The standard proof,the elegant proof,and the proof without words of tasks in geometry,and their dynamic investigation

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs are given under the headings: standard proof, elegant proof, and the proof without words. The solutions were obtained through a combination of mathematical tools and by dynamic investigation of the geometrical properties.     

Use of a Monte Carlo method in an algorithm which solves a set of functional inequalities

This paper shows how an existing algorithm, which is used in computer-aided design problems for solving a set of functional inequalities, may be modified by the inclusion of a Monte Carlo method to give a stochastic algorithm, which is easier to implement than its deterministic original and which solves the given set of functional inequalities almost surely.     

Relations among five radii of circles in a triangle,its sides and other segments

Students use GeoGebra to explore the mathematical relations among different radii of circles in a triangle (circumcircle, incircle, excircles) and the sides and other segments in the triangle. The more formal mathematical development of the relations that follows the explorations is based on known geometrical properties, different formulas relating the radii to the sides and the inequalities between the different averages. The activities described were conducted with pre-service teachers of mathematics, with empirical investigation of the relations using dynamical geometry software, and formal presentation of proofs.     

Convergence of the Finite Difference Scheme for a Fast Diffusion Equation in Porous Media

We are concerned with an implicit scheme for the finite difference solution to a nonlinear parabolic equation with a multivalued coefficient that describes the fast diffusion in a porous medium. The boundary conditions contain the multivalued function as well. We prove the stability and the convergence of the scheme, emphasizing the precise nature of convergence in this specific case, and compute the error level of the approximating solution. The method is aimed to simplify the numerical computations for the solutions to equations of this type, without performing an approximation of the multivalued function. The theory is illustrated by numerical results.     

Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves

The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinear parabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The main result of this paper is the proof of the previously formulated conjecture that, if a monotone initial function has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form to a system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves may depend on time. It is pointed out that the monotonicity condition can be replaced with the boundedness condition.     

Sharp estimates for the convergence rate of Fourier series in terms of orthogonal polynomials in L 2((a, b), p(x))

Sharp estimates are given for the convergence rate of Fourier series in terms of classical orthogonal polynomials in some classes of functions characterized by a generalized modulus of continuity in the space L ₂((a, b), p(x)). Expansions in terms of Laguerre, Hermite, and Jacobi polynomials are considered.