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For a given graph $G$, the metric and edge metric dimensions of $G$, $\dim(G)$ and ${\rm edim}(G)$, are the cardinalities of the smallest possible subsets of vertices in $V(G)$ such that they uniquely identify the vertices and the edges of $G$, respectively, by means of distances. It is already known that metric and edge metric dimensions are not in general comparable. Infinite families of graphs with pendant vertices in which the edge metric dimension is smaller than the metric dimension are already known. In this article, we construct a 2-connected graph $G$ such that $\dim(G)=a$ and ${\rm edim}(G)=b$ for every pair of integers $a,b$, where $4\le b, Comment: 12 pages

In this paper, the linearized Backward-Euler (BE) and backward-differential-formula (BDF) schemes are developed for the nonlinear and coupled prey-predator system with finite element method (FEM). The stability of the discrete solution in H 1 -norm is proved and used to deal with the quadratic term appeared in the system. Thanks to the boundness of the discrete solution obtained from the stability, the superclose results without time-step restriction are derived, in which a novel splitting technique is adopted to cope with the estimations related to the discrete time derivatives. Then, based on the high accuracy analysis, the unconditional superconvergence results of order O ( h 2 + τ ) and O ( h 2 + τ 2 ) in H 1 -norm are derived for BE and BDF schemes through post-processing approach, respectively. Here, h is the subdivision parameter and τ, the time step.

This study presents an adaptive binary-tree element subdivision method (BTSM) to evaluate the volume integrals with continuous or discontinuous kernels to facilitate automatic and high-quality patch generation. The BTSM, an essential technique implemented in boundary integral formulations, was proposed to guarantee a successful well-shaped element subdivision under any circumstances, improve integration accuracy, and reduce the sensitivity of integration calculation to element shape. In addition, the BTSM can be adopted to evaluate singular integrals and nearly singular integrals due to its high availability and reliability for arbitrary shape volume element subdivision. If the kernel function is discontinuous, a minority of sub-elements around the source or breakpoints need to be regenerated. The geometry-adaptive projection cavity construction algorithm and several comprehensive cavity projection techniques are implemented for facilitating patch generation. The improved cavity projection algorithms can unify the element segmentation to evaluate singular integrals and nearly singular integrals simultaneously. Numerical results demonstrate the efficiency and accuracy of the proposed method.

A cycle with four blocks C ( k 1 , k 2 , k 3 , k 4 ) is an oriented cycle formed of four blocks of lengths k 1 , k 2 , k 3 and k 4 respectively. We conjecture that for every positive integers k 1 , k 2 , k 3 , k 4 , there is an integer g ( k 1 , k 2 , k 3 , k 4 ) such that every strongly connected digraph with chromatic number greater than g ( k 1 , k 2 , k 3 , k 4 ) contains a subdivision of C ( k 1 , k 2 , k 3 , k 4 ) . As evidence, we prove this conjecture for k 2 = k 3 = k 4 = 1 .

In this paper we propose and analyze an hp-version of the composite Gauss-Legendre quadrature rule which includes local subdivision of the integration domain and local variation of the number of quadrature nodes employed on each subinterval. We derive an a priori error estimate that is fully explicit in the local length of each subinterval, in the local number of quadrature nodes employed on each subinterval, and in the local regularity of the integrand. In particular, for integrands with singularities at one or both endpoints, we prove that exponential convergence can be achieved by using geometric partition of the integration domain and linearly increasing of the number of quadrature nodes in successive subintervals. Numerical experiments are provided to illustrate the theoretical results.

Purpose Despite its well-founded criticism and lack of proper justification under core saturation conditions, the T-equivalent transformer model (Steinmetz scheme) is obviously championing in the literature. This educational paper aims to explain in a simple manner the limitations of the T-model of a low-frequency transformer and critically analyses some attempts to improve it. Design/methodology/approach Using a simplified examination of magnetic fluxes in the core and windings and using the modeling in ATPDraw, it is shown that transient transformer models with the indivisible leakage inductance allow circumventing the drawbacks of the T-model. Findings The authors show the absence of valid grounds for subdividing the leakage inductance of a transformer between its primary and secondary windings. The connection between the use of individual leakage inductances and inaccurate prediction of inrush current peaks is outlined as an important example. Practical implications The presented models can be used either as independent tools or serve as a reference for subsequent developments. Social implications Over generations, the habitual transformer T-equivalent is widely used by engineers and Electromagnetic Transients Program experts with no attention to its inadequacy under core saturation conditions. Having studied typical winding configurations, the authors have shown that neither of them has any relation to the T-equivalent. Originality/value This educational paper will contribute to the correct understanding of the transients occurring in a transformer under abnormal conditions such as inrush current or ferroresonance events, as well as during an out-of-phase synchronization of step-up generator transformers.

We propose Gauss–Seidel progressive iterative approximation (GS-PIA) for subdivision surface interpolation by combining the Gauss–Seidel iterative method for linear systems and progressive iterative approximation (PIA) for free-form curve and surface interpolation. We address the details of GS-PIA for Loop and Catmull–Clark surface interpolation and prove that they are convergent. In addition, GS-PIA may also be applied to surface interpolation for other stationary approximating subdivision schemes with explicit limit position formula/masks. GS-PIA inherits many good properties of PIA, such as having intuitive geometric meaning and being easy to implement. Compared with some other existing interpolation methods by approximating subdivision schemes, GS-PIA has three main advantages. First, it has a faster convergence rate than PIA and weighted progressive iterative approximation (W-PIA). Second, GS-PIA does not need to compute optimal weights while W-PIA does. Third, GS-PIA does not modify the mesh topology but some methods with fairness measures do. Numerical examples for Loop and Catmull–Clark subdivision surface interpolation illustrated in this paper show the efficiency and effectiveness of GS-PIA.

The entropy of a graph is an information-theoretic quantity which expresses the complexity of a graph. The generalized graph entropies result from applying information measures to a graph using various schemes for defining probability distributions over the elements of the graph. In this paper, the first Zagreb index of a class of composite graphs, namely, subdivision-vertex and subdivision-edge join graphs are investigated in order to evaluate the generalized graph entropies, and explicit formulae for the generalized graph entropy of subdivision-vertex and subdivision-edge join graphs are presented.

In this paper, a new nonlinear Hermite interpolatory subdivision scheme for curve interpolation is introduced. The scheme is constructed from the rational Bernstein–Bezier (RBB) spline. The limit function of the scheme interpolates both the function values and the derivatives. The work provides convergence analysis, polynomial reproduction, and shape preserving properties of the scheme. In particular, it is shown that the limit functions are globally C 1 and the scheme also reproduces quadratic polynomials. Moreover, the scheme preserves monotonicity and convexity. Several examples are provided to justify our claims.

In Zheng (J. Glob. Opt. 79:431-445;2021) the authors described a global optimization algorithm for multivariate continuous functions and applied it to an image processing problem. While the algorithm was shown to converge for all continuous functions, the convergence rate was not established. In this paper we assume that the objective function is smooth, and establish the asymptotic convergence rate when the algorithm is applied to such a function.

The Gale–Nikaido–Debreu lemma plays an important role in establishing the existence of competitive equilibrium. In this paper, we use Sperner’s lemma and basic elements of topology to prove the Gale–Nikaido–Debreu lemma.

International audience; Subdivision schemes are widely used in numerical mathematics such as signal/image approximation, analysis and control of data or numerical analysis. However, to develop their full power, subdivision schemes should be incorporated into a multiresolution analysis that, mimicking wavelet analyses, provides a multi-scale decomposition of a function, a curve, or a surface. The ingredients needed to define a multiresolution analysis associated to a subdivision scheme are a decimation scheme and detail operators. Their construction is not straightforward as soon as the subdivision scheme is non-interpolatory. This paper is devoted to the construction of decimation schemes and detail operators compatible with general subdivision schemes, including non-linear ones. Analysis of the performances of the constructed analyses is carried out. Some numerical applications are presented in the framework of image approximation.

In this paper, we propose a new family of non-stationary subdivision scheme, based on mixed trigonometric hyperbolic B-spline basis functions with two tension and one shape parameters. We prove tha...

To solve the Laplace-Beltrami eigenproblem on 3D models, we develop an efficient and fast computation method based on surface fitting and linear subspace. First, our method generates a finite subdivision surface to approximate the original high-resolution model. Then, we restrict the eigenproblem by constructing a subdivision linear subspace, whose basis is generated during the surface fitting process. Finally, we obtain the required eigenpairs by solving the restricted eigenproblem, whose scale is much smaller than that of the original model. Experimental results demonstrate that our eigenvalues and eigenvectors effectively approximate the ground truth. Especially, in the low-frequency band, the spectrum of our method performs much better than those of comparisons. Moreover, the eigenvectors mapped to the original mesh keep the orthogonality, making themselves a set of filtering basis on the original mesh. Meanwhile, our method also shows good performance on time and memory consumption.

This paper presents a phenomenon which sometimes occurs in tetravalent bipartite locally dart-transitive graphs, called a Base Graph–Connection Graph dissection. In this dissection, each white vertex is split into two vertices of valence 2 so that the connected components of the result are isomorphic. Given the Base Graph whose subdivision is isomorphic to each component, and the Connection Graph, which describes how the components overlap, we can, in some cases, provide a construction which can make a graph having such a decomposition. This paper investigates the general phenomenon as well as the special cases in which the connection graph has no more than one edge.

This paper presents an enhanced version of our previous work, hybrid non-uniform subdivision surfaces [19], to achieve optimal convergence rates in isogeometric analysis. We introduce a parameter $\lambda$ ($\frac{1}{4}

Molecular descriptors also known as topological indices are numeric quantities which characterize its topology and some properties of molecular graph. The 2-distance degree of a vertex u in a graph...

Topological index is a useful number associated with molecular graph and the number correlate certain physico-chemical properties of chemical compounds. The concept of line graph and subdivision gr...

Based on data from the European banking stress tests of 2014, 2016 and the transparency exercise of 2018 we demonstrate for the first time that the latent geometry of financial networks can be well-represented by geometry of negative curvature, i.e., by hyperbolic geometry. This allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos et al., which is based on the Poincar\'e disc model of hyperbolic geometry. We show that the latent dimensions of `popularity' and `similarity' in this model are strongly associated to systemic importance and to geographic subdivisions of the banking system. In a longitudinal analysis over the time span from 2014 to 2018 we find that the systemic importance of individual banks has remained rather stable, while the peripheral community structure exhibits more (but still moderate) variability., Comment: corrected typo in author's name

The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest number of edges a graph $G$ can have, such that for every edge-coloring of $G$ with $r$ colors there exists a monochromatic copy of $H$ in $G$. The notion of size-Ramsey numbers has been introduced by Erdős, Faudree, Rousseau and Schelp in 1978, and has attracted a lot of attention ever since. For a graph $H$, we denote by $H^q$ the graph obtained from $H$ by subdividing its edges with $q{-}1$ vertices each. In a recent paper of Kohayakawa, Retter and R{o}dl, it is shown that for all constant integers $q,r\geq 2$ and every graph $H$ on $n$ vertices and of bounded maximum degree, the $r$-size-Ramsey number of $H^q$ is at most $(\log n)^{20(q-1)}n^{1+1/q}$, for $n$ large enough. We improve upon this result using a significantly shorter argument by showing that $\hat{R}_r(H^q)\leq O(n^{1+1/q})$ for any such graph $H$.

This paper uses the continued fraction technique to construct a nonstationary 4-point ternary interpolatory subdivision scheme, which provides the user with a tension parameter that effectively handles cusps compared with a stationary 4-point ternary interpolatory subdivision scheme. Then, the continuous nonstationary 4-point ternary scheme is analyzed, and the limit curve is at least C 2 -continuous. Furthermore, the monotonicity preservation and convexity preservation are proved.

Based on a node group , the Newman type rational operator is constructed in the paper. The convergence rate of approximation to a class of non-smooth functions is discussed, which is regarding to X. Moreover, if the operator is constructed based on further subdivision nodes, the convergence rate is . The result in this paper is superior to the approximation results based on equidistant nodes, Chebyshev nodes of the first kind and Chebyshev nodes of the second kind.

Subdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having $C^{1}$ C 1 smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence ≥3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409–420, 1996) is a particular case. We can visualize the performance of the unified scheme by taking different values of the tension parameter. It provides an exact reproduction of parametric surfaces and is used in the processing of free-form surfaces in engineering.

Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper, we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of [Formula: see text] isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We replace the hypothesis of local linear independence for the basis of each level by a weaker assumption, which still ensures the linear independence of hierarchical splines. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by [Formula: see text] splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems.

In this paper, we use the finite element method to construct a new normalized basis of a univariate C 2 cubic spline space endowed with a specific subdivision of a real interval. Based on the polar forms, we introduce a new representation of the Hermite interpolant of any C 2 piecewise polynomial defined over this subdivision and we construct several superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.

By a subdivision of a polygon, we mean an orthogonal net such that the vertices of the polygon are nodes of the net, and the edges are composed of diagonals and sides of its cells. We study the subdivisions of convex polygons in which all edges have only diagonal directions. Such a polygon has four supporting vertices lying on different sides of the circumscribed rectangle. From each nonsupporting vertex, toward the interior of the polygon, there emanates a pair of broken lines in the directions of the orthogonal net. After a finite number of reflections in the boundary (the sum of the incidence and reflection angles is equal to 90°), the broken lines of such a pair can either get stuck at the supporting vertices or meet each other and form a closed orbit. It is proved that in the case of the pentagon, the second variant is not possible.

The problem of weighting a general n variables mean is solved by using a class of algorithms which involve iterated subdivisions (triangulations) of the $$\left( n-1\right) $$ -simplex $$\Delta _{n-1}$$ . The instance based on the barycentric subdivision is treated at length, while a more sketchy presentation is given to the algorithm based on the Freudenthal triangulation. Generalizing the weightings of 2 variables means based on Aczel iterations, the resulting weighting procedures turn out to be continuous and scale invariant, being geometric the rate of convergence of the algorithms on the class of scaled uniformly internal means.

This paper presents an algebraic characterisation of the polynomial reproduction property of Hermite subdivision of any order. A similar algebraic characterisation for Hermite schemes of order 3 which reproduce polynomial functions, their first and second derivatives was given in Conti and Huning (2019). This paper generalises the result in Conti and Huning (2019) to Hermite schemes of any order.

A propositional logic sentence in conjunctive normal form that has clauses of length two (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We first show that every such sentence that has been reduced, that is, which is unchanged under application of certain tautologies, is equisatisfiable to a 2-CNF whose associated multigraph is, in fact, a simple graph. Our main result is a complete characterization of graphs that can support unsatisfiable 2-CNF sentences. We show that a simple graph can support an unsatisfiable reduced 2-CNF sentence if and only if it contains any one of four specific small graphs as a topological minor. Equivalently, all reduced 2-CNF sentences supported on a given simple graph are satisfiable if and only if all subdivisions of those four graphs are forbidden as subgraphs of of the original graph. We conclude with a discussion of why the Robertson-Seymour graph minor theorem does not apply in our approach., Corrected error in Theorem 5 of previous version (Theorem 7 in this version). One direction of the result was incorrect. As a result, one more graph was included in the main result, Theorem 18 of previous version (Theorem 20 in this version)

A graph invariant is one of the tools of topological indices used in chemical graph theory. Recently, most of chemist study in Zagreb indices and Zagreb coindices related to the connectivity of ato...

Let G be a connected simple graph. A graph H is said to be a m-Super subdivision of G, if every edge of G is replaced by the complete bipartite graph Km,m with m > 2 in such a way that the end vert...

In this paper, superconvergence analysis of a mixed finite element method (MFEM) is proposed for the unsteady natural convection problem. The system is discretized by the Q 11 / Q 0 conforming finite elements in space and the linearized second-order BDF method in time. Based on the high accuracy characters of the elements, the superclose and global superconvergence results of order O ( h 2 + τ 2 ) for the velocity u, the temperature T in H 1 -norm and the pressure p in L 2 -norm are deduced rigorously. Here, h is the subdivision parameter and τ, the time step. The main difficulties caused by the complicate convection terms and the couplings between u, p and T are treated with new skillful techniques. Numerical results are provided to confirm the theoretical analysis.

An isogeometric approach for solving the Laplace–Beltrami equation on a two-dimensional manifold embedded in three-dimensional space using a Galerkin method based on Catmull–Clark subdivision surfaces is presented and assessed. The scalar-valued Laplace–Beltrami equation requires only$$C^0$$C0continuity and is adopted to elucidate key features and properties of the isogeometric method using Catmull–Clark subdivision surfaces. Catmull–Clark subdivision bases are used to discretise both the geometry and the physical field. A fitting method generates control meshes to approximate any given geometry with Catmull–Clark subdivision surfaces. The performance of the Catmull–Clark subdivision method is compared to the conventional finite element method. Subdivision surfaces without extraordinary vertices show the optimal convergence rate. However, extraordinary vertices introduce error, which decreases the convergence rate. A comparative study shows the effect of the number and valences of the extraordinary vertices on accuracy and convergence. An adaptive quadrature scheme is shown to reduce the error.

A novel element subdivision method based on the binary tree has been proposed for evaluation of singular domain integrals in BEM. In this paper, this element subdivision technique is called the Binary-Tree Subdivision Method (BTSM), which is applicable to arbitrary shape linear and curved volume elements with arbitrary locations of the source point. Compared to the Conventional Subdivision Method (CSM), a significant advantage of the BTSM is that it can handle singular domain integrals with continuous or discontinuous kernel and improve the accuracy of integration even with distorted elements. With the distinct feature that a single binary-tree data structure can efficiently handle volume element subdivision, it is flexible and convenient for the BTSM to be implemented in the formulation of the boundary integral equation which contains volume integrals. In addition, for the volume integrals with discontinuous kernel, an improved general projection algorithm based on Newton iteration has been proposed for curved boundary matching. Experiment results have demonstrated that the volume element is always subdivided by the BTSM in a fully automated manner and high-quality patch generation can be guaranteed in any situation. Several examples are given to verify the validity, robustness and accuracy of the proposed method.

In this paper, we investigate the cordiality of the isomorphic copies of paths mPn . We also give sufficient condition for mPn to admit the prime cordial labeling, product cordial labeling and tota...

A Roman dominating function (RDF) of a graph G is a labeling $$f:V(G)\longrightarrow \{0,1,2\}$$ such that every vertex with label 0 has a neighbor with label 2. The weight of an RDF is the sum of its functions values over all vertices, and the Roman domination number of G is the minimum weight of an RDF of G. The Roman domination subdivision number $$\mathrm {sd}_{\gamma _{R}}(G)$$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the Roman domination number of G. In this paper, we present a new upper bound on the Roman domination subdivision number by showing that for every connected graph G of order at least three, $$\begin{aligned} \mathrm {sd}_{\gamma _{R}}(G)\le 3+\min \{\deg _2(v)\mid v\in V\;\mathrm {and} \;d(v)\ge 2\}, \end{aligned}$$ where $$\deg _2(v)$$ is the number of vertices of G at distance 2 from vertex v. Moreover, we show that the decision problem associated with $$\mathrm {sd}_{\gamma _{R}}(G)$$ is NP-hard for bipartite graphs.

A numerical approach for solving second order singularly perturbed boundary value problems (SPBVPs) is introduced in this paper. This approach is based on the basis function of a 6-point interpolatory subdivision scheme. The numerical results along with the convergence, comparison and error estimation of the proposed approach are also presented.

In this paper, a hybridization of the classical stochastic Galerkin method (SGM) with two perturbative solution techniques is proposed to speed up the statistical analysis of nonuniform multiconductor transmission line (MTL) structures with parameters affected by uncertainty. The first method leverages a recently developed deterministic perturbation technique (PT) to deal with nonuniformity affecting the SGM-augmented MTL equations. This approach is proven to be computationally more efficient than the traditional solution based on line subdivision into uniform cascaded sections, yet its performance is still affected by the so-called "curse of dimensionality." To further mitigate this issue, a second method is proposed, which resorts to the solution of uncoupled MTLs having the same size as the original structure, and where the effects of both nonuniformity and stochasticity are iteratively included by means of distributed sources. The accuracy and computational efficiency of the proposed approaches are assessed based on the statistical prediction of the mixed-mode S-parameters of microstrip-line structures with different numbers of random parameters. The test cases demonstrate that the hybrid SGM-PT approach is applicable to problems with a few tens of random variables, which is an unprecedented result for state-of-the-art SGM implementations.

Knowing that the convergence of a multivariate subdivision scheme with a nonnegative mask can be characterized by whether or not some finite products of row-stochastic matrices induced by this mask have a positive column. However, the number of those products is exponential with respect to the size of matrices. For nonnegative univariate subdivision, this problem is completely solved. Thus, the convergence in this case can be checked in linear time with respect to the size of a square matrix. This paper will demonstrate the necessary and sufficient conditions for the convergence of some nonnegative bivariate subdivision schemes by means of the so-called connectivity of a square matrix, which is derived by a given mask. Moreover, the connectivity can be examined in linear time with respect to the size of this matrix.

In this paper, the properties of a new family of nonlinear dyadic subdivision schemes are presented and studied depending on the conditions imposed to the mean used to rewrite the linear scheme upon which the new scheme is based. The convergence, stability, and order of approximation of the schemes of the family are analyzed in general. Also, the elimination of the Gibbs oscillations close to discontinuities is proved in particular cases. It is proved that these schemes converge towards limit functions that are Holder continuous with exponent larger than 1. The results are illustrated with several examples.

In various wireless networking settings, node locations determine a network’s topology, allowing the network to be modelled by a geometric graph drawn in the plane. Without any additional information, local geometric routing algorithms can guarantee delivery to the target node only in restricted classes of geometric graphs, such as triangulations. In order to guarantee delivery on more general classes of geometric graphs (e.g., convex subdivisions or planar subdivisions), previous local geometric routing algorithms required [Formula: see text] state bits to be stored and passed with the message. We present the first local geometric routing algorithm using only one state bit to guarantee delivery on convex subdivisions and, when the algorithm has knowledge of the incoming port (the preceding node on the route), the first stateless local geometric routing algorithm that guarantees delivery on edge-augmented monotone subdivisions (including all convex subdivisions). We also show that [Formula: see text] state bits are necessary in planar subdivisions in which faces may have three or more reflex vertices.

The reformulated Zagreb index (or) RZ-invariant of a connected graph 𝒢 is defined as where d (a ) is the degree of a vertex a in 𝒢. In this paper, we study the RZ-invariants for vertex version of -...

In this paper, unconditional superconvergent estimate with a Galerkin finite element method (FEM) is presented for Ginzburg-Landau equation by conforming bilinear FE, while all previous works require certain time-step restrictions. First of all, a time-discrete system is introduced, with which the error function is split into a temporal error and a spatial error. On one hand, the regularity of the time-discrete system is deduced with a rigorous analysis. On the other hand, the error between the numerical solution and the solution of the time-discrete system is derived τ-independently with order O ( h 2 + h τ ) , where h is the subdivision parameter and τ, the time step. Then, the unconditional superclose result of order O ( h 2 + τ ) in the H 1 -norm is deduced directly based on the above estimates. Furthermore, the global superconvergent result is obtained through the interpolated postprocessing technique. At last, numerical results are provided to confirm the theoretical analysis.

To interpolate large drillhole data sets efficiently in implicit modeling of orebody, we optimize the solution of the RBF equation based on the two-level domain decomposition method. The solution performance is improved by balancing the two goals of the convergence rate and the iteration efficiency. The solution method converts the large domain into small subdomains which can be solved directly. The optimum division of subdomains can improve both the convergence of iteration and the efficiency of a single iteration, whereas the subdivision of a trivial subdomain leads the iteration failed to converge. For the efficiency of a single iteration, the kernel independent fast multipole method is used to calculate the matrix-vector product efficiently. Moreover, the initial solution of iteration and center reduction strategy are optimized for the dynamic updating of implicit surface. Combined with the preconditioned Krylov subspace method, the optimized solution method was implemented. The experimental results of several drillhole data sets show that the optimized solution method converges rapidly within a small number of iterations.

Surface convexity is a key issue in computer aided geometric design, which is widely applied in geometric modeling field, such as physical models, industrial design, automatic manufacturing, etc. In this paper, a sufficient convexity condition of the parametric Bezier surface over rectangles is proposed, which is firstly considered as a sufficient convexity condition for the Bezier control grid. The condition is proved by De Casteljau surface subdivision arithmetic, in which the recursive expressions elaborate that the control grid eventually converges to the surface. At last, two examples for the modeling of interpolation-type surface are discussed, one of which is a general surface and the other is a degenerate surface.

We show that for every graph $G$ and every graph $H$ obtained by subdividing each edge of $G$ at least $\Omega(\log |V(G)|)$ times, $H$ is nonrepetitively 3-colorable. In fact, we show that $\Omega(\log \pi'(G))$ subdivisions per edge are enough, where $\pi'(G)$ is the nonrepetitive chromatic index of $G$. This answers a question of Wood and improves a similar result of Pezarski and Zmarz that stated the existence of at least one 3-colorable subdivision with a linear number of subdivision vertices per edge.