R Function Research Articles

Construction of the Functional Voxel Model for a Spline Curve

Analytical models are the most accurate method of geometric information representation. Parameterized smooth curves cannot be used in the field of analytical geometry, which explains the necessity for finding of analytical representation of such curves. The article considered the construction of a smooth curve presented in an analytical form and some approaches to finding an analytical model for a parametric Bezier curve. А presentation of a function in the form of its functional areas was chosen as prototype of the analytical model. The selected representation formed on the basis of the De Casteljau's method of constructing the Bezier curve and set-theoretic modeling. The Rvachev functions (R-functions) are used as the mathematical apparatus of set-theoretic operations on function areas. The functional-voxel method makes it possible to simplify the computation of R-functional procedures. An algorithm for constructing the functional area of the Bezier curve is developed on the basis of the presented combined R-voxel approach. The obtained results allow for the conclusions about the adequacy of this approach and its development protentional to construct more complicated structures.

Functional-voxel Modeling of Navigation Algorithm ORCA

The article considers an example of modeling the ORCA algorithm using the functional-voxel method. An analytical description of the permissible collision zone using the set-theoretic apparatus of Rvachev functions (R-functions) is proposed. Based on graphical image models, an approach to constructing the area of permissible robot speeds on the plane and in space has been developed.

Spline R-function and Applications in FEM

Spline R-function and Applications in FEM

EQUATIONS AND INTERVAL COMPUTATIONS FOR SOME FRACTALS

Very few characteristic functions, or equations, are reported so far for fractals. Such functions, called Rvachev functions in function-based modeling, are zero on the boundary, negative for inside points and positive for outside points. This paper proposes Rvachev functions for some classical fractals. These functions are convergent series, which are bounded with interval arithmetic and interval analysis in finite time. This permits to extend the Recursive Space Subdivision (RSS) method, which is classical in Computer Graphics (CG) and Interval Analysis, to fractal geometric sets. The newly proposed fractal functions can also be composed with classical Rvachev functions today routinely used in Constructive Solid Geometry (CSG) trees of CG or function-based modeling.

METHOD OF KEY VECTORS EXTRACTION USING R-CLOUD CLASSIFIERS

A novel method for reducing a training data set in the context of nonparametric classification is proposed. The new method is based on the method of R-clouds. The advantages of the R-cloud classification method introduced recently are being investigated. The separating boundary of the R-cloud classifier is represented using Rvachev functions. The method of key vectors extraction uses the value of the R-cloud function to quantify the disturbance of the separating boundary, which is caused by removal of one data vector from the design dataset. The R-cloud method was found instructive and practical in a number of engineering problems related to pattern classification.

S and R functions for the display of Thompson–Howarth plots

S and R functions for the display of Thompson–Howarth plots

V.L. Rvachev's Functions in Problems of Computerized Tomography (Review)

V.L. Rvachev's Functions in Problems of Computerized Tomography (Review)

Modeling the pre-breakdown V-I characteristics of an electrostatic precipitator

A novel mathematical model for determining the electrical characteristics in a dc energized duct type electrostatic precipitator is described. The method is a quasi-analytical one, based on solving the current continuity equation by finite-difference method and Poisson's equation by variational principle with the help of Rvachev functions (R-functions). The methodology described represents a valuable design tool for simulating and comparing the voltage-current characteristics of different wire-plate precipitator configurations before optimizing the geometric parameters namely shape of the corona wire, shape of the collection electrodes, wire cross-section, wire-wire and wire-plate spacing. The proposed method will be useful in trying innovative ideas in the design aspect of a wire-duct precipitator. Other significant features of this method are reduced problem domain, less memory space, and faster convergence. The proposed method has been validated with published experimental results and the agreement is excellent. A comparison of electrical characteristics has been made for different sizes and shapes of corona wire and also for various configurations of the wire-plate precipitators.

Negative Flows of the potential KP-hierarchy

We construct a Grassmannian-like formulation for the potential KP-hierarchy including additional “negative” flows. Our approach will generalize the notion of a τ \tau -function to include negative flows. We compare the resulting hierarchy with results by Hirota, Satsuma and Bogoyavlenskii.

A certain extension of the concept of R-function

A certain extension of the concept of R-function

A New Approach To 2D Non-steady ThermalProblems

In the paper the generalization of the Rvachev's Functions Method (RFM) for spatial time domain is presented and at the same time the heat transfer processes for 2D problems are discussed. The proposed algorithm and numerical results are verified by the solutions obtained on the basis of FDM.