Sparse Subset Research Articles

Intersective sets for sparse sets of integers

Abstract For $E \subset \mathbb {N}$ , a subset $R \subset \mathbb {N}$ is E-intersective if for every $A \subset E$ having positive relative density, $R \cap (A - A) \neq \varnothing $ . We say that R is chromatically E-intersective if for every finite partition $E=\bigcup _{i=1}^k E_i$ , there exists i such that $R\cap (E_i-E_i)\neq \varnothing $ . When $E=\mathbb {N}$ , we recover the usual notions of intersectivity and chromatic intersectivity. We investigate to what extent the known intersectivity results hold in the relative setting when $E = \mathbb {P}$ , the set of primes, or other sparse subsets of $\mathbb {N}$ . Among other things, we prove the following: (1) the set of shifted Chen primes $\mathbb {P}_{\mathrm {Chen}} + 1$ is both intersective and $\mathbb {P}$ -intersective; (2) there exists an intersective set that is not $\mathbb {P}$ -intersective; (3) every $\mathbb {P}$ -intersective set is intersective; (4) there exists a chromatically $\mathbb {P}$ -intersective set which is not intersective (and therefore not $\mathbb {P}$ -intersective).

Data-Driven Sparse Sensor Placement Optimization on Wings for Flight-By-Feel: Bioinspired Approach and Application.

Flight-by-feel (FBF) is an approach to flight control that uses dispersed sensors on the wings of aircraft to detect flight state. While biological FBF systems, such as the wings of insects, often contain hundreds of strain and flow sensors, artificial systems are highly constrained by size, weight, and power (SWaP) considerations, especially for small aircraft. An optimization approach is needed to determine how many sensors are required and where they should be placed on the wing. Airflow fields can be highly nonlinear, and many local minima exist for sensor placement, meaning conventional optimization techniques are unreliable for this application. The Sparse Sensor Placement Optimization for Prediction (SSPOP) algorithm extracts information from a dense array of flow data using singular value decomposition and linear discriminant analysis, thereby identifying the most information-rich sparse subset of sensor locations. In this research, the SSPOP algorithm is evaluated for the placement of artificial hair sensors on a 3D delta wing model with a 45° sweep angle and a blunt leading edge. The sensor placement solution, or design point (DP), is shown to rank within the top one percent of all possible solutions by root mean square error in angle of attack prediction. This research is the first to evaluate SSPOP on a 3D model and the first to include variable length hairs for variable velocity sensitivity. A comparison of SSPOP against conventional greedy search and gradient-based optimization shows that SSPOP DP ranks nearest to optimal in over 90 percent of models and is far more robust to model variation. The successful application of SSPOP in complex 3D flows paves the way for experimental sensor placement optimization for artificial hair-cell airflow sensors and is a major step toward biomimetic flight-by-feel.

Local-structure-preservation and redundancy-removal-based feature selection method and its application to the identification of biomarkers for schizophrenia

Accurate diagnosis of mental disorders is expected to be achieved through the identification of reliable neuroimaging biomarkers with the help of cutting-edge feature selection techniques. However, existing feature selection methods often fall short in capturing the local structural characteristics among samples and effectively eliminating redundant features, resulting in inadequate performance in disorder prediction. To address this gap, we propose a novel supervised method named local-structure-preservation and redundancy-removal-based feature selection (LRFS), and then apply it to the identification of meaningful biomarkers for schizophrenia (SZ). LRFS method leverages graph-based regularization to preserve original sample similarity relationships during data transformation, thus retaining crucial local structure information. Additionally, it introduces redundancy-removal regularization based on interrelationships among features to exclude similar and redundant features from high-dimensional data. Moreover, LRFS method incorporates l2,1 sparse regularization that enables selecting a sparse and noise-robust feature subset. Experimental evaluations on eight public datasets with diverse properties demonstrate the superior performance of our method over nine popular feature selection methods in identifying discriminative features, with average classification accuracy gains ranging from 1.30 % to 9.11 %. Furthermore, the LRFS method demonstrates superior discriminability in four functional magnetic resonance imaging (fMRI) datasets from 708 healthy controls (HCs) and 537 SZ patients, with an average increase in classification accuracy ranging from 1.89 % to 9.24 % compared to other nine methods. Notably, our method reveals reproducible and significant changes in SZ patients relative to HCs across the four datasets, predominantly in the thalamus-related functional network connectivity, which exhibit a significant correlation with clinical symptoms. Convergence analysis, parameter sensitivity analysis, and ablation studies further demonstrate the effectiveness and robustness of our method. In short, our proposed feature selection method effectively identifies discriminative and reliable features that hold the potential to be biomarkers, paving the way for the elucidation of brain abnormalities and the advancement of precise diagnosis of mental disorders.

Dense and persistent odor representations in the olfactory bulb of awake mice.

Recording and analysis of neural activity is often biased toward detecting sparse subsets of highly active neurons, masking important signals carried in low magnitude and variable responses. To investigate the contribution of seemingly noisy activity to odor encoding, we used mesoscale calcium imaging from mice of both sexes to record odor responses from the dorsal surface of bilateral olfactory bulbs (OBs). The outer layer of the mouse OB is comprised of dendrites organized into discrete "glomeruli", which are defined by odor receptor-specific sensory neuron input. We extracted activity from a large population of glomeruli and used logistic regression to classify odors from individual trials with high accuracy. We then used add-in and drop-out analyses to determine subsets of glomeruli necessary and sufficient for odor classification. Classifiers successfully predicted odor identity even after excluding sparse, highly active glomeruli, indicating that odor information is redundantly represented across a large population of glomeruli. Additionally, we found that Random Forest feature selection informed by Gini Inequality (RFGI) reliably ranked glomeruli by their contribution to overall odor classification. RFGI provided a measure of "feature importance" for each glomerulus that correlated with intuitive features like response magnitude. Finally, in agreement with previous work, we found that odor information persists in glomerular activity after odor offset. Together, our findings support a model of olfactory bulb odor coding where sparse activity is sufficient for odor identification, but information is widely, redundantly available across a large population of glomeruli, with each glomerulus representing information about more than one odor.Significance statement This study leverages meso-scale imaging and machine learning to investigate how odor information is first represented in the brain. Typically, recordings of neuronal activity focus on active individual cells, potentially overlooking broader variations in neuronal responses across populations. Our results demonstrate that a considerable amount of olfactory information is redundantly distributed across a large proportion of olfactory bulb glomeruli. Even after excluding a majority of glomeruli, odor identification remained possible. These findings indicate that, although a few glomeruli are sufficient for odor recognition, an abundance of additional information is represented across a broad population. Understanding how the brain manages redundant olfactory information will shed light on its adaptive mechanisms for navigating diverse real-world circumstances and responding to fluctuating internal states.

A high-performance genetically encoded sensor for cellular imaging of PKC activity in vivo.

We report a genetically encoded fluorescence lifetime sensor for protein kinase C (PKC) activity, named CKAR3, based on Förster resonance energy transfer. CKAR3 exhibits a 10-fold increased dynamic range compared to its parental sensors and enables in vivo imaging of PKC activity during animal behavior. Our results reveal robust PKC activity in a sparse neuronal subset in the motor cortex during locomotion, in part mediated by muscarinic acetylcholine receptors.

A modeling framework for detecting and leveraging node-level information in Bayesian network inference.

Bayesian graphical models are powerful tools to infer complex relationships in high dimension, yet are often fraught with computational and statistical challenges. If exploited in a principled way, the increasing information collected alongside the data of primary interest constitutes an opportunity to mitigate these difficulties by guiding the detection of dependence structures. For instance, gene network inference may be informed by the use of publicly available summary statistics on the regulation of genes by genetic variants. Here we present a novel Gaussian graphical modeling framework to identify and leverage information on the centrality of nodes in conditional independence graphs. Specifically, we consider a fully joint hierarchical model to simultaneously infer (i) sparse precision matrices and (ii) the relevance of node-level information for uncovering the sought-after network structure. We encode such information as candidate auxiliary variables using a spike-and-slab submodel on the propensity of nodes to be hubs, which allows hypothesis-free selection and interpretation of a sparse subset of relevant variables. As efficient exploration of large posterior spaces is needed for real-world applications, we develop a variational expectation conditional maximization algorithm that scales inference to hundreds of samples, nodes and auxiliary variables. We illustrate and exploit the advantages of our approach in simulations and in a gene network study which identifies hub genes involved in biological pathways relevant to immune-mediated diseases.

Video Summarization using Attention Mechanisms and CNN

This Video summarization is a vital task in multimedia analysis, especially given the vast volume of video data in the digital age. Although deep learning methods have been extensively studied for this purpose, they often face challenges in efficiently processing long-duration videos. This paper tackles the issue of unsupervised video summarization by introducing a novel approach that selects a sparse subset of video frames to optimally represent the original video. The core concept involves training a deep summarizer network within a generative adversarial framework, which includes an autoencoder LSTM network as the summarizer and another LSTM network as the discriminator. The summarizer LSTM is designed to select key video frames and reconstruct the input video from these selected frames. Concurrently, the discriminator LSTM's role is to differentiate between the original video and its reconstruction. Through adversarial training between the summarizer and discriminator, combined with sparsity regularization, the network learns to produce optimal video summaries without requiring labeled data. Evaluations on several benchmark datasets indicate that this method delivers competitive performance compared to fully supervised state-of-the-art techniques, highlighting its effectiveness in unsupervised video summarization. Key Words: Event summarization, Critical information in videos, Surveillance systems ,Video analysis, Multimedia analysis, Deep learning, Unsupervised learning, Autoencoder LSTM, Long short-term memory network (LSTM)

Dense-sparse quantum Monte Carlo algebraic diagrammatic construction and importance ranking.

Quantum Monte Carlo Algebraic Diagrammatic Construction (QMCADC) has been proposed as a reformulation of the second-order ADC scheme for the polarization propagator within the projection quantum Monte Carlo formalism. Dense-sparse partitioning and importance ranking filtering strategies are now exploited to accelerate its convergence and to alleviate the signproblem inherent in such calculations. By splitting the configuration space into dense and sparse subsets, the corresponding projection operator is decomposed into four distinct blocks. Deterministic calculations handle the dense-to-dense and sparse-to-dense blocks, while the remaining blocks, dense-to-sparse and sparse-to-sparse, are stochastically evaluated. The dense set is efficiently stored in a fixed-size array, and the sparse set is represented through conventional floating random Monte Carlo walks. The stochastic projection is further refined through importance ranking criteria, enabling a reduction in the required number of walkers with a controllable bias. Our results demonstrate the integration of dense-sparse partitioning with importance ranking filtering to significantly enhance the efficiency of QMCADC, enabling large-scale molecular excited-state calculations. Furthermore, this novel approach maximizes the utilization of the sparsity of ADC(2), transforming QMCADC into a tailored framework for ADC calculations.

Transcriptomic contributions to a modern cytoarchitectonic parcellation of the human cerebral cortex.

Transcriptomic contributions to the anatomical, functional, and network layout of the human cerebral cortex (HCC) have become a major interest in cognitive and systems neuroscience. Here, we tested if transcriptomic differences support a modern, algorithmic cytoarchitectonic parcellation of HCC. Using a data-driven approach, we identified a sparse subset of genes that differentially contributed to the cytoarchitectonic parcellation of HCC. A combined metric of cortical thickness and myelination (CT/M ratio), as well as cell density, correlated with gene expression. Enrichment analyses showed that genes specific to the cytoarchitectonic parcellation of the HCC were related to molecular functions such as transmembrane transport and ion channel activity. Together, the relationship between transcriptomics and cytoarchitecture bridges the gap among (i) gradients at the macro-scale (including thickness and myelination), (ii) areas at the meso-scale, and (iii) cell density at the microscale, as well as supports the recently proposed cortical spectrum theory and structural model.

OWFD-UCPM: An open-world fault diagnosis scheme based on uncertainty calibration and prototype management

In real-world fault diagnosis tasks, it is never possible to enumerate all fault types beforehand since there are always unseen situations that may arise unexpectedly. Therefore, fault diagnosis is essentially an open-world recognition task. A desirable open-world fault diagnosis model must be able to perform open-set recognition (OSR) and incremental learning (IL). Classifier calibration and prototype replay are popular in two subtasks. However, existing calibration and replay strategies suffer from tampering with raw data and ignoring resource allocation, respectively. In this work, we formulate an open-world fault diagnosis scheme to address both issues. Firstly, we calibrate the uncertainty of the classifier’s prediction by assigning soft labels to the samples based on their distance from the class center. Then, Shannon entropy is employed to quantify the uncertainty as an estimate of the probability that the sample belongs to the unknown. Secondly, we adaptively manage the memory of old and new known classes according to the training accuracy. Then, based on this memory budget, a dissimilarity-based sparse subset selection algorithm is utilized to pick more diverse exemplars as prototypes for replay. Furthermore, we extend the open-set F-Measure (OSFM) to the weighted OSFM to make it applicable even when the classes are imbalanced. Experimental results on multiphase flow facility and wastewater treatment plant demonstrate the feasibility and superiority of the proposed method.

Behavioral functional networks underlying the effect of Alzheimer’s pathology on cognition

AbstractBackgroundTau pathology is associated with impaired cognitive function in both aging and Alzheimer’s disease (AD), but the mechanisms by which tau and cognition are linked remain unclear. We hypothesized that the integrity of behaviorally‐defined functional brain networks would help explain the relationship between tau and cognitive performance.MethodResting state fMRI, tau PET, and Ab PET were collected in two independent cohorts: 120 cognitively normal older adults from the Berkeley Aging Cohort Study (BACS; 77.6 ± 6.4 yrs), and 174 cognitively normal and MCI individuals from ADNI3 (72.8 ± 8.3 yrs). For each cohort, functional connectivity (FC) for each ROI‐ROI pair of a 246‐region atlas was computed, and LASSO+BIC regression of these values against episodic memory (EM) and executive function (EF) composite measures was used to define a sparse subset of connections positively associated with each cognitive measure. Overall network strength was computed as the sum of FC values for all selected ROI‐ROI pairs.ResultEM and EF networks were identified within each cohort, with Dice similarity coefficients indicating greater overlap of network regions within cohorts (BACS networks: 0.43; ADNI: 0.33) than within cognitive measures between cohorts (EM networks: 0.25; EF: 0.23; Figure 1). Adjusting for age and sex, lower EM network strength was associated with greater meta‐temporal ROI tau in both BACS (b = ‐4.05, p = .014) and ADNI (b = ‐2.20, p <.001), whereas lower EF network strength was related to meta‐ROI tau in ADNI only (b = ‐1.80, p <.001; Figure 2). In addition, causal mediation analysis revealed that the effect of meta‐ROI tau on EM performance was mediated by EM network strength in both BACS (p = .017) and ADNI (p = .002), but EF network strength was a significant mediator in ADNI only (p <.001; Figure 3).ConclusionGreater AD pathology is associated with reduced integrity of EM‐related networks in older adults, a relationship that may also extend to EF‐related networks as individuals progress from cognitively normal to MCI. Given that network strength was observed to mediate the relationship between tau pathology and cognition, behavioral brain networks may be a key mechanism in the effect of neuropathology on cognitive function in aging and AD.

Behavioral functional networks underlying the effect of Alzheimer’s pathology on cognition

AbstractBackgroundTau pathology is associated with impaired cognitive function in both aging and Alzheimer’s disease (AD), but the mechanisms by which tau and cognition are linked remain unclear. We hypothesized that the integrity of behaviorally‐defined functional brain networks would help explain the relationship between tau and cognitive performance.MethodResting state fMRI, tau PET, and Ab PET were collected in two independent cohorts: 120 cognitively normal older adults from the Berkeley Aging Cohort Study (BACS; 77.6 ± 6.4 yrs), and 174 cognitively normal and MCI individuals from ADNI3 (72.8 ± 8.3 yrs). For each cohort, functional connectivity (FC) for each ROI‐ROI pair of a 246‐region atlas was computed, and LASSO+BIC regression of these values against episodic memory (EM) and executive function (EF) composite measures was used to define a sparse subset of connections positively associated with each cognitive measure. Overall network strength was computed as the sum of FC values for all selected ROI‐ROI pairs.ResultEM and EF networks were identified within each cohort, with Dice similarity coefficients indicating greater overlap of network regions within cohorts (BACS networks: 0.43; ADNI: 0.33) than within cognitive measures between cohorts (EM networks: 0.25; EF: 0.23; Figure 1). Adjusting for age and sex, lower EM network strength was associated with greater meta‐temporal ROI tau in both BACS (b = ‐4.05, p = .014) and ADNI (b = ‐2.20, p <.001), whereas lower EF network strength was related to meta‐ROI tau in ADNI only (b = ‐1.80, p <.001; Figure 2). In addition, causal mediation analysis revealed that the effect of meta‐ROI tau on EM performance was mediated by EM network strength in both BACS (p = .017) and ADNI (p = .002), but EF network strength was a significant mediator in ADNI only (p <.001; Figure 3).ConclusionGreater AD pathology is associated with reduced integrity of EM‐related networks in older adults, a relationship that may also extend to EF‐related networks as individuals progress from cognitively normal to MCI. Given that network strength was observed to mediate the relationship between tau pathology and cognition, behavioral brain networks may be a key mechanism in the effect of neuropathology on cognitive function in aging and AD.

Cost-sensitive sparse subset selection

Cost-sensitive sparse subset selection

Beyond ℓ1 sparse coding in V1.

Growing evidence indicates that only a sparse subset from a pool of sensory neurons is active for the encoding of visual stimuli at any instant in time. Traditionally, to replicate such biological sparsity, generative models have been using the ℓ1 norm as a penalty due to its convexity, which makes it amenable to fast and simple algorithmic solvers. In this work, we use biological vision as a test-bed and show that the soft thresholding operation associated to the use of the ℓ1 norm is highly suboptimal compared to other functions suited to approximating ℓp with 0 ≤ p < 1 (including recently proposed continuous exact relaxations), in terms of performance. We show that ℓ1 sparsity employs a pool with more neurons, i.e. has a higher degree of overcompleteness, in order to maintain the same reconstruction error as the other methods considered. More specifically, at the same sparsity level, the thresholding algorithm using the ℓ1 norm as a penalty requires a dictionary of ten times more units compared to the proposed approach, where a non-convex continuous relaxation of the ℓ0 pseudo-norm is used, to reconstruct the external stimulus equally well. At a fixed sparsity level, both ℓ0- and ℓ1-based regularization develop units with receptive field (RF) shapes similar to biological neurons in V1 (and a subset of neurons in V2), but ℓ0-based regularization shows approximately five times better reconstruction of the stimulus. Our results in conjunction with recent metabolic findings indicate that for V1 to operate efficiently it should follow a coding regime which uses a regularization that is closer to the ℓ0 pseudo-norm rather than the ℓ1 one, and suggests a similar mode of operation for the sensory cortex in general.

Quantitative estimates for the size of an intersection of sparse automatic sets

A theorem of Cobham says that if k and ℓ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both k- and ℓ-automatic is eventually periodic. A multidimensional extension was later given by Semenov. In this paper, we give a quantitative version of the Cobham-Semenov theorem for sparse automatic sets, showing that the intersection of a sparse k-automatic subset of Nd and a sparse ℓ-automatic subset of Nd is finite with size that can be explicitly bounded in terms of data from the automata that accept these sets.

Demonstration of an AI-driven workflow for autonomous high-resolution scanning microscopy

Modern scanning microscopes can image materials with up to sub-atomic spatial and sub-picosecond time resolutions, but these capabilities come with large volumes of data, which can be difficult to store and analyze. We report the Fast Autonomous Scanning Toolkit (FAST) that addresses this challenge by combining a neural network, route optimization, and efficient hardware controls to enable a self-driving experiment that actively identifies and measures a sparse but representative data subset in lieu of the full dataset. FAST requires no prior information about the sample, is computationally efficient, and uses generic hardware controls with minimal experiment-specific wrapping. We test FAST in simulations and a dark-field X-ray microscopy experiment of a WSe2 film. Our studies show that a FAST scan of <25% is sufficient to accurately image and analyze the sample. FAST is easy to adapt for any scanning microscope; its broad adoption will empower general multi-level studies of materials evolution with respect to time, temperature, or other parameters.

Phase-dependent asymmetry of pattern masking in natural images explained by intrinsic position uncertainty.

A number of recent studies have been directed at measuring and modeling detection of targets at specific locations in natural backgrounds, a key subtask of visual search in natural environments. A useful approach is to bin natural background patches into joint histograms with bins along specific background dimensions. By measuring psychometric functions in a sparse subset of these bins, it is possible to estimate how the included dimensions jointly affect detectability over the whole space of natural backgrounds. In previous studies, we found that threshold is proportional to the product of the background luminance, contrast, and similarity; a result predicted by a simple template-matching observer with divisive normalization along each of the dimensions. The measure of similarity was the cosine similarity of the amplitude spectra of the target and background (SA)-a phase-invariant measure. Here, we investigated the effect of the cosine similarity of the target and background images (SI|A)-a phase-dependent measure. We found that threshold decreases monotonically with SI|A in agreement with a recent study (Rideaux et al., 2022). In contrast, the template-matching observer predicts threshold to be a U-shaped function of SI|A reaching a minimum when the target and background are orthogonal (SI|A = 0). Surprisingly, when the template-matching observer includes a small amount of intrinsic position uncertainty (measured in a separate experiment) the pattern of thresholds is explained.

Sparse attentional subsetting of item features and list-composition effects on recognition memory

Although knowledge is extremely high-dimensional, human episodic memory performance appears extremely low-dimensional, focused largely on stimulus-features that distinguish list items from one another. A cognitively plausible way this tension could be addressed is if selective attention selects a small number of features from each item. I consider an ongoing debate about whether stronger items (better encoded) interfere more than weaker items (less well encoded) with probe items during old/new episodic recognition judgements. This is called the list-strength effect, concerning whether or not effects of encoding strength are larger in lists of mixed strengths than in pure lists of a single strength. Analytic derivations with Anderson’s (1970) matched filter model show how storing only a small subset of features within high-dimensional representations, and assuming those same subsets tend to reiterate themselves item-wise at test, can support high recognition performance. In the sparse regime, the model produces a list-strength effect that is small in magnitude, resembling previous findings of so-called null list-strength effects. When the attended feature space is compact, such as for phonological features, attentional subsetting cannot be sparse. This introduces non-negligible cross-talk from other list items, producing a large-magnitude list-strength effect, similar to what is observed for the production effect (better recognition when reading aloud). This continuum-based account implies the existence of a continuous range of magnitudes of list-composition effects, including occasional inverted list-strength effects. This lays the foundation for propagating effects of task-relevant attention to sparse subsets of features through a broad range of models of memory behaviour.

Multiresolution functional characterization and correction of biofouling for improved biosensing efficacy

Multielectrode electrochemical biosensors promise on-the-spot inspection of target compounds in biofluids, reducing costs in personalized healthcare. However, sensor sensitivity may decrease after each use due to biofouling, where chemical attachments on sensor electrodes curtail sensing signals. Current biofouling characterization techniques rely on time-consuming offline tests and analysis, making them impractical for on-the-spot signal correction. Alternatively, we propose to statistically model and correct the biofouling-induced signal changes. However, in addition to biofouling, the signals are influenced by multiple sources of variation, each with different levels of impact. To effectively characterize and separate biofouling effects from the major sources of variability, we establish a multiresolution functional mixed-effect model based on domain knowledge. A biosensing signal is first decomposed into a smooth trend and local peaks. The smooth trend models the effects of population-level biofluid composition, as well as patient and electrode effects to isolate variability sources. Changes in local peak location and amplitude indicate biofouling. These local peaks are modeled using a sparse subset of high-order functional terms. By modeling the changes of those high-order terms, we can characterize and predict the biofouling between consecutive measurements. We propose a sequential parameter estimation procedure that ensures model identifiability. A nonparametric regression model is developed for biofouling prediction. The proposed strategy is validated through simulation and real case studies, effectively correcting biofouling-affected signals from new patients.

Lack-of-Corruption Goes with Liberal–Egalitarian Features of Democracy

Abstract This study aims at ferreting out a sparse subset, among the many democracy indicators, that best predicts lack-of-corruption (LoC). Two factors enable the authors to improve on previous research. First, the Varieties of Democracy (VD) data have become available over an extended time span, involving an unprecedented number of separate indicators. Second, the authors apply previously neglected methods, such as Gaussian standardization and measures to avoid overfitting. They also reduce random error on the LoC side by using the mean of two separate indices, VD and Transparency International, that correlate with R2 = 0.81. This results in an optimal set of 11 democracy indicators that predict the mean LoC with R2 = 0.87, close to what one could ideally expect to achieve. Among the five main categories VD distinguishes, the indicators that predict lack-of-corruption are mainly part of Liberal and Egalitarian Democracy, with lesser incidence of Electoral Democracy. This leaves out Participatory, and Deliberative Democracy.