Blending type approximation by operators of bivariate tensor product of -Bernstein-Kantorovich type.

In this paper, we introduce a family of operators of bivariate tensor product of -Bernstein-Kantorovich type. We estimate the rate of convergence of such operators for -continuous and -differentiable functions by using the mixed modulus of smoothness, establish the Voronovskaja type asymptotic formula for the bivariate -Bernstein-Kantorovich operators, as well as give some examples and their graphs to show the effect of convergence.

The Bézier variant of Kantorovich type -Bernstein operators.

In this paper, we introduce the Bézier variant of Kantorovich type -Bernstein operators with parameter [Formula: see text]. We establish a global approximation theorem in terms of second order modulus of continuity and a direct approximation theorem by means of the Ditzian-Totik modulus of smoothness. Finally, we combine the Bojanic-Cheng decomposition method with some analysis techniques to derive an asymptotic estimate on the rate of convergence for some absolutely continuous functions.

Approximation properties of -Bernstein operators.

In this paper, we introduce a new type -Bernstein operators with parameter [Formula: see text], we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Finally, we give some graphs and numerical examples to show the convergence of [Formula: see text] to [Formula: see text], and we see that in some cases the errors are smaller than [Formula: see text] to .

Shape-preserving properties of a new family of generalized Bernstein operators.

In this paper, we introduce a new family of generalized Bernstein operators based on integers, called -Bernstein operators, denoted by . We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of to any continuous functions . The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to .

Bivariate tensor product [Formula: see text]-analogue of Kantorovich-type Bernstein-Stancu-Schurer operators.

In this paper, we construct a bivariate tensor product generalization of Kantorovich-type Bernstein-Stancu-Schurer operators based on the concept of [Formula: see text]-integers. We obtain moments and central moments of these operators, give the rate of convergence by using the complete modulus of continuity for the bivariate case and estimate a convergence theorem for the Lipschitz continuous functions. We also give some graphs and numerical examples to illustrate the convergence properties of these operators to certain functions.

A basic problem of [Formula: see text]-Bernstein-type operators.

In this note, we give an elaboration of a basic problem on convergence theorem of [Formula: see text]-analogue of Bernstein-type operators. By some classical analysis techniques, we derive an exact class of [Formula: see text]-integer satisfying [Formula: see text] with [Formula: see text] and [Formula: see text] under [Formula: see text]. Our results provide an erratum to corresponding results on [Formula: see text]-analogue of Bernstein-type operators that appeared in recent literature.

GiPSiNet: an open source/open architecture network middleware for surgical simulations.

In this paper, we present the design and techniques of GiPSiNet, an open source/open architecture network middleware being developed for surgical simulations. GiPSiNet extends GiPSi (General Interactive Physical Simulation Interface), our framework for developing organ level surgical simulations, to network environments. This network extension is non-trivial, since the network settings pose several serious problems for distributed surgical virtual environments such as band-width limit, delays, and packet losses.

Evaluation methods of a middleware for networked surgical simulations.

Distributed surgical virtual environments are desirable since they substantially extend the accessibility of computational resources by network communication. However, network conditions critically affects the quality of a networked surgical simulation in terms of bandwidth limit, delays, and packet losses, etc. A solution to this problem is to introduce a middleware between the simulation application and the network so that it can take actions to enhance the user-perceived simulation performance.

Virtual environment-based training simulator for endoscopic third ventriculostomy.

A virtual environment-based endoscopic third ventriculostomy simulator is being developed for training neurosurgeons as a standardized method for evaluating competency. Magnetic resonance (MR) images of a patient's brain are used to construct the geometry model, realistic behavior in the surgical area is simulated by using physical modeling and surgical instrument handling is replicate by a haptic interface. The completion of the proposed virtual training simulator will help the surgeon to practice the techniques repeatedly and effectively, serving as a powerful educational tool.