Form Of Kernel Research Articles

Kernels, data & physics

Abstract Lecture notes from the course given by Professor Julia Kempe at the summer school ‘Statistical physics of Machine Learning’ in Les Houches. The notes discuss the so-called NTK approach to problems in machine learning, which consists of gaining an understanding of generally unsolvable problems by finding a tractable kernel formulation. The notes are mainly focused on practical applications such as data distillation and adversarial robustness, examples of inductive bias are also discussed.

Chaos and feasibility analysis of a healthy lifestyle’s sustainability using fractional-order polycystic ovarian syndrome’s effect on infertility

In this study, we developed a fractional-order model for polycystic ovarian syndrome issue transmission in women through four compartments with the Mittag-Leffler kernel. For biological feasibility, we discussed the positivity and boundedness of solutions with fixed point mappings in Banach spaces. Sensitivity analysis is performed using different variables as response functions each time, employing Latin Hyper-cube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) with 200 iterations. Further, the reproductive number is determined for the model’s biological feasibility with an analysis of parameter impact through Matlab software. The solutions are examined for global stability using the Lyapunov function with a first derivative. Chaos Control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium. Numerical solutions are derived by using a Newton polynomial in the generalized form of the Mittag-Leffler kernel, which illustrates the effect of the fractional operator and parameters involved in the system. Results are also analyzed with different assumptions of parameter values and initial conditions to observe the different impacts of treatment to overcome the risk of infertility in women. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies to overcome the risk of problems and best management care pre-detection of the issue and for treatment.

Application of first- and second-order adjoint methods to glacial isostatic adjustment incorporating rotational feedbacks

SUMMARY This paper revisits and extends the adjoint theory for glacial isostatic adjustment (GIA) of Crawford et al. (2018). Rotational feedbacks are now incorporated, and the application of the second-order adjoint method is described for the first time. The first-order adjoint method provides an efficient means for computing sensitivity kernels for a chosen objective functional, while the second-order adjoint method provides second-derivative information in the form of Hessian kernels. These latter kernels are required by efficient Newton-type optimization schemes and within methods for quantifying uncertainty for non-linear inverse problems. Most importantly, the entire theory has been reformulated so as to simplify its implementation by others within the GIA community. In particular, the rate-formulation for the GIA forward problem introduced by Crawford et al. (2018) has been replaced with the conventional equations for modelling GIA in laterally heterogeneous earth models. The implementation of the first- and second-order adjoint problems should be relatively easy within both existing and new GIA codes, with only the inclusions of more general force terms being required.

Benchmarking of quantum fidelity kernels for Gaussian process regression

Abstract Quantum computing algorithms have been shown to produce performant quantum kernels for machine-learning classification problems. Here, we examine the performance of quantum kernels for regression problems of practical interest. For an unbiased benchmarking of quantum kernels, it is necessary to construct the most optimal functional form of the classical kernels and the most optimal quantum kernels for each given data set. We develop an algorithm that uses an analog of the Bayesian information criterion to optimize the sequence of quantum gates used to estimate quantum kernels for Gaussian process models. The algorithm increases the complexity of the quantum circuits incrementally, while improving the performance of the resulting kernels, and is shown to yield much higher model accuracy with fewer quantum gates than a fixed quantum circuit ansatz. We demonstrate that quantum kernels thus obtained can be used to build accurate models of global potential energy surfaces (PES) for polyatomic molecules. The average interpolation error of the six-dimensional PES obtained with a random distribution of 2000 energy points is 16 cm−1 for H3O+, 15 cm−1 for H2CO and 88 cm−1 for HNO2. We show that a compositional optimization of classical kernels for Gaussian process regression converges to the same errors. This indicates that quantum kernels can achieve the same, though not better, expressivity as classical kernels for regression problems.

Mainardi smoothing homotopy method for solving nonlinear optimal control problems

This paper proposes a new type of smoothing homotopy method for solving general nonlinear optimal control problems via the indirect method, leveraging the Mainardi kernel as the smoothing kernel. The Mainardi kernel is firstly derived from the fundamental solution of the time-fractional diffusion-wave equation, representing a generalized form of the Gaussian kernel. By altering the fractional derivative order, the kernel can seamlessly switch between non-Gaussian and Gaussian forms. Then, parts of the two-point boundary value problems are convolved with the smoothing kernel, and the resulting surrogates are incorporated into the necessary conditions, replacing the terminal state and costate variables. The homotopy process is used for the smoothing parameter: increasing this parameter enhances the smoothing effect, simplifying the homotopy problems, while a parameter value of zero implies no smoothing, reverting to the original problems. Additionally, explicit expressions for the Mainardi kernel at specific derivative orders are derived, avoiding the inefficiencies associated with fractional derivatives. Simulation examples demonstrate that the proposed method offers significant advantages in flexibility and convergence compared to the traditional Gaussian smoothing homotopy method.

The SZ effect with anisotropic distributions and high energy electrons

Future observations of the Sunyaev-Zeldovich (SZ) effect promise ever improving measurements in terms of both sensitivity and angular resolution. As such, it is increasingly relevant to model `higher-order' contributions to the SZ effect. This work examines the effects of high-energy non-thermal electron distributions and those of anisotropic electron and photon distributions on the SZ signals. Analytic forms of the anisotropic scattering kernels for photons and electrons have been derived and investigated. We present a method for determining the anisotropic contributions through a spherical harmonic decomposition to arbitrary angular multipoles, and discuss the behaviour of these scattering kernels. We then carry out an exploration of various simplistic models of high energy non-thermal electron distributions, and examine their anisotropic behaviour. The kinematic SZ in the relativistic regime is studied using the kernel formulation allowing us to clarifying the role of kinematic corrections to the scattering optical depth. We finally present a release of an updated and refined version of SZpack including a new integrated Python interface and new modules for the calculation of various SZ signals, including those described in this paper.

A mathematical fractal-fractional model to control tuberculosis prevalence with sensitivity, stability, and simulation under feasible circumstances

Background:Tuberculosis, a global health concern, was anticipated to grow to 10.6 million new cases by 2021, with an increase in multidrug-resistant tuberculosis. Despite 1.6 million deaths in 2021, present treatments save millions of lives, and tuberculosis may overtake COVID-19 as the greatest cause of mortality. This study provides a six-compartmental deterministic model that employs a fractal–fractional operator with a power law kernel to investigate the impact of vaccination on tuberculosis dynamics in a population. Methods:Some important characteristics, such as vaccination and infection rate, are considered. We first show that the suggested model has positive bounded solutions and a positive invariant area. We evaluate the equation for the most important threshold parameter, the basic reproduction number, and investigate the model’s equilibria. We perform sensitivity analysis to determine the elements that influence tuberculosis dynamics. Fixed-point concepts show the presence and uniqueness of a solution to the suggested model. We use the two-step Newton polynomial technique to investigate the effect of the fractional operator on the generalized form of the power law kernel. Results:The stability analysis of the fractal–fractional model has been confirmed for both Ulam–Hyers and generalized Ulam–Hyers types. Numerical simulations show the effects of different fractional order values on tuberculosis infection dynamics in society. According to numerical simulations, limiting contact with infected patients and enhancing vaccine efficacy can help reduce the tuberculosis burden. The fractal–fractional operator produces better results than the ordinary integer order in the sense of memory effect at diffract fractal and fractional order values. Conclusion:According to our findings, fractional modeling offers important insights into the dynamic behavior of tuberculosis disease, facilitating a more thorough comprehension of their epidemiology and possible means of control.

Individual-level models of disease transmission incorporating piecewise spatial risk functions

Modelling epidemics is crucial for understanding the emergence, transmission, impact and control of diseases. Spatial individual-level models (ILMs) that account for population heterogeneity are a useful tool, accounting for factors such as location, vaccination status and genetic information.Parametric forms for spatial risk functions, or kernels, are often used, but rely on strong assumptions about underlying transmission mechanisms. Here, we propose a class of non-parametric spatial disease transmission model, fitted within a Bayesian Markov chain Monte Carlo (MCMC) framework, allowing for more flexible assumptions when estimating the effect on spatial distance and infection risk.We focus upon two specific forms of non-parametric spatial infection kernel: piecewise constant and piecewise linear. Although these are relatively simple forms, we find them to produce results in line with, or superior to, parametric spatial ILMs. The performance of these models is examined using simulated data, including under circumstances of model misspecification, and then applied to data from the UK 2001 foot-and-mouth disease.

Anomalous and ultraslow diffusion of a particle driven by power-law-correlated and distributed-order noises

We study the generalised Langevin equation (GLE) approach to anomalous diffusion for a harmonic oscillator and a free particle driven by different forms of internal noises, such as power-law-correlated and distributed-order noises that fulfil generalised versions of the fluctuation-dissipation theorem. The mean squared displacement and the normalised displacement correlation function are derived for the different forms of the friction memory kernel. The corresponding overdamped GLEs for these cases are also investigated. It is shown that such models can be used to describe anomalous diffusion in complex media, giving rise to subdiffusion, superdiffusion, ultraslow diffusion, strong anomaly, and other complex diffusive behaviours.

Effective mass approach to memory in non-Markovian systems.

Recent pioneering experiments on non-Markovian dynamics done, e.g., for active matter have demonstrated that our theoretical understanding of this challenging yet hot topic is rather incomplete and there is a wealth of phenomena still awaiting discovery. It is related to the fact that typically for simplification the Markovian approximation is employed and as a consequence the memory is neglected. Therefore, methods allowing to study memory effects are extremely valuable. We demonstrate that a non-Markovian system described by the Generalized Langevin Equation(GLE) for a Brownian particle of mass M can be approximated by the memoryless Langevin equationin which the memory effects are correctly reproduced solely via the effective mass M^{*} of the Brownian particle which is determined only by the form of the memory kernel. Our work lays the foundation for an impactful approach which allows one to readily study memory-related corrections to Markovian dynamics.

Fractional order model of MRSA bacterial infection with real data fitting: Computational Analysis and Modeling

Bacterial infections in the health-care sector and social environments have been linked to the Methicillin-Resistant Staphylococcus aureus (MRSA) infection, a type of bacteria that has remained an international health risk since the 1960s. From mild colonization to a deadly invasive disease with an elevated mortality rate, the illness can present in many different forms. A fractional-order dynamic model of MRSA infection developed using real data for computational and modeling analysis on the north side of Cyprus is presented in this paper. Initially, we tested that the suggested model had a positively invariant region, bounded solutions, and uniqueness for the biological feasibility of the model. We study the equilibria of the model and assess the expression for the most significant threshold parameter, called the basic reproduction number (ℛ0). The reproductive number’s parameters are also subjected to sensitivity analysis through mathematical methods and simulations. Additionally, utilizing the power law kernel and the fixed-point approach, the existence, uniqueness, and generalized Ulam–Hyers–Rassias stability are presented. Chaos Control was used to regulate the linear responses approach to bring the system to stabilize according to its points of equilibrium, taking into account a fractional-order system with a managed design where solutions are bound in the feasible domain. Finally, numerical simulations demonstrating the effects of different parameters on MRSA infection are used to investigate the impact of the fractional operator on the generalized form of the power law kernel through a two-step Newton polynomial method. The impact of fractional orders is emphasized in the study so that the numerical solutions support the importance of these orders on MRSA infection. With the application of fractional order, the significance of cognizant antibiotic usage for MRSA infection is verified.

Computational analysis and chaos control of the fractional order syphilis disease model through modeling

In this paper, we present a dynamic model of syphilis. The goal of this work is to provide a mathematical model for syphilis disease that contains information about the suspect, infected, recovered, and exposed. Boundedness, positivity, and unique solutions at equilibrium points are used in the qualitative and quantitative study of a proposed model using the power law kernel. The Routh-Hurwitz stability criteria are used to address the model's local and global stability. Chaos control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium. Using Lyapunov function tests, the global stability of free and endemic equilibrium points is confirmed. The Lipschitz condition and fixed point theory are utilized to satisfy the requirements for the existence and uniqueness of the exact solution. The generalized form of the power law kernel is solved using a two-step Lagrange polynomial method to investigate the impact of the fractional operator using numerical simulations that illustrate the effects of various parameters on the illness.

Recognition of nuts and seeds in children with/without food allergies and their mothers: A reflection of culinary culture.

Nuts and seeds are among the leading causes of food allergy. Effective food allergy management hinges on the ability to identify and avoid relevant foods. To evaluate the nut/seed recognition ability in both children and mothers. Primary caregivers (mothers) and their children (6-18 years old) with/without food allergies were shown photographs of nuts/seeds, and their products with visible/hidden allergens to assess their ability to recognize accurately. A total of 196 children and 184 mothers participated. The median ages of the children and mothers were 7.6 (6.8-10) and 37.8 (33.1-41.5) years, respectively. Over 75% of the children/adolescents and over 90% of the mothers accurately identified the kernel forms of nuts/seeds, except pine nuts. Walnuts, hazelnuts, almonds, and cashews were the most accurately recognized kernel forms by both populations. Generally, the kernel forms were recognized 5-20% more accurately than their in-shell forms, followed by products with visible and hidden forms, respectively. Some Turkish culinary-specific products with visible/hidden allergens were recognized as frequently as the kernel/in-shell forms by both study groups. Although there was a similar recognition pattern between study groups and subgroups (nut/seed allergy, other food allergy, controls), higher rates of recognition were found in mothers than in their children and adolescents than in schoolchildren. In Eastern Mediterranean region, nuts and sesame seeds are highly recognized by both mothers and their children. Accurate identification of these foods is likely a culinary feature, but not the result of increased awareness. More information is needed on whether this ability reduces the risk of exposure.

Analysis of Screw Press Machine Performance Using the Overall Equipment Effectiveness Method at PT Agro Synergy Nusantara

PT Agro Sinergi Nusantara PKS Aceh Seujahtera is a company engaged in the plantation and processing of oil palm fruit, where the production is in the form of crude palm oil (CPO), palm kernel and fiber. This factory has 2 horizontal type sterilizers with a capacity of 40 tons each. One machine that has a vital role in the production process is the screw press. Based on the results of observations and interviews with the company, it was found that the screw press machine was often damaged in the production process, as a result the production process was stopped and production results were not optimal, besides that the company would also experience losses. The purpose of this research is to measure the performance of the Screw Press Machine at PT Agro Sinergi Nusantara by calculating OEE through 3 main factors, namely Availability Ratio, Performance Effeciency Ratio, and Quality Ratio and determining corrective actions that can improve the performance of the Screw Press machine. The results of the performance analysis on the screw press machine obtained an Availability Ratio value of 84.52%, Performance Effeciency Ratio of 86.68%, Quality Ratio of 100%, and OEE of 73.37%. Based on a comparison with the Word Class Ideal Standard, only the Quality Ratio value meets the standard while the Availability Ratio, Performance Effeciency Ratio, and OEE have not met, meaning that the overall performance of the Screw Press Machine has decreased and has not met the Word Class Ideal Standard and it is necessary to improve the performance of the machine. Recommendations for improvements that can be made to improve the performance of the screw press machine again are to carry out routine maintenance on machine components so that damage can be prevented before severe damage occurs and cause longer downtime, conduct training for workers so that they can develop themselves and know more about the machine and the responsibilities that must be carried out, reinforce the implementation of routine and scheduled maintenance (preventive maintenance), and increase supervision during the FFB sorting process.

A general integration kernel formulation for immersed boundary method

This study proposes a new robust and accurate immersed boundary method for the immersion of solid bodies within a fluid with a Cartesian grid. The present method introduces the signed distance fields to recognize the immersed geometry contours, eliminating the need of Lagrangian points. To fully maximize the advantages offered by signed distance fields, a general integration kernel formulation is introduced into the direct forcing method to replace the conventional regularized delta function. With the combination of signed distance fields and kernel function, an interpolation along the radial direction instead of three-dimensional directions is feasible, which further reduces the extra calculation cost involved by immersed boundary method. The numerical results at low Reynolds numbers are compared to experimental and previous numerical results, which shows the efficiency and accuracy of this new method. Upon thorough validation, the proposed method in this paper demonstrates excellent performance across various scenarios, including static and moving cases as well as two- and three-dimensional configurations. And our method greatly reduces the cost of pretreatment of immersed geometry contours and apparently improves the convenience of the method.

Modelling cortical network dynamics

We have investigated the theoretical constraints of the interactions between coupled cortical columns. Each cortical column consists of a set of neural populations where each population is modelled as a neural mass. The existence of semi-stable states within a cortical column is dependent on the type of interaction between the neuronal populations, i.e., the form of the synaptic kernels. Current-to-current coupling has been shown, in contrast to potential-to-current coupling, to create semi-stable states within a cortical column. The interaction between semi-stable states of the cortical columns is studied where we derive the dynamics for the collected activity. For small excitations the dynamics follow the Kuramoto model; however, in contrast to previous work we derive coupled equations between phase and amplitude dynamics with the possibility of defining connectivity as a stationary and dynamic variable. The turbulent flow of phase dynamics which occurs in networks of Kuramoto oscillators would indicate turbulent changes in dynamic connectivity for coupled cortical columns which is something that has been recorded in epileptic seizures. We used the results we derived to estimate a seizure propagation model which allowed for inversions using the Laplace assumption (Dynamic Causal Modelling). The seizure propagation model was trialed on simulated data, and future work will investigate the estimation of the connectivity matrix from empirical data. This model can be used to predict changes in seizure evolution after virtual changes in the connectivity network, something that could be of clinical use when applied to epilepsy surgical cases.

Machine Learning of a Density Functional for Anisotropic Patchy Particles.

Anisotropic patchy particles have become an archetypical statistical model system for associating fluids. Here, we formulate an approach to the Kern-Frenkel model via the classical density functional theory to describe the positionally and orientationally resolved equilibrium density distributions in flat wall geometries. The density functional is split into a reference part for the orientationally averaged density and an orientational part in mean-field approximation. To bring the orientational part into a kernel form suitable for machine learning (ML) techniques, an expansion into orientational invariants and the proper incorporation of single-particle symmetries are formulated. The mean-field kernel is constructed via ML on the basis of hard wall simulation data. The results are compared to the well-known random-phase approximation, which strongly underestimates the orientational correlations close to the wall. Successes and shortcomings of the mean-field treatment of the orientational part are highlighted and perspectives are given for attaining a full-density functional via ML.

Determination of nonlocal kernels in nonlocal models for solving strain localization problems

ABSTRACT The strain localization during the damage and failure of materials can be accurately predicted by nonlocal continuum theories. Yet, the results of theoretical predictions are directly affected by the nonlocal kernel formulation. In this study, the correlation between the nonlocal kernel formulation and the strain localization simulation results is revealed. First, based on the nonlocal damage theory, a novel free energy functional was proposed with strain and damage parameters as independent variables, and a constitutive relation considering nonlocality and elastic damage was derived within the framework of continuum thermodynamics. Then, one-dimensional governing equations were obtained through the dual-linearization method. Finally, strain localization predictions obtained with several nonlocal kernel formulations were presented, and the relationship between the nonlocal kernels and the solution was analyzed, thereby determining an appropriate nonlocal kernel function for the strain localization problem.

Mathematical study of polycystic ovarian syndrome disease including medication treatment mechanism for infertility in women.

Among women of reproductive age, PCOS (polycystic ovarian syndrome) is one of the most prevalent endocrine illnesses. In addition to decreasing female fertility, this condition raises the risk of cardiovascular disease, diabetes, dyslipidemia, obesity, psychiatric disorders and other illnesses. In this paper, we constructed a fractional order model for polycystic ovarian syndrome by using a novel approach with the memory effect of a fractional operator. The study population was divided into four groups for this reason: Women who are at risk for infertility, PCOS sufferers, infertile women receiving therapy (gonadotropin and clomiphene citrate), and improved infertile women. We derived the basic reproductive number, and by utilizing the Jacobian matrix and the Routh-Hurwitz stability criterion, it can be shown that the free and endemic equilibrium points are both locally stable. Using a two-step Lagrange polynomial, solutions were generated in the generalized form of the power law kernel in order to explore the influence of the fractional operator with numerical simulations, which shows the impact of the sickness on women due to the effect of different parameters involved.

Analytical study of a Hepatitis B epidemic model using a discrete generalized nonsingular kernel

<abstract><p>Hepatitis B is a worldwide viral infection that causes cirrhosis, hepatocellular cancer, the need for liver transplantation, and death. This work proposed a mathematical representation of Hepatitis B Virus (HBV) transmission traits emphasizing the significance of applied mathematics in comprehending how the disease spreads. The work used an updated Atangana-Baleanu fractional difference operator to create a fractional-order model of HBV. The qualitative assessment and well-posedness of the mathematical framework were looked at, and the global stability of equilibrium states as measured by the Volterra-type Lyapunov function was summarized. The exact answer was guaranteed to be unique using the Lipschitz condition. Additionally, there were various analyses of this new type of operator to support the operator's efficacy. We observe that the explored discrete fractional operators will be $ \chi^2 $-increasing or decreasing in certain domains of the time scale $ \mathbb{N}_j: = {j, j + 1, ... } $ by looking at the fundamental characteristics of the proposed discrete fractional operators along with $ \chi $-monotonicity descriptions. For numerical simulations, solutions were constructed in the discrete generalized form of the Mittag-Leffler kernel, highlighting the impacts of the illness caused by numerous causes. The order of the fractional derivative had a significant influence on the dynamical process utilized to construct the HBV model. Researchers and policymakers can benefit from the suggested model's ability to forecast infectious diseases such as HBV and take preventive action.</p></abstract>