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## Introduction

Neural plasticity enables the human brain to adapt to the environment throughout the lifespan. From birth, we learn to associate sensory signals from different modalities when they arise from the same event in the external world1. Because sensory systems develop at different rates, some have suggested that multisensory coordination and integration build on intersensory calibration and continuous recalibration during development2,3. Presumably, the intersensory recalibrations should be extensive at younger ages and then decrease as maturation progresses, because with age, the systems become more specialized to native stimuli and neural plasticity decreases. Recently, however, Rohlf, Li, Bruns, and Röder4 demonstrated a reversed trend by showing that audio–visual integration in the spatial domain develops prior to the emergence of spatial recalibration to newly experienced spatial disparity. Here we examine this surprising developmental trend in the temporal domain among three sensory combinations—audio–visual, audio–tactile, and visual–tactile—by re-analyzing previously published data5,6,7.

The ability to integrate cross-modal signals supports our survival in a multisensory world: multisensory integration improves precision, accuracy, and processing speed of perception8,9,10,11,12,13. Temporal simultaneity provides one crucial cue in determining which cross-modal signals belong together14,15, while those integration benefits tend to decline when temporal asynchronies are introduced between the component signals16,17. In real world contexts, the relative timing of signals generated by an external event can vary significantly between sensory modalities. Specifically, the timing of audio–visual signals is dependent on source distance in audition but not in vision because sound travels much slower than light; on the other hand, a tactile stimulus is always applied to the body surface (i.e., directly to the sensory system). Once at the sensory periphery, there are internal factors such as differences in neural transduction times18: visual processing is the slowest19, and tactile processing time is positively correlated with the neural transduction distance20. Temporal asynchrony is therefore inevitable between stimulus signals arising from a common event.

To deal with the temporal variations of multisensory signals that originate from the same event, the multisensory system realigns cross-modal signals according to previous experience, a phenomenon known as temporal recalibration (see15, for a review). Temporal recalibration is manifested by shifts in the observer’s point of subjective simultaneity (PSS)—the time lag at which sensory signals are perceived as most likely to be simultaneous. For example, adapting to a visual-leading pair biases the PSS toward visual-leading presentations21,22,23,24,25,26,27,28,29. There are two forms of temporal recalibration at different time scales. Slow recalibration involves an exposure phase during which observers are presented repeatedly with a fixed time lag between signals (e.g., an auditory signal constantly lags a visual signal by 200 ms) for a period of time (e.g., several minutes) before demonstrating a change in tested simultaneity perception21,22. Following several minutes of adaptation, the recalibration effect lasts for about a minute23,24. Rapid recalibration, on the other hand, involves a change after a time lag on the single previous trial, but is influenced far less by the time lag of one further trial back25. Hence, slow and rapid recalibrations demonstrate quick adaptations and recoveries from stimulus variations of multisensory systems—such dynamic malleability is separate from plasticity which often involves experience-dependent changing of neural substrates. In adults, both rapid and slow recalibrations are more reliable for audio–visual than in audio–tactile or visual–tactile pairings23,30,31.

From a developmental perspective, the capability to detect temporal proximity between multisensory stimuli is observed at birth or even earlier in the fetal stage32,33,34. Nevertheless, precision of multisensory simultaneity perception is relatively low at these earlier stages, and its developmental trajectory is protracted and varies across sensory pairings (see Fig. 3 in7). Precision improves with age and reaches adult levels by 9 years for audio–visual pairings and by 11 years for visual–tactile and audio–tactile pairings. The PSS, however, is adult-like by the youngest ages tested (5 years in audio–visual pairings, and 7 years in visual–tactile and audio–tactile pairings5,6,7).

Given the early maturation of the PSS, it remains an open question whether young children exhibit temporal recalibration, and if so, how it might vary through development and for different sensory combinations. The sensitivity to temporal correspondence at very early ages and the early maturation of the PSS suggest that temporal recalibration may be present early in life. However, in practice, testing temporal recalibration would be an enormous procedural challenge with children, especially the long adaptation periods for the slow recalibration. On the other hand, the development of temporal processing throughout childhood suggests that rapid temporal recalibration may be present and linked to the maturation of simultaneity perception35 (see4,36 in the spatial domain). Rapid recalibration is easily uncovered by a sequential analysis of the data of simultaneity judgments25,35.

The current study examined the developmental trajectories of rapid temporal recalibration for three cross-modal pairs (i.e., audio–visual, audio–tactile, and visual–tactile; Fig. 1a), and their respective relations with age-related changes in the precision of simultaneity perception. To do so, we re-analyzed the datasets of three previous developmental studies5,6,7 that measured simultaneity perception in children and compared those data to adult groups. If cross-sensory calibration serves as the basis for the development of multisensory integration2, then we would expect rapid recalibration to emerge early and then decrease as multisensory simultaneity perception matures35. Alternatively, rapid recalibration might develop after the system matures, providing a form of malleability to accommodate the temporal variations among multisensory events4.

## Results

Two main measures were computed in our re-analysis, namely, the width of the temporal simultaneity window and the PSS shifts associated with rapid recalibration. Briefly, the width of the temporal simultaneity window of each participant was estimated using Gaussian distribution fits on the respective cross-modal pairing (Fig. 1b). To compute the magnitude of rapid temporal recalibration, each participant’s data for each cross-modal combination was first sorted into two bins based on the leading modality in the preceding trial. Gaussian distributions were then fitted to these subsets and the difference in distribution means yielded the PSS shift for the cross-modal combination (Fig. 1c). Further details about the analyses are provided in the Methods section.

### Rapid recalibration develops with age for audio–visual pairings

When each age group was compared to a hypothesized PSS shift of 0 ms (see Fig. 2a), audio–visual temporal recalibration was significant for 9 years of age or older (9-year-olds: t(18) = 3.15, p = 0.02, d = 0.72, bootstrapped p < 0.001; 11-year-olds: t(19) = 4.44, p = 0.001, d = 0.99, bootstrapped p < 0.001; adults: (t(19) = 3.23, p = 0.02, d = 0.72, bootstrapped p = 0.004), but not at 7 years of age (t(18) = 1.83, p = 0.08, d = 0.42, bootstrapped p = 0.053). For the audio–tactile or visual–tactile pairings, none of the age groups differed significantly from 0 ms (all ps > 0.05 for one-sample t-tests and bootstrap hypothesis tests), suggesting that rapid temporal recalibration processes remain weak for audio–tactile and visual–tactile pairings throughout development.

### Post-maturation audio–visual PSS shifts are larger than other combinations

Figure 2B compares the PSS shifts in audio–tactile and visual–tactile pairings against audio–visual pairings recorded for 7-, 9-, 11-year-olds, and adults. For the youngest age group, audio–visual PSS shifts were statistically comparable to those obtained with audio–tactile pairings (t(30) = 0.38, p = 0.71, d = 0.14, bootstrapped p = 0.65) and visual–tactile pairings (t(33) = 1.89, p = 0.07, d = 0.64, bootstrapped p = 0.05). Similarly, in the 9-year-olds, audio–visual PSS shifts did not differ significantly from audio-tactile (t(36) = 1.32, p = 0.19, d = 0.42, bootstrapped p = 0.18) and visual–tactile (t(36) = 0.79, p = 0.43, d = 0.25, bootstrapped p = 0.44) pairings. However, by 11 years of age, PSS shifts were significantly larger in audio–visual conditions than in audio–tactile (t(38) = 3.25, p = 0.002, d = 1.06, bootstrapped p < 0.001) and visual–tactile (t(38) = 3.12, p = 0.003, d = 0.99, bootstrapped p = 0.001) conditions. The larger PSS shifts in audio–visual than audio–tactile (t(38) = 2.00, p = 0.05, d = 0.63, bootstrapped p = 0.03) and visual–tactile (t(38) = 1.88, p = 0.07, d = 0.59, bootstrapped p = 0.04) pairings continued in adulthood.

### Post-maturation temporal precision correlates with audio–visual PSS shifts

The effect of age on the width of the simultaneity window is contrasted with its effect on the PSS shift in Fig. 2c. Overall, the width of the simultaneity window decreased with age for audio–visual pairings (F(3,74) = 5.63, p = 0.002, ηp2 = 0.19, bootstrapped p < 0.001), audio–tactile pairings (F(3,68) = 5.54, p = 0.002, ηp2 = 0.20, bootstrapped p = 0.001), and visual–tactile conditions (F(4,89) = 8.83, p < 0.001, ηp2 = 0.28, bootstrapped p < 0.001). Post-hoc pairwise comparisons showed that for audio–visual and audio–tactile presentations, main effects were driven by the 7-year-old groups, which had wider windows than the older age groups (ps < 0.05). Similarly, for visual–tactile pairings, 7- and 9-year-olds had larger window sizes than the other ages (ps < 0.05). In contrast, age did not have a significant effect on PSS shifts for the audio–visual pairing (F(3,74) = 1.44, p = 0.24, ηp2 = 0.06, bootstrapped p = 0.06), the audio–tactile pairing (F(3,68) = 0.17, p = 0.92, ηp2 = 0.01, bootstrapped p = 0.49), and the visual–tactile pairing (F(4,89) = 1.71, p = 0.15, ηp2 = 0.07, bootstrapped p = 0.20).

Further within-age-group analyses showed that the correlation between the PSS shifts and the widths of the simultaneity window was dependent on age and the modality combination. Specifically, positive correlations between larger PSS shifts and wider audio–visual simultaneity windows were demonstrated reliably by 11 years of age (Fig. 3a; 11-year-olds: ß = 0.36, t(18) = 3.47, p = 0.003, bootstrapped p = 0.009; adult: ß = 0.34, t(18) = 5.29, p < 0.001, bootstrapped p < 0.001) but not reliable in the younger age groups (7-year-olds: ß = 0.13, t(17) = 2.07, p = 0.05, bootstrapped p = 0.02; 9-year-olds: ß = −0.06, t(17) = -0.55, p = 0.59, bootstrapped p = 0.71). The other cross-modal pairings did not demonstrate a convincing relation between the PSS shift and the width of the simultaneity window (Table 1), showing instead a decrease in response variability with age (Fig. 3b–c). The only exception was the 7-year-olds in the visual–tactile condition (Fig. 3c, bottom-left panel), where an inverse correlation was observed between the PSS shift and the width of the simultaneity window (ß = -0.28, t(14) = 3.33, p = 0.005, bootstrapped p = 0.04).

## Discussion

We investigated the development of rapid recalibration in the temporal domain and its relation with the simultaneity window across three cross-modal pairings. The development of rapid recalibration was protracted and unique to audio–visual presentations: by 9 years of age, a significant shift of PSS attributed to rapid recalibration was observed (Fig. 2a, left panel). In contrast, none of the age groups demonstrated such PSS shifts for auditory–tactile or visual–tactile pairings (Fig. 2a, centre and right panels). By 11 years of age, PSS shifts for the audio–visual pairing were reliably larger than those for the other two cross-modal pairings, and a positive correlation between PSS shifts and the widths of the simultaneity window for the audio–visual pairing was observed reliably.

For the audio–visual pairing, rapid recalibration first appeared at age 9 and continued with age; in contrast, children’s precision of simultaneity perception (i.e., the width of the temporal window) improved with age up to age 9, at which point it was adultlike5. Combined, these results suggest that children can realign the temporal synchrony of recent audio–visual events rapidly only after audio–visual simultaneity perception matures, and not before. This result contrasts with that of Noel et al.35 who demonstrated that rapid recalibration peaked at 12.1 years of age and then decreased, while the precision of audio–visual simultaneity perception increased until late adolescence (17–18 years of age), suggesting a developmental trend of the audio–visual system from malleable to stable. However, Noel et al. had only nine participants in the age range of 7 to 10 years—the critical age range according to our data—and used a broad sliding window that averaged across 7–11 years of age. Doing so would miss the critical changes we observed with 20 children per age group for ages 7-, 9-, and 11-year-olds (see Footnote 1 in7).

The earlier development of the precision of audio–visual perception than rapid recalibration in both the temporal and spatial domains (the current study and4) suggests that plasticity/malleability of the audio–visual system does not decrease monotonically with age. Children’s wider simultaneity window5 and larger spatial ventriloquism effect4 indicate that they tend to integrate visual and auditory information originating from disparate timings or locations more often, and less selectively, than do adults. During development, the audio–visual system is plastic in order to accommodate physical growth of the body and the maturation of each sensory system. In addition, intersensory calibration based on daily audio–visual experience will lead to changes based on the more accurate (i.e., the less noisy) modality, the precision of each sensory system, and expectations about signals that ought to be integrated (i.e., the prior of common cause37,38). In turn, the audio–visual system develops a statistically optimal algorithm for signal integration in both spatial and temporal domains that is useful for most daily events8,39,40,41. During this developmental period, accommodating to particular audio–visual events which may occur occasionally or exceptionally (such as the rapid recalibration to the most recent event) might lead to a misrepresentation of the optimal window or even a prolonged developmental trajectory if the window keeps changing.

After the audio–visual system achieves optimal precision for integration, it remains malleable in both the temporal and spatial domains, even in adulthood. This is demonstrated in the phenomena of rapid recalibration25,42, slow recalibration21,22,43, and perceptual training with feedback43,44,45. Hence, the plasticity/malleability of the audio–visual system tends to pass through two developmental stages during which the system is susceptible to multisensory events at distinct time scales (see1). The first is a long-term scale beginning at birth and lasting until late childhood during which the multisensory system is tuned by general events from an overly broad system into optimal precision. This development of optimal integration is based on continuously improving sensory reliability and establishing the prior of common cause for multisensory signals4,46,47,48 (see49,50 for other cross-modal combinations). Afterward, when the system is relatively stable, the second time scale takes effect with mild short-term influences from recent experience. The rapid recalibration underpinned by short-term malleability may rely mainly on accommodating the prior of common cause and not depend as much on signal reliability51,52,53. For example, rapid recalibration occurs when the asynchronous audio–visual stimuli are supposed to originate from the same event (i.e., having a common cause), and thence they are realigned54.

We found that the development of rapid recalibration emerged specifically for the audio–visual pairing, and not for the audio–tactile and visual–tactile pairings at any age. These results are consistent with previous studies in adults23,24,25,30,31,55, suggesting that short-term malleability is more pronounced for audio–visual than for audio–tactile or visual–tactile pairings. Note that in experimental settings, like those we used, the locations of the audio and visual stimuli were very close5, whereas the locations of the audio–tactile and visual–tactile stimuli differed6,7. This difference in locations may have reduced the likelihood of multisensory integration9 and therefore the necessity of rapid recalibration. However, the potential influence of the locations of multisensory stimuli on rapid recalibration has been ruled out in prior studies23,24,55.

The need for rapid recalibration may stem from the nature of processing of each cross-modal combination. In cross-modal simultaneity perception, the arrival time difference between each signal is determined by their physical transmission time to receptors and neural transduction time to the associated brain areas56. For the audio–visual pairing, because both stimuli are distal and the speed of light is much faster than sound, the signal arrival time difference is distance-dependent. To accommodate rapidly such variations of arrival time differences, the audio–visual system can modify the PSS according to the estimated distance of the source57,58,59 or, even more straightforwardly, according to the asynchrony of the most recent experience (i.e., rapid recalibration). For audio–tactile and visual–tactile pairings, the tactile stimulus is proximal on the body’s surface, and therefore the variability of the arrival time differences is small, caused mainly by the different neural transduction times when stimulating different locations on the body23,31. Thus, the fact that rapid recalibration develops mainly in audio–visual simultaneity perception but only mildly, or not at all, in audio–tactile or visual–tactile simultaneity perception appears to indicate that greater malleability is maintained for the audio–visual pairing because of the greater variation in daily experience.

Our findings also reveal that the relation between rapid recalibration and precision in multisensory simultaneity perception is not straightforward through development. Contrary to Noel et al.35, who demonstrated a decrease of rapid recalibration and the width of audio–visual simultaneity window (i.e., an improvement of precision) from childhood to adulthood, we did not observe a similar effect in any cross-modal pairings (see Fig. 2c). Further analyses within each age group showed that the relation depended on age and the type of cross-modal presentation. For audio–visual presentations, rapid recalibration and the precision of simultaneity perception is correlated by 11 years of age (see25, for the first report such correlation in adults). This correlation suggests that the wider window (i.e., lower precision) of audio–visual simultaneity perception in older children and adults may be attributable partly to higher susceptibility to the recent audio–visual asynchrony (i.e., a higher malleability), indicating a need to rethink the ecological meaning of the precision of audio–visual simultaneity perception. The correlation in the audio–visual pairing contrasted with the other cross-modal presentations, which demonstrated a general reduction in variance and a null correlation in later stages of development. Interestingly, we found a significant inverse relation in the youngest visual–tactile condition, which indicates a repulsive rapid recalibration when the window of simultaneity perception is wide. This trend may be spurious because of the higher variance in the young children’s behavioural measures, the unique performance of three children, or an overall preference for touch in visual-tactile judgments by 8 years of age49.

A limitation in the present study was that we examined only the relation between the development of simultaneity perception and rapid recalibration in the temporal domain, while the development of slow recalibration was not studied. Recent studies in the spatial domain suggest that audio–visual integration and rapid recalibration share a common neural substrate36, whereas the rapid and slow recalibrations tend to be dissociable in terms of their underlying mechanisms60,61,62,63 and developmental trajectories4. Inspired by the results in the spatial domain, it would be interesting to measure the development of slow recalibration together with rapid recalibration and the simultaneity window in order to gain a full picture in the temporal domain. Van der Burg et al.24 have developed a novel method to measure the rapid and slow recalibrations together, which is promising for future developmental studies.

In conclusion, we demonstrate that cross-modal rapid recalibration in the temporal domain emerged after multisensory simultaneity perception achieved adult precision for audio–visual presentations. In contrast, rapid recalibration in the audio–tactile and visual–tactile pairings did not develop at any age31. The uniqueness of rapid recalibration in the audio–visual system highlights its malleability to accommodate the stimulus arrival time difference depending on the distance of the source. Future investigations should examine whether maturation of audio–visual simultaneity perception is a prerequisite for the development of rapid recalibration; for example, it would be interesting to examine rapid recalibration in patients treated for congenital cataract because they show abnormal audio–visual simultaneity perception later in life64. The result will provide a contrast to the spared audio–visual integration and rapid recalibration in the spatial domain after early visual deprivation65.

## Methods

### The dataset

Cross-modal simultaneity judgment data were re-analyzed from three previous developmental studies5,6,7. All experimental protocols of these studies were approved by the McMaster Research Ethics Board. Methods in each study conformed to the Canadian Tri-Council Statement on Ethical Conduct of Research Involving Humans and the Declaration of Helsinki. All three studies recruited twenty participants for each age group that were balanced approximately by sex at birth. All three studies recruited independent groups of participants. The audio–visual experiment tested 5-, 7-, 9-, 11-year-olds, and adults. The age groups recruited in the audio–tactile and visual–tactile studies were similar, in that 7-, 9-, 11-year-olds, and adults were tested, with the exception that 13-year-olds were also tested in the visual–tactile study. Written consent was obtained from adult participants. For child participants, verbal assent was obtained from the child in conjunction with informed, written consent from their parents.

The procedures used to acquire each dataset are described in detail in the respective studies, that is, audio–visual5, audio–tactile7, and visual–tactile6. The following, however, provides a brief description. In the audio–visual experiments, participants fixated on the middle of a visual grey ring (~ 2° inner diameter), where a visual white disk (~ 2° diameter) was presented for about 17 ms on each presentation. The auditory stimulus was a 17 ms white noise burst presented from speakers on either side of the monitor. Audio–tactile stimuli were a 10 ms white noise burst presented through closed-ear headphones and a 10 ms dull tap delivered to the right index finger using a solenoid-based mechanical device aligned with the body midline. The same visual stimulus from the audio–visual experiment was used to record the visual–tactile simultaneity judgments, and the tactile stimulus was a 17-ms tap delivered to the right index finger, situated 20° below the visual stimulus and aligned with the participant’s body midline. In all three studies, participants reported orally if they perceived the individual onsets of the cross-modal stimuli to be synchronous or asynchronous. An experimenter seated beside participants recorded their responses and ensured that each participant adhered to the task requirements. A total of 13 stimulus onset asynchronies (SOAs) were tested for each cross-modal pairing: ± 1200, ± 800, ± 400, ± 300, ± 200, ± 100 or 0 ms. Negative SOAs indicate an auditory leading stimulus in audio–visual and audio–tactile experiments and a tactile leading stimulus in visual–tactile experiments. A total of 130 trials (10 trials per SOA) were collected for each 5-year-old participant, whereas a total of 260 trials (20 trials per SOA) were collected for older participants.

### Analysis

#### Data preparation

In our reanalysis of the data, we estimated the magnitude of rapid temporal recalibration for each participant’s dataset (and for each cross-modal condition) by first dividing the data into two bins based on the leading modality in the preceding trial. This resulted in two subsets for a specific age group and cross-modal condition, where modality A (audition in audio–visual and audio–tactile pairings, and touch in visual–tactile pairings) or modality B was the leading stimulus in the previous trial (see Fig. 1). A Gaussian distribution was fitted to these subsets and the amplitude, mean, and standard deviation were free parameters. The shift in the PSS was computed by subtracting the mean of the B-leading subset from the mean of the A-leading subset (i.e., PSSA–PSSB). Following earlier studies, we label A-leading SOAs as negative and B-leading as positive. Using this convention, rapid temporal recalibration exhibits a positive relationship with SOA25. That is, if modality A led on the previous trial, then the current trial PSS is shifted toward negative SOAs, and if modality B led on the previous trial, then the current trial PSS is shifted toward positive SOAs.

To ensure that the estimated PSS shifts were representative of the participants’ responses, we selected age groups with the same number of trials and excluded individuals with poor Gaussian fits (R2 < 0.60) on either subset. This eliminated data from the 5-year-olds because of fewer trials. Because of poor Gaussian fits, data were excluded for one 7-year-old and one 9-year-old were excluded from the audio–visual condition. For the audio–tactile condition, data collected from seven 7-year-olds, and one 9-year-old were excluded from the analysis. Finally, for the visual–tactile condition, data from four 7-year-olds, one 9-year-old, and one 13-year-old were excluded.

#### Main analyses

We conducted three analyses. The purpose of the first analysis was to determine the age at which rapid temporal recalibration becomes significant. PSS shifts of each condition were compared against a hypothesized PSS shift of 0 ms, which represented no rapid temporal recalibration. These comparisons were conducted using two-tailed, one-sample t-tests and bootstrap hypothesis tests. To conduct the bootstrap tests, 1500 bootstrap samples were generated for each pairwise comparison by resampling individual PSS shifts with replacement. The mean PSS shift of each iteration was then compared against 0 ms. Statistical significance was determined using the formula:

$$p = 2\min \left( {\frac{1}{B}\mathop \sum \limits_{j = 1}^{B} {\text {I}}( \overline{\Delta PSS}_{j} > C),\;\frac{1}{B}\mathop \sum \limits_{j = 1}^{B} {\text {I}}( \overline{\Delta PSS}_{j} < C)} \right)$$

which computes the two-tailed probability of having a mean PSS shift greater or less than the comparison value (denoted by C, here 0 ms). B is the total number of bootstrap samples and I(.) is the indicator function that returns a value of 1 when the PSS shift is either smaller or larger than 0 ms. By conducting both forms of statistical analyses, we could verify if our results were driven by individual data or if they were specific to the type of statistical test. For example, a positive t-test with a large effect size (e.g., d = 0.8) and a negative bootstrap test may indicate a biased sample.

The second analysis aimed to evaluate the differences in PSS shifts among three cross-modal pairings at a given age. Adult observers experience larger audio–visual rapid temporal recalibration effects than audio–tactile or visual–tactile pairings23, but it is unclear if a similar advantage can be observed across the different age groups. To investigate, PSS shifts of audio–tactile and visual–tactile pairings were evaluated against the PSS shifts of audio–visual pairings at the age groups common to all three datasets (i.e., 7-, 9- and 11-years-olds, and adults). As before, these comparisons were performed using two-tailed, independent-sample t-tests and bootstrap hypothesis tests. Similar bootstrap procedures were performed, with the exceptions that the test statistic was the difference in mean PSS shift and the comparison value was a mean difference of 0 ms.

In the final analysis, we investigated the correlation between the widths of the simultaneity window and PSS shifts across and within age groups. Broader audio–visual temporal simultaneity windows have been linked to larger PSS shifts in adults25; however, it is unclear if this correlation applies to younger age groups and other cross-modal presentations. Studies have shown that younger children have a wider window in which they make cross-modal simultaneity judgments5,6,7,35, but the wider windows might result from factors independent of the larger PSS shifts, such as needing more practice, inattention, or fatigue. One-way ANOVAs were conducted on each type of cross-modal presentation to examine the effect of age on the width of the simultaneity window or PSS shift. Separate linear regression analyses were also performed within each age group for each cross-modal pairing, and this allowed us to test how well the PSS shifts were correlated with the width of the simultaneity window. Similar bootstrap procedures were performed, and the same sample selections were used for PSS shifts and the width of the simultaneity window. In the within-age-group analysis, regression slopes were estimated for each of the bootstrapped samples, after which the probability of samples with slopes greater or less than 0 was computed. Main effects were assessed using bootstrapped F ratios, and we estimated the p value by computing the probability of obtaining ratios smaller or equal to 1 (equal variances).


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# Viewpoint: How colloidal dispersions relax under stress

Physics 1, 22
A shear force can melt a colloidal glass, causing it to flow in a highly nonlinear fashion. Physicists have now found a way to put the description of this type of flow on a more formal theoretical footing.

Of the many physical systems that we can study, colloids and colloid dispersions may well be one of the most prevalent in our daily lives. Inks, paints, clays, emulsions, and milk, as well as surfactant aggregates (micelles) and nanoparticle suspensions are all colloid dispersions. The study of how these dispersions and other complex liquids flow—a field called rheology—is essentially a problem of nonequilibrium statistical mechanics that must account for many-body interactions. The response of colloid dispersions to a shear force is often nonlinear and can result in interesting physical phenomena. The application of a shearing force, for example, can increase the viscosity of a dispersion, an effect that can be harnessed to make personal protective devices [1].

Predicting how a colloid dispersion responds to an external stress and connecting this relationship to the microscopic interactions between the dispersed particles is a key goal of theoretical rheology. Writing in Physical Review Letters, Joseph Brader and Matthias Fuchs of the Universität Konstanz in Germany and Michael Cates of The University of Edinburgh report the development of a rheological constitutive equation for dense colloidal suspensions and glasses [2] (here, constitutive refers to the stress-response relationship). The work links a highly successful microscopic theory, which describes how colloidal particles become arrested at the glass transition of a colloid, to a formal description of suspension rheology. Although this is not the first effort in this direction [3], the paper by Brader et al. is an important contribution because it applies to both colloidal dispersion and glasses, can describe the response for nonlinear, arbitrary deformation fields, and satisfies the criterion that the stresses should be invariant for different reference frames.

The theory of the flow of colloidal dispersion under shear has a rich history that spans over a century. The field really begins with the seminal work of Einstein in 1906 [4, 5], in which he studied how single-particle motion contributes to the viscosity of a suspension. Around the same time, Smoluchowski developed a statistical description of Brownian motion of colloidal particles [6]. In the early 1970s, Batchelor formulated the stress tensor—a matrix whose components describe the forces acting on the material in all three spatial coordinates—for both hydrodynamic (that is, mediated by the fluid between the particles) and conserved interactions [7, 8, 9]. The past thirty years of theoretical development have largely focused on handling the fact that a concentrated colloid dispersion is a many-body problem, though the challenge of many-body hydrodynamic interactions is, in practice, largely resolved [10, 11, 12]. With advanced simulation methods, it is possible to obtain accurate solutions for sufficiently large ensembles of particles [13, 14], even if analytical calculations are intractable for all but regular arrays.

One way that many-body interactions affect particle dynamics is through excluded volume effects that result from the finite size of the dispersed particles (see Fig 1, left panel). In a sufficiently dense suspension of hard spheres, these effects can lead to the formation of a colloidal glass, a kinetic transition characterized by “dynamical arrest” of the particle motion [15]. However, when theories that are based on only pair-wise interactions between particles (rather than a many-body picture) are extended to concentrated dispersions, they do not capture this divergence in the viscosity at the ideal glass transition, but at much higher concentrations [16, 17].

A promising approach for describing colloidal dynamics just above and below the colloidal glass transition has proven to be mode-coupling theory (MCT). This theory is an ansatz proposed to handle highly correlated systems and can therefore account for the particle caging shown in the left panel of Fig. 1. The relaxation modes of the test particle (the red particle) are formally coupled to those of the neighbors (blue) comprising the cage, which themselves are caged by their neighbors (grey), ad infinitum. Indeed, MCT has been shown to accurately model the diffusion of particles, at equilibrium, and the viscoelasticity for charged sphere dispersions where hydrodynamic interactions are not dominant [3, 18, 19, 20] The theory also provides a qualitative framework for understanding the linkage between viscoelasticity and particle “caging” in hard-sphere dispersions [21, 22].

In their new work, Brader, Cates, and Fuchs put their earlier ansatz extending MCT for colloidal suspensions to nonlinear shear deformations on a more formal footing. A significant advantage of their approach is that they fold the successful MCT description of particle dynamics into a generalized formulation of the stress tensor (in particular, they use the Green-Kubo formalism [3]). Their identification of the full tensorial structure of the theory means it is not restricted to simple shear deformation. To illustrate their approach, Brader et al. consider a hard-sphere colloidal glass and calculate the dynamic yield stress, i.e., the force required to melt the glass and cause the dispersion to flow, under shear as well as uniaxial and planar extension flows.

The theory predicts what happens to the microstructure of a colloidal suspension under an applied deformation (an example of this sort of calculation is shown in the right panel of Fig. 1). The results could be directly compared with neutron scattering measurements of the density distribution in the colloidal suspension under shear flow [23], as well as against measurements of the viscoelastic properties [24]. The calculated stress tensor agrees with previous exact derivations based on micromechanics [25] in the dilute limit (i.e., when the many-body direct correlation function reduces back to the pair potential). Differences should arise for strongly correlated systems and they can now be rigorously tested against simulations and experimental measurements. It also remains to formally determine if the stress tensor and dynamical equation that Brader et al. have calculated for the nonequilibrium microstructure are thermodynamically consistent [26].

A further challenge for the theory will be to reproduce experimental measurements of the divergence in the viscosity for hard-sphere dispersions [27]. These dispersions have a large, but finite viscosity for volume fractions above the ideal glass transition, suggesting that particle motion is activated in a way that is not captured by MCT [28]. Finally, it will at some point be necessary to include in the theory the effects of hydrodynamic interactions, which are more significant in strongly shearing or deforming dispersions.

The phase behavior and dynamics of colloidal dispersions with attractive interactions are a current topic of great scientific interest but of even greater importance in technological practice [29]. The recent successes of MCT in describing the attractive driven glass transition in colloidal dispersions [30, 31, 32, 33] or the formation of a colloidal gel [34] suggest that the new formalism presented here might lead to accurate rheological constitutive equations for these systems, too. Further, with the formalism developed by Brader et al. it may be possible to describe time-dependent viscosities that occur in fluids exposed to changing shear (i.e., thixotropy) [35, 36] as well as ageing effects [37]. Thus, this theoretical advance opens numerous opportunities for further explorations of the rich behavior of concentrated colloidal dispersions.

## References

1. M. Decker, C. Halbach, C. Nam, N. Wagner, and E. Wetzel, Composites Science and Technology 67, 565 (2007)
2. J. M. Brader, M. E. Cates, and M. Fuchs, Phys. Rev. Lett. 101, 138301 (2008)
3. G. Nägele and J. Bergenholtz, J. Chem. Phys. 108, 9893 (1998)
4. A. Einstein, Ann. Phys. 324, 289 (1906)
5. A. Einstein, Ann. Phys. 339, 591 (1911)
6. M. von Smoluchowski, Ann. Phys. 326, 756 (1906)
7. G. K. Batchelor, J. Fluid Mech. 41, 545 (1970)
8. G. K. Batchelor and J. T. Green, J. Fluid Mech. 56, 401 (1972)
9. G. K. Batchelor and J. T. Green, J. Fluid Mech. 56, 375 (1972)
10. B. Cichocki, B. U. Felderhof, K. Hinsen, E. Wajnryb, and J. Bławzdziewicz, J. Chem. Phys. 100, 3780 (1994)
11. D R. Foss and J. F. Brady, J. Fluid Mech. 407, 167
12. A. Sierou and J. F. Brady, J. Rheol. 46, 1031 (2002)
13. A. J. Banchio and J. F. Brady, J. Chem. Phys. 118, 10323 (2003)
14. A. J. C. Ladd and R. Verberg, Journal of Statistical Physics 104, 1191
15. W. van Megen and P. N. Pusey, Phys. Rev. A 43, 5429 (1991)
16. J. F. Brady, J. Chem. Phys. 99, 567 (1993)
17. J. F. Brady, J. Chem. Phys. 98, 3335 (1993)
18. A. J. Banchio, G. Nägele, and J. Bergenholtz, J. Chem. Phys. 113, 3381 (2000)
19. A. J. Banchio, G. Nägele, and J. Bergenholtz, J. Chem. Phys. 111, 8721 (1999)
20. A. J. Banchio, J. Bergenholtz, and G. Nägele, Phys. Rev. Lett. 82, 1792 (1999)
21. T. G. Mason and D. A. Weitz, Phys. Rev. Lett. 75, 2770 (1995)
22. K. Miyazaki, et al., "Nonlinear viscoelasticity of metastable complex fluids," 2005 (to be published)
23. B. J. Maranzano and N. J. Wagner, J. Chem. Phys. 117, 10291 (2002)
24. J. J. Crassous, M. Siebenbürger, M. Ballauff, M. Drechsler, D. Hajnal, O. Henrich, and M. Fuchs, J. Chem. Phys. 128, 204902 (2008)
25. N. J. Wagner and B. J. Ackerson, J. Chem. Phys. 97, 1473 (1992)
26. N. Wagner, J. Non-Newton. Fluid 96, 177
27. Z. Cheng, J. Zhu, P. M. Chaikin, S-E. Phan, and W. B. Russel, Phys. Rev. E 65, 041405 (2002)
28. E. J. Saltzman and K. S. Schweizer, Phys. Rev. E 77, 051504 (2008)
29. R. Buscall, J. Rheol. 37, 621
30. E. Zaccarelli, J. Phys. Condens. Mat. 19, 323101 (2007)
31. P. J. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield, F. Sciortino, and D. A. Weitz, Nature 453, 499 (2008)
32. J. Bergenholtz and M. Fuchs, Phys. Rev. E 59, 5706 (1999)
33. J. Bergenholtz and M. Fuchs, J. Phys. Condens. Mat. 11, 10171
34. J. Bergenholtz, W.C.K. Poon, and M. Fuchs, Langmuir 19, 4493
35. P. C. F. Møller, J. Mewis, and D. Bonn, Soft Matter 2, 274 (2006)
36. K. Dullaert and J. Mewis, J. Rheol. 49, 1213 (2005)
37. F. Pignon, A. Magnin, and J.-M. Piau, J. Rheol. 42, 1349 (1998)

Norman Wagner leads an active research team at the University of Delaware that studies the rheology and nonequilibrium statistical mechanics of complex fluids. He also heads the Center for Neutron Science, a cooperative agreement with the National Center for Neutron Research at the National Institute of Standards and Technology (NIST). His research has led to inventions such as novel rheo-light scattering and rheo-SANS (small angle neutron scattering) devices, as well as STF-Armor™ (www.ccm.udel.edu/STF/).

Soft Matter

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# 1. Cable Theory and Compartmental Modeling

## Examples of different types of neurons

There are thalamic cells, pyramidal cells, double pyramidal cells, granule cells, purkinje cells. These all have different morphologies and perform different computations.

## Approach

1. Approximate dendrites as uniform membrane cylandiers
2. Synaptic inputs are approximated as 'injected currents'
3. Use the cable equation to create a system of differential equations for each cylinder.

## Important Equations

1. input conductance of semi-infinite cable
2. input conductance of infinite cable

### Unsealed end:

1. cable equation
2. input conductance of finite cable
3. s.s. voltage along cable

### Sealed end:

1. input conductance of the finite cable
2. s.s. voltage along cable

### Concatenated Cables:

1. Compute V(X) along branches, by determining $$G_{out}$$

### Definitions:

Know definitions and related biology of: - axial resistance, the resistance to current flow along the uniform cable (along the dendrite) - membrane resistance, the resistance of current flow out of the dendrite - membrance capacitance, the capacitance of a patch of membrane surface area - derive cable equation as a model of concatenated RC circuits

### Compartmental Models:

Be able to derive system of equations into a matrix form for a linear time-invariant system

Understand how transfer resistances work. Understand how distributing synaptic inputs works.

Understand how coincidence detection (AND operator), shunting inhibition (AND-NOT operator).

## Study

1. Definitions of cable theory and biological relations
2. Solving single cable equation and different boundary conditions
3. Derivation of cable equation under different boundary conditions
4. Deriving LTI system from compartmental models of cables
5. Derive transfer resistances
6. Derive AND operator (coincidence detection)
7. Derive AND-NOT operator (shunting inhibition)

# 2. Generalized Linear Models:

## General Form of Exponential Family Distribution

Be able to define all the terms, such as: natural parameters, sufficient statistic, natural link function, dispersion parameter

## Solving GLM Methods:

1. Penalized Quasi-likelihood
2. Laplace's Method

## Goodness of Fit

1. Chi-square test
2. Kolmogorov Statistic

# 3. Kalman Filter

## Stochastic State-Space Model

Here, list the linear, time-invariant system with a state evolution equation and measurement/observation equation.

## Assumptions

1. state and observation noises are independent, zero-mean Gaussian white processes with some defined covariances
2. initial state x_0 is a Gaussian R.V. independent of the state/observation noises

## Derivation

Derive the Kalman filter equations for the state update, covarariance estimates

1. Measurement update
2. Time update
3. Combined Update
4. Covariance Update

Review...

Review...

# 4. Expectation Maximization

## Basic Idea

By computing the likelihood of your unknown parameters, based on known outcomes. You can perform maximum likelihood estimation to get the best estimate of the unknown parameters

# 5. K-Means Algorithm

## Cost Function

The cost function is attempting to minimize the distortion (distance to centers) for every point in the set of data points, S.

## The Algorithm

''' Input: k clusters initialize centers: z_1, ..., z_k \in \real^d and clusters C_1, ..., C_k repeat until there is no further change in L(z, C): for each j (data point): C_j <- {x \in S whose closest center is z_j} for each j (data point): z_j <- mean(C_j) '''

Walk through for a small example to see how the algorithm works: (0, 0, 0) (0, 0.5, 1) (1, 3, 2) (4, 5, 6) (2, 3, 1) (5, 2, 0) (0, 1, 0) (1, 1, 0) (2, 1, 0)

## Initialization

Initialization of a k-means type algorithm initializes k "means", we call centers at the beginning, and then assigns points into these centers based on the cost function.

## Convergence

Show that the cost monotonically decreases, so the algorithm will converge at least in the sense of cost decreasing to a non-changing amount.

# 6. Sensory Pathways and Systems in Neuroscience

The different neuronal pathways in the central nervous system, vs the periphery nervous system.

Visual sensory pathway that leads from the retina all the way to the occipital lobe of the brain.

The somatosensory pathways that lead to the SI, and SII all with nerve sensors from the periphery of your body.

# 7. Motor Pathways and Systems in Neuroscience

How does motor movements get controlled in the brain?

The premotor cortex, and also the motor cortex are at the highest levels of control. We can also include the cerebellum when we talk about motor control.

# 8. General Topics in Neuroscience

Good things to know and remember are: - Hodgkin Huxley models: a biophysical model that models realistic neuronal firing based on concentration gradients, membrane conductances, membrane potentials and even more. - Nernst equation: a fundamental equation for determining the resting potential of a membrane in the presence of a concentration gradient of an ion. - Goldman-Katz equation: a model with multiple ions that determines the resting potential of a membrane (extension of Nernst) - Action potentials and the chronology of cell actions that are taken

# 9. Gaussian Mixture Models

This is the idea that some data distributions can be the combinations of many different Gaussian models with different means and variance, but superimposed on each other.

The goal of a gaussian mixture model is to identify the parameters of each of the separate gaussians by a process called Expectation Maximization. In this process, we switch between maximizing the likelihood by iterating on the parameters of the gaussian models and computing a new likelihood function given the new parameters.

# FAQ

1. Q: How long should I study for? A: I would study for about 2-3 weeks with nothing else going on. You want to be able to do a comprehensive overview of everything first, and then dive deeper into topics that are more likely to come up. You should also do test runs of walking through "open-ended" questions on a white board many times.

The goal here is to practice thinking out loud and being very clear in your communication.


text:
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In some packages like tabularx, I find the following syntax in many places {\ifnum0=`}\fi or even \ifnum0=`{\fi} But I can't figure out why this can be useful for? Is ` a register of some kind? ...
28k views

### What is \iffalse for?

So this is something I've wondered about for a while. What use is a conditional that always evaluates to false? I guess it's a neat way to "hide code from LaTeX" but apart from that, is it ever useful?...
2k views

### Where do I find \futurelet's nasty behaviour documented?

Until recently, I thought that execution of \futurelet\testtoken<token1><token2>... has the following effect: The control sequence \testtoken is \let to <token2> and \futurelet\...
2k views

### TeX macro idioms, or: understanding advanced macros

I often find that when I look at solutions from the experts here, the macro definitions are completely inscrutable to me. Trying to use standard techniques one would use to understand code in most ...
419 views

### When might it be helpful to use implicit braces instead of explicit braces?

We've covered the question "When can one use implicit braces instead of explicit braces?"; this leads immediately in my mind to the question "Why would one want to?" (Remember, implicit braces are e....
439 views

### Expansion of first token in a tabular cell

Background I currently do some work on the collcell package which collects the content of a tabular/array cell and provides it to a user defined macro, e.g. using >{\collectcell\mymacro}c<{\...
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### Replace chars with xstring. Catcode problem

I try to do something like this (minimal example): \documentclass[]{article} \usepackage{mathtools} \usepackage{xstring} \newcommand{\foo}[1]{ \StrSubstitute{#1}{:}{\string&}[\fooo] \...
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### how to lower last term in a continued fraction after a diagonal row of dots?

How can I lower the last term of a continued fraction when it follows a diagonal row of dots?


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# Probability of getting n heads when n is very large [closed]

How to show that the probability of getting n heads when 2n times coin is toss is very small? Moreover, how to show that the probability of more than n heads is close to 0.5?

## closed as off-topic by Em., user1551, user91500, Claude Leibovici, user223391 Jul 4 '16 at 21:17

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Em., user1551, user91500, Claude Leibovici, Community
If this question can be reworded to fit the rules in the help center, please edit the question.

• Is the coin fair? Are you familiar with the binomial distribution? – carmichael561 Jul 4 '16 at 2:35
• yes, the coin is fair. I am familiar with binomial distribution as well as stirling approximation. However, I am not getting any concrete answer. – meta_finance Jul 4 '16 at 2:39
• I think Stirling's formula should do the trick. For the second part, if $X$ denotes the number of heads, then $\mathbb{P}(X<n)+\mathbb{P}(X=n)+\mathbb{P}(X>n)=1$, $\mathbb{P}(X<n)=\mathbb{P}(X>n)$, and $\mathbb{P}(X=n)$ is small. – carmichael561 Jul 4 '16 at 2:41
• Oh. Got it. Thanks. I didn't give it a thought this way. – meta_finance Jul 4 '16 at 2:42
• @WillFisher clearly it is the probability of getting exactly, since that is the probability which is small as $n$ grows large. At least $n$ heads approaches $0.5$ which is not "small" in the context of probability. – JMoravitz Jul 4 '16 at 2:46

Let $X$ denote the random variable counting the number of times heads occurs in $2n$ independent flips of a fair coin.

Via the binomial distribution:

$Pr(X=k) = \dfrac{\binom{2n}{k}}{2^{2n}}$

More specifically, $Pr(X=n) = \dfrac{\binom{2n}{n}}{2^{2n}}$

Via stirling's approximation, $\binom{2n}{n}\sim \frac{4^n}{\sqrt{\pi n}}$ as $n$ grows large, so the above simplifies to:

$Pr(X=n)\sim \dfrac{4^n/\sqrt{\pi n}}{4^n}=\frac{1}{\sqrt{\pi n}}\to 0$ as $n$ grows large.

For the second half, as mentioned in comments above, we recognize that $Pr(X<n)=Pr(X>n)$ due to the coin being fair and that $Pr(X<n)+Pr(X=n)+Pr(X>n)=1$ as the three events form a partition of the sample space.

From the previous part, since $Pr(X=n)\to 0$ and the observation, this implies that $2Pr(X<n)\to 1$, and thus $Pr(X<n)\to 0.5$


text:
Class 12 Physics Modern Physics Nuclei

Why does the radiation C deflect more than A?
Solution: Radiation C is an electron beam whereas radiation A is an alpha particle beam. Since electron has very less mass as compared to alpha particle, so electron experience more force towards the plate than alpha particle and thus electron (i.e. radiation C) deflects more than alpha particle (i.e. radiation A).

Connecting you to a tutor in 60 seconds.


text:
Bits - Maple Programming Help

Home : Support : Online Help : Programming : Bits : Bits/Iff

Bits

 Iff
 compute bit-wise iff of the inputs

 Calling Sequence Iff(num1, num2) Iff(num1, num2, options)

Parameters

 num1 - an integer num2 - an integer options - optional arguments

Description

 • The Iff command computes the bit-wise logical if and only if of the inputs returning in terms of a number. All bits that are equal in both inputs will be set in the output.
 • Iff takes one optional argument, bits=number that specifies the number of bits to be considered in the input. All bits past the specified number are truncated.
 • Use with negative inputs requires that bits be set, either as an argument to Iff or globally via Settings. If both inputs are positive, no truncation need occur, so bits is computed as the largest most significant bit between the two inputs.

Examples

 > $\mathrm{with}\left(\mathrm{Bits}\right):$
 > $\mathrm{num1}≔\mathrm{Join}\left(\left[1,1,0,0,0,1,0,1,0,1,1,0,1,1,1\right]\right)$
 ${\mathrm{num1}}{:=}{30371}$ (1)
 > $\mathrm{num2}≔\mathrm{Join}\left(\left[0,1,1,0,0,1,0,0,1,0,0,0,1,0,1\right]\right)$
 ${\mathrm{num2}}{:=}{20774}$ (2)

The following number represents the Iff of num1,num2

 > $\mathrm{num3}≔\mathrm{Join}\left(\left[0,1,0,1,1,1,1,0,0,0,0,1,1,0,1\right]\right)$
 ${\mathrm{num3}}{:=}{22650}$ (3)

Compare:

 > $\mathrm{num3},\mathrm{Iff}\left(\mathrm{num1},\mathrm{num2}\right)$
 ${22650}{,}{22650}$ (4)

Negative inputs and outputs

 > $\mathrm{Settings}\left(\mathrm{defaultbits}=8,\mathrm{negativeout}=\mathrm{true}\right):$
 > $\mathrm{Iff}\left(-1,12\right)$
 ${12}$ (5)
 > $\mathrm{Iff}\left(5,-128\right)$
 ${122}$ (6)


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# 🏌 BGGP3 - How to crash a famous JS engine for fun

Contents

So this summer - and it has been this for the past 2 years - the famous Binary Golfing Grand Prix took place online for it’s 3rd edition, and of course I decided to give it a try.

After getting to try my best at golfing binaries (see my golfing posts), I had to race this competition for fun and learn a lot about fuzzing - as I have never fuzzed before.

## What is fuzzing?

Fuzzing a binary is like bruteforcing a binary’s input (cmd args, input file…) in order to get an odd behaviour, a bug or even the holy crash.

There are a lot of tools in order to help users to fuzz a binary and optimize the results (execs/s, payload mutation, harnesses…) and they are pretty easy to use.

The first tutorials online and my own knowledge led me to the famous AFL, or more exactly it’s big brother AFL++. So I will go with this tool for this grand prix in order to familiarize myself with it.

## Chosing target

Spoiler

Looking by curiosity what are all the known JavaScript engines (and there are a lot of them), one caught my attention: Espruino.

I thought this project was in the spirit of the minimalism, and thus wanted to see if it was enough to avoid crashes.

Especialy design for embeeded devices and IoT, still maintained after more than 7 years, this project was a good practice for fuzzing.

## Instrumentation

Note
For this fuzzing project, I used the AFL++ Docker container, mapping the code of the target in /src.

The first step for fuzzing, we have to prepare the target binary. ALF uses a custom GCC in order to optimize results and explore the binary. We have to build the binary from source and patch the Makefile.

 1 2 3 4 5  --- a/Makefile +++ b/Makefile @@ -1,8 +1,12 @@ - CC = gcc + CC = afl-gcc

In more complex project, you will have to also change the compiler to alf-clang-fast++, add some compilation flags for optimizations, and even have to patch the main() {} function for AFL.

On compilation you should see some warnings/infos from AFL:

I will copy the built binary in /usr/local/bin/, and create a new directory for this fuzzing:

 1 2 3  cd $HOME mkdir in # our inputs for the fuzzer mkdir out # outputs of the fuzzer (crashes, test queue...)  ## Minimizing test cases The next step is to get good test cases to speed up the process of fuzzing. The fuzzer will base its mutations on those test cases, so the more unique tests you have, the better. By chance, most projects have good tests in their /tests/ directory, so let’s copy them to our inputs:  1  cp /src/tests/*.js in/  Once again, for optimizations, it is a good practice to minimize the test cases, it can save a lot of time during fuzz.  1 2 3 4 5  mkdir min # our minimized inputs afl-cmin -i in -o min --$TARGET @@ # minify the tests for the target # the '@@' stands for the input file # ie. for the target /bin/cat, we would use: # afl-cmin -i in -o min -- /bin/cat @@

After running the command (this will take a while if you have hundreds of cases) we now have a lot more efficient test cases in our min dir.

## Fuzzing

Now that we have our test cases ready, let’s get to fuzing!

As simple as that:

### Act III: anger

After trying dozens of projects to fuzz, I started to be a bit angry, or more precisely frustrated. I was trying to fuzz a binary to get a crash, but I got nothing for the past week. Angry at myself that I couldn’t setup a simple fuzzing lab.

### Act IV: bargaining

In order to save myself of myself, I started to try a lot of things, with my newly gained experience in fuzzing, I decided to go back to my original idea: radare2.

With that, I also read the manual and all the options for AFL++ in order to get better results, I even asked online for help, of course SoEasY DMed me to help. He wasted some time with me, and we supposed that the low amount of exec/s was due to the fact that I start r2 in interactive mode, and the process limit itself to a certain amount of threads, so the time the process kill itself in order to spawn a new one, it limits a lot the number of execs.

He also asked me to try other binaries of the radare2 suite, even to write a harness for a specific function to fuzz.

### Act V: depression

Even with those tweaks and tries, there were no improvements in my results after 5 days of fuzzing. 😢

I decided to take a break for days.

### Act VI: testing

After my break I wanted to know why I did not have any results with my fuzzing. And for me the obvious reason was an issue of the famous 8th layer of the OSI model: the keyboard-chair interface.

For my previous choices, either it was a bad target, or I did things wrong.

### Act VII: acceptance

The only way forward for me was to find an other target, and see if I get a crash easily.

Now that you have the past history of the project, see the real fuzzing journey.

## Conclusion

I made 11th out of 33 and I am very happy about that

Here you will find the final results including all the submitted entries for this year!


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# Tag Info

9

DES actually demonstrated that a Feistel structure was not a guarantee against attacks. In "academic" terms, DES is broken by both differential and linear cryptanalysis, because they require, respectively, $2^{47}$ chosen plaintexts and $2^{43}$ known plaintexts, whereas the DES key is (effectively) 56 bits. Of course, for practical attacks, we would brute ...

7

If such a network had only a single round, then you might have a valid concern. This is why there needs to be least three rounds, so that every bit from L can potentially affect every other bit from L (via R from the second round). It isn't a structural flaw, because multiple rounds are assumed. Changing this round structure would mean that it was no longer ...

7

No, this is not a structural weakness of Feistel networks. For instance, we know it can't hurt diffusion properties. Actually, we know that it's not a structural weakness. How do we know that? Because we have a proof of security for Feistel networks (under certain conditions and assumptions). Those proofs imply that there is not a structural weakness in ...

6

Well, AES is not a Feistel cipher because it's a substitution-permutation network instead. If I were taking a test that asked me why AES was not a Feistel cipher, this would be my argument: namely, that the structure of substitution-permutation networks is fundamentally different from that of Feistel networks. (Here one could elaborate on invertibility and ...

4

The simple way to build authenticated encryption using a Feistel Network is to build a Feistel based block cipher, then use one of the many modes of operation that turn a block cipher into an authenticated encryption scheme (eg CCM,OCB,GCM). For a good survey on the subject of modes-of-operation I would recommend this paper by Rogaway. It does not cover the ...

3

By definition, a Feistel network uses a series of rounds that split the input block into two sides, uses one side to permute the other side, then swaps the sides. As always, Wikipedia has a nice diagram. AES doesn't do this. Performing a round necessarily permutes the entire state. Each round consists of the SubBytes, ShiftRows, MixColumns, and AddRoundKey ...

2

Your Formulas are alright, but there is some additional information from the exercise/setup: The exercise states, that $F$ should be considered as a blackbox (otherwise you could use the internal stages of $F$, as poncho already suggested). However, as I understand it you can stil evaluate $F$ on any input of your choice. At this point, you can do a couple ...

2

Yes, you certainly can. If you want a variable-length authenticated encryption mode, then simply take any Feistel cipher in the OCB mode. If fixed-length is fine, then the following idea should work. Build a wide Feistel-based permutation (fixed-key blockcipher) $G$ and encrypt $$C = G(P||N||K)\oplus K,$$ where $N$ is nonce, $P$ is plaintext, $C$ is ...

1

Ff1,2,3 are basically inspired by LubyRack off constructions . At the core they differ in their round functions and key scheduling FF1 supports greater range of lengths and a tweak FF2 generates subkey for each iteration to thwart any side channel attacks FF3 has tweaks is split and used in rounding function, also the reverse the sub-strings of given ...

1

As a page at ibm.com indicates, there could have been a bit of a "contra" attitude against Feistel ciphers thanks to DES having seen the first breaks in it's security etc. Down with the Feistel structure! In most ciphers, the round transformation has the well-known Feistel structure. In this structure typically part of the bits of the intermediate ...

1

A linear transformation in a block cipher is also considered an substitution, just not a non linear one. When they talk about active s-boxes, they are talking about not specifically about the nonlinear s-box, but a level of input being substituted with a different output. Combining these across multiple rounds results in what they call 'active' s-boxes. ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
Clickable tableofcontents [duplicate]

How to obtain clickable items in a table of contents? I mean, they should also appear in blue. I have the preamble:

``````\documentclass[12pt]{iopart}

\usepackage{graphicx,color}
\usepackage{iopams}
\usepackage{hyperref}
``````

As you can see, I'm using the `iopart` class (specific for IOP journals).

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marked as duplicate by Claudio Fiandrino, Mico, lockstep, Peter Grill, ThorstenMar 30 '13 at 9:12

This question was marked as an exact duplicate of an existing question.

Did you run `pdflatex` twice? The first time it is run an .aux file is read upon a subsequent run. – Peter Grill Mar 30 '13 at 8:25
Yes, sure, more than twice! After the abstract I simply put \tableofcontents, should I say something else? – martina Mar 30 '13 at 8:26
Ok, then it would be helpful if you composed a fully compilable MWE including `\documentclass` and the appropriate packages that reproduces the problem. – Peter Grill Mar 30 '13 at 8:28
@martina: no problems at all. I'll close it as a duplicate of the answers you pointed if your problem is fixed. – Claudio Fiandrino Mar 30 '13 at 8:42
It is, thank you. – martina Mar 30 '13 at 8:43


text:
# Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs?

Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so they must be important.

I always assumed that spectral graph theory extends graph theory by providing tools to prove things we couldn't otherwise, somewhat like how representation theory extends group theory. But most results I see in spectral graph theory seem to concern eigenvalues not not as means to an end, but as objects of study in their own right.

I also considered practical value as motivation, e.g. using a given set of eigenvalues to put bounds on essential properties of graphs, such as maximum vertex degree. But I can't imagine a situation in which I would have access to a graph's eigenvalues before I would know much more elementary information like maximum vertex degree.

(EDIT: for example, dtldarek points out that $\lambda_2$ is related to diameter, but then why would we need $\lambda_2$ when we already have diameter? Is it somehow conceptually beneficial?)

So, what is the meaning of graph spectra intuitively? And what are they used for? Why is finding the eigenvalues of a graph's adjacency/Laplacian matrices more than just a novel problem?

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There is a several-century old tradition of studying geometrical objects spectrally, through their Laplacians and other related objects; this was initiated by the greeks and their study of music, or something like that. Spectral graph theory is just the discretization of that. –  Mariano Suárez-Alvarez Feb 20 '13 at 9:37
Hi, in case of duplicated accounts, please visit our help page here. When the duplicated accounts are merged, you will regain ownership automatically of your question. –  Willie Wong Feb 20 '13 at 10:24

This question already has a number of nice answers; I want to emphasize the breadth of this topic.

Graphs can be represented by matrices - adjacency matrices and various flavours of Laplacian matrices. This almost immediately raises the question as to what are the connections between the spectra of these matrices and the properties of the graphs. Let's call the study of these connections "the theory of graph spectra". (But I am not entirely happy with this definition, see below.) It is tempting to view the map from graphs to eigenvalues as a kind of Fourier theory, but there are difficulties with this analogy. First, graphs in general are not determined by the their eigenvalues. Second, which of the many adjacency matrices should we use?

The earliest work on graph spectra was carried out in the context of the Hueckel molecular orbital theory in Quantum Chemistry. This lead among other things to work on the matching polynomial; this gives us eigenvalues without adjacency matrices (which is why I feel the above definition of the topic is unsatisfactory). A more recent manifestation of this stream of ideas is the work on the spectra of fullerenes.

The second source of the topic arises in Seidel's work on regular two-graphs, which started with questions about regular simplices in real projective space and lead to extraordinarily interesting questions about sets of equiangular lines in real space. The complex analogs of these questions are now of interest to quantum physicists - see SIC-POVMs. (It is not clear what role graph theory can play here.) In parallel with Seidel's work was the fundamental paper by Hoffman and Singleton on Moore graphs of diameter two. In both cases, the key observation was that certain extremal classes of graphs could be characterized very naturally by conditions on their spectra. This work gained momentum because a number of sporadic simple groups were first constructed as automorphism groups of graphs. For graph theorists it flowered into the the theory of distance-regular graphs, starting with the work of Biggs and his students, and still very active.

One feature of the paper of Hoffman and Singleton is that its conclusion makes no reference to spectra. So it offers an important graph theoretical result for which the "book proof" uses eigenvalues. Many of the results on distance-regular graphs preserve this feature.

Hoffman is also famous for his eigenvalue bounds on chromatic numbers, and related bounds on the maximum size of independent sets and cliques. This is closely related to Lovász's work on Shannon capacity. Both the Erdős-Ko-Rado theorem and many of its analogs can now be obtained using extensions of these techniques.

Physicists have proposed algorithms for graph isomorphism based on the spectra of matrices associated to discrete and continuous walks. The connections between continuous quantum walks and graph spectra are very strong.

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I can't speak much to what traditional Spectral Graph Theory is about, but my personal research has included the study of what I call "Spectral Realizations" of graphs. A spectral realization is a special geometric realization (vertices are not-necessarily-distinct points, edges are not-necessarily-non-degenerate line segments, in some $\mathbb{R}^n$) derived from the eigenvectors of a graph's adjacency matrix.

In particular, if the rows of a matrix constitute a basis for some eigenspace of the adjacency matrix a graph $G$, then the columns of that matrix are coordinate vectors of (a projection of) a spectral realization.

A spectral realization of a graph has two nice properties:

• It's harmonious: Every graph automorphism induces to a rigid isometry of the realization; you can see the graph's automorphic structure!
• It's eigenic: Moving each vertex to the vector-sum of its immediate neighbors is equivalent to scaling the figure; the scale factor is the corresponding eigenvalue.

Well, the properties are nice in theory. Usually, a spectral realization is a jumble of collapsed segments, or is embedded is high-dimensional space; such circumstances make a realization difficult to "see". Nevertheless, a spectral realization can be a helpful first pass at visualizing a graph. Moreover, a graph with a high degree of symmetry can admit some visually-interesting low-dimensional spectral realizations; for example, the skeleton of the truncated octahedron has this modestly-elaborate collection:

For a gallery of hundreds of these things, see the PDF linked at my Bloog post, "Spectral Realizations of Graphs".

Since many mathematical objects decompose into eigen-objects, it probably comes as no surprise that any geometric realization of a graph is the sum of spectral realizations of that graph. (Simply decomposing the realization's coordinate matrix into eigen-matrices gets most of the way to that result, although the eigen-matrices themselves usually represent "affine images" of properly-spectral realizations. The fact that affine images decompose into a sum of similar images takes an extension of a theorem of Barlotti.) There's likely something interesting to be said about how each spectral component spectral influences the properties of the combined figure.

Anyway ... That's why I care about the eigenvalues of graphs.

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That is very neat. I am going to leave the question open because I would still like to know the answer for traditional spectral graph theory, but I appreciate your post. I had actually come across your paper before and I think I will go revisit it and read it more carefully now. –  Alexander Gruber Feb 21 '13 at 20:20

Spectral graph theory is a discrete analogue of spectral geometry, with the Laplacian on a graph being a discrete analogue of the Laplace-Beltrami operator on a Riemannian manifold. The Laplacian $\Delta$ can be used to write down three important differential equations, both on a graph and a Riemannian manifold:

• The heat equation $\frac{\partial u}{\partial t} = \Delta u$, which describes how heat propagates on the graph / manifold,
• The wave equation $\frac{\partial^2 u}{\partial t^2} = \Delta u$, which describes how waves propagate on the graph / manifold,
• The Schrödinger equation $i \frac{\partial u}{\partial t} = \Delta u$, which describes how quantum particles propagate on the graph / manifold.

The behavior of solutions to these equations is controlled by the eigenvalues of the Laplacian. For the heat equation these eigenvalues control the rate at which a given heat distribution decays to a stationary distribution, and for the wave and Schrödinger equations these eigenvalues control the rate at which standing wave solutions oscillate (this gives some intuition as to why the eigenvalues should be related to the connectivity of the graph). So these eigenvalues describe some important physical properties of the graph / manifold.

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(revisiting after a year) Thanks again. Since asking this question, this is the answer that has stuck most in my head. I have a followup question. I can see how wave propogation on graphs could be useful in applied contexts, for computable approximations of continuous structures, and other things. Does wave propogation on (finite/discrete) graphs occur naturally in other nonapplied areas mathematics? –  Alexander Gruber Mar 18 '14 at 21:44
@Alexander: there are a number of results in spectral graph theory describing how the spectrum of the Laplacian controls random walks on a graph, and I think morally speaking the heat equation is the "reason" that these results exist. –  Qiaochu Yuan Mar 30 '14 at 1:31

Some random facts:

• The largest eigenvalue $\lambda_1$ is closely related to average degree.
• The second largest eigenvalue $\lambda_2$ is closely related to connectivity, that is graphs with small $\lambda_2$ have small diameter.
• A graph $G$ is bipartite if and only if for every eigenvalue $\lambda$, $-\lambda$ is also an eigenvalue.
• For connected $G$, the chromatic number of $G$, $\chi(G) \leq \lambda_1+1$, with equality for $G$ being complete or an odd cycle.
• Some use spectrum for clustering.

I hope it helps ;-)

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Expanding on a random fact mentioned by dtldarek: People frequently use spectral graph theory in clustering.

The general situation is that you have a collection of data points $x_1, \dots x_n$, that belong to some unknown clusters $A_1, \dots, A_k$. You want to identify which points lie in which cluster. You have some sort of connection graph, and the hope is that connections tend to occur within clusters -- e.g. the density of edges within $A_1$ is much higher than the density of edges connecting $A_1$ and $A_2$. One way to (try and) find a cluster, then can be thought of as searching for a subset $A$ such that $$\frac{\textrm{The number of edges between }A \textrm{ and the rest of the graph}}{\textrm{The total number of edges involving} A }$$ is unusually small.

More generally, we can imagine a weighted graph where between each pair of vertices we put a weight corresponding to how "close" the vertices are in some sense. The goal is then to find an $A$ such that the total weight of edges between $A$ and the rest of the graph is much smaller than the total weight of edges involving $A$.

The catch is that if $G$ has $n$ vertices, there's $2^n$ choices for $A$, so we can't just search them all if $n$ is large. But what often turns out to be a good approximation is to choose $A$ based on the eigenvectors for the Laplacian matrix of your (weighted) connection graph (in particular the eigenvectors corresponding to the smallest nonzero eigenvalues of $L$).

The idea is that if you take a vector $x$ and a graph having weights $w_{ij}$ on the eges, then $$x^T L x = \sum_{\textrm{edges } (i,j) } w_{ij} (x_i-x_j)^2$$ In the special case where $x_i=1$ for $i$ in $A$ and $0$ for $i$ not in $A$, this reduces down to $x^T L x$ is the total weight of edges connecting $A$ with the rest of the graph. Eigenvectors of $L$ with small eigenvalues correspond to choices of $x$ which make $x^T L x$ small.

For more on this, see this tutorial by Ulrike von Luxburg.

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Another reason the eigenvalues are used if for approximating certain functions in a graph that would be too complex to compute otherwise. For instance, in social networks one often wants to count the number of paths of length $k$ between two nodes $u$ and $v$. In a large graph, this may be too complex to compute exactly. One way to approximate it consists in considering the $k^{\mathrm{th}}$ power of the adjacency matrix $\mathbf A$. However, even if $\mathbf A$ is sparse, $\mathbf A^k$ is not for even small $k$ (remember the "six degrees of separation" – the diameter of a social network is low). Thus, a method to compute entries of $\mathbf A^k$ approximately uses the eigenvalue decomposition of $\mathbf A$. If the graph is undirected then we can write

$$\mathbf A = \mathbf U \mathbf Λ \mathbf U^{\mathrm T}$$ with $\mathbf U$ and $\mathbf Λ$ being $n \times n$ matrices. A rank-reduced eigenvalue can be computed efficiently, i.e., one where $\mathbf U$ is replaced by an $n×k$ matrix $\mathbf {\tilde U}$, and $\mathbf Λ$ by a $k×k$ matrix $\mathbf{\tilde Λ}$. Then, $\mathbf A^k$ can be approximated by

$$\mathbf A^k ≈ \mathbf{\tilde U} \mathbf{\tilde Λ}^k \mathbf{\tilde U}^{\mathrm T}$$

and to approximate only $(\mathbf A^k)_{uv}$ (i.e., the number of $k$-paths between nodes $u$ and $v$), we only need to compute $\mathbf{\tilde U}_{i\bullet} \mathbf{\tilde D}^k \mathbf{\tilde U}_{j\bullet}^{\mathrm T}$. This is a case where one is not interested in the eigenvalues of eigenvectors themselves, but uses the eigenvalue decomposition for approximation.

The same technique can be used to approximate other functions of $\mathbf A$, for instance the matrix exponential.

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text:
Opened 9 years ago

Closed 9 years ago

# Error Billing Reports

Reported by: Owned by: anonymous Russ Tyndall normal TimingAndEstimationPlugin normal 0.11

### Description

getting the following error when clicking on any of the Billing Reports on the Management page:

Report execution failed: ORDER BY column number 0 out of range - should be between 1 and 7

Using

Trac: 0.11rc1 Python: 2.5.2 (r252:60911, Apr 21 2008, 11:29:43) [GCC 4.2.3 (Ubuntu 4.2.3-2ubuntu7)] setuptools: 0.6c8 SQLite: 3.4.2 pysqlite: 2.4.0 Genshi: 0.5dev-r851 mod_python: 3.3.1 jQuery: 1.2.3

Any ideas?

Thanks!

### comment:1 follow-up:  2 Changed 9 years ago by Russ Tyndall

I am not sure having never seen that error, but try editing the report and changing the order by statement in the Ticket Work Summary report from:

ORDER BY __ticket__, _ord ASC, time ASC

to

ORDER BY __group__, _ord ASC, time ASC

and let me know if that fixes it. There may have been a change in how column aliasing is resolved in the order by (but I would sor of doubt it). If it doesnt work, just change it back so that this will not be the source of future errors.

### comment:2 in reply to:  1 Changed 9 years ago by anonymous

Thanks for the fast reply! But unfortunately that didn't help me. I tried the suggested changes to the report (of course with a backup copy), still the same error message. Also skipping the ORDER BY clause didn't change anything. I use custom workflows and other plugins in my trac project so I set up a new and clean trac project to rule out that the error comes from my own workflows and other plugins. But the error is the same there.

### comment:3 Changed 9 years ago by Russ Tyndall

It still gives you an error about the order-by column even when the order by clause is entirely removed! That is strange indeed.

I replicated the error on my end by upgrading trac, genshi and tande to the head revisions. Trying to figure it out now

### comment:4 Changed 9 years ago by Russ Tyndall

Resolution: → invalid new → closed

I finally tracked down what was going on. It seems the group column that trac reports use to group the results is broken in the trac head at the moment.

This is a bug in Trac11rc1 see: http://trac.edgewall.org/ticket/7179

Hopefully it will be fixed soon. There is a patch for this on the ticket if you are the adventurous type.

Thanks for the good bug report and excellent feedback,

Russ

### comment:5 Changed 9 years ago by Russ Tyndall

This seems to be fixed in the head of http://svn.edgewall.com/repos/trac/branches/0.11-stable/

### Modify Ticket

Change Properties


text:
# General formula for expanding wave function in terms of orthogonal states?

Given a wave function $\psi(x) = \langle \psi | x \rangle$. It can be expanded in terms of orthogonal states:

$$\langle \psi | x \rangle = \sum_n \langle \psi | n \rangle \langle n |x \rangle$$

Questions:

1. Can this always be done for an arbitary action?

2. Given a general action $S(x,\dot{x})$, is there a general formula to find $\langle \psi |n \rangle$ and $\langle x |n \rangle$ in terms of the action, S, and the wavefunction $\psi(x)$?

All I know is that the ground state can be given formally by the path integral:

$$\langle x | 0 \rangle = \int\limits^{y_0=x}_{y_{-\infty}=0} e^{iS(y,\dot{y})} Dy$$

and so (I think)

$$\langle \psi | 0 \rangle = \int \psi(y_0)e^{iS(y,\dot{y})} Dy$$

1. Are there any general formulae for $|n\rangle$ for $n>0$ ?

2. Also, can this be done in the case of full QFT or just for free fields?

• The expansion is just a matter of using the fact that the states |n> form a complete set, you can express that as the so-called closure relation: $$\sum_n |n\rangle\langle n| = 1$$ The path integral gives the time evolution, the matrix element you wrote down is to be interpreted in the Heisenberg picture, so it gives the amplitude for a particle at position 0 to evolve to position x. You can manipulate that via various closure relations and express it in terms of wavefunctions. – Count Iblis Oct 18 '15 at 19:42
• To get the ground state from the path integral, you can use the Gell-Mann and Low theorem – Count Iblis Oct 18 '15 at 19:42
• Hi, yes, I got the ground state using the path integral. I just wondered if there is a way to get the other states using the path integral? Maybe there is no general way of doing it? The states are eigenstates of the energy if this helps? Maybe there is no general formula? – zooby Oct 18 '15 at 19:52

(1) Can this always be done for an arbitrary action?

The answer to this was given by CountIblis in his comment. The expansion you are asking about follows from the closure formula for basis $\{|n\rangle\}$, $I = \sum_n{|n\rangle\langle n|}$: $$\langle \psi|x\rangle = \langle \psi|I|x\rangle = \sum_n{\langle\psi|n\rangle\langle n|x\rangle}$$ It is independent of the action that governs the evolution of $|\psi\rangle$.

(2) Given a general action $S(x,\dot{x})$, is there a general formula to find $\langle \psi|n\rangle$ and $\langle x|n\rangle$ in terms of the action, $S$, and the wavefunction $\psi(x)$?

It we know the wavefunctions $\langle x| \psi \rangle$ and $\langle x| n \rangle$, the overlap $\langle n |\psi\rangle = \langle \psi|n\rangle^*$ can be found (trivially) as $$\langle n |\psi\rangle = \int{dx \;\langle n |x\rangle \langle x|\psi\rangle} = \int{dx\; \langle x |n\rangle^* \langle x|\psi\rangle}$$
You can look at this as another application of the closure relation, this time for the $\{|x\rangle\}$ basis, $I = \int{dx\; |x\rangle \langle x|}$. Again, it is independent of the action for the evolution of $|\psi\rangle$.

The problem of finding the wavefunctions $\langle x| n \rangle$ for excited states in the path integral representation is another issue, and not a simple one. You may want to take a look at these papers to get an idea of what's involved:

• T. E. Sorensen & W. B. England, Molecular Physics 89, 1577 (1996)
• A.G. Ushveridze, Physics Letters A 110(4), 217–220 (1985)
• Unfortunately this did not answer the question. I did not say the $\langle x | n \rangle$ was known. This is one of the unknowns that we are trying to find! All that is known is S and the wavefunction. All you've done is express one unknown in terms of the other unknown! – zooby Oct 20 '15 at 13:25
• In answering (2) I said "If we know ...$\langle x|n\rangle$ ...", followed by (last paragraph) "The problem of finding the wavefunctions $\langle x|n\rangle$ ... is another issue". This means "we can calculate $\langle n| \psi\rangle$ provided we know $\langle x|n\rangle$ in addition to $\langle x|\psi\rangle$, but $\langle x|n\rangle$ itself is not so easy to calculate in the first place in path integral representation even if we know S exactly". However, I did provide 2 refs. that consider exactly this path integral calculation for excited states, closing the problem you asked about. – udrv Oct 20 '15 at 18:48

I don't think there is a general way. Also, probably the state basis is only applicable probably to free field theories. Just the quadratic part of the action.

In general it is a very difficult if not impossible task to diagonalise the Hamiltonian. That is why path integrals are used for interactions.


text:
Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > DETAIL:

### Paper:

TR15-174 | 18th October 2015 01:37

#### Complexity of distributions and average-case hardness

TR15-174
Authors: Dmitry Itsykson, Alexander Knop, Dmitry Sokolov
Publication: 5th November 2015 23:23
Keywords:

Abstract:

We address a natural question in average-case complexity: does there exist a language \$L\$ such that for all easy distributions \$D\$ the distributional problem \$(L, D)\$ is easy on the average while there exists some more hard distribution \$D'\$ such that \$(L, D')\$ is hard on the average? We consider two complexity measures of distributions: complexity of sampling and complexity of computing the distribution function. The most interesting measure is the complexity of sampling. We prove that for every \$b > a > 0\$ there exists a language \$L\$, an ensemble of distributions \$D\$ samplable in \$n^{\log^b n}\$ steps and a linear-time algorithm \$A\$ such that for every ensemble of distribution \$F\$ that samplable in \$n^{\log^a n}\$ steps, \$A\$ correctly decides \$L\$ on all inputs from \$\{0, 1\}^n\$ except of a set that has infinitely small \$F\$-measure, and for every algorithm \$B\$ there are infinitely many \$n\$ such that the set of all elements of \$\{0, 1\}^n\$ for which \$B\$ correctly decides \$L\$ has infinitely small \$D\$-measure.

In case of complexity of computing the distribution function we prove the following tight result: for every \$a > 0\$ there exists a language \$L\$, an ensemble of polynomial-time computable distributions \$D\$, and a linear-time algorithm \$A\$ such that for every computable in \$n^a\$ steps ensemble of distributions \$F\$, \$A\$ correctly decides \$L\$ on all inputs from \$\{0, 1\}^n\$ except a set that has \$F\$-measure at most \$2^{-n / 2}\$, and for every algorithm \$B\$ there are infinitely many \$n\$ such that the set of all elements of \$\{0, 1\}^n\$ for which \$B\$ correctly decides \$L\$ has \$D\$-measure at most \$2^{-n + 1}\$.

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