Search

We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation of a solution to the inverse problem. The minimization problem contains a regularization term which is given by a total variation penalization and is characterized by a regularization parameter. The discretization involves at the same time the boundary measurements, by the use of the complete electrode model, the unknown conductivity and the solution to the direct problem. The electrodes are characterized by a parameter related to their size, which in turn controls the number of electrodes to be used. The discretization of the unknown and of the solution to the direct problem is characterized by another parameter related to the size of the mesh involved. In our analysis we show how to precisely choose the regularization, electrodes size and mesh size parameters with respect to the noise level in such a way that the solution to the discrete regularized problem is meaningful. In particular we obtain that the electrodes and mesh size parameters should decay polynomially with respect to the noise level. [ABSTRACT FROM AUTHOR]

The Vertex Morphing method is a node-based shape parameterization that uses an explicit filtering approach to regularize the optimization problem and generate smooth shapes. It has been successfully applied to shape optimization problems of industrial size in recent years. This work investigates in detail how irregular discretizations, design surface boundaries, and complex geometries can influence the progress of a gradient-based optimization using the standard Vertex Morphing formulation. A sensitivity weighting approach based on the available shape morphing functions is presented, which eliminates all of the aforementioned influences. Subsequently, a design variable scaling strategy is developed that transforms the optimization problem into an alternative design space and allows the use of arbitrary, even highly irregular surface discretizations in combination with black-box optimization algorithms for shape optimization with the Vertex Morphing method. Illustrative academic examples and an application case of an additively manufactured part are presented to support the work. [ABSTRACT FROM AUTHOR]

Even for short space flights, such as manned flights to the Moon back in the 60s, it became extremely obvious that electric batteries are completely unsuitable because of their huge mass needed to store the required supply of onboard energy. Therefore, already in the 60s of the last century, the USA and the USSR took the path of using a hydrogen storage system using hydrogen fuel cells. Solar-hydrogen energy systems for spacecraft reliably provide energy for carrying out various kinds of unscheduled work on board and for eliminating emergency situations. Surplus energy is constantly accumulated on board in the form of the chemical energy of cryogenic hydrogen. The article presents too the result of the development of a solar-hydrogen power supply system for spacecraft, the onboard network of which provides slow and fast processes. This paper delves into designing digital control systems for power conditioning devices on the example of charging-discharging device based on a boost DC-DC converter. The device was implemented to condition a storage battery by properly charging and discharging it. A technique for constructing a small signal model of the converter has been developed. This involves constructing a block diagram using the discontinuous switching functions and analog models of electrical circuit components. The transfer functions of the analog controllers for the two-loop subordinate control system were derived. Then the procedure of a bilinear z-transform to the given transfer functions was applied. As a result, the discrete transfer functions of the digital controllers were obtained. Realization both analog and digital controllers in Matlab Simulink confirmed their identical behavior and stability within the power converter. Experimental studies of the static and dynamic waveforms of the digital control system showed controllability and stability of the charging-discharging device for the storage battery in typical operating modes. Thus, the method of synthesizing digital control systems for energy converters is proved valid and applicable for designing new power conditioning devices within modern power supplies. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the maximum value of the point‐polyserial correlation between a random variable with an assigned continuous distribution and an ordinal random variable with k $$ k $$ categories, which are assigned the first k $$ k $$ natural values 1 , 2 , … , k $$ 1,2,\dots, k $$, and arbitrary probabilities p i $$ {p}_i $$. For different parametric distributions, we derive a closed‐form formula for the maximal point‐polyserial correlation as a function of the p i $$ {p}_i $$ and of the distribution's parameters; we devise an algorithm for obtaining its maximum value numerically for any given k $$ k $$. These maximum values and the features of the corresponding k $$ k $$‐point discrete random variables are discussed with respect to the underlying continuous distribution. Furthermore, we prove that if we do not assign the values of the ordinal random variable a priori but instead include them in the optimization problem, this latter approach is equivalent to the optimal quantization problem. In some circumstances, it leads to a significant increase in the maximum value of the point‐polyserial correlation. An application to real data exemplifies the main findings. A comparison between the discretization leading to the maximum point‐polyserial correlation and those obtained from optimal quantization and moment matching is sketched. [ABSTRACT FROM AUTHOR]

In this paper, we use the distributionally robust approach to study stochastic variational inequalities under the ambiguity of the true probability distribution, which is referred to as distributionally robust variational inequalities (DRVIs). First of all, we adopt a relaxed value function approach to relax the DRVI and obtain its relaxation counterpart. This is mainly motivated by the robust requirement in the modeling process as well as the possible calculation error in the numerical process. After that, we investigate qualitative convergence properties as the relaxation parameter tends to zero. Considering the perturbation of ambiguity sets, we further study the quantitative stability of the relaxation DRVI. Finally, when the ambiguity set is given by the general moment information, the discrete approximation of the relaxation DRVI is discussed. [ABSTRACT FROM AUTHOR]

The traditional methods for predicting the distribution of albacore (Thunnus alalunga) fishing grounds have low performance and accuracy. Uneven sampling can result in unreasonable evaluation indicators. To address these issues, three methods, equi-frequency, K-means clustering algorithm, and 1-R split, were applied to discretize the catch per unit effort (CPUE) of albacore in the South Pacific from 2016 to 2021 and partition the fishing grounds into abundance levels. Eight machine learning models were used to predict the fishing grounds. In addition to the traditional evaluation index based on confusion matrix, top-k index was also used to evaluate the accuracy of fishery abundance predictions. The results showed that (1) When sampling is unbalanced, the reported accuracy does not fully represent the actual performance of the model in predicting the abundance of albacore in the fishing ground. F1 value can be used as the index of the model effect and stability. (2) In binary classification, the quartile stacking algorithm has the best stacking performance, with F1 0.89. (3) The top-1 prediction accuracy of three-category fishery forecasting is the highest at 0.74, and the top-1 prediction accuracy of five-category fishery forecasting is the highest at 0.54. (4) The top-k accuracy of classification of fisheries with multiple abundance using K-means is significantly better than that of equal frequency discretization (p < 0.001). The top-k evaluation index was used to predict the fishing grounds of albacore across multiple abundance levels for the first time in this study, which is significant for pioneering a new method for this application and which provides a demonstration of the development of artificial intelligence techniques for fisheries in the future. [ABSTRACT FROM AUTHOR]

In this work, we propose a novel family of procedures for deriving a discrete counterpart to a continuous probability distribution. They are based on a class of distances between cumulative distribution functions, including the Cramér, the Cramér-von Mises, and the Anderson-Darling distances as particular cases. The discrete counterpart is defined and derived as the random variable which minimizes its distance to the assigned continuous probability distribution among all the discrete random variables supported on the set of integers (or positive integers). Applications are provided with reference to the exponential and the normal distributions, among others; the discrete counterparts are derived, and their main properties are discussed, also in comparison with the one obtained through an existing discretization technique based on the preservation of the cumulative distribution function at integer values. Parameter estimation for these discrete analogs is discussed, along with an analysis of two real datasets, where they are compared in terms of goodness-of-fit with some popular discrete distributions. Furthermore, in order to highlight the effectiveness and the benefits derived from the proposed discretization procedures, we illustrate two practical applications in actuarial science and in reliability engineering. In the former case, the problem of determining the distribution of the total claims amount for a non-life insurance portfolio is considered, where the claim sizes can be modelled as iid random variables, and the number of claims is random as well. Actuaries use a recursive calculation method based on Panjer's formula, which requires an appropriate discretization of the individual claim distribution, and therefore the proposed procedures can be used. Since we consider two simple cases where the distribution of the total claims amount is analytically acquirable, the efficacy of the discretization procedures in the final approximation can be easily assessed and turns out to be satisfactory, especially when compared to the existing discretization. The latter case considers the determination of the reliability parameter for a complex stress-strength model. Here, the approximation by discretization is compared to Monte Carlo simulation and shown to be relevant: with a comparable if not smaller computational effort, discretization leads to similar results as simulation. Such discretizations can also naturally be applied to more complex problems such as scenario generation in stochastic programming. R code for this article is provided as supplementary material. [ABSTRACT FROM AUTHOR]

We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We discuss elliptic complexes and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry, and modelling software design. In particular, the ubiquitous concept of naturality is central. Natural differential operators have functorial analogues on the cochains of triangulated manifolds. In order to establish this correspondence, we derive formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan’s magic formula. This theorem is the main mathematical result of the paper. [ABSTRACT FROM AUTHOR]

We study the assortment optimization problem under general linear constraints, where the customer choice behavior is captured by the cross-nested logit model. In this problem, there is a set of products organized into multiple subsets (or nests), where each product can belong to more than one nest. The aim is to find an assortment to offer to customers so that the expected revenue is maximized. We show that, under the cross-nested logit model, the unconstrained assortment problem is NP-hard even when there are only two nests, and the problem is generally NP-hard to approximate to any constant factors. To tackle this challenging problem, we develop a new discretization mechanism to approximate the problem by a linear fractional program with a performance guarantee of (1 − ϵ) / (1 + ϵ) , for any accuracy level ϵ > 0. We then show that optimal solutions to the approximate problem can be obtained by solving mixed-integer linear programs. We further show that our discretization approach can also be applied to solve a joint assortment optimization and pricing problem, as well as an assortment problem under a mixture of cross-nested logit models to account for multiple classes of customers. Our empirical results on a large number of randomly generated test instances demonstrate that, under a performance guarantee of 90% (i.e., expected revenues are guaranteed to be at least 90% of the optimal revenue), the percentage gaps between the objective values obtained from our approximation methods and the optimal expected revenues are no larger than 1.2%. [ABSTRACT FROM AUTHOR]

In real-life data, count data are considered more significant in different fields. In this article, a new form of the one-parameter discrete linear-exponential distribution is derived based on the survival function as a discretization technique. An extensive study of this distribution is conducted under its new form, including characteristic functions and statistical properties. It is shown that this distribution is appropriate for modeling over-dispersed count data. Moreover, its probability mass function is right-skewed with different shapes. The unknown model parameter is estimated using the maximum likelihood method, with more attention given to Bayesian estimation methods. The Bayesian estimator is computed based on three different loss functions: a square error loss function, a linear exponential loss function, and a generalized entropy loss function. The simulation study is implemented to examine the distribution's behavior and compare the classical and Bayesian estimation methods, which indicated that the Bayesian method under the generalized entropy loss function with positive weight is the best for all sample sizes with the minimum mean squared errors. Finally, the discrete linear-exponential distribution proves its efficiency in fitting discrete physical and medical lifetime count data in real-life against other related distributions. [ABSTRACT FROM AUTHOR]

High-throughput chromosome conformation capture (Hi-C) technology captures spatial interactions of DNA sequences into matrices, and software tools are developed to identify topologically associating domains (TADs) from the Hi-C matrices. With structural information theory, SuperTAD adopted a dynamic programming approach to find the TAD hierarchy with minimal structural entropy. However, the algorithm suffers from high time complexity. To accelerate this algorithm, we design and implement an approximation algorithm with a theoretical performance guarantee. We implemented a package, SuperTAD-Fast. Using Hi-C matrices and simulated data, we demonstrated that SuperTAD-Fast achieved great runtime improvement compared with SuperTAD. SuperTAD-Fast shows high consistency and significant enrichment of structural proteins from Hi-C data of human cell lines in comparison with the existing six hierarchical TADs detecting methods. [ABSTRACT FROM AUTHOR]

In the statistical literature, several discrete distributions have been developed so far for modeling non-negative integer-valued phenomena, yet there is still room for new counting models that adequately capture the diversity of real data sets. Here, we first discuss a count distribution derived as a discrete analogue of the continuous half-logistic distribution, which is obtained by preserving the expression of its survival function at each non-negative integer support point. The proposed discrete distribution has a mode at zero and allows for over-dispersion; these two features make it suitable for modeling purposes in many fields (e.g., insurance and ecology), when these conditions are satisfied by the data. In order to widen its spectrum of applications, a discrete analogue is also presented of the type I generalized half-logistic distribution (obtained by adding a shape parameter to the simple one-parameter half-logistic), which allows us to model count data whose mode is not necessarily zero. For these new count distributions, the main statistical properties are outlined, and parameter estimation along with related issues is discussed. Their feasibility is proved on two real data sets taken from the literature, which have already been fitted by other well-established count distributions. Finally, a possible application is illustrated in the insurance field, related to the exact/approximate determination of the distribution of the total claims amount through the well-known Panjer's recursive formula, within the framework of collective risk models. [ABSTRACT FROM AUTHOR]

Automatic Guided Vehicles (AGVs) are widely used in flexible manufacturing systems for material handling inside the factory. Traffic management strategies, required to guarantee a conflict-free operation of the overall fleet, discretize the workspace of the AGVs and use the resulting graph model for route planning and execution. In zone control approaches, AGVs move from node to node on a permit basis, with limitations on the allowed number of AGVs at a time in each area of the graph to prevent and/or resolve deadlocks and conflicts. Hence, for an optimal implementation of traffic controllers in real manufacturing systems, it is essential to understand how the layout discretization influences the performance of the AGV network. This paper analyzes its effect in grid-like shaped workspaces by using a representative zone control algorithm and a recently developed improvement of it. Realistic numerical experiments on different layouts reveal that denser discretizations do not yield faster executions or increase in throughput, while lower control periods in the permit system entail significant performance uplifts. [ABSTRACT FROM AUTHOR]

In this paper we prove that the solutions of an implicit scheme converge to equilibrium points under certain conditions on the potential function $ F $ and the forcing term $ g $. For this purpose, we use a Lojasiewicz inequality and introduce a discretized energy and a Lyapunov function. In addition, we provide some numerical simulations that support our findings. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behaviour of the bounded solution of the following forced system and its discretization$ \varepsilon u''+\vert u'\vert^\alpha u'+\nabla F(u) = g(t) $with conditions on F and g. We present similar forced angle conditions for both continuous and discrete cases, and then we apply it to examples from the literature. [ABSTRACT FROM AUTHOR]

Warped time-frequency systems have recently been introduced as a class of structured continuous frames for functions on the real line. Herein, we generalize this framework to the setting of functions of arbitrary dimensionality. After showing that the basic properties of warped time-frequency representations carry over to higher dimensions, we determine conditions on the warping function which guarantee that the associated Gramian is well-localized, so that associated families of coorbit spaces can be constructed. We then show that discrete Banach frame decompositions for these coorbit spaces can be obtained by sampling the continuous warped time-frequency systems. In particular, this implies that sparsity of a given function f in the discrete warped time-frequency dictionary is equivalent to membership of f in the coorbit space. We put special emphasis on the case of radial warping functions, for which the relevant assumptions simplify considerably. [ABSTRACT FROM AUTHOR]

To delve into the mechanism behind wideband oscillation instability in voltage source converter based high voltage direct current （VSC-HVDC） transmission system and swiftly assess system stability with precision， a modeling method and a stability criterion based on z-domain impedance are proposed. Firstly， employing precise discretization techniques， a z-domain impedance model is established for voltage-source converters considering phase-locked loops （PLLs）， inner-current loop， and voltage feedforward control， along with a z-domain impedance model for AC grids in a global coordinate system. Subsequently， a criterion for generalized forbidden zone is proposed using Gershgorin circle theorem to analyze the impact of phase-locked loops and line impedances on system stability. Finally， simulation analysis is conducted on the VSC-HVDC system to demonstrate the effectiveness of the proposed generalized impedance criterion.

The generalized uncertainty principle provides a promising phenomenological approach to reconciling the fundamentals of general relativity with quantum mechanics. As a result, noncommutativity, measurement uncertainty, and the fundamental theory of quantum mechanics are subject to finite gravitational fields. The generalized uncertainty principle (RGUP) on curved spacetime allows for the imposition of quantum-induced elements on general relativity (GR) in four dimensions. In the relativistic regimes, the determination of a test particle's spacetime coordinates, 〈 x μ 〉 , becomes uncertain. There exists a specific range of coordinates where the accessibility to 〈 (x μ) 2 〉 is notably limited. Consequently, the spacetime coordinates lack both smoothness and continuity. The quantum-mechanical calculations of the spacetime coordinates are directly linked to their measurement through the expectation value 〈 x μ 〉. This expectation value is dependent on 〈 (x μ) 2 〉 , which is itself limited by a minimum measurable length Δ x m i n 2 . A crucial finding presented in this script is the existence of a lower bound for the Hamiltonian, which implies the stability of the quantum nature of spacetime. The direct correlation between Δ x m i n and − 〈 g 〉 μ ν is a significant discovery, suggesting a proportional relationship. The conjecture is made that the primary metric g carries all essential information regarding spacetime curvature and serves a role akin to the Jacobian determinant in general relativity. Moreover, the linear relationship between Δ x m i n and the Planck length ℓ p is established with a proportionality factor of − 〈 g 〉 μ ν β 2 , where β 2 denotes the RGUP parameter. The discretization of spacetime coordinates results in the discontinuity of the test particle's wavefunction ψ (x , t) , leading to an unrealistic Δ p. It is also noted that the lower limit of 〈 (p μ) 2 〉 is directly proportional to the fundamental tensor. [ABSTRACT FROM AUTHOR]

Data-driven models used for predictive classification and regression tasks are commonly computed using floating-point arithmetic and powerful computers. We address constraints in distributed sensor networks like the IoT, edge, and material-integrated computing, providing only low-resource embedded computers with sensor data that are acquired and processed locally. Sensor networks are characterized by strong heterogeneous systems. This work introduces and evaluates a virtual machine architecture that provides ML as a service layer (MLaaS) on the node level and addresses very low-resource distributed embedded computers (with less than 20 kB of RAM). The VM provides a unified ML instruction set architecture that can be programmed to implement decision trees, ANN, and CNN model architectures using scaled integer arithmetic only. Models are trained primarily offline using floating-point arithmetic, finally converted by an iterative scaling and transformation process, demonstrated in this work by two tests based on simulated and synthetic data. This paper is an extended version of the FedCSIS 2023 conference paper providing new algorithms and ML applications, including ANN/CNN-based regression and classification tasks studying the effects of discretization on classification and regression accuracy. [ABSTRACT FROM AUTHOR]

This article presents a parallel finite element discretization scheme for solving numerically the steady natural convection equations, where a fully overlapping domain decomposition technique is used for parallelization. In this scheme, each processor computes independently a local solution in its subdomain using a mesh that covers the entire domain. It has a small mesh size h around the subdomain and a large mesh size H away from the subdomain. The discretization scheme is easy to implement based on existing serial software. It can yield an optimal convergence rate for the approximate solutions with suitable algorithmic parameters. Compared with the standard finite element method, the scheme is able to obtain an approximate solution of comparable accuracy with considerable reduction in computational time. Theoretical and numerical results show the promise of the scheme, where numerical simulation results for some benchmark problems such as the buoyancy-driven square cavity flow, right-angled triangular cavity flow and sinusoidal hot cylinder flow are provided. [ABSTRACT FROM AUTHOR]

Copyright of Zhejiang Electric Power is the property of Zhejiang Electric Power Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Lymphatic Filariasis (LF) is a vector-borne ailment that creates disfiguring and disabling in the forbearer. This gruesome disease is generally acquired at the childhood stage through the intervention of a microfilariae (mf) parasite in the host's bloodstream, but its clinical manifestation arises subsequently in life. The conventional approach used for detecting mf within the blood smear is done by a pathologist through a microscope. Because the procedure of such inspection is manual, it might lead to inconsistent outcomes. This research study proposed a new OCR-NURBS image segmentation technique employing a curve-fitting algorithm for the microscopic image of the mf parasite. To achieve the aforementioned objective, five different procedures have been applied: (i) Filtering of image for removal of artifacts; (ii) Application of Dilation and Erosion morphological operation for image enrichment near edges of mf; (iii) A pixel clustering segmentation technique to segregate the parasite from the image backdrop; (iv) Smoothing and discretization on the boundary curve to preserve the edge points; (v) NURBS functional network for the construction of curves of mf from the border points. The results of the proposed method demonstrate phenomenal performance in terms of visual interpretation of segmented images. Additionally, the efficacy of the proposed model is assessed through the evaluation of the image quality index and morphological parameters of the image. Furthermore, a comparison is made between the parameters of manually segmented images and those generated by the proposed model in order to assess the error incurred by the model. The results demonstrate that the proposed model exhibits robustness in performing segmentation of mf parasites. [ABSTRACT FROM AUTHOR]

Assembly sequence planning is a critical facet of product design and manufacturing, influencing assembly effectiveness and product tolerances. Ineffectual sequences can hamper product quality and development efficiency. This study introduces a matrix-based evaluation function and an enhanced pollination algorithm for optimal assembly sequence determination. This approach accounts for various assembly factors and links the evaluation matrix to fuzzy mathematics through expert decision-making methods. The algorithm is tailored to assembly sequences, with defined rules for judgment, operation, assignment and repair. It incorporates Levy flights for broader search exploration and an adaptive mutation factor for refined local search. The hybrid algorithm, combining genetic algorithm principles, is employed for sequence planning and analysis. Comparative simulations highlight the hybrid algorithm's robust global search and convergence capabilities. This research offers practical significance in enhancing assembly performance and overall product reliability within the product development cycle. [ABSTRACT FROM AUTHOR]

The Roe algebra C * (X) is a noncommutative C * -algebra reflecting metric properties of a space X, and it is interesting to understand the correlation between the Roe algebra of X and the (uniform) Roe algebra of its discretization. Here, we perform a minor step in this direction in the simplest non-trivial example, namely X = R , by constructing a continuous field of C * -algebras over [ 0 , 1 ] , with the fibers over non-zero points constituting the uniform C * -algebra of the integers, and the fibers over 0 constituting a C * -algebra related to R. [ABSTRACT FROM AUTHOR]

A study of the possibilities of measuring current values, phase shift angles and other parameters of electrical networks using these data, when sampling with a frequency not multiple of the network frequency, was carried out. In the proposed algorithm, an intermediate linear approximation is used to calculate the instantaneous values of interpolated signals at equidistant moments of time during the period of the main frequency of the network. The cases when such sampling is performed on one device with sequential or simultaneous coding of input signals and the case when sampling frequencies do not match in each measurement channel are considered. A study of the use of these methods was conducted using the developed computer model and layout of the distributed measurement system. References 32, figures 19. [ABSTRACT FROM AUTHOR]

In some geomechanical treatments of induced seismicity, the fault surface is idealized to be a plane. We depart from this assumption by comparing a discretization model and a kriging model, both of which allow the incorporation of rugosity, roughness, and curvature into the fault surface and subsequent geomechanical models of hazard. We test the Hogback Flexural Faults of the San Juan Basin, which could potentially pose a problem for induced seismicity in a carbon sequestration project in the northwestern portion of the basin. The discretization model emmeshes data about the location of the fault surface in three-dimensional space into hexagonally close-packed spheres. Each sphere that contains enough data is termed a region and Bayes' Law is used to find a distribution of strikes and dips that describe the data within the region. The kriging model uses Gaussian processes to interpolate and extrapolate a surface through all data points. The results show that the discretized regions possess, in general, lower Coulomb failure functions, but the uncertainty in the distributions, i.e., the ranges, becomes greater as the discretization increases due to overfitting. The majority of the uncertainty in both the discretization model and the kriging model is contained in the geomechanical priors. Finally, the discretization and kriging of the fault surface elucidates locations with higher Coulomb failure functions. [ABSTRACT FROM AUTHOR]

This article describes the R package CorrToolBox, which is designed for modeling the correlation transitions under specified distributional assumptions within the realm of discretization in the context of the latency and threshold concepts. The practical utility and functionality of the package are demonstrated by several illustrative examples. In addition, the package's feasibility and performance are evaluated via simulation studies using synthetic mixed data with a range of marginal distributions. [ABSTRACT FROM AUTHOR]

In the study of the equations of motion of systems of various physical nature, there are problems in determining the qualitative indicators and properties of motion according to the known structure and properties of the equations under consideration. Such qualitative indicators for finite-dimensional systems are, in particular, integral invariants — integrals of some functions that retain their value during the system movement. They were introduced into analytical mechanics by A. Poincaré. In the future, the connection of integral invariants with a number of fundamental concepts of classical dynamics was established. The main purpose of this work is to extend some notions of the theory of integral invariants to broad classes of equations of motion of infinite-dimensional systems. Using a given Hamilton’s action, the equations of motion of potential systems with an infinite number of degrees of freedom are obtained, generalizing the well-known Birkhoff equations. A difference analog with discrete time is constructed for them. Based on it, a difference approximation of the corresponding integral invariant of the first order is found.

Discrete Laplacians are the basis for various tasks in geometry processing. While the most desirable properties of the discretization invariably lead to the so‐called cotangent Laplacian for triangle meshes, applying the same principles to polygon Laplacians leaves degrees of freedom in their construction. From linear finite elements it is well‐known how the shape of triangles affects both the error and the operator's condition. We notice that shape quality can be encapsulated as the trace of the Laplacian and suggest that trace minimization is a helpful tool to improve numerical behavior. We apply this observation to the polygon Laplacian constructed from a virtual triangulation [BHKB20] to derive optimal parameters per polygon. Moreover, we devise a smoothing approach for the vertices of a polygon mesh to minimize the trace. We analyze the properties of the optimized discrete operators and show their superiority over generic parameter selection in theory and through various experiments. [ABSTRACT FROM AUTHOR]

This work is devoted to investigate the global asymptotic stability of equilibriums for a reaction–diffusion susceptible-infected-susceptible (SIS) epidemic model with spatial heterogeneity and mass-action-type nonlinearity. By discretizing the spatial variables of the model, first, Lyapunov functions are constructed for the corresponding ordinary differential equations (ODEs) model of the original SIS PDEs model, and then the construction method is generalized to the PDEs model in which either the susceptible or the infectious individuals are spreading in spatial heterogeneity environment. For both the cases, we obtained the standard threshold dynamics results. [ABSTRACT FROM AUTHOR]