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Scalable Global Optimization via Local Bayesian Optimization

Bayesian optimization has recently emerged as a popular method for the sample-efficient optimization of expensive black-box functions. However, the application to high-dimensional problems with several thousand observations remains challenging, and on difficult problems Bayesian optimization is often not competitive with other paradigms. In this paper we take the view that this is due to the implicit homogeneity of the global probabilistic models and an overemphasized exploration that results from global acquisition. This motivates the design of a local probabilistic approach for global optimization of large-scale high-dimensional problems. We propose the $\texttt{TuRBO}$ algorithm that fits a collection of local models and performs a principled global allocation of samples across these models via an implicit bandit approach. A comprehensive evaluation demonstrates that $\texttt{TuRBO}$ outperforms state-of-the-art methods from machine learning and operations research on problems spanning reinforcement learning, robotics, and the natural sciences.

Learning Search Space Partition for Black-box Optimization using Monte Carlo Tree Search

High dimensional black-box optimization has broad applications but remains a challenging problem to solve. Given a set of samples $\{\vx_i, y_i\}$, building a global model (like Bayesian Optimization (BO)) suffers from the curse of dimensionality in the high-dimensional search space, while a greedy search may lead to sub-optimality. By recursively splitting the search space into regions with high/low function values, recent works like LaNAS shows good performance in Neural Architecture Search (NAS), reducing the sample complexity empirically. In this paper, we coin LA-MCTS that extends LaNAS to other domains. Unlike previous approaches, LA-MCTS learns the partition of the search space using a few samples and their function values in an online fashion. While LaNAS uses linear partition and performs uniform sampling in each region, our LA-MCTS adopts a nonlinear decision boundary and learns a local model to pick good candidates. If the nonlinear partition function and the local model fits well with ground-truth black-box function, then good partitions and candidates can be reached with much fewer samples. LA-MCTS serves as a \emph{meta-algorithm} by using existing black-box optimizers (e.g., BO, TuRBO) as its local models, achieving strong performance in general black-box optimization and reinforcement learning benchmarks, in particular for high-dimensional problems.

Adaptive Local Bayesian Optimization Over Multiple Discrete Variables

In the machine learning algorithms, the choice of the hyperparameter is often an art more than a science, requiring labor-intensive search with expert experience. Therefore, automation on hyperparameter optimization to exclude human intervention is a great appeal, especially for the black-box functions. Recently, there have been increasing demands of solving such concealed tasks for better generalization, though the task-dependent issue is not easy to solve. The Black-Box Optimization challenge (NeurIPS 2020) required competitors to build a robust black-box optimizer across different domains of standard machine learning problems. This paper describes the approach of team KAIST OSI in a step-wise manner, which outperforms the baseline algorithms by up to +20.39%. We first strengthen the local Bayesian search under the concept of region reliability. Then, we design a combinatorial kernel for a Gaussian process kernel. In a similar vein, we combine the methodology of Bayesian and multi-armed bandit,(MAB) approach to select the values with the consideration of the variable types; the real and integer variables are with Bayesian, while the boolean and categorical variables are with MAB. Empirical evaluations demonstrate that our method outperforms the existing methods across different tasks.

High-Dimensional Bayesian Optimization with Manifold Gaussian Processes

Bayesian optimization (BO) is a powerful approach for seeking the global optimum of expensive black-box functions and has proven successful for fine tuning hyper-parameters of machine learning models. The Bayesian optimization routine involves learning a response surface and maximizing a score to select the most valuable inputs to be queried at the next iteration. These key steps are subject to the curse of dimensionality so that Bayesian optimization does not scale beyond 10--20 parameters. In this work, we address this issue and propose a high-dimensional BO method that learns a nonlinear low-dimensional manifold of the input space. We achieve this with a multi-layer neural network embedded in the covariance function of a Gaussian process. This approach applies unsupervised dimensionality reduction as a byproduct of a supervised regression solution. This also allows exploiting data efficiency of Gaussian process models in a Bayesian framework. We also introduce a nonlinear mapping from the manifold to the high-dimensional space based on multi-output Gaussian processes and jointly train it end-to-end via marginal likelihood maximization. We show this intrinsically low-dimensional optimization outperforms recent baselines in high-dimensional BO literature on a set of benchmark functions in 60 dimensions.

Scaling Gaussian Process Regression with Derivatives

Gaussian processes (GPs) with derivatives are useful in many applications, including Bayesian optimization, implicit surface reconstruction, and terrain reconstruction. Fitting a GP to function values and derivatives at $n$ points in $d$ dimensions requires linear solves and log determinants with an ${n(d 1) \times n(d 1)}$ positive definite matrix-- leading to prohibitive $\mathcal{O}(n 3d 3)$ computations for standard direct methods. We propose iterative solvers using fast $\mathcal{O}(nd)$ matrix-vector multiplications (MVMs), together with pivoted Cholesky preconditioning that cuts the iterations to convergence by several orders of magnitude, allowing for fast kernel learning and prediction. Our approaches, together with dimensionality reduction, allows us to scale Bayesian optimization with derivatives to high-dimensional problems and large evaluation budgets. Papers published at the Neural Information Processing Systems Conference.


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# wikiHow to Find the Diagonal of a Square Using Its Area

A diagonal is a straight line that runs between opposite corners of a square.[1] A diagonal divides a square into two right triangles. This property, as well as the fact that a square has four equal side lengths, makes it possible to find the length of a square’s diagonal using the area.

### Part 1 Finding the Length of One Side

1. 1
Set up the formula for the area of a square. The formula is ${\displaystyle {\text{Area}}=s^{2}}$, where ${\displaystyle s}$ equals the length of one side of the square.[2] In order to find the length of the diagonal, you first need to find the side length of the square.
• The area of a rectangle is found by multiplying its length and width, but since a square rectangle has four equal side lengths, you can use this shortened formula for its area.[3]
2. 2
Plug the area into the formula. Make sure you put the value on the left side of the equation. Do not substitute for ${\displaystyle s}$. If you don’t know the area of the square, you cannot use this method.
• For example, you might need to find the diagonal length of a square that has an area of 169 square centimeters. So, your equation will look like this: ${\displaystyle 169=s^{2}}$.
3. 3
Find the square root of both sides of the equation. Taking the square root of both sides will give you the side length of the square. The easiest way to calculate a square root of a number is to use the ${\displaystyle {\sqrt {x}}}$ function on a calculator. You can also calculate a square root by hand. Remember that when finding the square root of a squared variable, the answer is simply the variable.
• For example:
${\displaystyle 169=s^{2}}$
${\displaystyle {\sqrt {169}}={\sqrt {s^{2}}}}$
${\displaystyle 13=s}$
So, a square with an area of 169 square centimeters has a side length of 13 cm.

### Part 2 Calculating the Length of the Diagonal

1. 1
Set up the formula for the diagonal of a square. The formula is ${\displaystyle {\text{Diagonal}}=s{\sqrt {2}}}$, where ${\displaystyle s}$ is the length of one of the square’s sides.[4]
• This formula is derived from the Pythagorean Theorem. The diagonal divides a square into two right triangles. So, using the side lengths and the Pythagorean Theorem, you get the formula ${\displaystyle s^{2}+s^{2}=c^{2}}$, which simplifies to ${\displaystyle s{\sqrt {2}}=c}$.
2. 2
Plug the side length into the formula. Substitute for the variable ${\displaystyle s}$. You should have calculated the side length previously.
• For example, if you found that the side length of a square is 13 cm, your equation will look like this: ${\displaystyle {\text{Diagonal}}=13{\sqrt {2}}}$
3. 3
Multiply the side length by the square root of 2. The square root of 2 is about 1.414. You can also use a calculator to find a more precise answer. Calculating the side length by ${\displaystyle {\sqrt {2}}}$ gives you the length of the square’s diagonal.
• For example:
${\displaystyle {\text{Diagonal}}=13{\sqrt {2}}}$
${\displaystyle {\text{Diagonal}}=13(1.414)}$
${\displaystyle {\text{Diagonal}}=18.382}$
So, the length of the square's diagonal is about 18.38 cm.

### Part 3 Completing Sample Problems

1. 1
Try this problem. A square has an area of 120 square centimeters. What is the length of its diagonal?
• Find the side length of the square by using the formula ${\displaystyle {\text{Area}}=s^{2}}$:
${\displaystyle 120=s^{2}}$
${\displaystyle {\sqrt {120}}={\sqrt {s^{2}}}}$
${\displaystyle 10.95=s}$
• Plug the side length into the formula ${\displaystyle {\text{Diagonal}}=s{\sqrt {2}}}$:
${\displaystyle {\text{Diagonal}}=10.95{\sqrt {2}}}$
${\displaystyle {\text{Diagonal}}=10.95(1.414)}$
${\displaystyle {\text{Diagonal}}=15.4833}$
So, the length of the diagonal is about 15.5 cm.
2. 2
Find the distance between opposite corners. The corners are at the northwest and southeast corners of a square field. The field has an area of 16,000 square feet.
• The distance between opposite corners of a square is the length of the diagonal. So first find the side length using the area formula:
${\displaystyle {\text{Area}}=s^{2}}$
${\displaystyle 16,000=s^{2}}$
${\displaystyle {\sqrt {16,000}}={\sqrt {s^{2}}}}$
${\displaystyle 126.49=s}$
• Plug the side length into the formula ${\displaystyle {\text{Diagonal}}=s{\sqrt {2}}}$:
${\displaystyle {\text{Diagonal}}=126.49{\sqrt {2}}}$
${\displaystyle {\text{Diagonal}}=126.49(1.414)}$
${\displaystyle {\text{Diagonal}}=178.86}$
So, the length of the diagonal is about 179 feet.
3. 3
Compare two diagonals. The diagonal of rhombus is 12 feet. How does its length compare to the diagonal of a square with an area of 72.25 square feet?
• Find the square’s side length by using the area formula:
${\displaystyle {\text{Area}}=s^{2}}$
${\displaystyle 72.25=s^{2}}$
${\displaystyle {\sqrt {72.25}}={\sqrt {s^{2}}}}$
${\displaystyle 8.5=s}$
• Plug the side length into the formula ${\displaystyle {\text{Diagonal}}=s{\sqrt {2}}}$:
${\displaystyle {\text{Diagonal}}=8.5{\sqrt {2}}}$
${\displaystyle {\text{Diagonal}}=8.5(1.414)}$
${\displaystyle {\text{Diagonal}}=12.02}$
So, the length of the square’s diagonal is about 12 feet. This means that the diagonal of the rhombus is about the same length.

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# Large-scale brain modes reorganize between infant sleep states and carry prognostic information for preterms

## Abstract

Sleep architecture carries vital information about brain health across the lifespan. In particular, the ability to express distinct vigilance states is a key physiological marker of neurological wellbeing in the newborn infant although systems-level mechanisms remain elusive. Here, we demonstrate that the transition from quiet to active sleep in newborn infants is marked by a substantial reorganization of large-scale cortical activity and functional brain networks. This reorganization is attenuated in preterm infants and predicts visual performance at two years. We find a striking match between these empirical effects and a computational model of large-scale brain states which uncovers fundamental biophysical mechanisms not evident from inspection of the data. Active sleep is defined by reduced energy in a uniform mode of neural activity and increased energy in two more complex anteroposterior modes. Preterm-born infants show a deficit in this sleep-related reorganization of modal energy that carries novel prognostic information.

## Introduction

Changes in behavioral and cognitive states arise from transitions in the configuration of functional brain networks1,2,3. Falling asleep and switching between sleep states are archetypal cortical state transitions that are reflected in the reorganization of patterns of functional interactions between remote brain regions4,5. Importantly, the nature of sleep architecture in the early stages of life shapes brain development and influences future behavior6,7. Understanding sleep patterns in neonates thus carries important clinical potential8,9. However, the neural principles underpinning sleep states early in life are poorly understood. Advances in our knowledge of these fundamental processes are therefore essential for improving brain health outcomes with broader implications for behavior across the lifespan.

Here, we map whole-cortex functional interactions associated with distinct sleep states in two groups of infants, born at term or extremely preterm. We measured cortical activity when infants were in states of active sleep (AS) and quiet sleep (QS). These distinct states of vigilance are key components of the infants’ sleep-wake cycle, which gradually transform with neurodevelopment into the alternating cycle of mature rapid eye movement (REM) and non-REM sleep states such as deep sleep10. Using high-density electroencephalography (EEG) and tools from network science, we assess frequency-resolved reconfigurations in whole-cortex functional connectivity as a function of sleep states in these two groups of infants. This analysis provides a first comprehensive characterization of abnormal sleep-related reconfigurations of cortical connectivity following extreme preterm birth. To interrogate the notion that infants’ sleep states shape the development of cortical pathways and related brain functions, we test if changes in connectivity as a function of sleep states in infants exposed to prematurity carry prognostic information regarding behavioral performance at two years.

Moving beyond the characterization of cortical connectivity patterns, we develop a novel biophysical model based on neural field theory11. This is a well-established approach for modeling brain dynamics at macroscopic scales12 in healthy states13,14 and following pharmacological manipulation15. This model allows study of the neural principles underpinning sleep-dependent changes in cortical dynamics. Our results show that the emergence of distinct sleep-related patterns of cortical connectivity between infants born at term and extremely preterm are caused by differences in the redistribution of energy in low-dimensional modes of spatiotemporal neural activity.

## Results

### Analysis of infant sleep EEG

We analyzed multi-channel scalp EEG data from two groups of infants: extremely preterm (EP, N = 42) and full-term healthy controls (HC, N = 52), with gestational age (median ± interquartile range) of 26.6 ± 1.4 and 40.4 ± 1.7 weeks, respectively. EEG data in both groups were acquired at term-equivalent age of 41.1 ± 2 weeks. Continuous epochs of EEG were recorded in states of AS and QS (Fig. 1a). Cortical source signals were reconstructed from these sensor data using a detailed, realistic infant head model4. The neuronal signals representing activity of cortical parcels were filtered into four frequency bands: delta (0.4–1.5 Hz), theta (4–8 Hz), alpha (8–13 Hz), and low beta (13–22 Hz). Following standard techniques for the study of functional networks derived from electrophysiological data, we studied a key “intrinsic mode” of connectivity, namely pair-wise amplitude covariation16. Frequency-specific functional interactions between cortical activity in these parcels were hence computed as pairwise correlations between amplitude envelopes of mutually orthogonalized neuronal signals17, resulting in functional connectivity matrices for each frequency band and sleep state (Fig. 1b). We statistically compared18 whole-brain functional networks between sleep states and groups and correlated these patterns of connectivity with neurocognitive outcomes (Fig. 1c). We then established a biophysical model to explain these cortical network configurations (Fig. 1d).

### Effects of sleep state and preterm birth on brain dynamics

Analysis of cortico-cortical functional connectivity shows that across both infant groups, sleep states differ significantly in the activity of two broadband networks comprising posterior and anterior brain regions (family wise error rate (FWER)-corrected pFWER < 0.023 for all paired two-tailed t-tests; Fig. 2a for alpha and left column on Supplementary Fig. 1 for other bands). AS is characterized by stronger connectivity within a network comprising occipital, central, and temporal regions (Fig. 2a, b red), extending to frontal cortex in the low beta band (Supplementary Fig. 1). QS shows a more widespread increase in connectivity throughout the cortex, with higher long-range connectivity and a notable involvement of frontal brain regions (Fig. 2a, b blue).

To further characterize the spectral fingerprints of these effects, we filtered the neural signals into (fast) carrier signals and their (slow) amplitude fluctuations across a staircase of relatively narrow bands (evenly spaced in logarithmic coordinates and covering the carrier-amplitude frequency space), then repeated the network-based contrasts. These analyses confirm the existence of functional network effects in both quiet and active sleep states, supported by fast carrier oscillations that are strongest in alpha and beta frequencies (Fig. 2c), and whose amplitudes fluctuate maximally on time scales of 5–11 seconds (0.09–0.19 Hz). The strongest effects are seen for higher carrier frequencies in QS than AS (~16 Hz vs. ~10 Hz). Interestingly, there exists a second effect in QS centered over the delta band, composed of relatively short-range edges that involve frontal regions and consistent with the enhanced power of slow-wave activity classically seen in quiet/deep sleep (Supplementary Fig. 1).

Across both sleep states, EP infants show significantly stronger connectivity compared with HC with functional networks in the alpha range encompassing occipital, temporal, central, and frontal regions (pFWER < 0.0001, for all paired two-tailed t-tests; Fig. 3a, b). These effects are also associated with stronger long-range cortico-cortical connectivity in EP relative to HC (Fig. 3b). Spectral analyses using narrow-band filters suggest this effect is strongest in the alpha and beta range (Fig. 3c), although long-range functional networks, significantly stronger in EP, are also present in the delta and theta bands (Supplementary Fig. 1). EP infants also show a reduction in connectivity between anterior and posterior cortical regions in the theta band (pFWER = 0.01, paired two-tailed t-test; Supplementary Fig. 1). When analyzed according to the bank of narrow-band filters, the reduction in functional networks is evident across the theta, alpha and beta frequencies (Fig. 3c). However, when analyzed according to the four pre-specified frequency bands, these effects only survive FWER correction in the theta range (Supplementary Fig. 1).

Analysis of interactions between groups and sleep states shows that long-range alpha connectivity between frontal and occipital cortices plays a key role in supporting distinct sleep states that differentiate EP from HC (violet in Fig. 4a, pFWER = 0.0004, paired two-tailed t-test). Moreover, overlapping patterns of occipital connectivity across a broad frequency range—including theta, alpha, and low beta bands—show a group-by-sleep interaction (orange in Fig. 4a and Supplementary Fig. 2a, all pFWER < 0.0046, paired two-tailed t-test). For both fronto-occipital and occipital connectivity patterns, the group-by-sleep interaction is driven by an attenuated modulation of connectivity strength between AS and QS in EP (Fig. 4a). Visual inspection of the functional connectivity patterns suggests largely symmetric (left-right) patterns (Figs. 2 and 4a). Formal comparison of the aggregate connectivity within each of the hemispheres confirms that there were indeed no statistically significant left-right asymmetries in either of the groups or sleep states (false discovery rate (FDR)-corrected pFDR > 0.2 for all four cases, paired two-tailed Wilcoxon signed-rank test; Supplementary Fig. 3). Comparable results hold true for the low beta band (Supplementary Fig. 2a).

### Sleep-related brain dynamics in ex-preterms preempt outcomes

We next asked if these group-by-sleep cortical networks carry prognostic information for key developmental outcomes in the EP group. Because of the central role of the occipital cortex in the sleep state differences (Fig. 2a) and the differentiation of HC from EP (Fig. 4a and Supplementary Fig. 2a), we tested whether changes in connectivity across sleep states are linearly associated with visual performance of EP at two years of age (online Methods). To benchmark this against more complex brain functions, we also tested the changes in connectivity against a standard measure of social-emotional performance at two years. These analyses reveal strong and significant (FDR-corrected) negative correlations (partial two-tailed Pearson test controlling for age at EEG recording) between sleep-induced changes in alpha (Fig. 4b; R = −0.514, p = 0.003, 95% confidence interval (CI) [−0.732, −0.202]) and beta (Supplementary Fig. 2b; R = −0.461, p = 0.009, 95% CI [−0.698, −0.134]) connectivity in posterior networks (comprising mostly occipital cortices) with visual performance. No significant associations between sleep-induced connectivity changes and visual behavior are detected for low-frequency bands. A negative correlation is also apparent for social-emotional performance, but the association is not statistically significant when accounting for multiple comparisons (all puncorrected > 0.039; Fig. 4b and Supplementary Fig. 2b). The regressions against socio-emotional indices and functional networks are not significant in any frequency band, nor are those for visual outcome and functional networks in the theta band (Supplementary Fig. 2b). These findings support the conjecture that preterm birth has a major effect on the development of large-scale neural networks underpinning behavior6, including later neurocognitive performance that relies on visual function19.

### Cortical eigenmodes shape early brain activity

It has been recently shown that large-scale patterns of neural activity can be modeled with cortical eigenmodes11,13,20. That is, that the geometry of the cortex constrains the decomposition of neural activity into spatiotemporal modes—analogous to the harmonics of a musical instrument—whose linear superposition forms the basis of whole-brain functional connectivity patterns. The dynamic activity that each spatial mode carries can be modeled through the application of neural field theory14,21. The functional networks that differ between AS and QS have the appearance of a long-wavelength anteroposterior pattern of activity that changes between the two sleep states (Fig. 2a). To assess if this broad anteroposterior pattern of cortical connectivity emerges from changes in the activity of key low-dimensional modes, we applied an eigenmode decomposition to a neonatal cortical mesh (see Methods). Similar to the eigenmodes calculated on the adult cortex11, these modes possess either a symmetric or antisymmetric appearance, explaining increasingly complex patterns of activity as the mode order increases (Fig. 5a and Supplementary Fig. 4). The first mode (m1) is uniform and global, whereas the second mode (m2) is a left-right pattern reflecting the symmetry of the brain’s hemispheres. The next two modes (m3 and m4) capture long-range anteroposterior activity patterns, which mirror the observed changes in cortical activity as a function of sleep state (Fig. 2a).

The dominant anteroposterior orientation and lack of fine spatial structure of the connectivity contrasts depicted in Figs. 2 and 4 suggests an explanation in terms of the relative weights of these four low order modes. This is plausible because low order modes are the most excitable, and hence carry the most energy22. To test the hypothesis that changes in sleep-related connectivity reflect changes in the weights of the eigenmodes, we developed a parsimonious neural field model of cortical activity in which simulated neural activity, Y, is a linear superposition of spatial eigenmodes, each of which supports fast carrier oscillations that fluctuate in amplitude on slower time scales. The functional connectivity of these oscillations across the cortex emerges from the spatial character of each corresponding mode. To achieve this, we express Y as,

$$Y = \mathop {\sum }\limits_j a_jm_j(x,y,z)\,{\mathrm{cos}}\,(\nu t)\,{\mathrm{cos}}\,(\omega t) + \sigma \eta (x,y,z,t),$$
(1)

where Y = Y(x, y, z, t) is the modeled spatiotemporal cortical activity at time t and position (x, y, z), aj are the mode weights, mj are the mode spatial patterns, $$\nu$$ is the (low) frequency of amplitude modulation, ω is the faster carrier frequency, η is spatiotemporal Gaussian white noise, and σ is its standard deviation (Fig. 5a). To model the effects in the alpha band, we set ω to 10 Hz, with amplitude modulated on the slow time scale of ~10 s ($$\nu$$ = 0.1 Hz). To match the sampling of the empirical data, cortical dynamics were simulated for 300 s at 100 Hz sampling frequency. We then determined the subject-specific values of σ and the aj mode weights such that the modeled activity best matches the empirical data. The best fitting parameters minimize differences in the first two moments of the empirical vs. the modeled distributions of functional connectivity values. Connectivity only depends on the relative sizes of the oscillations and the noise; the overall scaling of the model dynamics is set to match the standard deviation (SD) of the empirical time series. As there are no left-right asymmetries in our functional networks (Supplementary Fig. 3), we further simplified the model by setting the second mode to zero. We therefore focus on bilaterally symmetric modes, estimating the coefficients for the first (m1), third (m3), fourth (m4) modes (Fig. 5a). The face validity of our model is highlighted by a strong overlap between individual empirical and simulated patterns of cortico-cortical connectivity (Fig. 5b and Supplementary Fig. 5). These results indicate that the observed functional connectivity patterns across sleep states (Fig. 2) arise from the distribution of energy in oscillatory spatiotemporal modes and their ratio to unstructured noise.

### Sleep states reflect redistribution of energy in brain modes

Fitting the mode parameters to our empirical data reveals that sleep state transitions predominantly reflect a change in the weighting of the first mode with significant but weaker changes in the coefficients of the third and fourth modes (Fig. 5c). Compared to active sleep, quiet sleep is defined by higher energy in the first uniform mode (F1,92 = 1.6 × 102, p = 8.2 × 10−22, effect size η2 = 0.615, mixed ANOVA) and slightly lower energy in the two anteroposterior modes (mode three F1,92 = 7.98, p = 0.006, η2 = 0.08; mode four F1,92 = 5.51, p = 0.021, η2 = 0.057; mixed ANOVA; Fig. 5c). Hence, the model indicates a shift in the weighting of uniform to more complex antero-posterior activity across sleep states, and a re-weighting of these broadly coherent oscillations relative to the noise.

We found a significant sleep-by-group interaction confined purely to the first mode (mixed ANOVA, F1,92 = 7.76, p = 0.006, η2 = 0.03; Fig. 5c). This result suggests that dynamics in the first, uniform mode underpins the different patterns of high order-connectivity observed in HC vs. EP as a function of sleep states. That is, a decrease of energy in this mode reveals relatively greater energy in the higher order modes—and more so in HC than in EP—and hence the correspondingly more complex spatiotemporal patterns evident in Fig. 2. Importantly, this reduction in global cortical energy in EP is also reflected in spectral power changes across sleep states in empirical data (Supplementary Fig. 6).

## Discussion

We combined connectivity analyses of high-density EEG with biophysical modeling to study cortical dynamics associated with sleep states in pre-term and full-term infants. We found that sleep-related differences in connectivity patterns between infants born at term vs. those born extremely preterm correlate with later neurodevelopmental outcomes. Crucially, biophysical modeling pinpointed a specific deficit in large-scale neuronal dynamics underpinning these sleep-induced connectivity differences. Understanding the neural basis of infant sleep is an important endeavor for basic and translational neuroscience. Our work establishes several new systems-level insights in this endeavor, and holds potential for clinical translation.

We found that infant AS vs. QS states are characterized by broadband differences in cortical connectivity. AS is associated with increased functional connectivity in visual occipital regions whereas QS involves a broader increase in long-range cortical connectivity. These modulations of whole-cortex connectivity across different sleep states support the proposal that sleep-dependent patterns of brain activity play a key role in promoting the development of large-scale cortical networks that underlie a broad range of behaviors4,23. As a specific example of this, we find that changes in occipital connectivity between AS and QS are affected by extremely preterm birth, and the magnitude of this alteration correlates with visual behavioral outcomes at two years of age. That is, alterations in functional connectivity in visual cortex due to preterm birth may impact on the maturation of functional assemblies that are important for emerging visual function. Although we find enhanced correlations in lower frequencies during QS, as might be expected for activity during deep sleep24, these do not correlate significantly with neurobehavioral outcomes. Rather, our findings support the notion that synchronization within the visual system at alpha and beta frequencies is key for the development of perceptual function25,26. The correlation of occipital functional connectivity with social-emotional development at two years is not statistically significant, suggesting some specificity to these functional-developmental couplings. Intriguingly, the correlation between sleep-state related functional network changes in the EP infants and visual function at 2 years (Fig. 4b) is in the opposite direction to the between-group effect with the HC infants. Without experimental control over the exposure of EP infants to normal sensory experience in the neonatal intensive care unit, it is not possible to infer whether these effects are driven by preterm birth or the additional post-natal exposures of EP infants. Future work, incorporating longer-term (life-long) outcomes and other functional metrics could also help disambiguate those preterm-related functional network deficits that persist from those that can be modulated by environmental influences during childhood (e.g., inter-personal relationships, health interventions, education).

Computational modeling suggests that a redistribution of energy in low-dimensional modes of spatiotemporal neural activity supports the emergence of these sleep-related patterns of cortical functional connectivity. Specifically, the broad increase in cortical connectivity associated with QS emerges from an increase of oscillatory energy in a globally uniform mode. The regional increase in occipital connectivity in AS reflects a relatively greater proportion of energy in higher-order oscillatory modes following a reduction in the first mode. Of note, changes in the corresponding more complex antero-posterior modes—whose geometry mirrors the topology of the functional connectivity changes—are small in comparison. The use of computational modeling thus highlights changes in large-scale cortical activity that are not immediately obvious from inspection of functional connectivity changes viewed in isolation, but instead speak to more subtle changes in whole brain oscillatory and stochastic activity.

Scalp EEG is most sensitive to source activity in adjacent cortex. While sleep-related cortical reconfigurations are likely driven by subcortical structures27, our findings permit a non-invasive characterization of the cortical manifestations of these subcortical modulations. Moreover, these results relate whole-cortex dynamics to functional brain development. The physiological mechanism for the transitions themselves—i.e., the mechanisms for the dynamics of the mode weights—could in principle be described using models of the ascending arousal system28,29. These ascending neuromodulatory systems modulate neuronal gain, increasing neuronal signal to noise in a manner that supports abrupt phase transitions30. The present findings thus provide a foundation for the development of a unified theory of neural dynamics supporting sleep. The integration of invasive and non-invasive neurophysiological recordings in preclinical models of preterm birth could play an important role in overcoming the inevitable limitation of scalp-only EEG in human clinical data. Likewise, our eigenmode decomposition is currently derived from cortical geometry only: The incorporation of structural connectomes of infant brain into the decomposition13 could further refine and individualize the functional modes used for our biophysical model. Such work requires further advances in the accurate reconstruction of individual tractography data from neonates. Given that geometry contributes substantially to connectivity31, which shapes large-scale dynamics32, it is likely that such resulting modes would differ only slightly from those presently used.

Findings from the current work have substantial translational potential, opening a novel diagnostic and prognostic window into brain monitoring and prediction of developmental outcomes in extremely preterm infants33. This information could be harnessed to develop new bed-side tests needed for benchmarking interventions to improve the neurocognitive outcomes of preterm infants, the globally biggest neonatal risk factor34. More broadly, our framework shows how the reconfiguration of functional networks across brain states can be recast as changes in large-scale spatiotemporal modes of activity. Together with the rapid penetration of brain network theory across clinical disorders35, this suggests a fundamental role for neural field theory in translational neuroscience.

## Methods

### Ethics

The study design and procedures have been approved by the Ethics Committee of the Helsinki University Central Hospital (Finland). Informed written consent was received from a guardian before inclusion of an infant into the study.

### Subjects

Data were acquired at the Helsinki University Central Hospital (Finland). Some data related to infants included in this cohort have been used for previous independent publications4,7,36,37. Multi-channel EEG data were collected from two cohorts of infants: extremely preterm (EP, N = 46) and full-term healthy controls (HC, N = 67).

### EEG data

EEG data were recorded with a NicOne EEG amplifier (Cardinal Healthcare/Natus, USA) or a Cognitrace amplifier (ANT B.V., Enschede, The Netherlands). EEG caps (sintered Ag/AgCl electrodes; Waveguard, ANT-Neuro, Germany) had 19 or 28 scalp electrodes positioned according to the International 10–20 standard. The same 19 EEG channels (Fp1, Fp2, F7, F3, Fz, F4, F8, T7, C3, Cz, C4, T8, P7, P3, Pz, P4, P8, O1, O2) across all recordings were selected for analyses. Further details regarding EEG acquisition in newborns can be found elsewhere38.

### Sleep state selection

Each recording session continued until the infant had undergone two sleep (vigilance) states: active sleep (AS) and quiet sleep (QS). Vigilance state assessment was performed in our data through a combination of electrophysiological and behavioral measures. Polygraphic channels (chin electromyogram, electrocardiogram, electrooculogram, and respiratory sensors) were used for this purpose. EEG traces during AS exhibit continuous fluctuations, respiration is irregular, and occasional eye movements are present. Conversely, EEG during QS is characteristically discontinuous, and respiration is regular39 (Fig. 1a). Next, we selected 5-min-long artifact-free EEG epochs from the most representative periods of AS and QS. The selection of the epoch length was based on our previous studies demonstrating that a continuous 5 minute recording duration provides a reliable and stable estimate of the whole-brain functional network activity in infants4,7. To avoid transitions between vigilance states, representative sleep epochs were selected from within well-established patterns of corresponding behavior and brain activity, and not close to the transition between QS and AS. Subjects that lacked sufficient epoch lengths or had poor quality data (due to movement artifacts or loss of contact) were excluded from further analysis. The final sample included 42 neonates in the EP group and 52 neonates in the HC.

### Neurodevelopmental assessment

The neurodevelopment of infants was assessed at two years of corrected age using the structured Griffiths Mental Developmental Scales40. These scales were chosen because they measure the cognitively demanding domains of visual performance and social-emotional performance. The assessment of visual and social-emotional performance in EP was chosen because of their established, widespread clinical use plus their broad impact on lifelong neurocognitive performance and quality of life, hence improving the translational potential of our study41,42. While other outcomes such as gross motor development and hearing are also affected in some infants, studies in ex-preterm infants have shown that these are the most likely to be modified by a host of individual and treatment interventions43.

### EEG data pre-processing

EEG data were initially band-pass filtered into the 0.15–45 Hz frequency band, down-sampled to a sampling rate of 100 Hz and re-referenced to the common average reference montage. For further analysis we filtered the data into four frequency bands of interest: 0.4–1.5 Hz (delta, δ), 4–8 Hz (theta, θ), 8–13 Hz (alpha, α), and 13–22 Hz (low beta, β). For band-pass filtering, we applied in series a combination of low-pass and high-pass Butterworth filters with the corresponding cut-off frequencies and stop-band attenuation of 20 dB. Each filter was applied in both forward and backward directions to avoid introducing phase lags into the EEG signals. An overview of the entire analytic pipeline is shown in Supplementary Fig. 9.

### Source reconstruction

A realistic infant head model was used to compute cortical source signals from multi-channel EEG4. Briefly, we generated scalp, skull, and intracranial volume shells (2562 equidistant vertices in each) from segmented anatomical magnetic resonance imaging data of a healthy full-term infant. In the source space, we used a cortical template scaled to the infant size and spatially smoothed to match the brain folding at term-equivalent age. The source space comprised 8014 electrical dipoles (of fixed orientation orthogonal to the surface) approximating the local neuronal activity. Tissue conductivities for intracranial volume, skull, and scalp were taken as 1.79 S/m, 0.2 S/m, and 0.43 S/m, respectively44,45. The head model included 19 scalp EEG electrodes placed according to the empirical recordings. The forward solution (i.e., the operator that estimates the contribution of cortical sources to scalp EEG) was computed using a symmetric boundary element method implemented in the openMEEG package46. For the noise covariance matrix, we used the identity matrix, which assumes equal noise levels in EEG sensors. To compute cortical sources from EEG (the inverse solution), we applied dynamic statistical parametric mapping47 as implemented in the Brainstorm software package48. The 8014 cortical sources were clustered into 58 parcels including 125 ± 26 (mean ± SD) sources each (Fig. 1b). The adopted brain parcellation scheme was symmetric across hemispheres (29 parcels in each hemisphere). The activity of each parcel was taken as the weighted mean activity of all sources within it4.

### Connectivity analysis

Pairwise functional interactions between cortical parcels were estimated by computing Pearson correlation coefficients between amplitude envelopes of the corresponding cortical signals. Pairs of signals were first orthogonalized relative to each other in 2 s non-overlapping time windows17. This procedure was performed in both directions: signal X was orthogonalized relative to signal Y, and signal Y was orthogonalized relative to X. Two output correlation coefficients were then averaged and used as the functional interaction estimate between X and Y. The correlation coefficients between all possible parcel pairs led to a full adjacency matrix having (58 × 57)/2 = 1653 connections. The resulting matrices were corrected by excluding connections which cannot be reliably estimated using 19 recording EEG electrodes. To define such connections, we simulated artificial parcel activity, where pairs of specific regions were synchronized at one time, and that was used to compute synthetic EEG and to reconstruct back parcel signals. We repeated this in 500 iterations for each pair of parcels. Finally, we contrasted pairwise interactions of reconstructed parcels that were initially in synchrony to the set of surrogate values from all non-synchronous parcels from all iterations. This allowed us to generate a statistically based binary template where all interactions below the 99th percentile of surrogates were rejected (zeros) as non-reliable and all others (ones) were used for further analysis (for more details see elsewhere4). Here, we excluded 32% of interactions from all adjacency matrices (the same non-reliable interactions in all subjects), leaving 1128 connections in each for further analysis. Finally, to reduce inter-individual differences in total connectivity across subjects, each adjacency matrix was normalized by dividing each weight by the global sum of the magnitude of the weights in the matrix.

### Network-based statistics

The network-based statistic (NBS)18 was used to estimate group and sleep main effects, as well as group-by-sleep interactions. Analyses were carried out independently for each frequency band of interest. A search t-statistic threshold of 3 (equivalent to p < 0.001, uncorrected) was initially applied to all 1128 possible pairwise connections between cortical parcels. The size of surviving sets of pairwise connections was recorded. Permutation testing was used to estimate a corrected p-value for each pattern of connections (p < 0.05, FWER-corrected at the level of the whole pattern). In each permutation, the assignment of data to a given group or sleep state was randomized. For each of the 5000 permutations the size of the largest network was recorded, allowing for the generation of a null distribution and FWER correction.

### Assessment of effect size in the frequency domain

To study the spectral fingerprints of sleep and gestational age effects, pre-processed EEG data were filtered into 21 frequency bands covering the range 0.42–22 Hz. Central frequencies (f) of these filters were spaced such that the (k + 1)-th frequency was a fixed multiple of the k-th frequency given by fk+1 = 1.2 fk. The slowest carrier frequency f1 was set to 0.5 Hz and the ensuing highest frequency was f21 = 19.2 Hz. Cut-off frequencies were 0.85f and 1.15f, and stop-band frequencies were 0.5f and 1.5f. Band-pass filtering was implemented by applying pairs of low-pass and high-pass Butterworth filters. This yielded carrier oscillations with overlapping frequency bands of equal width on a logarithmic scale. The amplitudes of these carrier frequencies were then band-pass filtered into 15 frequency bands to estimate amplitude fluctuations at different time scales49. Amplitude filters were designed according to the same principles as for the carrier frequencies. The slowest amplitude frequency f1 was set to 0.015 Hz and the ensuing highest frequency was f15 = 0.19 Hz. This double filter bank led to 21 × 15 = 315 functional connectivity matrices per subject at each sleep state. Connectivity analyses of the main effects of group and sleep state were estimated using NBS with a search (height) t-statistic threshold of 2.5. Frequency combinations (carrier and amplitude) for which there were no suprathreshold edges were denoted null (Figs. 2c, 3c). Effect size was computed as Cohen's d (i.e., NBS-derived mean t-statistic divided by the square root of the total degrees of freedom).

### Brain-behavior correlations

For each sleep state (AS and QS) and EP infants with follow-up neurocognitive examination (N = 32), we computed the mean connectivity strength in cortical patterns showing a significant group-by-sleep interaction (Fig. 4a and Supplementary Fig. 2a). We then calculated subject-specific differences in mean connectivity (AS minus QS). Finally, we computed correlation coefficients (two-tailed Pearson’s correlation) indexing the linear association between changes in functional connectivity and scores of visual and social-emotional performance at two years of age. We used partial correlation analyses (partialcorr function in Matlab), controlling for conceptional age of infants at the time of EEG recording to account for neurodevelopment7. Resulting p-values from the six brain-behavior tests (two behavioral measures and three frequency bands) were corrected using a standard Benjamini-Hochberg false discovery rate (FDR) correction procedure.

### Eigenmode decomposition and computational model

Application of neural field theory has recently shown that large-scale resting-state activity patterns are well described by eigenmodes of the cortical surface11. Details regarding neural field theory14,22 and solutions of neural field equations on cortical surfaces to obtain eigenmodes20 have been extensively presented elsewhere11. Here, we briefly describe the specific aspects of neural field theory employed, and the practical steps to calculate eigenmodes.

Neural field theory is a continuum description of neural activity, built upon standard (mean-field) approaches from physics and used to average over the microscopic details of individual neurons50. This enables modeling of the dynamics of large populations of neurons, where the relevant physiological quantities are average membrane potentials and firing rates over local neural populations. For comparison with EEG, the main quantity of interest is the excitatory cortico-cortical activity field ϕee, representing the mean incoming spike rates at pyramidal neurons. We are interested in spatiotemporal dynamics and hence use a formulation in terms of partial differential equations, which retain spatial information across the cortical sheet modeled here as a two-dimensional surface. Importantly, the general form of the model adopted here can be used to accurately predict resting-state EEG spectra14,51, including sleep52. It has been shown that, in the absence of external inputs, resting-state cortical activity can be written as a sum of eigenmodes

$$\phi _{ee} = \mathop {\sum }\limits_j m_j({\boldsymbol{r}})e^{ - i\omega _jt},$$
(2)

each with eigenfrequency ωj, and amplitude mj(r) at position r satisfying the Helmholtz equation,

$$\nabla ^2m_j({\boldsymbol{r}}) = - k_j^2m_j({\boldsymbol{r}}),$$
(3)

where $$k_j^2$$ is the eigenvalue and 2 is the Laplace-Beltrami operator on the cortical surface. We used a cortical surface53 scaled to the infant size and spatially smoothed to match the brain folding at term-equivalent age. We solve Equation (3) for the eigenmodes using a finite element method implemented in Matlab54,55,56. The first six eigenmodes are shown in Supplementary Fig. 4.

The existence of amplitude-amplitude correlations in the data requires the oscillatory amplitudes to vary in time at each point. We assume the simplest form of oscillatory amplitude modulation in Equation (1) of the main text, where each mode is modulated with a (cosine) oscillation at a fixed low frequency $$\nu$$ (here, 0.1 Hz). This is equivalent to a beat, a linear superposition of two nearby frequencies. Thus, each amplitude-modulated mode mjcos($$\nu$$)cos(ωt) in our final signal Y is a sum of a pair of pure oscillatory modes $$\frac{1}{2}m_j\,{\mathrm{cos}}\,[(\omega - \nu )t] + \frac{1}{2}m_j\,{\mathrm{cos}}\,[(\omega + \nu )t]$$, where we have also assumed that each amplitude-modulated mode has the same carrier frequency ω (here, 10 Hz for alpha band oscillations).

We restricted our attention to the first four modes j = 1–4 (Supplementary Fig. 4) on the basis that the dominant feature in Fig. 2a is a clear anteroposterior pattern, with an absence of the finer spatial structure captured by modes j ≥ 5. Moreover, as detailed in the main text, formal testing for a left-right asymmetry in the functional connectivity patterns showed no significant effect (Supplementary Fig. 3). Hence, we further simplified the model by setting mode weight a2 = 0. This leaves four parameters in Equation (1) that need to be estimated from the data: a1, a3, a4, and σ.

We fitted the model using a two-step procedure. First, because the correlation dynamics only depend on the ratios of the aj to the noise amplitude σ, without loss of generality we fixed σ = 1 and calculated the model functional connectivity over a 3-D grid of values of $$a{\prime}_{\!\!1}$$, $$a{\prime}_{\!\!3}$$, and $$a{\prime}_{\!\!4}$$, where the primes denote the σ = 1 case. The gridded values of $$a{\prime}_{\!\!j}$$ spanned the range 0-0.5 (steps of 0.0025). This range was selected based on initial testing that showed that the best fits fall in the range $$0 < a{\prime}_{\!\!j} < 0.5$$ (Supplementary Fig. 7). We summarized each pattern of functional connectivity in the parameter space by calculating the first two moments (mean and SD) of the distribution of edge weights. Then, for each subject, we calculated the same summary statistics and found the corresponding parameters that minimized the cost function,

$$J(a{\prime}_{\!\!1} ,a\prime_{\!\!3} ,a\prime_{\!\!4} ) = [\mu _{{\mathrm{data}}} - \mu _{{\mathrm{model}}}(a\prime_{\!\!1} ,a\prime_{\!\!3} ,a\prime_{\!\!4} )]^2 + [\sigma _{{\mathrm{data}}} - \sigma _{{\mathrm{model}}}(a\prime_{\!\!1} ,a\prime_{\!\!3} ,a\prime_{\!\!4} )]^2,$$
(4)

where μdata and σdata are the mean and standard deviation of the functional connectivity weights across the empirical network, and μmodel and σmodel for the model. We finally estimated the subject-specific σ and $$a_j = \sigma a{\prime}_{\!\!j}$$ by finding the value of σ such that Y(t) and the empirical time series have the same standard deviation across time.

Data from all (N = 42) preterm infants were used in these analyses. For display purposes, data from one outlier was omitted from Fig. 5c (see Supplementary Fig. 8 for all data points). This data point was included in all formal analyses.

### Statistical analysis

Pairwise group comparisons for connectivity asymmetries (Supplementary Fig. 3) were performed using two-tailed signed-rank Wilcoxon tests. This non-parametric approach was adopted because data did not follow a normal distribution (Shapiro-Wilk test). P-values were FDR corrected with Benjamini-Hochberg procedure.

To test for significant differences in mode weights (Fig. 5c) and global amplitudes (Supplementary Fig. 6), we used a mixed analysis of variance (ANOVA).

### Analysis software

MRI segmentation was performed using FSL57 and the head model was computed using the openMEEG package46. Source reconstruction was done using algorithms implemented in Brainstorm48. Brain networks were visualized using the Matlab toolbox BrainNet Viewer58. Statistical analyses were performed using standard and custom Matlab functions and JASP software (https://jasp-stats.org).

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## Data availability

Data will be available upon reasonable request. The source data underlying Figs. 4, 5 and Supplementary Figs. 2, 3, and 6 are provided as a Source Data file.

## Code availability

Custom Matlab code implementing connectivity analysis, the biophysical model, and its fitting to empirical data can be found here: https://github.com/babyEEG/Infant-Sleep.

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## Acknowledgements

The authors thank Dr. Aulikki Lano and Dr. Mari Videman for assessing neurodevelopment of the infants, and A.J.K. Phillips for helpful comments. A.T. was supported by Finnish Cultural Foundation (Suomen Kulttuurirahasto; 00161034). A.T. and S.V. were also funded by Academy of Finland (276523 and 288220) and Sigrid Jusélius Foundation (Sigrid Juséliuksen Säätiö), as well as Finnish Pediatric Foundation (Lastentautien tutkimussäätiö). J.A.R., A.Z., M.B. and L.C. are supported by the Australian National Health and Medical Research Council (J.A.R. 1144936 and 1145168, A.Z. 1136649, M.B. 1037196, L.C. 1099082 and 1138711). This work was also supported by the Rebecca L. Cooper Foundation (J.A.R, PG2018109) and the Australian Research Council Centre of Excellence for Integrative Brain Function (M.B., CE140100007).

## Author information

A.T., J.A.R., S.V., M.B., L.C. designed the research; A.T., J.A.R., L.C. performed the analyses; A.Z. and X.Z. contributed analysis tools; S.V. organized data collection; and all authors wrote the paper.

Correspondence to Anton Tokariev or Luca Cocchi.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Journal peer review information: Nature Communications thanks Gustavo Deco and Xi Long for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


text:
# zbMATH — the first resource for mathematics

Pointwise multiplication of Besov and Triebel-Lizorkin spaces. (English) Zbl 0839.46026
Summary: It is shown that para-multiplication applies to a certain product $$\pi(u, v)$$ defined for appropriate $$u$$ and $$v$$ in $${\mathcal S}'(\mathbb{R}^n)$$. Boundedness of $$\pi(\cdot, \cdot)$$ is investigated for the anisotropic Besov and Triebel-Lizorkin spaces – i.e., for $$B^{M, s}_{p, q}$$ and $$F^{M, s}_{p, q}$$ with $$s\in \mathbb{R}$$ and $$p$$ and $$q$$ in $$]0, \infty]$$ (though $$p< \infty$$ in the $$F$$-case) – with a treatment of the generic as well as of various borderline cases.
For $$\max(s_0, s_1)> 0$$ the spaces $$B^{M, s_0}_{p_0, q_0}\oplus B^{M, s_1}_{p_1, q_1}$$ and $$F^{M, s_0}_{p_0, q_0}\oplus F^{M, s_1}_{p_1, q_1}$$ to which $$\pi(\cdot, \cdot)$$ applies are determined. For generic $$F^{s_0}_{p_0, q_0}\oplus F^{s_1}_{p_1, q_1}$$ the receiving $$F^s_{p, q}$$ spaces are characterized.
It is proved that $$\pi(f, g)= f\cdot g$$ holds for functions $$f$$ and $$g$$ when $$f\cdot g\in L_{1,\text{loc}}$$ roughly speaking. In addition, $$\pi(f, u)= fu$$ when $$f\in {\mathcal O}_M$$ and $$u\in {\mathcal S}'$$.
Moreover, for an arbitrary open set $$\Omega\subset \mathbb{R}^n$$, a product $$\pi_\Omega(\cdot, \cdot)$$ is defined by lifting to $$\mathbb{R}^n$$. Boundedness of $$\pi$$ on $$\mathbb{R}^n$$ is shown to carry over to $$\pi_\Omega$$ is general.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46F05 Topological linear spaces of test functions, distributions and ultradistributions
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##### References:
 [1] Multiplication in Sobolev and Besov Spaces, Nonlinear Analysis (Pisa), Scuola Normalc Supcriore, Pisa, 1991, pp. 27–50 · Zbl 0894.46026 [2] Bony, Ann. scient. Éc. Norm. Sup. 14 pp 209– (1981) [3] Colombeau, Comm. Part. Diff. Eq. 15 pp 905– (1990) [4] Franke, Math. Nachr. 125 pp 29– (1986) [5] Grubb, Diff. and Integr. Eg. [6] Grubb, Math. Scand. 69 pp 217– (1991) · Zbl 0766.35034 · doi:10.7146/math.scand.a-12380 [7] Hanouzet, Comm. Part. Diff. Eq. 10 pp 433– (1985) [8] Jawerth, Math. Scand. 40 pp 94– (1977) · Zbl 0358.46023 · doi:10.7146/math.scand.a-11678 [9] The Stationary Navier-Stokes Equations in Lp-Related Spaces, Ph. D. thesis, University of Copenhagen, Denmark, 1993, Ph. D.-series 1 [10] Kaminski, Studia Math. 74 pp 83– (1982) [11] Knuth, Amer. Math Monthly 99 pp 403– (1992) [12] and , Theory of Multipliers in Spaces of Differentiate Functions, Monographs and Studies in Mathematies, Vol. 23, Pitman, London, 1985 [13] Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics Series, Vol. 259, Longman Scientific & Technical, Harlow, UK, 1992 [14] Foundations af Non-linear Global Analysis, Benjamin, New York, 1968 [15] New Thoughts on Besov Spaces, Duke Univ. Math. Series, Durham, 1976 [16] On Pointwise Multipliers in Besov-Triebel-Lizorkin Spaces, Seminar Analysis of the Karl Weierstraß-Institute 1985/86 (Leipzig) [17] ( and ed.) Teubner-Texte zur Mathematik, Vol. 96, Teubner Verlagsgesellschaft, 1987, pp. 45–103 [18] Sickel, Forum Math. 5 pp 73– (1991) [19] and , Hölder Inequalities and Sharp Imbcddings in Function Spaces of Bsp,q and FsP,q type, (preprint) [20] Strichartz, J. Math. Mechanics 16 pp 1031– (1967) [21] Triebel, Ann. Mat. Pura Appl. 113 pp 33– (1977) [22] Spaces of Besov-Hardy-Sobolev Type, Teubner-Texte zur Mathematik, Vol. 15, Teubner Verlagsgesellschaft, Leipzig, 1978 [23] Theory of Function Spaces, Monographs in Mathematics, Vol. 78, Birkhauser Verlag, Basel, 1983 · Zbl 1235.46002 · doi:10.1007/978-3-0346-0416-1 [24] Theory of Function Spaces II, Monographs in Mathematics, Vol. 84, Birkhauser Verlag, Basel, 1992 · Zbl 1235.46003 · doi:10.1007/978-3-0346-0419-2 [25] Yamazaki, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 33 pp 131– (1986) [26] Yamazaki, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 33 pp 311– (1986) [27] Zolesio, Proc. R. S. E. (A) 78 pp 113– (1977) · Zbl 0382.46020 · doi:10.1017/S0308210500009872
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# Probability of duplicate free sample of iid discrete random sample

Let $\{X_1,\ldots,X_n\}$ be independent identically distributed discrete random variables. I am interested in computing the probability of the event that the sample is duplicate free: $$\mathbb{P}\left( \bigcap_{i<j} \{ X_i \not= X_j\}\right)$$ in terms of $p_2 = \mathbb{P}\left(X_1 = X_2\right)$, $p_3=\mathbb{P}\left(X_1 = X_2 = X_3\right)$, ..., $p_n = \mathbb{P}\left(X_1 = X_2 = X_3=\ldots=X_n\right)$.

Special case
If $X_k$ are uniformly distributed with the size of the sample space being $d$, this is a classic birthday problem with the answer: $$\mathbb{P}\left( \bigcap_{i<j} \{ X_i \not= X_j\}\right) = \frac{n!}{d^n} \binom{d}{n} = \sum_{k=1}^n \frac{s(n,k)}{d^{n-k}}$$ where $s(n,k)$ denotes the Stirling number of the first kind.

Motivation
Consider IEEE floating point number with mantissa $m$ encoded as $d$-tuple of significant binary digits (i.e. the first bit is always 1), and integer binary exponent $e$. For a random real $0<x<1$, bits are iid Bernoulli(1/2) random variables, and $-e$ is a Geometric(1/2) random variable. I am interested in computing the probability of the size $n$ sample having a duplicate.

My approach
Applying inclusion-exclusion principle, the complementary probability is $$\sum_{i<j} \mathbb{P}\left(X_i = X_j\right) - \sum_{i<j,i<p<q} \mathbb{P}\left(X_i = X_j, X_p=X_q\right)+\ldots = \\ \binom{n}{2} p_2 - 3 \binom{n}{4} p_2^2 - 2 \binom{n}{3} p_3 + \ldots$$

Solutions, ideas, references are welcome.

-

It uses quite a neat argument involving generating functions. First, note that $$r_n := \mathbb{P}\left(\bigcap_{1\leqslant i<j \leqslant n} \{X_i \not= X_j\}\right) = n! \sum_{1 \leqslant i_1 < i_2 < \ldots < i_{n-1} < i_n \leqslant n} \pi_{i_1} \pi_{i_2} \cdots \pi_{i_{n}} = n! [t^n] \prod_{k \geqslant 1} \left(1 + \pi_k t\right)$$ where $\pi_k = \mathbb{P}(X=k)$. Therefore $$1 + \sum_{n=1}^\infty \frac{r_n}{n!} t^n = \prod_{k \geqslant 1} \left(1 + \pi_k t\right) = \exp\left( \sum_{k \geqslant 1} \log\left(1+\pi_k t\right)\right) = \exp\left( \sum_{s=1}^\infty \frac{(-1)^{s-1}}{s} t^s \sum_{k \geqslant 1} \pi_k^s\right) = \exp\left( 1 + \sum_{s=2}^\infty \frac{(-1)^{s-1}}{s} t^s p_s\right)$$ This gives the expression for $r_n$ as the complete Bell polynomial: $$r_n = B_n\left(1, -p_2, \ldots, (-1)^{n-1} (n-1)! p_n\right) \tag{1}$$ In particular, assuming $r_0 =1$ the above result implies the recurrence equation for the non-occurrence probabilities $r_n$: $$r_n = \sum_{k=1}^n (-1)^{k-1} \frac{(n-1)!}{(n-k)!} p_k r_{n-k} \tag{2}$$
Large $n$ asymptotics (with $n \cdot \max\limits_{k \geqslant 1} \pi_k$ being small) is also worked out in the article: $$r_n \approx \exp\left(-\binom{n}{2} p_2 - \binom{n}{3}\left( 3 p_2^2 - 2 p_3 \right) + \cdots \right)$$


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Math Help - sin cos trig integral

1. sin cos trig integral

sin^3xcos^3x dx Kinda stuck because I took the integral both ways but get different answers.


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# How do I graph a vector [cos x, sin x]?

how do I do the following:

Consider the matrix $\begin{bmatrix} \cos x & -\sin x\\ \sin x & \cos x \end{bmatrix}$ and the vector $\begin{bmatrix} \cos y\\ \sin y \end{bmatrix}$. Show that the product of the $2\times 2$ matrix with $2\times 1$ matrix $\begin{bmatrix} \cos(x+y)\\ \sin(x+y) \end{bmatrix}$ graphically, by sketching.

I can show this using trig identities, but I am not sure how to graph the $2\times 1$ trig vector in first place, let alone explain the rest of the question graphically. All the help very much appreciated!

-

[cos(y),sin(y)] is the unique point on the unit circle with angle $y$ from the origin. Your $2x2$ matrix rotates the plane about the origin by angle $x$, which is why the result is a point on the same circle with angle $x+y$.

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Thank you very much! –  LinAlgStudent Jan 9 '11 at 0:51

Every point $(x,y)$ can be thought of as a $2 \times 1$ matrix.

If $(x,y)$ is a point in the 2D plane, then for a $2\times 2$ matrix $A$, the product of $A$ and $[x,y]^{T}$ gives another 2D point.

In this case, you $A$ is a rotation matrix, and the $2 \times 1$ matrix you have corresponds to a point on the unit circle.

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Thanks a lot, so I am visualizing it now, it makes sense that it's a unit circle. How do I go about showing that the 2x2 matrix rotates all of the points by angle x though? It doesn't seem apparent to me, even after I multiply the matrices out. –  LinAlgStudent Jan 9 '11 at 0:52
@user5157: The point (cos y, sin y) corresponds to an angle of y. By your trigonometry, the new point is (cos (x+y), sin (x+y)), which corresponds to an angle x+y... –  Aryabhata Jan 9 '11 at 0:55
Sorry, I wasn't clear. I am supposed to show that without using trig identities at first, just by sketching. My question is, how do I show it via sketching the unit circle and having that 2x2 matrix, that 2x2 matrix just shifts all of the points by angle x. –  LinAlgStudent Jan 9 '11 at 0:57
@User: Consider what happens to the point (1,0). That is enough to make the claim that all points on the circle get rotated (why?). –  Aryabhata Jan 9 '11 at 1:03


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### Session KG: Relativistic Heavy Ions: Au+Au

2:00 PM–5:15 PM, Thursday, September 22, 2005
Ritz-Carlton Hotel Room: Plantation 2

Chair: Hideki Hamagaki, University of Tokyo

Abstract ID: BAPS.2005.HAW.KG.1

### Abstract: KG.00001 : PHENIX Measurement of Particle Yields at High pT with Respect to Reaction Plane in Au+Au collisions at $\sqrt{s}$ = 200 GeV

2:00 PM–2:15 PM

Preview Abstract MathJax On | Off Abstract

#### Author:

David Winter
(Columbia University)

One of the most intriguing puzzles in RHIC physics is the origin of the azimuthal anisotropy of particle yields at high pT ($>$ 5 GeV/c). Traditional flow and parton energy loss pictures have failed to describe the magnitude of this anisotropy. Measurement of the azimuthal asymmetry v2 at high pT will shed light on the contributions from flow, recombination, and energy loss, as well as the transition from soft to hard production mechanisms. The PHENIX Run4 dataset provides a powerful opportunity for exploring the angular anisotropy of identified particle yields at high pT. Complementing traditional v2 measurements, we present pi0 and eta yields as a function of angle with reaction plane, up to pT of at least 10 GeV/c. The centrality dependence of the angular anisotropy allows us to probe the density and path-length dependence of the energy loss of hard-scattered partons. We will discuss various mechanisms for particle production in this high pT region.

To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2005.HAW.KG.1


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Home How to use Rails link_to to POST to the controller and have an HTML class attribute?
 I've looked up the link_to in the Apidock but simply can't get the proper Rails syntax to create the following: All Changes Complete  I have: <%= link_to t('view.benefits.all_changes_complete'), { controller: "benefits", action: "create", method: :post }, { class: "btn btn-primary"} %>  The above syntax kind of works, but the method: :post is being added to the href and clicking the link does a GET, resulting in the following: All Changes Complete


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# cosx(2sinx + sqrt3)(-sqrt2 cosx + 1) = 0??

Jul 25, 2016

Maybe you meant "how do I solve this equation". By the zero-product property, one or all of these factors can be equal to $0$.

$\cos x \left(2 \sin x + \sqrt{3}\right) \left(- \sqrt{2} \cos x + 1\right) = 0$

• When $\cos x = 0$, $x = \frac{\pi}{2} , \frac{3 \pi}{2} , . . .$, or:

$\textcolor{b l u e}{x = \frac{\pi}{2} \pm n \pi}$

• When $2 \sin x + \sqrt{3} = 0$, $\sin x = - \frac{\sqrt{3}}{2}$. That can occur for ${240}^{\circ}$, ${600}^{\circ}$, ${960}^{\circ}$, etc., and ${300}^{\circ}$, ${660}^{\circ}$, ${1020}^{\circ}$, etc. Or:

$\textcolor{b l u e}{x = \frac{4 \pi}{3} \pm 2 n \pi}$, where $n$ is an integer.
$\textcolor{b l u e}{x = \frac{5 \pi}{3} \pm 2 n \pi}$, where $n$ is an integer.

• When $1 - \sqrt{2} \cos x = 0$, $\cos x = \frac{1}{\sqrt{2}} \cdot \left(\frac{\sqrt{2}}{\sqrt{2}}\right) = \frac{\sqrt{2}}{2}$. This occurs when $x = {45}^{\circ} , {315}^{\circ} , {405}^{\circ} , {675}^{\circ} , . . .$. So $x = \frac{\pi}{4} , \frac{7 \pi}{4} , \frac{9 \pi}{4} , \frac{15 \pi}{4} , . . .$, or:

$\textcolor{b l u e}{x = 2 n \pi \pm \frac{\pi}{4}}$, where $n$ is an integer.

Note that written this way, this already includes both $\frac{\pi}{4}$ and $\frac{7 \pi}{4}$ for $0 < x < 2 \pi$ and other corresponding angles in subsequent cycles. For example:

$n = - 1 \to x = - \frac{9 \pi}{4} , - \frac{7 \pi}{4}$
$n = 0 \to x = - \frac{\pi}{4} , \frac{\pi}{4}$
$n = 1 \to x = \frac{7 \pi}{4} , \frac{9 \pi}{4}$
$n = 2 \to x = \frac{15 \pi}{4} , \frac{17 \pi}{4}$

If I took this at face value, I honestly don't think this is equal to $0$.

http://www.wolframalpha.com/input/?i=Is+cosx%282sinx%2Bsqrt3%29%28+-sqrt2cosx%2B1%29+%3D+0

If we tried, we would get:

cosx(2sinx+sqrt3)( -sqrt2cosx+1) stackrel(?)(=) 0

(2sinxcosx+sqrt3cosx)( -sqrt2cosx+1) stackrel(?)(=) 0

-2sqrt2sinxcos^2x + 2sinxcosx - sqrt6cos^2x + sqrt3cosx stackrel(?)(=) 0

2sinxcosx + sqrt3cosx stackrel(?)(=) 2sqrt2sinxcos^2x + sqrt6cos^2x

(2sinx + sqrt3)cosx stackrel(?)(=) (2sqrt2sinx + sqrt6)cos^2x

cancel((2sinx + sqrt3))cosx stackrel(?)(=) sqrt2cancel((2sinx + sqrt3))cos^2x

cosx stackrel(?)(=) sqrt2cos^2x

1 stackrel(?)(=) sqrt2cosx

$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \ne \cos x$ necessarily. Functions are not always the same as constants.

$\therefore$ with no value for $x$, this is an impossible proof.


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CServer Side ProgrammingProgramming

The linked list representation of a number is provided in such a way that the all nodes of the linked list are treated as one digit of the number. The node stores the number such that the first element of the linked list holds the most significant digit of the number, and the last element of the linked list holds the least significant bit of the number. For example, the number 202345 is represented in the linked list as (2->0->2->3->4->5).

And to add one to this linked list represented number we have to check the value of the least significant bit of the list. If it is less than 9 than it's ok otherwise the code will change the next digit and so on.

Now lets see an example to know how to do it, 1999 is represented as (1-> 9-> 9 -> 9) and adding 1 to it should change it to (2->0->0->0)

Input:1999
Output:2000

## Explanation

To add 1 to a given number represented as a linked list meaning to follow some steps that are,

• reversing the linked list: you need to reverse the linked list that this means changing the last digit to the first and the first to the last. For example, 1-> 9-> 9 -> 9 is converted to 9-> 9 -> 9 ->1.
• for this changed linked list now traverse the list, in the left-most node add one. if this node’s value is equal to 9 then propagate a carry to the next Node. Do the same procedure until the carry is there.
• reverse the string back as in original form and then returned the head to get the string printed.

## Example

#include <iostream>
using namespace std;
//n=next node ; d=data ; p= previous node; h=head node; c=current node
class Node {
public:
int d;
Node* n;
};
Node *newNode(int d) {
Node *new_node = new Node;
new_node->d = d;
new_node->n = NULL;
return new_node;
}
Node *reverse(Node *h) {
Node * p = NULL;
Node * c = h;
Node * n;
while (c != NULL) {
n = c->n;
c->n = p;
p = c;
c = n;
}
return p;
}
Node* res = h;
Node *temp, *p = NULL;
int carry = 1, sum;
while (h != NULL) {
sum = carry + h->d;
carry = (sum >= 10)? 1 : 0;
sum = sum % 10;
h->d = sum;
temp = h;
h = h->n;
}
if (carry > 0)
temp->n = newNode(carry);
return res;
}
h = reverse(h);
return reverse(h);
}
int main() {
Node *h = newNode(1);
h->n = newNode(9);
h->n->n = newNode(9);
h->n->n->n = newNode(9);
}


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## anonymous 3 years ago help! use the quadratic formula to solve x²+3=-4x

1. anonymous

x^2+4x+3 = 0 (x+1) (x+ 3) = 0 then x = -1 or x = -3

2. anonymous

no no use quadratic formula to solve @gerryliyana

3. anonymous

do you know what the quadratic formula is? or are you asking for that.

4. anonymous

i know what it is but i still dont know how 2 get the answer...

5. anonymous

set up equation so that you have: ax^2+bx+c=0 and use a, b and c correctly in your formula

6. anonymous

x²+4x+3=0?

7. anonymous

so a=?,b=?, and c=?

8. anonymous

yess ur all right

9. anonymous

smart boy!!

10. anonymous

a=o b=4 c=3

11. anonymous

nooo

12. anonymous

a = 1, b=4, c=3

13. anonymous

yes

14. anonymous

sorry i wasnt thinking

15. anonymous

so now what?

16. anonymous

use $x = \frac{ -b \pm \sqrt{(b^{2} - 4ac)} }{ 2a }$

17. anonymous

substitute all parameter, including a, b, and c

18. anonymous

to solve x²+4x + 3 =0 you should find the value of x, one way to do this by using formula above ...,

19. anonymous

$x = \frac{ -b \pm \sqrt{b^{2}-4ac} }{ 2a }$ $x = \frac{ -4 \pm \sqrt{4^{2}-(4)(1)(3)} }{ 2(1) }$

20. anonymous

ok

21. anonymous

Thanks both of you!


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# MoL-1999-01: Normal Gentzen Deductions in the Classical Case

MoL-1999-01: Palmigiano, Alessandra (1999) Normal Gentzen Deductions in the Classical Case. [Report]

## Abstract

In this thesis I define the notion of normality for deductions in a Gentzen
system for the classical case; I prove the normalization theorem for this
notion, and I build two direct correspondences G_n \to N*, N* \to G_n between
normal Gentzen deductions and natural deductions in *-normal form, i.e.
natural deductions in normal form that also satisfy other conditions.

Item Type: Report MoL-1999-01 Master of Logic Thesis (MoL) Series 1999 12 Oct 2016 14:38 12 Oct 2016 14:38 https://eprints.illc.uva.nl/id/eprint/704