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Quantifying and assessing changes in biological diversity are central aspects of many ecological studies, yet accurate methods of estimating biological diversity from sampling data have been elusive. Hill numbers, or the effective number of species, are increasingly used to characterize the taxonomic, phylogenetic, or functional diversity of an assemblage. However, empirical estimates of Hill numbers, including species richness, tend to be an increasing function of sampling effort and, thus, tend to increase with sample completeness. Integrated curves based on sampling theory that smoothly link rarefaction (interpolation) and prediction (extrapolation) standardize samples on the basis of sample size or sample completeness and facilitate the comparison of biodiversity data. Here we extended previous rarefaction and extrapolation models for species richness (Hill number q D , where q = 0) to measures of taxon diversity incorporating relative abundance (i.e., for any Hill number q D , q > 0) and present a unified approach for both individual-based (abundance) data and sample-based (incidence) data. Using this unified sampling framework, we derive both theoretical formulas and analytic estimators for seamless rarefaction and extrapolation based on Hill numbers. Detailed examples are provided for the first three Hill numbers: q = 0 (species richness), q = 1 (the exponential of Shannon's entropy index), and q = 2 (the inverse of Simpson's concentration index). We developed a bootstrap method for constructing confidence intervals around Hill numbers, facilitating the comparison of multiple assemblages of both rarefied and extrapolated samples. The proposed estimators are accurate for both rarefaction and short-range extrapolation. For long-range extrapolation, the performance of the estimators depends on both the value of q and on the extrapolation range. We tested our methods on simulated data generated from species abundance models and on data from large species inventories. We also illustrate the formulas and estimators using empirical data sets from biodiversity surveys of temperate forest spiders and tropical ants.

Multilinear PageRank is a variant of the PageRank algorithm that takes into account multiple relationships among nodes in a network. This algorithm can make web page ranking more efficient and accurate by considering multiple types of connections at once. The higher-order power method is commonly used to calculate the multilinear PageRank vector due to its ease of implementation and low storage needs, and because it is a natural extension of the traditional power method used in the PageRank algorithm. However, the convergence of this method is not guaranteed, and even when it occurs, the process is often slow. In this paper, we show how some vector extrapolation methods such as minimal polynomial extrapolation (MPE) and reduced rank extrapolation (RRE) could be used for accelerating the computation of the fixed-point multilinear PageRank.

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Human’s imagination capability provides recognition of unseen environment which should be improved in robots in order to have better mapping, planning, navigation and exploration capabilities in the fields where the robots are utilized such as military, disasters, and industry. The task of completion of a partial scene via estimating the unobserved parts relied on the known information is called scene extrapolation. It increases performance and satisfies a valid approximation of unseen content even if it is impossible or hard to obtain it due to the issues related with security, environment, etc. In this survey paper, the studies related to learning-based scene extrapolation in robotics are presented and evaluated taking the efficiencies and limitations of the methods into account to provide researchers in this field a general overview on this task and encourage them to improve the current studies for higher success. In addition, the methods which use common datasets and metrics are compared. To the best of our knowledge, there isn’t any survey on this essential topic and we hope this survey will compensate this.

We propose an integrated sampling, rarefaction, and extrapolation methodology to compare species richness of a set of communities based on samples of equal completeness (as measured by sample coverage) instead of equal size. Traditional rarefaction or extrapolation to equal-sized samples can misrepresent the relationships between the richnesses of the communities being compared because a sample of a given size may be sufficient to fully characterize the lower diversity community, but insufficient to characterize the richer community. Thus, the traditional method systematically biases the degree of differences between community richnesses. We derived a new analytic method for seamless coverage-based rarefaction and extrapolation. We show that this method yields less biased comparisons of richness between communities, and manages this with less total sampling effort. When this approach is integrated with an adaptive coverage-based stopping rule during sampling, samples may be compared directly without rarefaction, so no extra data is taken and none is thrown away. Even if this stopping rule is not used during data collection, coverage-based rarefaction throws away less data than traditional size-based rarefaction, and more efficiently finds the correct ranking of communities according to their true richnesses. Several hypothetical and real examples demonstrate these advantages.

It is vital to minimize the impact of errors for near-future quantum devices that will lack the resources for full fault tolerance. Two quantum error mitigation (QEM) techniques have been introduced recently, namely, error extrapolation [Y. Li and S. C. Benjamin, Phys. Rev. X 7, 021050 (2017); K. Temme et al., Phys. Rev. Lett. 119, 180509 (2017)] and quasiprobability decomposition [K. Temme et al., Phys. Rev. Lett. 119, 180509 (2017)]. To enable practical implementation of these ideas, here we account for the inevitable imperfections in the experimentalist’s knowledge of the error model itself. We describe a protocol for systematically measuring the effect of errors so as to design efficient QEM circuits. We find that the effect of localized Markovian errors can be fully eliminated by inserting or replacing some gates with certain single-qubit Clifford gates and measurements. Finally, having introduced an exponential variant of the extrapolation method we contrast the QEM techniques using exact numerical simulation of up to 19 qubits in the context of a “swap” test circuit. Our optimized methods dramatically reduce the circuit’s output error without increasing the qubit count.

Stochastic algorithms are critical in addressing complex rural pipe networks and non-convex stochastic optimization problems. With the development of artificial intelligence, large-scale optimization problems that cannot be solved effectively by traditional optimization methods have emerged. Therefore, stochastic optimization algorithms are rapidly developing in the field of optimization. This paper introduces an inertial extrapolation stochastic BFGS (IESBFGS) algorithm, an innovative amalgamation of the inertial extrapolation technique and the finite memory quasi-Newton algorithm to solve nonconvex stochastic optimization problems. Firstly, the inertial extrapolation technique is employed to track the iteration point to the optimal x-value. Second, it is combined with a finite-memory proposed Newton algorithm thereby increasing the convergence speed. Then, the superiority of IESBFGS is verified by comparing the experimental performance with other better algorithms on machine learning SVM model and ERM model. Finally, it is shown that the algorithm offers good prospects for solving nonconvex problems.

Data on the emotionality of words is important for the selection of experimental stimuli and sentiment analysis on large bodies of text. While norms for valence and arousal have been thoroughly collected in English, most languages do not have access to such large datasets. Moreover, theoretical developments lead to new dimensions being proposed, the norms for which are only partially available. In this paper, we propose a transformer-based neural network architecture for semantic and emotional norms extrapolation that predicts a whole ensemble of norms at once while achieving state-of-the-art correlations with human judgements on each. We improve on the previous approaches with regards to the correlations with human judgments by Δ r = 0.1 on average. We precisely discuss the limitations of norm extrapolation as a whole, with a special focus on the introduced model. Further, we propose a unique practical application of our model by proposing a method of stimuli selection which performs unsupervised control by picking words that match in their semantic content. As the proposed model can easily be applied to different languages, we provide norm extrapolations for English, Polish, Dutch, German, French, and Spanish. To aid researchers, we also provide access to the extrapolation networks through an accessible web application.

A comprehensive understanding of the dynamics of tribological interactions is essential for enhancing efficiency and durability in a multitude of technical domains. Conventional experimental techniques in tribology are frequently costly and time-consuming. In contrast, elastohydrodynamic lubrication (EHL) simulation models present a viable alternative for calculating frictional forces in sealing contacts. These calculations are based on the hydrodynamics within the sealing contact, as defined by the Reynolds equation, the deformation of the seal, and the contact mechanics. However, a significant drawback of these simulations is the time-consuming calculation process. To overcome these experimental and computational limitations, machine learning algorithms offer a promising solution. Physics-informed machine learning (PIML) improves on traditional data-driven models by incorporating physical principles. In particular, physics-informed neural networks (PINNs) are as effective hybrid solvers that combine data-driven and physics-based methods to solve the partial differential equations that drive EHL simulations. By integrating physical laws into the parameter optimization of the neural network (NN), PINNs provide accurate and fast solutions. Thus, unlike traditional NNs, PINNs have the potential to make accurate predictions beyond the limited training domain. The objective of this study is to demonstrate the feasibility of spatial and temporal extrapolation of the PINN and to analyze its reliability, both with and without consideration of cavitation. Two test cases are employed to examine the pressure and cavitation distribution within a sealing contact that extends beyond the spatial and temporal training range. The findings indicate that PINNs can surmount the typical constraints associated with NNs in the extrapolation of solution spaces, which represents a notable advancement in terms of computational efficiency and model flexibility.

The Poisson equation with arbitrary domains is considered in this paper using an extrapolation technique. The two-dimensional problem is primarily examined, and the extrapolation formulas O2 and O4 are thoroughly deduced. Not only the effect of different extrapolations on boundary point errors is analyzed, but a strategy for selecting extrapolation points is also provided. Finally, numerical accuracy results are presented.