Order of the canonical vector bundle over configuration spaces of spheres

Given a vector bundle, its (stable) order is the smallest positive integer t such that the t-fold self-Whitney sum is (stably) trivial. So far, the order and the stable order of the canonical vector bundle over configuration spaces of Euclidean spaces have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54], [F. R. Cohen, M. E. Mahowald and R. J. Milgram, The stable decomposition for the double loop space of a sphere, Algebraic and Geometric Topology (Stanford 1976), Proc. Sympos. Pure Math. 32 Part 2, American Mathematical Society, Providence 1978, 225–228], and [S.-W. Yang, Order of the Canonical Vector Bundle on {C_{n}(k)/\Sigma_{k}}, ProQuest LLC, Ann Arbor, 1978]. Moreover, the order and the stable order of the canonical vector bundle over configuration spaces of closed orientable Riemann surfaces with genus greater than or equal to one have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54]. In this paper, we mainly study the order and the stable order of the canonical vector bundle over configuration spaces of spheres and disjoint unions of spheres.

Remark on quasi-ideals of ordered semigroups

The aim is to correct part of the Remark 3 of my paper “On regular, intra-regular ordered semigroups” in Pure Math. Appl. (PU.M.A.) 4, no. 4 (1993), 447-461. On this occasion, some further results and the similarity between the po-semigroups and the le-semigroups are discussed.

NIELSEN EQUIVALENCE OF GENERATING PAIRS OF SL(2,q)

We present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

FRACTAL BOUNDARIES IN PARTIAL DIFFERENTIAL EQUATIONS AND IN PHYSICS

We will say that Nature reads an object as fractal when the object's physical properties depend upon its – not necessarily integral – fractal dimension df. We will review a number of problems in physics in which an object is thus recognized as a fractal, annotating the corresponding laws L=L(df) of dependence on df. Next we will find that, although these problems are obviously not related, many such laws L(df) have a certain common structure, which we will identify, and denote with Ldf. Finally, we will solve some problems in pure mathematics – P.D.E. in regions with fractal boundaries – finding again and again, our function Ldf. Again, these pure-math problems are not related… what is the significance of Ldf? Our conjecture: such strong formulistic connections are due to profound underlying yet-to-be-studied relationships – for which Ldf is the clue – among the experimental problems, among the various pure math problems, and between the latter and the former.