In this paper, we apply the theory of soft sets to partially ordered semigroup. First, several new notions such as left (right) filter soft partially ordered semigroup, filter soft partially ordered semigroup, whole left (right) filter soft partially ordered semigroup, whole filter soft partially ordered semigroup, prime left (right) ideal soft partially ordered semigroup, prime ideal soft partially ordered semigroup over partially ordered semigroups S are introduced. Further, with prime left (right) ideal soft partially ordered semigroup and prime ideal soft partially ordered semigroup over S, the necessary and sufficient conditions that the non-null soft set over S is a right (left) filter soft partially ordered semigroup and filter soft partially ordered semigroup over S are given separately. Finally, quotient ordered homomorphic images and inverse images under partially ordered homomorphic on left (right) filter soft partially ordered semigroup, filter soft partially ordered semigroup, whole left (right) filter soft partially ordered semigroup and whole filter soft partially ordered semigroup are studied,and some related conclusions are obtained.

Artificial intelligence generated content (AIGC) has received significant attention at present. As the numerous generative models proposed, the emerging diffusion model has attracted extensive attention due to its highly interpretable mathematical properties and the ability to generate high-quality and diverse results. Nowadays, diffusion models have achieved remarkable results in the field of condition-guided image generation. This achievement promotes the development of diffusion models in other conditional tasks and has various applications in areas such as movies, games, paintings, and virtual reality. For instance, the diffusion model can generate high-resolution images in text-guided image generation tasks while ensuring the quality of the generated images. In this paper, we first introduce the definition and background of diffusion models. Then, we present a review of the development history and latest progress of conditional image generation based on diffusion models. Finally, we conclude this survey with discussions on challenges and future research directions of diffusion models.

Algebraic analysis of fuzzy logic is one of the hot issues in the field of fuzzy logic research. In this paper, Fuzzy implication algebras and its ideals problem are further studied by using the method and principle of algebra and lattice theory. Firstly, some new properties of Fuzzy implication algebras are revealed by using pseudo-complement operators. Secondly, the concepts of ideal and generating ideal are introduced in Fuzzy implication algebras, and their properties and equivalent characterizations are investigated. Finally, the lattice theory characteristics of the set consisting of all ideals in a given Fuzzy implication algebra are discussed, it is proved that the set forms a distributive continuous (algebraic) lattice with respect to the set-inclusion order, in particular, it forms a complete Heyting algebra, and then forms a Frame.

In order to study the impact of cooperative behaviors between predators on the dynamic behavior of the food chain system, a top-level predator (which only feeds on the predator population) on the basis of the original cooperative hunting model is introduced, and a food chain model with cooperative hunting is established. The local stability of the equilibrium point is discussed by using the linearization method. By establishing an appropriate Lyapunov function, a sufficient condition for the global stability of the system at the equilibrium point is given. In addition, by using the central manifold reduction theorem, the explicit formulas of the branch periodic solution is studied. Numerical simulations are conducted to verify our theoretical analysis. The results show that when there is no cooperative relationship between predators, the positive equilibrium point is stable. With the increase of cooperative predation parameter \alpha, the system will have a stable limit cycle, and the limit cycle will continue to expand following the increase of cooperative predation parameter \alpha. Moreover, if the predator population density is too large, the system will produce continuous periodic oscillation, that is, the three species i.e. diet, mid-level predator, top-level predator, either coexist in the form of periodic oscillation, or the number of species eventually tends to be stable. Therefore, cooperative hunting is more conducive to maintaining ecological balance.

We construct a q-type Lupas-Kantorovich operators based on the q-type integer and q-type calculus, investigate the rate of convergence of q-type Lupas-Kantorovich operators by modulus continuity as well as the properties of statistical convergence and weighted statistical convergence of the operators.

Accurate and high-quality shape generation of individual teeth from cone-beam computerized tomography (CBCT) is essential for computer-aided dentistry. Existing methods need post-process to extract isosurfaces and the output meshes cannot be directly used as the input for most subsequent applications (such as finite element analysis (FEA). In this paper, we propose the network that directly learns the multi-resolution mesh guided by diffeomorphic deformation. Overall, our solution is a classic two-stage schema widely used in tooth reconstruction. Firstly, we adopt a revised anchor-free detector to locate each individual tooth with high precision. Then, we design the top-to-bottom flows from the multi-level features of each individual, referred to as pyramid flows, to predict diffeomorphic deformation from a sphere to a detailed tooth. Finally, we validate the effectiveness and efficiency of the proposed approach by comparing with the previous segmentation methods and other explicit surface learning-based methods in the experiment.

A class of coupled k-Hessian equations is considered. Firstly, by using the u_{\mathrm{0}}-sublinear theorem, it is proved that the equation has at most one radial k-admissible solution under the super-linear and sub-linear conditions. The uniqueness of radial k-admissible solution is verified by a numerical example. Finally, by incorporate with the monotone iterative technique, the uniform convergence of the solution is also discussed.

Reliability is the primary topic in the theoretical research of reliability models, especially the premise for discussing other indicators in the δ shock model. In this paper, three methods including multiple integration method, independent convolution method and memoryless method are used to prove the system reliability of Yule-Furry classical δ shock model, and the similarities and differences between the three methods are analyzed. Based on the proof process of the memoryless method, the distribution function of the lifetime T is obtained, and a simple proof of reliability is derived.

Let H be an infinite dimensional complex separable Hilbert space and B(H) be the algebra of all bounded linear operators on H. We call property (W_E) holds for T\in B(H) if \sigma (T)\backslash {\sigma }_{\mathrm{w}}(T)=E(T), where \sigma (T) and {\sigma }_{\mathrm{w}}(T) denote the spectrum and Weyl spectrum of T respectively, and E(T)=\{\lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\text{?}\sigma (T):\text{?}\mathrm{d}\mathrm{i}\mathrm{m}N(T-\lambda I)>\mathrm{0}\}. By using the property of topological uniform descent of operators in this paper, the necessary and sufficient conditions for which the property (W_E) holds for bounded linear operators and its functions are given. Furthermore, we show the applications of the main results, and explore the relationship of property (W_E) between the Drazin invertible operator and its Drazin inverse.

Due to the increasing complexity of data form, multi-view clustering algorithms emerge one after another. The main disadvantages of existing methods include: the computational complexity of these methods is high; the final clustering involves additional processing steps; the similarity graph constructed may not be the optimal graph. In order to solve the above problems, a clustering algorithm based on sparse consensus graph decomposition is proposed. The algorithm is first tested on single-view data, and then extends from single-view data to multi-view data. The algorithm takes into account different contributions of different views to the final result by giving each view appropriate weight, at the same time, makes use of the L_{\mathrm{2,1}} norm to obtain the consensus graph with better performance, learns the non-negative representation matrix on the basis of the consensus graph, and reveals the cluster result directly after alternation iteration. Finally, an update iterative algorithm is proposed and tested on a large number of data sets to verify the effectiveness of the algorithm.

This paper discusses the effect of ex-situ ecological combined remediation (ESECR) system on the purification of phosphorus of different forms in the polluted water of Pulin Creek, a tributary of Jiulong River, and each unit of ESECR treatment system is analyzed on the correlation between different phosphorus forms and environmental factors. Results showed that the average total removal rate of TP, DTP, DP and PP of ESECR treatment system were 84.3%, 84.4%, 90.8% and 78.9% respectively, among them the DP purification effect was the best. In each unit of ESECR treatment system, when the phosphorus content in the raw water was low, the sludge-membrane symbiotic (SSM) high efficiency coagulation water purification system had good TP, DTP and DP purification effect, the average removal rate were 81.4%, 76.0% and 84.8% respectively, the average removal rate of PP was 66.7%; when the content of phosphorus in the raw water was high, the average removal rate of TP, DTP and DP of the SSM high efficiency coagulation water purification system were 37.8%, 44.6% and 45.8%, nevertheless, the purification effect of PP was significant, and the average removal rate was 93.2%. When the processed water of the SSM high efficiency coagulation water purification system with low phosphorus content, the purification effect of ecological pond Ⅱ on TP and DTP was better than that of ecological pond Ⅰ, but the results on DP was opposite. For PP, the purification effect in ecological pond Ⅱ was better than that in ecological pond Ⅰ regardless of the impact of phosphorus content. The purification effect of ecological pond Ⅲ fluctuated, the purification effect on DP is relatively significant, and the average removal rate was 94.1%. In addition to the SSM high efficiency coagulation water purification system, the form of phosphorus in the polluted water of Pulin Creek and purified water by ESECR technology was mainly PP, and the percentage of each form in total phosphorus appeared as PP>DTP>DP. Pearson correlation analysis showed that different phosphorus forms were correlated with pH, water temperature and dissolved oxygen (DO) in ESECR treatment system.

In response to the low computational efficiency in the context of multi-scale structural topology design, an efficient asymptotic homogenization GPU parallel strategy is presented. The strategy leverages the graphics processing unit (GPU) and investigates parallel strategies for level set initialization, large sparse stiffness matrix equations solving and constitutive properties computing. Experimental results demonstrate that the computing efficiency of the asymptotic homogenization can be greatly improved by adopting the parallel strategies, in particular, when refining a three-dimensional unit cell grid to a resolution of 100 000, the GPU parallel strategy achieves a speedup of two orders of magnitude compared to the CPU serial.

Atherosclerosis is a critical cause of cardiovascular disease and stroke. Simulating and visualizing this process is crucial to relevant medical research. To tackle the problems of existing methods regarding the difficulties on realistic simulation and low efficiency, we propose a novel and highly efficient atherosclerotic plaque simulation solution based on a fluid-solid coupled incompressible smoothed particle hydrodynamics (SPH) method. Firstly, we discretize the blood into incompressible fluid particles using fluid-solid coupled in compressible SPH to control the stability of blood. Then, we adopt the plaque generation model to model blood, monocytes and other particles to control plaque generation based on pathological analysis of blood composition. Finally, we compute the physical properties of blood and plaque by coupling fluid-solid particles to simulate the plaque clogging effect. To make the simulation as in real-time as possible, parallel accelerated computation is implemented in CUDA architecture. Several realistic renderings of plaque simulations are provided.The results show that our method can achieves fast simulation of plaque generation in blood while avoiding the high computational cost associated with the partial differential equation model for plaque generation.

A multi-morphology design method based on conformal mapping is proposed to design triply periodic minimal surface (TPMS) microchannels with freeform boundary constraints. This method first maps the boundary of a freeform surface to a plane, allowing for channel topology design in the 2D parametric domain; Then, a Beta growth function algorithm based on loop is proposed to achieve smooth transitions of various TPMS morphological features; Finally, by mapping the designed microchannels to the 3D space constrained by the free surface, the microchannels meet the design requirements. Our results show that the microchannels constructed by this method have good adaptability to complex surface boundaries and can achieve the design goals of internal morphological features.

Marching cubes (MC) is a classic algorithm for isosurface extraction. However, it can only be used to reconstruct closed surfaces, as it requires dividing the space into inside and outside. To solve this problem, a neural marching cubes algorithm for open surfaces is proposed. The key to our method is to introduce a new sort of point position called irrelative point. Since no isosurface exists between irrelevant points and inside or outside points, thus connecting the inside and outside of the target shapes. The determination of irrelevant points does not require complex networks or calculations, and can be directly determined by the distance from points to the surface. Meanwhile, a residual module introducing attention mechanism is adopted to replace the original network. In addition, new tessellations are designed, and open surfaces can be reconstructed with the help of irrelative points. Finally, a smoothing process is incorporated to improve the reconstruction quality of the border. By testing on both closed and open surfaces under various metrics, our experiments show that the proposed method achieves high-quality reconstruction of open surfaces while maintaining the capability of reconstructing closed surfaces.

To address the problem that the performance of existing methods is unsatisfactory for detecting intrinsic symmetry in two-dimensional shapes, based on the flexible function mapping framework, we propose a spectral optimization method, named FM-2DSISD, to compute dense point maps for two-dimensional intrinsic symmetric shapes. Firstly, we design an algorithm robust to noise to extract sparse feature symmetry point maps. Secondly, using the feature symmetric point maps and the functional map framework, a mathematical model is developed whose optimization objective is to maintain the diagonality and orthogonality of each principal submatrix of the functional map matrix. We prove that the defined optimization objective can preserve the isometry of intrinsic symmetry maps. To solve the formula, we give an alternating iterative algorithm between the spatial and spectral domains via the spectral up-sampling technique. Numerical experiments show that the proposed algorithm performs better than the state-of-the-art methods on two-dimensional smooth shapes and noisy shapes.

In visualization field, Focus+Context techniques have been developed to visualize large, complex models on the display device with limited resolution. In this work, we propose a novel method for Focus+Context visualization based on optimal mass transportation. An optimal mass transportation map deforms a volume to itself, transforms the source measure (volumetric element) to the target measure with the minimal transportation cost. Solving the optimal mass transportation problem is equivalent to a convex optimization, and can be converted to computing power Voronoi diagrams in classical computational geometry. Comparing to existing approaches, the proposed method has solid theoretic foundation, which guarantees the existence, uniqueness and the smoothness of the solution. It allows the user to accurately control the target measure, and select multiple focus regions with irregular shapes. The resulting deformation is globally smooth and flipping free. Experiments with several volume data sets from medical applications and scientific simulations demonstrate the effectiveness and efficiency of our method.

The configurational transition of polyelectrolyte chain confined in a cylinder was studied by means of coarse-grained molecular dynamics simulation. Specifically, the effects on the size and configuration transition of polyelectrolyte chain under the parameters of counterion valence, chain stiffness and confinement radius were studied by calculating the average radius of gyration, average persistence length, and average shape factor. It has been found that there are stable rod-like, globular, folded-chain and ring structures in confined space. In the case of trivalent counterions, the polyelectrolyte chain with moderate stiffness forms a stable ring structure, which can be well characterized by a tangent correlation function. The mechanism of condensation of polyelectrolyte chain and counterion is further explored by statistical analysis of polyelectrolyte chain and counterions distribution. Finally, the kinetic characteristics of the nucleation and growth of the ring structure are discussed, and the results agree with the experimental results. The results can provide a deep insight about the configurational behaviors of biological macromolecules in confined spaces.

The fractional-order permanent magnet synchronization motor chaotic systems were studied based on sliding mode control theory and fractional-order calculus, using two self-adaptive sliding mode methods. Self-adaptive sliding mode synchronization of system was researched, by constructing sliding mode function, controllers and adaptive rules. Two sufficient conditions were established to ensure self-adaptive sliding mode synchronization of the systems. Numerical simulations verify the feasibility of the proposed method.

This paper proposes to employ matrix weighted NURBS surfaces to fit and fair quad meshes. For a quadrilateral mesh with given or estimated unit normals at vertices, a matrix weighted NURBS surface can be constructed by choosing the mesh vertices as control points and employing normals at each vertex for computing matrix weights. Compared with traditional NURBS surfaces, matrix weighted NURBS surfaces have quasi-cylindrical accuracy. When the input data is uniformly sampled from a smooth surface, the constructed matrix weighted NURBS surface has good smoothness and fits the mesh model well; if the input grid data contain noise, a fair fitting surface that approximates the original grid well can still be obtained by resampling control vertices on current fitting surfaces and re-calculating vertex normals based on the new quad meshes iteratively.