Exponential Kernel Research Articles

Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative

Abstract This study demonstrates the use of fractional calculus in the field of epidemiology, specifically in relation to dengue illness. Using noninteger order integrals and derivatives, a novel model is created to examine the impact of temperature on the transmission of the vector–host disease, dengue. A comprehensive strategy is proposed and illustrated, drawing inspiration from the first dengue epidemic recorded in 2009 in Cape Verde. The model utilizes a fractional-order derivative, which has recently acquired popularity for its adaptability in addressing a wide variety of applicable problems and exponential kernel. A fixed point method of Krasnoselskii and Banach is used to determine the main findings. The semi-analytical results are then investigated using iterative techniques such as Laplace-Adomian decomposition method. Computational models are utilized to support analytical experiments and enhance the credibility of the results. These models are useful for simulating and validating the effect of temperature on the complex dynamics of the vector–host interaction during dengue outbreaks. It is essential to note that the research draws on dengue outbreak studies conducted in various geographic regions, thereby providing a broader perspective and validating the findings generally. This study not only demonstrates a novel application of fractional calculus in epidemiology but also casts light on the complex relationship between temperature and the dynamics of dengue transmission. The obtained results serve as a foundation for enhancing our understanding of the complex interaction between environmental factors and infectious diseases, leading the way for enhanced prevention and control strategies to combat global dengue outbreaks.

Mathematical analysis of a time-fractional coupled tumour model using Laplace and finite Fourier transforms

A time-fractional diffusive tumour growth model is investigated in this paper. The model assumes two different tumour populations having diverse diffusion and proliferation rates. The classical integer model, coupled with the fractional operators with power-law, Mittag-Leffler and exponential kernels is solved analytically with the help of integral transforms Numerical investigations of the resulting solution are carried out to examine the effect of the different fractional operators. The spherical form of the model is considered, as this describes the tumour more adequately as compared to other coordinates system. From the numerical experiments, it is observed that, the kernel contained in the respective fractional operator could alter the dynamics of the tumour concentration under consideration.

Communicability cosine distance: similarity and symmetry in graphs/networks

A distance based on the exponential kernel of the adjacency matrix of a graph and representing how well two vertices connect to each other in a graph is defined and studied. This communicability cosine distance (CCD) is a Euclidean spherical distance accounting for the cosine of the angles spanned by the position vectors of the graph vertices in this space. The Euclidean distance matrix (EDM) of CCD is used to quantify the similarity between vertices in graphs and networks as well as to define a local vertex invariant—a closeness centrality measure, which discriminate very well vertices in small graphs. It allows to distinguish all nonidentical vertices, also characterizing all identity (asymmetric) graphs–those having only the identity automorphism–among all connected graphs of up to 9 vertices. It also characterizes several other classes of identity graphs. We also study real-world networks in term of both the discriminating power of the new centrality on their vertices as well as in ranking their vertices. We analyze some dictionary networks as well as the network of copurshasing of political books, remarking some of the main advantages of the new approaches studied here.

Investigation of Magnetized Casson Nanofluid Flow along Wedge: Gaussian Process Regression

An unsteady two-dimensional magnetized Casson nanofluid flow model is constructed over a wedge under the effect of thermal radiation and chemical reaction. The multiple slip effects are also assumed near the surface of the wedge along with the convective boundary restrictions. This study investigates the application of soft computing techniques to address the challenges posed by the complexity of problem modeling and numerical methods. Traditional approaches incorporating various model factors may struggle to provide accurate solutions. To resolve this issue, Gaussian process regression (GPR) is employed to predict the solution of the proposed flow model. With the help of the numerical shooting method together with Runge–Kutta–Fehlberg fourth-fifth-order (RKF-45) reference data, the GPR model is trained. The numerical simulation illustrated that the Casson fluid parameter β and the unsteadiness parameter S strengthen the friction factor, and the heat transfer rate is enhanced as the radiation parameter Rd becomes larger. In addition, the Biot numbers Bi1 & Bi2 lead to strengthen nanoparticle temperature; an opposite behavior is noticed with the skin friction coefficient S˜fxRex0.5, heat transfer rate H˜tx Rex0.5, and nanoparticle transfer rate C˜txRex0.5. The GPR model with the exponential Kernel function provided better performance than other functions on both training and checking datasets to predict S˜fxRex0.5,H˜tx Rex0.5, and C˜txRex0.5. Statistical metrics including RMSE, MAE, MAPE, MSE, R2, and R are employed to check the accuracy and convergences of the predicted and numerical solutions obtained through GPR and RKF-45. It is observed that all three GPR models had an R2 value of higher than 0.9. The proposed study demonstrates the advantages of employing soft computing methods (GPR) to effectively analyse the behavior of complex flow models.

Degeneration of kernel regression with Matern kernels into low-order polynomial regression in high dimension.

Kernel methods such as kernel ridge regression and Gaussian process regression with Matern-type kernels have been increasingly used, in particular, to fit potential energy surfaces (PES) and density functionals, and for materials informatics. When the dimensionality of the feature space is high, these methods are used with necessarily sparse data. In this regime, the optimal length parameter of a Matern-type kernel may become so large that the method effectively degenerates into a low-order polynomial regression and, therefore, loses any advantage over such regression. This is demonstrated theoretically as well as numerically in the examples of six- and fifteen-dimensional molecular PES using squared exponential and simple exponential kernels. The results shed additional light on the success of polynomial approximations such as PIP for medium-size molecules and on the importance of orders-of-coupling-based models for preserving the advantages of kernel methods with Matern-type kernels of on the use of physically motivated (reproducing) kernels.

An asymptotic formulation for boundary value problems in a non-local elastic half-space

The primary purpose of this study is to develop an asymptotic formulation for boundary value problems in a non-local elastic half-space. For the sake of simplicity, the non-locality is limited to the vertical direction, which is represented by a one-dimensional exponential kernel, and the problem is formulated within the framework of Eringen’s theory. The proposed asymptotic approach is based on the assumption that the internal characteristic length is significantly smaller than a typical wavelength. This assumption allows for the development of an asymptotic formulation that expresses the considered boundary value problem in terms of local stresses. Additionally, the formulation includes explicit correction terms to the classical boundary conditions, which arise from the non-local effects. As an example application of the derived formulation, the Rayleigh surface waves in a plane strain problem are considered. Finally, numerical results are presented for certain specific values of elastic parameters to illustrate the effects of non-locality on the analyzed system.

A Fractional Study of MHD Casson Fluid Motion With Thermal Radiative Flux and Heat Injection/Suction Mechanism Under Ramped Wall Condition: Application of Rabotnov Exponential Kernel

Abstract The primary objective of this research is to extend the concept of fractionalized Casson fluid flow. In this study, a comprehensive analysis of magnetohydrodynamic (MHD) natural convective flow of Casson fluid is conducted, focusing on obtaining analytical solutions using the non-integer-order derivative known as the Yang–Abdel-Aty–Cattani (YAC) operator. The YAC operator utilized in this research possesses a more generalized exponential kernel. The fluid flow is examined in the vicinity of an infinitely vertical plate with a characteristic velocity denoted as u 0. The mathematical modelling of the problem incorporates partial differential equations, incorporating Newtonian heating and ramped conditions. To facilitate the analysis, a suitable set of variables is introduced to transform the governing equations into a dimensionless form. The Laplace transform (LT) is then applied to the fractional system of equations, and the obtained results are presented in series form and also expressed in terms of special functions. The study further investigates the influence of relevant parameters, such as α, β, Pr , Q, Gr, M, Nr and K, on the fluid flow to reveal interesting findings. A comparison of different approaches reveals that the YAC method yields superior results compared to existing operators found in the literature. Graphs are generated to illustrate the outcomes effectively. Additionally, the research explores the limiting cases of the Casson and viscous fluid models to derive the classical form from the YAC fractionalized Casson fluid model.

Fractional heat transfer DPL model incorporating an exponential Rabotnov kernel to study an infinite solid with a spherical cavity

<abstract> <p>The objective of this study was to investigate the thermodynamic reactions of thermoelastic materials by utilizing a modified mathematical fractional thermoelastic model. This model combines a fractional derivative with Rabotnov's exponential kernel and the idea of a two-phase delay, which makes it possible to show thermoelastic behavior more accurately. The model was utilized to investigate an unbounded material with a spherical cavity subjected to a decreasing and shifting heat flux on its inner surface. The problem was solved using analytical approaches, with a strong focus on the Laplace transform. The transform was numerically inverted to provide time-domain results. The study presented graphs that compared the outcomes of utilizing a single kernel fractional derivative with the results obtained using the Rabotnov kernel and fractional order. These graphs showed how the Rabotnov kernel and fractional order affected the physical fields under investigation. This novel theoretical framework has the potential to be advantageous in diverse domains, including engineering, solid mechanics, and materials science.</p> </abstract>

Time-inhomogeneous Hawkes processes and its financial applications

<abstract><p>We consider time-inhomogeneous Hawkes processes with an exponential kernel, and we analyze some properties of the model. Time-inhomogeneity for the Hawkes process is indispensable for short rate models or for other calibration purposes, while financial applications for the time-homogeneous case already well known. Distributional properties for such a model generate computational tractability for a financial application. In this paper, moments and the Laplace transform of time-inhomogeneous Hawkes processes are obtained from the distributional properties of the underlying processes. As an applications to finance, we investigate the pricing formula for zero-coupon bonds when short-term interest rates are governed by the time-inhomogeneous Hawkes process. Numerical illustrations are also provided. As an illustrative example, we apply the derived moments and Laplace transform of time-inhomogeneous Hawkes processes to the pricing of zero-coupon bonds within a financial context. By considering the short-term interest rate as driven by inhomogeneous Hawkes processes, we develop explicit formulae for valuing zero-coupon bonds. This application is particularly relevant for modeling interest rate dynamics in real-world scenarios, allowing for a more nuanced understanding of pricing dynamics. Through numerical illustrations, we demonstrate the computational tractability of our approach, showcasing its practical utility for financial practitioners and providing insights into the intricate interplay between time-inhomogeneous Hawkes processes and bond pricing in dynamic markets.</p></abstract>

A NEW FRACTIONAL DERIVATIVE MODEL FOR THE NON-DARCIAN SEEPAGE

In this paper, a new fractional derivative model for the non-Darcian seepage within the exponential decay kernel is addressed for the first time. The new fractional derivative model is for high-speed non-Darcian and low-speed non-Darcian seepage, in which the applied zone is enlarged.

Unveiling Fringe Social Network Dynamics via Parameter Estimation with Hawkes Processes

Fringe social networks, e.g., 4chan or Truth, position themselves as “free speech” alternatives to their mainstream counterparts like Facebook or X (formerly Twitter). Due to their very lax moderation policies, they however tend to become a hotbed for misinformation or otherwise malicious content, which then tends to spread towards the general public. In order to effectively counter such a process, it is important to properly understand and model how content appears and spreads over fringe social networks. Accordingly, in this study we focus on the now-defunct Parler social network, and conduct a statistical analysis over 183 million posts dating from August 2018 to January 2021. The primary objective is to comprehensively analyze hashtag cascades related to the first impeachment of U.S. President Donald Trump. Our aim is to (i) uncover how external actors inject malicious and hateful tendencies into the network and (ii) quantify the levels of attention within these communities. We find that the hashtag cascade can be effectively modeled using the Hawkes process framework, specifically, employing an exponential decay kernel. Rigorous parameter estimation and statistical tools are applied to substantiate this assertion and evaluate the model’s goodness of ft. Importantly, the analysis reveals correlations between levels of hate, the dissemination of misleading information, and the attention garnered within these fringe social communities.

Backlight and dim space object detection based on a novel event camera.

For space object detection tasks, conventional optical cameras face various application challenges, including backlight issues and dim light conditions. As a novel optical camera, the event camera has the advantages of high temporal resolution and high dynamic range due to asynchronous output characteristics, which provides a new solution to the above challenges. However, the asynchronous output characteristic of event cameras makes them incompatible with conventional object detection methods designed for frame images. Asynchronous convolutional memory network (ACMNet) for processing event camera data is proposed to solve the problem of backlight and dim space object detection. The key idea of ACMNet is to first characterize the asynchronous event streams with the Event Spike Tensor (EST) voxel grid through the exponential kernel function, then extract spatial features using a feed-forward feature extraction network, and aggregate temporal features using a proposed convolutional spatiotemporal memory module ConvLSTM, and finally, the end-to-end object detection using continuous event streams is realized. Comparison experiments among ACMNet and classical object detection methods are carried out on Event_DVS_space7, which is a large-scale space synthetic event dataset based on event cameras. The results show that the performance of ACMNet is superior to the others, and the mAP is improved by 12.7% while maintaining the processing speed. Moreover, event cameras still have a good performance in backlight and dim light conditions where conventional optical cameras fail. This research offers a novel possibility for detection under intricate lighting and motion conditions, emphasizing the superior benefits of event cameras in the realm of space object detection.

Supervised Manifold Learning Based on Multi-Feature Information Discriminative Fusion within an Adaptive Nearest Neighbor Strategy Applied to Rolling Bearing Fault Diagnosis.

Rolling bearings are a key component for ensuring the safe and smooth operation of rotating machinery and are very prone to failure. Therefore, intelligent fault diagnosis research on rolling bearings has become a crucial task in the field of mechanical fault diagnosis. This paper proposes research on the fault diagnosis of rolling bearings based on an adaptive nearest neighbor strategy and the discriminative fusion of multi-feature information using supervised manifold learning (AN-MFIDFS-Isomap). Firstly, an adaptive nearest neighbor strategy is proposed using the Euclidean distance and cosine similarity to optimize the selection of neighboring points. Secondly, three feature space transformation and feature information extraction methods are proposed, among which an innovative exponential linear kernel function is introduced to provide new feature information descriptions for the data, enhancing feature sensitivity. Finally, under the adaptive nearest neighbor strategy, a novel AN-MFIDFS-Isomap algorithm is proposed for rolling bearing fault diagnosis by fusing various feature information and classifiers through discriminative fusion with label information. The proposed AN-MFIDFS-Isomap algorithm is validated on the CWRU open dataset and our experimental dataset. The experiments show that the proposed method outperforms other traditional manifold learning methods in terms of data clustering and fault diagnosis.

Correction of Italian under-reporting in the first COVID-19 wave via age-specific deconvolution of hospital admissions

When the COVID-19 pandemic first emerged in early 2020, healthcare and bureaucratic systems worldwide were caught off guard and largely unprepared to deal with the scale and severity of the outbreak. In Italy, this led to a severe underreporting of infections during the first wave of the spread. The lack of accurate data is critical as it hampers the retrospective assessment of nonpharmacological interventions, the comparison with the following waves, and the estimation and validation of epidemiological models. In particular, during the first wave, reported cases of new infections were strikingly low if compared with their effects in terms of deaths, hospitalizations and intensive care admissions. In this paper, we observe that the hospital admissions during the second wave were very well explained by the convolution of the reported daily infections with an exponential kernel. By formulating the estimation of the actual infections during the first wave as an inverse problem, its solution by a regularization approach is proposed and validated. In this way, it was possible to compute corrected time series of daily infections for each age class. The new estimates are consistent with the serological survey published in June 2020 by the National Institute of Statistics (ISTAT) and can be used to speculate on the total number of infections occurring in Italy during 2020, which appears to be about double the number officially recorded.

A Bayesian method to mitigate the effects of unmodelled time-varying systematics for 21-cm cosmology experiments

ABSTRACT Radio observations of the neutral hydrogen signal from the Cosmic Dawn and Epoch of Reionization have helped to provide constraints on the properties of the first stars and galaxies. Since this global 21-cm cosmological signal from the Cosmic Dawn is effectively constant on observing time-scales and since effects resulting from systematics will vary with time, the effects of these systematics can be mitigated without the need for a model of the systematic. We present a method to account for unmodelled time-varying systematics in 21-cm radio cosmology experiments using a squared exponential Gaussian process kernel to account for correlations between time bins in a fully Bayesian way. We find by varying the model parameters of a simulated systematic that the Gaussian process method improves our ability to recover the signal parameters by widening the posterior in the presence of a systematic and reducing the bias in the mean fit parameters. When varying the amplitude of a model sinusoidal systematic between 0.25 and 2.00 times the 21-cm signal amplitude and the period between 0.5 and 4.0 times the signal width, we find on average a 5 per cent improvement in the root mean squared error of the fitted signal. We can use the fitted Gaussian process hyperparameters to identify the presence of a systematic in the data, demonstrating the method’s utility as a diagnostic tool. Furthermore, we can use Gaussian process regression to calculate a mean fit to the residuals over time, providing a basis for producing a model of the time-varying systematic.

Modeling of thermophysical properties and vapor-liquid equilibrium using Gaussian process regression

Thermophysical properties are essential in the numerical simulation of fluid flows and combustion in propulsion and power-generation systems. Existing property evaluation tools, such as NIST REFPROP, analytical methods based on cubic equations of state (EOSs), and deep neural network (DNN) models, tend to be inaccurate against experimental data in certain thermodynamic regions. The objective of this paper is to develop data-driven models based on Gaussian process regression (GPR), which correlates directly with experimental data in a statistical manner and can accurately predict thermodynamic and transport properties and vapor-liquid equilibrium (VLE) behavior across the entire range of thermodynamic regions. Different kernel functions in GPR training are examined for various data types. As specific examples, experimental property data of carbon dioxide and VLE behavior of four binary mixtures are collected extensively from literature for training and validation of GPR models. Besides, physical constraint is incorporated during the training process of VLE behavior via logit transformation. The results show that GPR with exponential kernel presents excellent performance for evaluation of density, constant-pressure specific heat, speed of sound, and viscosity, with prediction errors substantially lower than those of Peng-Robinson (PR) EOS, NIST REFPROP, and DNN models. GPR with Gaussian kernel shows minimal errors (less than 0.6%) for determination of saturation pressure and gas-phase composition against experimental counterparts for VLE of hydrocarbon/nitrogen binary mixtures. The developed GPR models can be programmed and incorporated into large-scale simulations to achieve more reliable calculations of practical fluid flows where real-gas effect is important.

Ostrowski-Type Fractional Integral Inequalities: A Survey

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, (ζ,m)-convex functions, s-convex functions, (s,r)-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, MT-convex functions, P-convex functions, m-convex functions, (s,m)-convex functions, exponentially s-convex functions, (β,m)-convex functions, exponential-convex functions, ζ¯,β,γ,δ-convex functions, quasi-geometrically convex functions, s−e-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented.

Direction of Arrival Estimation with Nested Arrays in Presence of Impulsive Noise: A Correlation Entropy-Based Infinite Norm Strategy

Direction of arrival (DOA) estimation with nested arrays has been widely investigated in the field of array signal processing, but most studies assume that the noise is Gaussian white noise. In practical situations, there may exist impulsive noise (a kind of heavy-tailed noise), wherein the performance of traditional subspace-based DOA estimation algorithms deteriorates significantly. In this paper, we propose a correlation entropy-based infinite norm preprocessing algorithm, which can be applicable to any type of impulsive noise. Each snapshot of the sensor array data is processed by an exponential kernel function with the infinite norm, which can effectively combat the outliers. Furthermore, we construct the equivalent second-order covariance matrix and perform DOA estimation using classical subspace methods. Simulation results demonstrate the effectiveness of the proposed method for both symmetric α-stable distribution and the Gaussian mixture model.

Discrete fractional calculus with exponential memory: Propositions, numerical schemes and asymptotic stability

A new fractional difference with an exponential kernel function is proposed in this study. First, a difference operator is defined by the exponential function. From the Cauchy problem of the nth-order difference equation, new fractional-order sum and differences are presented. The propositions between each other and numerical schemes are derived. Finally, fractional linear difference equations are presented, and exact solutions are given by using a new discrete Mittag-Leffler function.

Discrete fractional calculus with exponential memory: Propositions, numerical schemes and asymptotic stability

A new fractional difference with an exponential kernel function is proposed in this study. First, a difference operator is defined by the exponential function. From the Cauchy problem of the nth-order difference equation, new fractional-order sum and differences are presented. The propositions between each other and numerical schemes are derived. Finally, fractional linear difference equations are presented, and exact solutions are given by using a new discrete Mittag-Leffler function.