8,138 research outputs found
A Short Proof that Minimal Sets of Planar Ordinary Differential Equations are Trivial
We present a short proof, relaying on the divergence theorem, verifying that
minimal sets in the plane are trivial
The Zassenhaus filtration, Massey Products, and Representations of Profinite Groups
We consider the p-Zassenhaus filtration (G_n) of a profinite group G. Suppose
that G=S/N for a free profinite group S and a normal subgroup N of S contained
in S_n. Under a cohomological assumption on the n-fold Massey products (which
holds e.g., if the p-cohomological dimension of G is at most 1), we prove that
G_{n+1} is the intersection of all kernels of upper-triangular unipotent
(n+1)-dimensional representations of G over \mathbb F_p. This extends earlier
results by Minac, Spira, and the author on the structure of absolute Galois
groups of fields.Comment: Added more references, strengthened Lemma 2.3, added Remark 12.
The lower -central series of a free profinite group and the shuffle algebra
For a prime number and a free profinite group on the basis , let
, be the lower -central filtration of . For
, we give a combinatorial description of
in terms of the Shuffle algebra on
On the Construction of Polar Codes for Channels with Moderate Input Alphabet Sizes
Current deterministic algorithms for the construction of polar codes can only
be argued to be practical for channels with small input alphabet sizes. In this
paper, we show that any construction algorithm for channels with moderate input
alphabet size which follows the paradigm of "degrading after each polarization
step" will inherently be impractical with respect to a certain "hard"
underlying channel. This result also sheds light on why the construction of
LDPC codes using density evolution is impractical for channels with moderate
sized input alphabets.Comment: 9 page
Filtrations of free groups as intersections
For several natural filtrations of a free group S we express the n-th term of
the filtration as the intersection of all kernels of homomorphisms from S to
certain groups of upper-triangular unipotent matrices. This generalizes a
classical result of Grun for the lower central filtration. In particular, we do
this for the n-th term in the lower p-central filtration of S
On the marginal deformations of general (0,2) non-linear sigma-models
In this note we explore the possible marginal deformations of general (0,2)
non-linear sigma-models, which arise as descriptions of the weakly-coupled
(large radius) limits of four-dimensional compactifications of
the heterotic string, to lowest order in and first order in conformal
perturbation theory. The results shed light from the world-sheet perspective on
the classical moduli space of such compactifications. This is a contribution to
the proceedings of String-Math 2012.Comment: LaTeX2e, 11 pages, no figures, published in Proc.Symp.Pure Math. 90
(2015) 171-17
An estimation of Hempel distance by using Reeb graph
Let be Heegaard surfaces of a closed orientable 3-manifold. In this
paper, we introduce a method for giving an upper bound of Hempel distance of
by using the Reeb graph derived from a certain horizontal arc in the
ambient space of the Rubinstein-Scharlemann graphic derived
from and . This is a refinement of a part of Johnson's arguments used
for determining stable genera required for flipping high distance Heegaard
splittings.Comment: 17 pages, 22 figure
The Cohomology of canonical quotients of free groups and Lyndon words
For a prime number and a free profinite group , let be the
th term of its lower -central filtration, and the
corresponding quotient. Using tools from the combinatorics of words, we
construct a canonical basis of the cohomology group
, which we call the Lyndon basis, and use it to
obtain structural results on this group. We show a duality between the Lyndon
basis and canonical generators of . We prove that the
cohomology group satisfies shuffle relations, which for small values of
fully describe it.Comment: Several minor issues fixed and a few references added. To appear in
Documenta Mathematic
Tight Bounds for Averaging Multi-Frequency Differential Inclusions, Applied to Control Systems
We present new tight bounds for averaging differential inclusions, which we
apply to multi-frequency inclusions consisting of a sum of time periodic
set-valued mappings. For this family of inclusions we establish an a tight
estimate of order O\left(\epsilon\right) on the approximation error. These
results are then applied to control systems consisting of a sum of
time-periodic functions
Gravitational Waves in Bouncing Cosmologies from Gauge Field Production
We calculate the gravitational waves (GW) spectrum produced in various Early
Universe scenarios from gauge field sources, thus generalizing earlier
inflationary calculations to bouncing cosmologies. We consider generic
couplings between the gauge fields and the scalar field dominating the energy
density of the Universe. We analyze the requirements needed to avoid a
backreaction that will spoil the background evolution. When the scalar is
coupled only to term, the sourced GW spectrum is exponentially
enhanced and parametrically the square of the vacuum fluctuations spectrum,
, giving an even bluer spectrum than the
standard vacuum one. When the scalar field is also coupled to term, the
amplitude is still exponentially enhanced, but the spectrum can be arbitrarily
close to scale invariant (still slightly blue), , that is
distinguishable form the slightly red inflationary one. Hence, we have a proof
of concept of observable GW on CMB scales in a bouncing cosmology.Comment: Added Figure, matches the published versio
- …