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The Sombor index of a graph $ G $, introduced by Ivan Gutman, is defined as the sum of the weights $ \sqrt{d_G(u)^2+d_G(v)^2} $ of all edges $ uv $ of $ G $, where $ d_G(u) $ denotes the degree of vertex $ u $ in $ G $. The Sombor coindex was recently defined as $ \overline{SO}(G) = \sum_{uv\notin E(G)}\sqrt{d_G(u)^2+d_G(v)^2} $. As a new vertex-degree-based topological index, the Sombor index is important because it has been proved to predict certain physicochemical properties. Two-trees are very important structures in complex networks. In this paper, the maximum and second maximum Sombor index, the minimum and second minimum Sombor coindex of two-trees and the extremal two-trees are determined, respectively. Besides, some problems are proposed for further research.

How to create realistic shapes by interpolating two known shapes for facial blendshapes has not been investigated in the existing literature. In this paper, we propose a physics-based mathematical model and its analytical solutions to obtain more realistic facial shape changes. To this end, we first introduce the internal force of elastic beam bending into the equation of motion and integrate it with the constraints of two known shapes to develop the physics-based mathematical model represented with dynamic partial differential equations (PDEs). Second, we propose a unified mathematical expression of the external force represented with linear and various nonlinear time-dependent Fourier series, introduce it into the mathematical model to create linear and various nonlinear dynamic deformations of the curves defining a human face model, and derive analytical solutions of the mathematical model. Third, we evaluate the realism of the obtained analytical solutions in interpolating two known shapes to create new shape changes by comparing the shape changes calculated with the obtained analytical solutions and geometric linear interpolation to the ground-truth shape changes and conclude that among linear and various nonlinear PDE-based analytical solutions named as linear, quadratic, and cubic PDE-based interpolation, quadratic PDE-based interpolation creates the most realistic shape changes, which are more realistic than those obtained with the geometric linear interpolation. Finally, we use the quadratic PDE-based interpolation to develop a facial blendshape method and demonstrate that the proposed approach is more efficient than numerical physics-based facial blendshapes.

Water hypoxia intensification in lakes and reservoirs has become a global problem under global warming. The Mn/Fe ratio is frequently employed to reconstruct historical redox conditions, but interpretation of this ratio may be problematic when the accumulation of Fe and Mn is governed by factors in addition to redox processes. We tested the Fe and Mn contents and bacterial diversity of a 250 cm sediment column in a monomictic reservoir. The deposition time frame and sedimentary rate (approximately 60 cm/y) were determined by integrating the Fe and Mn contents with the hydrological time series, providing a sufficient archive to reconstruct the intra-year variation in water qualities. The inapplicability of the Mn/Fe ratio observed because redox processes, mixing patterns, and biological activity jointly affected the net accumulation of Fe and Mn in the sediment. The causes of the vertical variations in the sediment Fe and Mn contents are discussed, considering the runoff sediment input, hydrodynamic and thermodynamic characteristics of this reservoir, and bacterial distributions in the sediment column. High-speed deposition intensively occurred in summer and autumn; thus, during this period, fluvial sediment was the main source of Fe and Mn. Another source was biochemical sedimentation due to mixing and oxygenation in this reservoir, which played a larger role in spring and winter. Our study showed that the easily accessible current hydrological data of the reservoir provide a reference time frame for identifying historical environmental events and that the use of the Fe/Mn ratio alone is inconclusive for interpreting the historic oxygenation regimes of reservoirs. Future applications of this method should consider the individual reservoir characteristics that impact the mobility and net accumulation of Fe and Mn in the sediment.

Creating high-quality polygonal meshes which represent the membrane surface of neurons for both visualization and numerical simulation purposes is an important yet nontrivial task, due to their irregular and complicated structures. In this paper, we develop a novel approach of constructing a watertight 3D mesh from the abstract point-and-diameter representation of the given neuronal morphology. The membrane shape of the neuron is reconstructed by progressively deforming an initial sphere with the guidance of the neuronal skeleton, which can be regarded as a digital sculpting process. To efficiently deform the surface, a local mapping is adopted to simulate the animation skinning. As a result, only the vertices within the region of influence (ROI) of the current skeletal position need to be updated. The ROI is determined based on the finite-support convolution kernel, which is convolved along the line skeleton of the neuron to generate a potential field that further smooths the overall surface at both unidirectional and bifurcating regions. Meanwhile, the mesh quality during the entire evolution is always guaranteed by a set of quasi-uniform rules, which split excessively long edges, collapse undersized ones, and adjust vertices within the tangent plane to produce regular triangles. Additionally, the local vertices density on the result mesh is decided by the radius and curvature of neurites to achieve adaptiveness.

There has emerged substantial research in addressing single-view 3D reconstruction and the majority of the state-of-the-art implicit methods employ CNNs as the backbone network. On the other hand, transformers have shown remarkable performance in many vision tasks. However, it is still unknown whether transformers are suitable for single-view implicit 3D reconstruction. In this paper, we propose the first end-to-end single-view 3D reconstruction network based on the Pyramid Vision Transformer (PVT), called ED2IF2-Net, which disentangles the reconstruction of an implicit field into the reconstruction of topological structures and the recovery of surface details to achieve high-fidelity shape reconstruction. ED2IF2-Net uses a Pyramid Vision Transformer encoder to extract multi-scale hierarchical local features and a global vector of the input single image, which are fed into three separate decoders. A coarse shape decoder reconstructs a coarse implicit field based on the global vector, a deformation decoder iteratively refines the coarse implicit field using the pixel-aligned local features to obtain a deformed implicit field through multiple implicit field deformation blocks (IFDBs), and a surface detail decoder predicts an enhanced displacement field using the local features with hybrid attention modules (HAMs). The final output is a fusion of the deformed implicit field and the enhanced displacement field, with four loss terms applied to reconstruct the coarse implicit field, structure details through a novel deformation loss, overall shape after fusion, and surface details via a Laplacian loss. The quantitative results obtained from the ShapeNet dataset validate the exceptional performance of ED2IF2-Net. Notably, ED2IF2-Net-L stands out as the top-performing variant, exhibiting the highest mean IoU, CD, EMD, ECD-3D, and ECD-2D scores, reaching impressive values of 61.1, 7.26, 2.51, 6.08, and 1.84, respectively. The extensive experimental evaluations consistently demonstrate the state-of-the-art capabilities of ED2IF2-Net in terms of reconstructing topological structures and recovering surface details, all while maintaining competitive inference time.

Let G be a graph of order n and μ be an adjacency eigenvalue of G with multiplicity k≥1. A star complement for μ in G is an induced subgraph of G of order n−k with no eigenvalue μ. In this paper, we characterize the maximal graphs with the bipartite graph K2,s as a star complement for eigenvalues μ=−2,1 and study the cases of other eigenvalues for further research.

Curve network reconstruction from a set of unorganized points is an important problem in reverse engineering and computer graphics. In this paper, we propose an automatic method to extract curve segments and reconstruct curve networks from unorganized spatial points. Our proposed method divides reconstruction of curve networks into two steps: 1) detecting nodes of curve segments and 2) reconstructing curve segments. For detection of nodes of curve segments, we present a principal component analysis-based algorithm to obtain candidate nodes from unorganized spatial points and a Euclidean distance-based iterative algorithm to remove peripheral nodes and find the actual nodes. For reconstruction of curve segments, we propose an extraction algorithm to obtain the points on each of curve segments. We present quite a number of examples which use our proposed method to reconstruct curve networks from unorganized spatial points. The results demonstrate the effectiveness of our proposed method and its advantages of good automation and high reconstruction efficiency.

Abstract In this paper, we obtain a sharp upper bound on the spectral radius of a nonnegative k-uniform tensor and characterize when this bound is achieved. Furthermore, this result deduces the main result in [X. Duan and B. Zhou, Sharp bounds on the spectral radius of a nonnegative matrix, Linear Algebra Appl. 439:2961–2970, 2013] for nonnegative matrices; improves the adjacency spectral radius and signless Laplacian spectral radius of a uniform hypergraph for some known results in [D.M. Chen, Z.B. Chen and X.D. Zhang, Spectral radius of uniform hypergraphs and degree sequences, Front. Math. China 6:1279–1288, 2017]; and presents some new sharp upper bounds for the adjacency spectral radius and signless Laplacian spectral radius of a uniform directed hypergraph. Moreover, a characterization of a strongly connected k-uniform directed hypergraph is obtained.

In this paper, we define $k$-uniform tensors for $k\geq 2$, which are more closely related to the $k$-uniform hypergraphs than the general tensors, and introduce the parameter $r^{(q)}_{i}(\mathbb{A})$ for a tensor $\mathbb{A}$, which is the generalization of the $i$-th slice sum $r_ {i}(\mathbb{A})$ (also the $i$-th average 2-slice sum $m_{i}(\mathbb{A})$). By using $r^{(q)}_{i}(\mathbb{A})$ for $q\geq1$, we obtain a general result on the sharp upper bound for the spectral radius of a nonnegative $k$-uniform tensor. When $k=2, q=1, 2, 3$, this result deduces the main results for nonnegative matrices in [1,8,27]; when $k\geq 3, q=1$, this result deduces the main results in [5,20]. We also find that the upper bounds obtained from different $q$ can not be compared. Furthermore, we can obtain some known or new upper bounds by applying the general result to $k$-uniform hypergraphs and $k$-uniform directed hypergraphs, respectively.

BackgroundThe incidence of thyroid cancer (THCA) continues to increase in recent decades. Accumulating evidence showed that the unbalanced alternative splicing (AS) promotes the occurrence of cancers and leads to poor prognosis of patients. However, the research on alternative splicing events in THCA is lacking, and its underlying mechanism is not fully understood. This study identifies a novel prognostic signature based on AS events to reveal the relationship of AS with tumor immune microenvironment.MethodsBased on the AS data, transcriptional data, and clinical information, the differentially expressed alternative splicings (DEASs) were screened out. Least absolute shrinkage and selection operator (LASSO) regression and multi-Cox regression analyses were employed to identify prognostic results related to AS events and establish a prognostic signature. The predictive ability of the signature was assessed by Kaplan-Meier (K-M) survival curve, risk plots, and receiver operating characteristic (ROC) curves. Furthermore, correlations between tumor-infiltrating immune cells, immune checkpoints, immune score and prognostic signature were analyzed.ResultsAccording to the LASSO regression analysis, a total of five AS events were selected to construct the signature. K-M survival curve showed that the higher the risk score, the worse the OS of the patients. Risk plots further confirmed this result. ROC curves indicated the high predictive efficiency of the prognostic signature. As for tumor immune microenvironment, patients in the high-risk group had a higher proportion of immune cells, including plasma cell, CD8+ T cell, macrophages (M0 and M2), and activated dendritic cell. Immune checkpoint proteins, such as PDCD1LG2, HAVCR2, CD274, etc., were significantly higher in the high-risk group. We also found that the ESTIMATE score, stromal score, and immune score were lower in the high-risk group, while the result of tumor purity was the opposite.ConclusionsCollectively, a prognostic signature consisting of five AS events in THCA was established. Furthermore, there was an inextricable correlation between immune cell infiltration, immune checkpoint proteins, and AS events. This study will provide a basis for THCA immunotherapy in the future.

Simulation with position-based dynamics is very popular due to its high efficiency. However, it has the weaknesses of lacking details when too few vertices are involved in simulation and inefficiency when too many vertices are used for simulation. To tackle this problem, in this paper, we propose a new method of reconstructing dynamic 3D models with small data. The core elements of the proposed approach are a curve-represented geometric model and a physics-based mathematical model defined by dynamic partial differential equations. We first use the simulation method of position-based dynamics to generate a group of keyframe poses, which are used to create the deformation animation of a 3D model. Then, wireframe curves are extracted from skin deformation shapes of the 3D model at different keyframe poses. A physics-based mathematical model defined by dynamic partial differential equations is proposed. Its closed-form solution is obtained to represent the extracted curves, which are used to reconstruct the deformation models at different keyframe poses. Experimental examples and comparisons made in this paper indicate that the proposed method of reconstructing dynamic 3D models can greatly reduce data size while keeping good details.

Partial differential equation (PDE) based surfaces own a lot of advantages, compared to other types of 3D representation. For instance, fewer variables are required to represent the same 3D shape; the position, tangent, and even curvature continuity between PDE surface patches can be naturally maintained when certain conditions are satisfied, and the physics-based nature is also kept. Although some works applied implicit PDEs to 3D surface reconstruction from images, there is little work on exploiting the explicit solutions of PDE to this topic, which is more efficient and accurate. In this paper, we propose a new method to apply the explicit solutions of a fourth-order partial differential equation to surface reconstruction from multi-view images. The method includes two stages: point clouds data are extracted from multi-view images in the first stage, which is followed by PDE-based surface reconstruction from the obtained point clouds data. Our computational experiments show that the reconstructed PDE surfaces exhibit good quality and can recover the ground truth with high accuracy. A comparison between various solutions with different complexity to the fourth-order PDE is also made to demonstrate the power and flexibility of our proposed explicit PDE for surface reconstruction from images.

In this paper, we present a new modelling method to create 3D models. First, characteristic cross section curves are generated and approximated by generalized elliptic curves. Then, a vector-valued sixth-order partial differential equation is proposed, and its closed form solution is derived to create PDE surface patches from cross section curves where two adjacent PDE-surface patches are automatically stitched together. With the approach presented in this paper, C2 continuity between adjacent surface patches is well-maintained. Since surface creation of the model is transformed into the generation of cross sectional curves and few undetermined constants are required to describe cross sectional curves accurately, the proposed approach can save manual operations, reduce information storage, and generate 3D models quickly.

3D printing, regarded as the most popular additive manufacturing technology, is finding many applications in various industrial sectors. Along with the increasing number of its industrial applications, reducing its material consumption and increasing the strength of 3D printed objects have become an important topic. In this paper, we introduce unidirectionally and bidirectionally stiffened structures into 3D printing to increase the strength and stiffness of 3D printed objects and reduce their material consumption. To maximize the advantages of such stiffened structures, we investigated finite element analysis, especially for general cases of stiffeners in arbitrary positions and directions, and performed optimization design to minimize the total volume of stiffened structures. Many examples are presented to demonstrate the effectiveness of the proposed finite element analysis and optimization design as well as significant reductions in the material costs and stresses in 3D printed objects stiffened with unidirectional and bidirectional stiffeners.

Partial differential equation (PDE)-based geometric modelling and computer animation has been extensively investigated in the last three decades. However, the PDE surface-represented facial blendshapes have not been investigated. In this paper, we propose a new method of facial blendshapes by using curve-defined and Fourier series-represented PDE surfaces. In order to develop this new method, first, we design a curve template and use it to extract curves from polygon facial models. Then, we propose a second-order partial differential equation and combine it with the constraints of the extracted curves as boundary curves to develop a mathematical model of curve-defined PDE surfaces. After that, we introduce a generalized Fourier series representation to solve the second-order partial differential equation subjected to the constraints of the extracted boundary curves and obtain an analytical mathematical expression of curve-defined and Fourier series-represented PDE surfaces. The mathematical expression is used to develop a new PDE surface-based interpolation method of creating new facial models from one source facial model and one target facial model and a new PDE surface-based blending method of creating more new facial models from one source facial model and many target facial models. Some examples are presented to demonstrate the effectiveness and applications of the proposed method in 3D facial blendshapes.

In order to reduce the data size and simplify the process of creating characters’ 3D models, a new and interactive ordinary differential equation (ODE)-based C2 continuous surface creation algorithm is introduced in this paper. With this approach, the creation of a three-dimensional surface is transformed into generating two boundary curves plus four control curves and solving a vector-valued sixth order ordinary differential equation subjected to boundary constraints consisting of boundary curves, and first and second partial derivatives at the boundary curves. Unlike the existing patch modeling approaches which require tedious and time-consuming manual operations to stitch two separate patches together to achieve continuity between two stitched patches, the proposed technique maintains the C2 continuity between adjacent surface patches naturally, which avoids manual stitching operations. Besides, compared with polygon surface modeling, our ODE C2 surface creation method can significantly reduce and compress the data size, deform the surface easily by simply changing the first and second partial derivatives, and shape control parameters instead of manipulating loads of polygon points.

We show that an order m dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and then we obtain the characterization of order m dimension 2 strongly primitive tensors and the bound of the strongly primitive degree. Furthermore, we study the properties of strongly primitive tensors with n≥3 and propose some problems for further research.

We obtain the sharp bounds for the spectral radius of a nonnegative matrix and then obtain some known results or new results by applying these bounds to a graph or a digraph and revise and improve two known results.

Since non-automatic rigging requires heavy human involvements, and various automatic rigging algorithms are less efficient in terms of computational efficiency, especially for current curve-based skin deformation methods, identifying the iso-parametric curves and creating the animation skeleton requires tedious and time-consuming manual work. Although several automatic rigging methods have been developed, but they do not aim at curve-based models. To tackle this issue, this paper proposes a new rigging algorithm for automatic generation of dynamic skin deformation to quickly identify iso-parametric curves and create an animation skeleton in a few milliseconds, which can be seamlessly used in curve-based skin deformation methods to make the rigging process fast enough for highly efficient computer animation applications.

In 2012, Abdo and Dimitrov defined the total irregularity of a graph G=(V,E) as irrtG=1/2∑u,v∈VdGu-dGv, where dGu denotes the vertex degree of a vertex u∈V. In this paper, we investigate the total irregularity of bicyclic graphs and characterize the graph with the maximal total irregularity among all bicyclic graphs on n vertices.

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we apply the coefficient theorem on the characteristic polynomial of a signed graph and give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graph Γ∞p,q,l, obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.