46 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
Algebraic characteristic classes for idempotent matrices
This paper contains the algebraic analog for idempotent matrices of the Chern-Weil theory of characteristic classes. This is used to show, algebraically, that the canonical line bundle on the complex projective space is not stably trivial. Also a theorem is proved saying that for any smooth manifold there is a canonical epimorphism from the even dimensional algebraic de Rham cohomology of its algebra of smooth functions onto the standard even dimensional de Rham cohomology of the manifold