51,158 research outputs found
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 is greater
than that of the perfect insulator . By contrast, if , that
magnetoresistance keeps increasing as without ever saturating. This
abrupt change in the macroscopic response, which occurs when , 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
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
Stability Conditions and Branes at Singularities
I use Bridgeland's definition of a stability condition on a triangulated
category to investigate the stability of D-branes on Calabi-Yau cones given by
the canonical line bundle over a del Pezzo surface. In this context, I prove
the existence of the decay of a D3-brane into a set of fractional branes. This
is an important aspect of the derivation of quiver gauge theories from branes
at singularities via the technique of equivalences of categories. Some
important technical aspects of this equivalence are discussed. I also prove
that the representations corresponding to skyscraper sheaves supported off the
zero section are simple.Comment: 22 pages, uses utarticle.cls, dcpic.sty, v2: published versio
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