508 research outputs found
A Note on the Unsolvability of the Weighted Region Shortest Path Problem
Let S be a subdivision of the plane into polygonal regions, where each region
has an associated positive weight. The weighted region shortest path problem is
to determine a shortest path in S between two points s, t in R^2, where the
distances are measured according to the weighted Euclidean metric-the length of
a path is defined to be the weighted sum of (Euclidean) lengths of the
sub-paths within each region. We show that this problem cannot be solved in the
Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one
can compute exactly any number that can be obtained from the rationals Q by
applying a finite number of operations from +, -, \times, \div, \sqrt[k]{}, for
any integer k >= 2. Our proof uses Galois theory and is based on Bajaj's
technique.Comment: 6 pages, 1 figur
Equations solvable by radicals in a uniquely divisible group
We study equations in groups G with unique m-th roots for each positive
integer m. A word equation in two letters is an expression of the form w(X,A) =
B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as
fixed coefficients, and X in G as the unknown. Certain word equations, such as
XAXAX=B, have solutions in terms of radicals, while others such as XXAX = B do
not. We obtain the first known infinite families of word equations not solvable
by radicals, and conjecture a complete classification. To a word w we associate
a polynomial P_w in Z[x,y] in two commuting variables, which factors whenever w
is a composition of smaller words. We prove that if P_w(x^2,y^2) has an
absolutely irreducible factor in Z[x,y], then the equation w(X,A)=B is not
solvable in terms of radicals.Comment: 18 pages, added Lemma 5.2. To appear in Bull. Lon. Math. So
Manufacturing a mathematical group: a study in heuristics
I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley (Sect. 2). In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures (Sect. 3), that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forwar
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