15 research outputs found
Tiling by rectangles and alternating current
This paper is on tilings of polygons by rectangles. A celebrated physical
interpretation of such tilings due to R.L. Brooks, C.A.B. Smith, A.H. Stone and
W.T. Tutte uses direct-current circuits. The new approach of the paper is an
application of alternating-current circuits.
The following results are obtained:
- a necessary condition for a rectangle to be tilable by rectangles of given
shapes;
- a criterion for a rectangle to be tilable by rectangles similar to it but
not all homothetic to it;
- a criterion for a generic polygon to be tilable by squares.
These results generalize the ones of C. Freiling, R. Kenyon, M. Laczkovich,
D. Rinne and G. Szekeres.Comment: In English and in Russian; 21 pages; 6 figures; minor improvement of
exposition, Russian translation adde
Theory of differential inclusions and its application in mechanics
The following chapter deals with systems of differential equations with
discontinuous right-hand sides. The key question is how to define the solutions
of such systems. The most adequate approach is to treat discontinuous systems
as systems with multivalued right-hand sides (differential inclusions). In this
work three well-known definitions of solution of discontinuous system are
considered. We will demonstrate the difference between these definitions and
their application to different mechanical problems. Mathematical models of
drilling systems with discontinuous friction torque characteristics are
considered. Here, opposite to classical Coulomb symmetric friction law, the
friction torque characteristic is asymmetrical. Problem of sudden load change
is studied. Analytical methods of investigation of systems with such
asymmetrical friction based on the use of Lyapunov functions are demonstrated.
The Watt governor and Chua system are considered to show different aspects of
computer modeling of discontinuous systems