Calculus Model Research Articles

A continuous power law wave equation for biomedical ultrasound

The attenuation coefficient in most biological media is a power law with respect to frequency. Measurements indicate that the power law exponent y is near or exactly one for many classes of tissue. In J. Acous. Soc. Am. [124 2861–2872 (2008)], a power law wave equation (PLWE) was proposed to model this power law dependence using Riemann-Liouville fractional derivatives. The PLWE, like most other fractional calculus models for attenuation in the time-domain, is invalid for y =1 due to a discontinuity in the phase velocity. To address this problem, a continuous power law wave equation (CPLWE) is proposed that is valid for all power law exponents between zero and two. The CPLWE utilizes the recently developed Zolotarev fractional derivative of order y, which is a nonlocal operator even for y = 1. The 3D Green's function for the CPLWE that is valid for homogeneous media is derived using stable probability density functions in the Zolotarev M-parameterization. Solutions to the CPLWE that are valid for inhomoge...

Does a Fractal Microstructure Require a Fractional Viscoelastic Model?

The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity.

A NOTE ON PROCESS MODELLING: COMBINING SITUATION CALCULUS AND PETRI NETS

The situation calculus logic model is convenient for modelling the actions that can occur in an information system application. The interplay of pre-conditions and post-conditions determines a semantically justified partial order of the defined actions and serves to enforce integrity constraints. This form of specification allows the use of plan-generation algorithms to investigate, before the system is adopted, whether the proposed specification allows all desirable use cases, and effectively disallows undesirable ones. Especially for legacy applications, implemented without a prior specification, Process Mining techniques were employed to derive an implicit Petri net model from the analysis of a large number of traces registered in an execution log. However, if the system just begins to be used, and has a still empty execution log, this sort of process mining discovery would not be feasible. This paper explains how the Petri net model can be directly derived from the situation calculus specification rules. The main gist is to provide evidence that the two models are complementary, not only because the Petri net model is derivable from the situation calculus model, but also in view of the distinct advantages of the two models. While the situation calculus model leads to planning and simulated execution prior to implementation, the Petri net model can be designed to run in a restrictive mode, allowing an intuitive visualization of the workable sequences. As proof of concept, the paper describes a prototype to demonstrate the methods and applies it to two examples: a published request processing application used to introduce process mining notions; and an analogously structured trial by combat application taken from a popular movie. The prototype includes an interactive dramatization component, which enacts the second application.

Carbon Dioxide Emissions Monitoring in Romania in the Context of Greenhouse Gases Reduction

The weather changes we are currently witnessing, characterised by dynamism and extreme phenomena, are the direct and indirect result of human activities which are determining the global atmosphere change in composition. The paperwork follows the evaluation and comparison of carbon dioxide emission coefficient in case of liquid oil fuels burning. The calculus model used for the carbon dioxide emission coefficient evaluation has been developed based on mathematical models from specific publications. It was applied in the case study based on data from specific literature. The results obtained following the evaluation allowed certain comparisons in the field of carbon dioxide emissions in case of the complete burning of certain fluid hydrocarbons use in industry.

Research on Rock Creep Characteristics Based on the Fractional Calculus Meshless Method

The application of fractional calculus in the rheological problems has been widely accepted. In this study, the constitutive relationship of the generalized Kelvin model based on fractional calculus was studied, and the meshless method was introduced so as to derive a new meshless algorithm formula based on the fractional calculus of the generalized Kelvin model. By using the MTS815.02 hydraulic servo rock mechanics test system, the creep test of mudstones is carried out, and the related data of the creep process were obtained. Based on the generalized Kelvin model of fractional calculus, the related creep parameters of the argillaceous sandstone under compression were fitted. The results showed that the solution of the generalized Kelvin model based on fractional calculus was greatly consistent with the numerical method solution. Meanwhile, the meshless algorithm based on fractional calculus had a favorable stability and accuracy.

Research on Slope Deformation Prediction Based on Fractional‐Order Calculus Gray Model

Slope deformation prediction has important significance for slope prevention and control. Based on historical time series, the trend of displacement variation can be predicted in advance, and according to the development trend, risk warnings and treatment measures are proposed. The use of the mathematical model to predict slope deformation has been proved to be feasible by many studies; therefore, the choice of the predictive model and the practicability of the model are crucial issues in the prediction of slope deformation, and the mathematical prediction model used should be less complicated considering the practicality of the model. In view of slope deformation prediction, a fractional‐order calculus gray model based on the coupling of gray theory and the fractional derivative method is proposed, which takes a deep foundation pit slope in Chongqing, Southwest China, as the study object. The fractional‐order gray model is compared with the traditional gray models; therefore, the results show that the accuracy of slope deformation prediction based on the gray coupling model of cumulative displacement and fractional calculus is significantly higher than that of the conventional gray model, and its error is in the acceptable range compared with the actual monitoring data, which can meet the needs of engineering application. Compared with the traditional gray theory method, the gray coupling model of fractional‐order calculus only increases the fractional derivative order, which is verified to be feasible, and can be used as a reference method for slope deformation prediction. It has a certain theoretical basis and a good application prospect in slope deformation prediction.

Differintegration with Respect to Functions in Fractional Models Involving Mittag-Leffler Functions

We consider what it means to fractionally differentiate or integrate one function with respect to another function, in models of fractional calculus which involve Mittag-Leffler kernels. Using our presented definitions, we derive series expressions for such differ-integrals, which we then use to prove some important results such as an elegant formulation of the fractional chain rule.

Solution of space time fractional generalized KdV equation, KdV burger equation and Bona-Mahoney-Burgers equation with dual power-law nonlinearity using complex fractional transformation

Fractional calculus is a rising subject in the current research field. The researchers of different disciplines are using fractional calculus models to investigate different practical problems. In this paper, we found the exact solutions of space-time fractional generalized KdV equation, KdV Burger equation and Benjamin-Bona-Mahoney-Burgers equation with dual power-law nonlinearity. The solutions are expressed in terms of hyperbolic, trigonometric and rational functions.

Jackson and Hahn Difference Models as Discrete Bittner Operational Calculus Representations

Abstract In the paper, there have been constructed discrete models of the non-classical Bittner operational calculus that are related to the notions of Jackson and Hahn derivatives, both known from the quantum calculus. To achieve it, we have used the Bittner operational calculus representation for the forward difference.

Representing, Running, and Revising Mental Models: AComputational Model.

People use commonsense science knowledge to flexibly explain, predict, and manipulate the world around them, yet we lack computational models of how this commonsense science knowledge is represented, acquired, utilized, and revised. This is an important challenge for cognitive science: Building higher order computational models in this area will help characterize one of the hallmarks of human reasoning, and it will allow us to build more robust reasoning systems. This paper presents a novel assembled coherence (AC) theory of human conceptual change, whereby people revise beliefs and mental models by constructing and evaluating explanations using fragmentary, globally inconsistent knowledge. We implement AC theory with Timber, a computational model of conceptual change that revises its beliefs and generates human-like explanations in commonsense science. Timber represents domain knowledge using predicate calculus and qualitative model fragments, and uses an abductive model formulation algorithm to construct competing explanations for phenomena. Timber then (a) scores competing explanations with respect to previously accepted beliefs, using a cost function based on simplicity and credibility, (b) identifies a low-cost, preferred explanation and accepts its constituent beliefs, and then (c) greedily alters previous explanation preferences to reduce global cost and thereby revise beliefs. Consistency is a soft constraint in Timber; it is biased to select explanations that share consistent beliefs, assumptions, and causal structure with its other, preferred explanations. In this paper, we use Timber to simulate the belief changes of students during clinical interviews about how the seasons change. We show that Timber produces and revises a sequence of explanations similar to those of the students, which supports the psychological plausibility of AC theory.

An Operational Calculus Model for the nTH – Order Forward Difference

Abstract In this paper, there has been constructed such a model of a non-classical Bittner operational calculus, in which the derivative is understood as a forward difference Δn{x(k)}:={x(k+n)-x(k)}. Next, considering the operation Δn,b{x(k)}:={x(k+n)-bx(k)}, the presented model has been generalized.

Simplified model for compressive response of RC column footing with square cross-section

In this paper, a simplified calculus model for the prediction of the compressive response of RC column footing with a square cross-section is presented. As it is well-known RC concrete footing are designed adopting uniform contact pressures on the substrate and assuming a strut and tie model in deep members and a cantilever beam or slab model in flexible members. Deep and flexible members are distinguished in literature only based on the tangent of the angle expressed as the ratio between the depth and the shear span of the footing.In this paper, several subgrade contact pressures distribution for column footings (rigid or soft soils) were considered in developing a mechanical model able to derive the complete load displacement curves of RC deep and flexible footing in compression. The objective of the research was the more appropriate choice of the calculus model to predict the compressive response of single column footing. In particular, if the angle, expressed as the ratio between the depth and the shear span of the footing, is higher than 45° the strut and tie model was suggested for best prediction, while, if the angle is lower than 45°, the beam or the slab models with punching shear were adopted. For both cases simplified loading soil profiles corresponding to rigid, flexible or winker model were adopted. Effects of main parameters such as geometry (depth, width) and shape of footing (single or shaped), the mechanical ratio of longitudinal reinforcement and type of soil are investigated both numerically and analytically. Numerical results and available experimental results were utilized to verify the model. The comparison between analytical and numerical results allows one to validate the proposed model and to define the range of application of the strut and tie or beam model depending on the footing geometry, the mechanical ratio of longitudinal reinforcement and type of soil. Finally, the comparison between analytical and experimental results available in the literature gives a further confirmation on the reliability of the proposed model.

Impediments to Information Systems Replacement: A Calculus of Discontinuance

Relatively little attention has been directed toward examining information system (IS) obsolescence and replacement despite the significant organizational resources invested in such systems. Current understanding is, as a result, both incomplete and relatively preliminary in nature. In particular, there is an important need for research that identifies impediments to system replacement. We therefore draw on protection motivation theory to formulate a model of IS discontinuance that explains how the replacement of obsolete systems is constrained by the risks associated with replacing these systems, the resources that have been invested in them, their complexity, and institutionalized norms that reflect dominant views concerning preferred IS solutions. Partial least squares (PLS) and covariance based structural equation modeling (SEM) of survey data obtained from 112 senior-level IS managers indicates that replacement risks contribute to decreased replacement intentions while system investments and complexity are linked to increased replacement risks. Institutional norms were also found to indirectly contribute to replacement risks through their positive influence on system investments. The proposed model of discontinuance calculus thus demonstrates the importance of managing risks, system complexity, and system investments to ensure that obsolete systems do not become a significant impediment to organizational effectiveness.

Non-Gaussian diffusion imaging with a fractional order calculus model to predict response of gastrointestinal stromal tumor to second-line sunitinib therapy.

To demonstrate the clinical value of a non-Gaussian diffusion model using fractional order calculus (FROC) for early prediction of the response of gastrointestinal stromal tumor to second-line sunitinib targeted therapy. Fifteen patients underwent sunitinib treatment after imatinib resistance. Diffusion-weighted imaging with multiple b-values was performed before treatment (baseline) and 2 weeks (for early prediction of response) after initiating sunitinib treatment. Conventional MRI images at 12 weeks were used to determine the good and poor responders according to the modified Choi criteria for MRI. Diffusion coefficient D, fractional order parameter β (which correlates to intravoxel tissue heterogeneity), and a microstructural quantity µ were calculated using the FROC model. The FROC parameters and the longest diameter of the lesion, as well as their changes after 2 weeks of treatment, were compared between the good and poor responders. Additionally, the pretreatment FROC parameters were individually combined with the change in D (ΔD) using a logistic regression model to evaluate response to sunitinib treatment with a receiver operating characteristic analysis. Forty-two good-responding and 32 poor-responding lesions were identified. Significant differences were detected in pretreatment β (0.67 versus 0.74, P = 0.011) and ΔD (45.7% versus 12.4%, P = 0.001) between the two groups. The receiver operating characteristic analysis showed that ΔD had a significantly higher predictive power than the tumor size change (area under the curve: 0.725 versus 0.580; 0.95 confidence interval). When ΔD was combined with pretreatment β, the area under the curve improved to 0.843 with a predictive accuracy of 75.7% (56 of 74). The non-Gaussian FROC diffusion model showed clinical value in early prediction of gastrointestinal stromal tumor response to second-line sunitinib targeted therapy. The pretreatment FROC parameter β can increase the predictive accuracy when combined with the change in diffusion coefficient during treatment. Magn Reson Med 79:1399-1406, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

On a fractional order calculus model in diffusion weighted breast imaging to differentiate between malignant and benign breast lesions detected on X-ray screening mammography.

ObjectiveTo evaluate a fractional order calculus (FROC) model in diffusion weighted imaging to differentiate between malignant and benign breast lesions in breast cancer screening work-up using recently introduced parameters (βFROC, DFROC and μFROC).Materials and methodsThis retrospective analysis within a prospective IRB-approved study included 51 participants (mean 58.4 years) after written informed consent. All patients had suspicious screening mammograms and indication for biopsy. Prior to biopsy, full diagnostic contrast-enhanced MRI examination was acquired including diffusion-weighted-imaging (DWI, b = 0,100,750,1500 s/mm2). Conventional apparent diffusion coefficient Dapp and FROC parameters (βFROC, DFROC and μFROC) as suggested further indicators of diffusivity components were measured in benign and malignant lesions. Receiver operating characteristics (ROC) were calculated to evaluate the diagnostic performance of the parameters.Results29/51 patients histopathologically revealed malignant lesions. The analysis revealed an AUC for Dapp of 0.89 (95% CI 0.80–0.98). For FROC derived parameters, AUC was 0.75 (0.60–0.89) for DFROC, 0.59 (0.43–0.75) for βFROC and 0.59 (0.42–0.77) for μFROC. Comparison of the AUC curves revealed a significantly higher AUC of Dapp compared to the FROC parameters DFROC (p = 0.009), βFROC (p = 0.003) and μFROC (p = 0.001).ConclusionIn contrast to recent description in brain tumors, the apparent diffusion coefficient Dapp showed a significantly higher AUC than the recently proposed FROC parameters βFROC, DFROC and μFROC for differentiating between malignant and benign breast lesions. This might be related to the intrinsic high heterogeneity within breast tissue or to the lower maximal b-value used in our study.

A Bayesian methodology to update the probabilistic seismic hazard assessment

This paper presents a Bayesian methodology for updating the seismic hazard curves. The methodology is based on the comparison of predictive exceedance rates of a fixed acceleration level (given by the seismic hazard curves) and the observed exceedance rates in some selected sites. The application of the methodology needs, firstly, the definition of a prior probabilistic seismic hazard assessment based in a logic tree. Each main branch corresponds to a probabilistic model of calculus of seismic hazard. The method considers that, initially (or a priori), the weights of all branches of the logic tree are equivalent. Secondly, the method needs to compile the observations in the region. They are introduced in a database containing the recorded acceleration data (during the instrumental period). Nevertheless, the instrumental period in stable zones (as France) shows only very low acceleration levels recorded during a short observation period. Then, a method to enlarge the REX (number of observations) is presented taking into account the historical data and defining “synthetic” accelerations in the sites of observation. The synthetic REX allows to expand the period of observation and to increase the acceleration thresholds used in the Bayesian updating process. The application of the Bayesian approach leads to a new and more objective definition of the weights of each branch of the logic tree and, therefore, to new seismic hazard curves (mean and centiles). The Bayesian approach doesn’t change the probabilistic models (seismic hazard curves). It only modifies the weights of each branch of the logic tree.

Language-Theoretic and Finite Relation Models for the (Full) Lambek Calculus

We prove completeness for some language-theoretic models of the full Lambek calculus and its various fragments. First we consider syntactic concepts and syntactic concepts over regular languages, which provide a complete semantics for the full Lambek calculus $${\mathbf {FL}}_\perp $$FLź. We present a new semantics we call automata-theoretic, which combines languages and relations via closure operators which are based on automaton transitions. We establish the completeness of this semantics for the full Lambek calculus via an isomorphism theorem for the syntactic concepts lattice of a language and a construction for the universal automaton recognizing the same language. Finally, we use automata-theoretic semantics to prove completeness of relation models of binary relations and finite relation models for the Lambek calculus without and with empty antecedents (henceforth: $$\mathbf L $$L and $$\mathbf L1 $$L1), thus solving a problem left open by Pentus (Ann Pure Appl Log 75:179---213, 1995).

FracFit: A robust parameter estimation tool for fractional calculus models

AbstractAnomalous transport cannot be adequately described with classical Fickian advection‐dispersion equations (ADE) with constant coefficients. Rather, fractional calculus models may be used, which capture salient features of anomalous transport (e.g., skewness and power law tails). FracFit is a parameter estimation tool based on space‐fractional and time‐fractional models used by the hydrology community. Currently, four fractional models are supported: (1) space‐fractional advection‐dispersion equation (sFADE), (2) time‐fractional dispersion equation with drift (TFDE), (3) fractional mobile‐immobile (FMIM) equation, and (4) temporally tempered Lévy motion (TTLM). Model solutions using pulse initial conditions and continuous injections are evaluated using stable distributions or subordination integrals. Parameter estimates are extracted from measured breakthrough curves (BTCs) using a weighted nonlinear least squares (WNLS) algorithm. Optimal weights for BTCs for pulse initial conditions and continuous injections are presented. Two sample applications are analyzed: (1) pulse injection BTCs in the Selke River and (2) continuous injection laboratory experiments using natural organic matter. Model parameters are compared across models and goodness‐of‐fit metrics are presented, facilitating model evaluation.

Geometric modelling of the contact point between the bushing and sprocket in chain drives

An important problem of the bush chains dynamics is represented by the calculus of the normal and transversal forces on all the contacts; these forces are producing vibrations in the chain and due to this, the chain is affected by the wear. One aspect of that dynamics is referring directly on the sprockets geometry and on the bushing and sprocket contact. The paper presents a calculus method for the contact angle between the bushing and the sprocket; this angle is a variable one depending on the bushing’s number being in contact (i) and on the specific elongation of the chain (x) due to the functioning of it. Based on the presented calculus model, a comparative analysis is proposed for these factors by using sprockets with different teeth numbers and different specific elongations of the chain. The results of the numerical simulations allow the dissemination of recommendations regarding the contact angle’s evolution, from the beginning to the end of the contact and regarding the influence of the chain’s specific elongations on the out of use of it.

Communicating Personal Health Information in Virtual Health Communities: An Integration of Privacy Calculus Model and Affective Commitment

Communicating Personal Health Information in Virtual Health Communities: An Integration of Privacy Calculus Model and Affective Commitment