Randomly Charged Polymers, Random Walks, and Their Extremal Properties

Motivated by an investigation of ground state properties of randomly charged polymers, we discuss the size distribution of the largest Q-segments (segments with total charge Q) in such N-mers. Upon mapping the charge sequence to one--dimensional random walks (RWs), this corresponds to finding the probability for the largest segment with total displacement Q in an N-step RW to have length L. Using analytical, exact enumeration, and Monte Carlo methods, we reveal the complex structure of the probability distribution in the large N limit. In particular, the size of the longest neutral segment has a distribution with a square-root singularity at l=L/N=1, an essential singularity at l=0, and a discontinuous derivative at l=1/2. The behavior near l=1 is related to a another interesting RW problem which we call the "staircase problem". We also discuss the generalized problem for d-dimensional RWs.Comment: 33 pages, 19 Postscript figures, RevTe

Enumerating fundamental normal surfaces: Algorithms, experiments and invariants

Computational knot theory and 3-manifold topology have seen significant breakthroughs in recent years, despite the fact that many key algorithms have complexity bounds that are exponential or greater. In this setting, experimentation is essential for understanding the limits of practicality, as well as for gauging the relative merits of competing algorithms. In this paper we focus on normal surface theory, a key tool that appears throughout low-dimensional topology. Stepping beyond the well-studied problem of computing vertex normal surfaces (essentially extreme rays of a polyhedral cone), we turn our attention to the more complex task of computing fundamental normal surfaces (essentially an integral basis for such a cone). We develop, implement and experimentally compare a primal and a dual algorithm, both of which combine domain-specific techniques with classical Hilbert basis algorithms. Our experiments indicate that we can solve extremely large problems that were once though intractable. As a practical application of our techniques, we fill gaps from the KnotInfo database by computing 398 previously-unknown crosscap numbers of knots.Comment: 17 pages, 5 figures; v2: Stronger experimental focus, restrict attention to primal & dual algorithms only, larger and more detailed experiments, more new crosscap number

Measures induced by units

The half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop -equivalently, in the enveloping lattice-ordered abelian group- amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoop H induces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional on H. Since H is representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals -in this context usually called states- amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group of H), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.Comment: 24 pages, 1 figure. Revised version according to the referee's suggestions. Examples added, proof of Lemma 2.6 simplified, Section 7 expanded. To appear in the Journal of Symbolic Logi

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page