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A de Branges space $ \mathcal {B} $ B is regular if the constants belong to its space of associated functions and is symmetric if it is isometrically invariant under the map $ F(z) \mapsto F(-z) $ F (z) ↦ F (− z). Let $ K_\mathcal {B}(z,w) $ K B (z , w) be the reproducing kernel in $ \mathcal {B} $ B and $ S_\mathcal {B} $ S B be the operator of multiplication by the independent variable with maximal domain in $ \mathcal {B} $ B . Loosely speaking, we say that $ \mathcal {B} $ B has the $ \ell _p $ ℓ p -oversampling property relative to a proper subspace $ \mathcal {A} $ A of it, with $ p\in (2,\infty ] $ p ∈ (2 , ∞ ] , if there exists $ J_{\mathcal {A}\mathcal {B}}:{\mathbb C}\times {\mathbb C}\to {\mathbb C} $ J A B : C × C → C such that $ J(\cdot,w)\in \mathcal {B} $ J (⋅ , w) ∈ B for all $ w\in {\mathbb C} $ w ∈ C , $$\begin{align*} &\sum_{\lambda\in\sigma(S_{\mathcal{B}}^{\gamma})} \left(\frac{\left\lvert J_{\mathcal{A}\mathcal{B}}(z,\lambda) \right\rvert}{K_\mathcal{B}(\lambda,\lambda)^{1/2}}\right)^{p/(p-1)} ∑ λ ∈ σ (S B γ) (| J A B (z , λ) | K B (λ , λ) 1 / 2 ) p / (p − 1) < ∞ and F (z) = ∑ λ ∈ σ (S B γ) J A B (z , λ) K B (λ , λ) F (λ) , for all $ F\in \mathcal {A} $ F ∈ A and almost every self-adjoint extension $ S_{\mathcal {B}}^{\gamma } $ S B γ of $ S_\mathcal {B} $ S B . This definition is motivated by the well-known oversampling property of Paley-Wiener spaces. In this paper, we provide sufficient conditions for a symmetric, regular de Branges space to have the $ \ell _p $ ℓ p -oversampling property relative to a chain of de Branges subspaces of it. [ABSTRACT FROM AUTHOR]

We establish convergence analysis for Hermite-type interpolations for L 2 (R) -entire functions of exponential type whose linear canonical transforms (LCT) are compactly supported. The results bridges the theoretical gap in implementing the derivative sampling theorems for band-limited signals in the LCT domain. Both complex analysis and real analysis techniques are established to derive the convergence analysis. The truncation error is also investigated and rigorous estimates for it are given. Nevertheless, the convergence rate is O (1 / N) , which is slow. Consequently the work on regularization techniques is required. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of recovery of a burst-like forcing term in an initial value problem (IVP) in the framework of dynamical sampling. We introduce an idea of using two particular classes of samplers that allow one to predict the solution of the IVP over a time interval without a burst. This leads to two different algorithms that stably and accurately approximate the burst-like forcing term even in the presence of a measurement acquisition error and a large background source. [ABSTRACT FROM AUTHOR]

Current literature dealing with the theory of sampling process flows is limited in scope, a situation addressed by papers presented at conferences by the author. This paper provides a summary of methods to quantify process sampling variance in a sound statistical manner. Two methods of calculation of the process sampling variance are presented here with models for robust estimation of the process covariance/variogram. The strategies for process sampling to acquire the data for process characterisation are described. The method for process characterisation can be based on special sampling campaigns, on the analysis of on-line measurement data, or on the analysis of historical data, which may be based on composite samples, such as shift samples. These scenarios are all dealt with as well as methods based on analysis of process dynamics. The latter analysis demonstrates that use of covariance functions drawn from geostatistical practice can be inappropriate for process analysis. The work is supported with plant data examples and quantitative analyses of process conditions. [ABSTRACT FROM AUTHOR]

The linear canonical transform (LCT) plays an important role in signal and image processing from both theoretical and practical points of view. Various sampling representations for band-limited and non-band-limited signals in the LCT domain have been established. We focus in this paper on the derivative sampling reconstruction, where the reconstruction procedure utilizes samples of both the signal and its first derivative. Our major aim was to incorporate the reconstruction sampling operator with a Gaussian regularization kernel, which on the one hand is applicable for not necessarily band-limited signals and on the other hand hastens the convergence of the reconstruction procedure. The amplitude error is also considered with deriving rigorous estimates. The obtained theoretical results are tested through various simulated experiments. [ABSTRACT FROM AUTHOR]

Unevenly spaced samples from a periodic function are common in signal processing and can often be viewed as a perturbed equally spaced grid. In this paper, the question of how the uneven distribution of the samples impacts the quality of interpolation and quadrature is analyzed. Starting with equally spaced nodes on [ - π , π) with grid spacing h, suppose the unevenly spaced nodes are obtained by perturbing each uniform node by an arbitrary amount ≤ α h , where 0 ≤ α < 1 / 2 is a fixed constant. A discrete version of the Kadec-1/4 theorem is proved, which states that the nonuniform discrete Fourier transform associated with perturbed nodes has a bounded condition number independent of h, for any α < 1 / 4 . Then, it is shown that unevenly spaced quadrature rules converge for all continuous functions and interpolants converge uniformly for all differentiable functions whose derivative has bounded variation when 0 ≤ α < 1 / 4 . Though, quadrature rules at perturbed nodes can have negative weights for any α > 0 , a bound on the absolute sum of the quadrature weights is provided, which shows that perturbed equally spaced grids with small α can be used without numerical woes. While the proof techniques work primarily when 0 ≤ α < 1 / 4 , it is shown that a small amount of oversampling extends our results to the case when 1 / 4 ≤ α < 1 / 2 . [ABSTRACT FROM AUTHOR]

In this paper, the connections between the Sampling Kantorovich model and the sampling process are highlighted and exploited. Based on the theoretical framework of the Sampling Kantorovich operators, a sampling paradigm, here named Sampling Kantorovich by Difference (SKD), is introduced. In line of principle, SKD allows for overcoming the technical limitation due to the fact that the resolution of a signal/image is strictly connected with the size of the used sensors. We analyze the paradigm in the case of a simulated super resolution type problem. The same mathematical model, being extendable to other signal reconstruction procedures, suggests a theoretical way for new technical solutions in the sampling procedures. [ABSTRACT FROM AUTHOR]

The aim of this research is to help inspect the motion of cell life by applying electrical engineering scientific techniques to the cellular evolution of human neural networks. Using a mathematically rigorous theory of cellular biological progression, the hypothesis will demonstrate that cell life evolves toward increasing the organism's resonant energy transfer or "exposing points" with its natural environment. This increases the sampling points of electromagnetic radiation frequencies available for transferal into its biological system to stabilize its cellular reproduction cycle as a function of physical motion measurable in volt-ampere or calories/second as an experimental measure of fitness. [ABSTRACT FROM AUTHOR]

Given n samples of a function f : D → C in random points drawn with respect to a measure ϱS we develop theoretical analysis of the L2(D, ϱT )-approximation error. For a parituclar choice of ϱS depending on ϱT, it is known that the weighted least squares method from finite dimensional function spaces Vm, dim(Vm) = m < ∞ has the same error as the best approximation in Vm up to a multiplicative constant when given exact samples with logarithmic oversampling. If the source measure ϱS and the target measure ϱT differ we are in the domain adaptation setting, a subfield of transfer learning. We model the resulting deterioration of the error in our bounds. Further, for noisy samples, our bounds describe the bias-variance trade off depending on the dimension m of the approximation space Vm. All results hold with high probability. For demonstration, we consider functions defined on the d-dimensional cube given in unifom random samples. We analyze polynomials, the half-period cosine, and a bounded orthonormal basis of the non-periodic Sobolev space H2 mix. Overcoming numerical issues of this H²mix basis, this gives a novel stable approximation method with quadratic error decay. Numerical experiments indicate the applicability of our results. [ABSTRACT FROM AUTHOR]

The aim of this research is to help inspect the motion of cell life by applying electrical engineering scientific techniques to the cellular evolution of human neural networks. Using a mathematically rigorous theory of cellular biological progression, the hypothesis will demonstrate that cell life evolves toward increasing the organism's resonant energy transfer or “exposing points” with its natural environment. This increases the sampling points of electromagnetic radiation frequencies available for transferal into its biological system to stabilize its cellular reproduction cycle as a function of physical motion measurable in volt-ampere or calories/second as an experimental measure of fitness.

In this paper, we derive upper bounds for the covariance of functions of random variables in different cases, which is the extension of the results established in [4]. Some applications in statistics are also provided. Comparing to other results, ours can be used in more general cases involved more complicated statistical dependency. [ABSTRACT FROM AUTHOR]

We study the problem of stable reconstruction of the short-time Fourier transform from samples taken from trajectories in R 2 . We first investigate the interplay between relative density of the trajectory and the reconstruction property. Later, we consider spiraling curves, a special class of trajectories, and connect sampling and uniqueness properties of these sets. Moreover, we show that for window functions given by a linear combination of Hermite functions, it is indeed possible to stably reconstruct from samples on some particular natural choices of spiraling curves. [ABSTRACT FROM AUTHOR]

The Shannon-Whittaker Sampling Theorem states that a frequency bounded signal can be completely determined by its sampled values at a countable number of points. Thus, the theorem allows us to convert analog signals to digital signals by sampling (or evaluating) the signal at these points. In prior work, it was shown that if a signal is oversampled, and if some of the sampled values are lost when transmitting the signal, then it is still possible to reconstruct the signal. However, in certain situations, the reconstruction algorithm is very unstable. In this paper, we provide stability bounds on the reconstruction algorithm and determine when it is not feasible to perform the reconstruction. [ABSTRACT FROM AUTHOR]

With an increasing influx of classical signal processing methodologies into the field of graph signal processing, approaches grounded in discrete linear canonical transform have found application in graph signals. In this paper, we initially propose the uncertainty principle of the graph linear canonical transform (GLCT), which is based on a class of graph signals maximally concentrated in both vertex and graph spectral domains. Subsequently, leveraging the uncertainty principle, we establish conditions for recovering bandlimited signals of the GLCT from a subset of samples, thereby formulating the sampling theory for the GLCT. We elucidate interesting connections between the uncertainty principle and sampling. Further, by employing sampling set selection and experimental design sampling strategies, we introduce optimal sampling operators in the GLCT domain. Finally, we evaluate the performance of our methods through simulations and numerical experiments across applications. • Uncertainty principle in GLCT domain and its connection with sampling are introduced. • Optimal sampling and perfect recovery conditions in GLCT domain are established. • GLCT outperforms GFT and GFRFT in signal recovery, classification and clustering. [ABSTRACT FROM AUTHOR]

Premise of the studyCalifornia's vascular flora is the most diverse and threatened in temperate North America. Previous studies of spatial patterns of Californian plant diversity have been limited by traditional metrics, non-uniform geographic units, and distributional data derived from floristic descriptions for only a subset of species.MethodsWe revisited patterns of sampling intensity, species richness, and relative endemism in California based on equal-area spatial units, the full vascular flora, and specimen-based distributional data. We estimated richness, weighted endemism (inverse range-weighting of species), and corrected weighted endemism (weighted endemism corrected for richness), and performed a randomization test for significantly high endemism.Key resultsPossible biases in herbarium data do not obscure patterns of high richness and endemism at the spatial resolution studied. High species richness was sometimes associated with significantly high endemism (e.g., Klamath Ranges) but often not. In Stebbins and Major's (1965) main endemism hotspot, Southwestern California, species richness is high across much of the Peninsular and Transverse ranges but significantly high endemism is mostly localized to the Santa Rosa and San Bernardino mountains. In contrast, species richness is low in the Channel Islands, where endemism is significantly high, as also found for much of the Death Valley region.ConclusionsMeasures of taxonomic richness, even with greater weighting of range-restricted taxa, are insufficient for identifying areas of significantly high endemism that warrant conservation attention. Differences between our findings and those in previous studies appear to mostly reflect the source and scale of distributional data, and recent analytical refinements.

Time-vertex graph signal (TVGS) models describe time-varying data with irregular structures. The bandlimitedness in the joint time-vertex Fourier spectral domain reflects smoothness in both temporal and graph topology. In this paper, we study the critical sampling of three types of TVGS including continuous-time signals, infinite-length sequences, and finite-length sequences in the time domain for each vertex on the graph. For a jointly bandlimited TVGS, we prove a lower bound on sampling density or sampling ratio, which depends on the measure of the spectral support in the joint time-vertex Fourier spectral domain. We also provide a lower bound on the sampling density or sampling ratio of each vertex on sampling sets for perfect recovery. To demonstrate that critical sampling is achievable, we propose the sampling and reconstruction procedures for the different types of TVGS. Finally, we show how the proposed sampling schemes can be applied to numerical as well as real datasets. • We prove the necessary conditions for the stable reconstruction of JBL TVGS. • We prove lower bounds on sampling densities (or ratios) of the signals on subsets of vertices to be sampled. • We construct a multi-band sampling scheme to prove that critical sampling is achievable for any JBL TVGS. [ABSTRACT FROM AUTHOR]

In this paper, the connections between the Sampling Kantorovich model and the sampling process are highlighted and exploited. Based on the theoretical framework of the Sampling Kantorovich operators, a sampling paradigm, here named Sampling Kantorovich by Difference (SKD), is introduced. In line of principle, SKD allows for overcoming the technical limitation due to the fact that the resolution of a signal/image is strictly connected with the size of the used sensors. We analyze the paradigm in the case of a simulated super resolution type problem. The same mathematical model, being extendable to other signal reconstruction procedures, suggests a theoretical way for new technical solutions in the sampling procedures.

Abel-Whittaker series (1934) is viewed as a two-point Taylor series. It solves the interpolation problem f (2 n) (1) = a n , f (2 n + 1) (0) = b n , n ≥ 0 , for an appropriate analytic function f (·) . The remainder of the truncated series is investigated and sharp estimates for it are established. A key result is a Cauchy-type representation of the remainder that is established in terms of Euler's polynomials. The results are illustrated through numerical tables and sketched graphs, introducing the two-point theorem as an efficient approximation tool. Comparisons with both Taylor series and the classical sampling theorem of Whittaker (1915) are carried out. Dually, we derive and investigate the remainder associated with the dual Abel-Whittaker interpolation problem f (2 n + 1) (1) = a n , f (2 n) (0) = b n , n ≥ 0. [ABSTRACT FROM AUTHOR]

Recovering a signal (function) from finitely many binary or Fourier samples is one of the core problems in modern imaging, and by now there exist a plethora of methods for recovering a signal from such samples. Examples of methods which can utilize wavelet reconstruction include generalized sampling, infinite-dimensional compressive sensing, the parameterized-background data-weak (PBDW) method, etc. However, for any of these methods to be applied in practice, accurate and fast modeling of an N × M section of the infinite-dimensional change-of-basis matrix between the sampling basis (Fourier or Walsh-Hadamard samples) and the wavelet reconstruction basis is paramount. Building on the work of Gataric and Poon [SIAM J. Sci. Comput., 38 (2016), pp. A1075-A1099], we derive an algorithm which bypasses the NM storage requirement and the scrO (NM) computational cost of matrix-vector multiplication with this matrix and its adjoint when using Walsh-Hadamard samples and wavelet reconstruction. The proposed algorithm computes the matrix-vector multiplication in scrO (N logN) operations and has a storage requirement of (2q), where N = 2dqM (usually q and d = 1, 2 is the dimension. As matrix-vector multiplications are the computational bottleneck for iterative algorithms used by the mentioned reconstruction methods, the proposed algorithm speeds up the reconstruction of wavelet coefficients from Walsh-Hadamard samples considerably. [ABSTRACT FROM AUTHOR]

Aim: To derive null models for the expected number of species shared among multiple samples or habitat patches, allowing exploration of the geometric effects of subdivision on species diversity. Location: Global. Major taxa studied: Predominantly sessile organisms. Methods: The occurrence probability of a species in a subdivided area depends on its abundance and spatial pattern over a known habitat extent. The joint probability that two subareas share a species is the product of the probability of species occurrence in each subarea provided that the latter probability is independent. The sum of this probability over all species is the number of species the two subareas share, or zeta diversity of order 2. Generalizing from 2 to m subareas yields a null model for zeta diversity of order m. From zeta diversity, many metrics (e.g., beta and gamma diversity) for the m habitat patches can be calculated, revealing the effects of increasing habitat fragmentation. Results: The null models show the geometric effects of subdivision depend on patterns of spatial distribution of species within a landscape and evenness of species abundance distribution. For aggregated assemblages, increasing subdivision decreases shared species, increases beta diversity and results in higher total species richness in subdivided habitat than an equal contiguous area. Main conclusions: To correctly interpret diversity patterns in fragmented habitat the geometric effects of subdivision must be considered. Our models explain why fragmented habitat could have higher diversity than continuous habitat of equal area but predict a threshold patch‐size above which this will not occur (here c. 100 ha). Apparently positive diversity effects of subdivision, including more species in groups of small patches, are probable outcomes of spatial aggregation of assemblages. The shared species null models offer an analytical tool for exploring the geometric effects of subdivision on diversity while controlling for total habitat area. [ABSTRACT FROM AUTHOR]

• A demixing method is proposed for decomposing blurred, textured images into cartoon, tex- ture and noise parts. • This high-order partial differential approach is defined by directional mean curvature, G-norm and curvelet space. • A connection between the proposed variational method and filter banks in harmonic analysis is set up. • This article addresses a fundamental problem in image processing: signal is sparse under some transformed domains. • Both theoretical results and examples with natural images are provided to illustrate its per- formance. Approximation theory plays an important role in image processing, especially image deconvolution and decomposition. For piecewise smooth images, there are many methods that have been developed over the past thirty years. The goal of this study is to devise similar and practical methodology for handling textured images. This problem is motivated by forensic imaging, since fingerprints, shoeprints and bullet ballistic evidence are textured images. In particular, it is known that texture information is almost destroyed by a blur operator, such as a blurred ballistic image captured from a low-cost microscope. The contribution of this work is twofold: first, we propose a mathematical model for textured image deconvolution and decomposition into four meaningful components, using a high-order partial differential equation approach based on the directional mean curvature. Second, we uncover a link between functional analysis and multiscale sampling theory, e.g., harmonic analysis and filter banks. Both theoretical results and examples with natural images are provided to illustrate the performance of the proposed model. [ABSTRACT FROM AUTHOR]

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We consider the recovery of square-integrable signals from discrete, equidistant samples of their Gabor transform magnitude and show that, in general, signals can not be recovered from such samples. In particular, we show that for any lattice, one can construct functions in L 2 (R) which do not agree up to global phase but whose Gabor transform magnitudes sampled on the lattice agree. These functions have good concentration in both time and frequency and can be constructed to be real-valued for rectangular lattices. [ABSTRACT FROM AUTHOR]

In this paper, we study the sampling theory for non-decaying signals in mixed Lebesgue spaces L p , q (R d + 1). First, we give some Riesz-type bounds, which connect the weighted discrete norms of the samples to the weighted continuous norms of original and interpolated signals. Then, we prove that the sampling and reconstruction through the Riesz-type bounds are stable when the generated kernel is in the appropriate mixed Wiener amalgam spaces. [ABSTRACT FROM AUTHOR]

Questions: Ecologists collecting field samples of biological data have a keen interest in addressing the following question: how many rare species are there in as‐yet unsurveyed additional samples? Depending on the size of a targeted additional sample, statistical models for estimating the number of rare species have not been systematically established and compared. Location: Global. Methods: For fairly comparing and predicting rare‐species richness at the same sample‐size baseline, we systematically developed and compared four estimators for rarefaction and extrapolation of rare‐species richness with a given specific abundance. These four estimators included a uniformly minimum variance unbiased (UMVUE), Bayesian‐weighted, Chao‐derived unweighted and naïve estimator. Results: After extensive numerical tests, for conducting rarefaction of rare‐species richness (i.e., when additional sample size was not larger than the original one) it is recommended to implement UMVUE, as it has zero bias and coverage percentage closest to 0.95. However, the performance of Bayesian‐weighted and Chao‐derived estimators is also satisfactory. By contrast, for conducting extrapolation of rare‐species richness (i.e., when the additional sample size is larger than the original one), the Bayesian‐weighted estimator is recommended, as it has the best performance among the four estimators (here UMVUE is inapplicable). Conclusions: There was no absolute winner, as the different estimators have their own merits and are recommended under different settings. When conducting rarefaction of rare‐species richness, UMVUE, which has the highest accuracy, is recommended. By contrast, when conducting extrapolation of rare‐species richness, the Bayesian‐weighted estimator is recommended, as it has the overall best performance. To facilitate the potential application in the comparison and prediction of rare‐species diversity using rarefaction and extrapolation techniques, an R package (fRSE) has been developed; it is freely distributed at the following URL: https://github.com/ecomol/fRSE. [ABSTRACT FROM AUTHOR]

In this paper, we study the construction of classical geometry from the quantum entanglement structure by using information geometry. In the information geometry of classical spacetime, the Fisher information metric is related to a blurred metric of a classical physical space. We first show that a local information metric can be obtained from the entanglement contour in a local subregion. This local information metric measures the fine structure of entanglement spectra inside the subregion, which suggests a quantum origin of the information-geometric blurred space. We study both the continuous and the classical limits of the quantum-originated blurred space by using the techniques from the statistical sampling algorithms, the sampling theory of spacetime and the projective limit. A scheme for going from a blurred space with quantum features to a classical geometry is also explored. [ABSTRACT FROM AUTHOR]

This paper introduces a novel framework and corresponding methods for sampling and reconstruction of sparse signals in shift-invariant (SI) spaces. We reinterpret the random demodulator, a system that acquires sparse bandlimited signals, as a system for the acquisition of linear combinations of the samples in the SI setting with the box function as the sampling kernel. The sparsity assumption is exploited by the compressive sensing (CS) paradigm for a recovery of the SI samples from a reduced set of measurements. The SI samples are subsequently filtered by a discrete-time correction filter to reconstruct expansion coefficients of the observed signal. Furthermore, we offer a generalization of the proposed framework to other compactly supported sampling kernels that span a wider class of SI spaces. The generalized method embeds the correction filter in the CS optimization problem which directly reconstructs expansion coefficients of the signal. Both approaches recast an inherently continuous-domain inverse problem in a set of finite-dimensional CS problems in an exact way. Finally, we conduct numerical experiments on signals in polynomial B-spline spaces whose expansion coefficients are assumed to be sparse in a certain transform domain. The coefficients can be regarded as parametric models of an underlying continuous-time signal, obtained from a reduced set of measurements. Such continuous signal representations are particularly suitable for signal processing without converting them into samples. [ABSTRACT FROM AUTHOR]

The approximation of probability measures on compact metric spaces and in particular on Riemannian manifolds by atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of point measures we are concerned with the approximation by measures supported on Lipschitz curves. Special attention is paid to push-forward measures of Lebesgue measures on the unit interval by such curves. Using the discrepancy as distance between measures, we prove optimal approximation rates in terms of the curve's length and Lipschitz constant. Having established the theoretical convergence rates, we are interested in the numerical minimization of the discrepancy between a given probability measure and the set of push-forward measures of Lebesgue measures on the unit interval by Lipschitz curves. We present numerical examples for measures on the 2- and 3-dimensional torus, the 2-sphere, the rotation group on R 3 and the Grassmannian of all 2-dimensional linear subspaces of R 4 . Our algorithm of choice is a conjugate gradient method on these manifolds, which incorporates second-order information. For efficient gradient and Hessian evaluations within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate and discuss the common points shared, in their line of development, by both Sampling Theory and Design of Experiments. In fact, Sampling Theory adopts the main optimality criterion of the Optimal Design of Experiments, the minimization of variance, i.e. D-optimality. There is also an approach based on c-optimality, as far as ratio estimates are concerned, in Design of Experiments, and the A-optimality involved in a proposed Sampling technique. It is pointed out that the L2 norm is mainly applied as a distance measure. [ABSTRACT FROM AUTHOR]

Infrastructure monitoring is critical for safe operations and sustainability. Like many networked systems, water distribution networks (WDNs) exhibit both graph topological structure and complex embedded flow dynamics. The resulting networked cascade dynamics are difficult to predict without extensive sensor data. However, ubiquitous sensor monitoring in underground situations is expensive, and a key challenge is to infer the contaminant dynamics from partial sparse monitoring data. Existing approaches use multi-objective optimization to find the minimum set of essential monitoring points but lack performance guarantees and a theoretical framework. Here, we first develop a novel Graph Fourier Transform (GFT) operator to compress networked contamination dynamics to identify the essential principal data collection points with inference performance guarantees. As such, the GFT approach provides the theoretical sampling bound. We then achieve under-sampling performance by building auto-encoder (AE) neural networks (NN) to generalize the GFT sampling process and under-sample further from the initial sampling set, allowing a very small set of data points to largely reconstruct the contamination dynamics over real and artificial WDNs. Various sources of the contamination are tested, and we obtain high accuracy reconstruction using around 5%–10% of the network nodes for known contaminant sources, and 50%–75% for unknown source cases, which although larger than that of the schemes for contaminant detection and source identifications, is smaller than the current sampling schemes for contaminant data recovery. This general approach of compression and under-sampled recovery via NN can be applied to a wide range of networked infrastructures to enable efficient data sampling for digital twins. [ABSTRACT FROM AUTHOR]

• This research analyzes the content of the online short video head accounts analyzed by social networks. • Large number of online video samples and the time dimension, this research uses random theory to conduct random sampling according to the range of 15% of the empirical sample size, and selects different types of short videos as the final effective analysis samples. • The sampling results in this article show that Douyin popular science short video accounts account for 18.99% of science, technology and inventions, 16.59% of medical care and health and nature, and 15.38% of animals and plants. • The media environment based on sampling theory in this paper also provides an extremely favorable network platform for the effective dissemination of online video and film. The media landscape has been evolving at a dizzying pace in recent years, with short video platforms emerging as a significant public information conduit; yet, it must be acknowledged that the impact of these platforms on transmission is still rather limited. Both the width and the depth are not up to par. Improving the impact, increasing the reach and impact, and making the most of the Douyin platform for scientific communication activity is an immediate concern. Douyin, China's most popular video-sharing platform, with 400 million daily users. Douyin provides regular individuals with the resources they need to start their own video art careers; it is popular outside of tier 1 cities. There are also inventive contests on the site where individuals can try their hand at being famous overnight. In this study, we look at how social media platforms evaluate short video head accounts online. Research is conducted to separate out the communication features of the head accounts based on popular scientific themes, the length of the form, and the use of other platform functions in the operation process. This study use random theory to pick various sorts of short videos as the final effective analysis samples, with a random sampling range of 15% of the empirical sample size, in order to deal with the temporal dimension and the enormous number of online video samples. Among the topics covered by Douyin's popular scientific short video accounts, the percentages for science, technology, innovations, medical care, health, and nature stand at 18.99%, 16.59%, and 15.38%, respectively, according to the article's sample data. Online video and film may be effectively disseminated in this research's media ecosystem, which is based on sampling theory. [ABSTRACT FROM AUTHOR]

Many engineering, social, and biological complex systems consist of dynamical elements connected via a large-scale network. Monitoring the network's dynamics is essential for a variety of maintenance and scientific purposes. Whilst we understand how to optimally sample and discretize a single continuous dynamic element or a non-dynamic graph, we do not possess a theory on how to optimally sample networked dynamical elements. Here, we study nonlinear dynamic graph signals on a fixed complex network. We define the necessary conditions for optimal sampling in the combining time- and graph-domain to fully recover the networked dynamics. We firstly interpret the networked dynamics into a linearized matrix. Then, we prove that the dynamic signals can be sampled and fully recovered if the networked dynamics is stable and their initialization and inputs (which are unknown) are bandlimited in the graph spectral domain. This new theory directly maps optimal sampling locations and rates to the graph properties and governing nonlinear dynamics. This can inform the placement of experimental probes and sensors on dynamical networks especially for the case where the employed sensors are difficult to change. Also, this guides the design of each sensor's optimal sampling rate for further digital signal processing. We motivate the reader with two examples of recovering the networked dynamics for: social population growth and networked protein biochemical interactions with both bandlimited and arbitrary initialization and inputs.

We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of undersampling that compressible signals admit when sampled along spirals. More precisely, we derive a lower bound on the condition number for the reconstruction of functions of bounded variation, and for functions that are sparse in the Haar wavelet basis. [ABSTRACT FROM AUTHOR]

This paper is devoted to the establishment of two-dimensional sampling theorems for discrete transforms, whose kernels arise from second order partial difference equations. We define a discrete type partial difference operator and investigate its spectral properties. Green's function is constructed and kernels that generate orthonormal basis of eigenvectors are defined. A discrete Kramer-type lemma is introduced and two sampling theorems of Lagrange interpolation type are proved. Several illustrative examples are depicted. The theory is extendible to higher order settings. [ABSTRACT FROM AUTHOR]

Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclear Magnetic Resonance (NMR), radio interferometry, helium atom scattering etc., the theory of compressed sensing with Fourier transform measurements has reached a mature level. However, for binary measurements via the Walsh transform, the theory has long been merely non-existent, despite the large number of applications such as fluorescence microscopy, single pixel cameras, lensless cameras, compressive holography, laser-based failure-analysis etc. Binary measurements are a mainstay in signal and image processing and can be modelled by the Walsh transform and Walsh series that are binary cousins of the respective Fourier counterparts. We help bridging the theoretical gap by providing non-uniform recovery guarantees for infinite-dimensional compressed sensing with Walsh samples and wavelet reconstruction. The theoretical results demonstrate that compressed sensing with Walsh samples, as long as the sampling strategy is highly structured and follows the structured sparsity of the signal, is as effective as in the Fourier case. However, there is a fundamental difference in the asymptotic results when the smoothness and vanishing moments of the wavelet increase. In the Fourier case, this changes the optimal sampling patterns, whereas this is not the case in the Walsh setting. [ABSTRACT FROM AUTHOR]

This work studies the problem of estimating a two-dimensional superposition of point sources or spikes from samples of their convolution with a Gaussian kernel. Our results show that minimizing a continuous counterpart of the l1-norm exactly recovers the true spikes if they are suficiently separated, and the samples are suficiently dense. In addition, we provide numerical evidence that our results extend to non-Gaussian kernels relevant to microscopy and telescopy. [ABSTRACT FROM AUTHOR]

Geoscientific datasets can contain individual data for more than 50 different chemical elements. The association between these variables is as important as their individual values. However, it is commonly overlooked that the observed covariance may be overestimated due to correlated errors. Dependent errors arise from many sources, such as the segregation process of minerals associated with these variables during delimitation, extraction, and preparation steps. This study extends a classical model composed of grade-independent (additive) and grade-proportional (multiplicative) errors to a generalised multivariate model that can estimate the real variance, covariance, and correlation from observations affected by shared errors. The use of estimates of the real covariance is recommended when the study objective is to evaluate or estimate the association between processes instead of the association between observations. A numerical example illustrates the bias in statistics and discusses the relevance of considering shared errors in linear regression and kriging. [ABSTRACT FROM AUTHOR]

This work investigates the role of the filters implemented on analog-to-digital converters for the reconstruction of magnetic resonance images. We analyze the effects of these filters both from a theoretical and an experimental point of view and demonstrate how it may lead to severe degradation of the reconstructed images when the distance between consecutive samples is larger than Shannon's limit. Based on these findings, we propose a mathematical model and a numerical algorithm that allow to mitigate such filtering effects both for linear and nonlinear reconstructions. Experiments on simulated and real data on a 7 Tesla scanner show that the proposed ideas allow to significantly improve the overall image quality. These findings are particularly relevant for high resolution imaging and for recent sampling schemes saturating the maximum gradient amplitude. They also open new challenges in sampling theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of reconstructing an infinite-dimensional signal from a limited number of linear measurements. In particular, we show that for binary measurements (modelled with Walsh functions and Hadamard matrices) and wavelet reconstruction the stable sampling rate is linear. This implies that binary measurements are as efficient as Fourier samples when using wavelets as the reconstruction space. Powerful techniques for reconstructions include generalized sampling and its compressed versions, as well as recent methods based on data assimilation. Common to these methods is that the reconstruction quality depends highly on the subspace angle between the sampling and the reconstruction space, which is dictated by the stable sampling rate. As a result of the theory provided in this paper, these methods can now easily use binary measurements and wavelet reconstruction bases. [ABSTRACT FROM AUTHOR]

In today's age, where data is of prime importance and rate at which data generated and sources of data generation is quite huge. The attacks on the data have increased manifold due to which its security is of prime importance. The traditional defense mechanisms are proving to be inadequate. In this paper, a new method to analyses network logs using R and Big data technologies is introduced. The approach detects fraudulent ip address in the network logs based on a set of conditions using pig scripting language. The results of using this method to detect the attack on the dataset is also presented. [ABSTRACT FROM AUTHOR]