31,229 research outputs found
On the number of distinct block sizes in partitions of a set
AbstractThe average number of distinct block sizes in a partition of a set of n elements is asymptotic to e log n as n → ∞. In addition, almost all partitions have approximately e log n distinct block sizes. This is in striking contrast to the fact that the average total number of blocks in a partition is ∼n(log n)−1 as n → ∞
Unipotent classes in the classical groups parameterized by subgroups
This paper describes how to use subgroups to parameterize unipotent classes
in the classical algebraic group in characteristic 2. These results can be
viewed as an extension of the Bala-Carter Theorem, and give a convenient way to
compare unipotent classes in a group with unipotent classes of a subgroup
where is exceptional and is a Levi subgroup of classical type.Comment: 14 pages. Minor changes and rewriting compared to version
Stable Frank-Kasper phases of self-assembled, soft matter spheres
Single molecular species can self-assemble into Frank Kasper (FK) phases,
finite approximants of dodecagonal quasicrystals, defying intuitive notions
that thermodynamic ground states are maximally symmetric. FK phases are
speculated to emerge as the minimal-distortional packings of space-filling
spherical domains, but a precise quantitation of this distortion and how it
affects assembly thermodynamics remains ambiguous. We use two complementary
approaches to demonstrate that the principles driving FK lattice formation in
diblock copolymers emerge directly from the strong-stretching theory of
spherical domains, in which minimal inter-block area competes with minimal
stretching of space-filling chains. The relative stability of FK lattices is
studied first using a diblock foam model with unconstrained particle volumes
and shapes, which correctly predicts not only the equilibrium {\sigma} lattice,
but also the unequal volumes of the equilibrium domains. We then provide a
molecular interpretation for these results via self-consistent field theory,
illuminating how molecular stiffness regulates the coupling between
intra-domain chain configurations and the asymmetry of local packing. These
findings shed new light on the role of volume exchange on the formation of
distinct FK phases in copolymers, and suggest a paradigm for formation of FK
phases in soft matter systems in which unequal domain volumes are selected by
the thermodynamic competition between distinct measures of shape asymmetry.Comment: 40 pages, 22 figure
Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees
We introduce the notion of a restricted exchangeable partition of
. We obtain integral representations, consider associated
fragmentations, embeddings into continuum random trees and convergence to such
limit trees. In particular, we deduce from the general theory developed here a
limit result conjectured previously for Ford's alpha model and its extension,
the alpha-gamma model, where restricted exchangeability arises naturally.Comment: 35 pages, 5 figure
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