A new mathematical model of memristive neural networks described by the partly diffusive reaction-diffusion equations with weak synaptic coupling is proposed and investigated. Under rather general conditions it is proved that there exists an absorbing set showing the dissipative dynamics of the solution semiflow in the energy space and multiple ultimate bounds. Through uniform estimates and maneuver of integral inequalities and sharp interpolation inequalities on the interneuron differencing equations, it is rigorously proved that exponential synchronization of the neural network solutions at a uniform convergence rate occurs if the coupling strength satisfies a threshold condition expressed by the system parameters. Applications with numerical simulation to the memristive diffusive Hindmarsh-Rose neural networks and FitzHugh-Nagumo neural networks are also shown.

Mathematics serves as a fundamental intelligent theoretic basis for computation, and mathematical analysis is very useful to develop computational methods to solve various problems in science and engineering. Integral transforms such as Laplace Transform have been playing an important role in computational methods. In this paper, we will introduce Sumudu Transform in a new computational approach, in which effective computational methods will be developed and implemented. Such computational methods are straightforward to understand, but powerful to incorporate into computational science to solve different problems automatically. We will provide computational analysis and essentiality by surveying and summarizing some related recent works, with additional automatic proof details by applying system built-in functions. Applications include the computation of coefficients of Taylor's expansions, calculation of generating functions, mathematical identity proofs, solving differential equations and integral equations. For demonstration purposes, some of the methods were implemented in Maple with demonstrational results matching the expected values.

A new mathematical model of neural networks described by diffusive FitzHugh-Nagumo equations with memristors and linear synaptic coupling is proposed and investigated. The existence of absorbing set for the solution semiflow in the energy space is proved and global dynamics of the memristive neural networks are dissipative. Through uniform estimates and maneuver of integral inequalities on the interneuron difference equations, it is shown that exponential synchronization of the neural network at a uniform convergence rate occurs if the coupling strength satisfies a threshold condition explicitly expressed by the system parameters, which is illustrated by an example and numerical simulation experiments.

Enhancement is an important step in post-processing digital images for personal use, in medical imaging, and for object recognition. Most existing manual techniques rely on region selection, similarity, and/or thresholding for editing, never really considering the topological structure of the image. In this paper, we leverage the contour tree to extract a hierarchical representation of the topology of an image. We propose 4 topology-aware transfer functions for editing features of the image using local topological properties, instead of global image properties. Finally, we evaluate our approach with grayscale and color images.

With the popularization of Topological Data Analysis, the Reeb graph has found new applications as a summarization technique in the analysis and visualization of large and complex data, whose usefulness extends beyond just the graph itself. Pairing critical points enables forming topological fingerprints, known as persistence diagrams, that provides insights into the structure and noise in data. Although the body of work addressing the efficient calculation of Reeb graphs is large, the literature on pairing is limited. In this paper, we discuss two algorithmic approaches for pairing critical points in Reeb graphs, first a multipass approach, followed by a new single-pass algorithm, called Propagate and Pair.

Stochastic virus dynamics modeled by a system of stochastic differential equations with Beddington-DeAngelis functional response and driven by white noise is investigated. The global existence of positive solutions and the existence of stationary distribution are proved. Upper and lower bounds of the pathwise and asymptotic moments for the positive solutions are sharply estimated. The absorbing property in time average is shown and the moment Lyapunov exponents are proved to be nonpositive.

With the restriction of the diameter and feed direction of the cutting tool in milling process, electric discharge machining (EDM) is the only effective machining technology for the uncut regions with internal sharp corner.Automatic design of the electrode is of great significance for the CAD/CAM integration of EDM technology.In current CAD/CAM system the electrode design is done manually by technologists based on experience and knowledge.The procedure is tedious and timeconsuming.In this paper, a novel approach is proposed to automatically generate the electrode CAD model taking the topological vertices of uncut region as the hint.The hint feature points are innovatively defined and classified into three types: internal-sharp points, cutting-into points and interacting points.Based on this, our approach firstly determines the faces and the type of uncut region.Secondly, the interacting region is decomposed into the isolated region by reconstructing the topological structure, patching the split face and partitioning the shared face.Thirdly, the modeling parameters are extracted from the isolated region.Finally, the electrode CAD model is created by executing a set of generic modeling operations.The electrode CAD model can be directly used in the process planning, so as to promote the integration of CAD and CAM.

Assessing the quality of 3D printed models before they are printed remains a challeng- ing problem, particularly when considering point cloud-based models. This paper introduces an approach to quality assessment, which uses techniques from the field of Topological Data Analy- sis (TDA) to compute a topological abstraction of the eventual printed model. Two main tools of TDA, Mapper and persistent homology, are used to analyze both the printed space and empty space created by the model. This abstraction enables investigating certain qualities of the model, with respect to print quality, and identifies potential anomalies that may appear in the final product.

Asymptotic dynamics of stochastic Brusselator system with multiplicative noise is investigated in this work. The existence of random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compactness here is more rigorous through the bootstrap pullback estimations than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the attracting regularity to be an (L-2 x L-2, H-1 x H-1) random attractor.

In this work, the existence and properties of a global attractor for the weak solution semiflow of the Boissonade system are proved. A parameter adjusting and grouping estimation method is developed to show the absorbing property and asymptotic compactness of the solution trajectories of this reaction-diffusion system with quadratic and cubic nonlinearity. The upper-semicontinuity of the global attractors in the \(H^1\) product space for the solution semiflow with respect to the quadratic term coefficient converging to zero is proved. The barrier of the perturbed singularity between zero quadratic term coefficient and non-zero one is overcome by the uniform adjusting parameter.

In this article, we prove the existence of a random attractor of the stochastic three-component reversible Gray-Scott system on infinite lattice with additive noise. We use a transformation of addition involved with Ornstein-Uhlenbeck process, for proving the pullback absorbing property and the pullback asymptotic compactness of the reaction diffusion system with cubic nonlinearity.

This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with nonlinear damping and multiplicative white noise defined on an unbounded domain. By showing the pullback asymptotic compactness of the cocycle in a certain parameter region, we prove the existence of a random attractor when the intensity of noise is sufficiently small. For the stochastic wave equation with rapidly oscillating external force we prove that the Hausdorff distance between the random attractor Aϵ of the original equation and the random attractor A0 of the averaged equation is in the order of O(ϵ).

We explore when Hirota bilinear equations possess linear subspaces of solutions. First, we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to Hirota bilinear equations. Second, we show that multivariate polynomials whose zeros form a vector space can generate the desired Hirota bilinear equations with given linear subspaces of solutions, and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero. Third, applying an algorithm using weights, we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired Hirota bilinear equations.

In this paper, we obtain two kinds of sufficient conditions consisting of systems of linear partial differential equations, which guarantee that the corresponding Wroskian determinant solves the (3 + 1)-dimensional Jimbo–Miwa equation in the Hirota bilinear form. Our results suggest that more general conditions could be derived by further study.

The explicit expression of the flow equations of the noncommutative Kadomtsev–Petviashvili (ncKP) hierarchy is derived. Compared with the flow equations of the KP hierarchy, our result shows that the additional terms in the flow equations of the ncKP hierarchy indeed consist of commutators of dynamical coordinates {ui}. The recursion operator for the flow equations under n-reduction is presented. Further, under 2-reduction, we calculate a nonlocal recursion operator Φ(2) of the noncommutative Korteweg–de Vries(ncKdV) hierarchy, which generates a hierarchy of local, higher-order flows. Thus we solve the open problem proposed by Olver and Sokolov (1998 Commun. Math. Phys. 193 245–68).

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