32 research outputs found
The structure of the inverse to the Sylvester resultant matrix
AbstractGiven polynomials a(λ) of degree m and b(λ) of degree n, we represent the inverse to the Sylvester resultant matrix of a and b, if this inverse exists, as a canonical sum of m+n dyadic matrices each of which is a rational function of zeros of a and b
Minimum Perimeter Rectangles That Enclose Congruent Non-Overlapping Circles
We use computational experiments to find the rectangles of minimum perimeter
into which a given number n of non-overlapping congruent circles can be packed.
No assumption is made on the shape of the rectangles. In many of the packings
found, the circles form the usual regular square-grid or hexagonal patterns or
their hybrids. However, for most values of n in the tested range n =< 5000,
e.g., for n = 7, 13, 17, 21, 22, 26, 31, 37, 38, 41, 43...,4997, 4998, 4999,
5000, we prove that the optimum cannot possibly be achieved by such regular
arrangements. Usually, the irregularities in the best packings found for such n
are small, localized modifications to regular patterns; those irregularities
are usually easy to predict. Yet for some such irregular n, the best packings
found show substantial, extended irregularities which we did not anticipate. In
the range we explored carefully, the optimal packings were substantially
irregular only for n of the form n = k(k+1)+1, k = 3, 4, 5, 6, 7, i.e., for n =
13, 21, 31, 43, and 57. Also, we prove that the height-to-width ratio of
rectangles of minimum perimeter containing packings of n congruent circles
tends to 1 as n tends to infinity.Comment: existence of irregular minimum perimeter packings for n not of the
form (10) is conjectured; smallest such n is n=66; existence of irregular
minimum area packings is conjectured, e.g. for n=453; locally optimal
packings for the two minimization criteria are conjecturally the same (p.22,
line 5); 27 pages, 12 figure