124,160 research outputs found
Six Constructions of Difference Families
In this paper, six constructions of difference families are presented. These
constructions make use of difference sets, almost difference sets and disjoint
difference families, and give new point of views of relationships among these
combinatorial objects. Most of the constructions work for all finite groups.
Though these constructions look simple, they produce many difference families
with new parameters. In addition to the six new constructions, new results
about intersection numbers are also derived
Constructions of biangular tight frames and their relationships with equiangular tight frames
We study several interesting examples of Biangular Tight Frames (BTFs) -
basis-like sets of unit vectors admitting exactly two distinct frame angles
(ie, pairwise absolute inner products) - and examine their relationships with
Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one
frame angle.
We demonstrate a smooth parametrization BTFs, where the corresponding frame
angles transform smoothly with the parameter, which "passes through" an ETF
answers two questions regarding the rigidity of BTFs. We also develop a general
framework of so-called harmonic BTFs and Steiner BTFs - which includes the
equiangular cases, surprisingly, the development of this framework leads to a
connection with the famous open problem(s) regarding the existence of Mersenne
and Fermat primes. Finally, we construct a (chordally) biangular tight set of
subspaces (ie, a tight fusion frame) which "Pl\"ucker embeds" into an ETF.Comment: 19 page
High-rate self-synchronizing codes
Self-synchronization under the presence of additive noise can be achieved by
allocating a certain number of bits of each codeword as markers for
synchronization. Difference systems of sets are combinatorial designs which
specify the positions of synchronization markers in codewords in such a way
that the resulting error-tolerant self-synchronizing codes may be realized as
cosets of linear codes. Ideally, difference systems of sets should sacrifice as
few bits as possible for a given code length, alphabet size, and
error-tolerance capability. However, it seems difficult to attain optimality
with respect to known bounds when the noise level is relatively low. In fact,
the majority of known optimal difference systems of sets are for exceptionally
noisy channels, requiring a substantial amount of bits for synchronization. To
address this problem, we present constructions for difference systems of sets
that allow for higher information rates while sacrificing optimality to only a
small extent. Our constructions utilize optimal difference systems of sets as
ingredients and, when applied carefully, generate asymptotically optimal ones
with higher information rates. We also give direct constructions for optimal
difference systems of sets with high information rates and error-tolerance that
generate binary and ternary self-synchronizing codes.Comment: 9 pages, no figure, 2 tables. Final accepted version for publication
in the IEEE Transactions on Information Theory. Material presented in part at
the International Symposium on Information Theory and its Applications,
Honolulu, HI USA, October 201
Lagrangian Cobordisms via Generating Families: Constructions and Geography
Embedded Lagrangian cobordisms between Legendrian submanifolds are produced
from isotopy, spinning, and handle attachment constructions that employ the
technique of generating families. Moreover, any Legendrian with a generating
family has an immersed Lagrangian filling with a compatible generating family.
These constructions are applied in several directions, in particular to a
non-classical geography question: any graded group satisfying a duality
condition can be realized as the generating family homology of a connected
Legendrian submanifold in R^{2n+1} or in the 1-jet space of any compact
n-manifold with n at least 2.Comment: 34 pages, 11 figures. v2: corrected a referenc
Phenomenological viability of orbifold models with three Higgs families
We discuss the phenomenological viability of string multi-Higgs doublet
models, namely a scenario of heterotic orbifolds with two Wilson lines,
which naturally predicts three supersymmetric families of matter and Higgs
fields. We study the orbifold parameter space, and discuss the compatibility of
the predicted Yukawa couplings with current experimental data. We address the
implications of tree-level flavour changing neutral processes in constraining
the Higgs sector of the model, finding that viable scenarios can be obtained
for a reasonably light Higgs spectrum. We also take into account the tree-level
contributions to indirect CP violation, showing that the experimental value of
can be accommodated in the present framework.Comment: 31 pages, 12 figures. Comments and references added. Final version to
be published in JHE
Some explicit constructions of sets with more sums than differences
We present a variety of new results on finite sets A of integers for which
the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums
than differences) sets. First we show that there is, up to affine
transformation, a unique MSTD subset of {\bf Z} of size 8. Secondly, starting
from some examples of size 9, we present several new constructions of infinite
families of MSTD sets. Thirdly we show that for every fixed ordered pair of
non-negative integers (j,k), as n -> \infty a positive proportion of the
subsets of {0,1,2,...,n} satisfy |A+A| = (2n+1) - j, |A-A| = (2n+1) - 2k.Comment: 21 pages, no figures. Section 4 has been rewritten and Theorem 8 is a
strengthening of Theorem 9 in previous version. Reference list updated, plus
some other cosmetic change
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