Discrete Scale Research Articles

MODELING OF VIBRATIONAL PROCESSES IN DISCRETE MATRIX STRUC-TURES APPROACH

In the article are presented general principles of modeling vibrations in discrete structures formed in the form of special matrix forms of the Latin square (Sudoku type) are presented. The signs of structural and functional self-similarity for the matrix structures of standard Sudoku grids are formulated. It is shown that the structural principle can be interpreted as the implementation of the second iteration in the scale scaling algorithm characteristic of fractal objects. The signs of functional self-similarity of structures include the property of additive conservation of grid shapes to the requirements of Sudoku, which is formulated as a theorem. It is proved that the matrix sums of Sudoku constants and grids of arbitrary sizes, obtained taking into account the introduced cyclic ranking rule, will satisfy the three required Sudoku requirements. It is determined that by performing a given sequence of group shift operators, it is possible to establish a specific scenario for dynamically changing the state of a structure on a discrete time scale. It has been established that the evolution operators of linear-type group translations lead to matrix transformations of Sudoku grids from the set of equivalent ones (concerning the original ones), and the vortex-type group shifts operators to matrix transformations from many essentially different networks. The modes of harmonic, chaotic, and hybrid vibrations for structures of arbitrary size are considered. The requirements for transformations of the operators of the evolution of structures that provide the implementation of the considered modes are formulated. The results of modeling chaotic oscillatory processes by cycles of states of a discrete system that form similarities of attractor paths are analyzed. The principle of synchronization of chaotic states of matrix structures is established. The possibility of simulating the modes of beatings of oscillations in discrete cellular structures organized in the form of two-level matrix forms is substantiated. Specific examples show the results of simulating beatings of oscillations in cycles of changing states of a discrete system for two types of beats: similar to the result of a superposition of harmonic vibrations at multiple frequencies in the theory of radio signals, as well as noise-like beats.

GIS approaches to creating maps based on vegetation indices for forest management

The article considers the main approaches to providing information support for improving forest management on the territory, taking into account the most up-to-date experience, modern approaches and implementing sustainable forestry management based on the data of remote sensing and geoinformational modelling. To identify the state of vegetation in the area under analysis, the authors of the article propose to introduce the definition of a normalized differential vegetation index. A characteristic feature of vegetation and its state is spectral reflectivity, which is characterized by substantial differences in reflecting radiation from different wavelengths. To obtain the necessary MODIS packages, we used the LAADS DAAC service supported by NASA. The data of this service are freely accessible, convenient for the user, and they are taken from a large number of satellites and their spectra. The optimal option for solving the tasks of compiling NDVI maps is using the geographic information system QGIS. Among the main advantages of this GIS we can consider user-friendliness and its multi-platform basis. For proper visualization of the NDVI map, a special discrete scale of values is used, the purpose of which is to differentiate between organic and non-organic nature, as well as determine the density of vegetation. The introduction of the proposed technology will significantly increase the mobility and effectiveness of forest management.

Counting, measuring, and the fractional cardinalities puzzle

According to what I call the Traditional View, there is a fundamental semantic distinction between counting and measuring, which is reflected in two fundamentally different sorts of scales: discrete cardinality scales and dense measurement scales. Opposed to the Traditional View is a thesis known as the Universal Density of Measurement: there is no fundamental semantic distinction between counting and measuring, and all natural language scales are dense. This paper considers a new argument for the latter, based on a puzzle I call the Fractional Cardinalities Puzzle: if answers to ‘how many’-questions always designate cardinalities, and if cardinalities are necessarily discrete, then how can e.g. ‘2.38’ be a correct answer to the question ‘How many ounces of water are in the beaker?’? If cardinality scales are dense, then the answer is obvious: ‘2.38’ designates a fractional cardinality, contra the Traditional View. However, I provide novel evidence showing that ‘many’ is not uniformly associated with the dimension of cardinality across contexts, and so ‘how many’-questions can ask about other kinds of measures, including e.g. volume. By combining independently motivated analyses of cardinal adjectives, measure phrases, complex fractions, and degrees, I develop a semantics intended to defend the Traditional View against purported counterexamples like this and others which have received a fair amount of recent philosophical attention.

Discrete-time survival forests with Hellinger distance decision trees

Random survival forests (RSF) are a powerful nonparametric method for building prediction models with a time-to-event outcome. RSF do not rely on the proportional hazards assumption and can be readily applied to both low- and higher-dimensional data. A remaining limitation of RSF, however, arises from the fact that the method is almost entirely focussed on continuously measured event times. This issue may become problematic in studies where time is measured on a discrete scale t = 1, 2, ..., referring to time intervals [0,a_1), [a_1,a_2), ldots . In this situation, the application of methods designed for continuous time-to-event data may lead to biased estimators and inaccurate predictions if discreteness is ignored. To address this issue, we develop a RSF algorithm that is specifically designed for the analysis of (possibly right-censored) discrete event times. The algorithm is based on an ensemble of discrete-time survival trees that operate on transformed versions of the original time-to-event data using tree methods for binary classification. As the outcome variable in these trees is typically highly imbalanced, our algorithm implements a node splitting strategy based on Hellinger’s distance, which is a skew-insensitive alternative to classical split criteria such as the Gini impurity. The new algorithm thus provides flexible nonparametric predictions of individual-specific discrete hazard and survival functions. Our numerical results suggest that node splitting by Hellinger’s distance improves predictive performance when compared to the Gini impurity. Furthermore, discrete-time RSF improve prediction accuracy when compared to RSF approaches treating discrete event times as continuous in situations where the number of time intervals is small.

A Bayesian Nonparametric Latent Approach for Score Distributions in Test Equating

Equating is a family of statistical models and methods used to adjust scores on different test forms so that they can be comparable and used interchangeably. Equated scores are obtained estimating the equating transformation function, which maps the scores on the scale of one test form into their equivalents on the scale of other one. All the statistical models that have been proposed for estimating this function are based on continuous approximations of the score distributions, leading to equated scores lying on a continuous scale even though score scales are usually subsets of the integer numbers (e.g., the total number of correct answers). In this article, we develop a new equating method from which equated scores defined on the original discrete scale are obtained. Considering scores as ordinal random variables, we propose a continuous latent variable formulation to perform an equipercentile-like equating based on a Bayesian nonparametric model for score distributions. The proposed model is applied to simulated and real data collected under an equivalent group design. Some methods to assess the performance of our model are also discussed. Compared with discrete versions of equated scored obtained from traditional equating methods, the results show that the proposed method has better performance.

Entropy-based analysis of rhythmic tapping for the quantitative assessment of cerebellar ataxia

The severity of appendicular ataxia is currently assessed by neurological specialists observing the motor deficits during the performance of bedside tests. These tests are traditionally restricted to discrete subjective scales and prescribed by constraints that may in turn affect the subject's performance. To overcome these limitations and facilitate uniform monitoring of cerebellar ataxia, an instrumented assessment system using a wearable sensor has been developed for the rhythmic finger and foot tapping tasks. Departing from the conventional obligation to spatiotemporal parameters that are described clinically, the proposed approach investigates features based on information theory to evaluate the variability of the tapping dynamics. From a clinical trial of 24 healthy subjects and 34 patients diagnosed with cerebellar ataxia (CA), the findings revealed not only are there more variations in movement patterns and timing of tapping events in CA but that movement complexity was also degraded across multiple entropy scales. The system using combined models of different features was able to achieve an average classification accuracy of 81% and 79% in finger and foot tapping respectively. The outcomes strongly agreed with the clinical scale widely-used for the assessment and rating of ataxia (SARA) to a correlation level of 0.71 (p < 0.001) in finger tapping and 0.59 (p < 0.001) in foot tapping.

Scalar self-force for high spin black holes

We semianalytically investigate the scalar self-force experienced in the final stages of extreme mass ratio inspirals of nonspinning scalar particles into supermassive nearly extremal Kerr black holes. We exploit the near-horizon conformal symmetry to find the self-force for general corotating equatorial geodesics. The angular component of the self-force is shown to be universal at leading order in the high spin limit. We verify that the energy and angular momentum losses of the scalar particle match with the asymptotic fluxes of scalar radiation. In particular, we relate the previously described persistent oscillations in the asymptotic energy and angular momentum fluxes with the local self-force. Such oscillations arise from traveling waves that prevent the near-horizon and the asymptotic region to fully decouple in the extremal limit. Conformal invariance is therefore reduced to discrete scale invariance with associated logarithmic periodicity.

Naturalness in nuclear effective field theories

Nuclear effective field theories (EFTs) have been developed over the last quarter-century with considerable impact on the description of light and even medium-mass nuclei. At the core of any EFT is a systematic expansion of observables, which is usually obtained from a rule based on an assumption of naturalness. I discuss naturalness in the context of the relatively weak binding of nuclei, where discrete scale invariance plays a role in the emergence of complexity.

Topography scale effects on groundwater-surface water exchange fluxes in a Canadian Shield setting

Exchange fluxes occur when hydrological processes transfer water across the topographic surface between the groundwater and surface water systems. Many hydrological models are capable of estimating exchange fluxes at point-scales, however as models, including integrated hydrological models, are upscaled to encompass larger areas using non-point scale discretizations, the accuracy of hydrological process estimation declines such that the results can become empirical. Through analyses of topographic indices (e.g., slope, roughness, wetness index), hydraulic heads, and surface water/groundwater exchange fluxes at multiple horizontal spatial scales, this study indicates that it is possible to identify the scale dependencies and inherent empiricism in shallow groundwater hydrological models which contribute to variations in exchange flux estimates, and hence groundwater recharge/discharge rates, at different spatial scales. By identifying the scale dependencies and empiricism related to horizontal spatial scale choice, it should be possible to improve conceptual design choices when attempting to simulate locations with shallow water tables. This is important as a better understanding of the spatial variability of exchange fluxes will allow for better predictions of some catchment behaviours. The characterization of the models in this study are a reasonable approximation of a Canadian Shield site based upon field collected parameters. An analysis of the impact of model discretization scale on exchange fluxes, and specifically recharge, should be a component of all model studies.

MANAGING CONSENSUS BY MULTI-STAGE OPTIMIZATION MODELS WITH LINGUISTIC PREFERENCE ORDERINGS AND DOUBLE HIERARCHY LINGUISTIC PREFERENCES

Preference ordering structures are useful and popular tools to represent experts’ preferences in the decision making process. In the existing preference orderings, they lack the research on the precise relationship between any two adjacent alternatives in the preference orderings, and the decision making methods are unreasonable. To overcome these issues, this paper establishes a novel concept of linguistic preference ordering (LPO) in which the ordering of alternatives and the relationships between two adjacent alternatives should be fused well, and develops two transformation models to transform each LPO into the corresponding double hierarchy linguistic preference relation with complete consistency. Additionally, to fully respect the experts’ expression habits and provide more refined solutions to experts, this paper establishes a multi-stage consensus optimization model by considering the suggested preferences represented in both the continuous scale and the discrete scale, and develops a multi-stage interactive consensus reaching algorithm to deal with multi-expert decision making problem with LPOs. Furthermore, some numerical examples are presented to illustrate the developed methods and models. Finally, some comparative analyses between the proposed methods and models and some existing methods have been made to show the advantages of the proposed methods and models.

A Comparative Assessment of Graphic and 0–10 Rating Scales Used to Measure Entrepreneurial Competences

This article presents an empirical comparative assessment of the measurement quality of two instruments commonly used to measure fuzzy characteristics in computer-assisted questionnaires: a graphic scale (a line production scale using a slider bar) and an endecanary scale (a 0–10 rating scale using radio buttons). Data are analyzed by means of multitrait–multimethod models estimated as structural equation models with a mean and covariance structure. For the first time in such research, the results include bias, valid variance, method variance, and random error variance. The data are taken from a program that assesses entrepreneurial competences in undergraduate Economics and Business students by means of questionnaires administered on desktop computers. Neither of the measurement instruments was found to be biased with respect to the other, meaning that their scores are comparable. While both instruments achieve valid and reliable measurements, the reliability and validity are higher for the endecanary scale. This study contributes to the still scarce literature on fuzzy measurement instruments and on the comparability and relative merits of graphic and discrete rating scales on computer-assisted questionnaires.

Measuring well-being dispersion on discrete rating scales

Average levels of well-being across countries and time periods have been reported widely over the years as one of the measures that go beyond the Gross Domestic Product. It is relevant for policy makers to also have information on well-being dispersion to identify groups that lag behind with respect to their ability to pursue happiness. In addition, there is an inherent moral appeal to not only maximize wellbeing but also equalize well-being among people. In this paper, we try to answer the following research question: which measure(s) should be used to gain insight into wellbeing dispersion? We review sixteen measures and their properties, study their behavior in over 92 thousand simulated distributions, and apply them to 4 years of the Dutch Social Cohesion and Well-Being Survey, using bootstrapping to quantify their precision. Our inventory shows that when applied to a discrete ordinal rating scale, common measures such as the standard deviation and Gini coefficient do not show any advantage over the less restrictive index of ordinal variation (IOV). Only the generalized entropy and Atkinson index adhere to additional principles, most notably of diminishing transfers, at the expense of full scale invariance. The simulation study illustrates that dispersion measures are positively correlated but do not rank distributions the same. The field study shows that only the Atkinson index with high inequality aversion provides additional insight. We recommend using the index of ordinal variation, supplemented if needed with the Atkinson index using a high value for the inequality aversion parameter.

The Impact of Imperfect Spectrum Sensing on the Performance of LTE Licensed Assisted Access Scheme

The energy detection technology is adopted as the detection method of the unlicensed channel in LTE release 13. However, the detection may be imperfect in actual scenario due to the simplicity of the detection method. In order to eliminate the controversy over LTE licensed assisted access (LAA) scheme, the paper deeply investigates the performance of the scheme under imperfect spectrum sensing (ISS). Considering possible ISS during the enhanced clear channel assessment (eCCA), the LAA scheme is modeled as a new two dimensional discrete time markov chain. In order to capture ISS, different from the definition in Bianchi's model, the discrete time scale is defined as the end of backoff slot time in the proposed model. The definition enables LAA small base stations to enter the defer state from backoff state when the channel is detected to be busy, and further possible ISS in the defer period and backoff slot time (BST) of eCCA can be considered. Based on the proposed model, the expressions of collision probabilities and throughput under three backoff mechanisms based LAA schemes are derived to deeply analyze their performance by comparisons. A large number of experimental and analytical results prove the validity of the proposed model. The analytical results also show that not only the backoff mechanisms but also ISS can greatly affect the network performance. In the respect of fairness (i.e. the impact on WiFi users), the WiFi collision probability always increases with the decrease of the false alarm probability (FAP). This means the WiFi collision probabilities under three LAA schemes may be higher than the baseline level for the graceful coexistence when FAP is very little. In this sense, the LAA schemes cannot be referred to as being graceful though they have been proven to be deterministic graceful coexistence schemes under perfect spectrum sensing (PSS). In the respect of throughput, LAA throughput always decreases and WiFi throughput increases with the growth of FAP, and total throughput always decreases. Therefore, based on the fixed SNR and sampling rate in actual scenario, the network performance can be tuned by energy threshold, as is consistent with the viewpoint in other literatures.

Feature extraction of point clouds based on region clustering segmentation

This paper proposes a feature extraction method for scattered point clouds. First, a clustering algorithm is used to divide point clouds into different regions that represent the original features. In each sub-region, we calculate the angles between the directed line segments from sampling points to the neighborhood points and set the angle threshold to identify edge feature points of uniform distribution. For the edge points of non-uniform distribution, we introduce the local neighborhood size as a discrete scale parameter for edge point detection, and then accurately identify and record the detected edge points. Then, according to the mean curvature of point clouds, the local feature weights of sampling points in the sub-region are calculated so that potential sharp feature points in a local area are detected. Finally, a minimum spanning tree of feature points is established to construct connected regions and generate feature point sets. A Bidirectional Principal Component Analysis (BD-PCA) search method is used to trim and break the small branches and multiline segments to generate feature curves. We carried out experiments on point cloud models with different densities to verify the effectiveness and superiority of our method. Results show that the edge features and sharp features are effectively extracted, and our method is not affected by the noise, neighborhood scale, or quality of sampling.

Nonequilibrium Fixed Points of Coupled Ising Models.

Driven-dissipative systems are expected to give rise to nonequilibrium phenomena that are absent in their equilibrium counterparts. However, phase transitions in these systems generically exhibit an effectively classical equilibrium behavior in spite of their nonequilibrium origin. In this paper, we show that multicritical points in such systems lead to a rich and genuinely nonequilibrium behavior. Specifically, we investigate a driven-dissipative model of interacting bosons that possesses two distinct phase transitions: one from a high- to a low-density phase—reminiscent of a liquid-gas transition—and another to an antiferromagnetic phase. Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) symmetry. However, they coalesce at a multicritical point, giving rise to a nonequilibrium model of coupled Ising-like order parameters described by a symmetry. Using a dynamical renormalization-group approach, we show that a pair of nonequilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system. We elucidate various exotic features of these NEFPs. In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance. This further results in complex-valued critical exponents and spiraling phase boundaries, and it is also accompanied by a complex Liouvillian gap even close to the phase transition. As direct evidence of the nonequilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes “hotter” and “hotter” at longer and longer wavelengths. Finally, we argue that this nonequilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.

Maximal operators for cube skeletons

We study discretized maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally, $k$-skeletons in $\mathbb{R}^n$. Although these operators are known not to be bounded on any $L^p$, we obtain nearly sharp $L^p$ bounds for every small discretization scale. These results are motivated by, and partially extend, recent results of T. Keleti, D. Nagy and P. Shmerkin, and of R. Thornton, on sets that contain a scaled $k$-sekeleton of the unit cube with center in every point of $\mathbb{R}^n$.

Fast and Unbiased Estimation of Volume Under Ordered Three-Class ROC Surface (VUS) With Continuous or Discrete Measurements

Receiver Operating Characteristic (ROC) surfaces have been studied in the literature essentially during the last decade and are considered as a natural generalization of ROC curves in three-class problems. The volume under the surface (VUS) is useful for evaluating the performance of a trichotomous diagnostic system or a three-class classifier’s overall accuracy when the possible disease condition or sample belongs to one of three ordered categories. In the areas of medical studies and machine learning, the VUS of a new statistical model is typically estimated through a sample of ordinal and continuous measurements obtained by some suitable specimens. However, discrete scales of the prediction are also frequently encountered in practice. To deal with such scenario, in this paper, we proposed a unified and efficient algorithm of linearithmic order, based on dynamic programming, for unbiased estimation of the mean and variance of VUS with unidimensional samples drawn from continuous or non-continuous distributions. Monte Carlo simulations verify our theoretical findings and developed algorithms.

An adaptive quasicontinuum approach for modeling fracture in networked materials: Application to modeling of polymer networks

Materials with network-like microstructure, including polymers, are the backbone for many natural and human-made materials such as gels, biological tissues, metamaterials, and rubbers. Fracture processes in these networked materials are intrinsically multiscale, and it is computationally prohibitive to adopt a fully discrete approach for large scale systems. To overcome such a challenge, we introduce an adaptive numerical algorithm for modeling fracture in this class of materials, with a primary application to polymer networks, using an extended version of the Quasicontinuum method that accounts for both material and geometric nonlinearities. In regions of high interest, for example near crack tips, explicit representation of the local topology is retained where each polymer chain is idealized using the worm like chain model. Away from these imperfections, the degrees of freedom are limited to a fraction of the network nodes and the network structure is computationally homogenized, using the micro-macro energy consistency condition, to yield an anisotropic material tensor consistent with the underlying network structure. A nonlinear finite element framework including both material and geometric nonlinearities is used to solve the system where dynamic adaptivity allows transition between the continuum and discrete scales. The method enables accurate modelling of crack propagation without a priori constraint on the fracture energy while maintaining the influence of large-scale elastic loading in the bulk. We demonstrate the accuracy and efficiency of the method by applying it to study the fracture in different examples of network structures. We further use the method to investigate the effects of network topology and disorder on its fracture characteristics. We discuss the implications of our method for multiscale analysis of fracture in networked material as they arise in different applications in biology and engineering.

Horizon molecules in causal set theory

We propose a new definition of "horizon molecules" in Causal Set Theory following pioneering work by Dou and Sorkin. The new concept applies for any causal horizon and its intersection with any spacelike hypersurface. In the continuum limit, as the discreteness scale tends to zero, the leading behaviour of the expected number of horizon molecules is shown to be the area of the horizon in discreteness units, up to a dimension dependent factor of order one. We also determine the first order corrections to the continuum value, and show how such corrections can be exploited to obtain further geometrical information about the horizon and the spacelike hypersurface from the causal set.

Stoichiometric knife-edge model on discrete time scale

Ecological stoichiometry is the study of the balance of multiple elements in ecological interactions and processes (Sterner and Elser in Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, 2002). Modeling under this framework enables us to investigate the effect nutrient content on organisms whether the imbalance involves insufficient or excess nutrient content. This phenomenon is called the “stoichiometric knife-edge”. In this paper, a discrete-time predator–prey model that captures this phenomenon is established and qualitatively analyzed. We systematically expound the similarities and differences between our discrete model and the corresponding continuous analog. Theoretical and numerical analyses show that while the discrete and continuous models share many properties, differences also exist. Under certain parameter sets, the models exhibit qualitatively different dynamics. While the continuous model shows limit cycle, Hopf bifurcation, and saddle-node bifurcation, the discrete-time model exhibits richer dynamical behaviors, such as chaos. By comparing the dynamics of the continuous and discrete model, we can conclude that stoichiometric effects of low food quality on predators are robust to the discretization of time. This study can possibly serve as an example for pointing to the importance of time scale in ecological modeling.