On the Equivalence of the Riemann-Liouville and the Caputo Fractional Order Derivatives in Modeling of Linear Viscoelastic Materials

Mathematics Subject Classification: 26A33

On a Differential Equation with Left and Right Fractional Derivatives

Mathematics Subject Classification: 26A33; 70H03, 70H25, 70S05; 49S05

Inversion Formulas for the q-Riemann-Liouville and q-Weyl Transforms Using Wavelets

Mathematics Subject Classification: 42A38, 42C40, 33D15, 33D60

On q–Analogues of Caputo Derivative and Mittag–Leffler Function

Mathematics Subject Classification: 33D60, 33E12, 26A33

Design of Fractional Order Digital Differentiators and Integrators Using Indirect Discretization

Mathematics Subject Classification: 26A33, 93B51, 93C95

Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equations

Mathematics Subject Classification: 44A05, 44A35

Fractional Integration and Fractional Differentiation of the M-Series

Mathematics Subject Classification: 26A33, 33C60, 44A15

Project Proposals Development – Entrepreneurship and Reflection in Teaching Information Technology

This work presents a model for development of project proposals by students as an approach to teaching information technology while promoting entrepreneurship and reflection. In teams of 3 to 5 participants, students elaborate a project proposal on a topic they have negotiated with each other and with the teacher. The project domain is related to the practical application of state-of-theart information technology in areas of substantial public interest or of immediate interest to the participants. This gives them ample opportunities for reflection not only on technical but also on social, economic, environmental and other dimensions of information technology. This approach has long been used with students of different years and programs of study at the Faculty of Mathematics and Informatics, Plovdiv University “Paisiy Hilendarski”. It has been found to develop all eight key competences for lifelong learning set forth in the Reference Framework and procedural skills required in real life.

Construction and evaluation of achievement tests In English

Доклад, поместен в сборника на Националната конференция "Образованието в информационното общество", Пловдив, май, 2011 г.

Computing with the Square Root of NOT

To the two classical reversible 1-bit logic gates, i.e. the identity gate (a.k.a. the follower) and the NOT gate (a.k.a. the inverter), we add an extra gate, the square root of NOT. Similarly, we add to the 24 classical reversible 2-bit circuits, both the square root of NOT and the controlled square root of NOT. This leads to a new kind of calculus, situated between classical reversible computing and quantum computing.

Development and Implementation of NURBS Models of Quadratic Curves and Surfaces

This article goes into the development of NURBS models of quadratic curves and surfaces. Curves and surfaces which could be represented by one general equation (one for the curves and one for the surfaces) are addressed. The research examines the curves: ellipse, parabola and hyperbola, the surfaces: ellipsoid, paraboloid, hyperboloid, double hyperboloid, hyperbolic paraboloid and cone, and the cylinders: elliptic, parabolic and hyperbolic. Many real objects which have to be modeled in 3D applications possess specific features. Because of this these geometric objects have been chosen. Using the NURBS models presented here, specialized software modules (plug-ins) have been developed for a 3D graphic system. An analysis of their implementation and the primitives they create has been performed.

Stochastic Arithmetic Theory and Experiments

Stochastic arithmetic has been developed as a model for exact computing with imprecise data. Stochastic arithmetic provides confidence intervals for the numerical results and can be implemented in any existing numerical software by redefining types of the variables and overloading the operators on them. Here some properties of stochastic arithmetic are further investigated and applied to the computation of inner products and the solution to linear systems. Several numerical experiments are performed showing the efficiency of the proposed approach.

Quasi-Monte Carlo Methods for some Linear Algebra Problems. Convergence and Complexity

We present quasi-Monte Carlo analogs of Monte Carlo methods for some linear algebra problems: solving systems of linear equations, computing extreme eigenvalues, and matrix inversion. Reformulating the problems as solving integral equations with a special kernels and domains permits us to analyze the quasi-Monte Carlo methods with bounds from numerical integration. Standard Monte Carlo methods for integration provide a convergence rate of O(N^(−1/2)) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)^k)N^(−1)). We have shown theoretically and through numerical tests that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the considered Monte Carlo methods. We also analyze the complexity of considered quasi-Monte Carlo algorithms and compare them to the complexity of the analogous Monte Carlo and deterministic algorithms.

Computer-Assisted Proofs and Symbolic Computations

We discuss some main points of computer-assisted proofs based on reliable numerical computations. Such so-called self-validating numerical methods in combination with exact symbolic manipulations result in very powerful mathematical software tools. These tools allow proving mathematical statements (existence of a fixed point, of a solution of an ODE, of a zero of a continuous function, of a global minimum within a given range, etc.) using a digital computer. To validate the assertions of the underlying theorems fast finite precision arithmetic is used. The results are absolutely rigorous. To demonstrate the power of reliable symbolic-numeric computations we investigate in some details the verification of very long periodic orbits of chaotic dynamical systems. The verification is done directly in Maple, e.g. using the Maple Power Tool intpakX or, more efficiently, using the C++ class library C-XSC.

High-Order Control Variations and Small-Time Local Controllability

The importance of “control variations” for obtaining local approximations of the reachable set of nonlinear control systems is well known. Heuristically, if one can construct control variations in all possible directions, then the considered control system is small-time locally controllable (STLC). Two concepts of control variations of higher order are introduced for the case of smooth control systems. The relation between these variations and the small-time local controllability is studied and a new sufficient STLC condition is proved.