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M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for subspaces S of an Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing S. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing S. This last one allows traces of pullbacks of singular strata in the interior of the domain of a singular simplex. In this work, we prove that the two corresponding intersection homologies are isomorphic for Siebenmann's CS sets. In terms of King's paper, this means that polyhedral dimension is a "reasonable" dimension. The proof uses a Mayer-Vietoris argument which needs an adapted subdivision. With the polyhedral dimension, that is a subtle issue. General position arguments are not sufficient and we introduce strong general position. With it, a stability is added to the generic character and we can do an inductive cutting of each singular simplex. This decomposition is realised with pseudo-barycentric subdivisions where the new vertices are not barycentres but close points of them. [ABSTRACT FROM AUTHOR]

Let be a simple connected graph consisting of n vertices and m edges. In a proper l -coloring, an r -dynamic coloring of a graph is one in which each vertex's neighbors are provided at least min { r , d (k) } different colors. The r -dynamic chromatic number of graph , given as χ r () , is the minimal l such that the graph has r -dynamic l coloring of . In this study, we combine a few common graphs to provide the r -dynamic coloring of the subdivision-edge neighborhood corona of path graph P s and star graph K 1 , s . These graphs include path graph P q , cycle graph C q , complete graph K q , star graph K 1 , q , and double star graph K 1 , q , q . [ABSTRACT FROM AUTHOR]

Mathematical science is the human recognition on the evolution laws of things that we can understand with the principle of logical consistency by mathematics, i.e., math-ematical reality. So, is the mathematical reality equal to the reality of thing? The answer is not because there always exists contradiction between things in the eyes of human, which is only a local or conditional conclusion. Such a situation enables us to extend the mathe-matics further by combinatorics for the reality of thing from the local reality and then, to get a combinatorial reality of thing. This is the combinatorial conjecture for mathematical science, i.e., CC conjecture that I put forward in my postdoctoral report for Chinese Acade-my of Sciences in 2005, namely any mathematical science can be reconstructed from or made by combinatorialization. After discovering its relation with Smarandache multi-spaces, it is then be applied to generalize mathematics over 1-dimensional topological graphs, namely the mathematical combinatorics that I promoted on science internationally for more than 20 years. This paper surveys how I proposed this conjecture from combinatorial topology, how to use it for characterizing the non-uniform groups or contradictory systems and furthermore, why I introduce the continuity ow GL as a mathematical element, i.e., vectors in Banach space over topological graphs with operations and then, how to apply it to generalize a few of important conclusions in functional analysis for providing the human recognition on the reality of things, including the subdivision of substance into elementary particles or quarks in theoretical physics with a mathematical supporting. [ABSTRACT FROM AUTHOR]

Alon and Krivelevich proved that for every n -vertex subcubic graph H and every integer q ≥ 2 there exists a (smallest) integer f = f (H , q) such that every K f -minor contains a subdivision of H in which the length of every subdivision-path is divisible by q. Improving their superexponential bound, we show that f (H , q) ≤ 21 2 q n + 8 n + 14 q , which is optimal up to a constant multiplicative factor. [ABSTRACT FROM AUTHOR]

Loop subdivision is a significant surface scheme with wide applications in fields like computer graphics and wavelet. As a type of stationary scheme, Loop subdivision cannot adjust the limit surface directly. In this paper, we present a new way to solve this problem by proposing a symmetric non-stationary Loop subdivision based on a suitable iteration. This new scheme can be used to adjust the limit surfaces freely and thus can generate surfaces with different shapes. For this new scheme, we show that it is C 2 convergent in the regular part of mesh and is at least tangent plane continuous at the limit positions of the extraordinary points. Additionally, we present a non-uniform generalization of this new symmetric non-stationary subdivision so as to locally control the shape of the limit surfaces. More interestingly, we present the limit positions of the initial points, both for the symmetric non-stationary Loop subdivision and its non-uniform generalization. Such limit positions can be used to interpolate the initial points with different valences, generalizing the existing result. Several numerical examples are given to illustrate the performance of the new schemes. [ABSTRACT FROM AUTHOR]

This study proposes an uncertainty quantification method based on deep neural networks and Catmull–Clark subdivision surfaces for vibroacoustic problems. The deep neural networks are utilized as a surrogate model to efficiently generate samples for stochastic analysis. The training data are obtained from numerical simulation by coupling the isogeometric finite element method and the isogeometric boundary element method. In the simulation, the geometric models are constructed with Catmull–Clark subdivision surfaces, and meantime, the physical fields are discretized with the same spline functions as used in geometric modelling. Multiple deep neural networks are trained to predict the sound pressure response for various parameters with different numbers and dimensions in vibroacoustic problems. Numerical examples are provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In multifunctional rural regions, strong commercial incentives exist for agricultural landholders to convert their land from farming to residential, lifestyle, and tourist land uses. In Australia, those regions mainly comprise coastal, peri‐urban, and high‐amenity rural areas. The pace and pattern of land‐use conversion in these regions is shaped by the interaction of landholders with land‐use planning regulations, notably minimum lot size (MLS) controls. This paper examines that interface in a deep dive into the role of land‐use planning controls in shaping the future of farming in an area of rapid rural change, the Ballina‐Lismore region in northern New South Wales. We argue that although planning controls in the region are designed to protect land for agriculture by curbing pressures for suburbanisation, they have also inadvertently contributed to the proliferation of unplanned rural living. This proliferation has occurred because MLS controls have incentivised agricultural landholders to sidestep restrictions on subdivision by exploiting concessions and flexibility in some of the controls and have forced some nonagricultural buyers of rural land to acquire bigger holdings than they may have otherwise desired, hence "sterilising" these agricultural‐zoned landholdings from farming. We conclude that to better protect agriculture in multifunctional rural regions, land‐use planning needs to look beyond deterrence mechanisms, such as MLS restrictions, and towards planning incentives to promote farming on agricultural‐zoned land. [ABSTRACT FROM AUTHOR]

• Extension of the s-version FE method to non-linear material problems was proposed. • A strategy to improve efficiency of constructing stiffness matrices was introduced. • Accuracy and efficiency were validated with a 3D stress concentration problem. • This study simplifies modelling structures with defects and material nonlinearity. This study intended to extend the s-version of the finite element method (s-version FEM) to cope with elastic–plastic problems. Compared with the conventional FEM, the s-version FEM, which overlays a set of local mesh with fine element size representing irregular features over the conventional FE mesh with coarse element size, can considerably simplify issues in domain discretisation with fewer degrees of freedom and provide accurate numerical predictions. However, most applications of the s-version FEM were limited to elastic problems only, leaving its applications to elastic–plastic problems almost blank owing to insufficient instructions on stress update when plasticity sets in. To address these issues, detailed instructions and formulations to cope with plasticity problems with the s-version FEM were presented for its first time. The recursive element subdivision technique was implemented to generalise its application where local elements intersect with global element edges. A 3D stress concentration problem with linear/non-linear material properties was analysed; their numerical results were compared with the exact solutions and those obtained from the conventional FEM with very fine elements. The comparison highlights the flexibility of the s-version FEM in domain discretisation and its superior accuracy in numerical calculations, thereby exhibiting high potential applications in structural integrity assessment with complex structures, geometric defects, and material non-linearity. [ABSTRACT FROM AUTHOR]

In this paper, the b -coloring problem is solved for every subdivision-edge neighborhood corona of paths, cycles, stars and complete graphs. In addition, some exact values and sharp bounds are established for the b -chromatic number of subdivision-edge neighborhood coronas of these graphs with any other graph. [ABSTRACT FROM AUTHOR]

The inflation of hyperelastic thin shells is a highly nonlinear problem that arises in multiple important engineering applications. It is characterised by severe kinematic and constitutive nonlinearities and is subject to various forms of instabilities. To accurately simulate this challenging problem, we present an isogeometric approach to compute the inflation and associated large deformation of hyperelastic thin shells following the Kirchhoff–Love hypothesis. Both the geometry and the deformation field are discretized using Catmull–Clark subdivision bases which provide the required C 1 -continuous finite element approximation. To follow the complex nonlinear response exhibited by hyperelastic thin shells, inflation is simulated incrementally, and each incremental step is solved using the Newton–Raphson method enriched with arc-length control. An eigenvalue analysis of the linear system after each incremental step assesses the possibility of bifurcation to a lower energy mode upon loss of stability. The proposed method is first validated using benchmark problems and then applied to engineering applications, where the ability to simulate large deformation and associated complex instabilities is clearly demonstrated. [ABSTRACT FROM AUTHOR]

The centrality index on a graph is a real-valued function on the nodes, which provides a ranking of vital nodes in the graph. Each node could be important from an angle, depending on how the concept of importance is defined. There are various measures of centrality, and each one defines a node's importance from a different perspective and provides relevant analytical information about the graph. Leverage centrality of nodes in a graph was defined by Joyce et al. in 2010 as a means to analyze connections within the brain. The definition of this measure shows that it is unique among existing measures in that it counts not just a node's degree, but also its neighbor's degrees. In this paper, we study the leverage centrality of nodes in the kth barycentric subdivision of some classes of graphs. This is a new concept in literature. The process can make regular graphs irregular, and the leverage center of the edge-subdivided graphs under study was investigated. [ABSTRACT FROM AUTHOR]

We prove that for every digraph F $F$ and every assignment of pairs of integers (re,qe)e∈A(F) ${({r}_{e},{q}_{e})}_{e\in A(F)}$ to its arcs there exists an integer N $N$ such that every digraph D $D$ with dichromatic number greater than N $N$ contains a subdivision of F $F$ in which e $e$ is subdivided into a directed path of length congruent to re ${r}_{e}$ modulo qe ${q}_{e}$, for every e∈A(F) $e\in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs, and at the same time yields a novel short proof of his result. [ABSTRACT FROM AUTHOR]

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the (k × k) -wall or the line graph of a subdivision of the (k × k) -wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows. 1. For t ≥ 2 , a t-theta is a graph consisting of two nonadjacent vertices and three internally vertex-disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v , a triangle B disjoint from v and three paths starting at v and vertex-disjoint otherwise, each joining v to a vertex of B , and each of length at least t. We prove that for all k , t and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a t -theta, or a t -pyramid, or the line graph of a subdivision of the (k × k) -wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t -theta for some t ≥ 2). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every Δ and subcubic subdivided caterpillar T , every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph. [ABSTRACT FROM AUTHOR]

In this paper we prove several results on Ramsey numbers R(H,F) for a fixed graph H and a large graph F, in particular for F = Kn. These results extend earlier work of Erd\H os, Faudree, Rousseau, and Schelp and of Balister, Schelp, and Simonovits on so-called Ramsey sizelinear graphs. Among other results, we show that if H is a subdivision of K4 with at least six vertices, then R(H,F) =O(v(F)+e(F)) for every graph F. We also conjecture that if H is a connected graph with e(H) v(H) (k+1/2)2, then R(H,Kn) = O(nk). The case k = 2 was proved by Erd\H os, Faudree, Rousseau, and Schelp. We prove the case k = 3. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n. For $ \alpha \in [0,1] $ α ∈ [ 0 , 1 ] , the $ A_{\alpha } $ A α -matrix of G is defined as $ A_{\alpha }(G)=\alpha D(G)+ (1-\alpha)A(G) $ A α (G) = α D (G) + (1 − α) A (G) , where $ A(G) $ A (G) and $ D(G) $ D (G) are the adjacency matrix and the diagonal degree matrix of G, respectively. The eigenvalues of the $ A_{\alpha } $ A α -matrix of G are called the $ A_{\alpha } $ A α -eigenvalues of G. In this paper, we first study the properties on $ A_{\alpha } $ A α -eigenvalues, i.e. how the $ A_{\alpha } $ A α -eigenvalues behave under some kinds of graph transformations including vertex deletion, vertex contraction, edge deletion and edge subdivision. Moreover, we also present the relationships between the $ A_{\alpha } $ A α -eigenvalues of G and its k-domination number, independence number, chromatic number and circumference, respectively. [ABSTRACT FROM AUTHOR]

This paper presents a new trigonometric composite Hermite interpolation method for solving Fredholm linear integral equations. This operator approximates locally both the function and its derivative, which is known on the subdivision nodes. Then we derive a class of quadrature rules with endpoint corrections based on integrating the composite Hermite interpolant. We also provide error estimation and numerical examples to illustrate that this new operator can provide highly accurate results. [ABSTRACT FROM AUTHOR]

We propose a global optimization method based on simplex subdivision and Lagrange dual bound to solve a specific type of problem known as linear multiplicative problem (LMP). The method utilizes the Lagrangian duality theory to solve the equivalent problem (ELMP) and obtain a lower bound by solving a linear programming problem (LP). To improve the convergence speed of the algorithm, we introduce two deleting rules that efficiently prune the feasible region without an optimal solution. These rules help accelerate the algorithm’s convergence by reducing unnecessary computations. The convergence of the proposed algorithm is theoretically proven, ensuring its reliability and effectiveness. To validate its feasibility and performance, we conduct numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient method for constructing symmetric subdivision schemes reproducing high-degree polynomials, without solving a system of linear equations. Original symmetric subdivision and its deduced subdivisions have similar increasing characteristics to the family of pseudo-splines from the polynomial reproduction point of view. Several examples are given to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

An odd graph is a finite graph all of whose vertices have odd degrees. A graph G $G$ is decomposable into k $k$ odd subgraphs if its edge set can be partitioned into k $k$ subsets each of which induces an odd subgraph of G $G$. The minimum value of k $k$ for which such a decomposition of G $G$ exists is the odd chromatic index, χo′(G) ${\chi }_{o}^{^{\prime} }(G)$, introduced by Pyber. For every k≥χo′(G) $k\ge {\chi }_{o}^{^{\prime} }(G)$, the graph G $G$ is said to be odd k $k$‐edge‐colorable. Apart from two particular exceptions, which are, respectively, odd 5‐ and odd 6‐edge‐colorable, the rest of connected loopless graphs are odd 4‐edge‐colorable, and moreover one of the color classes can be reduced to size ≤2 $\le 2$. In addition, it has been conjectured that an odd 4‐edge‐coloring with a color class of size at most 1 is always achievable. Atanasov et al. characterized the class of loopless subcubic graphs in terms of the value χo′(G)≤4 ${\chi }_{o}^{^{\prime} }(G)\le 4$. In this paper, we extend their result to a characterization of all loopless subdivisions of odd graphs in terms of the value of the odd chromatic index. This larger class S ${\mathscr{S}}$ is of a particular interest as it collects all "least instances" of nonodd graphs. As a prelude to our main result, we show that every connected graph G∈S $G\in {\mathscr{S}}$ requiring the maximum number of four colors, becomes odd 3‐edge‐colorable after removing a certain edge. Thus, we provide support for the mentioned conjecture by proving it for all subdivisions of odd graphs. The paper concludes with few problems for possible further work. [ABSTRACT FROM AUTHOR]

Stem volume is a very important parameter in forestry inventory and carbon storage. The stem volume estimated by most existing methods deviates from its true value because the irregularity of the stem is usually overlooked. In this study, we propose a stem segment volume estimation based on the tetrahedron model using TLS data. First, the initial stem segment surface model, including the lower, upper, and outer triangular surface models, was gradually reconstructed. Next, the outer surface model was subdivided based on the edge subdivision. Then, a closed triangular surface model without self-intersection was obtained. Afterward, a tetrahedron model of the stem segment was generated using TetGen software (Version 1.6.0) for the triangular surface model. Finally, the stem segment volume was calculated by summing the volumes of all the tetrahedrons in the tetrahedron model. An experiment with 76 stem segments from different tree species with different parameters showed that the reconstructed stem segment surface model effectively reflected the geometrical features of the stem segment surface. Compared to the volume based on the simulated sectional measurement, the MAPE of the volume based on the tetrahedron model was 2.12%. The results demonstrated the validity of the presented method for stem surface reconstruction and stem volume estimation, and the volume based on the tetrahedron model was closer to the true value than that based on the sectional measurement. [ABSTRACT FROM AUTHOR]

Due to properties such as interpolation, smoothness, and spline connections, Hermite subdivision schemes employ fast iterative algorithms for geometrically modeling curves/surfaces in CAGD and for building Hermite wavelets in numerical PDEs. In this paper, we introduce a notion of generalized Hermite (dyadic) subdivision schemes and then we characterize their convergence, smoothness and underlying matrix masks with or without interpolation properties. We also introduce the notion of linear-phase moments for achieving the polynomial-interpolation property. For any given integer m ∈ N , we constructively prove that there always exist convergent smooth generalized Hermite subdivision schemes with linear-phase moments such that their basis vector functions are spline functions in C m (R d) and have linearly independent integer shifts. As by-products, our results resolve convergence, smoothness, and existence of Lagrange, Hermite, or Birkhoff subdivision schemes. Even in dimension one our results significantly generalize and extend many known results on extensively studied univariate Hermite subdivision schemes. To illustrate the theoretical results in this paper, we provide examples of convergent generalized Hermite subdivision schemes with symmetric matrix masks having short support and smooth basis vector functions with or without interpolation property. [ABSTRACT FROM AUTHOR]

For an arbitrary invariant ρ(G) of a graph G the ρ-subdivision number sdρ(G) is the minimum number of edges of G whose subdivision results in a graph H with ρ(H) 6= ρ(G). Set sdρ(G) = |E(G)| if such an edge set does not exist. In the first part of this paper we give some general results for the ρ-subdivision number. In the second part we study this parameter for the chromatic number, for the chromatic index, and for the total chromatic number. We show among others that there is a strong relationship to the ρ-edge stability number for these three invariants. In the last part we consider a modification, namely the ρ-multiple subdivision number where we allow multiple subdivisions of the same edge. [ABSTRACT FROM AUTHOR]

The main objective of this research is to analyze some geometrical properties of a quaternary four-point interpolatory subdivision scheme ( Q 4 -scheme) with a shape parameter and then find the precise range of the shape parameter of the Q 4 -scheme to generate fractal and convexity of limit curves. That allows the curve to change easily and be more flexible without altering its control points. Therefore, by taking the various values of tension parameters, the curve still preserves its characteristics and geometrical configuration. These geometric modeling examples show that our techniques can be easily performed, and they can also provide us with an alternative strong strategy for the modeling of complex figures. [ABSTRACT FROM AUTHOR]

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into four triangles by joining the midpoints of its edges. We show the existence of a uniform δ > 0 such that, at any step of the subdivision, all the triangle angles lie in the interval (δ , π - δ) . Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses. [ABSTRACT FROM AUTHOR]

In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let \mathcal {C}(G) be the set of cycle lengths in a graph G and let \mathcal {C}_{\mathrm {odd}}(G) be the set of odd numbers in \mathcal {C}(G). We prove that, if G has chromatic number k, then \sum _{\ell \in \mathcal {C}_{\mathrm {odd}}(G)}1/\ell \geq (1/2-o_k(1))\log k. This solves Erdős and Hajnal's odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erdős asked whether there is some d such that each graph with chromatic number at least d (or perhaps even only average degree at least d) has a cycle whose length is a power of 2. We show that an average degree condition is sufficient for this problem, solving it with methods that apply to a wide range of sequences in addition to the powers of 2. Finally, we use our methods to show that, for every k, there is some d so that every graph with average degree at least d has a subdivision of the complete graph K_k in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984. [ABSTRACT FROM AUTHOR]

Because of hyper-complexity, a difficulty to define, multiple stakeholders with conflicting perspectives, and a lack of clear-cut solutions, wicked problems necessitate innovative and adaptive strategies. Operations research (OR) has been a valuable tool for managers to make informed decisions for years. However, as we face increasingly complex and messy problems, it has become apparent that relying solely on either hard or soft OR approaches is no longer sufficient. We need to explore more innovative methodologies to address these wicked problems effectively. This study has bridged the research gap by proposing a structured process encompassing a subdivision-based problem structuring method for defining the wicked problem, a multi-attribute decision-making (MADM) for prioritizing subproblems, and a hard OR technique, data envelopment analysis (DEA) for tackling one of the most critical subdivisions. The proposed methodology, the subdivision-based problem structuring method (SPSM), implemented in a case study, focuses on a higher education institution experiencing a decline in student admissions and involves five steps. First, a diverse group of stakeholders is formed to ensure the comprehensive consideration of perspectives. Second, the wicked problem is defined, considering long-term consequences, multiple stakeholders, and qualitative stakeholder opinions. Third, a hierarchical structure is created to break down the wicked problem into manageable subproblems. Fourth, a multi-criteria decision-making (MCDM) method prioritizes subproblems. Finally, the subproblems are addressed one by one using a combination of soft and hard OR tools. The findings highlight the benefits of integrating hard and soft OR approaches. The study concludes with reflections on the implications of using a combined OR approach to tackle wicked problems in higher education and beyond. [ABSTRACT FROM AUTHOR]

Applications of algebraic theory can be found in different logical fields, including materials science, hereditary qualities, design, as well as in hypothetical and applied math, including logarithmic calculation, cryptography, game hypothesis, and analysis of symphonies. Generally, binary operations are utilized to characterize the algebraic structure of a nonempty set. Group is a fundamental algebraic structure that frames the establishment of different structures in algebra. Further, the concept of a group has produced a lot of good ideas, such as subgroups, quotient groups, and simple groups, to divide larger groups into more understandable parts. Neutrosophic set is not only used in groups, but it is also applied in computing, network systems, processing systems, and computer technology. Neutrosophic logic is adopted in mobile ad hoc network (MANET). A mobile ad hoc network (MANET) is a wireless ad hoc network, and is often a self-configuring net of mobile devices with no requirement for added infrastructure. Further, we discuss the anti-fuzzy (Δϝ) subgroup. A Δϝ subgroup means a fuzzy subdivision of a category if the degree of membership of the combination of two components is larger than or equal to the minimum of their separate degrees. Moreover, in this study, we have examined the concepts of a implication-based (𝒥→ρ) neutrosophic Δϝ subgroup ⱺⱱꬲꞧ а finite group and a 𝒥⃗𝜌 neutrosophic Δϝ normal subgroup ⱺⱱꬲꞧ а finite group. Finally, we demonstrate some of its fundamental properties. [ABSTRACT FROM AUTHOR]

This paper presents the advanced classes of linear symmetric subdivision schemes for the fitting of data and the creation of geometric shapes. These schemes are derived from the B-spline and Lagrange's blending functions. The important characteristics of the derived schemes, including continuity, support, and the impact of parameters on the magnitude of the artifact and Gibbs oscillations are discussed. Schemes additionally generalize various subdivision schemes. Linear symmetric subdivision schemes can produce Gibbs oscillations when the initial data is taken from discontinuous functions. Additionally, these schemes may generate unwanted artifacts in the limit curve that do not exist in the original polygon. One solution is to use non-linear schemes, but this approach increases the computational complexity of the scheme. An alternative approach is proposed that involves modifying the linear symmetric schemes by introducing parameters into the linear rules. The suitable values of these parameters reduce or eliminate Gibbs oscillations and artifacts while still using linear symmetric schemes. Our approach provides a balance between reducing or eliminating Gibbs oscillations and artifacts while maintaining computational efficiency. In the second half, the piecewise parametric polynomial curves by using the blending polynomials used in the symmetric schemes are also presented. The majority of the properties of uniform quadratic and cubic B-splines with  G 2  geometric continuities are inherited by these polynomial curves. These curves can also be used for local interpolation of the control points with  G 2  continuity. Furthermore, by adjusting the value of the shape parameter, uniform cubic and quadratic B-spline curves can also be produced. These polynomial curves also satisfy the shape preserving properties of initial data. [ABSTRACT FROM AUTHOR]

Let G$G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy‐to‐describe bijections between spanning trees of G$G$ and (σ,σ∗)$(\sigma ,\sigma ^*)$‐compatible orientations, where the (σ,σ∗)$(\sigma ,\sigma ^*)$‐compatible orientations are the representatives of equivalence classes of orientations up to cycle–cocycle reversal that are determined by a cycle signature σ$\sigma$ and a cocycle signature σ∗$\sigma ^*$. Their bijections are geometric because the construction comes from zonotopal subdivisions. In this paper, we extend the geometric bijections to subgraph‐orientation correspondences. Moreover, we extend the geometric constructions accordingly. Our proofs are combinatorial, which do not make use of the zonotopes. We also provide geometric proofs for partial results, which make use of zonotopal tiling, relate to Backman, Baker, and Yuen's method, and motivate our combinatorial constructions. Finally, we explain that the main results hold for regular matroids. [ABSTRACT FROM AUTHOR]

The present paper provides a new definition of the dual interpolation curve in a geometric-intuitive way based on adaptive curve refinement techniques. The dual interpolation curve is an implementation of the interpolatory subdivision scheme for curve modeling, which comprises polynomial segments of different degrees. Dual interpolation curves maintain various desirable properties of conventional curve modeling methods, such as local adaptive subdivision, high interpolation accuracy and convergence, and continuous and discontinuous boundary representation. In addition, the dual interpolation curve is mainly applied to solve the difficult geometry defeaturing problems for curve modeling in existing computer-aided technology. By adding fictitious and intrinsic nodes inside or at the vertices of interpolation elements, the dual interpolation curve is flexible and convenient for characterizing a set of ordered points or discrete segments. Combined with the Lagrange interpolation polynomial and meshless method, the proposed approach is capable of characterizing the non-smooth boundary for geometry defeaturing. Experimental results are given to verify the validity, robustness, and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

We have derived functions of the lowest possible degree that enable us to evaluate curvature monotonicity for any 2D and 3D rational Bézier curves. We proved that the degree of the function is at most 8 n − 12 for planar rational Bézier curves of degree n , and is at most 11 n − 18 for space rational Bézier curves of degree n. These functions are derived in the Bernstein basis, allowing for efficient checking of curvature monotonicity using subdivision or Bézier clipping. As an application, we present real-time visualization of the region of a particular control point that guarantees monotonic variation of curvature over the entire segment of the rational Bézier curve. This allows users to identify where to move the control point to ensure that the curvature changes monotonically. • Curvature Monotonicity Evaluation Functions (CMEFs) are derived for Bézier curves • These CMEFs cover polynomial/rational planar/space Bézier curves of any degree • Derived CMEFs have succinct structures in Bernstein basis • Exact degree reduction mechanism in the derivation process is clarified • CMEFs can be used to interactive curve design with monotonically varying curvature [ABSTRACT FROM AUTHOR]

The idea of improving multi‐sided piecewise polynomial surfaces, by explicitly prescribing their behavior at a central surface point, allows for decoupling shape finding from enforcing local smoothness constraints. Quadratic‐Attraction Subdivision determines the completion of a quadratic expansion at the central point to attract a differentiable subdivision surface towards bounded curvature, with good shape also in‐the‐large. [ABSTRACT FROM AUTHOR]

We introduce HexBox, an intuitive modeling method and interactive tool for creating and editing hexahedral meshes. Hexbox brings the major and widely validated surface modeling paradigm of surface box modeling into the world of hex meshing. The main idea is to allow the user to box‐model a volumetric mesh by primarily modifying its surface through a set of topological and geometric operations. We support, in particular, local and global subdivision, various instantiations of extrusion, removal, and cloning of elements, the creation of non‐conformal or conformal grids, as well as shape modifications through vertex positioning, including manual editing, automatic smoothing, or, eventually, projection on an externally‐provided target surface. At the core of the efficient implementation of the method is the coherent maintenance, at all steps, of two parallel data structures: a hexahedral mesh representing the topology and geometry of the currently modeled shape, and a directed acyclic graph that connects operation nodes to the affected mesh hexahedra. Operations are realized by exploiting recent advancements in grid‐based meshing, such as mixing of 3‐refinement, 2‐refinement, and face‐refinement, and using templated topological bridges to enforce on‐the‐fly mesh conformity across pairs of adjacent elements. A direct manipulation user interface lets users control all operations. The effectiveness of our tool, released as open source to the community, is demonstrated by modeling several complex shapes hard to realize with competing tools and techniques. [ABSTRACT FROM AUTHOR]

"Barberpole" tempo illusions are a family of auditory illusions based on the synchronization of faded rhythmic streams playing at different rates, often manufacturing experiences of seemingly eternal acceleration or deceleration. The forefather of all such illusions, based on layers whose rates are powers of two apart ("octaves"), was studied by Jean-Claude Risset in the late seventies and is now known as Risset rhythm. This article provides a mathematical framework for barberpole tempo illusions, generalizing Risset rhythms for arbitrary numbers of subdivisions, non-integer proportions, arbitrary rate modulation, and increasingly accelerating tempi. Furthermore, this article describes a new illusion of eternal rallentando/accelerando based on the full harmonic spectrum of rates. This construction shows that Risset rhythms are related to barberpole variable-rate polyrhythms. A notable application of the study of divisional structures that barberpole illusions underpin is the construction of bistable auditory figures (accelerating or decelerating depending on the stream being focused). [ABSTRACT FROM AUTHOR]

A classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least C k 2 has a subdivision of K k , the complete graph on k vertices. We study two directions extending this result. • Verstraëte conjectured that a quadratic bound guarantees in fact two vertex-disjoint isomorphic copies of a K k -subdivision. • Thomassen conjectured that for each k ∈ N there is some d = d (k) such that every graph with average degree at least d contains a balanced subdivision of K k. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on d (k) remains open. In this paper, we show that a quadratic lower bound on average degree suffices to force a balanced K k -subdivision. This gives the right order of magnitude of the optimal d (k) needed in Thomassen's conjecture. Since a balanced K m k -subdivision trivially contains m vertex-disjoint isomorphic K k -subdivisions, this also confirms Verstraëte's conjecture in a strong sense. [ABSTRACT FROM AUTHOR]

We prove that a graph has an r -bounded subdivision of a wheel if and only if it does not have a graph-decomposition of locality r and width at most two. [ABSTRACT FROM AUTHOR]

The PL geometric category of a polyhedron P, denoted plgcat(P), is a combinatorial notion which provides a natural upper bound for the Lusternik--Schnirelmann category, and it is defined as the minimum number of PL collapsible subpolyhedra of P that cover P. In dimension 2 the PL geometric category is at most 3. It is easy to characterize/recognize 2-polyhedra P with plgcat(P) = 1. Borghini provided a partial characterization of 2-polyhedra with plgcat(P) = 2. We complement his result by showing that it is NP-hard to decide whether plgcat(P) ≤ 2. Therefore, we should not expect much more than a partial characterization, at least in an algorithmic sense. Our reduction is based on the observation that 2-dimensional polyhedra P admitting a shellable subdivision satisfy plgcat(P) ≤ 2 and a (nontrivial) modification of the reduction of Goaoc, Paták, Patáková, Tancer and Wagner showing that shellability of 2-complexes is NP-hard. [ABSTRACT FROM AUTHOR]

Subdivision schemes are equipped with some rules that take a polygon as an input and produce smooth curves or surfaces as an output. This presents the issue of how accurately the polygon approximates the limit curve and surface. What number of iterations/levels would be necessary to achieve the required shape at a user-specified error tolerance? In fact, several methods have been introduced in the case of stationary schemes to address the issue in terms of the error bounds (distance between polygon/polyhedron and limiting shape) and subdivision depth (the number of iterations required to obtain the result at a user-specified error tolerance). However, in the case of non-stationary schemes, this topic needs to be further studied to meet the requirements of new practical applications. This paper highlights a new approach based on a convolution technique to estimate error bounds and subdivision depth for non-stationary schemes. The given technique is independent of any condition on the coefficient of the non-stationary subdivision schemes, and it also produces the best results with the least amount of computational effort. In this paper, we first associated constants with the vectors generated by the given non-stationary schemes, then formulated an expression for the convolution product. This expression gives real values, which monotonically decrease with the increase in the order of the convolution in both the curve and surface cases. This convolution feature plays an important role in obtaining the user-defined error tolerance with fewer iterations. It achieves a trade-off between the number of iterations and user-specified errors. In practice, more iterations are needed to achieve a lower error rate, but we achieved this goal by using fewer iterations. [ABSTRACT FROM AUTHOR]

Subdivision algorithm is an efficient way to reconstruct curves and surfaces for discrete data points which can reconstruct feature and edge contour for edge extraction. By introducing local pushback operation, a brand-new interproximate subdivision framework is established which bridges the gap between interpolating schemes and approximating ones. This subdivision framework not only contains most existing schemes, but also includes some new schemes. Then, a class of nonstationary interproximate subdivision schemes are proposed. Properties of the schemes such as continuity and convexity are further discussed. With the help of the framework, limit curves can freely interpolate given points and approximate the other ones if data points have been divided into interpolating points and approximating ones by certain rules. Numerical examples show that the proposed approach can faithfully interpolate feature data points and obtain better visual effects than the ones obtained via other known schemes. [ABSTRACT FROM AUTHOR]

In this paper we define and study for a finite partially ordered set P a class of simplicial complexes on the set P r of r-element multichains of P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show that there are exactly 2 r such functions which yield subdivisions of the order complex of P, of which 2 r - 1 are pairwise different. Within this class are, for example, the order complexes of the intervals in P, the zig-zag poset of P, and the r th edgewise subdivision of the order complex of P. We also exhibit a large subclass for which our simplicial complexes are order complexes and homotopy equivalent to the order complex of P. [ABSTRACT FROM AUTHOR]

The unsolved problem in three-dimensional surgical planning for patients with facial deformity, dysgnathia, or asymmetry is the lack of a normative database of "norm skulls" that can be used as treatment objectives. A study was conducted on 90 Eurasian persons (46 male and 44 female adults) for whom cone beam-computed tomography images were available. Inclusion criteria were adult patients with a skeletal Class I pattern, proper interincisal relationship with normal occlusion, the absence of an open bite both in the anterior and posterior region, and a normal and balanced facial appearance; patients with dysgnathia and malformations were excluded. A total of 18 landmarks were digitized and 3D cephalometric measurements were performed and analyzed by means of proportions calculated from the landmarks. Male and female skulls were analyzed, as well as subdivisions revealed by cluster analysis. The data showed that four subtypes of skulls were distinguishable with statistical significance (p < 0.05). A male and a female type subdivided in a brachiocephalic and dolichocephalic phenotype could be identified. For each type, a mean shape was calculated by a Procrustes transformation, which, in turn, was used to create four template skulls from a male and a female skull. This was accomplished by fitting the polygon models of the two skulls to each of the two subtypes based on the landmarks marked on them using a thin plate spline transformation. The normative data of the subtypes can individually serve as a guide for orthodontic surgery in the Eurasian population, which is especially helpful in 3D planning and the execution of craniofacial operations. [ABSTRACT FROM AUTHOR]

Complex objects are omnipresent in different domains such as automotive, robotics, aeronautics, industrial design and medical. When such objects are no longer available, inexpensive similar or enhanced objects should be created; the Reverse Engineering is used to produce them from a points cloud by several methods. Delaunay triangulation is one of the oldest and most fundamental reconstruction methods. It consists of connecting the points to create the requested model. Three types of algorithms are used to build Delaunay triangulations: (i) Divide-and-conquer paradigm, (ii) Sweep line algorithms and (iii) Incremental insertion algorithms. With the appearance of machines with several processors, the Divide-and-conquer method has become very popular. It consists of splitting the points cloud into sub-clouds (partitions), triangulating the partitions independently of each other, and finally merging the sub-triangulations to obtain the solution. In this work, we are interested in the stage of splitting the points cloud. Several buckets or stripes splitting (partitioning) techniques have been proposed, such as octree and cells partitioning. So far, the best method for partitioning a points set into subsets with appropriate sizes has not been developed. To face this issue, this paper deals with partitioning the part space represented by the points cloud into regions via pooling points into sub-sets. For this purpose, the Basic K-means (BK) method and its derivative method, the Fast Global K-means (FGK), have been introduced for the first time in this research field. Obtained partitions, points sub-cloud, are divided over the processor cores. Independently, these partitions are further triangulated simultaneously by parallelizing the computations on several processors to improve the load balance and thus reduce the processing times. Finally, these methods are evaluated on a non-uniform points cloud to demonstrate their performances. Further, a comparative study is established to determine the most appropriate partitioning method. Results demonstrated that FGK achieved better performances in terms of partitions homogeneity, balanced workload at different processors, and consequently on the computation times compared to cell, octree and BK methods. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic has become a worldwide concern and has caused great frustration in the human community. Governments all over the world are struggling to combat the disease. In an effort to understand and address the situation, we conduct a thorough study of a COVID-19 model that provides insights into the dynamics of the disease. For this, we propose a new L S H S E A I H R COVID-19 model, where susceptible populations are divided into two sub-classes: low-risk susceptible populations, L S , and high-risk susceptible populations, H S . The aim of the subdivision of susceptible populations is to construct a model that is more reliable and realistic for disease control. We first prove the existence of a unique solution to the purposed model with the help of fundamental theorems of functional analysis and show that the solution lies in an invariant region. We compute the basic reproduction number and describe constraints that ensure the local and global asymptotic stability at equilibrium points. A sensitivity analysis is also carried out to identify the model's most influential parameters. Next, as a disease transmission control technique, a class of isolation is added to the intended L S H S E A I H R model. We suggest simple fixed controls through the adjustment of quarantine rates as a first control technique. To reduce the spread of COVID-19 as well as to minimize the cost functional, we constitute an optimal control problem and develop necessary conditions using Pontryagin's maximum principle. Finally, numerical simulations with and without controls are presented to demonstrate the efficiency and efficacy of the optimal control approach. The optimal control approach is also compared with an approach where the state model is solved numerically with different time-independent controls. The numerical results, which exhibit dynamical behavior of the COVID-19 system under the influence of various parameters, suggest that the implemented strategies, particularly the quarantine of infectious individuals, are effective in significantly reducing the number of infected individuals and achieving herd immunity. [ABSTRACT FROM AUTHOR]

To overcome the well‐known shape deficiencies of bi‐cubic subdivision surfaces, Evolving Guide subdivision (EG subdivision) generalizes C2bi‐quartic (bi‐4) splines that approximate a sequence of piecewise polynomial surface pieces near extraordinary points. Unlike guided subdivision, which achieves good shape by following a guide surface in a two‐stage, geometry‐dependent process, EG subdivision is defined by five new explicit subdivision rules. While formally only C1at extraordinary points, EG subdivision applied to an obstacle course of inputs generates surfaces without the oscillations and pinched highlight lines typical for Catmull‐Clark subdivision. EG subdivision surfaces join C2with bi‐3 surface pieces obtained by interpreting regular sub‐nets as bi‐cubic tensor‐product splines and C2with adjacent EG surfaces. The EG subdivision control net surrounding an extraordinary node can have the same structure as Catmull‐Clark subdivision: two rings of 4‐sided facets around each extraordinary nodes so that extraordinary nodes are separated by at least one regular node. [ABSTRACT FROM AUTHOR]

A subdivision transformation method evaluated for weakly singular integrals is proposed in this paper, and the method is implemented as follows: based on the position of the source points, the shape information of the elements and the nearest distance from the source point to the integral element, a subdivision technology is constructed at first. With this subdivision technology, the integral element can be divided into several integral blocks with good shapes. And then, a simpler coordinate transformation method is constructed to remove the weak singularities of the integral blocks obtained by the subdivision technology. Compared with the conventional polar coordinate transformation method, the present transformation method does not need to calculate their integral interval, which is simpler and more effective to implement. Finally, the paper gives three numerical examples to verify the accuracy and validity of the present method. [ABSTRACT FROM AUTHOR]

Objectives: Voronoi diagram is a fundamental structure in geo-computing, but it still encounters the problem of exactness and the challenge of an exact algorithm comparable to planar Voronoi diagrams in the topographic space. Methods: The geodesic distance field of computational geometry is introduced into the triangulated irregular network in the discrete topographic surface. The hyperbolic curves representing the bisector are gradually grown from the singularity of the distance field on the edge of the grid. The precise division of the discrete surface is obtained by the arrangement of the hyperbolic curves, and the exact geodesic Voronoi diagram (GVD) is obtained by clustering the divided patches. Then, the exact Voronoi diagrams are tested quantitatively and qualitatively. Results and Conclusions: It is found that the exact GVD can bring a basic improvement for the spatial analysis of topographic surface. The direct algorithm based on singular growth and hyperbolic arrangement is intuitive and easy to implement, which avoids the excessive subdivision and preprocessing of grid patches by the existing algorithms, and provides a useful exploration for the development of Voronoi diagram analysis in digital topographic analysis. [ABSTRACT FROM AUTHOR]

The Polish part of the Western Outer Carpathian lineament network was analysed based on the Global Multi-resolution Terrain Elevation Data 2010 (GMTED2010) digital elevation model. Lineaments were identified in the visual screening of the hillshade model. To the best of our knowledge, no one has studied the geometrical properties of the network in relation to the topological ones. The NetworkGT QGIS toolbox was applied to identify the nodes and branches of the network as well as to calculate the topology parameters. Our aim was to find differences between the western and eastern parts of the Western Outer Carpathians; therefore, the analyses were carried out in six sectors chosen based on the geographical subdivision in the geological context: three in the north, mainly the Silesian unit, and three in the south, mainly the Magura unit. We found general agreement of the identified network with the photo-lineament map; however, some of the photo-lineaments are not confirmed by a digital elevation model (DEM). We found that the topological parameters of the networks change from west to east but not from north to south. There are areas of increased interconnectivity, especially the Nowy Sącz Basin, where the lineament network may reflect a complicated system of cross-cutting, deep-rooted fault zones in the basement. [ABSTRACT FROM AUTHOR]

In 1984, Thomassen conjectured that for every constant fc∈N, there exists d such that every graph with average degree at least d contains a balanced subdivision of a complete graph on k vertices, i.e., a subdivision in which each edge is subdivided the same number of times. Recently, Liu and Montgomery confirmed Thomassen's conjecture. We show that for every constant 0 < c < 1/2, every graph with average degree at least d contains a balanced subdivision of a complete graph of size at least Ω(dc). Note that this bound is almost optimal. Moreover, we show that every sparse expander with minimum degree at least d contains a balanced subdivision of a complete graph of size at least Ω(d). [ABSTRACT FROM AUTHOR]