 1.9% per year over the same period (OECD, 2021[35]). This has primarily been achieved through more efficient use of inputs, such as fertilisers, animal feed and land, which are significant sources of emissions. Efficiency gains have also allowed many countries to reduce their use of synthetic nitrogen fertilisers and pesticides, while steadily expanding agricultural production. Advances in genomic science and precision agriculture can strengthen sustainable productivity by allowing for a more judicious application of environmentally harmful inputs. Globally, some 45% of nitrogen added to fields is not taken up by crops, implying that there is considerable scope to decrease emissions and reduce nutrient surpluses without compromising productivity and food security (Blandford and Hassapoyannes, 2018[81]). Pesticide use can often be decreased without affecting the productivity and profitability of farms, resulting in reduced health and environmental risks (Lechenet et al., 2017[82]). Similarly, evidence suggests that with more sanitary farming practices, the use of antibiotics on animal farms for growth-promoting purposes can be eliminated with little or no adverse impact on the economic or technical performance of farms (Ryan, 2019[83]). A comprehensive approach to resilience and risk management can contribute to productivity and sustainability by enhancing the long-term stability of food systems. Resilience implies strengthening the agricultural sector’s capacity to prepare and plan for adverse events, absorb the impacts of negative shocks, adapt in response to an evolving risk environment, and transform if current processes and systems are no longer sustainable (OECD, 2020[52]). Developing a diverse portfolio of risk management instruments is necessary to tackle food security risks, and can strengthen farmers’ capacities to innovate and adapt to climate change (OECD, 2013[19]). Public funding for R&D can support the development of new innovations such as drought-resistant seeds and water management technologies, which allow farmers to manage risks more effectively and maintain more sustainable production practices (OECD, 2019[31]). Box 1.5 outlines the principles for effective disaster risk management for resilience. Potential trade-offs between policies to promote productivity, sustainability and resilience also deserve special attention. For example, improvements in total factor productivity often lead to lower prices and increased food demand. In some cases, this may trigger an expansion of production, resulting in higher GHG emissions (Blandford and Hassapoyannes, 2018[81]). Productivity-driven increases in production have also been associated with large-scale reductions in biodiversity on farms, with fewer varieties and breeds of plants and animals being cultivated. This loss in genetic diversity undermines the resilience of food systems to pests, pathogens and climate-related shocks (IPBES, 2019[90]). Measures to strengthen resilience by building redundancies into supply chains may involve some trade-offs with productivity performance (at least in the short-term). Efforts to strengthen total factor productivity in livestock production (e.g. through advances in herd genetics, feed and pasture quality, farm and animal management) have translated into declining emissions intensities over time. However, enteric fermentation from ruminant livestock production remains the leading source of direct emissions from agriculture worldwide, with beef having the largest emissions footprint by a wide margin (in terms of CO2eq per 100 g of protein produced) (Blandford and Hassapoyannes, 2018[81]). Generally, countries with a high livestock density (per hectare) have high nitrogen and phosphorus surpluses and high GHG emissions from agriculture, thus making it difficult to achieve sustainable productivity (Lankoski and Thiem, 2020[76]). Policy choices to reduce GHG emissions from agriculture also invoke trade-offs. Emission taxes can significantly reduce emissions by reallocating production towards less emission-intensive commodities, but may raise production costs and increase food prices. They could also induce carbon leakage if applied unilaterally by specific countries. Abatement subsidies used to reward carbon sequestration require government expenditures, and are half as effective in mitigating GHG emissions, but have a much lower impact on agricultural production and per capita food consumption, and would eliminate potential carbon leakage. A shift to lower emission diets by consumers is assessed to have a much smaller impact on reducing agricultural emissions than any emission tax (Henderson et al., 2021[91]; OECD, 2019[92]). Governments responded swiftly to the COVID-19 pandemic, with measures that were required to keep food and agriculture markets functioning, and that were mostly co-operative at the international level. As a result, most shocks were absorbed rapidly, with trade and markets recovering during the year. Average gross farm receipts for OECD and emerging economies actually increased in 2020, and in several large countries the sector was the best performing or least affected economically. That said, income shocks have affected the food security of many poorer consumers. Moreover, the virus remains active in many countries. An estimated 776 unique policy response measures were adopted by governments of countries covered in this report, covering all categories of measures, highlighting the breadth and responsiveness of public actions to address the impact of the crisis. While 19% of these measures were urgent responses to ensure supply and keep the sector functioning, just under 70% of measures took the form of temporary relief, and should be phased out as the crisis recedes. Ten per cent of measures are “no regrets”, in the sense that they improve market functioning and thereby contribute to improved resilience. These measures, such as trade facilitation, should be maintained or even scaled up after the crisis. The remaining 2% of measures did not fit this classification. At the same time, 11% of measures, mostly introduced as temporary relief, were identified as potentially market distorting or environmentally harmful. In particular, these included export bans, other trade restrictions, and regulatory flexibilities. Some of these were applied temporarily, and the remainder need to be rescinded. A first and partial assessment of budgetary expenditures in response to the COVID-19 crisis suggests that a minimum of USD 157 billion was earmarked in funding or offered in financing means (subsidised loans or credit) to the sector. Close to half of this amount (USD 75 billion) was allocated to support for agriculture and food sector actors, and a further USD 55 billion to food assistance programmes, with the remaining USD 27 billion directed towards general services or labour and biosecurity measures. These amounts do not include the share of economy-wide recovery packages adopted in these countries (which exceeds USD 5.6 trillion) from which the agriculture sector may have benefited. OECD countries favoured relief measures for the agro-food sector and food assistance, largely via earmarked funds, while emerging economies used more non-support measures and allocated relatively more loans and credits towards the agriculture and food sector. While extensive contingencies were made for the agricultural sector, the fact that overall economic effects were in many cases less serious than those faced in other sectors means that actual financial disbursements may turn out to be substantially lower than allocations. Although the COVID-19 pandemic dominated policy responses, a number of other policy reforms or initiatives were introduced in 2020. In addition to revised agricultural policy frameworks, and changes or reforms to existing support measures and policies, two important developments relate to strengthened agri-environmental policies and the continued trend of new bilateral or regional free trade agreements. New steps aimed at enhancing the environmental performance of agriculture and food systems include the European Union’s Green Deal together with the Farm to Fork and Biodiversity Strategies, the Carbon Neutral Strategy, New Zealand’s 2019 Zero Carbon Act and complementing strategies in 2020, Canada’s A Healthy Environment and A Healthy Economy plan, Japan’s Green Growth Strategy, and new strategies on reducing agricultural GHG emissions in several other countries. A number of initiatives also focused on making water management systems more sustainable, and on tackling food loss and waste. On the trade side, the existing trend towards bilateral and regional trade agreements continued in 2020. With the Regional Comprehensive Economic Partnership, the world’s largest Free Trade Agreement was signed in 2020, including the ten members of the Association of South-East Asian Nations and five other countries in Asia-Pacific. Smaller trade agreements also continued to be put in place, including a number of agreements signed by the United Kingdom to ensure continued trade relations after the country’s departure from the European Union. In 2018-20, agricultural support policies across the 54 countries covered in this report generated USD 720 billion per year in transfers to agriculture, which in nominal terms is more than twice the aggregate level of transfers observed in 2000-02, but nevertheless lower when expressed relative to agricultural value added. About three-quarters of this support, USD 540 billion, was directed to individual producers, either in the form of higher prices or through direct payments. Reforms in OECD countries have stalled in the past ten years, with little change in the level or composition of support. Indeed, some countries have rolled back earlier reform efforts. Across the 54 countries, two-thirds of support is still provided in ways that are potentially most market and trade distorting, likely to harm the environment including by raising GHG emissions. This is reflected in a weakened sectoral performance in terms of delivering sustainable productivity growth. Overall, total net support to the sector (TSE) costs the economy 0.8% of combined GDP across the 54 countries, down from 1.0% at the beginning of the century. When measured relative to the size of the agricultural sector, total net support amounted to 23% of agricultural value added in 2018-20, compared with 35% in 2000-02. Producer support as a share of gross farm receipts (%PSE) has been declining over the past two decades, from 18% in 2000-02 to 11% in 2018-20. While producer support in OECD countries declined from 28% of gross farm receipts (GFR) in 2000-02 to 18% in 2018-20, it almost doubled in emerging economies from 3.8% in 2000-02 to 7.4% in 2018-20. To some extent, the decline in the overall average %PSE also reflects higher levels of negative market price support in some emerging economies. A central element of many countries’ support policies continues to be market price support. Total positive price support amounted to USD 272 billion per year in 2018-20, corresponding to 7% of the combined GFR. In contrast, a small number of countries implicitly taxed their farmers by suppressing domestic prices of some or all commodities, for instance through export restrictions. This resulted in a transfer of USD 104 billion per year away from producers. Of payments to farmers, USD 66 billion was linked to output or unconstrained input use, and has a similar tendency to create distortions as market price support. Added to the positive price transfer, this gives a total of USD 338 billion of potentially most distorting support to producers. A larger amount, USD 202 billion, was more decoupled from production decisions. Of this, only a small element, USD 1.5 billion, was conditional on the provision of clearly identified public goods, such as ecosystem services. Payments to agriculture as a whole, “general services” (GSSE), amounted to USD 102 billion, or 14% of total net support. This category includes investments in public goods, such as R&D and innovation, off-farm infrastructure and biosecurity (USD 76 billion). It also includes payments with a potential to distort markets, in the form of marketing and promotion and support for public stockholding (USD 42 billion). Subsidies for consumers (such as food assistance programmes) amounted to USD 78 billion per year, or 11% of all positive transfers to agriculture. Nonetheless, on average consumers were taxed by agricultural policies, as these subsidies remained small relative to the higher food expenditures, due to the persistent market price support in many countries. The variation in support levels across countries remains significant, however. Levels of producer support in 2018-20 ranged from less than 3% of GFR in New Zealand, Brazil, Ukraine, Australia and Chile to between 40% and 60% in Japan, Korea, Switzerland, Norway and Iceland, while net producer support was negative in Argentina, Viet Nam and India. High levels of producer support continue to be underpinned by a strong focus on market price support, but the importance of budgetary payments to producers varies strongly as well. Iceland, Norway, India, Turkey and Kazakhstan directed most-distorting output and input support to their producers at rates of between 4% and 12% of GFR in 2018-20, while less distorting payments worth more than 10% of GFR were provided in the European Union and the United Kingdom, as well as in Iceland, Switzerland and Norway. Across the dimensions of the triple challenge – ensuring food security and nutrition for all, providing livelihoods to farmers and others along the food chain, and using natural resources sustainably while reducing greenhouse gas emissions – food systems are sometimes accused of “systems failure”. Such an assessment overlooks important achievements, not least that of feeding a world population that has grown from 3 billion in 1960 to about 7.5 billion today, predominantly through improved yields and productivity rather than increased agricultural area. Even so, policies have not managed to address rapid structural change across food systems and the problems these changes have induced, be they a rising incidence of obesity, continued adjustment pressures on farmers, or mounting resource pressures and GHG emissions. The USD 272 billion of positive market price intervention and USD 104 billion of implicit taxation both have negative implications for food security at the global level, because they impede the efficient allocation of domestic resources and weaken the balancing role of trade in getting food from surplus to deficit regions. By constraining trade, they also contribute to increased price volatility on international food markets. The USD 338 billion of potentially most distorting support, comprising market price support and payments linked to output or the unconstrained use of inputs, is inefficient at transferring income to farmers, as a large share of the benefits are capitalised into land values or leak in the form of higher prices for inputs. It also tends to be inequitable, to the extent that support is linked directly to production. Finally, through its direct incentive to increase production, it contributes to increased resource pressures, including through impacts on water quality, biodiversity, and can raise GHG emissions. At the global level, however, the widespread adoption of such policies may constrain emissions by lowering production as a result of resource inefficiencies. The USD 202 billion of producer support that is decoupled from production decisions creates fewer distortions at the margin and therefore has less adverse impacts on global food security. It also has a reduced tendency to contribute to additional resource pressures and GHG emissions. While the effects on farmers’ incomes may still be inequitably distributed, there tend to be lower rates of leakage to non-farm landowners or input suppliers. Two important rationales for farm support are to provide social transfers in order to redress problems of low incomes, and to support the provision of environmental public goods. However, little of the budgetary support that is extended to producers is based on an assessment of their overall income from all sources, while just USD 1.5 billion of the USD 268 billion of budgetary payments to producers was linked clearly to the provision of environmental public goods. Instruments with potentially more positive effects on food security, incomes and resource use mostly fall within the category of general services for the sector (GSSE), and include investments in R&D, biosecurity and infrastructure. However the USD 102 billion of expenditure in this category represented just 16.5% of total net support (TSE) in 2018-20, a slight decline from the 17.2% estimated for 2000-02. Across the OECD, this share was even lower at 13.5% in the most recent period. Relative to the size of the agricultural sector, support to general services declined even more strongly, from 6% of agricultural value added in 2000-02 to 3.8% in 2018-20. Despite evidence of high returns, spending on agricultural knowledge and innovation systems was just USD 26 billion per year (1.0% of agricultural value added), while spending on the development and maintenance of infrastructure for the sector amounted to USD 42 billion per year (1.5% of agricultural value added). The foremost ways in which agricultural policies can contribute to improved food systems performance are through sustainable productivity growth and system-wide resilience. The former is necessary to reconcile the objective of ensuring food security (i.e. availability and access at affordable prices) with resource constraints. It also contributes to income generation, albeit while imposing a burden on those producers who do not participate in productivity gains (and which may require flanking policies). The latter will be required to confront new sources of risk caused by a changing climate, unanticipated changes in policy, or the economy-wide effects of shocks external to the agricultural sector, such as the global COVID-19 pandemic. As policy reforms have stalled, progress in achieving sustainable productivity growth has also deteriorated. For the 48 countries for which data are available, only five countries achieved strong sustainable productivity growth (improvements in all environmental indicators and TFP growth relative to the OECD median) between 2007 and 2016, compared with 13 countries between 1997 and 2006. Similarly, the disproportionately low allocation of resources to policies that enhance the sector’s capacity to absorb risks has undermined its capacity to adapt and transform in response to those risks. Trade plays an essential role in maintaining stability and fostering resilience in the global food system. By allowing produce to flow from food surplus areas to food deficit areas, trade helps to absorb the impacts of local and regional supply shocks. This generally results in lower price volatility, reduced uncertainty of supply, and greater integration of global and regional markets. Where production variability is weakly correlated among countries, trade can help to mitigate supply volatility and manage domestic food shortages driven by poor harvests, droughts, floods and other catastrophic events. The stabilising role of trade is only likely to increase in importance, as domestic production shocks become more frequent due to climate change. The continued use of price policies – in the form of both positive and negative market price support – and associated use of border measures undermines this critical aspect of resilience. Phase out price interventions and market distorting producer support. These policies have the most negative overall impact on food security and the environment. They are also an inefficient way of supporting livelihoods, with poor targeting in terms of either who is paying for the policy or who is receiving the benefit. The withdrawal of positive market price support and associated trade protection would nevertheless imply a loss of income by producers that may need to be accompanied by transitional assistance and social safety nets. Conversely, the removal of policies that suppress domestic prices would raise prices, with a potential need for targeted income transfers to low-income households and consumers. Target income support to farm households most in need; where possible shift its role away from agricultural budgets, and towards economy-wide social policies and safety nets. In many countries, income support predominantly benefits large farm households with comparatively high income and wealth. A move to more targeted support would bring gains in efficiency and equity, but require deeper investments in data collection, in particular on the total incomes and assets of agricultural households. Agricultural policy would still have an important role in underwriting those aspects of agricultural risk management that cannot be covered by farmers themselves or by risk markets, and in fostering greater resilience to future shocks. Re-orient public expenditures towards investments in public goods with the potential to deliver sustainable productivity growth and improved sectoral resilience. Specifically, investments in innovation systems should be made central to agricultural support policies. However, innovation – which encompasses not just new technologies, but improved practices and systems – is currently marginal, with just 6% of all budgetary support going to research and innovation directly, 9% to public investments in infrastructure and 2% to biosecurity. These shares could be almost doubled by a redirection of market distorting payments, and raised further still by a reallocation of income support away from farmers whose incomes from farm and off-farm sources would be above average even without support. Public goods can also be generated by individual agricultural producers in the form of ecosystem services and other environmental amenities demanded by societies. Targeted and tailored payments to producers can foster the availability of such goods, and provide additional income opportunities for farm households. Food systems around the world face a formidable triple challenge of providing food security and nutrition to a growing global population, providing livelihoods to those along the food supply chain, and contributing to environmental sustainability. Effective agricultural policies can make an important contribution to each of these goals, but they will not be sufficient. A wider food systems approach means mobilising policies in a wide range of areas that go beyond primary agriculture, for example via targeted policies to encourage healthier dietary choices, broad policies to ensure balanced rural and economic development, and economy-wide plans to curb GHG emissions. It also means exploiting synergies and managing trade-offs between the different dimensions of the triple challenge. A “food systems approach” to addressing these challenges requires that agricultural policymakers take a holistic view of the performance of policies related to multiple objectives, and co-ordinate to avoid incoherent policies. References [28] Alston, J. et al. (2010), A Meta-Analysis of Rates of Return to Agricultural R&D: Ex Pede Herculem?, International Food Policy Research Institute (IFPRI) Research Report 113, https://www.ifpri.org/publication/meta-analysis-rates-return-agricultural-r-d. [51] Anderson, K. and S. Nelgen (2012), “Trade Barrier Volatility and Agricultural Price Stabilization”, World Development, Vol. 40/1, pp. 36-48, https://doi.org/10.1016/j.worlddev.2011.05.018. [24] Anderson, K. and E. Valenzuela (2021), “What impact are subsidies and trade barriers abroad having on Australasian and Brazilian agriculture?”, Australian Journal of Agricultural and Resource Economics, Vol. 65/2, https://doi.org/10.1111/1467-8489.12413. [6] Baragwanath, T. (2021), “Digital opportunities for demand-side policies to improve consumer health and the sustainability of food systems”, OECD Food, Agriculture and Fisheries Papers, No. 148, OECD Publishing, Paris, https://dx.doi.org/10.1787/bec87135-en. [44] Beghin, J. and H. Jensen (2008), “Farm policies and added sugars in US diets”, Food Policy, Vol. 33/6, pp. 480-488, https://doi.org/10.1016/j.foodpol.2008.05.007. [70] Benton, T. et al. (2021), Food system impacts on biodiversity loss: Three levers for food system transformation in support of nature, Chatham House, London, https://www.chathamhouse.org/sites/default/files/2021-02/2021-02-03-food-system-biodiversity-loss-benton-et-al_0.pdf. [81] Blandford, D. and K. Hassapoyannes (2018), “The role of agriculture in global GHG mitigation”, OECD Food, Agriculture and Fisheries Papers, No. 112, OECD Publishing, Paris, http://dx.doi.org/10.1787/da017ae2-en. [40] Bouis, H. (2000), “Improving Human Nutrition through Agriculture: The Role of International Agricultural Research. Conference Summary and Recommendations.”, Food and Nutrition Bulletin, Vol. 21/4, pp. 550-567, https://doi.org/10.1177/156482650002100441. [46] Bowman, M. and D. Zilberman (2013), “Economic Factors Affecting Diversified Farming Systems”, Ecology and Society, Vol. 18/1, https://www.jstor.org/stable/26269286. [16] Brink, L. (2018), “Two indicators, little in common, same name: Market Price Support”, CAP Reform, http://capreform.eu/two-indicators-little-in-common-same-name-market-price-support/ (accessed on 25 March 2019). [48] Brooks, J. (ed.) (2012), Agricultural Policies for Poverty Reduction, OECD Publishing, Paris, https://doi.org/10.1787/9789264112902-en. [23] Brooks, J. and A. Matthews (2015), “Trade Dimensions of Food Security”, OECD Food, Agriculture and Fisheries Papers No. 77, http://dx.doi.org/10.1787/5js65xn790nv-en. [66] Busch, J. and K. Ferretti-Gallon (2017), “What Drives Deforestation and What Stops It? A Meta-Analysis”, Review of Environmental Economics and Policy, Vol. 11/1, pp. 3-23, http://dx.doi.org/10.1093/reep/rew013. [64] Dasgupta, P. (2021), The Economics of Biodiversity: The Dasgupta Review. Abridged Version, HM Treasury, London, https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/957292/Dasgupta_Review_-_Abridged_Version.pdf. [59] de Frahan, B., J. Dong and R. De Blander (2017), “Farm Household Incomes in OECD Member Countries over the Last 30 Years of Public Support”, LIS Working Paper Series 700, https://www.econstor.eu/bitstream/10419/169260/1/700.pdf. [75] DeBoe, G. (2020), “Impacts of agricultural policies on productivity and sustainability performance in agriculture: A literature review”, OECD Food, Agriculture and Fisheries Papers, No. 141, OECD Publishing, Paris, http://dx.doi.org/10.1787/6bc916e7-en. [80] DeBoe, G. et al. (2020), “Reforming Agricultural Policies Will Help to Improve Environmental Performance”, EuroChoices, Vol. 19/1, pp. 30-35, https://doi.org/10.1111/1746-692X.12247. [49] Deuss, A. (2017), “Impact of agricultural export restrictions on prices in importing countries”, OECD Food, Agriculture and Fisheries Papers No. 105, http://dx.doi.org/10.1787/1eeeb292-en. [50] Deuss, A. (2015), “Review of the performance and impacts of recent stockholding policies”, in Issues in Agricultural Trade Policy: Proceedings of the 2014 OECD Global Forum on Agriculture, OECD Publishing, Paris, https://doi.org/10.1787/9789264233911-5-en. [60] Dewbre, J., J. Antón and W. Thompton (2001), “The Transfer Efficiency and Trade Effects of Direct Payments”, American Journal of Agricultural Economics, Vol. 83/5, pp. 1204-1214, https://doi.org/10.1111/0002-9092.00268. [14] Diakosavvas, D. (2002), “Agricultural Policies in China after WTO Accession”, in China in the Global Economy, OECD Publishing, Paris, https://doi.org/10.1787/9789264158894-en. [15] Effland, A. (2011), “Classifying and Measuring Agricultural Support: Identifying Differences Between the WTO and OECD Systems”, Economic Information Bullentin 74, http://www.ers.usda.gov/ (accessed on 19 April 2019). [17] FAO (2020), Hunger and food insecurity, http://www.fao.org/hunger/en/ (accessed on 1 March 2021). [95] FAO (2006), “Food Security”, Policy Brief, Issue 2, http://www.fao.org/fileadmin/templates/faoitaly/documents/pdf/pdf_Food_Security_Cocept_Note.pdf (accessed on 29 April 2021). [68] FAO and UNEP (2020), The State of the World’s Forests 2020. Forests, biodiversity and people., FAO, Rome, https://doi.org/10.4060/ca8642en. [89] FAO, IFAD and WFP (2019), Strengthening resilience for food security and nutrition: A Conceptual Framework for Collaboration and Partnership among the Rome-based Agencies, Food and Agriculture Organization of the United Nations (FAO), the International Fund for Agricultural Development (IFAD) and the World Food Programme (WFP), https://docs.wfp.org/api/documents/WFP-0000062320/download/. [18] FAO, IFAD, UNICEF, WFP and WHO (2020), The State of Food Security and Nutrition in the World 2020. Transforming food systems for affordable healthy diets, FAO, Rome, https://doi.org/10.4060/ca9692en. [9] FEBA (2020), FEBA EU Monitoring - November-December 2020, European Food Banks Federation, https://lp.eurofoodbank.org/wp-content/uploads/2020/12/FEBA_EU_Monitoring_NovDec2020.pdf. [33] Filipski, M. and K. Covarrubias (2012), Distributional Impacts of Commodity Prices in Developing Countries, OECD Publishing, Paris, https://doi.org/10.1787/9789264112902-en. [84] G7 Agriculture Ministers (2017), , G7 Bergamo Agriculture Ministers’ Meeting Communiqué 14-15 October 2017 - Empowering Farmers, Developing Rural Areas and Enhancing Cooperation to Feed the Planet, http://www.g7italy.it/en/documenti-ministeriali. [38] GBD 2019 Risk Factors Collaborators (2020), “Global burden of 87 risk factors in 204 countries and territories, 1990–2019: a systematic analysis for the Global Burden of Disease Study 2019”, Global Health Metrics, Vol. 396/10258, pp. 1223-1249, https://doi.org/10.1016/S0140-6736(20)30752-2. [7] GFN (2020), Strengthening Food Donation Operations During COVID-19: Key issues and Best Practices for Governments Around the Globe, The Global FoodBanking Network, https://www.foodbanking.org/wp-content/uploads/2020/06/Global-Food-Donation-Policy-Atlas-COVID19-Issue-Brief.pdf. [43] Giner, C. and J. Brooks (2019), “Policies for encouraging healthier food choices”, OECD Food, Agriculture and Fisheries Papers No. 137, http://dx.doi.org/10.1787/11a42b51-en. [45] Global Panel on Agriculture and Food Systems for Nutrition (2020), Foresight 2.0: Future Food Systems: For people, our planet, and prosperity, https://foresight.glopan.org/. [25] Greenville, J. et al. (2019), “Influencing GVCs through Agro-Food Policy and Reform”, OECD Food, Agriculture and Fisheries Papers No. 125, http://dx.doi.org/10.1787/9ce888e0-en. [57] Greenville, J., K. Kawasaki and M. Jouanjean (2019), “Employment in Agriculture and Food Trade: Assessing the Role of GVCs”, OECD Food, Agriculture and Fisheries Papers, No. 124, OECD Publishing, Paris, http://dx.doi.org/10.1787/5ed3b181-en. [4] Gruère, G. and J. Brooks (2021), “Viewpoint: Characterising early agricultural and food policy responses to the outbreak of COVID-19”, Food Policy, Vol. 100, p. 102017, http://dx.doi.org/10.1016/j.foodpol.2020.102017. [11] Gruère, G., M. Shigemitsu and S. Crawford (2020), “Agriculture and water policy changes: Stocktaking and alignment with OECD and G20 recommendations”, OECD Food, Agriculture and Fisheries Papers, No. 144, OECD Publishing, Paris, https://dx.doi.org/10.1787/f35e64af-en. [78] Guerrero, S. (2018), “Farmland Birds under Pressure”, EuroChoices, Vol. 17/3, pp. 24-25, https://doi.org/10.1111/1746-692X.12204. [47] Hattersley, L. et al. (2020), “Countering common arguments against taxes on sugary drinks”, Health, Nutrition and Population Knowledge Brief, World Bank, Washington, DC, http://hdl.handle.net/10986/34361. [42] Hazell, P. (2009), “The Asian Green Revolution”, IFPRI Discussion Paper, International Food Policy Research Institute, https://www.ifpri.org/publication/asian-green-revolution. [32] Heisey, P. and K. Fuglie (2018), “Public agricultural R&D in high-income countries: Old and new roles in a new funding environment”, Global Food Security, Vol. 17, pp. 92-102, https://doi.org/10.1016/j.gfs.2018.03.008. [91] Henderson, B. et al. (2021), “Policy strategies and challenges for climate change mitigation in the Agriculture, Forestry and Other Land Use (AFOLU) sector”, OECD Food, Agriculture and Fisheries Papers, No. 149, OECD Publishing, Paris, https://doi.org/10.1787/47b3493b-en. [73] Henderson, B. and J. Lankoski (2020), “Assessing the Environmental Impacts of Agricultural Policies”, Applied Economic Perspectives and Policy, pp. 1-16, https://doi.org/10.1002/aepp.13081. [72] Henderson, B. and J. Lankoski (2019), “Evaluating the environmental impact of agricultural policies”, OECD Food, Agriculture and Fisheries Papers, No. 130, OECD Publishing, Paris, http://dx.doi.org/10.1787/add0f27c-en. [67] Hosonuma, N. et al. (2012), “An assessment of deforestation and forest degradation drivers in developing countries”, Environmental Research Letters, Vol. 7/4, https://doi.org/10.1088/1748-9326/7/4/044009. [2] IMF (2021), Primary Commodity Prices, https://www.imf.org/en/Research/commodity-prices (accessed on 20 April 2021). [90] IPBES (2019), Summary for policymakers of the global assessment report on biodiversity and ecosystem services of the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services, IPBES secretariat, Bonn, Germany, https://www.ipbes.net/sites/default/files/2020-02/ipbes_global_assessment_report_summary_for_policymakers_en.pdf. [63] IPCC (2019), “Summary for Policymakers”, in Climate Change and Land: an IPCC special report on climate change, desertification, land degradation, sustainable land management, food security, and greenhouse gas fluxes in terrestrial ecosystems, https://www.ipcc.ch/srccl/chapter/summary-for-policymakers/. [34] Ivanic, M. and W. Martin (2008), “Implications of Higher Global Food Prices for Poverty in Low-Income Countries”, Agricultural Economics, Vol. 39/s1, pp. 405-416, https://doi.org/10.1111/j.1574-0862.2008.00347.x. [41] Kataki, P. (2002), “Shifts in Cropping System and Its Effect on Human Nutrition: Case Study from India”, Journal of Crop Production, Vol. 6/1-2, pp. 119-144, https://doi.org/10.1300/J144v06n01_08. [30] Kristkova, Z., M. van Dijk and H. van Meijl (2017), “Assessing the Impact of Agricultural R&D Investments on Long-Term Projections of Food Security”, World Agricultural Resources and Food Security (Frontiers of Economics and Globalization), Vol. 17, pp. 1-17, https://doi.org/10.1108/S1574-871520170000017001. [77] Laborde, D. et al. (2021), “Agricultural subsidies and global greenhouse gas emissions”, Nature Communications, Vol. 12/2601, https://doi.org/10.1038/s41467-021-22703-1. [36] Laborde, D. et al. (2020), “COVID-19 risks to global food security”, Science, Vol. 369/6503, pp. 500-502, https://doi.org/10.1126/science.abc4765. [76] Lankoski, J. and A. Thiem (2020), “Linkages between agricultural policies, productivity and environmental sustainability”, Ecological Economics, Vol. 178, https://doi.org/10.1016/j.ecolecon.2020.106809. [82] Lechenet, M. et al. (2017), “Reducing pesticide use while preserving crop productivity and profitability on arable farms”, Nature Plants, Vol. 3, https://doi.org/10.1038/nplants.2017.8. [54] Lowder, S., J. Skoet and T. Raney (2016), “The Number, Size, and Distribution of Farms, Smallholder Farms, and Family Farms Worldwide”, World Development, Vol. 87, pp. 16-29, https://doi.org/10.1016/j.worlddev.2015.10.041. [12] OECD (2021), Agriculture and water policies: characteristics and evolution from 2009 to 2019, OECD country profiles, OECD Publishing, Paris, http://www.oecd.org/agriculture/topics/water-and-agriculture#profiles. [62] OECD (2021), “Digital opportunities for Sanitary and Phytosanitary (SPS) Systems and the trade facilitation effects of SPS Electronic Certification”, OECD Food, Agriculture and Fisheries Papers, No. 152, OECD Publishing, Paris, https://dx.doi.org/10.1787/cbb7d0f6-en. [35] OECD (2021), Making Better Policies for Food Systems, OECD Publishing, Paris, https://doi.org/10.1787/ddfba4de-en. [94] OECD (2021), OECD Economic Outlook, Interim Report March 2021, OECD Publishing, Paris, https://dx.doi.org/10.1787/34bfd999-en. [93] OECD (2021), Policies for the Future of Farming and Food in Norway, OECD Agriculture and Food Policy Reviews, OECD Publishing, Paris, https://doi.org/10.1787/20b14991-en. [3] OECD (2020), Agricultural Policy Monitoring and Evaluation 2020, OECD Publishing, Paris, https://dx.doi.org/10.1787/928181a8-en. [74] OECD (2020), “Exploring the Linkages between Agricultural Policies, Productivity and Environmental Sustainability”, https://one.oecd.org/document/COM/TAD/CA/ENV/EPOC(2019)4/FINAL/en/pdf. [5] OECD (2020), “Measuring distortions in international markets: The agriculture sector”, Agriculture Policy Brief, https://issuu.com/oecd.publishing/docs/measuring_distortions_in_internatio. [96] OECD (2020), “OECD Agro-Food Productivity-Sustainability-Resilience Policy Framework: Revised Framework”, https://one.oecd.org/document/TAD/CA/APM/WP(2019)25/FINAL/en/pdf. [1] OECD (2020), OECD Economic Outlook, Volume 2020 Issue 2, OECD Publishing, Paris, https://dx.doi.org/10.1787/39a88ab1-en. [52] OECD (2020), Strengthening Agricultural Resilience in the Face of Multiple Risks, OECD Publishing, Paris, https://doi.org/10.1787/2250453e-en. [92] OECD (2019), Enhancing Climate Change Mitigation through Agriculture, OECD Publishing, Paris, https://dx.doi.org/10.1787/e9a79226-en. [31] OECD (2019), Innovation, Productivity and Sustainability in Food and Agriculture: Main Findings from Country Reviews and Policy Lessons, OECD Food and Agricultural Reviews, OECD Publishing, Paris, https://doi.org/10.1787/c9c4ec1d-en. [37] OECD (2019), The Heavy Burden of Obesity: The Economics of Prevention, OECD Health Policy Studies, OECD Publishing, Paris, https://doi.org/10.1787/67450d67-en. [21] OECD (2016), Evolving Agricultural Policies and Markets: Implications for Multilateral Trade Reform, OECD Publishing, Paris, http://dx.doi.org/10.1787/9789264264991-en. [13] OECD (2016), OECD’S Producer Support Estimate and Related Indicators of Agricultural Support - Concepts, Calculations, Interpretation and Use (The PSE Manual), https://www.oecd.org/agriculture/topics/agricultural-policy-monitoring-and-evaluation/documents/producer-support-estimates-manual.pdf. [88] OECD (2014), Boosting Resilience through Innovative Risk Governance, OECD Reviews of Risk Management Policies, OECD Publishing, Paris, https://dx.doi.org/10.1787/9789264209114-en. [19] OECD (2013), Global Food Security: Challenges for the Food and Agricultural System, OECD Publishing, Paris, http://dx.doi.org/10.1787/9789264195363-en. [86] OECD (2011), Managing Risk in Agriculture: Policy Assessment and Design, OECD Publishing, Paris, https://dx.doi.org/10.1787/9789264116146-en. [85] OECD (2009), Managing Risk in Agriculture: A Holistic Approach, OECD Publishing, Paris, https://dx.doi.org/10.1787/9789264075313-en. [58] OECD (2003), Farm Household Income: Issues and Policy Responses, OECD Publishing, Paris, https://doi.org/10.1787/9789264099678-en. [27] OECD (2002), Agricultural Policies in OECD Countries: A Positive Reform Agenda, OECD Publishing, Paris, https://doi.org/10.1787/9789264199682-en. [20] OECD/FAO (2020), OECD-FAO Agricultural Outlook 2020-2029, FAO, Rome/OECD Publishing, Paris, https://doi.org/10.1787/1112c23b-en. [29] Piesse, J. and C. Thirtle (2010), “Agricultural R&D, technology and productivity”, Philosophical transactions of the Royal Society B, Vol. 365/1554, pp. 3035-3047, https://doi.org/10.1098/rstb.2010.0140. [22] Pingali, P. (2015), “Agricultural policy and nutrition outcomes – getting beyond the preoccupation with staple grains”, Food Security, Vol. 7, pp. 583–591, https://doi.org/10.1007/s12571-015-0461-x. [39] Pingali, P. (2012), “Green Revolution: Impacts, limits, and the path ahead”, Proceedings of the National Academy of Sciences of the United States of America (PNAS), Vol. 109/31, pp. 12302-12308, https://doi.org/10.1073/pnas.0912953109. [10] Placzek, O. (2021), “Socio-economic and demographic aspects of food security and nutrition”, OECD Food, Agriculture and Fisheries Papers, No. 150, OECD Publishing, Paris, https://dx.doi.org/10.1787/49d7059f-en. [56] Punthakey, J. (2020), “Foreign direct investment and trade in agro-food global value chains”, OECD Food, Agriculture and Fisheries Papers, No. 142, OECD Publishing, Paris, https://dx.doi.org/10.1787/993f0fdc-en. [71] Raven, P. and D. Wagner (2021), “Agricultural intensification and climate change are rapidly decreasing insect biodiversity”, Proceedings of the National Academy of Sciences of the United States of America (PNAS), Vol. 118/2, https://doi.org/10.1073/pnas.2002548117. [61] Ricciardi, V. et al. (2018), “How much of the world’s food do smallholders produce?”, Global Food Security, Vol. 17, pp. 64-72, https://doi.org/10.1016/j.gfs.2018.05.002. [83] Ryan, M. (2019), “Evaluating the economic benefits and costs of antimicrobial use in food-producing animals”, OECD Food, Agriculture and Fisheries Papers, No. 132, OECD Publishing, Paris, http://dx.doi.org/10.1787/f859f644-en. [79] Sud, M. (2020), “Managing the Biodiversity Impacts of Fertiliser and Pesticide Use: Overview and insights from trends and policies across selected OECD countries”, OECD Environment Working Papers, No. 155, OECD Publishing, Paris, https://dx.doi.org/10.1787/63942249-en. [26] Thompson-Lipponen, C. and J. Greenville (2019), “The Evolution of the Treatment of Agriculture in Preferential Trade Agreements”, OECD Food, Agriculture and Fisheries Papers No. 126, http://dx.doi.org/10.1787/751d274f-en. [69] Tilman, D. et al. (2017), “Future threats to biodiversity and pathways to their prevention”, Nature, Vol. 546, pp. 73-81, https://doi.org/10.1038/nature22900. [55] Townsend, R. et al. (2017), Future of food: Shaping the food system to deliver jobs, World Bank Group, Washington, DC, http://documents.worldbank.org/curated/en/406511492528621198/Future-of-food-shaping-the-food-system-to-deliver-jobs. [87] UNISDR (2015), Sendai Framework for Disaster Risk Reduction 2015 - 2030, United Nations Office for Disaster Risk Reduction, https://www.unisdr.org/files/43291_sendaiframeworkfordrren.pdf (accessed on 22 August 2018). [65] United Nations (2021), The United Nations World Water Development Report 2021: Valuing Water, UNESCO, Paris, http://www.unesco.org/reports/wwdr/2021/en. [8] USDA (2021), USDA Announces Continuation of the Farmers to Families Food Box Program, Fifth Round of Food Purchases, U.S. Department of Agriculture, Washington, DC, https://www.usda.gov/media/press-releases/2021/01/04/usda-announces-continuation-farmers-families-food-box-program-fifth#:~:text=(Washington%2C%20D.C.%2C%20January%204,to%20Families%20Food%20Box%20Program (accessed on 4 January). [53] World Bank (2021), “Employment in agriculture (% of total employment) (modeled ILO estimate)”, World Development Indicators, https://data.worldbank.org/indicator/SL.AGR.EMPL.ZS (accessed on 28 March 2021). Producer Support Estimate (PSE): The annual monetary value of gross transfers from consumers and taxpayers to agricultural producers, measured at the farm gate level, arising from policy measures that support agriculture, regardless of their nature, objectives or impacts on farm production or income. It includes market price support, budgetary payments and budget revenue foregone, i.e. gross transfers from consumers and taxpayers to agricultural producers arising from policy measures based on: current output, input use, area planted/animal numbers/receipts/incomes (current, non-current), and non-commodity criteria. PSE categories are defined in Box 1 A.1. Market Price Support (MPS): The annual monetary value of gross transfers from consumers and taxpayers to agricultural producers arising from policy measures that create a gap between domestic market prices and border prices of a specific agricultural commodity, measured at the farm gate level. MPS is available by commodity, and sums of negative and positive components are reported separately where relevant along with the total MPS. Producer Single Commodity Transfers (producer SCT): The annual monetary value of gross transfers from consumers and taxpayers to agricultural producers, measured at the farm gate level, arising from policies linked to the production of a single commodity such that the producer must produce the designated commodity in order to receive the payment. This includes broader policies where transfers are specified on a per-commodity basis. Producer SCT is also available by commodity. Group Commodity Transfers (GCT): The annual monetary value of gross transfers from consumers and taxpayers to agricultural producers, measured at the farm gate level, arising from policies whose payments are made on the basis that one or more of a designated list of commodities is produced, i.e. a producer may produce from a set of allowable commodities and receive a transfer that does not vary with respect to this decision. All Commodity Transfers (ACT): The annual monetary value of gross transfers from consumers and taxpayers to agricultural producers, measured at the farm gate level, arising from policies that place no restrictions on the commodity produced but require the recipient to produce some commodity of their choice. Other Transfers to Producers (OTP): The annual monetary value of gross transfers from consumers and taxpayers to agricultural producers, measured at the farm gate level, arising from policies that do not require any commodity production at all. Consumer Single Commodity Transfers (consumer SCT): The annual monetary value of gross transfers from (to) consumers of agricultural commodities, measured at the farm gate level, arising from policies linked to the production of a single commodity. Consumer SCT is also available by commodity. Consumer Support Estimate (CSE): The annual monetary value of gross transfers from (to) consumers of agricultural commodities, measured at the farm gate level, arising from policy measures that support agriculture, regardless of their nature, objectives or impacts on consumption of farm products. If negative, the CSE measures the burden (implicit tax) on consumers through market price support (higher prices), that more than offsets consumer subsidies that lower prices to consumers. General Services Support Estimate (GSSE): The annual monetary value of gross transfers arising from policy measures that create enabling conditions for the primary agricultural sector through development of private or public services, institutions and infrastructure, regardless of their objectives and impacts on farm production and income, or consumption of farm products. The GSSE includes policies where primary agriculture is the main beneficiary, but does not include any payments to individual producers. GSSE transfers do not directly alter producer receipts or costs or consumption expenditures. GSSE categories are defined below. Total Support Estimate (TSE): The annual monetary value of all gross transfers from taxpayers and consumers arising from policy measures that support agriculture, net of the associated budgetary receipts, regardless of their objectives and impacts on farm production and income, or consumption of farm products. Total Budgetary Support Estimate (TBSE): The annual monetary value of all gross budgetary transfers from taxpayers arising from policy measures that support agriculture, regardless of their objectives and impacts on farm production and income, or consumption of farm products. Percentage PSE (%PSE): PSE transfers as a share of gross farm receipts (including support in the denominator). Percentage SCT (%SCT): Single Commodity Transfers as a share of gross farm receipts for the specific commodity (including support in the denominator). Share of SCT in total PSE (%): Share of Single Commodity Transfers in the total PSE. This indicator is also calculated by commodity. Producer Nominal Protection Coefficient (producer NPC): The ratio between the average price received by producers (at farm gate), including payments per tonne of current output, and the border price (measured at farm gate). The Producer NPC is also available by commodity. Producer Nominal Assistance Coefficient (producer NAC): The ratio between the value of gross farm receipts including support and gross farm receipts (at farm gate) valued at border prices (measured at farm gate). Percentage CSE (%CSE): CSE transfers as a share of consumption expenditure on agricultural commodities (at farm gate prices), net of taxpayer transfers to consumers. The %CSE measures the implicit tax (or subsidy, if CSE is positive) placed on consumers by agricultural price policies. Consumer Nominal Protection Coefficient (consumer NPC): The ratio between the average price paid by consumers (at farm gate) and the border price (measured at farm gate). The Consumer NPC is also available by commodity. Consumer Nominal Assistance Coefficient (consumer NAC): The ratio between the value of consumption expenditure on agricultural commodities (at farm gate) and that valued at border prices. Percentage TSE (%TSE): TSE transfers as a percentage of GDP. Percentage TBSE (%TBSE): TBSE transfers as a percentage of GDP. Percentage GSSE (%GSSE): Share of expenditures on general services in the Total Support Estimate (TSE). Share of potentially most distorting transfers in cumulated gross producer transfers (%): represents the sum of positive MPS, the absolute value of negative MPS, payments based on output and payments based on unconstrained use of variable inputs, relative to the sum of positive MPS, the absolute value of negative MPS, and all budgetary payments to producers. Per cent change in PSE: Per cent change in the nominal value of the PSE expressed in national currency. The per cent change is calculated using the two most recent years in the series. Contribution of MPS to per cent change in PSE: Per cent change in nominal PSE if all variables other than MPS are held constant. Contribution of price gap to per cent change in the PSE: Per cent change in nominal PSE if all variables other than gap between domestic market prices and border prices are held constant. Contribution of quantity produced to per cent change in the PSE: Per cent change in nominal PSE if all variables other than quantity produced are held constant. Contribution of budgetary payments (BP) to per cent change in PSE: Per cent change in nominal PSE if all variables other than BP are held constant. Contribution of BP elements to per cent change in PSE: Per cent change in nominal PSE if all variables other than a given BP element are held constant. BP elements include Payments based on output, Payments based on input use, Payments based on current A/An/R/I, production required, Payments based on non-current A/An/R/I, production required, Payments based on non-current A/An/R/I, production not required, Payments based on non-commodity criteria and Miscellaneous payments. Per cent change in Producer Price: Per cent change in Producer Price (at farm gate) expressed in national currency. The per cent change is calculated using the two most recent years in the series. Per cent change in Border Price: Per cent change in Border Price (at farm gate) expressed in national currency. The per cent change is calculated using the two most recent years in the series. Contribution of Exchange Rate to per cent change in Border Price: Per cent change in the Border Price (at farm gate) expressed in national currency if all variables other than Exchange Rate between national currency and USD are held constant. Contribution of Border Price expressed in USD to per cent change in Border Price: Per cent change in the Border Price (at farm gate) expressed in national currency if all variables other than Border Price (at farm gate) expressed in USD are held constant. Note: The producer price change and the border price change are not calculated when the negative price gap occurs at the commodity level for the current or previous year. • Agricultural knowledge generation: Budgetary expenditure financing research and development (R&D) activities related to agriculture, and associated data dissemination, irrespective of the institution (private or public, ministry, university, research centre or producer groups) where they take place, the nature of research (scientific, institutional, etc.), or its purpose. • Agricultural knowledge transfer: Budgetary expenditure financing agricultural vocational schools and agricultural programmes in high-level education, training and advice to farmers that is generic (e.g. accounting rules, pesticide application), not specific to individual situations, and data collection and information dissemination networks related to agricultural production and marketing. • Agricultural product safety and inspection: Budgetary expenditure financing activities related to agricultural product safety and inspection. This includes only expenditures on inspection of domestically produced commodities at first level of processing and border inspection for exported commodities. • Pest and disease inspection and control: Budgetary expenditure financing pest and disease control of agricultural inputs and outputs (control at primary agriculture level) and public funding of veterinary services (for the farming sector) and phytosanitary services. • Input control: Budgetary expenditure financing the institutions providing control activities and certification of industrial inputs used in agriculture (e.g. machinery, industrial fertilisers, pesticides, etc.) and biological inputs (e.g. seed certification and control). • Hydrological infrastructure: Budgetary expenditure financing public investments into hydrological infrastructure (irrigation and drainage networks). • Storage, marketing and other physical infrastructure: Budgetary expenditure financing investments to off-farm storage and other market infrastructure facilities related to handling and marketing primary agricultural products (silos, harbour facilities – docks, elevators; wholesale markets, futures markets), as well as other physical infrastructure related to agriculture, when agriculture is the main beneficiary. • Institutional infrastructure: Budgetary expenditure financing investments to build and maintain institutional infrastructure related to the farming sector (e.g. land cadastres; machinery user groups, seed and species registries; development of rural finance networks; support to farm organisations, etc.). • Farm restructuring: Budgetary payments related to reform of farm structures financing entry, exit or diversification (outside agriculture) strategies. • Collective schemes for processing and marketing: Budgetary expenditure financing investment in collective, mainly primary, processing, marketing schemes and marketing facilities, designed to improve marketing environment for agriculture. • Promotion of agricultural products: Budgetary expenditure financing assistance to collective promotion of agro-food products (e.g. promotion campaigns, participation on international fairs). • Cost of public stockholding: Budgetary expenditure covering the costs of storage, depreciation and disposal of public storage of agricultural products. • Miscellaneous: Budgetary expenditure financing other general services that cannot be disaggregated and allocated to the above categories, often due to a lack of information. More detailed information on the indicators, their use and limitations is available in OECD’s Producer Support Estimate and Related Indicators of Agricultural Support: Concepts, Calculation, Interpretation and Use (the PSE Manual) available on the OECD public website (http://www.oecd.org/agriculture/topics/agricultural-policy-monitoring-and-evaluation/documents/producer-support-estimates-manual.pdf). Notes ← 1. More recent estimates from OECD (2021[94]) suggest a slightly smaller GDP decline at -3.4% globally. Data provided in this section are based on the regular report from December 2020. ← 2. Publicly supported short-time work schemes allow companies to temporarily reduce the work time of employees by up to 100%, while wage differences are partly or fully subsidised by the government. ← 3. Three main types of impacts were observed on the agriculture and food sector (OECD, 2020[96]). First, there were impacts on agricultural production, due to the unavailability of labour, restrictions on access to intermediate agricultural inputs, and impacts on farmers’ income in affected subsectors that could not sell their output. Second, there have been impacts on consumer demand, with increased food insecurity led by unemployment and income shocks associated with containment measures, reduced demand for high value products, shifts in consumer demand towards retail, over food consumed away from home, and decline in biofuel demand due to transportation restrictions. Third, supply chain disruptions were observed in many countries, due in part to contamination in processing firms, transport and logistic issues, and difficulties in obtaining inputs. ← 4. Some of the early responses, such as the declaration of agriculture and food as being an essential sector that were reported in the 2020 report, have not been repeated in all country chapters for this year’s edition; however, they are also included in the analysis to ensure a full coverage of measures. ← 5. This categorisation can be further separated into 20 sub categories of measures (OECD, 2020[3]). ← 6. This grouping was also used in Gruère and Brooks (2021[4]) to characterise early policy responses to the COVID--19. Efforts were made to ensure a consistent and unique attribution of a group to each of the policy measures, although the attribution of some measures to a specific group could be subjective. ← 7. While targeted food assistance for low income households can also be considered urgent, the implemented measures essentially aim to cushion consumers from the economic impacts rather than cope with the urgency of the crisis for the delivery of agriculture and food products. ← 8. The majority of measures in this group could be considered market distorting and potentially environmentally harmful if maintained for long enough to affect producers’ decisions. ← 9. For a discussion on agriculture and water management progress, see Box 1.4. ← 10. The OECD also collaborates with other international organisations (FAO, IDB, the World Bank and IFPRI) in the Consortium for Measuring the Policy Environment for Agriculture (www.ag-incentives.org), which provides estimates for countries not covered by the OECD. ← 11. Food availability refers to the supply of sufficient quantities of food (either through domestic production or imports), while Access to food refers to the ability of individuals to access adequate resources to acquire appropriate foods for a nutritious diet (FAO, 2006[95]). ← 12. In the case of New Zealand, market price support for eggs and poultry arises as an unintended impact of science-based SPS measures whose sole purpose is to keep out diseases. ← 13. Possibly due to underreporting of other components of the GSSE (e.g. infrastructure and inspection and control). ← 14. The composition of a “healthy diet” varies according to individual characteristics, cultural contexts, local availability of foods and dietary customs. Healthy diets reflect global guidelines and ensure that a person’s needs for macronutrients (proteins, fats and carbohydrates including dietary fibres) and essential micronutrients (vitamins and minerals) are met (FAO, IFAD, UNICEF, WFP and WHO, 2020[18]). ← 15. Dietary risks as defined by the Global Burden of Diseases, Injuries, and Risk Factors Study include diets “low in whole grains, fruit, fibre, legumes, nuts and seeds, omega-3 fatty acids, Polyunsaturated fatty acids (PUFAs), vegetables, milk, and calcium”; and diets “high in sodium, trans fats, red or processed meat, and sugar-sweetened beverages” (GBD 2019 Risk Factors Collaborators, 2020[38]). ← 16. Enteric fermentation is a digestive process that occurs in cattle, sheep, goats and other ruminant livestock, whereby methane (CH4) emissions are produced in the rumen through a process of microbial fermentation. Metadata, Legal and Rights This document, as well as any data and map included herein, are without prejudice to the status of or sovereignty over any territory, to the delimitation of international frontiers and boundaries and to the name of any territory, city or area. Extracts from publications may be subject to additional disclaimers, which are set out in the complete version of the publication, available at the link provided. © OECD 2021 The use of this work, whether digital or print, is governed by the Terms and Conditions to be found at http://www.oecd.org/termsandconditions.	2021-10-28 04:42:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22445493936538696, "perplexity": 7264.806264898513}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588257.34/warc/CC-MAIN-20211028034828-20211028064828-00538.warc.gz"}
http://www.researchgate.net/publication/45869665_Aligned_dipolar_Bose-Einstein_condensate_in_a_double-well_potential_From_cigar-shaped_to_pancake-shaped	Article # Aligned dipolar Bose-Einstein condensate in a double-well potential: From cigar-shaped to pancake-shaped • ##### D. Blume Physical Review A (Impact Factor: 3.04). 08/2009; DOI: 10.1103/PhysRevA.80.053622 Source: arXiv ABSTRACT We consider a Bose-Einstein condensate (BEC), which is characterized by long-range and anisotropic dipole-dipole interactions and vanishing s-wave scattering length, in a double-well potential. The properties of this system are investigated as functions of the height of the barrier that splits the harmonic trap into two halves, the number of particles (or dipole-dipole strength) and the aspect ratio $\lambda$, which is defined as the ratio between the axial and longitudinal trapping frequencies $\omega_z$ and $\omega_{\rho}$. The phase diagram is determined by analyzing the stationary mean-field solutions. Three distinct regions are found: a region where the energetically lowest lying stationary solution is symmetric, a region where the energetically lowest lying stationary solution is located asymmetrically in one of the wells, and a region where the system is mechanically unstable. For sufficiently large aspect ratio $\lambda$ and sufficiently high barrier height, the system consists of two connected quasi-two-dimensional sheets with density profiles whose maxima are located either at $\rho=0$ or away from $\rho=0$. The stability of the stationary solutions is investigated by analyzing the Bogoliubov de Gennes excitation spectrum and the dynamical response to small perturbations. These studies reveal unique oscillation frequencies and distinct collapse mechanisms. The results derived within the mean-field framework are complemented by an analysis based on a two-mode model. Comment: 21 pages, 16 figures 1 Bookmark · 93 Views • ##### Article: Symmetry breaking and a dynamical property of a dipolar Bose–Einstein condensate in a double-well potential [Hide abstract] ABSTRACT: We investigate the properties of a three-dimensional (3D) dipolar Bose–Einstein condensate (BEC) in a double-well potential (DWP). The symmetry breaking and self-trapping (SBST) phenomena that the original symmetric ground state is replaced by a new asymmetric one and localized in one of the wells are demonstrated for Dy164 atoms in the 3D DWP by means of numerical solutions of the Gross–Pitaevskii (GP) equation. The results show that the SBST properties are affected dramatically by the magnetization direction for purely dipolar BEC. The SBST under various scattering lengths are also studied. In addition, the dynamical picture of the SBST induced by a gradual transformation of the single-well potential into a double-well is also illustrated. Physics Letters A 01/2014; 378(s 1–2):48–52. · 1.77 Impact Factor • Source ##### Article: Roton excitations in a trapped dipolar Bose-Einstein condensate [Hide abstract] ABSTRACT: We consider the quasi-particle excitations of a trapped dipolar Bose-Einstein condensate. By mapping these excitations onto radial and angular momentum we show that the roton modes are clearly revealed as discrete fingers in parameter space, whereas the other modes form a smooth surface. We examine the properties of the roton modes and characterize how they change with the dipole interaction strength. We demonstrate how the application of a perturbing potential can be used to engineer angular rotons, i.e. allowing us to controllably select modes of non-zero angular momentum to become the lowest energy rotons. Physical Review A 08/2013; 88(4). · 3.04 Impact Factor • Source ##### Article: Dipolar Bose-Einstein condensates in triple-well potentials [Hide abstract] ABSTRACT: Dipolar Bose-Einstein condensates in triple-well potentials are well-suited model systems for periodic optical potentials with important contributions of the non-local and anisotropic dipole-dipole interaction, which show a variety of effects such as self-organisation and formation of patterns. We address here a macroscopic sample of dipolar bosons in the mean-field limit. This work is based on the Gross-Pitaevskii description of dipolar condensates in triple-well potentials by Peter et al 2012 J. Phys. B: At. Mol. Opt. Phys. 45 225302. Our analysis goes beyond the calculation of ground states presented there and clarifies the role of excited and metastable states in such systems. In particular, we find the formation of phases originating from the interplay of several states with distinct stability properties. As some of the phases are formed by metastable states special attention is paid to the characteristics of phase transitions in real-time and the dynamical stabilisation of the condensate. Journal of Physics B Atomic Molecular and Optical Physics 10/2013; 46(23):235301. · 2.03 Impact Factor	2014-08-20 07:36:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6778754591941833, "perplexity": 1101.5051260557084}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500800767.23/warc/CC-MAIN-20140820021320-00024-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.cuemath.com/numbers/shapes/	# Shapes Go back to 'Pre-number Math' 1 Introduction to Shapes 2 Shapes around us 3 What are Shapes? 4 Different Types of Shapes 5 List of Geometric Shapes 6 Two-Dimensional Geometric Shapes 7 Important Notes on Shapes 8 Solved Examples on Shapes 9 Practice Questions on Shapes 10 Challenging Questions on Shapes 11 Maths Olympiad Sample Papers 12 Frequently Asked Questions (FAQs) ## Introduction to Shapes Shapes in math is studied as a part of Geometry. Shapes are all around us. Here is a short video lesson on how shapes are taught the Cuemath way. ## Shapes around us Have you noticed the shape of a pizza? It is round in shape. If we cut out a slice from the pizza, the slice is triangular in shape. The handkerchief you carry is square in shape. ## What are Shapes? Shapes can be defined as the boundary or outline of an object. It is the surface we see. It does not depend on the size or colour. There are many types of shapes. Everything we see has a shape. Even Earth has a shape. Have you seen the model of the Earth called the Globe? What is the shape of this object? It looks like a sphere. Now, look at this pencil box. It looks like a cuboid. ## Different Types of Shapes Shapes in math can be defined in many ways based on their properties. Shapes define the boundary of an object. Shapes can be of the following types: 1. open or closed 2. two dimensional $$(2\text{D})$$ or three dimensional $$(3\text{D})$$ 3. curved or straight Closed shapes can be traced without any break. It starts and ends in the same place. Open Shapes are not continuous. Let's look at these examples below: • Closed Shapes • Open Shapes We learn about these open and closed shapes in Maths- Shapes Grade 3. A shape can also be two dimensional ($$2\text{D}$$) or three dimensional ($$3\text{D}$$). A square is a $$2\text{D}$$ shape whereas a cube is a $$3\text{D}$$ shape. $$3\text{D}$$ shapes have length, width and height. You can learn more about it here. $$2\text{D}$$ shapes, as the name suggests, have only two of these measurements. The common $$2\text{D}$$ and $$3\text{D}$$ shapes are listed here in Table - 1 and Table - 2. We learn about these $$2\text{D}$$ and $$3\text{D}$$ shapes in Maths- Shapes Grade 4. ## List of Geometric Shapes We know that shapes are made of straight lines or curved lines and they can be open or closed. Lines are nothing but a collection of points. Many points put together form a line. If it is straight, it is called a straight line and if it is curved, it is called a curved line. Geometric shapes are closed shapes which are created by joining lines together. Look at this shape below; It is made of four straight lines. Can you think of another quadrilateral shape? Next, look at this shape here: It has no straight line, or corner, or side to it. It is a closed shape made of curved lines. This shape is called an ellipse. Can you think of another shape with no edges? This is a three-dimensional shape. It has length, breadth and height. This shape is called a cube. There are several other $$3\text{D}$$ shapes. Few are listed here in the table: Table - 1 $$3\text{D}$$ Shapes Images Cuboid Cone Pyramid Cylinder Look around you and identify more $$3\text{D}$$ shapes. ## Two-Dimensional Geometric Shapes Have you seen a sheet of paper? It consists of length and breadth. A two-dimensional shape does not have any depth. The common $$2\text{D}$$ shapes we see are listed below: Table - 2 $$2\text{D}$$ Shapes Number of sides Image Triangle $$3$$ Square $$4$$ Rhombus $$4$$ Kite $$4$$ Rectangle $$4$$ Trapezium $$4$$ Pentagon $$5$$ Hexagon $$6$$ Heptagon $$7$$ Octagon $$8$$ Decagon $$10$$ Dodecagon $$12$$ Hexadecagon $$16$$ Circle $$0$$ Can you draw a few more $$2\text{D}$$ shapes? Important Notes 2. Polygons are closed shapes made of straight lines. They are named after the number of sides they have. ## Solved Examples Example 1 Is the given figure a two-dimensional or a three-dimensional shape? How many sides does it have? Solution: The football looks like a sphere in shape and has $$0$$ sides The figure is a three-dimensional shape Example 2 What $$3\text{D}$$ shape does the ice-cream resemble? Solution: It looks like a cone Example 3 How many sides are there in the given shape ? Is it a closed or open shape? Solution: Count the number of straight lines in the star. There are $$10$$ straight lines. We can trace the outline of this star without any break. Hence, it is a closed shape. It is a closed shape with $$10$$ sides Example 4 What shape is the given figure? How many sides does it have? Solution: The given shape has $$4$$ sides. It is a Quadrilateral It is a Quadrilateral with $$4$$ sides ## Practice Questions Challenging Questions 1. Is the given shape a Quadrilateral? 2. What is the name of the closed shape which has 100 sides? IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective. ## 1. What are the types of shapes? There are many different types of shapes. They are grouped together based on their properties. A shape can be: • closed or open • two dimensional or three dimensional • regular or irregular The properties of shapes define what type they are. ## 2. What are the 16 basic shapes? The 16 basic shapes are : Triangle, Square, Rectangle, Kite, Rhombus, Trapezium, Pentagon, Hexagon, Nanogen, Decagon, Dodecagon, Hexadecogon, Star, Oval, Ellipse and Circle. ## 3. What are the 4 basic shapes? There are many types of shapes but we come across 4 basic shapes very often. The 4 basic shapes are: circle, square, triangle, and rectangle. ## 4. What are the 5 basic shapes? The five basic shapes which we come across often are: circle, rectangle, square, triangle, and oval. More Important Topics Numbers Algebra Geometry Measurement Money Data Trigonometry Calculus	2020-09-21 10:38:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3620073199272156, "perplexity": 1761.6092726471018}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400201601.26/warc/CC-MAIN-20200921081428-20200921111428-00272.warc.gz"}
https://www.gamedev.net/forums/topic/388866-thread-synchronisation-problems/	This topic is 4443 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts Platform: WindowsXP Development Program: Visual Studio .NET 2003 Application type: Win32 Hello all, I got a class who got his own thread running, to do some on-goinging processing (keyboard and mouse). At some point when I want exit my program, the thread of course has to exit too. I'm trying to do this with the help of mutexes. First the baseclass looks the following. typedef enum { CTL_KEYBOARD, CTL_MOUSE, }INPUT_CONTROL, PINPUT_CONTROL; class __declspec(dllexport) CObjectInput { public: CObjectInput(HWND hWnd, INPUT_CONTROL Control); ~CObjectInput(); LPDIRECTINPUT8 GetObject(); LPDIRECTINPUTDEVICE8 GetDevice(); void Acquire(); void Unacquire(); virtual void Process() = 0; private: HANDLE Occupied; LPDIRECTINPUT8 Object; LPDIRECTINPUTDEVICE8 Device; }; CObjectInput::CObjectInput(HWND hWnd, INPUT_CONTROL Control) : Object(NULL), Device(NULL) { Occupied = CreateMutex(NULL, false, "Occupied"); // Mutex starting in signaled state, according to documentation / Lots of uninteresting inits here / } CObjectInput::~CObjectInput() { WaitForSingleObject(Occupied, INFINITE); CloseHandle(Occupied); if(Object) { if(Device) { Device->Release(); } Object->Release(); } } { CObjectInput Myself = (CObjectInput*)Parameters; while(1) { switch(WaitForSingleObject(Myself->Occupied, INFINITE)) { case WAIT_OBJECT_0: Myself->Process(); // Function that need some resources in the baseclass ReleaseMutex(Myself->Occupied); break; case WAIT_FAILED: case WAIT_ABANDONED: break; default: break; } } return 0; } The constructor and destructor are always called in the process execution, so no different thread is executing them. When exiting my program, I sometimes get the following Runtime - error (seen in the image below). Afbreken = Abort, Opnieuw = Try Again, Negeren = Ignore It is being triggered by calling the Myself->Process() function, in the thread execution. That it suddenly becomes an pure virtual function is caused by the class destruction. I am trying to solve this problem by using a mutex. But is not working correctly. I hoped it would work like the following (this is the current situation): [User pressed 'ESC'] Baseclass Waits for mutex Run Threadfunction Baseclass retrieves mutex Threadfunction waits for mutex Baseclass destroys and closes Threadfunction exits, waiting for mutex mutex resulted into abandoned Sometimes this works and sometimes it doesn't. I cannot explain why this happens. According to documentation, only one thread can own the mutex. Both threads are waiting forever (INFINITE milliseconds). See Remarks for more info on Mutexes and WaitForSingleObject Does anyone see what I am doing wrong here? If you need more info, don't hasitate to ask. Best regards, Xeile [Edited by - Xeile on April 22, 2006 8:58:15 AM] ##### Share on other sites In C++ an object is considered to be destroyed as soon as execution enters the destructor. Objects in C++ are also destroyed outside in, and so by the time the base class destructor is called the derived class destructor will already have been called. This means by the time the base class destructor is called your vtable is already wrong. Now if the objects derived class destructor is called in between the ThreadProc thread aquiring the mutex and calling the virtual function, then you will get the error you describe. The locks do not protect that from happening. ##### Share on other sites Aha, that makes sence. Thank you, I can continue now :D Regards, Xeile ##### Share on other sites It's a really good idea to let your threads terminate properly, and wait on each other (but don't use mutexes for this!). Use a proper platform API mechanism (WaitThread) to ensure a thread has terminated before thinking it has. I usually trigger thread termination by setting a flag in an object associated with the thread. The thread itself sees this and exits it's ThreadMain, allowing the thread to die gracefully. This is a neat way of doing this without resorting to signals / events and cleaning up undue mess from premature terminations. • 18 • 11 • 16 • 9 • 50 • ### Forum Statistics • Total Topics 631396 • Total Posts 2999783 ×	2018-06-21 14:54:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1720782369375229, "perplexity": 4812.3834065632755}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864172.45/warc/CC-MAIN-20180621133636-20180621153636-00065.warc.gz"}
http://luizsol.com/i-am-a-bad-programmer/	# I've realized that I'm a bad programmer 2019-03-17 TL;DR: Until recently I’ve always believed that Object Oriented Programing is the best and professional way of solving most problems, but I’m probably wrong. From now on I’ll change this behavior by first thinking if OOP is the right tool for the problem I’m facing. It’s safe to say that I’ve been a programmer since 2012. It started by learning some markup here and there (HTML, LaTeX), then I tried a bit of R. It was only when I dropped out of my Economics undergrad program and (briefly) enrolled in the Computer Sciences (CS) program that I started to do some real programming. As it was with many people my age, I started learning the basic CS concepts in C (I still have nightmares where I’m debugging my unreadable code in Code::Blocks using a bunch of printfs). After a while I started learning Java, and then Python, R (for real this time), C++, Assembly, MatLab and many others. With time I started noticing that the code I was producing was getting more complex. A natural consequence of more complex projects, I thought to myself. But recently, listening to the opinion of other programmers made me realize that maybe the cause of my ever increasing code complexity is that all this time I’ve been believing in a lie. ## The OOP indoctrination machine The first Programming Language (PL) that I learned with support for real Object-Oriented Programming (OOP) was Java. I still remember the example used by the teacher in the class to explain OOP concepts: cars. He talked about how car is just a concept (class), and that you can have many instances of this concept such as an specific chevy impala (object), and how you can interact with this car (methods), you can change it’s properties (attributes) and you can interact with it’s console and pedals to change it’s state (setters, getters and interfaces), and the list goes on and on. The fact is that almost every place that teaches programming preaches that OOP is the most professional, efficient and safe way to do it, and most people will go through their life drinking the OOP Kool-Aid. Once I learned to program in this mindset I was hooked. The ideia of mapping real world concepts to code and encapsulating everything seems like a good idea at first, specially for those who don’t have a lot of programming experience. But what they never tell you while you’re learning this paradigm is how much complexity and fragility you’re adding to your code by doing it. ## The project I’ve started working as an intern at Investment One Partners and my first task there was to finish implementing a system in which the analysts could share and visualize each other’s analysis and market data. It’s first version was already halfway finished by my boss using R and Shiny. Given his short experience with programming I’d say that my boss is a very good programmer and has a good knack for systems development (no, I’m not writing this just to publicly kissing his ass), so when I took charge of the project it wasn’t too messy. The first thing I did was to modularize and organize the code, as to improve the development process. Then I went on to implement the missing screens and vois là: the first “production” version was finished in less than 3 months. But even though the project was working as intended some things bothered me: 1. It’s performance was abysmal 2. Almost 40% of the code was just functions that executed SQL Procs and Queries. 3. There wasn’t any access control 4. No task support: all asynchronous tasks were being executed by a polling script running in the same machine 5. Expanding the system’s functionalities was becoming exponentially difficult 6. It was implemented in R, for god’s sake As the OOP minion that I am I thought to myself: All problems will go away if I rewrite the system using a REAL web framework, implemented in a REAL programming language that uses REAL OOP, so I made the following suggestion to my clients (the analysts): give me 4 months to rewrite the system; I’ll spend 1 month learning Django and React and the other 3 rewriting the system, this way I’ll solve all the system’s problems. They agreed, and this was almost 1 year ago. Even controlling for the planning fallacy, I grossly underestimated how long it would take to finish the rewrite. By the month 7 of the rewrite I was completely convinced that doing this was a big mistake, but I didn’t understand why exactly. Was I a crappy programmer? Was the project just too complex? Was it normal for a project like this to take this amount of time and I just underestimated the time it would take to be finished? Since then I’ve been beating myself over my (apparent) absurd lack of programming and planning skills. ## The proverbial straw Then, in one recent day, two things happened: 1. My boss was able to implement a feature using Node.js and 20 lines of code that I thought that would take me at least 200 (he’d never programmed in JavaScript before) 2. I watched Object-Oriented Programming is Bad1 and What Programming is Never About (Informal Lecture) Watching both videos and witnessing my boss kick my ass productivity-wise made one thing clear to me: OOP nudges me to overengineer software. I won’t go over all the points made in both videos here, but as the Object-Oriented Programming is Bad video argues and I’ve experienced in my professional life, the OOP’s encapsulation promise simply doesn’t work. And by trying to abstract and encapsulate complexity in the preached way I end up creating a huge number of abstraction classes, which end up just creating complexity of their own. Looking at all the projects in which I worked on I get this strong impression that every time I used an OOPL to solve a problem I ended up spending a lot of mental energy trying to fit the problem into the OO paradigm, when in reality I should have been questioning myself whether using the OO paradigm was the right tool for the problem in the first place. This conclusion gets even clearer when I see that every time I was forced to tackle a problem using a non OOPL (such as MatLab and C) I ended up solving it in a much faster and simpler manner than I usually do. Now I can see that all this time I was the hammer that interpret every problem as a nail, and seeing the What Programming is Never About (Informal Lecture) only confirms my suspicions. This video made me remember that programming is never about the code, it’s beauty or level of abstraction; programming is about moving data around to solve a problem. It’s not the case that such things as code organization and readability doesn’t matter, but they should never be the programmer’s first concerns, and I guess that I forgot this somewhere while I was diving deep into Design Patterns, code style guides and PEP’s. ## What now? Knowing these things about myself is both a blessing and a curse. It opens my eyes to other approaches (functional, pure imperative programming, logical etc) and stops me from thinking that the only “enterprise level” solutions are OOP. At the same time it will make finishing my company’s project a much harder endeavour since now I’ll be forced to see and work with my past mistakes on a daily basis (rerewriting is NOT an option). I’m not saying that I’ll avoid OOPL like the plague (and even if I wanted it would be impossible). I’m only saying that, from now on, I’ll always make sure to first take some time to think if the OO paradigm is the right tool for the job at hand. I know that my road ahead won’t be a walk in the park: by leaving OOPL as my only go-to tool I’m trading a set of problems for another one, but I have to say that it feels good to take the blindfold off. The sad part is that just as I thought that I was becoming a really good programmer because I was getting really good at Python I had a wake up call and now that my horizons expanded I find myself as a rookie once again. At least climbing this ladder is kind of fun. As for my company’s project, I guess that I’ll just have to keep on grinding to the V1 finish line. After that maybe I’ll implement it’s new features using a microservices architecture if its upsides are greater that its downsides. We’ll see. 1. I strongly suggest watching at least the first video to whomever programming is a relevant part of his life.	2020-05-30 15:27:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3175630271434784, "perplexity": 1135.638828509304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347409337.38/warc/CC-MAIN-20200530133926-20200530163926-00159.warc.gz"}
http://www.universalpantograph.com/aggregator/sources/37	# Lambda the Ultimate Updated: 5 days 7 hours ago ## Seemingly impossible programs Wed, 10/22/2014 - 09:57 ### Categories: Engineering In case this one went under the radar, at POPL'12, Martín Escardó gave a tutorial on seemingly impossible functional programs: Programming language semantics is typically applied to prove compiler correctness and allow (manual or automatic) program verification. Certain kinds of semantics can also be applied to discover programs that one wouldn't have otherwise thought of. This is the case, in particular, for semantics that incorporate topological ingredients (limits, continuity, openness, compactness). For example, it turns out that some function types (X -> Y) with X infinite (but compact) do have decidable equality, contradicting perhaps popular belief, but certainly not (higher-type) computability theory. More generally, one can often check infinitely many cases in finite time. I will show you such programs, run them fast in surprising instances, and introduce the theory behind their derivation and working. In particular, I will study a single (very high type) program that (i) optimally plays sequential games of unbounded length, (ii) implements the Tychonoff Theorem from topology (and builds finite-time search functions for infinite sets), (iii) realizes the double-negation shift from proof theory (and allows us to extract programs from classical proofs that use the axiom of countable choice). There will be several examples in the languages Haskell and Agda. A shorter version (coded in Haskell) appears in Andrej Bauer's blog. ## EATCS Award 2014: Gordon Plotkin Sun, 10/12/2014 - 22:54 ### Categories: Engineering Gordon Plotkin is renowned for his groundbreaking contributions to programming language semantics, which have helped to shape the landscape of theoretical computer science, and which have im-pacted upon the design of programming languages and their verification technologies. The in-fluence of his pioneering work on logical frameworks pervades modern proof technologies. In addition, he has made outstanding contributions in machine learning, automated theorem prov-ing, and computer-assisted reasoning. He is still active in research at the topmost level, with his current activities placing him at the forefront of fields as diverse as programming semantics, applied logic, and systems biology. Well deserved, of course. Congrats! ## CFP: Off-the-Beaten-Track (OBT) workshop at POPL 2015 Sun, 10/05/2014 - 15:59 ### Categories: Engineering Announcing the 2015 edition of the OBT workshop, to be co-located with POPL 2015, in Mumbai, India. Two-page paper submissions are due November 7, 2014. From the web page (http://www.cs.rice.edu/~sc40/obt15/): Programming language researchers have the principles, tools, algorithms and abstractions to solve all kinds of problems, in all areas of computer science. However, identifying and evaluating new problems, particularly those that lie outside the typical core PL problems we all know and love, can be a significant challenge. This workshop's goal is to identify and discuss problems that do not often show up in our top conferences, but where programming language research can make a substantial impact. We hope fora like this will increase the diversity of problems that are studied by PL researchers and thus increase our community's impact on the world. While many workshops associated with POPL have become more like mini-conferences themselves, this is an anti-goal for OBT. The workshop will be informal and structured to encourage discussion. We are at least as interested in problems as in solutions. ## Domain settings Sat, 10/04/2014 - 11:24 ### Categories: Engineering I am about to make some changes to the name server definitions. Since changes take time to propagate, you may have trouble reaching the site for awhile. If this happens, try using the .com domain instead of the preferred .org domain. ## sml-family.org Tue, 09/30/2014 - 19:27 ### Categories: Engineering In his blog, Bob Harper, in joint effort with Dave MacQueen and Lars Bergstrom, announces the launch of sml-family.org: The Standard ML Family project provides a home for online versions of various formal definitions of Standard ML, including the "Definition of Standard ML, Revised" (Standard ML 97). The site also supports coordination between different implementations of the Standard ML (SML) programming language by maintaining common resources such as the documentation for the Standard ML Basis Library and standard test suites. The goal is to increase compatibility and resource sharing between Standard ML implementations. The site includes a history section devoted to the history of ML, and of Standard ML in particular. This section will contain a collection of original source documents relating to the design of the language. ## Inferring algebraic effects Sat, 09/27/2014 - 23:16 ### Categories: Engineering We present a complete polymorphic effect inference algorithm for an ML-style language with handlers of not only exceptions, but of any other algebraic effect such as input & output, mutable references and many others. Our main aim is to offer the programmer a useful insight into the effectful behaviour of programs. Handlers help here by cutting down possible effects and the resulting lengthy output that often plagues precise effect systems. Additionally, we present a set of methods that further simplify the displayed types, some even by deliberately hiding inferred information from the programmer. Pretnar and Bauer's Eff has made previous appearances here on LtU. Apart from the new fangled polymorphic effect system, this paper also contains an Eff tutorial. ## LtU's new server Tue, 09/23/2014 - 07:26 ### Categories: Engineering Lambda the Ultimate is now running on a new, faster, more reliable server. The old one is now, uh... pining for the fjords. This should resolve the increasingly frequent outages we've seen recently. Because the old server had started failing, we didn't have time to do as much quality control on the migration as we would have liked. If anyone notices any issues with the site, please comment in this thread. Currently known issues: • Non-Latin UTF-8 characters apparently didn't survive the database migration correctly. This is a particular issue if you have a username containing non-Latin characters - you may not be able to log in currently. • It's possible that some comments posted later on Monday don't appear on the new site. • New user signup emails are not yet working. • Due to DNS propagation, not everyone will see the new site immediately. The above issues should be resolved sometime on Tuesday. ## Breaking the Complexity Barrier of Pure Functional Programs with Impure Data Structures Mon, 09/22/2014 - 14:10 ### Categories: Engineering Breaking the Complexity Barrier of Pure Functional Programs with Impure Data Structures by Pieter Wuille and Tom Schrijvers: Pure functional programming language offer many advantages over impure languages. Unfortunately, the absence of destructive update, imposes a complexity barrier. In imperative languages, there are algorithms and data structures with better complexity. We present our project for combining existing program transformation techniques to transform inefficient pure data structures into impure ones with better complexity. As a consequence, the programmer is not exposed to the impurity and retains the advantages of purity. This paper is along the same lines a question I asked a couple of years ago. The idea here is to allow programming using immutable interfaces, and then automatically transform it into a more efficient mutable equivalent. ## Inside the Wolfram Language Sun, 09/21/2014 - 02:20 ### Categories: Engineering Video of Stephen Wolfram showing off the Wolfram Language and sharing his perspective on the design of the language at Strange Loop conference. ## What's in store for the most widely used language by discerning hackers? Wed, 09/17/2014 - 05:14 ### Categories: Engineering Or, in other words, what's the future of Emacs Lisp (and unavoidable HN discussion). The original message contains some interesting tidbits. I am not sure how the discussion on emacs-devel will develop. But speculating about things such as Guile elisp is, of course, our bailiwick. ## An operational and axiomatic semantics for non-determinism and sequence points in C Sun, 09/14/2014 - 09:36 ### Categories: Engineering In a recent LtU discussion, naasking comments that "I always thought languages that don't specify evaluation order should classify possibly effectful expressions that assume an evaluation order to be errors". Recent work on the C language has provided reasonable formal tools to reason about evaluation order for C, which has very complex evaluation-order rules. The C11 standard of the C programming language does not specify the execution order of expressions. Besides, to make more effective optimizations possible (e.g. delaying of side-effects and interleav- ing), it gives compilers in certain cases the freedom to use even more behaviors than just those of all execution orders. Widely used C compilers actually exploit this freedom given by the C standard for optimizations, so it should be taken seriously in formal verification. This paper presents an operational and ax- iomatic semantics (based on separation logic) for non-determinism and sequence points in C. We prove soundness of our axiomatic se- mantics with respect to our operational semantics. This proof has been fully formalized using the Coq proof assistant. One aspect of this work that I find particularly interesting is that it provides a program (separation) logic: there is a set of inference rules for a judgment of the form $$\Delta; J; R \vdash \{P\} s \{Q\}$$, where $$s$$ is a C statement and $$P, Q$$ are logical pre,post-conditions such that if it holds, then the statement $$s$$ has no undefined behavior related to expression evaluation order. This opens the door to practical verification that existing C program are safe in a very strong way (this is all validated in the Coq theorem prover). ## Luca Cardelli Festschrift Fri, 09/12/2014 - 10:10 ### Categories: Engineering Earlier this week Microsoft Research Cambridge organised a Festschrift for Luca Cardelli. The preface from the book: Luca Cardelli has made exceptional contributions to the world of programming languages and beyond. Throughout his career, he has re-invented himself every decade or so, while continuing to make true innovations. His achievements span many areas: software; language design, including experimental languages; programming language foundations; and the interaction of programming languages and biology. These achievements form the basis of his lasting scientific leadership and his wide impact. ... Luca is always asking "what is new", and is always looking to the future. Therefore, we have asked authors to produce short pieces that would indicate where they are today and where they are going. Some of the resulting pieces are short scientific papers, or abridged versions of longer papers; others are less technical, with thoughts on the past and ideas for the future. We hope that they will all interest Luca. Hopefully the videos will be posted soon. ## Re-thinking Prolog Thu, 09/11/2014 - 14:31 ### Categories: Engineering A recent paper by Oleg Kiselyov and Yukiyoshi Kameyama at the university of Tsukuba discusses weaknesses and areas for improvement to Prolog. Quite many computations and models are mostly deterministic. Implementing them in Prolog with any acceptable performance requires the extensive use of problematic features such as cut. Purity is also compromised when interfacing with mainstream language libraries, which are deterministic and cannot run backwards. Divergence is the constant threat, forcing the Prolog programmers to forsake the declarative specification and program directly against the search strategy. All in all, Classical Prolog is the exquisite square peg in the world with mostly round holes The strong points of Prolog can be brought into an ordinary functional programming language. Using OCaml as a representative, we implement lazy guessing as a library, with which we reproduce classical Prolog examples. Furthermore, we demonstrate parser combinators that use committed choice (maximal munch) and can still be run forwards and backwards. They cannot be written in Classical Prolog. Logic variables, unification, and its WAM compilation strategy naturally emerge as a "mere optimization" of the Herbrand universe enumeration. The paper mentions the strength of the approach used by miniKanren (which embeds logic programming with fairer search strategy than normal Prolog into Scheme) and Hansei (which embeds probability based nondeterminism into Ocaml using delimited continuations to allow direct-style expression of monadic code). After motivating some choices by studying the prototypical example of running append backwards they cover running parsers with "maximal munch" rule backwards - something that cannot be (declaratively) expressed in prolog. A very interesting paper on logic programming! It also thanks Tom Schrijvers of CHR fame at the end. ## Scratch jr Sat, 09/06/2014 - 17:45 ### Categories: Engineering Scratch jr is an iPad version of the Scratch environment, designed with young kids in mind. It is the best kid-oriented programming tool I tried so far, and my five year old has great fun making "movies" with it. As I noted on twitter an hour after installing, the ability to record your own voice and use it for your sprites is a killer feature. Check it out! ## Scala woes? Sat, 09/06/2014 - 12:12 ### Categories: Engineering A fork in the back? See discussion over at HN. People in the know are encouraged to shed light on the situation. ## Howard on Curry-Howard Sat, 08/30/2014 - 16:07 ### Categories: Engineering Philip Wadler posts his exchange with William Howard on history of the Curry-Howard correspondence. Howard on Curry-Howard. ## Cost semantics for functional languages Thu, 08/14/2014 - 11:53 ### Categories: Engineering There is an ongoing discussion in LtU (there, and there) on whether RAM and other machine models are inherently a better basis to reason about (time and) memory usage than lambda-calculus and functional languages. Guy Blelloch and his colleagues have been doing very important work on this question that seems to have escaped LtU's notice so far. A portion of the functional programming community has long been of the opinion that we do not need to refer to machines of the Turing tradition to reason about execution of functional programs. Dynamic semantics (which are often perceived as more abstract and elegant) are adequate, self-contained descriptions of computational behavior, which we can elevate to the status of (functional) machine model -- just like "abstract machines" can be seen as just machines. This opinion has been made scientifically precise by various brands of work, including for example implicit (computational) complexity, resource analysis and cost semantics for functional languages. Guy Blelloch developed a family of cost semantics, which correspond to annotations of operational semantics of functional languages with new information that captures more intentional behavior of the computation: not only the result, but also running time, memory usage, degree of parallelism and, more recently, interaction with a memory cache. Cost semantics are self-contained way to think of the efficiency of functional programs; they can of course be put in correspondence with existing machine models, and Blelloch and his colleagues have proved a vast amount of two-way correspondences, with the occasional extra logarithmic overhead -- or, from another point of view, provided probably cost-effective implementations of functional languages in imperative languages and conversely. This topic has been discussed by Robert Harper in two blog posts, Language and Machines which develops the general argument, and a second post on recent joint work by Guy and him on integrating cache-efficiency into the model. Harper also presents various cost semantics (called "cost dynamics") in his book "Practical Foundations for Programming Languages". In chronological order, three papers that are representative of the evolution of this work are the following. Parallelism In Sequential Functional Languages Guy E. Blelloch and John Greiner, 1995. This paper is focused on parallelism, but is also one of the earliest work carefully relating a lambda-calculus cost semantics with several machine models. This paper formally studies the question of how much parallelism is available in call-by-value functional languages with no parallel extensions (i.e., the functional subsets of ML or Scheme). In particular we are interested in placing bounds on how much parallelism is available for various problems. To do this we introduce a complexity model, the PAL, based on the call-by-value lambda-calculus. The model is defined in terms of a profiling semantics and measures complexity in terms of the total work and the parallel depth of a computation. We describe a simulation of the A-PAL (the PAL extended with arithmetic operations) on various parallel machine models, including the butterfly, hypercube, and PRAM models and prove simulation bounds. In particular the simulations are work-efficient (the processor-time product on the machines is within a constant factor of the work on the A-PAL), and for P processors the slowdown (time on the machines divided by depth on the A-PAL) is proportional to at most O(log P). We also prove bounds for simulating the PRAM on the A-PAL. Space Profiling for Functional Programs Daniel Spoonhower, Guy E. Blelloch, Robert Harper, and Phillip B. Gibbons, 2011 (conference version 2008) This paper clearly defines a notion of ideal memory usage (the set of store locations that are referenced by a value or an ongoing computation) that is highly reminiscent of garbage collection specifications, but without making any reference to an actual garbage collection implementation. We present a semantic space profiler for parallel functional programs. Building on previous work in sequential profiling, our tools help programmers to relate runtime resource use back to program source code. Unlike many profiling tools, our profiler is based on a cost semantics. This provides a means to reason about performance without requiring a detailed understanding of the compiler or runtime system. It also provides a specification for language implementers. This is critical in that it enables us to separate cleanly the performance of the application from that of the language implementation. Some aspects of the implementation can have significant effects on performance. Our cost semantics enables programmers to understand the impact of different scheduling policies while hiding many of the details of their implementations. We show applications where the choice of scheduling policy has asymptotic effects on space use. We explain these use patterns through a demonstration of our tools. We also validate our methodology by observing similar performance in our implementation of a parallel extension of Standard ML Cache and I/O efficient functional algorithms Guy E. Blelloch, Robert Harper, 2013 (see also the shorter CACM version) The cost semantics in this last work incorporates more notions from garbage collection, to reason about cache-efficient allocation of values -- in that it relies on work on formalizing garbage collection that has been mentioned on LtU before. The widely studied I/O and ideal-cache models were developed to account for the large difference in costs to access memory at different levels of the memory hierarchy. Both models are based on a two level memory hierarchy with a fixed size primary memory (cache) of size $$M$$, an unbounded secondary memory, and assume unit cost for transferring blocks of size $$B$$ between the two. Many algorithms have been analyzed in these models and indeed these models predict the relative performance of algorithms much more accurately than the standard RAM model. The models, however, require specifying algorithms at a very low level requiring the user to carefully lay out their data in arrays in memory and manage their own memory allocation. In this paper we present a cost model for analyzing the memory efficiency of algorithms expressed in a simple functional language. We show how many algorithms written in standard forms using just lists and trees (no arrays) and requiring no explicit memory layout or memory management are efficient in the model. We then describe an implementation of the language and show provable bounds for mapping the cost in our model to the cost in the ideal-cache model. These bound imply that purely functional programs based on lists and trees with no special attention to any details of memory layout can be as asymptotically as efficient as the carefully designed imperative I/O efficient algorithms. For example we describe an $$O(\frac{n}{B} \log_{M/B} \frac{n}{B})$$ cost sorting algorithm, which is optimal in the ideal cache and I/O models. ## Stream Processing with a Spreadsheet Wed, 08/13/2014 - 04:10 ### Categories: Engineering ECOOP 2014 paper (distinguished) by Vaziri et. al, abstract: Continuous data streams are ubiquitous and represent such a high volume of data that they cannot be stored to disk, yet it is often crucial for them to be analyzed in real-time. Stream processing is a programming paradigm that processes these immediately, and enables continuous analytics. Our objective is to make it easier for analysts, with little programming experience, to develop continuous analytics applications directly. We propose enhancing a spreadsheet, a pervasive tool, to obtain a programming platform for stream processing. We present the design and implementation of an enhanced spreadsheet that enables visualizing live streams, live programming to compute new streams, and exporting computations to be run on a server where they can be shared with other users, and persisted beyond the life of the spreadsheet. We formalize our core language, and present case studies that cover a range of stream processing applications. ## Safely Composable Type-Specific Languages Mon, 08/11/2014 - 06:27 ### Categories: Engineering Cyrus Omar, Darya Kurilova, Ligia Nistor, Benjamin Chung, Alex Potanin, and Jonathan Aldrich, "Safely Composable Type-Specific Languages", ECOOP14. Programming languages often include specialized syntax for common datatypes (e.g. lists) and some also build in support for specific specialized datatypes (e.g. regular expressions), but user-defined types must use general-purpose syntax. Frustration with this causes developers to use strings, rather than structured data, with alarming frequency, leading to correctness, performance, security, and usability issues. Allowing library providers to modularly extend a language with new syntax could help address these issues. Unfortunately, prior mechanisms either limit expressiveness or are not safely composable: individually unambiguous extensions can still cause ambiguities when used together. We introduce type-specific languages (TSLs): logic associated with a type that determines how the bodies of generic literals, able to contain arbitrary syntax, are parsed and elaborated, hygienically. The TSL for a type is invoked only when a literal appears where a term of that type is expected, guaranteeing non-interference. We give evidence supporting the applicability of this approach and formally specify it with a bidirectionally typed elaboration semantics for the Wyvern programming language. ## A Next Generation Smart Contract and Decentralized Application Platform Wed, 07/23/2014 - 17:12 ### Categories: Engineering A Next Generation Smart Contract and Decentralized Application Platform, Vitalik Buterin. When Satoshi Nakamoto first set the Bitcoin blockchain into motion in January 2009, he was simultaneously introducing two radical and untested concepts. The first is the "bitcoin", a decentralized peer-to-peer online currency that maintains a value without any backing, intrinsic value or central issuer. So far, the "bitcoin" as a currency unit has taken up the bulk of the public attention, both in terms of the political aspects of a currency without a central bank and its extreme upward and downward volatility in price. However, there is also another, equally important, part to Satoshi's grand experiment: the concept of a proof of work-based blockchain to allow for public agreement on the order of transactions. Bitcoin as an application can be described as a first-to-file system: if one entity has 50 BTC, and simultaneously sends the same 50 BTC to A and to B, only the transaction that gets confirmed first will process. There is no intrinsic way of determining from two transactions which came earlier, and for decades this stymied the development of decentralized digital currency. Satoshi's blockchain was the first credible decentralized solution. And now, attention is rapidly starting to shift toward this second part of Bitcoin's technology, and how the blockchain concept can be used for more than just money. Commonly cited applications include using on-blockchain digital assets to represent custom currencies and financial instruments ("colored coins"), the ownership of an underlying physical device ("smart property"), non-fungible assets such as domain names ("Namecoin") as well as more advanced applications such as decentralized exchange, financial derivatives, peer-to-peer gambling and on-blockchain identity and reputation systems. Another important area of inquiry is "smart contracts" - systems which automatically move digital assets according to arbitrary pre-specified rules. For example, one might have a treasury contract of the form "A can withdraw up to X currency units per day, B can withdraw up to Y per day, A and B together can withdraw anything, and A can shut off B's ability to withdraw". The logical extension of this is decentralized autonomous organizations (DAOs) - long-term smart contracts that contain the assets and encode the bylaws of an entire organization. What Ethereum intends to provide is a blockchain with a built-in fully fledged Turing-complete programming language that can be used to create "contracts" that can be used to encode arbitrary state transition functions, allowing users to create any of the systems described above, as well as many others that we have not yet imagined, simply by writing up the logic in a few lines of code. Includes code samples.	2014-11-01 13:45:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32922476530075073, "perplexity": 2396.181009153562}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414637906909.58/warc/CC-MAIN-20141030025826-00218-ip-10-16-133-185.ec2.internal.warc.gz"}
https://dealii.org/developer/doxygen/deal.II/code_gallery_MultipointFluxMixedFiniteElementMethods.html	Reference documentation for deal.II version Git 65001fc 2018-06-18 17:06:20 -0400 The 'Higher Order Multipoint Flux Mixed Finite Element (MFMFE) methods' code gallery program This program was contributed by Eldar Khattatov <[email protected]>. It comes without any warranty or support by its authors or the authors of deal.II. This program is part of the deal.II code gallery and consists of the following files (click to inspect): # Introduction This program presents the implementation of an arbitrary order multipoint flux mixed finite element method for the Darcy equation of flow in porous medium and illustrates the use case of the new enhanced Raviart-Thomas finite element for the purposes of local elimination of velocity degrees of freedom. # Higher order Multipoint Flux Mixed Finite Element methods Mixed finite element (MFE) methods are commonly used for modeling of fluid flow and transport, as they provide accurate and locally mass conservative velocities and robustness with respect to heterogeneous, anisotropic, and discontinuous coefficients. A main disadvantage of the MFE methods in their standard form is that they result in coupled velocity-pressure algebraic systems of saddle-point type, which restricts the use of efficient iterative solvers (see step-20 for example). One way to address this issue, a special MFE method, the multipoint flux mixed finite element (MFMFE) method was developed, which reduces to cell-centered finite differences on quadrilateral, hexahedral and simplicial grids, and exhibits robust performance for discontinuous full tensor coefficients. The method was motivated by the multipoint flux approximation (MPFA) method, which was developed as a finite volume method. The method utilizes the trapezoidal quadrature rule for the velocity mass matrix, which reduces it to a block-diagonal form with blocks associated with mesh vertices. The velocities can then be easily eliminated, resulting in a cell-centered pressure system. The aforementioned MFMFE methods are limited to the lowest order approximation. In a recent work we developed a family of arbitrary order symmetric MFMFE methods on quadrilateral and hexahedral grids, that uses the enhanced Raviart-Thomas finite element space and the tensor-product Gauss-Lobatto quadrature rule to achieve a block-diagonal velocity mass-matrix with blocks corresponding to the nodes associated with the veloicty DOFs. # Formulation of the method The method is defined as follows: find $$(\mathbf{u}_h,p_h) \in \mathbf{V}^k_h\times W^{k-1}_h$$, where $$k\ge 1$$, \begin{align} \left(\mathbf{K}^{-1}\mathbf{u}_h, \mathbf{v} \right)_Q -\left(p_h,\nabla\cdot\mathbf{v}\right) &= -\left\langle\mathcal{R}^{k-1}_h g, \mathbf{v}\right\rangle_{\Gamma_D}, &&\quad \mathbf{v}\in\mathbf{V}^k_h, \nonumber\\ \left(\nabla\cdot\mathbf{u}_h, w\right) &= \left(f,w\right), &&\quad w\in W_h^{k-1}. \nonumber \end{align} Here, $$(\cdot,\cdot)_Q$$ indicates that the term is to be assembled with the use of Gauss-Lobatto quadrature rule with $$k+1$$ points. Note that this leads to non-exact integration, however the optimal order of convergence is maintained. Another important point is related to the Dirichlet boundary data $$g$$, that has to be projected to $$Q^{k-1}(\Gamma_D)$$, a space of piecewise polynomials of order at most $$k-1$$. This requirement is needed to obtain the optimal order of convergence both in theory and practice. While this might look like an extra complication for the implementation, one can use the fact that the function evaluated in $$k$$ Gaussian points is $$\mathcal{O}(h^{k+1})$$ close to its $$L^2$$-projection onto the space $$Q^k$$, hence for the assembling of the RHS we will be using Gaussian quadrature rule of degree $$k$$. For this method, enhanced Raviart-Thomas space FE_RT_Bubbles of order $$k$$ is used for the velocity space $$\mathbf{V}^k_h$$ and the space of discontinuous piecewise polynomials FE_DGQ of order $$k-1$$ is used for the pressure space $$W_h^{k-1}$$. ## Reduction to a pressure system and its stencil Since the DOFs of $$\mathbf{V}_h^k(K)$$ are chosen as the dim vector components at the tensor-product Gauss-Lobatto quadrature points, in the velocity mass matrix obtained from the bilinear form $$(\mathbf{K}^{-1} \mathbf{u}_h,\mathbf{v})_Q$$, the dim DOFs associated with a quadrature point in an element $$K$$ are completely decoupled from other DOFs in $$K$$. Due to the continuity of normal components across faces, there are couplings with DOFs from neighboring elements. We distinguish three types of velocity couplings. • The first involves localization of degrees of freedom around each vertex in the grid. Only this type occurs in the lowest order case $$k=1$$. The number of DOFs that are coupled around a vertex equals the number of faces $$n_v$$ that share the vertex. • The second type of coupling is around nodes located on faces, but not at vertices. In 2d, these are edge DOFs. In 3d, there are two cases to consider for this type of coupling. One case is for nodes located on faces, but not on edges. The second case in 3d is for nodes located on edges, but not at vertices. • The third type of coupling involves nodes interior to the elements, in which case only the dim DOFs associated with the node are coupled. Due to the localization of DOF interactions described above, the velocity mass matrix obtained from the bilinear form $$(\mathbf{K}^{-1} \mathbf{u}_h,\mathbf{v})$$, is block-diagonal with blocks associated with the Gauss-Lobatto quadrature points. In particular, in 2d, there are $$n_v \times n_v$$ blocks at vertices ( $$n_v$$ is the number of neighboring edges), $$3 \times 3$$ blocks at edge points, and $$2 \times 2$$ blocks at interior points. In 3d, there are $$n_v \times n_v$$ blocks at vertices ( $$n_v$$ is the number of neighboring faces), $$2n_e \times 2n_e$$ blocks at edge points ( $$n_e$$ is the number of neighboring elements), $$5 \times 5$$ blocks at face points, and $$3 \times 3$$ blocks at interior points. ## Elimination procedure The local elimination procedure is done as follows (it is very similar to the Schur complement approach, except everything is done locally). Having a system of equations corresponding to a particular node $$i$$ \begin{align} \begin{pmatrix} A_i & B_i \\ -B_i^T & 0 \end{pmatrix} \begin{pmatrix} u \\ p \end{pmatrix}= \begin{pmatrix} f_i \\ g_i \end{pmatrix},\nonumber \end{align} we first write the velocity in terms of pressure from the first equation in the system, i.e. \begin{align} u = A_i^{-1}f - A_i^{-1}B_i p.\nonumber \end{align} Here, $$A_i$$ are small local matrices (full matrices), that are cheap to invert. We also store their inverses as they are further used in velocity solution recovery. With this, the second equation in the system above yields \begin{align} B_i^TA_i^{-1}B_i p = g_i - B_i^TA_i^{-1}f,\nonumber \end{align} where $$B_i^TA_i^{-1}B_i$$ is a local node's contribution to the global pressure system. By following the above steps, one gets the global cell-centered SPD pressure matrix with a compact stencil. After solving for the pressure variable, we use the expression for local velocities above in order to recover the global velocity solution. ## Convergence properties While the proposed schemes can be defined and are well posed on general quadrilateral or hexahedra, for the convergence analysis we need to impose a restriction on the element geometry. This is due to the reduced approximation properties of the MFE spaces on arbitrarily shaped quadrilaterals or hexahedra that our new family of elements inherits as well. However, introducing the notion of $$h^2$$-parallelograms in 2d and regular $$h^2$$-parallelepipeds in 3d, one can show that there is no reduction in accuracy. A (generalized) quadrilateral with vertices $$\mathbf{r}_i$$, $$i=1,\dots,4$$, is called an $$h^2$$-parallelogram if \begin{align} \|\mathbf{r}_{34} - \mathbf{r}_{21}\|_{\mathbb{R}^{dim}} \le Ch^2,\nonumber \end{align} and a hexahedral element is called an $$h^2$$-parallelepiped if all of its faces are $$h^2$$-parallelograms. Furthermore, an $$h^2$$-parallelepiped with vertices $$\mathbf{r}_i,\, i=1,\dots,8$$, is called regular if \begin{align} \|(\mathbf{r}_{21} - \mathbf{r}_{34}) - (\mathbf{r}_{65} - \mathbf{r}_{78})\|_{\mathbb{R}^{dim}} \le Ch^3.\nonumber \end{align} With the above restriction on the geometry of an element, the $$k$$-th order MFMFE method converges with order $$\mathcal{O}(h^{k})$$ for all variables in their natural norms, i.e. $$H_{div}$$ for the velocity and $$L^2$$ for pressure. The method also exhibits superconvergence of order $$\mathcal{O}(h^{k+1})$$ for pressure variable computed in $$k$$ Gaussian points. # Numerical results We test the method in 2d on a unit square domain. We start with initial grid with $$h = \frac14$$, and then distort it randomly using GridTools::distort_random() function. The pressure analytical solution is chosen to be \begin{align} p = x^3y^4 + x^2 + \sin(xy)\cos(xy), \nonumber \end{align} and the full permeability tensor coefficient is given by \begin{align} \mathbf{K} = \begin{pmatrix} (x+1)^2 + y^2 & \sin{(xy)} \\ \sin{(xy)} & (x+1)^2 \end{pmatrix}.\nonumber \end{align} The problem is then solved on a sequence of uniformly refined grids, with the errors and convergence rates for the case $$k=2$$ shown in the following table. Cycle Cells # DOFs $$\\|\mathbf{u} - \mathbf{u}_h\\|_{L^2}$$ Rate $$\\|\nabla\cdot(\mathbf{u} - \mathbf{u}_h)\\|_{L^2}$$ Rate $$\\|p - p_h\\|_{L^2}$$ Rate $$\\|\mathcal{Q}_h^{1}p - p_h\\|_{L^2}$$ Rate 0 16 280 1.24E-01 - 8.77E-01 - 9.04E-03 - 7.95E-04 - 1 64 1072 3.16E-02 2.0 2.21E-01 2.0 2.24E-03 2.0 1.07E-04 2.9 2 256 4192 7.87E-03 2.0 5.55E-02 2.0 5.59E-04 2.0 1.43E-05 2.9 3 1024 16576 1.96E-03 2.0 1.39E-02 2.0 1.40E-04 2.0 1.87E-06 2.9 4 4096 65920 4.89E-04 2.0 3.47E-03 2.0 3.49E-05 2.0 2.38E-07 3.0 5 16384 262912 1.22E-04 2.0 8.68E-04 2.0 8.73E-06 2.0 3.01E-08 3.0 We are also interested in performance of the method, hence the following table summarizes the wall time cost of the different parts of the program for the finest grid (i.e., $$k=2$$, $$h=\frac{1}{128}$$): Section wall time % of total Compute errors 0.734s 13% Make sparsity pattern 0.422s 7.5% Nodal assembly 0.965s 17% Output results 0.204s 3.6% Pressure CG solve 1.89s 33% Pressure matrix assembly 0.864s 15% Velocity solution recovery 0.0853s 1.5% Total time 5.64s 100% So one can see that the method solves the problem with 262k unknowns in about 4.5 seconds, with the rest of the time spent for the post-processing. These results were obtained with 8-core Ryzen 1700 CPU and 9.0.0-pre version of deal.II in release configuration. # Annotated version of data.h / * --------------------------------------------------------------------- * * This file is part of the deal.II Code Gallery. * * --------------------------------------------------------------------- * * Author: Ilona Ambartsumyan, Eldar Khattatov, University of Pittsburgh, 2018 * / #ifndef MFMFE_DATA_H #define MFMFE_DATA_H #include <deal.II/base/function.h> #include <deal.II/base/tensor_function.h> ### Data and exact solution. This file declares the classes for the given data, i.e. right-hand side, exact solution, permeability tensor and boundary conditions. For simplicity only 2d cases are provided, but 3d can be added straightforwardly. namespace MFMFE { using namespace dealii; template <int dim> class RightHandSide : public Function<dim> { public: RightHandSide () : Function<dim>(1) {} virtual double value (const Point<dim> &p, const unsigned int component = 0) const; }; template <int dim> double RightHandSide<dim>::value (const Point<dim> &p, const unsigned int / component /) const { const double x = p[0]; const double y = p[1]; switch (dim) { case 2: return -(x(yyyy)6.0-(yy)sin(xy2.0)2.0+2.0)(x2.0+xx+yy+1.0)-sin(xy)(cos(xy2.0)+(xx)(yyy)1.2E1 -xysin(xy2.0)2.0)2.0-(x2.0+2.0)(x2.0+(xx)(yyyy)3.0+ycos(xy2.0))+(xx)(sin(xy2.0) -x(yy)6.0)pow(x+1.0,2.0)2.0-xcos(xy)(x2.0+(xx)(yyyy)3.0+y(pow(cos(xy),2.0)2.0-1.0)) -xycos(xy)((xx)(yyy)4.0+pow(cos(xy),2.0)2.0-1.0); default: Assert(false, ExcMessage("The RHS data for dim != 2 is not provided")); } } template <int dim> class PressureBoundaryValues : public Function<dim> { public: PressureBoundaryValues () : Function<dim>(1) {} virtual double value (const Point<dim> &p, const unsigned int component = 0) const; }; template <int dim> double PressureBoundaryValues<dim>::value (const Point<dim> &p, const unsigned int / component /) const { const double x = p[0]; const double y = p[1]; switch (dim) { case 2: return (xxx)(yyyy)+cos(xy)sin(xy)+xx; default: Assert(false, ExcMessage("The BC data for dim != 2 is not provided")); } } template <int dim> class ExactSolution : public Function<dim> { public: ExactSolution () : Function<dim>(dim+1) {} virtual void vector_value (const Point<dim> &p, Vector<double> &value) const; virtual void vector_gradient (const Point<dim> &p, }; template <int dim> void ExactSolution<dim>::vector_value (const Point<dim> &p, Vector<double> &values) const { Assert (values.size() == dim+1, ExcDimensionMismatch (values.size(), dim+1)); const double x = p[0]; const double y = p[1]; switch (dim) { case 2: values(0) = -(x2.0+(xx)(yyyy)3.0+ycos(xy2.0))(x2.0+xx+yy+1.0)-xsin(xy)(cos(xy2.0)+(xx)(yyy)4.0); values(1) = -sin(xy)(x2.0+(xx)(yyyy)3.0+ycos(xy2.0))-x(cos(xy2.0)+(xx)(yyy)4.0)pow(x+1.0,2.0); values(2) = (xxx)(yyyy)+cos(xy)sin(xy)+xx; break; default: Assert(false, ExcMessage("The exact solution for dim != 2 is not provided")); } } template <int dim> void { const double x = p[0]; const double y = p[1]; switch (dim) { case 2: +(xx)(yyy)1.2E1-xysin(xy2.0)2.0)-(x2.0+2.0)(x2.0+(xx)(yyyy)3.0 +ycos(xy2.0))-xycos(xy)((xx)(yyy)4.0+pow(cos(xy),2.0)2.0-1.0); -y(x2.0+(xx)(yyyy)3.0+ycos(xy2.0))2.0-(xx)cos(xy)((xx)(yyy)4.0 +pow(cos(xy),2.0)2.0-1.0)+(xx)sin(xy)(sin(xy2.0)-x(yy)6.0)2.0; +(xx)(yyy)1.2E1-xysin(xy2.0)2.0)-x(cos(xy2.0)+(xx)(yyy)4.0)(x2.0+2.0) -ycos(xy)(x2.0+(xx)(yyyy)3.0+y(pow(cos(xy),2.0)2.0-1.0)); -x(yy)6.0)pow(x+1.0,2.0)2.0-xcos(xy)(x2.0+(xx)(yyyy)3.0 +y(pow(cos(xy),2.0)2.0-1.0)); break; default: Assert(false, ExcMessage("The exact solution's gradient for dim != 2 is not provided")); } } template <int dim> class KInverse : public TensorFunction<2,dim> { public: KInverse () : TensorFunction<2,dim>() {} virtual void value_list (const std::vector<Point<dim> > &points, std::vector<Tensor<2,dim> > &values) const; }; template <int dim> void KInverse<dim>::value_list (const std::vector<Point<dim> > &points, std::vector<Tensor<2,dim> > &values) const { Assert (points.size() == values.size(), ExcDimensionMismatch (points.size(), values.size())); for (unsigned int p=0; p<points.size(); ++p) { values[p].clear (); const double x = points[p][0]; const double y = points[p][1]; switch (dim) { case 2: values[p][0][0] = pow(x+1.0,2.0)/(x4.0+(xx)(yy)-pow(sin(xy),2.0)+x(yy)2.0+(xx)6.0+(xxx)4.0+xxxx+yy+1.0); values[p][0][1] = -sin(xy)/(x4.0+(xx)(yy)-pow(sin(xy),2.0)+x(yy)2.0+(xx)6.0+(xxx)4.0+xxxx+yy+1.0); values[p][1][0] = -sin(xy)/(x4.0+(xx)(yy)-pow(sin(xy),2.0)+x(yy)2.0+(xx)6.0+(xxx)4.0+xxxx+yy+1.0); values[p][1][1] = (x2.0+xx+yy+1.0)/(x4.0+(xx)(yy)-pow(sin(xy),2.0)+x(yy)2.0+(xx)6.0+(xxx)4.0+xxxx+yy+1.0); break; default: Assert(false, ExcMessage("The inverse of permeability tensor for dim != 2 is not provided")); } } } } #endif //MFMFE_DATA_H # Annotated version of mfmfe.cc / --------------------------------------------------------------------- * * This file is part of the deal.II Code Gallery. * * --------------------------------------------------------------------- * * Author: Ilona Ambartsumyan, Eldar Khattatov, University of Pittsburgh, 2018 * / ### Include files As usual, the list of necessary header files. There is not much new here, the files are included in order base-lac-grid-dofs-numerics followed by the C++ headers. #include <deal.II/base/convergence_table.h> #include <deal.II/base/logstream.h> #include <deal.II/base/timer.h> #include <deal.II/base/work_stream.h> #include <deal.II/lac/full_matrix.h> #include <deal.II/lac/solver_cg.h> #include <deal.II/lac/block_sparse_matrix.h> #include <deal.II/lac/block_vector.h> #include <deal.II/lac/precondition.h> #include <deal.II/grid/grid_generator.h> #include <deal.II/grid/grid_tools.h> #include <deal.II/grid/grid_in.h> #include <deal.II/grid/tria.h> #include <deal.II/dofs/dof_renumbering.h> #include <deal.II/dofs/dof_tools.h> #include <deal.II/fe/fe_dgq.h> #include <deal.II/fe/fe_system.h> #include <deal.II/fe/fe_tools.h> #include <deal.II/numerics/vector_tools.h> #include <deal.II/numerics/matrix_tools.h> #include <deal.II/numerics/data_out.h> #include <fstream> #include <unordered_map> This is a header needed for the purposes of the multipoint flux mixed method, as it declares the new enhanced Raviart-Thomas finite element. #include <deal.II/fe/fe_rt_bubbles.h> For the sake of readability, the classes representing data, i.e. RHS, BCs, permeability tensor and the exact solution are placed in a file data.h which is included here #include "data.h" As always the program is in the namespace of its own with the deal.II classes and functions imported into it namespace MFMFE { using namespace dealii; ### Definition of multipoint flux assembly data structures The main idea of the MFMFE method is to perform local elimination of the velocity variables in order to obtain the resulting pressure system. Since in deal.II assembly happens cell-wise, some extra work needs to be done in order to get the local mass matrices $$A_i$$ and the corresponding to them $$B_i$$. namespace DataStructures { This will be achieved by assembling cell-wise, but instead of placing the terms into a global system matrix, they will populate node-associated full matrices. For this, a data structure with fast lookup is crucial, hence the hash table, with the keys as Point<dim> template <int dim> struct hash_points { size_t operator()(const Point<dim> &p) const { size_t h1,h2,h3; h1 = std::hash<double>()(p[0]); switch (dim) { case 1: return h1; case 2: h2 = std::hash<double>()(p[1]); return (h1 ^ h2); case 3: h2 = std::hash<double>()(p[1]); h3 = std::hash<double>()(p[2]); return (h1 ^ (h2 << 1)) ^ h3; default: } } }; Here, the actual hash-tables are defined. We use the C++ STL unordered_map, with the hash function specified above. For convenience these are aliased as follows template <int dim> using PointToMatrixMap = std::unordered_map<Point<dim>, std::map<std::pair<types::global_dof_index,types::global_dof_index>, double>, hash_points<dim>>; template <int dim> using PointToVectorMap = std::unordered_map<Point<dim>, std::map<types::global_dof_index, double>, hash_points<dim>>; template <int dim> using PointToIndexMap = std::unordered_map<Point<dim>, std::set<types::global_dof_index>, hash_points<dim>>; Next, since this particular program allows for the use of multiple threads, the helper CopyData structures are defined. There are two kinds of these, one is used for the copying cell-wise contributions to the corresponging node-associated data structures... template <int dim> struct NodeAssemblyCopyData { PointToMatrixMap<dim> cell_mat; PointToVectorMap<dim> cell_vec; PointToIndexMap<dim> local_pres_indices; PointToIndexMap<dim> local_vel_indices; std::vector<types::global_dof_index> local_dof_indices; }; ... and the other one for the actual process of local velocity elimination and assembling the global pressure system: template <int dim> struct NodeEliminationCopyData { FullMatrix<double> node_pres_matrix; Vector<double> node_pres_rhs; FullMatrix<double> pressure_matrix; Vector<double> velocity_rhs; Vector<double> vertex_vel_solution; }; Similarly, two ScratchData classes are defined. One for the assembly part, where we need FEValues, FEFaceValues, Quadrature and storage for the basis fuctions... template <int dim> struct NodeAssemblyScratchData { NodeAssemblyScratchData (const FiniteElement<dim> &fe, const Triangulation<dim> &tria, NodeAssemblyScratchData (const NodeAssemblyScratchData &scratch_data); FEValues<dim> fe_values; FEFaceValues<dim> fe_face_values; std::vector<unsigned int> n_faces_at_vertex; const unsigned long num_cells; std::vector<Tensor<2,dim>> k_inverse_values; std::vector<double> rhs_values; std::vector<double> pres_bc_values; std::vector<Tensor<1,dim> > phi_u; std::vector<double> div_phi_u; std::vector<double> phi_p; }; template <int dim> NodeAssemblyScratchData<dim>:: NodeAssemblyScratchData (const FiniteElement<dim> &fe, const Triangulation<dim> &tria, : fe_values (fe, fe_face_values (fe, num_cells(tria.n_active_cells()), phi_u(fe.dofs_per_cell), div_phi_u(fe.dofs_per_cell), phi_p(fe.dofs_per_cell) { n_faces_at_vertex.resize(tria.n_vertices(), 0); for (; face != endf; ++face) for (unsigned int v=0; v<GeometryInfo<dim>::vertices_per_face; ++v) n_faces_at_vertex[face->vertex_index(v)] += 1; } template <int dim> NodeAssemblyScratchData<dim>:: NodeAssemblyScratchData (const NodeAssemblyScratchData &scratch_data) : fe_values (scratch_data.fe_values.get_fe(), fe_face_values (scratch_data.fe_face_values.get_fe(), n_faces_at_vertex(scratch_data.n_faces_at_vertex), num_cells(scratch_data.num_cells), k_inverse_values(scratch_data.k_inverse_values), rhs_values(scratch_data.rhs_values), pres_bc_values(scratch_data.pres_bc_values), phi_u(scratch_data.phi_u), div_phi_u(scratch_data.div_phi_u), phi_p(scratch_data.phi_p) {} ...and the other, simpler one, for the velocity elimination and recovery struct VertexEliminationScratchData { VertexEliminationScratchData () = default; VertexEliminationScratchData (const VertexEliminationScratchData &scratch_data); FullMatrix<double> velocity_matrix; Vector<double> pressure_rhs; Vector<double> local_pressure_solution; Vector<double> tmp_rhs1; Vector<double> tmp_rhs2; Vector<double> tmp_rhs3; }; VertexEliminationScratchData:: VertexEliminationScratchData (const VertexEliminationScratchData &scratch_data) : velocity_matrix(scratch_data.velocity_matrix), pressure_rhs(scratch_data.pressure_rhs), local_pressure_solution(scratch_data.local_pressure_solution), tmp_rhs1(scratch_data.tmp_rhs1), tmp_rhs2(scratch_data.tmp_rhs2), tmp_rhs3(scratch_data.tmp_rhs3) {} } ### The MultipointMixedDarcyProblem class template The main class, besides the constructor and destructor, has only one public member run(), similarly to the tutorial programs. The private members can be grouped into the ones that are used for the cell-wise assembly, vertex elimination, pressure solve, vertex velocity recovery and postprocessing. Apart from the MFMFE-specific data structures, the rest of the members should look familiar. template <int dim> class MultipointMixedDarcyProblem { public: MultipointMixedDarcyProblem (const unsigned int degree); ~MultipointMixedDarcyProblem (); void run (const unsigned int refine); private: void assemble_system_cell (const typename DoFHandler<dim>::active_cell_iterator &cell, DataStructures::NodeAssemblyScratchData<dim> &scratch_data, DataStructures::NodeAssemblyCopyData<dim> &copy_data); void copy_cell_to_node(const DataStructures::NodeAssemblyCopyData<dim> &copy_data); void node_assembly(); void make_cell_centered_sp (); void nodal_elimination(const typename DataStructures::PointToMatrixMap<dim>::iterator &n_it, DataStructures::VertexEliminationScratchData &scratch_data, DataStructures::NodeEliminationCopyData<dim> &copy_data); void copy_node_to_system(const DataStructures::NodeEliminationCopyData<dim> &copy_data); void pressure_assembly (); void solve_pressure (); void velocity_assembly (const typename DataStructures::PointToMatrixMap<dim>::iterator &n_it, DataStructures::VertexEliminationScratchData &scratch_data, DataStructures::NodeEliminationCopyData<dim> &copy_data); void copy_node_velocity_to_global(const DataStructures::NodeEliminationCopyData<dim> &copy_data); void velocity_recovery (); void reset_data_structures (); void compute_errors (const unsigned int cycle); void output_results (const unsigned int cycle, const unsigned int refine); const unsigned int degree; Triangulation<dim> triangulation; DoFHandler<dim> dof_handler; SparsityPattern cell_centered_sp; SparseMatrix<double> pres_system_matrix; Vector<double> pres_rhs; std::unordered_map<Point<dim>, FullMatrix<double>, DataStructures::hash_points<dim>> pressure_matrix; std::unordered_map<Point<dim>, FullMatrix<double>, DataStructures::hash_points<dim>> A_inverse; std::unordered_map<Point<dim>, Vector<double>, DataStructures::hash_points<dim>> velocity_rhs; DataStructures::PointToMatrixMap<dim> node_matrix; DataStructures::PointToVectorMap<dim> node_rhs; DataStructures::PointToIndexMap<dim> pressure_indices; DataStructures::PointToIndexMap<dim> velocity_indices; unsigned long n_v, n_p; Vector<double> pres_solution; Vector<double> vel_solution; ConvergenceTable convergence_table; TimerOutput computing_timer; }; #### Constructor and destructor, reset_data_structures In the constructor of this class, we store the value that was passed in concerning the degree of the finite elements we shall use (a degree of one would mean the use of FE_RT_Bubbles(1) and FE_DGQ(0)), and then construct the vector valued element belonging to the space $$V_h^k$$ described in the introduction. The constructor also takes care of initializing the computing timer, as it is of interest for us how well our method performs. template <int dim> MultipointMixedDarcyProblem<dim>::MultipointMixedDarcyProblem (const unsigned int degree) : degree(degree), fe(FE_RT_Bubbles<dim>(degree), 1, FE_DGQ<dim>(degree-1), 1), dof_handler(triangulation), computing_timer(std::cout, TimerOutput::summary, TimerOutput::wall_times) {} The destructor clears the dof_handler and all of the data structures we used for the method. template <int dim> MultipointMixedDarcyProblem<dim>::~MultipointMixedDarcyProblem() { reset_data_structures (); dof_handler.clear(); } This method clears all the data that was used after one refinement cycle. template <int dim> void MultipointMixedDarcyProblem<dim>::reset_data_structures () { pressure_indices.clear(); velocity_indices.clear(); velocity_rhs.clear(); A_inverse.clear(); pressure_matrix.clear(); node_matrix.clear(); node_rhs.clear(); } #### Cell-wise assembly and creation of the local, nodal-based data structures First, the function that copies local cell contributions to the corresponding nodal matrices and vectors is defined. It places the values obtained from local cell integration into the correct place in a matrix/vector corresponging to a specific node. template <int dim> void MultipointMixedDarcyProblem<dim>::copy_cell_to_node(const DataStructures::NodeAssemblyCopyData<dim> &copy_data) { for (auto m : copy_data.cell_mat) { for (auto p : m.second) node_matrix[m.first][p.first] += p.second; for (auto p : copy_data.cell_vec.at(m.first)) node_rhs[m.first][p.first] += p.second; for (auto p : copy_data.local_pres_indices.at(m.first)) pressure_indices[m.first].insert(p); for (auto p : copy_data.local_vel_indices.at(m.first)) velocity_indices[m.first].insert(p); } } Second, the function that does the cell assembly is defined. While it is similar to the tutorial programs in a way it uses scrath and copy data structures, the need to localize the DOFs leads to several differences. template <int dim> void MultipointMixedDarcyProblem<dim>:: assemble_system_cell (const typename DoFHandler<dim>::active_cell_iterator &cell, DataStructures::NodeAssemblyScratchData<dim> &scratch_data, DataStructures::NodeAssemblyCopyData<dim> &copy_data) { copy_data.cell_mat.clear(); copy_data.cell_vec.clear(); copy_data.local_vel_indices.clear(); copy_data.local_pres_indices.clear(); const unsigned int dofs_per_cell = fe.dofs_per_cell; const unsigned int n_q_points = scratch_data.fe_values.get_quadrature().size(); const unsigned int n_face_q_points = scratch_data.fe_face_values.get_quadrature().size(); copy_data.local_dof_indices.resize(dofs_per_cell); cell->get_dof_indices (copy_data.local_dof_indices); scratch_data.fe_values.reinit (cell); const KInverse<dim> k_inverse; const RightHandSide<dim> rhs; const PressureBoundaryValues<dim> pressure_bc; const FEValuesExtractors::Vector velocity (0); const FEValuesExtractors::Scalar pressure (dim); const unsigned int n_vel = dimpow(degree+1,dim); std::unordered_map<unsigned int, std::unordered_map<unsigned int, double>> div_map; One, we need to be able to assemble the communication between velocity and pressure variables and put it on the right place in our final, local version of the B matrix. This is a little messy, as such communication is not in fact local, so we do it in two steps. First, we compute all relevant LHS and RHS for (unsigned int q=0; q<n_q_points; ++q) { for (unsigned int k=0; k<dofs_per_cell; ++k) { scratch_data.phi_u[k] = scratch_data.fe_values[velocity].value(k, q); scratch_data.div_phi_u[k] = scratch_data.fe_values[velocity].divergence (k, q); scratch_data.phi_p[k] = scratch_data.fe_values[pressure].value (k, q); } for (unsigned int i=0; i<dofs_per_cell; ++i) { for (unsigned int j=n_vel; j<dofs_per_cell; ++j) { double div_term = (- scratch_data.div_phi_u[i] scratch_data.phi_p[j] - scratch_data.phi_p[i] * scratch_data.div_phi_u[j]) * scratch_data.fe_values.JxW(q); if (std::abs(div_term) > 1.e-12) div_map[i][j] += div_term; } double source_term = -scratch_data.phi_p[i] * scratch_data.rhs_values[q] * scratch_data.fe_values.JxW(q); if (std::abs(scratch_data.phi_p[i]) > 1.e-12 \|\| std::abs(source_term) > 1.e-12) copy_data.cell_vec[p][copy_data.local_dof_indices[i]] += source_term; } } Then, by making another pass, we compute the mass matrix terms and incorporate the divergence form and RHS accordingly. This second pass, allows us to know where the total contribution will be put in the nodal data structures, as with this choice of quadrature rule and finite element only the basis functions corresponding to the same quadrature points yield non-zero contribution. for (unsigned int q=0; q<n_q_points; ++q) { std::set<types::global_dof_index> vel_indices; for (unsigned int k=0; k<dofs_per_cell; ++k) { scratch_data.phi_u[k] = scratch_data.fe_values[velocity].value(k, q); scratch_data.div_phi_u[k] = scratch_data.fe_values[velocity].divergence (k, q); scratch_data.phi_p[k] = scratch_data.fe_values[pressure].value (k, q); } for (unsigned int i=0; i<dofs_per_cell; ++i) for (unsigned int j=i; j<dofs_per_cell; ++j) { double mass_term = scratch_data.phi_u[i] * scratch_data.k_inverse_values[q] * scratch_data.phi_u[j] * scratch_data.fe_values.JxW(q); if (std::abs(mass_term) > 1.e-12) { copy_data.cell_mat[p][std::make_pair(copy_data.local_dof_indices[i], copy_data.local_dof_indices[j])] += mass_term; vel_indices.insert(i); copy_data.local_vel_indices[p].insert(copy_data.local_dof_indices[j]); } } for (auto i : vel_indices) for (auto el : div_map[i]) if (std::abs(el.second) > 1.e-12) { copy_data.cell_mat[p][std::make_pair(copy_data.local_dof_indices[i], copy_data.local_dof_indices[el.first])] += el.second; copy_data.local_pres_indices[p].insert(copy_data.local_dof_indices[el.first]); } } The pressure boundary conditions are computed as in step-20, std::map<types::global_dof_index,double> pres_bc; for (unsigned int face_no=0; face_no<GeometryInfo<dim>::faces_per_cell; ++face_no) if (cell->at_boundary(face_no)) { scratch_data.fe_face_values.reinit (cell, face_no); for (unsigned int q=0; q<n_face_q_points; ++q) for (unsigned int i = 0; i < dofs_per_cell; ++i) { double tmp = -(scratch_data.fe_face_values[velocity].value(i, q) * scratch_data.fe_face_values.normal_vector(q) * scratch_data.pres_bc_values[q] * scratch_data.fe_face_values.JxW(q)); if (std::abs(tmp) > 1.e-12) pres_bc[copy_data.local_dof_indices[i]] += tmp; } } ...but we distribute them to the corresponding nodal data structures for (auto m : copy_data.cell_vec) for (unsigned int i=0; i<dofs_per_cell; ++i) if (std::abs(pres_bc[copy_data.local_dof_indices[i]]) > 1.e-12) copy_data.cell_vec[m.first][copy_data.local_dof_indices[i]] += pres_bc[copy_data.local_dof_indices[i]]; } Finally, node_assembly() takes care of all the local computations via WorkStream mechanism. Notice that the choice of the quadrature rule here is dictated by the formulation of the method. It has to be degree+1 points Gauss-Lobatto for the volume integrals and degree for the face ones, as mentioned in the introduction. template <int dim> void MultipointMixedDarcyProblem<dim>::node_assembly() { TimerOutput::Scope t(computing_timer, "Nodal assembly"); dof_handler.distribute_dofs(fe); std::vector<types::global_dof_index> dofs_per_component (dim+1); DoFTools::count_dofs_per_component (dof_handler, dofs_per_component); n_v = dofs_per_component[0]; n_p = dofs_per_component[dim]; pres_rhs.reinit(n_p); dof_handler.end(), this, &MultipointMixedDarcyProblem::assemble_system_cell, &MultipointMixedDarcyProblem::copy_cell_to_node, DataStructures::NodeAssemblyCopyData<dim>()); } #### Making the sparsity pattern Having computed all the local contributions, we actually have all the information needed to make a cell-centered sparsity pattern manually. We do this here, because SparseMatrixEZ leads to a slower solution. template <int dim> void MultipointMixedDarcyProblem<dim>::make_cell_centered_sp() { TimerOutput::Scope t(computing_timer, "Make sparsity pattern"); DynamicSparsityPattern dsp(n_p, n_p); std::set<types::global_dof_index>::iterator pi_it, pj_it; unsigned int i, j; for (auto el : node_matrix) for (pi_it = pressure_indices[el.first].begin(), i = 0; pi_it != pressure_indices[el.first].end(); ++pi_it, ++i) for (pj_it = pi_it, j = 0; pj_it != pressure_indices[el.first].end(); ++pj_it, ++j) dsp.add(pi_it - n_v, pj_it - n_v); dsp.symmetrize(); cell_centered_sp.copy_from(dsp); pres_system_matrix.reinit (cell_centered_sp); } #### The local elimination procedure This function finally performs the local elimination procedure. Mathematically, it follows the same idea as in computing the Schur complement (as mentioned in the introduction) but we do so locally. Namely, local velocity DOFs are expressed in terms of corresponding pressure values, and then used for the local pressure systems. template <int dim> void MultipointMixedDarcyProblem<dim>:: nodal_elimination(const typename DataStructures::PointToMatrixMap<dim>::iterator &n_it, DataStructures::VertexEliminationScratchData &scratch_data, DataStructures::NodeEliminationCopyData<dim> &copy_data) { unsigned int n_edges = velocity_indices.at((n_it).first).size(); unsigned int n_cells = pressure_indices.at((n_it).first).size(); scratch_data.velocity_matrix.reinit(n_edges,n_edges); copy_data.pressure_matrix.reinit(n_edges,n_cells); copy_data.velocity_rhs.reinit(n_edges); scratch_data.pressure_rhs.reinit(n_cells); { std::set<types::global_dof_index>::iterator vi_it, vj_it, p_it; unsigned int i; for (vi_it = velocity_indices.at((n_it).first).begin(), i = 0; vi_it != velocity_indices.at((n_it).first).end(); ++vi_it, ++i) { unsigned int j; for (vj_it = velocity_indices.at((n_it).first).begin(), j = 0; vj_it != velocity_indices.at((n_it).first).end(); ++vj_it, ++j) { if (j != i) } for (p_it = pressure_indices.at((n_it).first).begin(), j = 0; p_it != pressure_indices.at((n_it).first).end(); ++p_it, ++j) copy_data.velocity_rhs(i) += node_rhs.at((n_it).first)[vi_it]; } for (p_it = pressure_indices.at((n_it).first).begin(), i = 0; p_it != pressure_indices.at((n_it).first).end(); ++p_it, ++i) scratch_data.pressure_rhs(i) += node_rhs.at((n_it).first)[p_it]; } copy_data.Ainverse.reinit(n_edges,n_edges); scratch_data.tmp_rhs1.reinit(n_edges); scratch_data.tmp_rhs2.reinit(n_edges); scratch_data.tmp_rhs3.reinit(n_cells); copy_data.Ainverse.invert(scratch_data.velocity_matrix); copy_data.node_pres_matrix.reinit(n_cells, n_cells); copy_data.node_pres_rhs = scratch_data.pressure_rhs; copy_data.node_pres_matrix = 0; copy_data.node_pres_matrix.triple_product(copy_data.Ainverse, copy_data.pressure_matrix, copy_data.pressure_matrix, true, false); copy_data.Ainverse.vmult(scratch_data.tmp_rhs1, copy_data.velocity_rhs, false); copy_data.pressure_matrix.Tvmult(scratch_data.tmp_rhs3, scratch_data.tmp_rhs1, false); copy_data.node_pres_rhs = -1.0; copy_data.node_pres_rhs += scratch_data.tmp_rhs3; copy_data.p = (n_it).first; } Each node's pressure system is then distributed to a global pressure system, using the indices we computed in the previous stages. template <int dim> void MultipointMixedDarcyProblem<dim>:: copy_node_to_system(const DataStructures::NodeEliminationCopyData<dim> &copy_data) { A_inverse[copy_data.p] = copy_data.Ainverse; pressure_matrix[copy_data.p] = copy_data.pressure_matrix; velocity_rhs[copy_data.p] = copy_data.velocity_rhs; { std::set<types::global_dof_index>::iterator pi_it, pj_it; unsigned int i; for (pi_it = pressure_indices[copy_data.p].begin(), i = 0; pi_it != pressure_indices[copy_data.p].end(); ++pi_it, ++i) { unsigned int j; for (pj_it = pressure_indices[copy_data.p].begin(), j = 0; pj_it != pressure_indices[copy_data.p].end(); ++pj_it, ++j) pres_system_matrix.add(pi_it - n_v, pj_it - n_v, copy_data.node_pres_matrix(i, j)); pres_rhs(pi_it - n_v) += copy_data.node_pres_rhs(i); } } } The WorkStream mechanism is again used for the assembly of the global system for the pressure variable, where the previous functions are used to perform local computations. template <int dim> void MultipointMixedDarcyProblem<dim>::pressure_assembly() { TimerOutput::Scope t(computing_timer, "Pressure matrix assembly"); pres_rhs.reinit(n_p); WorkStream::run(node_matrix.begin(), node_matrix.end(), this, &MultipointMixedDarcyProblem::nodal_elimination, &MultipointMixedDarcyProblem::copy_node_to_system, DataStructures::VertexEliminationScratchData(), DataStructures::NodeEliminationCopyData<dim>()); } #### Velocity solution recovery After solving for the pressure variable, we want to follow the above procedure backwards, in order to obtain the velocity solution (again, this is similar in nature to the Schur complement approach, see step-20, but here it is done locally at each node). We have almost everything computed and stored already, including inverses of local mass matrices, so the following is a relatively straightforward implementation. template <int dim> void MultipointMixedDarcyProblem<dim>:: velocity_assembly (const typename DataStructures::PointToMatrixMap<dim>::iterator &n_it, DataStructures::VertexEliminationScratchData &scratch_data, DataStructures::NodeEliminationCopyData<dim> &copy_data) { unsigned int n_edges = velocity_indices.at((n_it).first).size(); unsigned int n_cells = pressure_indices.at((n_it).first).size(); scratch_data.tmp_rhs1.reinit(n_edges); scratch_data.tmp_rhs2.reinit(n_edges); scratch_data.tmp_rhs3.reinit(n_cells); scratch_data.local_pressure_solution.reinit(n_cells); copy_data.vertex_vel_solution.reinit(n_edges); std::set<types::global_dof_index>::iterator p_it; unsigned int i; for (p_it = pressure_indices[(n_it).first].begin(), i = 0; p_it != pressure_indices[(n_it).first].end(); ++p_it, ++i) scratch_data.local_pressure_solution(i) = pres_solution(p_it - n_v); pressure_matrix[(n_it).first].vmult(scratch_data.tmp_rhs2, scratch_data.local_pressure_solution, false); scratch_data.tmp_rhs2 = -1.0; scratch_data.tmp_rhs2+=velocity_rhs[(n_it).first]; A_inverse[(n_it).first].vmult(copy_data.vertex_vel_solution, scratch_data.tmp_rhs2, false); copy_data.p = (n_it).first; } Copy nodal velocities to a global solution vector by using local computations and indices from early stages. template <int dim> void MultipointMixedDarcyProblem<dim>:: copy_node_velocity_to_global(const DataStructures::NodeEliminationCopyData<dim> &copy_data) { std::set<types::global_dof_index>::iterator vi_it; unsigned int i; for (vi_it = velocity_indices[copy_data.p].begin(), i = 0; vi_it != velocity_indices[copy_data.p].end(); ++vi_it, ++i) vel_solution(vi_it) += copy_data.vertex_vel_solution(i); } Use WorkStream to run everything concurrently. template <int dim> void MultipointMixedDarcyProblem<dim>::velocity_recovery() { TimerOutput::Scope t(computing_timer, "Velocity solution recovery"); vel_solution.reinit(n_v); WorkStream::run(node_matrix.begin(), node_matrix.end(), this, &MultipointMixedDarcyProblem::velocity_assembly, &MultipointMixedDarcyProblem::copy_node_velocity_to_global, DataStructures::VertexEliminationScratchData(), DataStructures::NodeEliminationCopyData<dim>()); solution.reinit(2); solution.block(0) = vel_solution; solution.block(1) = pres_solution; solution.collect_sizes(); } #### Pressure system solver The solver part is trivial. We use the CG solver with no preconditioner for simplicity. template <int dim> void MultipointMixedDarcyProblem<dim>::solve_pressure() { TimerOutput::Scope t(computing_timer, "Pressure CG solve"); pres_solution.reinit(n_p); SolverControl solver_control (2.0n_p, 1e-10); SolverCG<> solver (solver_control); solver.solve(pres_system_matrix, pres_solution, pres_rhs, identity); } ### Postprocessing We have two postprocessing steps here, first one computes the errors in order to populate the convergence tables. The other one takes care of the output of the solutions in .vtk format. #### Compute errors The implementation of this function is almost identical to step-20. We use ComponentSelectFunction as masks to use the right solution component (velocity or pressure) and integrate_difference to compute the errors. Since we also want to compute Hdiv seminorm of the velocity error, one must provide gradients in the ExactSolution class implementation to avoid exceptions. The only noteworthy thing here is that we again use lower order quadrature rule instead of projecting the solution to an appropriate space in order to show superconvergence, which is mathematically justified. template <int dim> void MultipointMixedDarcyProblem<dim>::compute_errors(const unsigned cycle) { TimerOutput::Scope t(computing_timer, "Compute errors"); ExactSolution<dim> exact_solution; Vector<double> cellwise_errors (triangulation.n_active_cells()); QTrapez<1> q_trapez; VectorTools::integrate_difference (dof_handler, solution, exact_solution, const double p_l2_error = cellwise_errors.l2_norm(); VectorTools::integrate_difference (dof_handler, solution, exact_solution, const double p_l2_mid_error = cellwise_errors.l2_norm(); VectorTools::integrate_difference (dof_handler, solution, exact_solution, const double u_l2_error = cellwise_errors.l2_norm(); VectorTools::integrate_difference (dof_handler, solution, exact_solution, const double u_hd_error = cellwise_errors.l2_norm(); const unsigned int n_active_cells=triangulation.n_active_cells(); const unsigned int n_dofs=dof_handler.n_dofs(); } #### Output results This function also follows the same idea as in step-20 tutorial program. The only modification to it is the part involving a convergence table. template <int dim> void MultipointMixedDarcyProblem<dim>::output_results(const unsigned int cycle, const unsigned int refine) { TimerOutput::Scope t(computing_timer, "Output results"); std::vector<std::string> solution_names(dim, "u"); solution_names.push_back ("p"); std::vector<DataComponentInterpretation::DataComponentInterpretation> DataOut<dim> data_out; data_out.build_patches (); std::ofstream output ("solution" + std::to_string(dim) + "d-" + std::to_string(cycle) + ".vtk"); data_out.write_vtk (output); convergence_table.set_precision("Velocity,L2", 3); convergence_table.set_precision("Velocity,Hdiv", 3); convergence_table.set_precision("Pressure,L2", 3); convergence_table.set_precision("Pressure,L2-nodal", 3); convergence_table.set_scientific("Velocity,L2", true); convergence_table.set_scientific("Velocity,Hdiv", true); convergence_table.set_scientific("Pressure,L2", true); convergence_table.set_scientific("Pressure,L2-nodal", true); convergence_table.set_tex_caption("cells", "\\# cells"); convergence_table.set_tex_caption("dofs", "\\# dofs"); convergence_table.set_tex_caption("Velocity,L2", " \\\|\\u - \\u_h\\\|_{L^2} "); convergence_table.set_tex_caption("Velocity,Hdiv", " \\\|\\nabla\\cdot(\\u - \\u_h)\\\|_{L^2} "); convergence_table.set_tex_caption("Pressure,L2", " \\\|p - p_h\\\|_{L^2} "); convergence_table.set_tex_caption("Pressure,L2-nodal", " \\\|Qp - p_h\\\|_{L^2} "); convergence_table.set_tex_format("cells", "r"); convergence_table.set_tex_format("dofs", "r"); convergence_table.evaluate_convergence_rates("Velocity,L2", ConvergenceTable::reduction_rate_log2); convergence_table.evaluate_convergence_rates("Velocity,Hdiv", ConvergenceTable::reduction_rate_log2); convergence_table.evaluate_convergence_rates("Pressure,L2", ConvergenceTable::reduction_rate_log2); convergence_table.evaluate_convergence_rates("Pressure,L2-nodal", ConvergenceTable::reduction_rate_log2); std::ofstream error_table_file("error" + std::to_string(dim) + "d.tex"); if (cycle == refine-1) { convergence_table.write_text(std::cout); convergence_table.write_tex(error_table_file); } } ### Run function The driver method run() takes care of mesh generation and arranging calls to member methods in the right way. It also resets data structures and clear triangulation and DOF handler as we run the method on a sequence of refinements in order to record convergence rates. template <int dim> void MultipointMixedDarcyProblem<dim>::run(const unsigned int refine) { Assert(refine > 0, ExcMessage("Must at least have 1 refinement cycle!")); dof_handler.clear(); triangulation.clear(); convergence_table.clear(); for (unsigned int cycle=0; cycle<refine; ++cycle) { if (cycle == 0) { We first generate the hyper cube and refine it twice so that we could distort the grid slightly and demonstrate the method's ability to work in such a case. GridGenerator::hyper_cube (triangulation, 0, 1); triangulation.refine_global(2); GridTools::distort_random (0.3, triangulation, true); } else triangulation.refine_global(1); node_assembly(); make_cell_centered_sp(); pressure_assembly(); solve_pressure (); velocity_recovery (); compute_errors (cycle); output_results (cycle, refine); reset_data_structures (); computing_timer.print_summary (); computing_timer.reset (); } } } ### The main function In the main functione we pass the order of the Finite Element as an argument to the constructor of the Multipoint Flux Mixed Darcy problem, and the number of refinement cycles as an argument for the run method. int main () { try { using namespace dealii; using namespace MFMFE; MultipointMixedDarcyProblem<2> mfmfe_problem(2); mfmfe_problem.run(6); } catch (std::exception &exc) { std::cerr << std::endl << std::endl << "----------------------------------------------------" << std::endl; std::cerr << "Exception on processing: " << std::endl << exc.what() << std::endl << "Aborting!" << std::endl << "----------------------------------------------------" << std::endl; return 1; } catch (...) { std::cerr << std::endl << std::endl << "----------------------------------------------------" << std::endl; std::cerr << "Unknown exception!" << std::endl << "Aborting!" << std::endl << "----------------------------------------------------" << std::endl; return 1; } return 0; }	2018-06-19 22:05:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 8, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5785767436027527, "perplexity": 9413.989888365892}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863206.9/warc/CC-MAIN-20180619212507-20180619232507-00318.warc.gz"}
http://clay6.com/qa/4563/find-the-equation-of-the-plane-passing-through-the-line-of-intersection-of-	Browse Questions # Find the equation of the plane passing through the line of intersection of the planes $2x+y-z=3 \: and \: 5x-3y+4z+9=0$ and parallel to the line $\large \frac{x-1}{2}=\frac{y-3}{4}=\frac{z-5}{5}.$ Toolbox: • Vector equation of a line passing through a point and parallel to a given vector is $\overrightarrow r=\overrightarrow a+\lambda \overrightarrow b$ Step 1: Equation of the planes are $2x+y-z=3$ and $5x-3y+4z+9=0$ The equation of the plane passing through the line of intersection of these planes is $(2x+y-z-3)+\lambda(5x-3y+4z+9)=0$ $x(2+5\lambda)+y(1-3\lambda)+2(4\lambda-1)+9\lambda-3=0$-----(1) Step 2: The plane is parallel to the line $\large\frac{x-1}{2}=\frac{y-3}{4}=\frac{z-5}{5}$ $\therefore 2(2+5\lambda)+4(1-3\lambda)+5(4\lambda-1)=0$ (i.e) $18\lambda+3=0$ $\Rightarrow \lambda=-\large\frac{1}{6}$ Step 3: Put the value of $\lambda$ in emu(1) we obtain $x(2-\large\frac{5}{6})+y(1+\large\frac{3}{6})+z(-\large\frac{4}{6}-1)-\large\frac{9}{6}-3=0$ $\Rightarrow 7x+9y-10z-27=0$ This is the equation of the required plane.	2017-06-22 14:21:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5161570906639099, "perplexity": 141.12752499027343}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128319575.19/warc/CC-MAIN-20170622135404-20170622155404-00279.warc.gz"}
https://www.gradesaver.com/textbooks/math/trigonometry/CLONE-68cac39a-c5ec-4c26-8565-a44738e90952/chapter-4-graphs-of-the-circular-functions-section-4-2-translations-of-the-graphs-of-the-sine-and-cosine-functions-4-2-exercises-page-162/37	# Chapter 4 - Graphs of the Circular Functions - Section 4.2 Translations of the Graphs of the Sine and Cosine Functions - 4.2 Exercises - Page 162: 37 Refer to the blue graph below. #### Work Step by Step RECALL: The graph of $y=\cos{(x-d)}$ involves a horizontal shift of the parent function $y=\cos{x}$. The shift is $d$ units to the right when $d \gt 0$ and $\|d\|$ units to the left when $d\lt0$. The given function has $d=\frac{\pi}{2}$, which is positive. Thus, the given function involves a $\frac{\pi}{2}$-unit shift to the right of the parent function $y=\cos{x}$. To graph the given function, perform the following steps: (1) Graph the parent function $y=\cos{x}$ over a two-period interval, which is $[0, 4\pi]$. (Refer to the red graph in the answer part above.) (2) Shift the graph of the parent function $\frac{\pi}{2}$ units to the right. (Refer to the blue graph in the answer part above.) After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2021-03-05 22:50:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5580377578735352, "perplexity": 367.37183065041154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178373761.80/warc/CC-MAIN-20210305214044-20210306004044-00333.warc.gz"}
https://math.stackexchange.com/questions/3385093/tricky-epsilon-delta-proof	# Tricky epsilon-delta proof $$\lim_{x\to -1}\frac{x-1}{x^2-x+1}=-\frac{2}{3}$$ What I've got so far is that: $$\forall\epsilon>0,\exists\delta>0\text{ s.t. }0<\|x-(-1)\|< \delta\implies\left\|\frac{x-1}{x^2-x+1}-(-\frac{2}{3})\right\|<\epsilon\\ \forall\epsilon>0,\exists\delta>0\text{ s.t. }0<\| x+1\|< \delta\implies\left\|\frac{(x+1)(2x-1)}{3x^2-3x+3}\right\|<\epsilon$$ How do I go about finding a value for delta from here? Thanks. • @Mason apologies, that was a typo on my part. Oct 8 '19 at 7:23 Take the case where $$x\in [-2,0]$$ (i.e. $$\delta < 1$$). Then we have the following inequality $$\left\|\frac{(x+1)(2x-1)}{3(x^2-x+1)}\right\| < \frac{5}{3}\|x+1\|$$ by maximizing the numerator and minimizing the denominator. So set $$\delta = \min(\frac{3}{5}\epsilon,1)$$ and the proof step for the limit follows in both cases. If $$\epsilon > \frac{5}{3}$$: $$\|x+1\|<1\implies \left\|\frac{(x+1)(2x-1)}{3(x^2-x+1)}\right\| < \frac{5}{3}\|x+1\| < \frac{5}{3} < \epsilon$$ If $$\epsilon \leq \frac{5}{3}$$: $$\|x+1\|<\frac{3}{5}\epsilon \implies \left\|\frac{(x+1)(2x-1)}{3(x^2-x+1)}\right\| < \frac{5}{3}\|x+1\| < \epsilon$$ • Everything seems great except I don't understand where you get the interval of [-2,0]. Could you elaborate on that, please? Thank you. Oct 8 '19 at 15:53 • Edit: I've figured out how you got [-2,0] but why do we need to get the greatest possible value out of the fraction if we want the minimum for our delta anyway? Oct 8 '19 at 16:18 • @mathgeek101 I wrote out the proof step because that's what makes it clear why we maximize the fraction. Oct 8 '19 at 18:38 Note that you have $$0<\|x+1\|<\delta$$, and you have $$(x+1)$$ in your numerator in the second line. So we make this more explicit: $$\left\|\frac{(x+1)(2x-1)}{3x^2-3x+3}\right\|=\|x+1\|\cdot \left\|\frac{2x-1}{3x^2-3x+3}\right\|<\epsilon$$ This is what we want to hold. Now, in order to get a handle on this, we have $$\|x+1\|\cdot \left\|\frac{2x-1}{3x^2-3x+3}\right\|<\delta\cdot \left\|\frac{2x-1}{3x^2-3x+3}\right\|$$ and $$\delta$$ we can control. So we want to pick a $$\delta$$ small enough that the right-hand side here is less than $$\epsilon$$, since that will automatically make the left-hand side smaller than $$\epsilon$$. Thus we have to see how large that second factor on the right-hand side could possibly become. I claim that it's always smaller than $$1$$. Which in turn gives us: $$\delta\cdot \left\|\frac{2x-1}{3x^2-3x+3}\right\|<\delta$$ So as long as we pick $$\delta= \epsilon$$, putting together all these inequalities gives us $$\left\|\frac{(x+1)(2x-1)}{3x^2-3x+3}\right\|<\epsilon$$, which is what we want, and we have proven our limit. Proof of claim: We can either do calculus to find max and min, or we do some algebra and split into cases. I'll go with the algebra option. Note that the numerator $$3x^2-3x+3$$ is always positive, so we may remove the absolute value signs from it. This gives us $$\left\|\frac{2x-1}{3x^2-3x+3}\right\| = \frac{\|2x-1\|}{3x^2-3x+3}<1\\ \|2x-1\|<3x^2-3x+3$$ For $$x\leq \frac12$$, this turns into $$1-2x<3x^2-3x+3$$, which is easily verified by the quadratic formula. For $$x\geq\frac12$$, it turns into $$2x-1<3x^2-3x+3$$, which is also easily verified with the quadratic formula. So we get that $$\left\|\frac{2x-1}{3x^2-3x+3}\right\|<1$$, and we are done.	2021-10-23 17:26:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 29, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9486476182937622, "perplexity": 105.45183173423118}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585737.45/warc/CC-MAIN-20211023162040-20211023192040-00096.warc.gz"}
https://www.aimsciences.org/article/doi/10.3934/dcdsb.2016.21.417	# American Institute of Mathematical Sciences March 2016, 21(2): 417-436. doi: 10.3934/dcdsb.2016.21.417 ## Stefan problem, traveling fronts, and epidemic spread 1 Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen Received November 2014 Revised April 2015 Published November 2015 The scalar reaction diffusion equation with a nonlinearity of logistic type has a minimal speed $c_0$ for standard traveling fronts. It is shown that also for speeds $0 < c < c_0$ there are traveling fronts but these are solutions to free boundary value (Stefan) problems. Furthermore, these speeds depend in a monotone way on the Stefan coefficient which links the loss of matter at the free boundary to the displacement per time. The results are extended to correlated random walks, Cattaneo systems and, in particular, to models for epidemic spread. In the epidemic problems a dichotomy phenomenon shows up: For small values of the Stefan coefficient there are no fronts indicating that for such values and certain data the free boundary stays bounded. Citation: Karl P. Hadeler. Stefan problem, traveling fronts, and epidemic spread. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 417-436. doi: 10.3934/dcdsb.2016.21.417 ##### References: show all references ##### References: [1] Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357 [2] Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519 [3] Samuel Herrmann, Nicolas Massin. Exit problem for Ornstein-Uhlenbeck processes: A random walk approach. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020058 [4] Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 789-799. doi: 10.3934/dcdsb.2014.19.789 [5] Harunori Monobe, Hirokazu Ninomiya. Traveling wave solutions with convex domains for a free boundary problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 905-914. doi: 10.3934/dcds.2017037 [6] Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769 [7] Mauro Garavello. Boundary value problem for a phase transition model. Networks & Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89 [8] Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337 [9] Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013 [10] Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212 [11] Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011 [12] Byung-Hoon Hwang, Seok-Bae Yun. Stationary solutions to the boundary value problem for the relativistic BGK model in a slab. Kinetic & Related Models, 2019, 12 (4) : 749-764. doi: 10.3934/krm.2019029 [13] Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667 [14] Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045 [15] Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122 [16] Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128 [17] Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234 [18] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [19] Norbert Požár, Giang Thi Thu Vu. Long-time behavior of the one-phase Stefan problem in periodic and random media. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 991-1010. doi: 10.3934/dcdss.2018058 [20] Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281 2018 Impact Factor: 1.008	2020-02-24 06:29:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6216232180595398, "perplexity": 4105.071353249238}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145897.19/warc/CC-MAIN-20200224040929-20200224070929-00445.warc.gz"}
https://socratic.org/questions/what-is-the-relationship-between-diatomic-molecules-and-diatomic-elements	# What is the relationship between diatomic molecules and diatomic elements? So ${O}_{2}$ is a diatomic element (and thus a diatomic molecule) $H C l$ is a diatomic molecule, but not a diatomic element.	2020-01-29 20:09:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7772149443626404, "perplexity": 1320.2249595090705}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251802249.87/warc/CC-MAIN-20200129194333-20200129223333-00162.warc.gz"}
https://www.ias.ac.in/listing/bibliography/pram/C_L_Mehta	• C L Mehta Articles written in Pramana – Journal of Physics • Hölder’s inequality for matrices We prove that for arbitraryn×n matricesA1,A2,…,Am and for positive real numbersp1,p2,…,pm withp1−1+p2−1+…+pm/−1=1, the inequality$$\|Tr(A_1 A_2 ...A_m )^2 \|< \mathop {II}\limits_{k = 1}^m [Tr(A_k^\dag A_k )^{p_k } ]P_k^{ - 1}$$ holds. • Evanescent waves and the van Cittert Zernike theorem in cylindrical geometry The cylindrical angular spectrum of the wavefield is introduced. In this representation the field consists of homogeneous as well as evanescent waves. The representation is applied to propagation problems and an analogue of van Cittert Zernike theorem is obtained in cylindrical geometry. • Extremum uncertainty product and sum states We consider the states with extremum products and sums of the uncertainties in non-commuting observables. These are illustrated by two specific examples of harmonic oscillator and the angular momentum states. It shows that the coherent states of the harmonic oscillator are characterized by the minimum uncertainty sum 〈(Δq)2〉 + 〈(Δp)2〉. The extremum values of the sums and products of the uncertainties of the components of the angular momentum are also obtained. • Intensity fluctuations in thermal light with orthogonally polarised multiple-peak spectrum The moment generating function of the integrated light intensity of thermal radiation having multiple peak spectrum is obtained. Cases of two-peak and three-peak spectra where different peaks are in orthogonal states of polarisation are considered. The moment generating function is shown to be the product of two simpler generating functions. • Two-mode para-Bose number states Two-mode para-Bose number states are discussed. The two-mode system has been chosen as it is a representative of the multi-mode system. Salient properties like normalization, orthogonality and degeneracy of these states have also been discussed. • Squeezed vacuum as an eigenstate of two-photon annihilation operator We introduce the inverse annihilation and creation operatorsâ−1 andâ+-1 by their actions on the number states. We show that the squeezed vacuum exp(1/2;ξâ+2]\|0> and squeezed first number state exp[1/2;ξâ+2]\|n=1> are respectively the eigenstates of the operators (â†−1â) and (ââ+-1) with the eigenvalue ξ. • # Pramana – Journal of Physics Volume 94, 2020 All articles Continuous Article Publishing mode • # Editorial Note on Continuous Article Publication Posted on July 25, 2019	2020-09-20 11:47:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8001424670219421, "perplexity": 1528.073284285456}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400197946.27/warc/CC-MAIN-20200920094130-20200920124130-00535.warc.gz"}
http://math.stackexchange.com/questions/215646/if-i-have-n-choose-k-how-can-i-rewrite-it-to-be-something-that-ends-in-choose	# If I have (n choose k) how can I rewrite it to be something that ends in “choose k+1”? See title for question; just trying to rewrite a combinatoric - Suppose that $k\ne n$. Then $$\binom{n}{k}=\frac{n!}{k!(n-k)!}=(k+1)\frac{n!}{(k+1)!(n-k)!}=\frac{k+1}{n-k}\frac{n!}{(k+1)!(n-k-1)!}.$$ It follows that $$\binom{n}{k}=\frac{k+1}{n-k}\binom{n}{k+1}.$$	2014-11-22 16:31:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9704585075378418, "perplexity": 377.57288505377187}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400378429.52/warc/CC-MAIN-20141119123258-00221-ip-10-235-23-156.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-geometry/118213-upper-half-plane.html	1. ## upper half plane Compute the following integral. $\displaystyle \int^{\infty}_{0} \frac{x^2}{x^8+1}dx.$ I think that I should get $\displaystyle \frac{1}{4} \sqrt{1-\frac{1}{\sqrt 2}} \cdot \pi$. I see we know this is even. Then there is work to show that $\displaystyle \int^{\infty}_{-\infty}$ is$\displaystyle 2\pi i$ times the sum of the residues of $\displaystyle \frac{z^2}{z^8+1}$ in the upper half plane. Then I found that $\displaystyle c_k = \text{exp}[i(\frac{\pi}{8}+\frac{2k\pi}{8}]$ where $\displaystyle k=0,1, \ldots, 7.$ Then the zeros in the upper half plane occur at $\displaystyle k=0,1,2,3$. Then I used $\displaystyle \text{Res}_{z=c_k}=\frac{c_k^2}{8c_k^7}=\frac{1}{8 c_k^5}=\frac{1}{8}c_k^{-5}$ where $\displaystyle k=0,1,2,3.$ Then when I add these residues up I did not get the answer above, I got $\displaystyle \frac{\sqrt{\sqrt{2}+2} \cdot \pi}{16} - \frac{\sqrt{2-\sqrt{2}} \cdot \pi }{16}i.$ 2. Originally Posted by xboxlive89128 Compute the following integral. $\displaystyle \int^{\infty}_{0} \frac{x^2}{x^8+1}dx.$ I think that I should get $\displaystyle \frac{1}{4} \sqrt{1-\frac{1}{\sqrt 2}} \cdot \pi$. I see we know this is even. Then there is work to show that $\displaystyle \int^{\infty}_{-\infty}$ is$\displaystyle 2\pi i$ times the sum of the residues of $\displaystyle \frac{z^2}{z^8+1}$ in the upper half plane. Then I found that $\displaystyle c_k = \text{exp}[i(\frac{\pi}{8}+\frac{2k\pi}{8}]$ where $\displaystyle k=0,1, \ldots, 7.$ Then the zeros in the upper half plane occur at $\displaystyle k=0,1,2,3$. Then I used $\displaystyle \text{Res}_{z=c_k}=\frac{c_k^2}{8c_k^7}=\frac{1}{8 c_k^5}=\frac{1}{8}c_k^{-5}$ where $\displaystyle k=0,1,2,3.$ Then when I add these residues up I did not get the answer above, I got $\displaystyle \frac{\sqrt{\sqrt{2}+2} \cdot \pi}{16} - \frac{\sqrt{2-\sqrt{2}} \cdot \pi }{16}i.$ That seems to be pretty much correct, but it's not obvious how to transform your answer to the given one. I would take the residues as $\displaystyle e^{ik\pi/8}$, where k = 1, 3, 5, 7. Their negative-fifth powers are $\displaystyle e^{ik\pi/8}$, where k = 1, 7, –3, –5. The sum of the first two of these is $\displaystyle 2i\sin(\pi/8)$, and the sum of the other two is $\displaystyle -2i\sin(3\pi/8)$. So the sum of the four residues is $\displaystyle \tfrac i4(\sin(\pi/8) - \sin(3\pi/8)) = -\tfrac i2\cos(\pi/4)\sin(\pi/8)$ (addition formula for trig functions!). Now use $\displaystyle \cos(\pi/4) = 1/\sqrt2$ and $\displaystyle \sin(\pi/8) = \frac1{\sqrt2}\sqrt{1-\frac1{\sqrt2}}$ to see that $\displaystyle 2\pi i$ times the sum of the residues is $\displaystyle \frac\pi2\sqrt{1-\frac1{\sqrt2}}$. That is the integral from $\displaystyle -\infty$ to $\displaystyle \infty$, so you need to divide by 2 to get the integral from 0 to $\displaystyle \infty$.	2018-06-22 17:55:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9755666851997375, "perplexity": 87.0963164203005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864740.48/warc/CC-MAIN-20180622162604-20180622182604-00507.warc.gz"}
https://cartesianproduct.wordpress.com/tag/software/	## Failing software – again The line above is from a real (and current at time-of-posting) job advertisement for a software developer. I’m not positing it because I think it is bad, shocking or dangerous, but mainly because it is illustrative of the real world: developers are expected to be “pragmatic” when it comes to testing the software they make for correctness. So when, as happened today with British Airways, we see a major software failure (or at least what looks like a major software failure), maybe we should not be too surprised. There is more to it, of course, than the fact developers are expected to compromise on testing their code: there is the simple fact that most functions are not computable at all and that for many others we don’t know how to compute them. For an example of something uncomputable there is Hilbert’s tenth problem: For any given a Diophantine equation is there a general algorithm that tells us whether or not there is a solution with the unknowns taking integer values… • A Diophantine equation (named after Diophantus of Alexandria) is a polynomial (e.g., $x = y$, $y^2 + 5x + z^{67} = 0$, $3x + 2y +5x^2y^3 + 90 = 7z$ and so on) equation where all the coefficients of the unknowns are integer (counting number) values and where there are a finite number of unknowns. • An algorithm is a set of rules of simple mathematical procedures to solve a problem So the task is to take a Diophantine equation and not to find a solution but to determine whether such a solution exists – but no such algorithm exists and therefore we cannot write any sort of computer program that would do this for us, no matter how powerful a computer we were using. And for an example of something for which a solution does exist but which we will struggle to find, there is the famoustravelling salesman problem” – namely given a set of cities all connected by roads (or railways, or air routes) what is the most efficient way of visiting all the cities. This problem is an example of what is known as an “NP” problem – in other words one for which (we believe) no general algorithm exists to produce a solution in “polynomial time” (meaning in a time related to the length of the input – in this case the number of cities). Plainly a solution does exist – there is a quickest way to travel to all the cities – but the only immediately available way to be certain of finding this is to try all the solutions. But that will take time: for even moderately sized problems quite possibly more time than we have before the Sun swallows the Earth. To get round this we have to use heuristics (essentially educated guesswork) or approximations. In the travelling salesman case those approximations are powerful and efficient – that’s why modern logistics works – but the general point remains the same: in many cases we are building software that approximates the solution to our problem but may not answer it fully correctly and, unless there is a breakthrough which eliminates NP problems as a class, we are always going to be stuck with that. Steve McNamara, general secretary of the Licensed Taxi Drivers’ Association in London, quoted in the Guardian on the prospect of driverless buses in the capital: “We don’t have a a lot of confidence in anything that comes out of TfL [Transport for London], to be honest, and the fact that they’re suggesting it means it’s almost certainly likely not to happen. “Who knows with technology, but some of the simplest things, they still can’t do. The best example is voice recognituion technology. “If you’ve got it on your car… it’s rubbish. If you’ve got it on your phone, it’s probably worse. They’re all crap, aren’t they? None of them work, and they can’t even get that right. And they expect people to get into driverless cars?” Where do you begin with this? Firstly, we should note that the Mayor’s office ran a million miles away from the suggestion – in their own paper – that at some point between now and 2050 driverless buses will be on London’s streets. To make it worse they – plainly less than truthfully – tried to claim that references in their own paper to driverless vehicles were a reference to tube trains. The disappointing thing is that instead of actually once again pioneering a public transport technology – London gave the world underground railways and once had the world’s most admired bus network too – London’s public admisitrators are not willing to lead. Before anyone on the left says “what about the jobs”, my reply is “what about them?” Is not the left meant to be about freeing human creativity from the realm of necessity? The issue is the distribution of the opportunities freed by the removal of the need to drive buses – it cannot be about preserving relatively low-skilled jobs that are no longer required. As for Steve McNamara, I am amused by the fact he thinks speech recognition is the “simplest thing”. Should we reply that three billion years of evolution produced only one species that can speak so it can’t be that simple? Or perhaps ask McNamara how many languages he can speak given that speech recognition is so simple? In fact, my guess would be that speech recognition is probably many more times more difficult, computationally speaking, than driving a bus. However the risk of human injury means that speech recognition software is socially more acceptible than driverless vehicles – for now. But I don’t expect that to last. ## Time to write a signal handler? I am trying to execute some self-written pieces of software that require a lot of wall clock time – around three weeks. I run them on the University of York‘s compute server which is rebooted on the first Tuesday of every month, so the window for the software is limited. I have until about 7 am on 5 August before the next reboot. To add to the complication the server runs Kerberos which does not seem to play well with the screen/NFS combination I am using. And – I keep killing the applications in error – this time, just half an hour ago I assume I was on a dead terminal session (ie an ssh login which had long since expired) and pressed ctrl-C, only to my horror to discover it was a live screen (it had not responded to ctrl-A, ctrl-A for whatever reason). Time to add a signal handler to catch ctrl-C to at least give me the option of changing my mind! ## President Obama’s campaign and free software It seems a row has broken out between staff on President Barack Obama’s re-election campaign over the fate of the free software it produced (the article linked here refers to it all as “open source” but on this issue I tend to side with RMS and not ESR). Actually I do not blame the DNC at all for not wanting to release any source (if that is what they want to do – it is not entirely clear). It would simply be foolish to surrender an advantage they have over their opponents if there is no need to do so. Nor does there appear to be any ethical issue involved: the core idea of the free software movement is surely that any user of software should have access to the source code out of which it is built. If the DNC does not distribute the software then they are under no moral or any other obligation to hand out the source code. By far the worst idea the article talks of is selling the software: that would truly be a breach of the ethics of free software – because plainly trying to use the built software as a revenue stream means keeping the software hidden or forcing users, 1970s Unix-style, to sign NDAs. Either of those is worse than keeping a piece of private software private. There is a wider question, of course, could distributing the software help build a better world. But if the distribution helps the US republican party, then surely the answer for the DNC is no? ## Here we go again I used to have a blog. It was meant to be about “politics and free software” (not the politics of free software) but ended up being mainly about politics. I wrote the last entry on that in January 2008 and subsequently took it off line (the content is still on my server at home and it was amusing to read it again just now, but it’s not going back up). My politics haven’t changed – so if you want to do something to make Britain a better place to live I still recommend you start here. But I am not going to write about politics here. The geeky title ought to give the game away – this one is about computing (and, I suppose, mathematics to an extent). My inspiration came from this: generally speaking I am in the n – log(n) part of this matrix and while I am not interested in pursuing a career in computing I am passionate about improving my skills and competency, so the comment that a log(n) programmer “maintains a blog in which personal insights and thoughts on programming are shared”, left me with little choice. Of course I’ll actually have to demonstrate some insights and thoughts too.	2017-10-22 15:26:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45393985509872437, "perplexity": 901.0679501727085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825308.77/warc/CC-MAIN-20171022150946-20171022170946-00209.warc.gz"}
https://projecteuclid.org/euclid.agt/1535594417	## Algebraic & Geometric Topology ### The factorization theory of Thom spectra and twisted nonabelian Poincaré duality Inbar Klang #### Abstract We give a description of the factorization homology and $E n$ topological Hochschild cohomology of Thom spectra arising from $n$–fold loop maps $f : A → B O$, where $A = Ω n X$ is an $n$–fold loop space. We describe the factorization homology $∫ M Th ( f )$ as the Thom spectrum associated to a certain map $∫ M A → B O$, where $∫ M A$ is the factorization homology of $M$ with coefficients in $A$. When $M$ is framed and $X$ is $( n − 1 )$–connected, this spectrum is equivalent to a Thom spectrum of a virtual bundle over the mapping space $Map c ( M , X )$; in general, this is a Thom spectrum of a virtual bundle over a certain section space. This can be viewed as a twisted form of the nonabelian Poincaré duality theorem of Segal, Salvatore and Lurie, which occurs when $f : A → B O$ is nullhomotopic. This result also generalizes the results of Blumberg, Cohen and Schlichtkrull on the topological Hochschild homology of Thom spectra, and of Schlichtkrull on higher topological Hochschild homology of Thom spectra. We use this description of the factorization homology of Thom spectra to calculate the factorization homology of the classical cobordism spectra, spectra arising from systems of groups and the Eilenberg–Mac Lane spectra $H ℤ ∕ p$, $H ℤ ( p )$ and $H ℤ$. We build upon the description of the factorization homology of Thom spectra to study the ($n = 1$ and higher) topological Hochschild cohomology of Thom spectra, which enables calculations and a description in terms of sections of a parametrized spectrum. If $X$ is a closed manifold, Atiyah duality for parametrized spectra allows us to deduce a duality between $E n$ topological Hochschild homology and $E n$ topological Hochschild cohomology, recovering string topology operations when $f$ is nullhomotopic. In conjunction with the higher Deligne conjecture, this gives $E n + 1$–structures on a certain family of Thom spectra, which were not previously known to be ring spectra. #### Article information Source Algebr. Geom. Topol., Volume 18, Number 5 (2018), 2541-2592. Dates Revised: 2 May 2018 Accepted: 20 May 2018 First available in Project Euclid: 30 August 2018 https://projecteuclid.org/euclid.agt/1535594417 Digital Object Identifier doi:10.2140/agt.2018.18.2541 Mathematical Reviews number (MathSciNet) MR3848394 Zentralblatt MATH identifier 06935815 #### Citation Klang, Inbar. The factorization theory of Thom spectra and twisted nonabelian Poincaré duality. Algebr. Geom. Topol. 18 (2018), no. 5, 2541--2592. doi:10.2140/agt.2018.18.2541. https://projecteuclid.org/euclid.agt/1535594417 #### References • M Ando, A J Blumberg, D Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map, preprint (2011) • M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, An $\infty$–categorical approach to $R$–line bundles, $R$–module Thom spectra, and twisted $R$–homology, J. Topol. 7 (2014) 869–893 • M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, Units of ring spectra, orientations and Thom spectra via rigid infinite loop space theory, J. Topol. 7 (2014) 1077–1117 • D Ayala, J Francis, Zero-pointed manifolds, preprint (2014) • D Ayala, J Francis, Factorization homology of topological manifolds, J. Topol. 8 (2015) 1045–1084 • M Barratt, S Priddy, On the homology of non-connected monoids and their associated groups, Comment. Math. Helv. 47 (1972) 1–14 • J Beardsley, Relative Thom spectra via operadic Kan extensions, Algebr. Geom. Topol. 17 (2017) 1151–1162 • J Beardsley, Topological Hochschild homology of $X(n)$, preprint (2017) • A J Blumberg, Topological Hochschild homology of Thom spectra which are $E_\infty$ ring spectra, J. Topol. 3 (2010) 535–560 • A J Blumberg, R L Cohen, C Schlichtkrull, Topological Hochschild homology of Thom spectra and the free loop space, Geom. Topol. 14 (2010) 1165–1242 • I Bobkova, A Lindenstrauss, K Poirier, B Richter, I Zakharevich, On the higher topological Hochschild homology of $\mathbb F_p$ and commutative $\mathbb F_p$–group algebras, from “Women in topology: collaborations in homotopy theory” (M Basterra, K Bauer, K Hess, B Johnson, editors), Contemp. Math. 641, Amer. Math. Soc., Providence, RI (2015) 97–122 • F R Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978) 99–110 • F R Cohen, J P May, L R Taylor, $K({\bf Z},0)$ and $K(Z\sb{2},0)$ as Thom spectra, Illinois J. Math. 25 (1981) 99–106 • R L Cohen, Stable proofs of stable splittings, Math. Proc. Cambridge Philos. Soc. 88 (1980) 149–151 • R L Cohen, J D S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002) 773–798 • R L Cohen, J D S Jones, Gauge theory and string topology, Bol. Soc. Mat. Mex. 23 (2017) 233–255 • E S Devinatz, M J Hopkins, J H Smith, Nilpotence and stable homotopy theory, I, Ann. of Math. 128 (1988) 207–241 • B I Dundas, A Lindenstrauss, B Richter, Towards an understanding of ramified extensions of structured ring spectra, Math. Proc. Cambridge Philos. Soc. (online publication March 2018) • A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, Amer. Math. Soc., Providence, RI (1997) • J Francis, The tangent complex and Hochschild cohomology of $\mathscr E_n$–rings, Compos. Math. 149 (2013) 430–480 • V Franjou, J Lannes, L Schwartz, Autour de la cohomologie de Mac Lane des corps finis, Invent. Math. 115 (1994) 513–538 • V Franjou, T Pirashvili, On the Mac Lane cohomology for the ring of integers, Topology 37 (1998) 109–114 • D Gepner, M Groth, T Nikolaus, Universality of multiplicative infinite loop space machines, Algebr. Geom. Topol. 15 (2015) 3107–3153 • G Ginot, T Tradler, M Zeinalian, Higher Hochschild cohomology, Brane topology and centralizers of $E_n$–algebra maps, preprint (2012) • G Ginot, T Tradler, M Zeinalian, Higher Hochschild homology, topological chiral homology and factorization algebras, Comm. Math. Phys. 326 (2014) 635–686 • K Gruher, P Salvatore, Generalized string topology operations, Proc. Lond. Math. Soc. 96 (2008) 78–106 • K Hess, P-E Parent, J Scott, CoHochschild homology of chain coalgebras, J. Pure Appl. Algebra 213 (2009) 536–556 • G Horel, Operads, modules and topological field theories, preprint (2014) • G Horel, Higher Hochschild cohomology of the Lubin–Tate ring spectrum, Algebr. Geom. Topol. 15 (2015) 3215–3252 • G Horel, Factorization homology and calculus à la Kontsevich Soibelman, J. Noncommut. Geom. 11 (2017) 703–740 • P Hu, Higher string topology on general spaces, Proc. London Math. Soc. 93 (2006) 515–544 • P Hu, I Kriz, A A Voronov, On Kontsevich's Hochschild cohomology conjecture, Compos. Math. 142 (2006) 143–168 • B Knudsen, Betti numbers and stability for configuration spaces via factorization homology, Algebr. Geom. Topol. 17 (2017) 3137–3187 • A Kupers, J Miller, $E_n$–cell attachments and a local-to-global principle for homological stability, Math. Ann. 370 (2018) 209–269 • L G Lewis, Jr, J P May, M Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer (1986) • J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009) • J Lurie, Higher algebra, book project (2017) \setbox0\makeatletter\@url http://www.math.harvard.edu/~lurie/papers/HA.pdf {\unhbox0 • M Mahowald, D C Ravenel, P Shick, The Thomified Eilenberg–Moore spectral sequence, from “Cohomological methods in homotopy theory” (J Aguadé, C Broto, C Casacuberta, editors), Progr. Math. 196, Birkhäuser, Basel (2001) 249–262 • E J Malm, String topology and the based loop space, preprint (2011) • J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer (1972) • J P May, What precisely are $E_\infty$ ring spaces and $E_\infty$ ring spectra?, from “New topological contexts for Galois theory and algebraic geometry” (A Baker, B Richter, editors), Geom. Topol. Monogr. 16, Geom. Topol. Publ., Coventry (2009) 215–282 • D McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975) 91–107 • J Miller, Nonabelian Poincaré duality after stabilizing, Trans. Amer. Math. Soc. 367 (2015) 1969–1991 • S Priddy, $K({\bf Z}/2)$ as a Thom spectrum, Proc. Amer. Math. Soc. 70 (1978) 207–208 • D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 2nd edition, AMS Chelsea Ser. 347, Amer. Math. Soc., Providence, RI (2004) • P Salvatore, Configuration spaces with summable labels, from “Cohomological methods in homotopy theory” (J Aguadé, C Broto, C Casacuberta, editors), Progr. Math. 196, Birkhäuser, Basel (2001) 375–395 • P Salvatore, Configuration spaces on the sphere and higher loop spaces, Math. Z. 248 (2004) 527–540 • P Salvatore, N Wahl, Framed discs operads and Batalin–Vilkovisky algebras, Q. J. Math. 54 (2003) 213–231 • C Schlichtkrull, Units of ring spectra and their traces in algebraic $K$–theory, Geom. Topol. 8 (2004) 645–673 • C Schlichtkrull, Higher topological Hochschild homology of Thom spectra, J. Topol. 4 (2011) 161–189 • S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511 • G Segal, Locality of holomorphic bundles, and locality in quantum field theory, from “The many facets of geometry” (O García-Prada, J P Bourguignon, S Salamon, editors), Oxford Univ. Press (2010) 164–176 • T Veen, Detecting periodic elements in higher topological Hochschild homology, Geom. Topol. 22 (2018) 693–756	2019-06-24 21:26:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 24, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7190632224082947, "perplexity": 2456.4048787496795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999740.32/warc/CC-MAIN-20190624211359-20190624233359-00063.warc.gz"}
https://socratic.org/questions/531e867a02bf342492a08b07	# Question #08b07 Mar 11, 2014 A spectator ion is an ion that exists as both a reactant and a product in a chemical equation. #### Explanation: Consider the reaction $\text{NaCl(aq)" +"AgNO"_3"(aq)" → "AgCl(s)" + "NaNO"_3"(aq)}$ The ionic equation is: $\text{Na"^+"(aq)" + "Cl"^(-)"(aq)" + "Ag"^+"(aq)" + "NO"_3^(-)"(aq)" →"AgCl(s)" + "Na"^+"(aq)" + "NO"_3^(-)"(aq)}$ The ${\text{Na}}^{+}$ and ${\text{NO}}_{3}^{-}$ ions are spectator ions because they remain unchanged on both sides of the equation. They just "watch" the other ions react, hence the name. We usually cancel the spectator ions from each side of an ionic equation to form a net ionic equation: $\cancel{\text{Na⁺(aq)") + "Cl"^(-)"(aq)" + "Ag"^+"(aq)" + cancel("NO₃⁻(aq)") →"AgCl(s)" + cancel("Na⁺(aq)") + cancel("NO₃⁻(aq)}}$ $\text{Cl"^(-)"(aq)" + "Ag"^+"(aq)" → "AgCl(s)}$ EXAMPLE: Identify the spectator ions and write the net ionic equation for the reaction: $\text{HCl(aq)" + "NaHCO"_3"(aq)" → "NaCl(aq)" + "H"_2"O(l)" + "CO"_2"(g)}$ SOLUTION: Ionic Equation: $\text{H"^+"(aq)" + "Cl"^(-)"(aq)" + "Na"^+"(aq)" + "HCO"_3^(-)"(aq)" → "Na"^+"(aq)" + "Cl"^(-)"(aq)" + "H"_2"O(l)" + "CO"₂"(g)}$ The spectator Ions are ${\text{Cl}}^{-}$ and ${\text{Na}}^{+}$. Net Ionic Equation: $\text{H"^+"(aq)" + cancel("Cl⁻(aq)") + cancel("Na⁺(aq)") + "HCO"_3^(-)"(aq)" → cancel("Na⁺(aq)") + cancel("Cl⁻(aq)") + "H"_2"O(l)" + "CO"_2"(g)}$ $\text{H"^+"(aq)" + "HCO"_3^(-)"(aq)" → "H"_2"O(l)" + "CO"_2"(g)}$ Here is a video on spectator ions	2019-03-18 14:14:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7625601887702942, "perplexity": 4220.199424930087}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201329.40/warc/CC-MAIN-20190318132220-20190318153628-00047.warc.gz"}
https://cs.stackexchange.com/questions/115429/example-of-a-code-that-can-be-decoded-using-bounded-distance-decoder	# Example of a code that can be decoded using bounded distance decoder In the book Information theory, Inference and Learning Algorithm, in chapter 13, MacKay defines bounded distance decoding A bounded distance decoder is a decoder that returns the closest codeword to a received binary vector $$\mathbf{r}$$ if the distance from $$\mathbf{r}$$ to that codeword is less than or equal to $$t$$; otherwise it returns a failure message. Could anybody provide an example of a simple linear code that can be decoded by such a bounded distance decoder? The simplest example is a repetition code. You encode the bit 0 as 000 and the bit 1 as 111. The decoder outputs the majority bit, which is always at distance at most $$t = 1$$. As another example, suppose we encode 0 as 0000 and 1 as 1111. We can still correct up to one error (by taking the majority), but we cannot accommodate $$t = 2$$, since given 0011 we can't tell whether it came from 0000 or from 1111. Therefore we need to declare failure. Any perfect code will do. A perfect code is a $$t-$$error correcting code where the spheres of Hamming radius $$t$$ around the codewords cover the whole space. Edit: In response to OP's comment, if the $$t-$$spheres around codewords overlap, the code, by definition, is not a $$t-$$error correcting code. In general, all decoding algorithms used in practice are bounded distance decoding algorithms, since except for very short lengths, it is not feasible to do a brute force comparison of the received word with all codewords. The Syndrome decoding of a linear code is one such example, see any coding textbook, as well as Wikipedia entry here • thanks, I was looking for examples of nonperfect codes for which this becomes necessary – user2723984 Oct 9 '19 at 17:01 • what becomes necessary, sorry? – kodlu Nov 3 '19 at 0:30 • in a perfect code the distance from the received codeword to the nearest codeword is never more than the minimum distance. An error can be erroneously corrected, but the decoder will never know, e.g. in the repetition code, two errors might turn $000$ in $110$, which will be corrected into $111$, but the distance from $110$ to the nearest codeword is still $1$, hence the decoder will never "give up". You might construct a perfect code of distance $5$ and artificially give up on correcting more than $1$ error, but [...] – user2723984 Nov 3 '19 at 10:03 • I was asking if there are codes where the spheres do overlap and to avoid this we bound the maximum distance of the decoding – user2723984 Nov 3 '19 at 10:04	2021-05-15 09:16:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7044679522514343, "perplexity": 469.3929313232476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991378.48/warc/CC-MAIN-20210515070344-20210515100344-00460.warc.gz"}
https://mathematica.stackexchange.com/questions/linked/86972	Linked Questions 11 questions linked to/from Iteration variable in variable 2answers 819 views What alternative notation of subscripts/superscripts that are also treated as SYMBOLS in Mathematica? I tried to avoid using the subscript $A_0$ in the Module, and get an error with $A[0]$. ... 5answers 3k views How to create symbols from strings and set values for them? I am trying to convert a list of string names into symbols, which will then be used to store data. I have 24 files (where the name of each file is a member of the list mentioned above) that I need to ... 4answers 1k views Create an adaptive amount of local variables for error propagation I intend to write a function which calculates the result and the error for any formula with any amount of variables using the Gauß Error Propagation. The error $\mathrm{d}R$ for a function $A(a,b,c)$ ... 3answers 7k views Table of Variables I use a list of variables {x1, x2, x3} to Solve a particular set of equations. I am now trying to generalise this depending on ... 3answers 413 views How can I test properties of a symbol from the string name without the symbol completely evaluating Suppose I have a few symbols, one of which has a value: {abc1, abc2 = 5, abc3}; I can use Names to get the list of names, as ... 2answers 5k views Generating a vector of dummy variables So I'm the situation of needing analytical solutions to a family of equations of the form Ax=b, where A is an nxn matrix. I've written a function that does what I want, but I'm currently using a bit ... 2answers 858 views How to replace subscript symbols I have an expression with subscripted variables. I would like to replace all those terms with other symbols. For example, $P_1 + P_2^3 \rightarrow P1 + P2^3$ I used the following code but it doesn't ... 3answers 308 views How to 'merge' a list like FromDigits, but with a mixture of numbers and symbols? How do I go from {C,4,G,5,S,7} to C4G5S7 2answers 470 views Automatically compute r[1] = r1; r[2] = r2; … I have three variables like this, r1 = 4; r2 = 3; r3 = 1; and I want to insert them into a function use them in a function definition like so: ... 1answer 1k views Proper use of arbitrary number of variables So, I'm working on a project where the number of independent variables is not fixed. Consider a problem of $N$ independent variables, $\boldsymbol{r}$. I want to perform different things with them. ... 1answer 206 views Arbitrary number of variables in Module, Block, etc How do I specify an arbitrary number of variable names and initialize them inside of a Module or Block? For example, I want to do something like: ...	2021-01-27 23:19:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.765412449836731, "perplexity": 679.062063682516}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704833804.93/warc/CC-MAIN-20210127214413-20210128004413-00269.warc.gz"}
https://www.groundai.com/project/analysis-of-the-axialvector-doubly-heavy-tetraquark-states-with-qcd-sum-rules/	1 Introduction ## Abstract In this article, we construct the axialvector-diquark-scalar-antidiquark type currents to interpolate the axialvector doubly heavy tetraquark states, and study them with the QCD sum rules in details by carrying out the operator product expansion up to the vacuum condensates of dimension 10. Analysis of the axialvector doubly heavy tetraquark states with QCD sum rules [2mm] Zhi-Gang Wang 1 Department of Physics, North China Electric Power University, Baoding 071003, P. R. China PACS number: 12.39.Mk, 12.38.Lg Key words: Tetraquark state, QCD sum rules ## 1 Introduction The scattering amplitude for one-gluon exchange is proportional to taijtakl = −13(δijδkl−δilδkj)+16(δijδkl+δilδkj), (1) where , the is the Gell-Mann matrix. The negative sign in front of the antisymmetric antitriplet indicates the interaction is attractive while the positive sign in front of the symmetric sextet indicates the interaction is repulsive, the attractive interaction favors formation of the diquarks in color antitriplet [1]. The color antitriplet diquarks with or only have two structures in Dirac spinor space, where and for the axialvector and tensor diquarks, respectively. The axialvector diquarks are more stable than the tensor diquarks . Recently, the LHCb collaboration observed the doubly charmed baryon state in the mass spectrum in a data sample collected by LHCb at with a signal yield of , and measured the mass, but did not determine the spin [2]. The maybe have the spin or , we can take the diquark as basic constituent to construct the current JΞcc(x) = εijkcTi(x)Cγμcj(x)γ5γμuk(x), (2) or JμΞcc(x) = εijkcTi(x)Cγμcj(x)uk(x), (3) to study it with the QCD sum rules [3]. In this article, we choose the axialvector diquarks to construct the currents to interpolate the doubly heavy tetraquark states. Up to now, no experimental candidates for the quark configurations or have been observed. There have been several works on the doubly heavy tetraquark states, such as potential quark models or simple quark models [4, 5], QCD sum rules [6, 7, 8], heavy quark symmetry [9, 10], lattice QCD [11, 12], etc. Although the doubly heavy tetraquark states have been studied with the QCD sum rules, the energy scale dependence of the QCD sum rules has not been studied yet. In Refs.[13, 14, 15, 16, 17], we observe that in the QCD sum rules for the hidden charm (or bottom) tetraquark states and molecular states, the integrals ∫s04m2Q(μ)dsρQCD(s,μ)exp(−sT2), (4) are sensitive to the heavy quark masses , where the denotes the QCD spectral densities and the denotes the Borel parameters. Variations of the heavy quark masses or the energy scales lead to changes of integral ranges of the variable besides the QCD spectral densities , therefore changes of the Borel windows and predicted masses and pole residues. In this article, we revisit the QCD sum rules for the axialvector doubly heavy tetraquark states and choose the optimal energy scales to extract the masses. The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the axialvector doubly heavy tetraquark states in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusion. ## 2 The QCD sum rules for the axialvector doubly heavy tetraquark states In the following, we write down the two-point correlation functions and in the QCD sum rules, ΠJ/ημν(p) = i∫d4xeip⋅x⟨0\|T{J/ημ(x)J/η†ν(0)}\|0⟩, (5) where Jμ(x) = εijkεimnQTi(x)CγμQj(x)¯um(x)γ5C¯sTn(x), (6) ημ(x) = εijkεimnQTi(x)CγμQj(x)¯um(x)γ5C¯dTn(x), (7) , the , , , , are color indexes, the is the charge conjugation matrix. On the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators and into the correlation functions and respectively to obtain the hadronic representation [18, 19], and isolate the ground state contributions, ΠJ/ημν(p) = λ2ZM2Z−p2(−gμν+pμpνp2)+⋯ (8) = ΠJ/η(p2)(−gμν+pμpνp2)+⋯, where the pole residues are defined by , the are the polarization vectors of the axialvector tetraquark states . In the following, we briefly outline the operator product expansion for the correlation functions and in perturbative QCD. We contract the , , and quark fields in the correlation functions and with Wick theorem, and obtain the results: ΠJμν(p) = −2iεijkεimnεi′j′k′εi′m′n′∫d4xeip⋅x (9) Tr[γμSkk′Q(x)γνCSTjj′Q(x)C]Tr[γ5Um′m(−x)γ5CSTn′n(−x)C], Πημν(p) = −2iεijkεimnεi′j′k′εi′m′n′∫d4xeip⋅x (10) Tr[γμSkk′Q(x)γνCSTjj′Q(x)C]Tr[γ5Um′m(−x)γ5CDTn′n(−x)C], where the , , and are the full , , and quark propagators, respectively [19, 20], U/Dij(x) = iδij⧸x2π2x4−δij⟨¯qq⟩12−δijx2⟨¯qgsσGq⟩192−igsGaαβtaij(⧸xσαβ+σαβ⧸x)32π2x2 (11) −18⟨¯qjσμνqi⟩σμν+⋯, Sij(x) = iδij⧸x2π2x4−δijms4π2x2−δij⟨¯ss⟩12+iδij⧸xms⟨¯ss⟩48−δijx2⟨¯sgsσGs⟩192+iδijx2⧸xms⟨¯sgsσGs⟩1152 (12) −igsGaαβtaij(⧸xσαβ+σαβ⧸x)32π2x2−18⟨¯sjσμνsi⟩σμν+⋯, SijQ(x) = i(2π)4∫d4ke−ik⋅x{δij⧸k−mQ−gsGnαβtnij4σαβ(⧸k+mQ)+(⧸k+mQ)σαβ(k2−m2Q)2 (13) −g2s(tatb)ijGaαβGbμν(fαβμν+fαμβν+fαμνβ)4(k2−m2Q)5+⋯⎫⎬⎭, fλαβ = (⧸k+mQ)γλ(⧸k+mQ)γα(⧸k+mQ)γβ(⧸k+mQ), fαβμν = (⧸k+mQ)γα(⧸k+mQ)γβ(⧸k+mQ)γμ(⧸k+mQ)γν(⧸k+mQ). (14) Then we compute the integrals both in coordinate space and in momentum space, and obtain the correlation functions at the quark level, therefore the QCD spectral densities through dispersion relation. limϵ→0ImΠJ/η(s+iϵ)π = ρJ/η(s). (15) In Eqs.(11-12), we retain the terms and come from the Fierz re-ordering of the and to absorb the gluons emitted from other quark lines to form and to extract the mixed condensates and , respectively. In this article, we carry out the operator product expansion to the vacuum condensates up to dimension-10, and take into account the vacuum condensates which are vacuum expectations of the operators of the orders with in a consistent way [13, 14, 15, 16, 17]. Once the analytical expressions of the QCD spectral densities are obtained, we can take the quark-hadron duality below the continuum thresholds and perform Borel transform with respect to the variable to obtain the following QCD sum rules, λ2Zexp(−M2ZT2)=∫s04m2QdsρJ/η(s)exp(−sT2), (16) where ρJ(s) = ρ0(s)+ρ3(s)+ρ4(s)+ρ5(s)+ρ6(s)+ρ8(s)+ρ10(s), (17) ρη(s) = ρJ(s)∣ms→0,⟨¯ss⟩→⟨¯qq⟩,⟨¯sgsσGs⟩→⟨¯qgsσGq⟩, (18) ρ0(s) = 1512π6∫yfyidy∫1−yzidzyz(1−y−z)2(s−¯¯¯¯¯m2Q)3(5s−¯¯¯¯¯m2Q) (19) +m2Q128π6∫yfyidy∫1−yzidz(1−y−z)2(s−¯¯¯¯¯m2Q)3, ρ3(s) = ms[⟨¯ss⟩−2⟨¯qq⟩]32π4∫yfyidy∫1−yzidzyz(s−¯¯¯¯¯m2Q)(3s−¯¯¯¯¯m2Q) (20) +msm2Q[⟨¯ss⟩−2⟨¯qq⟩]16π4∫yfyidy∫1−yzidz(s−¯¯¯¯¯m2Q), ρ4(s) = −m2Q384π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(zy2+yz2)(1−y−z)2(2s−¯¯¯¯¯m2Q) (21) −m4Q384π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(1y3+1z3)(1−y−z)2 +m2Q128π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(1y2+1z2)(1−y−z)2(s−¯¯¯¯¯m2Q) −11536π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(1−y−z)2(s−¯¯¯¯¯m2Q)(5s−3¯¯¯¯¯m2Q) +1256π4⟨αsGGπ⟩∫yfyidy∫1−yzidzyz(s−¯¯¯¯¯m2Q)(3s−¯¯¯¯¯m2Q) +m2Q128π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(s−¯¯¯¯¯m2Q), ρ5(s) = ms[3⟨¯qgsσGq⟩−⟨¯sgsσGs⟩]48π4∫yfyidyy(1−y)s, (22) ρ6(s) = ⟨¯qq⟩⟨¯ss⟩3π2∫yfyidyy(1−y)s, (23) ρ8(s) = Missing or unrecognized delimiter for \left ρ10(s) = ⟨¯qgsσGq⟩⟨¯sgsσGs⟩48π2∫yfyidyy(1−y)(sT2+2s2T4+s3T6)δ(s−˜m2Q) (25) −11⟨¯qgsσGq⟩⟨¯sgsσGs⟩6912π2∫yfyidy(1+s2T2)δ(s−˜m2Q), , , , , , , when the functions and appear. We derive Eq.(16) with respect to , then eliminate the pole residues to obtain the QCD sum rules for the masses, M2Z=−ddτ∫s04m2QdsρJ/η(s)e−τs∫s04m2QdsρJ/η(s)e−τs. (26) ## 3 Numerical results and discussions We take the standard values of the vacuum condensates , , , , , at the energy scale [18, 19, 21], and choose the masses , , from the Particle Data Group [22]. Furthermore, we take into account the energy-scale dependence of the input parameters, ⟨¯qq⟩(μ) = ⟨¯qq⟩(Q)[αs(Q)αs(μ)]49, ⟨¯ss⟩(μ) = ⟨¯ss⟩(Q)[αs(Q)αs(μ)]49, ⟨¯qgsσGq⟩(μ) = ⟨¯qgsσGq⟩(Q)[αs(Q)αs(μ)]227, ⟨¯sgsσGs⟩(μ) = ⟨¯sgsσGs⟩(Q)[αs(Q)αs(μ)]227, mc(μ) = mc(mc)[αs(μ)αs(mc)]1225, mb(μ) = mb(mb)[αs(μ)αs(mb)]1223, ms(μ) = ms(2GeV)[αs(μ)αs(2GeV)]49, αs(μ) = 1b0t[1−b1b20logtt+b21(log2t−logt−1)+b0b2b40t2], (27) where , , , , , and for the flavors , and , respectively [22], and evolve all the input parameters to the optimal energy scales to extract the masses of the . In Refs.[13, 14, 15, 16, 17], we study the acceptable energy scales of the QCD spectral densities for the hidden-charm (hidden-bottom) tetraquark states and molecular states in the QCD sum rules in details for the first time, and suggest an energy scale formula to determine the optimal energy scales, which enhances the pole contributions remarkably and works well. The energy scale formula also works well in studying the hidden-charm pentaquark states [23]. We can assign the and to be the axialvector tetraquark states with the quark constituents and respectively, and choose the currents, JQ¯Qμ(x) = εijkεimn√2{uTj(x)Cγ5Qk(x)¯dm(x)γμC¯QTn(x)−uTj(x)CγμQk(x)¯dm(x)γ5C¯QTn(x)}, with to study them with the QCD sum rules [13, 16]. If we take the updated values of the effective heavy quark masses and [24], the optimal energy scales of the QCD spectral densities of the and are and , respectively. There are no experimental candidates for the doubly heavy tetraquark states. Firstly, we suppose that the ground state type axialvector tetraquark states and have degenerate masses, and study the masses of the ground state axialvector tetraquark states at the same energy scales of the QCD spectral densities as the ones for the ground state axialvector tetraquark states . In Fig.1, we plot the predicted masses of the () and () with variations of the Borel parameter for the continuum threshold parameter () and the energy scale () [13, 16, 24]. From the figure, we can see that the experimental values of the masses of the and can be well reproduced, there appear platforms for the masses of the tetraquark states, which lie slightly below the corresponding masses of the and , respectively. If we choose the Borel windows as and for the tetraquark states and , respectively, the pole contributions are and , respectively, it is reliable to extract the masses. Furthermore, the continuum threshold parameters satisfy the relation and , respectively, which are consistent with our naive expectation that the mass gaps of the ground states and the first radial excited states of the tetraquark states are about [25, 26]. The energy scales and work well. In Ref.[5], Karliner and Rosner obtain the masses and for the type axialvector tetraquark states and respectively based on a simple potential quark model, which can reproduce the mass of the doubly charmed baryon state . In Ref.[10], Eichten and Quigg obtain the masses and for the type axialvector tetraquark states and respectively based on the heavy quark symmetry, where the mass of the doubly charmed baryon state is taken as input parameter in the charm sector, while in the bottom sector, there are no experimental candidates for the baryon states and . From Fig.1, we can see that if we take the same parameters, such as the energy scales, continuum threshold parameters, etc, in the charm sector, the predicted mass is slightly smaller than the value from a simple potential quark model [5] and much smaller than the value from the heavy quark symmetry [10], in the bottom sector, the predicted mass is much larger than the value from a simple potential quark model [5] and slightly larger than the value from the heavy quark symmetry [10]. Now we revisit the subject of how to choose the energy scales of the QCD spectral densities. In calculation, we neglect the perturbative corrections to the currents , which can be taken into account in the leading logarithmic approximation through an anomalous dimension factor, , the are the anomalous dimension of the interpolating currents , ⟨0\|J/ηα(0;μ)\|ZQQ(p)⟩ = [αs(μ0)αs(μ)]γJ⟨0\|J/ηα(0;μ0)\|ZQQ(p)⟩ (29) = [αs(μ0)αs(μ)]γJλZ(μ0)εα=λZ(μ)εα. The pole residues are energy scale dependent quantities, at the leading order approximation, we can set . At the QCD side, the correlation functions can be written as Π(p2) = ∫s04m2Q(μ)dsρJ/η(s,μ)s−p2+∫∞s0dsρJ/η(s,μ)s−p2, (30) through dispersion relation, and they are energy scale independent according to the approximation or , ddμΠ(p2) = 0, (31) which does not mean the pole contributions are energy scale independent, ddμ∫s04m2Q(μ)dsρJ/η(s,μ)s−p	2018-11-20 07:18:32	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9283871650695801, "perplexity": 1568.0431258842218}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039746301.92/warc/CC-MAIN-20181120071442-20181120093442-00022.warc.gz"}
http://reuters-finance.com/2019/12/13/premier-stochastic-oscillator-explained/	# Premier Stochastic Oscillator Explained Investing The premier stochastic oscillator (PSO) is a technical indicator based on George Lane’s stochastic oscillator. The PSO differs in that it is normalized to register neutral values at zero, resulting in greater sensitivity to recent, short-term price moves. Additionally, the PSO is calculated using a double exponential moving average that creates a smoother and more even response to market changes. Figure 1 illustrates how the two stochastic oscillators respond differently to market changes. Figure 1: This chart shows both the premier stochastic oscillator and a standard stochastic oscillator applied to the e-mini Russell 2000 futures contract. ### History of the PSO The PSO was first introduced by technical analyst Lee Leibfarth in the August 2008 issue of the journal Technical Analysis of Stocks & Commodities. Stochastic oscillators have long been used to help traders and investors identify areas where trend changes are likely. Leibfarth developed the PSO to take advantage of a standard stochastic oscillator’s strengths while enhancing it to become more reactive to market activity. The result is a faster indicator that provides earlier signals for potential trend changes. ### Calculating the PSO Before looking into the calculations of the PSO, it is helpful to understand the logic behind a standard stochastic oscillator. The classic stochastic oscillator measures price momentum by comparing a trading instrument’s current price to a price range specified in a lookback period (the number of periods from which price data are collected). For example, if the range is between $60 and$70 and the current price is \$67.50, then the price is at 75% of the range. The goal of a stochastic oscillator is to figure out where price has been and anticipate where price is headed. This is achieved by determining if price bars are closing close to their highs or lows. When prices are closing nearer to bar highs, it is indicative of an uptrending market. Conversely, when prices are closing nearer to bar lows, it signifies a downtrending market. The basic calculation for the main value of a standard stochastic oscillator (%K) is: begin{aligned} &text{%K} = 100 times left [ frac { text{C} – text{L}_n }{ text{H}_n – text{L}_n } right ] \ &textbf{where:} \ &text{C} = text{most recent closing price} \ &n = text{lookback period} \ &text{L}_n = text{low of the } n text{ previous price bars} \ &text{H}_n = text{highest price during the same } n text{ period} \ end{aligned} %K=100×[HnLnCLn]where:C=most recent closing pricen=lookback periodLn=low of the n previous price barsHn=highest price during the same n period The premier stochastic oscillator normalizes the standard stochastic oscillator by applying a five-period double exponential smoothing average of the %K value, resulting in a symmetric scale of 1 to -1. The PSO calculation, then, is: begin{aligned} &text{PSO} = frac { text{Exponential Value (S)} – 1 }{ text{Exponential Value (S)} + 1 } \ &textbf{where:} \ &text{S} = text{5-period double smoothed exponential EMA} ((% text{K} – 50) times .1) \ &text{%K} = text{8-period stochastic oscillator} \ end{aligned} PSO=Exponential Value (S)+1Exponential Value (S)1where:S=5-period double smoothed exponential EMA((%K50)×.1)%K=8-period stochastic oscillator (Note: The TradeStation EasyLanguage code for the premier stochastic oscillator is available at www.PowerZoneTrading.com.) ### Interpreting the PSO The PSO appears as a curving line with four horizontal lines that represent threshold levels. These threshold levels are customizable; that is, the levels can be changed by the user to adapt to individual trading styles and instruments. Figure 2 shows the PSO, appearing on a sub-chart below the price chart, with the four different threshold levels. Figure 2: A chart of the e-mini Russell 2000 futures contract showing the PSO and its four threshold levels. The threshold levels are important to the indicator because they can be used to identify areas where market reversals are expected to occur. As the curved line meanders up and down, it crosses above and below the threshold levels. The “outer” thresholds, at the very top and very bottom, represent the extremes, or areas that are overbought (the top line) or oversold (the bottom line). When the PSO moves above the upper or below the lower, price will be expected to pull back. The “inner” thresholds are placed near the zero line and can be utilized as a transitional area to spot pullbacks and short-term reversals. As the PSO returns from overbought and oversold areas, price has a tendency to accelerate toward the zero line and reverse. This transitional area (between the inner thresholds) can be useful in spotting short-term reversals. The PSO can be used to anticipate changes in market direction. With the ability to change where the threshold levels appear, the PSO is adaptable to different trading styles. The PSO can easily be incorporated into a countertrend-type strategy since it is used to identify changes in market direction. The following are suggested uses for the PSO, understanding that each trader or investor would need to adjust the indicator to suit his or her needs. Outer Threshold Setups Outer threshold setups form when the PSO crosses out of the outer limits and then returns. As previously mentioned, price has a tendency to pull back and then return to overbought or oversold areas. This can provide a good entry point to: • Go long when the PSO crosses below the upper threshold (0.9 in this example) after it has already crossed above the threshold. A short-term reversal may occur where price returns to the extreme overbought territory. • Go short when the PSO crosses above the lower threshold (-0.9 in this instance) after it has already penetrated the lower threshold. Again, a short-term reversal may occur as prices make another push lower. Figure 3: This chart of the e-mini Russell 2000 futures contract shows potential long (buying) positions using both the outer and inner thresholds. Inner Threshold Setups Inner threshold setups that can be identified when the PSO comes from the outer thresholds and accelerates toward the center (zero) line. This can present an opportunity to: • Go long when the PSO comes from overbought areas (0.9 in this instance) and crosses the inner threshold level (0.2 in this example). Unlike the outer threshold setups, the PSO does not need to re-cross the threshold level to trigger the setup. • Go Short when the PSO returns from an oversold region (-0.9 in Figure 3) to the inner threshold level (in this example, -0.2). (Note: the Go Short example is not shown in Figure 3.) Figure 3 shows a chart with long setups highlighted, using both the outer and inner threshold examples. For short trades, the logic can be reversed. Please note that the PSO is not a strategy—rather, it is an indicator that can be used as part of a trader’s or investor’s toolbox. As with any market analysis tool, this indicator needs to be optimized to fit each trader’s style and preferred trading instrument. ### The Bottom Line The classic stochastic oscillator has been used since the 1950s by traders and investors to anticipate areas where the market may change direction. The classic and premier stochastic oscillator are based on price movement that occurs within the price bar itself—whether bars are closing nearer to their highs or lows—to determine which way the market is heading. The premier stochastic oscillator creates a smoother, faster-reacting stochastic that can help traders and investors determine areas where direction changes are probable—sooner than a standard stochastic—enabling participants to catch a bigger part of a move. The author, Jean Folger, is co-founder of PowerZone Trading with the aforementioned founder, Lee Leibfarth.	2020-06-06 14:26:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.41478797793388367, "perplexity": 2767.0067513708404}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348513321.91/warc/CC-MAIN-20200606124655-20200606154655-00255.warc.gz"}
http://math.stackexchange.com/questions/7578/two-metrics-induce-the-same-topology-but-one-is-complete-and-the-other-isnt	# Two metrics induce the same topology, but one is complete and the other isn't I'm looking for an example of two metrics that induce the same topology, but so that one metric is complete and the other is not (Since it is known that completeness isn't a topological invariant). Thanks in advance for any hints or ideas. - The real line and (0,1) are topologically equivalent but the second is not complete. d(x,y) and d(f(x),f(y)) where f is a topological equivalence (homeomorphism) between a complete and an incomplete space will be two metrics that "induce the same topology but one is complete and the other isn't". – T.. Oct 23 '10 at 17:46 It may be worth noting that there can be no compact counterexample, so open intervals and infinite discrete sets, used in all of the examples so far, are natural places to look. – Jonas Meyer Oct 23 '10 at 19:10 The metric space $\{\frac {1}{n} \mid n\in \mathbb{N} \}$ with the usual metric is incomplete (since we don't have zero), and it has the discrete topology. the same space with the metric $d(x,y)=1 \iff x\neq y$ also has a discrete topology but is complete, since any cauchy sequence will eventually be constant. - I like this! This is in some sense a minimal counterexample, since every finite metric space has the discrete topology and is complete. – Qiaochu Yuan Oct 23 '10 at 12:29 There is a lot to be said for minimal counterexamples, but it doesn't provide much insight into why the image of a complete space under a homeomorphism can fail to be complete. You could say many fancy words such as "forgetful functor" and "completely metrisable/uniformisable", but in simple terms if you don't mention the words "uniform continuity", you won't have pointed out the central reason why a homeomorphism can fail to preserve completeness. – kahen Oct 23 '10 at 15:27 +1 ; And this is an example for the fact that nonequivalent metric may determine same topology. – HK Lee Nov 10 '15 at 11:07 The interval $M:=]-1,1[$ with the usual line element $ds:=\|dx\|$ is an incomplete metric space: The sequence $x_n:=1-{1\over n} \ (n\to\infty)$ converges in $\mathbb R$, so it is a Cauchy sequence, but it diverges in $M$. On the other hand, the "hyperbolic metric" defined by $ds:=\|dx\|/(1-x^2)$ induces the same topology on $M$ but is complete. The latter statement needs of course a proof. Suffice it here to say that now the endpoints $\pm1$ are "infinitely far away". - Look for a homeomorphism $f: \mathbb R \to \mathbb R_+$ (you should know a good one). Then take the usual metric on $\mathbb R$ and use it to define a metric $d$ on $\mathbb R_+$ by $d(x,y) = \|f(x)-f(y)\|$ (). Now what can you say about the relationship between $d$ and the usual metric on $\mathbb R_+$? EDIT: Made the question here correct and more illustrative. So the identity $\iota: (\mathbb R_+,d) \to (\mathbb R_+,\|\cdot\|)$ is a homeomorphism. Now just ask yourself which property, that continuous functions between metric spaces can have, $\iota$ doesn't have? Can you formulate a theorem about what property a homeomorphism must have to preserve completeness? (): Pullback and pushforward might be of interest. - If $(X,d)$ is a connected, complete metric space and $G$ is a nonempty proper open subset of $X$, then $G$ is not complete with the restricted metric (because it is not closed), but $G$ is homeomorphic to a complete metric space, because the map $x\mapsto(x,1/d(x,X\setminus G))$ embeds $G$ onto a closed subspace of $X\times \mathbb{R}$. (This is more or less Lemma 3.1.1 of Arveson's Invitation.) A topological space that is homeomorphic to a separable complete metric space is called a Polish space. A subspace of a Polish space is Polish if and only if it is a $G_\delta$, while $G_\delta$s in complete metric spaces are not typically complete with the restricted metric. For example, there is a complete metric inducing the usual topology on the set of irrational numbers. - As many others have pointed out, the way to do this is to think about pairs of metric spaces that are homeomorphic but not isometric. In many cases, the metric will be complete on one space but not the other. For example: • The set $\{1/n \mid n \in \mathbb{N}\}$ is incomplete, but the natural numbers $\mathbb{N}$ are complete. • The ray $(0,\infty)$ and the open interval $(0,1)$ are incomplete, but the real line is complete. • An open disc in the plane is incomplete, but the entire plane is complete. • The punctured plane $\mathbb{R}-\{0\}$ is incomplete, but the infinite cylinder $\mathbb{R} \times S^1$ (where $S^1$ is the circle) is complete. • The real line minus the $x$-axis is incomplete, but a disjoint pair of planes is complete. • The twice-punctured plane $\mathbb{R}^2 - \{(-1,0),(1,0)\}$ is incomplete, but the graph of $$z = \frac{1}{(x^2-1)^2 + y^2}$$ in $\mathbb{R}^3$ is complete. (This graph has asymptotic "cusps" at $(-1,0)$ and $(1,0)$.) In each case, the homeomorphism between the two spaces can be used to define a nonstandard metric on the latter space that makes it incomplete. - Consider one complete normed vector space $(V,\\|\cdot\\|)$, that is, a Banach space, and let $$B = \{ x \in V : \\|x\\| < 1 \}$$ be the unit ball centered at the origin. Define $f: V \rightarrow B$ by $x \mapsto x /(1+ \\|x\\|)$. Then f is continuous and is easy to show that the inverse of $f$ is also continuous, so f is a homeomorphism. Now $B$ isn't complete, since if $x \in V$ and $\\| x \\| = 1$, then $x_n = (1-1/n)x$ is a cauchy sequence in $B$ that doesn't converge. As Kahen referred, if we have two metric spaces $X$ and $Y$ with the latter complete and a uniformly continuous homeomorphism $g : X \rightarrow Y$, then $X$ is complete. To prove this you'll need to observe that since $g$ is uniformly continuous then it sends Cauchy sequences to Cauchy sequences. - With the usual metric, $(0,1]$ is not complete. Let us define another metric $\rho$ on this space using the homeomorphism $\phi: [1,\infty) \to (0,1]$. Define $\rho(x,y) = \vert\frac{1}{x}-\frac{1}{y}\vert$. It is easy to check that $\rho$ is indeed a metric. Also, note that the usual metric and $\rho$ are equivalent and hence induce the same topology. Now consider a Cauchy sequence in $((0,1],\rho)$. $\rho(x_n,x_m) < \epsilon$ for all $n \geq N$ or, $\vert\frac{1}{x}-\frac{1}{y}\vert < \epsilon$ for all $n \geq N$. Hence $(\frac{1}{x_n})$ is a Cauchy sequence in ($[1,\infty),$usual metric), a complete metric space. Let us say, it converges to $y \in [1,\infty)$. It is easy to check that $(x_n)$ converges to $x = \frac{1}{y}$ in $((0,1],\rho)$. -	2016-05-25 15:02:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9571444988250732, "perplexity": 135.17178082326697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049274994.48/warc/CC-MAIN-20160524002114-00173-ip-10-185-217-139.ec2.internal.warc.gz"}
https://socratic.org/questions/how-do-you-solve-2x-5-3-4x-5-1-5#159449	# How do you solve (2x/5) + 3 - (4x/5) = 1/5? $x = 7$ Rearrange to make $x$ the subject: $\left(2 \frac{x}{5}\right) + 3 - \left(4 \frac{x}{5}\right) = \frac{1}{5}$ => $- 2 \frac{x}{5} = - \frac{14}{5}$ => $x = 7$	2021-10-17 12:53:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9085005521774292, "perplexity": 2144.528828898978}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585177.11/warc/CC-MAIN-20211017113503-20211017143503-00020.warc.gz"}
https://cds.ismrm.org/protected/16MProceedings/PDFfiles/2897.html	Localized, gradient-reversed ultrafast z-spectroscopy in vivo at 7T Neil Wilson1, Kevin D'Aquilla1, Catherine Debrosse1, Hari Hariharan1, and Ravinder Reddy1 1Department of Radiology, Center for Magnetic Resonance and Optical Imaging (CMROI), University of Pennsylvania, Philadelphia, PA, United States ### Synopsis Ultrafast z-spectroscopy can be collected by saturating the nuclear spins with an RF pulse in the presence of a gradient, effectively encoding the offset frequency spatially across a voxel and allowing full z-spectra to be collected in a single shot. When asymmetry analysis is applied, frequencies on one physical side of the voxel are compared with those on the other physical side. This can be a problem if there is inhomogeneity or partial voluming. By acquiring an additional z-spectrum with the gradient polarity reversed, mixed z-spectra can be created in which the positive and negative offset frequencies come from the same side of the voxel. This method is more robust to inhomogeneity and partial voluming typically found in vivo as demonstrated here with studies on 7T in human brain. ### Purpose To collect ultrafast z-spectra in vivo at 7T where voxel homogeneity cannot be assured. ### Introduction CEST contrast is generated by selective saturation of a labile proton group and measurement of the subsequent reduction in water signal.1 Standard methods collect a single offset frequency per acquisition which leads to a trade-off between spectral resolution and scan time. Saturating in the presence of a gradient spreads the offset frequency spatially across a voxel. Effectively, each voxel is split into slivers that are encoded for each offset frequency. This encoding is resolved by applying a similar gradient during readout and allows for ultrafast2 collection of a full z-spectrum in a single shot and has been applied in vitro3,4,5 and recently, localized in vivo in human brain (UCEPR)6. Spectral resolution is determined by the number of readout points and does not affect overall scan time. In conventional z-spectroscopy, inhomogeneity and partial volume effects are averaged over the entire voxel for each frequency, whereas in ultrafast z-spectroscopy (UFZS), different offsets experience different effects, resulting in unpredictable errors in analysis. Here, we demonstrate an extension to UCEPR6 that makes it more robust to voxel inhomogeneity and partial volume effects by acquiring an additional scan with the saturating gradient polarity reversed. Reversing the gradient polarity reverses the direction of offset frequencies relative to isocenter. Physical locations that were saturated at a positive offset originally are saturated at a negative offset during reversal. We call this approach gradient-reversed ultrafast z-spectroscopy (GRUFZS). ### Methods The pulse sequence is a modified version of PRESS in which excitation was performed with a hard, nonselective 90o pulse followed by three slice selective refocusing pulses and is shown in Fig 1. Saturation used Hanning-filtered rectangular pulses with a total saturation time of 800 ms with peak B1 power 220 Hz. The saturation pulse was frequency shifted such that the voxel center was on resonance and the saturation bandwidth was $\pm$5 ppm over the voxel. The readout was oversampled by a factor of 10 to ensure adequate sampling of T2*-weighted gradient echo. TR/TE = 8000/30 ms. Voxel size was 15x15x15 mm3. Scans without saturation were acquired and used to normalize for proton density differences across the voxel. All scans were taken on a Siemens 7T whole body scanner. Time domain data was filtered using a binary threshold at the noise level since the readout was much longer than the actual echo duration. Data was Fourier transformed to the spatial/z-spectral domain and normalized with the unsaturated scan. Mixed z-spectra were created by taking the positive offset frequencies from the positive polarity scan and combining with the negative offset frequencies from the negative polarity scan and vice versa. $\text{MTR}_{asym}$ values were calculated and normalized by the negative offset intensity. ### Results and Discussion Figure 2 shows an example reconstruction for a cortical white matter voxel. The white matter voxel shown is fairly homogeneous with little partial voluming. Nevertheless, the asymmetry plots from the mixed z-spectra in Fig 2g are much closer than those from the acquired spectra in Fig 2f. For an inhomogeneous voxel with significant partial voluming, UFZS results in widely different, sometimes negative asymmetry values depending on the gradient polarity as shown in Fig 3f. The mixed asymmetry plots calculated from GRUFZS in Fig 3g show much better agreement, especially farther from water. Table 1 shows results of different voxel placements as well as different ultrafast directions and compares to conventional z-spectroscopy. It is clear that regular UFZS fails in many of the cases since $\text{MTR}_{asym}$ can be positive or negative depending on the gradient polarity choice, while GRUFZS values are much closer and consistently reflect positive asymmetry values. Conventional z-spectroscopy at the same digital spectral resolution as GRUFZS would take 25x longer, and though there is an SNR penalty associated with ultrafast scanning, the high sensitivity of CEST along with the large voxel sizes here mean that SNR is not a limiting factor. Compared to previously-reported UFZS, GRUFZS requires additional scans. However, there is no associated penalty in scanning efficiency since the mixed z-spectra can be averaged to give a single $\text{MTR}_{asym}$ value. Alternatively, the mixed z-spectra can be kept separate, as they are effectively acquired from two adjacent half voxels. ### Conclusion We have presented a method to acquire ultrafast z-spectra in vivo that requires only an additional scan with the gradient polarity reversed. The improved tolerance to inhomogeneity and partial voluming is evident in the asymmetry plots of the mixed z-spectra compared to the originally acquired ones. This method offers a fast, robust way to record full z-spectra in vivo. ### Acknowledgements This work was supported by the National Institute of Health through grant number P41-EB015893 and the National Institute of Neurological Disorders and Stroke through Award Number R01NS087516. ### References [1] Wol ff SD, Balaban RS. NMR imaging of labile proton exchange. Journal of Magnetic Resonance 1990;86:164-169. [2] Frydman L, Scherf T, Lupulescu A. The acquisition of multidimensional NMR spectra withina single scan. Proceedings of the National Academy of Sciences 2002;99:15858-15862. [3] Swanson SD. Broadband excitation and detection of cross-relaxation NMR spectra. Journal of Magnetic Resonance 1991;95:615-618. [4] Xu X, Lee JS, Jerschow A. Ultrafast Scanning of Exchangeable Sites by NMR Spectroscopy. Angewandte Chemie International Edition 2013;52:8281-8284. [5] Dopfert J, Witte C, Schroder L. Slice-selective gradient-encoded CEST spectroscopy for monitoringdynamic parameters and high-throughput sample characterization. Journal of Magnetic Resonance 2013;237:34-39. [6] Liu Z, Dimitrov IE, Lenkinski RE, Hajibeigi A, Vinogradov E. UCEPR: Ultrafast localized CEST-spectroscopy with PRESS in phantoms and in vivo. Magnetic Resonance in Medicine 2015;Early View. ### Figures Table 1: Comparison of $\text{MTR}_{asym}$ values between mixed z-spectra (dubbed Left and Right) and acquired z-spectra (Positive and Negative) in GRUFZS as well as with conventional z-spectroscopy (Conv) for white matter (WM), thalamus, and prefrontal cortex voxel placement. GRUFZS is shown with the ultrafast direction varied (x,y,z).	2021-06-12 20:35:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5292282700538635, "perplexity": 4513.094962880104}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487586390.4/warc/CC-MAIN-20210612193058-20210612223058-00123.warc.gz"}
https://www.physicsforums.com/threads/dimensional-analysis-i-know-my-equations-are-right.554212/	# Homework Help: Dimensional Analysis. I know my equations are right 1. Nov 26, 2011 ### flyingpig 1. The problem statement, all variables and given/known data A birchwood table company has an individual who does all its finishing work and it wishes to use him in this capacity at least 36 hours each week. By union contract, the assembly area can be used at most 48 hours each week. The company has three models of birch tables, T1, T2 and T3. T1 requires 1 hour for assembly, 2 hours for finishing, and 9 board feet of birch. T2 requires 1 hour for assembly, 1 hour for finishing and 9 board feet of birch. T3 requires 2 hours for assembly, 1 hour for finishing and 3 board feet of birch. Write a LOP that will compute how many of each model should be made in order to minimize the board feet of birchwood used. 2. What I want to do with this problem Here is the thing, I wrote out the equations, but my variables don't mean a thing. I tried to make some sense out of it Here is the equations $$1x_1 + 1x_2 +2x_3 \leq 48$$ $$2x_1 + 1x_2 +1x_3 \geq 36$$ $$P = 9x_1 + 9x_2 + 3x_3$$ $$x_1, x_2, x_3 \geq 0$$ For instance the first equation is $$1x_1 + 1x_2 +2x_3 \leq 48$$ Right hand side is hours, so I expect the units on $$1x_1 + 1x_2 +2x_3$$ cancel out so that it gives me hours too Look at the coefficients of $$1x_1 + 1x_2 +2x_3$$ The "1" in front of $$x_1$$ represents "hour for assembly" or "hour/assembly". So to make things work out, $$x_1$$ has units "assembly for T_1[/tex]" But that doesn't work for the second equation because I will need $$x_1$$ to have units "finishing for T_1[/tex]" Last edited: Nov 26, 2011 2. Nov 26, 2011 ### I like Serena x1 is the number of tables of type T1 that is produced per week. For assembly, the 1 in front of x1, is the hours of assembly/table of type T1. 3. Nov 26, 2011 ### flyingpig I don't see how those units could cancel out... x1 would change for finishing work, but I need a consistent unit 4. Nov 26, 2011 ### Ray Vickson If x1 is the number of pieces of T1 to produce, x1 is a dimensional number. The number of assembly hours per piece of T1 is 1, so x1 units need 1x1 hours. Note that 1x1 is dimensionless; the "hours" occurs outside the expression, because we take 1 as the number of hours, not a time of 1 hour. Similarly, the number of hours we have available is 48; the '48' is dimensionless. I avoided saying the available time is 48 hours, in favor of saying the number of hours available is 48. See the difference? RGV 5. Nov 26, 2011 ### I like Serena How would x1 change? For finishing work we have: x1 is still the "number of tables of type T1" The 1 in front of x1 is the "hours of finishing/table of type T1". The "number of tables of type T1" cancels out, and the result is "hours of finishing". The total "hours of finishing" is supposed to exceed 36 "hours of finishing".	2018-06-18 02:31:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44222351908683777, "perplexity": 2036.1040647534737}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267859923.59/warc/CC-MAIN-20180618012148-20180618032148-00631.warc.gz"}
https://www.educator.com/mathematics/multivariable-calculus/hovasapian/double-integrals.php	Enter your Sign on user name and password. Forgot password? • Follow us on: Start learning today, and be successful in your academic & professional career. Start Today! Loading video... This is a quick preview of the lesson. For full access, please Log In or Sign up. For more information, please see full course syllabus of Multivariable Calculus • ## Related Books Lecture Comments (9) 1 answerLast reply by: Professor HovasapianFri Aug 22, 2014 8:27 PMPost by Denny Yang â™• [Moderator] on August 19, 2014Also in example 3, you made a minor calculation error. You wrote: The integral of(2x^3 - 12x^2 +19x, x, (3/2), 3 ). It should have been (2x^3 - 12x^2 +18x, x, (3/2), 3 ). The [ x, (3/2), 3 ] is the variable we are integrating and the lower bound and upper bound. Typing it just like you would on a Ti-89 Calculator. 1 answerLast reply by: Professor HovasapianFri Aug 22, 2014 8:23 PMPost by Denny Yang â™• [Moderator] on August 19, 2014Q. ii) Integrate âˆ« âˆ’ Ï€\protect\mathord/ \protect phantom Ï€2 / 20 ey dx.What is this? 1 answerLast reply by: Professor HovasapianMon May 13, 2013 12:53 AMPost by Josh Winfield on May 12, 2013Hello Raffi,Can you just clarify why in example 3 you put the limit of integration for dx as (3-y) on the bottom and (y) on the top. I had them the other way around but im wrong but i cant figure out why? 0 answersPost by Justin Dehorty on February 23, 2013How do I find the Lower and Upper Sums? Apparently this is another way to calculate double integrals, but I don't understand how to set up the sigmas properly. 1 answerLast reply by: Professor HovasapianSat Dec 29, 2012 5:01 PMPost by Alexander Rekitsanov Alexander Rekitsanov on December 15, 2012I think that the region the prof is integrating is wrong in example 3. in the question it is stating that region that is bounded by y=0 ,y=x and y=-x+3 but the prof ignores the y=0 part. isn't the region suppose to be the left triangle in the drawing ? please if you could clarify it, thank you. ### Double Integrals Find intervals of integration for dydx over the shaded region and set up the double integral. • To set up the integral pay close attendtion to the order of the differentials, dy and dx. • Since dy is prior to dx we set up the integral so that we first cover the intervals for y and then those for x. That is, the intervals of y are a function of x. • Our shaded region is the area between the curves y = x2 and y = 1. Hence x2 ≤ y ≤ 1. Note that this area runs from - 1 ≤ x ≤ 1. Thus our intervals of integration yield the double integral ∫ − 11x21 f dydx. Note the order of integration corresponds to the opposite order of the differentials. Find intervals of integration for dxdy over the shaded region and set up the double integral. • To set up the integral pay close attendtion to the order of the differentials, dy and dx. • Since dx is prior to dy we set up the integral so that we first cover the intervals for x and then those for y. That is, the intervals of x are a function of y. • Our shaded region is the left side of the circle x2 + y2 = 1. Solving for x yields x = ±√{1 − y2} , we only want the left side, so we take the negative result. • Hence − √{1 − y2} ≤ x ≤ 0 and this area runs from − 1 ≤ y ≤ 1. Thus our intervals of integration yield the double integral ∫ − 11 − √{1 − y2}0 f dxdy. Note the order of integration corresponds to the opposite order of the differentials. i) Integrate ∫ − 21 5xy dy. • Taking the integral in respect to y takes any other variable as a constant. So ∫ − 21 5xy dy = [5/2]xy2 \| − 21 = [5/2]x(1)2 − [5/2]x( − 2)2 = [5/2]x − 10x = − [15/2]x. ii) Integrate ∫13 5xy dx. • Taking the integral in respect to x takes any other variable as a constant. So ∫13 5xy dx = [5/2]x2y \|13 = [5/2](3)2y − [5/2](1)2y = [45/2]y − [5/2]y = 20x iii) Integrate ∫13 − 21 5xy dydx. • We take the integral in respect to the order of the differentials. So ∫13 − 21 5xy dydx = ∫13− [15/2]x dx = − [15/4]x2 \|13 = − [15/4]( 9 − 1 ) = − 30 iv) Integrate ∫ − 2113 5xy dxdy. • We take the integral in respect to the order of the differentials. So ∫ − 2113 5xy dxdy = ∫ − 21 20x dy = 10x2 \| − 21 = 10( 1 − 4 ) = − 30. Note that ∫13 − 21 5xy dydx = ∫ − 2113 5xy dxdy. i) Integrate ∫ − 20 cos(x) dy. • Taking the integral in respect to y takes any other variable as a constant. So ∫ − 20 cos(x) dy = \|ycos(x) \| − 20 = (0)(cos(x)) − ( − 2)cos(x) = 2cos(x) ii) Integrate ∫ − π\protect\mathord/ \protect phantom π2/ 20 ey dx. • Taking the integral in respect to x takes any other variable as a constant. So ∫ − π\mathord/ \protect phantom π2 20 ey dx = xey \| − π\mathord/ \protect phantom π2 20 = (0)ey − ( − [(π)/2] )ey = [(πey)/2] iii) Integrate ∫ − π\mathord/ \protect phantom π2 20 − 20 eycos(x) dydx. • We take the integral in respect to the order of the differentials. So ∫ − π\mathord/ \protect phantom π2 20 − 20 eycos(x) dydx = ∫ − π\mathord/ \protect phantom π2 20 ( eycos(x) \| − 20 ) dx = ∫ − π\mathord/ \protect phantom π2 20 ( cos(x) − e − 2cos(x) ) dx = sin(x) \| − π\mathord/ \protect phantom π2 20 − e − 2sin(x) \| − π\mathord/ \protect phantom π2 20 = 1 − e − 2 iv) Integrate ∫ − 20 − π\mathord/ \protect phantom π2 20 eycos(x) dxdy. • We take the integral in respect to the order of the differentials. So ∫ − 20 − π\mathord/ \protect phantom π2 20 eycos(x) dxdy = ∫ − 20 ( eysin(x) \| − π\mathord/ \protect phantom π2 20 ) dy = ∫ − 20 ey dy = ey \| − 20 = 1 − e − 2. Note that ∫ − π\mathord/ \protect phantom π2 20 − 20 eycos(x) dydx = ∫ − 20 − π\mathord/ \protect phantom π2 20 eycos(x) dxdy. Integrate given that y ∈ [ − 1,1], x ∈ [0,2] and f(x,y) = [1/2]xy3. • We set up our double integral in respect to the differentials dy and dx. Note that we have dydx. • So our double integral has the intervals of x followed by those of y. Then Integrating yields ∫02 − 11 [1/2]xy3 dydx = ∫02 ( [1/8]xy4 \| − 11 ) dx = ∫02 0dx = 0 Integrate given that y ∈ [0,π], x ∈ [ − π,π] and g(x,y) = cos(x)sin(y). • We set up our double integral in respect to the differentials dy and dx. Note that we have dydx. • So our double integral has the intervals of x followed by those of y. Then Integrating yields ∫ − ππ0π cos(x)sin(y) dydx = ∫ − ππ ( − cos(x)cos(y) \|0π ) dx = ∫ − ππ 2cos(x)dx = 2sin(x) \| − ππ = 0 Integrate given that x ∈ [ − [1/2],[1/2] ], y ∈ [0,5] and h(x,y) = yex. • We set up our double integral in respect to the differentials dy and dx. Note that we have dydx. • So our double integral has the intervals of x followed by those of y. Then Integrating yields ∫ − 1 \mathord/ \protect phantom 1 2 21 \mathord/ \protect phantom 1 2 205 yex dydx = ∫ − 1 \mathord/ \protect phantom 1 2 21 \mathord/ \protect phantom 1 2 2 ( [1/2]y2ex \|05 ) dx = ∫ − 1 \mathord/ \protect phantom 1 2 21 \mathord/ \protect phantom 1 2 2 ( [25/2]ex )dx = [25/2]ex \| − 1 \mathord/ \protect phantom 1 2 21 \mathord/ \protect phantom 1 2 2 = [25/2]( e1 \mathord/ \protect phantom 1 2 2 − e − 1 \mathord/ \protect phantom 1 2 2 ) Let f(x,y) = [y/(x2)] and R be the shaded region described by: Find dA. • Note that dA is the area covered by the differentials dx and dy. We shall integrate in respect to that order, that is dxdy. • Our region is the area between the curve y = x3 and y = x. The intervals of integration for y ∈ [x3,x]. • To find the intervals of integration for x, we note that x runs from 0 to 1 along our shaded region and so x ∈ [0,1]. Then dA = ∫01x3x [y/(x2)]dydx = ∫01 ( [(y2)/(2x2)] \|x3x ) dx = ∫01 ( [((x)2)/(2x2)] − [((x3)2)/(2x2)] ) dx = ∫01 ( [1/2] − [(x4)/2] )dx = ( [1/2]x − [1/10]x5 ) \|01 = [2/5] Let f(x,y) = k where k a constant, k > 0 and R be the shaded region described by: Find dA. • Note that dA is the area covered by the differentials dx and dy. We shall integrate in respect to that order, that is dxdy. • Our region is the area between the curve y = 0 and y = sin(x). The intervals of integration for y ∈ [0,sin(x)]. • To find the intervals of integration for x, we note that x runs from 0 to p along our shaded region and so x ∈ [0,p]. Then dA = ∫0p0sin(x) kdydx = ∫0p ( ky \|0sin(x) ) dx = ∫0p ( ksin(x) − ksin(0) ) dx = ∫0p ksin(x)dx = − kcos(x) \|0p = 2k Let f(x,y) = − 2x3y2 and R be the region bounded by x = 0, y = 1 and y = x. i) Find dydx. Do not integrate. • Sketching our region yields • Note the blue line demonstrating our intervals of integration for y, that is y ∈ [x,1]. We can also see that x ∈ [0,1]. Hence dydx = ∫01x1− 2x3y2 dydx. Let f(x,y) = − 2x3y2 and R be the region bounded by x = 0, y = 1 and y = x. ii) Find dxdy. Do not integrate. • Sketching our region yields • Note the blue line demonstrating our intervals of integration for x, that is x ∈ [0,y]. We can also see that y ∈ [0,1]. Hence dydx = ∫010y− 2x3y2 dxdy. Let f(x,y) = − 2x3y2 and R be the region bounded by x = 0, y = 1 and y = x. iii) Verify that ∫01x1− 2x3y2 dydx = ∫010y− 2x3y2 dxdy . • We integrate each side of the equation and see if they are equal. • First ∫01x1− 2x3y2 dydx = ∫01 ( − [2/3]x3y3 \|x1 )dx = ∫01 ( − [2/3]x3 + [2/3]x6 ) dx = ( − [1/6]x4 + [2/21]x7 ) \|01 = − [1/14] • Second ∫010y− 2x3y2 dxdy = ∫01 ( − [1/2]x4y2 \|0y )dy = ∫01 ( − [1/2]y6 ) dy = − [1/14]y7 \|01 = − [1/14] Since both double integrals yielded the same value, then ∫01x1− 2x3y2 dydx = ∫010y− 2x3y2 dxdy . Note that the second double integral was simpler to compute. *These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer. Answer ### Double Integrals Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. • Intro 0:00 • Double Integrals 0:52 • Introduction to Double Integrals • Function with Two Variables • Example 1: Find the Integral of xy³ over the Region x ϵ[1,2] & y ϵ[4,6] • Example 2: f(x,y) = x²y & R be the Region Such That x ϵ[2,3] & x² ≤ y ≤ x³ • Example 3: f(x,y) = 4xy over the Region Bounded by y= 0, y= x, and y= -x+3 ### Transcription: Double Integrals Hello and welcome back to educator.com and multivariable calculus.0000 Today we are going to start on a new topic. We are going to be talking about double integrals.0004 The nice thing about this particular topic is you do not have to learn anything new.0008 There is going to be some new notation, but in face the notation itself is not even new. It is exactly the same as what you saw in single variable calculus, just picked up from another dimension -- dimension 1, dimension 2.0012 So for all practical purposes, you have done this stuff that we are going to be doing over and over and over again.0025 Rather than belabor the point with a lot of theory, I am actually just going to give a couple of quick definitions, and then we are just going to launch right into examples.0032 We just want to be able to handle these problems and not worry about what is going on behind the scenes too much.0041 You already know a lot of what is going on. Okay. Let us just jump right on in.0048 Okay. Now, in single variable calculus what we did was we defined this integral of f(x)/some interval from a to b.0054 We had something that looked like this... the integral from a to b of f(x) dx, and this was some number and we learned a bunch of techniques for evaluating this.0063 Well, let us go ahead and talk about what some of these things mean.0077 This right here is just a value of f at a given x on the particular interval, and this ab is of course the closed interval from a to b, just something like this.0080 If there is some function, this is a, this is b, this is the function, so we are just integrating along that length. That is the important part, we are just integrating along a length.0095 Now, this dx here it is just a differential length element -- a differential is just a fancy word for small -- so differential length element, and this symbol right here... that is just the symbol for an infinite sum.0105 So, what we have done is we have a particular function, we have a bunch of x's in between a and b, we evaluate them at each of those x's, and then we multiply by the actual length, some length element along x and then we take these numbers that we get and we just add them up with this fancy technique called integration.0137 Then we get some number. That is the integral, that is all we have done.0159 Well, we can do the same thing with a function of two variables, except now, a function of 2 variables is not defined just over the x axis, it is defined over x and y.0162 So, instead of a differential length element, what you are going to have is a differential area element.0173 Because again, now you are going to have an interval from a to b along x, but you are also going to have an interval c to d along y, so now we are going to basically break up the a to b the way we did before in single variable calculus, but we are also going to break up c to d and what you end up having is a bunch of little rectangles.0180 Instead of integrating over a length, we are integrating over a length and another length.0205 Well, length × length is area, so we are integrating over the entire area. Everything else is exactly the same, the notation is exactly the same.0212 Let us go ahead and write this out. So, for a function of two variables, f(x,y), we can integrate this function over an area, which is just length × length.0220 You will see in a minute that is exactly what you are going to be doing. You are just going to be doing one integral at a time, in an iterated fashion, in a row.0272 The symbol for it is exactly the same, except you are going to use two integral signs instead of one.0282 So, double integral, f(x,y) dy dx.0288 Also written as double integral of f, I will leave off the x,y, da.0299 dy × dx, well, differential can be x, differential in the y-direction, that gives me a little bit of a square, so this square is a differential area element. That is why we have da for dy/dx, that is it.0310 This is the one that we want to concentrate on, but you will see this when you see certain theorems, certain statements along as you understand what is happening.0327 This is a symbolic definition, so let us talk about what these mean. This is the value of f at the point (x,y).0337 This is a differential area element.0349 So, instead of having a very, very, tiny length, like single variable calculus, what we have is a really, really tiny area. One of these small rectangles or squares.0356 This, of course, is the symbol for the infinite sum -- the symbol for infinite sum -- and we use two of them to differentiate from the single variable because we are talking about two variables.0367 Later, when we do functions of three variables, you are going to see triple integral, three integral signs. It actually exists that you can keep going.0382 Okay, that is it. That is literally all that is going on here.0391 Let me go ahead and draw out one more time this thing just to make sure we completely understand.0396 So, now, because our domain is now in two dimensions, we are going to split up the x length, we are going to split up the y length, and we are just going to add the value of the function over all of these little rectangles, that is what this is.0405 Now, I am going to write out how we actually evaluate this. The practical way of finding the integral, and we do it with a single integral at a time.0423 So, let us go here. The double integral of f(x,y) dy/dx, is evaluated as -- let me go ahead and write over here -- so from a to b, from c to d, so, our interval along the x-axis is a to b, our interval along the y axis is c to d.0434 It is equivalent to taking the integral from a to b, taking the integral from c to d of f(x,y) dy, and then dx.0478 What this symbol means is that we are going to be working from the inside out, just like we do in mathematics.0495 What we are going to do is we are going back to one variable first, in this particular case, dy, f(x,y) dy, and when we do this we have to remember that y is the variable we are integrating with respect to so we keep x constant.0500 When we do this, and we evaluate this integral from c to d, we are going to get a function x.0515 Now this function of x is what we integrate from a to b and we get our final answer. That is it, this is called iterated integration and you can actually do the integral in either order.0519 You can either do f(x,y) dy dx, or you can do f(x,y) dx dy.0531 The problem itself, the particular domain you are going to be working with, will decide for you which you are going to integrate first.0536 Again, you are just integrating one variable at a time. let us just go ahead and jump into some examples, and hopefully it will all make sense.0544 Again, you have done this over and over and over again. Now instead of just stopping after one integral, you are doing another integral after that. The only thing that you have to watch out for is keeping track of the variables.0553 If you are going to be integrating with respect to y, make sure to keep x constant and carry it forward.0563 If you are integrating with respect to x, hold y constant, and carry it forward and do the evaluation. That is actually going to be the biggest stumbling block in this. It is not going to be the mathematics itself, it is going to be keeping track of what is going on.0571 So, example 1. Find the integral of xy3 over the region x goes from 1 to 2, and y goes from 4 to 6.0585 We have this little bit of a rectangle, we have 1, we have 2, let us say 4 is here. Let us say 6 is here, so we have this little rectangle that we are going to be evaluating this function, integrating this function over this domain.0622 Well, the integral, I will just call it i. In this particular case I have a choice since I am given the endpoints of the intervals explicitly, I can do dx dy, or I can do dy dx.0642 I am going to do dy first and save dx for last. Personal choice.0657 So I am going to integrate from 1 to 2. I am going to integrate from 4 to 6 -- we are working inside out -- xy3 dy dx.0660 okay. So, I am going to do the first integral. The inside integral first. This is going to equal the integral from 1 to 2, and it is always great to write out everything, do not keep things in your head, do not write things shorthand, write everything out.0672 It is not going to hurt if you have a couple of extra lines of mathematics, at least that way if there is a problem, you can follow it -- working your way back.0686 Now, when you integrate this, this symbol stays, we are doing this integration. We are integration with respect to y, x stays constant.0696 Well, integrate y3, the integral of y3 is y4/4, so it becomes xy4/4 evaluated from 4 to 6 and then dx... so far so good.0705 That equals the integral from 1 to 2, well when I put 6 into y and evaluate this, I am going to end up with so 64/4.0721 I get the integral of 324x -... and of course I put 4 in for y, take the fourth power, divide by 4, multiply by 6, I am going to get -64x dx.0736 That is equal to the integral from 1 to 2 of 260x dx.0752 I am going to pull the 260 out because that is just my personal preference. I like to keep the constants outside and multiply them at the end.0759 260, integral of x, dx = 260 × x2/2 evaluated from 1 to 2.0770 I just integrated this now with respect to x. I got x2/2, and when I run through this evaluation putting 2 in, subtracting putting 1 in, multiplying by 260, I end up with 390.0781 That is it. Nice and simple. Okay.0796 Let us go ahead and give a physical interpretation of what this means and it might help, it might not. In single variable calculus, we had some function and we had the interval from a to b. Well, we interpret the integral physically as the area underneath the graph.0801 Okay. We have the same thing in a function of two variables. We said in previous lessons that a function of two variables, since it is defined over a region in the x,y plane, the value of the function can actually be used as a third variable.0821 We can graph 3-space. Basically, a function of two variables is a surface in 3-space, so let us go ahead and draw a little -- so this is the x, and this is the y, and this is the z -- so there are some surface above the x,y plane.0836 Well, basically, the integral of a function of two variables over a particular region can be interpreted as the volume of everything above that region up to the surface. That is it.0861 Again, the area underneath the graph, the volume underneath the graph. We are just moving up one dimension. That is a physical interpretation that is there for you. If you want to think about it that way, that is fine.0884 I think it helps to -- you know -- in certain circumstances, but it is important to remember that an integral is a number. It is an algebraic property, not necessarily a physical property. It can be interpreted as such, but that is not what it is.0894 Let us go ahead and do another example here. So, example 2.0908 This time, f -- let us let f equal to x2y -- and r be the region such that x runs from 2 to 3 and y is the region that is above, well between, the functions x2 and x3.0918 Let us go ahead and draw this out. Again, we do not need a precise drawing... that is the x2. that is the x3, let us say this is 2 and this is 3.0962 We have that, we have that, so the region that we are looking at is that region right there. That is the area over which we are going to be integrating this particular function.0975 Okay. So, in this particular case, we are constrained by the nature of the problem to do our... to differentiate with respect to y first and then differentiate with respect to x. It is simply easier that way.0990 So, let us go ahead and do it.1005 So, the integral is equal to the integral from 2 to 3, of the integral, so the lower function is the x2, the upper in this case is the x3.1008 So, we are going to go from x2 to x3, this is perfectly acceptable.1023 The function that we are integrating is x2y dy dx, so we just need to make sure that we keep the order correct.1027 Well, this is going to equal the integral 2 to 3. Now when we integrate x2y with respect to y, we are holding x2 constant, so it is going to be y2/2... x2 y2/2.1041 We are evaluating it from x2 to x3, and then we are going to do the dx.1055 This is going to be the integral from 2 to 3 -- oops, let us see if we can eliminate as many of these stray lines as possible -- the integral from 2 to 3, so when we put x3 in for here, we get x into y because we are evaluating with respect to y.1060 This is going to be x6 × x2, so we are going to get x8/2 - ... and when we put x2 in for here, it is going to be x4 × x2, it is going to be x6/2 dx.1079 Right? So far so good. Let us go ahead and go to the next page.1099 When we do that integration, it is going to equal -- actually, let me write it again so we have it on this page -- 2 to 3 x8/2 - x6/2 dx = x9/18 - x7/14.1103 Right? 6, 7 × 2 is 14, 9 × 2 is 18. We are going to evaluate that from 2 to 3, and then when we do this evaluation we are going to end up somewhere in the neighborhood -- ahh its okay, I am just going to do an approximate... it is going to be somewhere in the neighborhood of about 918 when you actually run that.1130 The number itself, I mean it is important, but it is not that important. It is the process that is important. This is where we want to come to.1153 Now, let us go ahead and do a third example. Example 3.1163 We have f(x,y) is equal to... this time, 4xy is our function, over the region bounded by y = 0, y = x and y = -x + 3.1172 Let us go ahead and draw this region out and go ahead and see what it is we are looking at.1200 So, y = 0, that is the x axis. y = x, that is this line right here, y = -x + 3, so let us go up 3, and let us go down that way in a 45 degree angle.1207 So this is going to hit at 3 and let us go ahead and put a half-way mark here. This is actually going to be 3/2 because these have the same slope.1224 In this particular case, our region is right here, this triangle. We are going to be integrating this function over this particular triangle.1235 Of course you remember from first variable calculus, sometimes regions have to be split up simply to make the integral a little bit easier to handle.1242 In this particular case, I am going to decide to save the integration with respect to x last, the outside integral.1249 Since that is the case, I am actually going to split this into two regions. This region r, I am going to split it up into R1 and R2.1257 I am going to integrate up to here, and up to there. Then I am going to add those 2 together. The reason that I do that is the upper function from here to here is different. Here it is 0 to x, here it is 0 to -x + 3, which is why I am splitting it up.1265 Let me erase this thing. So, in this particular case, our integral over R is going to equal the integral over R1 + the integral over R2 because the integrals are additive, that is exactly right.1282 Let us go ahead and deal with the integral of -- I am going to move to the next page -- so, the integral over R1 is going to equal the integral from 0 to 3/2, that is the first half and the y value, let me draw my domain again so I have it here.1300 It is going to be that, and that, so the y value is going to go from 0 to x, and my function is 4xy dy dx.1324 That is -- excuse me -- equal to the integral from 0 to 3/2 of 2xy2, I am integrating with respect to y, I am holding that constant, y2/2, the 2 and the 4 cancel leaving me 2xy2, and I am evaluating that from 0 to x dx.1339 That is going to equal, well, 2 × the integral from 0 to 3/2... when I put x in for here, it is x2 × x is x3, so it is x3 dx.1359 This is 0, so it goes away and now that is going to equal 2 × x4/4 evaluated from 0 to 3/2.1374 When I go ahead and do that, I get 81/32, so that is the integral with respect to the first region.1389 Now we will do the integral over the second region, this region right here.1399 So, the integral over region 2 is equal to -- now we are integrating from 3/2 to 3 -- so, 3/2 to 3, and the y value is going to be 0 to -x + 3.1404 Because now the upper function is -x + 3 and again our function is 4xy dy dx.1423 Well, that is going to equal 3/2 to 3, it is going to be again 2xy2, evaluated from 0 to -x + 3.1432 When I put this in here, let us see if I have enough room, no I probably do not so let me come down here -- the integral from 0... nope, doing 3/2 to 3 now -- again, keep track of our numbers.1449 From 3/2 to 3, it is going to be 2x × -x + 32, because the 0 goes away and I am just putting this into that, dx.1464 Ohh... crazy... we do not want these crazy lines all over the place.1480 Okay. 0 dx, when I multiply this out, actually I am going ot multiply everything out here so it is fine. We will do 3/2 over 3, it is going to be 2x × x2 - 6x + 9 dx = the integral from 3/2 to 3 of 2x3 - 12 x2 + 19x dx.1490 When I go ahead and evaluate that, I use mathematical software to evaluate that, I ended up with 243 over 32, therefore the integral of f over our original region R is equal to the 81/32, the integral over the first region + 243/32, the integral of the other region = 324/32.1532 That is it. Okay. So, in this particular case, I took this region and I divided it into 2 regions and I did 2 separate integrals.1567 There is a way to actually do this as one integral by just reversing the order of integration by doing dx first and then doing the dy.1575 I will go ahead and show you what that is. I will not do the integration, but I will show you how to approach it.1584 So, let me draw the domain again. We had that, and we had this, okay? So this was 3, this was 0, we had 3/2 here, this was 3, this was y = x, or if I wrote it in terms of x, x = y.1590 Okay. This graph right here, that is the y = -x + 3. If I write it in terms of x, it is x = 3 - y.1619 Now, I am thinking about it this way. I am going to do my final integration with respect to y, which means I am going to go from 0 all the way up to 3.1630 In this particular case, there is no interference. The difference -- it says if I turn the graph this way -- the difference, this length right here, okay, that I am going to be integrating this way... this length is the difference between this function and this function.1644 In this particular case, I can go ahead and take the difference between this function and this function, and then integrate with respect to y this way, so the integral looks like this.1662 The integral from 0 to -- this is not 3, this is actually 3/2 -- 0 to, this value right here is 3/2.1681 If you solve simultaneously, this is x is 3/2, this is y is 3/2, so this y value is 3/2.1692 So 0 to 3/2, the integral... well, it is going to be... you want to keep this positive, so we are taking this function, this function, this is the lower that is the upper, so let us go 3 - y to y, 4xy, this time we are going to do dx dy.1699 That is it. So, you can do this as a single integral as long as you change your perspective and if you are integrating finally along this direction, the first integration that you do, which is going to be this way, well, this difference always stays the same in the sense that it is always going to be a difference between this function and this function.1726 That is it. The outer integral covers the entire region, I did not actually have to break this up, but the way that I chose to do it, simply because I like doing x last.1752 I had to break this region up into 2, and the reason I broke it up into 2 is because the y value is actually different depending on which function I am working with past this 3/2 mark.1764 That is it. That is our introduction to double integrals. Thank you for joining us here at educator.com, we will see you next time. 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https://www.scienceforums.net/topic/44123-g%C3%B6dels-completeness-theorem/	# Gödel's Completeness Theorem ## Recommended Posts I'm currently reading this book called Set Theory and the Continuum Hypothesis, written by Paul Cohen, which is a model-theoretic investigation of the topics. I'm trying to rediscover the proof of Gödel's Completeness theorem for myself, but I'm kind of stuck on certain details of the proof provided in the book. In the preface, the author mentioned that he did not "polish up" the final draft of the book, so many important details are left out. Although it is written for people with little to no background in propositional logic, the book assumes that one has a background in abstract mathematics, namely in Model theory. I'm only an undergraduate student not yet knowledgeable enough to understand the methods Cohen uses to prove his arguments. So I'm wondering if someone here can help me understand what certain things mean in the proofs. The parts that I've put in red are the parts I'm stuck on. I will quote directly from the book so as to give you a full view of what I'm asking. $c_\alpha$ and $R_\beta$ are constant and relation symbols, respectively. Theorem 3. (Completeness of the Propositional Calculus). If $S$ contains no quantifiers and is consistent, then there is a model $M$ for $S$ in which every element of $M$ is of the form $\overline{c}_\alpha$ for some $c_\alpha$ appearing in $S$. We need one lemma. LEMMA. If $T$ is a consistent set of statements, $A$ an arbitrary statement, either $T \cup \{A\}$ or $T \cup \{\neg A\}$ is consistent. PROOF. If $T \cup \{A\}$ is inconsistent, then for some $B_i$ in $T, A \land B_1 \land \cdots \land B_n \rightarrow C \land \neg C$ for some $C$, is valid. If $T \cup \{\neg A\}$ is inconsistent, then for some $B'_i$ in $T$, $\neg A \land B'_1 \land \cdots \land B'_m \rightarrow C \land \neg C$ is valid. The propositional calculus now implies that $B_1 \land \cdots \land B_n \land B'_1 \land \cdots \land B'_m \rightarrow C \land \neg C$ is valid, so that $T$ must be inconsistent. Now to prove Theorem 3. Let $S$ be well-ordered. This induces a well-ordering on all the constant and relation symbols which appear in $S$. This in turn induces a well-ordering of all possible statements of the form $c_i = c_j$ and $R_\beta (c_1,...,c_n)$ where $c_i$ and $R_\beta$ are constant and relation symbols occurring in $S$. Call these statements $F_\alpha$. We now define statements $G_\alpha$ by induction on $\alpha$. If $F_\alpha$ is consistent with $S \cup \{G_\beta \| \beta < \alpha \}$ we put $G_\alpha = F_\alpha$, otherwise put $G_\alpha = \neg F_\alpha$. By our lemma and by induction on $\alpha$, it follows that $S \cup \{G_\beta \| \beta \leq \alpha \}$ is consistent for all $\alpha$. Since any contradiction must be derived from a finite number of statements, we see that $H = S \cup \{G_\alpha \}$ is a consistent system. For each $c_\alpha$ occurring in S, define $\overline{c}_\alpha = c_\beta$ where $\beta$ is the least index such that $c_\alpha = c_\beta$ belongs to $H$. (There must be some such since $c_\alpha = c_\alpha$ belongs to $H$.) Let $M$ be the set of $\overline{c}_\alpha$, and define $\overline{R}_\beta$ as the set of all $\langle \overline{c}_\alpha 1 ,..., \overline{c}_\alpha n \rangle$ such that $R_\beta (c_\alpha ,..., c_\alpha n)$ is in $H$. Thus our model consists of a subset of the formal symbols $c_\alpha$. It is easy to see that every statement in $H$, or its negation, must be a consequence of the $G_\alpha$ since every atomic relation occurring in a statement of $S$ or its negation appears among the $G_\alpha$. A negation of a statement in S cannot be such a consequence since in that case $H$ would not be consistent. Since we have defined $M$ so that all the $G_\alpha$ are true, it follows that in $M$ all the statements of $H$ and hence of $S$ are also true. Thus Theorem 3 is proved. I have a very vague idea, but I'm not entirely sure what those parts mean exactly. If someone could help me out here by explaining those parts to me in more detail I would appreciate it. Also, if there are any questions regarding the excerpt from the book I will do my best to answer them. Thank you! Jeffrey Edited by Abstract_Logic ##### Share on other sites Regarding Completeness: You may try going to the source. Godel actually published (2) papers: 1) 'On the completeness of the calculus of logic (1929)' ; 2) 'The completeness of the axioms of the functional calculus of logic (1930)'. See 'Kurt Godel COLLECTED WORKS Volume I Publications 1929-1936'. Regarding The Continuum Hypothesis: As the story goes, Godel believed it to be false. In 1947 he published an 'expository essay' showing that the CH cannot be proved false. Curiously, he stopped there. One almost gets the sense that he did not attempt to prove it true, as this would contradict his intuition. Who knows? Cohen, on the other hand, proved that the CH cannot be proved true (1963). Though still debatable, the two results essentially showed the CH to be undecidable. Let me know if this helps. Good Luck! liarliarpof ## Create an account or sign in to comment You need to be a member in order to leave a comment ## Create an account Sign up for a new account in our community. It's easy! Register a new account	2023-03-31 11:58:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.816288948059082, "perplexity": 165.00568031336928}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949642.35/warc/CC-MAIN-20230331113819-20230331143819-00074.warc.gz"}
https://www.bhardwajclass.com/2019/10/class-10-maths-trigonomtery-test-paper-pdf.html	## TRIGONOMETRY CLASS 10 TEST PAPER As we all know "Practice makes a man perfect" so, more you practice , better you perform. for the sake of your practice we prepared Practice Papers,Test Papers, Important Questions strictly based on Latest CBSE Curriculum and new Pattern.This Trigonometry Class 10 Test Paper is based on new pattern and has all type of Questions as objective type questions, short questions and long questions. Practice Class 10 Pair of linear equations in two variables Test Paper Click Here ONE MARK QUESTIONS 1. If ${\mathrm{sin}}^{2}A=\frac{1}{2}{\mathrm{tan}}^{2}{45}^{0},\text{ }where\text{ }A$ is an acute angle, thenthe value of $A$ $a) 45 0 b) 30 0 c) 60 0 d) 90 0$ 2. Express $\mathrm{sin}{67}^{0}+\mathrm{cos}{75}^{0}$ in terms of trigonometric ratios of angles between ${0}^{0}\text{ }and\text{ }{45}^{0}.$ 3. If ${\mathrm{sec}}^{2}\theta \left(1+\mathrm{sin}\theta \right)\left(1-\mathrm{sin}\theta \right)=k$ , then find the value of $k.$ 4. $\mathrm{sin}\theta +\mathrm{cos}\theta$ is always greater then 1. (True/False) TWO MARKS QUESTIONS 5. IF $\mathrm{sin}\theta +\mathrm{cos}\theta =\sqrt{3},$ then prove that $\mathrm{tan}\theta +\mathrm{cot}\theta =1.$ 6. Find the value of $3\mathrm{cos}{68}^{0}.\mathrm{cos}ec{22}^{0}-\frac{1}{2}\mathrm{tan}{43}^{0}.\mathrm{tan}{47}^{0}.\mathrm{tan}{12}^{0}.\mathrm{tan}{60}^{0}.\mathrm{tan}{78}^{0}$ 7. Prove that ${\left(\mathrm{cos}ec\theta -\mathrm{cot}\theta \right)}^{2}=\frac{1-\mathrm{cos}\theta }{1+\mathrm{cos}\theta }$ . THREE MARKS QUESTIONS 8. $\frac{\mathrm{cos}ec\theta +\mathrm{cot}\theta }{\mathrm{cos}ec\theta -\mathrm{cot}\theta }=1+2{\mathrm{cot}}^{2}\theta +2\mathrm{cos}ec\theta \mathrm{cot}\theta$ 9. Find an acute angle $\theta ,\text{ }when\text{ }\frac{\mathrm{cos}\theta -\mathrm{sin}\theta }{\mathrm{cos}\theta +\mathrm{sin}\theta }=\frac{1-\sqrt{3}}{1+\sqrt{3}}.$ 10. If $\mathrm{cot}\theta =\frac{7}{8},\text{ }find\text{ }\frac{\left(1+\mathrm{sin}\theta \right)\left(1-\mathrm{sin}\theta \right)}{\left(1+\mathrm{cos}\theta \right)\left(1-\mathrm{cos}\theta \right)}$ FOUR MARKS QUESTIONS. 11. Prove that $\frac{1}{\left(\mathrm{cos}ec\text{ }x+\mathrm{cot}x\right)}-\frac{1}{\mathrm{sin}x}=\frac{1}{\mathrm{sin}x}-\frac{1}{\left(\mathrm{cos}ec\text{ }x-\mathrm{cot}x\right)}$ 12. Prove that $\frac{\mathrm{tan}\theta }{1-\mathrm{cot}\theta }+\frac{\mathrm{cot}\theta }{1-\mathrm{tan}\theta }=1+\mathrm{sec}\theta \mathrm{cos}ec\theta =1+\mathrm{tan}\theta +\mathrm{cot}\theta$	2021-04-18 14:43:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5555711984634399, "perplexity": 2004.073774442827}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038492417.61/warc/CC-MAIN-20210418133614-20210418163614-00381.warc.gz"}
http://mathhelpforum.com/calculus/176810-does-make-sense.html	# Thread: Does this make sense? 1. ## Does this make sense? I'm working on past finals and for one of the integrals I don't understand how they got to the second step in the solution listed: ImageShack® - Online Photo and Video Hosting I don't understand how they went from x^2/(1+x^2) to 1 and 4/(1+x^2) to 3/(1+x^2) 2. Originally Posted by DannyMath I'm working on past finals and for one of the integrals I don't understand how they got to the second step in the solution listed: ImageShack® - Online Photo and Video Hosting I don't understand how they went from x^2/(1+x^2) to 1 and 4/(1+x^2) to 3/(1+x^2) The either did polynomial long division or rewrote the problem as follows $\displaystyle \frac{x^2-2x+4}{x^2+1}=\frac{(x^2+1)-2x+3}{x^2+1}=\frac{x^2+1}{x^2+1}+\frac{-2x}{x^2+1}+\frac{3}{x^2+1}$ 3. It looks like polynomial long division has been applied here. $\displaystyle (x^2-2x+4)\div (1+x^2)= 1- \frac{2x+3}{1+x^2}$	2017-06-24 09:41:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45119965076446533, "perplexity": 755.4658751163324}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320243.11/warc/CC-MAIN-20170624082900-20170624102900-00540.warc.gz"}
https://www.authorea.com/users/5705/articles/62619-the-complexity-of-probability-distribution-functions/_show_article	# The complexity of probability distribution functions AbstractThis is the abstract section. The abstract should be one section and count less than 200 words. # Introduction Main text paragraph. Citing a journal paper (Lastname 2008). And now citing a book reference (Lastname 2007). Main text paragraph.	2017-07-22 08:52:36	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9885864853858948, "perplexity": 7232.04171277518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423927.54/warc/CC-MAIN-20170722082709-20170722102709-00091.warc.gz"}
https://tex.stackexchange.com/questions/20425/z-level-in-tikz	# “Z-level” in TikZ This is an idle question, but neverthless, I think it's interesting. Does TikZ have a simple way to implement a "z-level" specification. By this, I don't mean 3D drawing, I mean specifying which parts of the picture are on top of what. So I'd write something like this: \draw[fill, blue,zlevel=1] (0,0) circle (2); \draw[fill,red,zlevel=2] (0,0) circle (1); and have the red circle appear on top of the blue one (since it has a higher z-level). I know I could just reverse the order, but I can imagine that being able to specify things this way round might make incremental changes to a TikZ picture in beamer easier by simplifying the overlay specifications... I suppose you could do something with remember picture, but it seems a "z-level" key would be the easiest solution (from the end user perspective). Is there anything like this? Is it possible? • It should be noted that if you need only two levels, then having the part "below" in between \begin{scope}[on background layer]…\end{scope} is enough. The solutions below are to accommodate more than 2 layers. – Clément Feb 4 at 18:23 You can use the layers describe in section 108 Layered Graphics of the pgfmanual. You need to declare the layers first using \pgfdeclarelayer and then use them with \pgfsetlayers outside the picture. With the {pgfonlayer}{<layer name>} environment you can can wrap TikZ or PGF commands to be drawn on this layer. The background library provides a on background layer style for scopes, so it should be possible to define a zlevel style as well: \tikzset{zlevel/.style={% execute at begin scope={\pgfonlayer{#1}}, execute at end scope={\endpgfonlayer} }} But you need to put all the commands into a scope. • Turns out your last sentence isn't wholly correct! (See my updated answer) – Loop Space Jun 13 '11 at 14:02 • @Andrew: Nice! I was thinking about if this might be possible, but didn't had the time to investigate. – Martin Scharrer Jun 13 '11 at 14:13 • @MartinScharrer Please, I would so use a MWE here now, since I have no idea as how scopes work. – yo' May 26 '13 at 21:32 • @tohecz: TikZ scope are normal environments \begin{scope}[zlevel=...] ... \end{scope}. – Martin Scharrer May 27 '13 at 5:36 Yes. They are called "layers". I've used this a few times in various solutions: As have various others: https://tex.stackexchange.com/search?q=pgfonlayer Update: In a desperate attempt to "out-TikZ" Martin, here's a version that works as a key on a single path. It works by a bit of sneaky hackery together with careful observation as to when particular commands get executed. We need to install some code at the beginning and the end of the path. The beginning is okay as TikZ options get parsed early on. The end is a little trickier and that's where the sneaky hackery comes in. Basically, the \tikz@path@do@at@end macro gets done, as it says, at the end. It occurs after an \endgroup and the keys on the path are examined inside that group, so at first sight it seems that there's no way for a key inside the group to hook in to that \tikz@path@do@at@end without making a global assignment (something I'm reluctant to do). But if we introduce a new \begingroup then the original \endgroup at the end of the TikZ path-setting routine no longer ends the main grouping but ends our internal grouping, thus allowing us access to \tikz@path@do@at@end. So long as we are careful and end the outer group, followed again by \tikz@path@do@at@end (which is now back to its original meaning) then we can put our layer ending code there. \documentclass{article} \usepackage{tikz} \pgfdeclarelayer{back} \pgfsetlayers{back,main} \makeatletter \pgfkeys{% /tikz/on layer/.code={ \def\tikz@path@do@at@end{\endpgfonlayer\endgroup\tikz@path@do@at@end}% \pgfonlayer{#1}\begingroup% }% } \makeatother \begin{document} \begin{tikzpicture} \fill[green!50!white,opacity=.5] (0,0) rectangle (8,8); \begin{pgfonlayer}{back} \draw[ultra thick,red,line width=1cm] (-1,-1) -- (9,9); \end{pgfonlayer} \draw[ultra thick,red,line width=1cm,on layer=back] (-1,9) -- (9,-1); \end{tikzpicture} \end{document} with the result that both the red lines are behind the green rectangle. (As with many of my hackery answers, this comes with a "use at own risk" warning, and is also as much about the hack as the specific application. Indeed, I'd been looking for a way to hook in to that \tikz@path@do@at@end macro for something else and this might just be how I can do it.) Edit: (again) I don't know how "safe" modifying \tikz@path@do@at@end is so here's an alternative that uses \aftergroup instead. It uses the same idea of the hack: start a new group so that the existing \endgroup is taken as the end of the inner group. This is a replacement for the \pgfkeys bit in the previous example: \pgfkeys{% /tikz/on layer/.code={ \pgfonlayer{#1}\begingroup \aftergroup\endpgfonlayer \aftergroup\endgroup }, /tikz/node on layer/.code={ \pgfonlayer{#1}\begingroup \expandafter\def\expandafter\tikz@node@finish\expandafter{\expandafter\endgroup\expandafter\endpgfonlayer\tikz@node@finish}% } } Update: As Bruce pointed out in the comments, on layer doesn't work within the \node command (though \path[on layer=back] node {node}; works). That's because \node[options] becomes \path node[options] and not \path[options] node so the hooks are in different places. After a bit of detective work, I found a hook to hang the closing bit of the layer command on. So \node[node on layer=back] {on back layer}; works. I still need to look a bit further to find something that works with \draw (0,0) -- node[put this on the back layer but not the path] {back} (8,0);. (and no longer any need for \makeatletter!) Update (2012-03-07): Revisiting this, I found that the node on layer was no longer working (if it ever did - I'm dubious now as I don't think that the TikZ code has changed). I have found a new set of hooks to hook into for the nodes. One significant advantage of the new approach is that it is possible to put nodes on a different level to their surrounding path. Rather neat, I thought. Here's the current code: \documentclass{article} %\url{https://tex.stackexchange.com/q/46957/86} \usepackage{tikz} %\usepackage[tracelevel=silent]{trace-pgfkeys} \pgfdeclarelayer{back} \pgfdeclarelayer{front} \pgfsetlayers{back,main,front} \makeatletter \pgfkeys{% /tikz/on layer/.code={ \pgfonlayer{#1}\begingroup \aftergroup\endpgfonlayer \aftergroup\endgroup }, /tikz/node on layer/.code={ \gdef\node@@on@layer{% \setbox\tikz@tempbox=\hbox\bgroup\pgfonlayer{#1}\unhbox\tikz@tempbox\endpgfonlayer\egroup} \aftergroup\node@on@layer }, /tikz/end node on layer/.code={ \endpgfonlayer\endgroup\endgroup } } \def\node@on@layer{\aftergroup\node@@on@layer} \makeatother \begin{document} \begin{tikzpicture} \draw[line width=1cm,red] (2,1) -- (2,-1); \draw[ultra thick,white,preaction={on layer=back,line width=1cm,blue,draw}] (0,0) -- (4,0); \draw[line width=1cm,red] (2,-2) -- (2,-4); \draw[ultra thick,white,postaction={on layer=back,line width=1cm,blue,draw}] (0,-3) -- (4,-3); \begin{scope}[xshift=5cm] \draw[line width=1cm,red] (2,1) -- (2,-1); \draw[ultra thick,white,preaction={line width=1cm,blue,draw}] (0,0) -- (4,0); \draw[line width=1cm,red] (2,-2) -- (2,-4); \draw[ultra thick,white,postaction={line width=1cm,blue,draw}] (0,-3) -- (4,-3); \end{scope} \begin{scope}[yshift=-5.2cm,xshift=2cm] \draw[line width=1cm,red] (0,1) -- (0,-1); \path[on layer=back] node[draw] (a) {At the back of beyond}; \node[node on layer=front] at (0,-2) (b) {At the front of beyond}; \draw[line width=1cm,red] (b.north) -- (b.south); \draw[line width=1cm,red] node[thin,black,draw,node on layer=back] at (0,-4) (c) {At the side of beyond} (c.north) -- (c.south); \begin{scope}[xshift=5cm] \draw[line width=1cm,red] (0,1) -- (0,-1); \path node[draw] (a) {At the back of beyond}; \node at (0,-2) (b) {At the front of beyond}; \draw[line width=1cm,red] (b.north) -- (b.south); \draw[line width=1cm,red] node[thin,black,draw] at (0,-4) (c) {At the side of beyond} (c.north) -- (c.south); \end{scope} \end{scope} \end{tikzpicture} \end{document} (NB the commented out package is left in deliberately as it was very useful in figuring out what was going on. See How do I debug pgfkeys? for more details on that package.) Here's the result of the above. In each case, the layering is happening on the left and not on the right. The fact that the left and right are different proves that something is being shifted from one layer to another. • +1 for taking the "out-TikZ Martin" challenge :-) and another impressive lesson for what one can achieve with TikZ by "just" defining some extra styles. – Daniel Jun 14 '11 at 7:48 • @Daniel: Thanks! The more I look at the code, the more opportunities I see for doing "weird" stuff like this. Turns out my last sentence was right: this was just what I needed. There's lots of unseen "hooks" that one can use to mess with TikZ's routine and make it do something completely different to what was intended. – Loop Space Jun 14 '11 at 7:51 • @Andrew: This looks brilliant. But your code doesn't seem to work for nodes. If I type \node[on layer=back] {This should be on back layer} at (3,3); then it doesn't put it on the back layer. – Bruce Bartlett Jun 15 '11 at 17:49 • @Bruce: True. Looking at the code, I can't see any easy hooks for the \node command. Although \node is "an abbreviation for \path node" the options to the \node don't get put on the \path. A way around it is \path[on layer=back] node {This should be on the back layer} at (3,3);. – Loop Space Jun 15 '11 at 18:55 • @Andrew: Ok good, that's not a bad workaround. – Bruce Bartlett Jun 15 '11 at 19:24 This is merely a supplement to Loop Space's fantastic answer, which makes it work as I'd expect with the TikZ backgrounds library. The only reason I've renamed the background and foreground layers is for consistency with the backgrounds library and with my own extension of that. \documentclass[border=10pt,multi,tikz]{standalone} \usetikzlibrary{backgrounds} \pgfdeclarelayer{foreground} \pgfsetlayers{background,main,foreground} \makeatletter \tikzset{% on foreground layer/.style={% execute at begin scope={% \pgfonlayer{foreground}% \let\tikz@options=\pgfutil@empty% \tikzset{every on foreground layer/.try,#1}% \tikz@options% }, execute at end scope={\endpgfonlayer} }, % addasu o ateb Loop Space: https://tex.stackexchange.com/a/20426/ %\url{https://tex.stackexchange.com/q/46957/86} on layer/.code={% \pgfonlayer{#1}\begingroup \aftergroup\endpgfonlayer \aftergroup\endgroup }, node on layer/.code={% \gdef\node@@on@layer{% \setbox\tikz@tempbox=\hbox\bgroup\pgfonlayer{#1}\unhbox\tikz@tempbox\endpgfonlayer\egroup} \aftergroup\node@on@layer }, } \def\node@on@layer{\aftergroup\node@@on@layer} \makeatother \begin{document} \begin{tikzpicture} \draw[line width=1cm,red] (2,1) -- (2,-1); \draw[ultra thick,white,preaction={on layer=background,line width=1cm,blue,draw}] (0,0) -- (4,0); \draw[line width=1cm,red] (2,-2) -- (2,-4); \draw[ultra thick,white,postaction={on layer=background,line width=1cm,blue,draw}] (0,-3) -- (4,-3); \begin{scope}[xshift=5cm] \draw[line width=1cm,red] (2,1) -- (2,-1); \draw[ultra thick,white,preaction={line width=1cm,blue,draw}] (0,0) -- (4,0); \draw[line width=1cm,red] (2,-2) -- (2,-4); \draw[ultra thick,white,postaction={line width=1cm,blue,draw}] (0,-3) -- (4,-3); \end{scope} \begin{scope}[yshift=-5.2cm,xshift=2cm] \draw[line width=1cm,red] (0,1) -- (0,-1); \path[on layer=background] node[draw] (a) {At the background of beyond}; \node[node on layer=foreground] at (0,-2) (b) {At the front of beyond}; \draw[line width=1cm,red] (b.north) -- (b.south); \draw[line width=1cm,red] node[thin,black,draw,node on layer=background] at (0,-4) (c) {At the side of beyond} (c.north) -- (c.south); \begin{scope}[xshift=5cm] \draw[line width=1cm,red] (0,1) -- (0,-1); \path node[draw] (a) {At the background of beyond}; \node at (0,-2) (b) {At the front of beyond}; \draw[line width=1cm,red] (b.north) -- (b.south); \draw[line width=1cm,red] node[thin,black,draw] at (0,-4) (c) {At the side of beyond} (c.north) -- (c.south); \end{scope} \end{scope} \path (current bounding box.north west) -- (current bounding box.north east) \foreach \i [count=\j] in {0,.225,.275,.5,.725,.775,1} {coordinate [pos=\i] (p\j)}; \begin{scope}[on background layer] \fill [yellow, fill opacity=.3] (current bounding box.north west) rectangle (current bounding box.south -\| p2) (p7) rectangle (p6 \|- current bounding box.south); \end{scope} \begin{scope}[on foreground layer] \fill [green, fill opacity=.3] (p2) rectangle (current bounding box.south -\| p3) (p6) rectangle (p5 \|- current bounding box.south); \end{scope} \fill [magenta, fill opacity=.3] (p3) rectangle (current bounding box.south -\| p4) (p5) rectangle (p4 \|- current bounding box.south); \end{tikzpicture} \end{document}	2019-09-18 23:49:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.888948380947113, "perplexity": 3766.710986691534}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573385.29/warc/CC-MAIN-20190918234431-20190919020431-00496.warc.gz"}

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