Generating infinite symmetric groups

Let S=Sym(\Omega) be the group of all permutations of an infinite set \Omega. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, respectively as a monoid, then there exists a positive integer n such that every element of S may be written as a group word, respectively a monoid word, of length \leq n in the elements of U. Several related questions are noted, and a brief proof is given of a result of Ore's on commutators that is used in the proof of the above result.Comment: 9 pages. See also http://math.berkeley.edu/~gbergman/papers To appear, J.London Math. Soc.. Main results as in original version. Starting on p.4 there are references to new results of others including an answer to original Question 8; "sketch of proof" of Lemma 11 is replaced by a full proof; 6 new reference

Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N. G. de Bruijn showed that the symmetric group Sym(\Omega) on an infinite set \Omega contains a free subgroup on 2^{card(\Omega)} generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality \leq card(\Omega), Sym(\Omega) contains a coproduct of 2^{card(\Omega)} copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras \bf{V}, and formulated as a statement about functors Set --> \bf{V}. From this one easily obtains analogs of the results stated above with "group" and Sym(\Omega) replaced by "monoid" and the monoid Self(\Omega) of endomaps of \Omega, by "associative K-algebra" and the K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis \Omega, and by "lattice" and the lattice Equiv(\Omega) of equivalence relations on \Omega. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(\Omega), Self(\Omega) and End_K (V) contains a coproduct of 2^{card(\Omega)} copies of itself. That paper also gave an example of a group of cardinality 2^{card(\Omega)} that was {\em not} embeddable in Sym(\Omega), and R. McKenzie subsequently established a large class of such examples. Those results are shown to be instances of a general property of the lattice of solution sets in Sym(\Omega) of sets of equations with constants in Sym(\Omega). Again, similar results -- this time of varying strengths -- are obtained for Self(\Omega), End_K (V), and Equiv(\Omega), and also for the monoid \Rel of binary relations on \Omega. Many open questions and areas for further investigation are noted.Comment: 37 pages. Copy at http://math.berkeley.edu/~gbergman/papers is likely to be updated more often than arXiv copy Revised version includes answers to some questions left open in first version, references to results of Wehrung answering some other questions, and some additional new result

Critical point for the strong field magnetoresistance of a normal conductor/perfect insulator/perfect conductor composite with a random columnar microstructure

A recently developed self-consistent effective medium approximation, for composites with a columnar microstructure, is applied to such a three-constituent mixture of isotropic normal conductor, perfect insulator, and perfect conductor, where a strong magnetic field {\bf B} is present in the plane perpendicular to the columnar axis. When the insulating and perfectly conducting constituents do not percolate in that plane, the microstructure-induced in-plane magnetoresistance is found to saturate for large {\bf B}, if the volume fraction of the perfect conductor $p_S$ is greater than that of the perfect insulator $p_I$. By contrast, if $p_S<p_I$, that magnetoresistance keeps increasing as ${\bf B}^2$ without ever saturating. This abrupt change in the macroscopic response, which occurs when $p_S=p_I$, is a critical point, with the associated critical exponents and scaling behavior that are characteristic of such points. The physical reasons for the singular behavior of the macroscopic response are discussed. A new type of percolation process is apparently involved in this phenomenon.Comment: 4 pages, 1 figur

On the Consistency of Orbifolds

Modular invariance is a necessary condition for the consistency of any closed string theory. In particular, it imposes stringent constraints on the spectrum of orbifold theories, and in principle determines their spectrum uniquely up to discrete torsion classes. In practice, however, there are often ambiguities in the construction of orbifolds that are a consequence of the fact that the action of the orbifold elements on degenerate ground states is not unambiguous. We explain that there exists an additional consistency condition, related to the spectrum of D-branes in the theory, which eliminates these ambiguities. For supersymmetric orbifolds this condition turns out to be equivalent to the condition that supersymmetry is unbroken in the twisted sectors, but for non-supersymmetric orbifolds it appears to be a genuinely new consistency condition.Comment: 10 pages, LaTex. The sign ambiguities in the GSO-projection are clarified in the abstract and the introduction, and revised in sections 3 and 4. In particular we clarify that modular invariance fixes all the ambiguities in principle, but in practice this is hard to do. The final conclusion regarding the spectrum of the non-supersymmetric orbifold remains unchange

On the Bergman-Milton bounds for the homogenization of dielectric composite materials

The Bergman-Milton bounds provide limits on the effective permittivity of a composite material comprising two isotropic dielectric materials. These provide tight bounds for composites arising from many conventional materials. We reconsider the Bergman-Milton bounds in light of the recent emergence of metamaterials, in which unconventional parameter ranges for relative permittivities are encountered. Specifically, it is demonstrated that: (a) for nondissipative materials the bounds may be unlimited if the constituent materials have relative permittivities of opposite signs; (b) for weakly dissipative materials characterized by relative permittivities with real parts of opposite signs, the bounds may be exceedingly large

Tachyon condensation in unstable type I D-brane systems

Type I string theory provides eight classes of unstable D-brane systems. We determine the gauge group and tachyon spectrum for each one, and thereby describe the gauge symmetry breaking pattern in the low-energy world-volume field theory. The topologies of the resulting coset vacuum manifolds are related to the real K-theory groups KO^{-n}, extending the known relations between the Type II classifying spaces BU and U and the complex K-theory groups K^0 and K^{-1}. We also comment on the role of the background D9-branes.Comment: 14 pages, LaTex; footnote regarding bosonic D-branes has been corrected, references adde

Stable Non-BPS Dyons in N=2 SYM

As a novel application of string junctions, we provide evidence for the existence of stable non-BPS dyons with magnetic charge greater than 1 in (the semiclassical regime of) N=2 SU(2) Super-Yang-Mills theory. In addition, we find a new curve of marginal stability. Moduli space is therefore divided into four regions, each containing a different stable particle spectrum.Comment: 12 pages LaTex, 5 figures; Added comments in both section 3 and the conclusions regarding the applicability of the string web picture in the field theory limi

Pathological changes in seals in Swedish waters: the relation to environmental pollution

This thesis concerns the disease situation for the three seal species that inhabit the Swedish coastal waters; the grey seal (Halichoerus grypus), the ringed seal (Phoca hispida botnica) and the harbour seal (Phoca vitulina). A severe decline of the populations of Baltic grey and ringed seals took place during the second half of the 1960s. It was suggested to be caused by the contamination by industrial chemicals, above all organochlorines such as PCB and DDT. High concentrations of these substances were found in the Baltic biota. The author has performed necropsy or examination of organ samples from animals, which were found dead on shore, by caught at fishery or killed by hunting during 1977-2002. Multiple chronic organ lesions were found most prominent in the female reproductive organs (uterine stenoses and occlusions), intestines (colonic ulcers) and adrenals (cortical hyperplasia). Severe lesions were present also in the skeleton, integument and kidneys. The character and distribution of the lesions was regular and the disease picture tentatively was named the Baltic Seal Disease Complex (BSDC). The changes in the female reproductive organs indicate that reproductive failure is an important factor behind the decline of the Baltic seal populations. Adrenocortical hyperplasia was a regular and striking component of the BSDC. It is a common feature of prolonged stress in animals and man. The animals in this study have suffered from severe inflammatory processes in connection with more or less advanced malnutrition due to hampered ingestion and digestion of food. This is in the author’s opinion the most probable explanation of the adrenal changes. Inflammatory changes were most prominent in the intestines with deep ulcerations, in several cases leading to perforation of the intestinal wall. Bacteriological investigation revealed opportunistic or pathogenic micro-organisms but a common bacterial aetiology could not be suggested. The severity and wide dispersion of the lesions are interpreted as signs of a defective immune response. Minor lesions in the ileocaeco-colonic region caused by hookworms are regarded as the primary event of the ulcerous processes facilitating the establishment of secondary bacterial infections. Harbour seals showed less developed pathological changes but instead were victims of two Distemper epizootics with high mortality (c60%), during 1988 and 2002. During the 14- year-period after 1988 the Swedish harbour seal population gradually attained to the preepizootic size; a fast recover compared with the situation in Baltic grey and ringed seal populations suffering from the BSDC problems. A decrease in the prevalence of the lesions of the BSDC has been demonstrated concurrent with a decreased contamination of the Baltic biota towards the end of the 1900s. This is a strong indication of the role of pollutants as the main factor behind the BSDC. Other factors may also be involved, however, as indicated by the observation that the prevalence of intestinal ulcers still is high in Baltic grey seals