Discrete Scale Research Articles

A Gompertz distribution for time scales

We investigate a family of probability distributions, with three parameters associated with the dynamic Gompertz function. We prove its existence for various parameter sets and discuss the existence of its time scale moments. Afterwards, we investigate the special case of discrete time scales, where it is shown that the discrete Gompertz distribution is a $q$-geometric distribution of the second kind. Further, we find their $q$-binomial moments, we bound their expected value, and we show how a classical Gompertz distribution is obtained from them.

Efimov effect at the Kardar-Parisi-Zhang roughening transition.

Surface growth governed by the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than two undergoes a roughening transition from smooth to rough phases with increasing the nonlinearity. It is also known that the KPZ equation can be mapped onto quantum mechanics of attractive bosons with a contact interaction, where the roughening transition corresponds to a binding transition of two bosons with increasing the attraction. Such critical bosons in three dimensions actually exhibit the Efimov effect, where a three-boson coupling turns out to be relevant under the renormalization group so as to break the scale invariance down to a discrete one. On the basis of these facts linking the two distinct subjects in physics, we predict that the KPZ roughening transition in three dimensions shows either the discrete scale invariance or no intrinsic scale invariance.

Crown-scheduling of sets of parallelizable tasks for robustness and energy-elasticity on many-core systems with discrete dynamic voltage and frequency scaling

Crown scheduling is a static scheduling approach for sets of parallelizable tasks with a common deadline, aiming to minimize energy consumption on parallel processors with frequency scaling. We demonstrate that crown schedules are robust, i. e. that the runtime prolongation of one task by a moderate percentage does not cause a deadline transgression by the same fraction. In addition, by speeding up some tasks scheduled after the prolonged task, the deadline can still be met at a moderate additional energy consumption. We present a heuristic to perform this re-scaling online and explore the tradeoff between additional energy consumption in normal execution and limitation of deadline transgression in delay cases. We evaluate our approach with scheduling experiments on synthetic and application task sets. Finally, we consider influence of heterogeneous platforms such as ARM’s big.LITTLE on robustness.

THE SUMUDU TRANSFORM ON DISCRETE TIME SCALES

The study of dynamic equations on time scale is an area of mathematics that has recently received a lot of attention. This type of calculus has been created in order to unify the study of discrete and continuous analysis. Integral transforms play a crucial role in analysis in solving differential and difference equations. In this paper, we introduce the Sumudu transform on two different time scales by using the theory of time scale calculus. Finally, we employ these definitions to derive the results like, convolution, delay(shift) and inversion on these discrete time scales. Further, these results coincide with those in continuous case.

Piecewise Multi-Parameter Tangent/Cotangent Function and Its Application in Chaotic Pseudorandom Bit Generator

To increase the number of parameters of 1-dimensional discrete chaotic mappings, employing translation & scaling, this letter creatively designs a piecewise 5-parameters Tangent/Cotangent Function. Both bifurcation diagram and Lyapunov exponent spectrum illustrate that the proposed function seems to be inappropriate for cryptographic applications, due to its narrow chaotic area. However, after applying it to devising chaotic pseudorandom bit generator, it outperforms skew tent mapping, whose chaotic area is full. It might be applied to some other fields as well, such as cryptographic hash functions, which will possibly be paid attention to in the future.

Extended log periodic approach in analysing local critical behaviour–case study for Covid -19 spread in Albania.

Log-periodic (LP) functions of the general form y = y 0 + A(t − t c ) m (1 + Bcos(ω ∗ log(t − t c ) + ϕ 1)) have been demonstrated effective in the analysis of the processes characterized by the discrete scale of invariance (DSI) structure and also self-organization behaviour. If other self-organization processes opposing or supporting the principal activity would be present, a multilevel DSI structure is expected to develop and the resulting dynamics would depart from the log-periodic shape. When discussing the processes characterising the daily new positive records for the COVID-19 cases in the country (Albania) we have identified the elements that are responsible to generate modified LP behaviour. The new records that initially represented simply the findings of the state laboratories, are modified each successive days by the pressure for more tests form anxious individuals and other effects which produce a herding behaviour which emergence a LP dynamics. Meanwhile, the reactive behaviour aiming to oppose undesired occurrence would generate additional LP sub-processes that can be trapped by a modified LP function of the form y = y 0 + A(t − t c ) m + B(t − t c ) m cos (ω ∗ log(t − t c ) + ϕ 1) + C cos( (ω − ω 1 ) log(t − t c ) + ϕ 2)+D(t − t c ) m cos ((ω + ω 1) log(t − t c ) + ϕ 3). During the period of the self-organization behaviour the process is highly nonlinear and therefore the classical models (SIR) based on the mean field assumption and the corresponding ODE equations are not effective to represent the system dynamics. In our case the LP fit to the Covid-19 new cases data series, has challenged the ODE models for the time interval of initial appearance of the positive cases in the country (as of 2 March 2020) up to 3 month later. Also, the LP function has predicated the multiphase waving behaviour and we have forecasted two peak, each of them weeks before recurrence, respectively at 28 April and 10-12 June. Those peaks have been confirmed later within 2 days uncertainty. It resulted that the first regime have been succeeded by another new self-organization regime due to drastic condition changes as result of the socio-economic opening which started in the end of the May 2020. As result, the new regime is juxtaposed over the old one and the LP dynamics remained characteristic and another critical time has appeared. The new critical time has been reproduced with good certainty (1 August) and also the magnitude of the new cases. We observed that our empirical LP function is effective in the describing long term dynamics whereas local techniques involving neural networks approaches have reproduced very well the new cases after the first regime. We concluded that after identification of the LP regimes, we can adopt successfully short time forecasting for new occurrence when working away from the critical time identified by the first method.

Multi-Scale Graph Convolutional Network With Spectral Graph Wavelet Frame

Graph neural networks have achieved impressive progress in solving large and complex graph-structured problems. However, existing methods cannot sufficiently explore the advantages of multi-scale information by enlarging the receptive fields through stacking deep layers or multiplying exponentiated graph Laplacian. They are restricted by the discrete scales rigidly dependent on node-hops or suffer from the over-smoothing issue incurred by long-range low-pass filtering. In this paper, we propose a multi-scale graph convolutional network based on the spectral graph wavelet frame to improve multi-scale representation learning. The proposed network flexibly leverages multi-scale neighboring information with continuous scaling to enhance the discrimination ability of the learned multi-scale representations and offer stable feature extraction ensured by frame bounds in theory. The multi-scale spectral convolutional layer is constructed with a low-pass filter and a sequence of dilated band-pass filters to achieve well-established localization in both vertex and frequency domains. The graph convolutional filters can be realized based on spectral graph wavelet kernel functions analytically formulated using parameterized band-limited filters or adaptively learned from training data. Furthermore, fast approximations of exact spectral filtering are developed for memory efficient operations on sparse tensors with guaranteed prediction accuracy and numerical stability. The proposed network is shown to achieve state-of-the-art performance in extensive experiments on citation networks, bioinformatics graphs, and social networks.

Establishment of a multilevel linear model to analyse the factors affecting piglet litter performance at birth.

This study aimed to establish a feasible model for analysing factors affecting piglet litter performance at birth. Data of 61,984 litters were collected from 16 herds, and general linear model (GLM), multilevel Poisson regression model (MPM) and multilevel linear model (MLM) were established to compare their goodness of fit for these data. Influencing factors of piglet litter performance at birth were analysed using the established optimal model. Results showed the intraclass correlation coefficients of total born piglets (TBP), piglets born alive (PBA), low-birth-weight piglets (LBW), and average birth weight of piglets (ABW) reached 27.89%, 23.88%, 24.66% and 22.27%, respectively (p<.05). Akaike's information criterion and Bayesian information criterion in MLM of TBP, PBA, LBW and ABW were lower than those in GLM. Pearson residuals in MPM increased to nearly 1 after introduction of a discrete scale factor, and the p values in MPM were similar to those in MLM. Analyses of MLM indicated crossbred sows with good management supplemented with oregano essential oil and farrowing at warm season had higher TBA, PBA and ABW, but lower LBW than other sows (p<.05). In conclusion, MLM is superior to GLM and can replace MPM in analysing discrete data with hierarchical structure in pig production. More importantly, other potential influencing factors of litter performance at birth can be analysed using the established MLM in the future.

Bait-lamina test for assessment of polluted soils: Rough vs. Precise scales

The bait-lamina test is one of the few available methods for measuring the functional activity of soil animals, which was standardized for soil health assessment by ISO 18311. The bait consumption is measured visually on discrete scales. A two-point scale (0 or 1) is used more often, but the threshold for bait consumption scores varies from study to study: 1 point is assigned either to a hole that has been perforated to any extent, or to a hole that has been at least half perforated, or to a fully empty hole. Less often, a three-point scale (0, 0.5, and 1) or a five-point scale (0, 0.25, 0.5, 0.75, and 1) can be used. We investigated how much the use of different scales can influence statistical inferences.We have established that a point on a five-point scale is directly proportional to the proportion of the mass consumption of bait, so in this scale the value of feeding activity is an acceptable surrogate for the bait consumption rate. We analyzed outcomes of the bait-lamina test for 7 measurement rounds at the control area and an area highly polluted with metals near the copper smelter. We assumed the most accurate five-point scale to be the reference and compared it to six less precise scales with fewer points (two three-point and four two-point scales). The difference between the reference and such rough scales depends on the percentage of intermediate points (0.25, 0.5, and 0.75). It turned out that intermediate values are by no means rare (from 11% to 100% of all non-zero values), and in the polluted area they are almost twice as common as in the control area (66% and 28%, respectively). Because of this, the relative difference between the reference scale and rough scales is 2–4 times greater at the polluted site than at the control site. This can lead to a bias in the effect size index: for the rough scales, the log Response Ratio can either double or decrease by one-third relative to the reference scale. This increases the probability of type I and II errors in statistical hypothesis testing.Thus, the outcomes of the bait-lamina test are not invariant relative to the measuring procedure. The three-point scale and ISO 18311 two-point scale are the least biased relative to the reference scale. The use of other rough scales carries a risk of artifacts and should be avoided.

Modelling imperfections in unreinforced masonry structures: Discrete element simulations and scale model experiments of a pavilion vault

The structural assessment of doubly-curved vaulted masonry structures, such as pavilion vaults, poses challenges specific to their high degree of indeterminacy. Two-dimensional equilibrium analysis methods may provide a lower bound of load or displacement capacity, but they do not accurately describe the three-dimensional (3D) behaviour of these structures, particularly when shear deformation (e.g. sliding) is important. Therefore, discrete element modelling (DEM) methods, which can effectively simulate 3D load re-distribution, have been used to investigate support displacement capacity and corresponding 3D collapse mechanisms. DEM analyses are usually conducted on perfect digital geometries. Meanwhile, both real structures and small-scale physical models have implicit assembly and fabrication imperfections, which may significantly alter their response. The present paper aims to investigate the influence of geometrical and mechanical imperfections by comparing DEM analyses with the results obtained from tests on a scale model. In particular, a new method to simulate imperfections within the DEM framework is proposed, and a DEM parametric analysis is compared to the measured behaviour of a 3D-printed scale model of a pavilion (or cloister) vault on spreading supports. The influence of both mechanical imperfections andgeometrical imperfections, due to element geometry deviations or imprecision of the assembly process, have been investigated. Based on these analyses, the three-dimensional behaviour of a pavilion vault subjected to horizontal displacement of the supports is described, and the variability of results due to imperfections is demonstrated. A good agreement between DEM analyses and 3D-printed scale model tests is achieved, in terms of crack patterns and mechanisms. Geometrical imperfections did change the load paths within the vault, as expected, and also influenced the displacement capacity.

OP206 Expert Elicitation Of Probabilistic Distributions to Inform Survival Modelling of CD19 Chimeric Antigen Receptor T-Cell Therapies

IntroductionIn 2018, the National Centre for Pharmacoeconomics (NCPE) was commissioned to conduct a health technology assessment (HTA) of one of the first commercially available chimeric antigen receptor (CAR) T-cell therapies, tisagenlecleucel. CAR T-cells are a major advance in personalized cancer treatment, demonstrating promising outcomes in relapsed/refractory pediatric acute lymphoblastic leukemia (pALL). However, the results are based on short-term follow up, limiting their value in predicting long-term survival and leading to uncertainty about the most appropriate survival modeling method to employ. This study aimed to address these limitations by means of expert elicitation.MethodsAn expert elicitation method, the histogram technique, was employed. A predefined discrete numerical scale was presented in Microsoft Excel® and the expert was asked to place twenty crosses on a frequency chart. These crosses represented the expert's beliefs about the distribution of particular quantities. Each cross represented five percent of the probabilistic distribution. Individual distributions were then aggregated across experts using linear pooling.ResultsA total of seventeen experts were invited to take part; eight agreed to participate and five completed the exercise. Three experts did not consider tisagenlecleucel to be a “curative” therapy because patients had a higher risk of death, compared with the age- and sex-matched general population. The aggregated distributions indicated the five-year overall survival rate to be thirty-three percent (95% CI 8.65–56.88) in patients who do not receive a subsequent stem cell transplant and twenty percent (95% CI 2.38 -52.04) in those who do.ConclusionsThe results of this study will be used to calibrate CD19 CAR T-cell therapy survival estimates presented in HTA submissions to the NCPE to ensure more robust assessments. They will also be used to inform the construction of a de novo cost-utility model for examining the cost effectiveness of CD19 CAR T-cell therapies for relapsed/refractory pALL in the Irish healthcare setting.

Tunable discrete scale invariance in transition-metal pentatelluride flakes

Log-periodic quantum oscillations discovered in transition-metal pentatelluride give a clear demonstration of discrete scale invariance (DSI) in solid-state materials. The peculiar phenomenon is convincingly interpreted as the presence of two-body quasi-bound states in a Coulomb potential. However, the modifications of the Coulomb interactions in many-body systems having a Dirac-like spectrum are not fully understood. Here, we report the observation of tunable log-periodic oscillations and DSI in ZrTe5 and HfTe5 flakes. By reducing the flakes thickness, the characteristic scale factor is tuned to a much smaller value due to the reduction of the vacuum polarization effect. The decreasing of the scale factor demonstrates the many-body effect on the DSI, which has rarely been discussed hitherto. Furthermore, the cut-offs of oscillations are quantitatively explained by considering the Thomas-Fermi screening effect. Our work clarifies the many-body effect on DSI and paves a way to tune the DSI in quantum materials.

Random-mass disorder in the critical Gross-Neveu-Yukawa models

An important yet largely unsolved problem in the statistical mechanics of disordered quantum systems is to understand how quenched disorder affects quantum phase transitions in systems of itinerant fermions. In the clean limit, continuous quantum phase transitions of the symmetry-breaking type in Dirac materials such as graphene and the surfaces of topological insulators are described by relativistic (2+1)-dimensional quantum field theories of the Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched random-mass disorder, both uncorrelated or with long-range power-law correlations. Using the replica method combined with a controlled triple epsilon expansion below four dimensions, we find a variety of new finite-randomness critical and multicritical points with nonzero Yukawa coupling between low-energy Dirac fields and bosonic order parameter fluctuations, and compute their universal critical exponents. Analyzing bifurcations of the renormalization-group flow, we find instances of the fixed-point annihilation scenario—continuously tuned by the power-law exponent of long-range disorder correlations and associated with an exponentially large crossover length—as well as the transcritical bifurcation and the supercritical Hopf bifurcation. The latter is accompanied by the birth of a stable limit cycle on the critical hypersurface, which represents the first instance of fermionic quantum criticality with emergent discrete scale invariance.

Additions to Philippine slender skinks of the Brachymeles bonitae complex (Reptilia: Squamata: Scincidae) IV: Resurrection and redescription of Brachymeles burksi

The diversity of Philippine amphibians and reptiles has increased over the last few decades, in part due to re-evaluation of species formerly believed to be widespread. Many of these investigations of widespread species have uncovered multiple closely related cryptic lineages comprising species complexes, each restricted to individual Pleistocene Aggregate Island Complexes (PAICs). One group in particular for which widespread cryptic diversity has been common is the clade of Philippine skinks of the genus Brachymeles. Recent phylogenetic studies of the formerly recognized widespread species Brachymeles bonitae have indicated that this species is actually a complex distributed across several major PAICs and smaller island groups in the central and northern Philippines, with numerous species that exhibit an array of digit loss and limb reduction patterns. Despite the recent revisions to the B. bonitae species complex, studies suggest that unique cryptic lineages still exist within this group. In this paper, we resurrect the species Brachymeles burksi Taylor 1917, for a lineage of non-pentadactyl, semi-fossorial skink from Mindoro and Marinduque islands. First described in 1917, B. burksi was synonymized with B. bonitae in 1956, and has rarely been reconsidered since. Evaluation of genetic and morphological data (qualitative traits, meristic counts, and mensural measurements), and comparison of recently-obtained specimens to Taylor’s original description support this species’ recognition, as does its insular distribution on isolated islands in the central portions of the archipelago. Morphologically, B. burksi is differentiated from other members of the genus based on a suite of unique phenotypic characteristics, including a small body size, digitless limbs, a high number of presacral vertebrae, the absence of auricular openings, and discrete (non-overlapping) meristic scale counts. The recognition of this central Philippine species further increases the diversity of non-pentadactyl members of the B. bonitae complex, and reinforces the biogeographic uniqueness of the Mindoro faunal region. KEYWORDS: biodiversity, endemism, faunal region, fossoriality, limb reduction

MobileVisFixer: Tailoring Web Visualizations for Mobile Phones Leveraging an Explainable Reinforcement Learning Framework.

We contribute MobileVisFixer, a new method to make visualizations more mobile-friendly. Although mobile devices have become the primary means of accessing information on the web, many existing visualizations are not optimized for small screens and can lead to a frustrating user experience. Currently, practitioners and researchers have to engage in a tedious and time-consuming process to ensure that their designs scale to screens of different sizes, and existing toolkits and libraries provide little support in diagnosing and repairing issues. To address this challenge, MobileVisFixer automates a mobile-friendly visualization re-design process with a novel reinforcement learning framework. To inform the design of MobileVisFixer, we first collected and analyzed SVG-based visualizations on the web, and identified five common mobile-friendly issues. MobileVisFixer addresses four of these issues on single-view Cartesian visualizations with linear or discrete scales by a Markov Decision Process model that is both generalizable across various visualizations and fully explainable. MobileVisFixer deconstructs charts into declarative formats, and uses a greedy heuristic based on Policy Gradient methods to find solutions to this difficult, multi-criteria optimization problem in reasonable time. In addition, MobileVisFixer can be easily extended with the incorporation of optimization algorithms for data visualizations. Quantitative evaluation on two real-world datasets demonstrates the effectiveness and generalizability of our method.

Discrete scaling based on operator theory

Signal scaling is a fundamental operation of practical importance in which a signal is made wider or narrower along the coordinate direction(s). Scaling, also referred to as magnification or zooming, is complicated for signals of a discrete variable since it cannot be accomplished simply by moving the signal values to new coordinate points. Most practical approaches to discrete scaling involve interpolation. We propose a new approach based on hyperdifferential operator theory that does not involve conventional interpolation. This approach provides a self-consistent and pure definition of discrete scaling that is fully consistent with discrete Fourier transform theory. It can potentially be applied to other problems in signal theory and analysis such as transform designs. Apart from its theoretical elegance, it also provides a basis for numerical implementation.

Development of probabilistic assessment framework for pedestrian wind environment using Bayesian technique

This study extends the assessment framework proposed by Murakami et al. (1986) for the pedestrian wind environment to a fully probabilistic method by implementing Bayesian modeling. The method quantifies the uncertainties in constructed models based on the measured wind data and the results of the assessment. To model the probabilities of the daily maximum mean wind speed and wind direction, we employed the Weibull distribution and categorical distribution, respectively. The parameters defining the probability distributions were probabilistically modeled using Bayesian techniques. Using the wind data measured at the Meteorological Observatory of Tokyo, we demonstrated the effectiveness of the proposed method. The results showed that the wind direction probability and each parameter of the Weibull distribution could be estimated in the form of a posterior probability density function. Using the constructed models, we predicted the exceedance probability of the daily maximum instantaneous wind speed and evaluated the wind environment index (rank) in a city model. We provided a discrete rank scale in the form of a probability distribution, which enables us to quantify the evaluation uncertainties. Additionally, we clarified the effect of varying the amount of data used for the model construction. The uncertainty of the exceedance probability decreased with the amount of data. When only 1 year of data was used, some evaluation points possibly changed over three ranks. Even when 5-year observation data was used, the evaluated rank of some points varied within the range of uncertainty, thereby highlighting the importance of uncertainty quantification in the wind environment assessment.

A Hybrid Mesh DGTD Algorithm Based on Virtual Element for Tetrahedron and Hexahedron

In the discontinuous Galerkin time-domain (DGTD) method, tetrahedral meshes are typically used to discretize the computational domain. However, due to their different topological structures, when tetrahedrons or hexahedrons of the same discrete scale are used to divide the same computational domain, there are more tetrahedral mesh units, resulting in higher temporal and spatial complexity. In this article, a hybrid mesh DGTD algorithm based on virtual elements for tetrahedra and hexahedra is proposed to improve the computational efficiency compared with the traditional DGTD method. The fine-structured computational domain and the outer region are computed using tetrahedral and hexahedral meshes, respectively. During calculation, the numerical flux of each interface element requires data from the other subdomain. In this article, virtual elements are established at each subdomain interface to enable the exchange of field values for ease of processing. Edge assignment or field interpolation is applied to sample points to create corresponding virtual units to calculate fringe fields. Finally, the field value of each virtual element is assigned to the corresponding real element for the numerical flux calculation. Numerical results verify the correctness and efficiency of the proposed method.

Static Scheduling of Moldable Streaming Tasks With Task Fusion for Parallel Systems With DVFS

We consider the problem of statically scheduling a task graph of moldable streaming tasks (i.e., the actor network) to a multicore or many-core CPU with discrete dynamic voltage and frequency scaling (DVFS). We employ an integer linear programming (ILP) approach that combines allocating cores to tasks, mapping tasks to core subsets, selecting a DVFS level for each task, and considering all options for task fusion as provided by a cost model, given data throughput and latency requirements and targeting low energy consumption. We also propose a partly decoupled approach that applies greedy prefusion before running an ILP-based scheduler considering the other three subproblems together. We use microbenchmarking on an ARM big.LITTLE architecture to quantify the advantage of task fusion in the above setting, and evaluate the use of task fusion in terms of energy savings, latency improvement, and scheduling time for three real-world applications. We confirm the scheduling results by running the applications with and without task fusions on the ARM big.LITTLE. Results indicate that streaming applications can profit from task fusion, as we achieve a significant reduction of energy consumption in most cases, while scheduling time is only moderately increased.

Global bifurcation results for nonlinear dynamic equations on time scales

We prove a global bifurcation result for a parameterized dynamic equation on time scales. The approach is topological and based on a notion of topological degree for compact perturbations on nonlinear Fredholm maps in Banach spaces. Also, we provide several examples considering discrete, continuous and hybrid time scales in order to illustrate our main results.