Abstract:
Midpoint subdivision generalizes the Lane-Riesenfeld algorithm for uniform tensor product splines and can also be applied to non regular meshes. For example, midpoint subdivision of degree 2 is a specific Doo-Sabin algorithm and midpoint subdivision of degree 3 is a specific Catmull-Clark algorithm. In 2001, Zorin and Schroeder were able to prove C1-continuity for midpoint subdivision surfaces analytically up to degree 9. Here, we develop general analysis tools to show that the limiting surfaces under midpoint subdivision of any degree >= 2 are C1-continuous at their extraordinary points.

Abstract:
In this paper we study scalar multivariate subdivision schemes with general integer expanding dilation matrix. Our main result yields simple algebraic conditions on the symbols of such schemes that characterize their polynomial reproduction, i.e. their capability to generate exactly the same polynomials from which the initial data is sampled. These algebraic conditions also allow us to determine the approximation order of the associated refinable functions and to choose the "correct" parametrization, i.e. the grid points to which the newly computed values are attached at each subdivision iteration. We use this special choice of the parametrization to increase the degree of polynomial reproduction of known subdivision schemes and to construct new schemes with given degree of polynomial reproduction.

Abstract:
We study scalar multivariate non-stationary subdivision schemes with a general dilation matrix. We characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols. These algebraic conditions provide a useful theoretical tool for checking the reproduction properties of existing schemes and for constructing new schemes with desired reproduction capabilities and other enhanced properties. We illustrate our results with several examples.

Abstract:
The main purpose of this paper is to use a variant of Gr"uss inequality to obtain some perturbed midpoint inequalities. Moreover, we show that our results are sharp and precisely characterize the functions for which equality holds. Thus we provide improvements of some recent results for perturbed midpoint inequalities.

Abstract:
In this paper, we present a new convergence analysis and error estimates for the Midpoint method in Banach spaces by using Newton-Kantorovich-type assumptions and a technique based on a new system of recurrence relations. Finally, we give three examples where we improve the error bounds are better given by other authors.

Abstract:
Numerical algorithms based on variational and symplectic integrators exhibit special features that make them promising candidates for application to general relativity and other constrained Hamiltonian systems. This paper lays part of the foundation for such applications. The midpoint rule for Hamilton's equations is examined from the perspectives of variational and symplectic integrators. It is shown that the midpoint rule preserves the symplectic form, conserves Noether charges, and exhibits excellent long--term energy behavior. The energy behavior is explained by the result, shown here, that the midpoint rule exactly conserves a phase space function that is close to the Hamiltonian. The presentation includes several examples.

Abstract:
In this paper, both general and exponential bounds of the distance between a uniform Catmull-Clark surface and its control polyhedron are derived. The exponential bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Based on the exponential bound, we can predict the depth of subdivision within a user-specified error tolerance. This is quite useful and important for pre-computing the subdivision depth of subdivision surfaces in many engineering applications such as surface/surface intersection, mesh generation, numerical control machining and surface rendering.

Abstract:
We consider the question when the so--called spectral condition} for Hermite subdivision schemes extends to spaces generated by polynomials and exponential functions. The main tool are convolution operators that annihilate the space in question which apparently is a general concept in the study of various types of subdivision operators. Based on these annihilators, we characterize the spectral condition in terms of factorization of the subdivision operator.

Abstract:
Subdivision has become a staple of the geometric modeling community allowing coarse, polygonal shapes to represent highly refined, smooth shapes with guaranteed continuity properties. Unlike regular surface spines, such as NURBS, subdivision surface can handle shapes of arbitrary topology in a unified framework which is important in designing aesthetically pleasing shapes. This study presents a scheme of combined quad/triangle subdivision surface that has many important features resulting it s superiority over some other existing methods like individual triangle or individual quadrilateral subdivision schemes. In general, the scheme produces nicer surface for combined quad/triangle meshes.

Abstract:
The concept of midpoint percolation has recently been applied to characterize the double percolation transitions in negatively curved structures. Regular $d$-dimensional hypercubic lattices are in the present work investigated using the same concept. Specifically, the site-percolation transitions at the critical thresholds are investigated for dimensions up to $d=10$ by means of the Leath algorithm. It is shown that the explicit inclusion of the boundaries provides a straightforward way to obtain critical indices, both for the bulk and surface parts. At and above the critical dimension $d=6$, it is found that the percolation cluster contains only a finite number of surface points in the infinite-size limit. This is in accordance with the expectation from studies of lattices with negative curvature. It is also found that the number of surface points, reached by the percolation cluster in the infinite limit, approaches 2d for large dimensions $d$. We also note that the size dependence in proliferation of percolating clusters for $d\ge 7$ can be obtained by solely counting surface points of the midpoint cluster.