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 lower pp-central series of a free profinite group and the shuffle algebra

For a prime number $p$ and a free profinite group $S$ on the basis $X$, let $S^{(n,p)}$, $n=1,2,\ldots$ be the lower $p$-central filtration of $S$. For $p>n$, we give a combinatorial description of $H^2(S/S^{(n,p)},\mathbb{Z}/p)$ in terms of the Shuffle algebra on $X$

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.

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 $\mathcal{N}= 1$ compactifications of the heterotic string, to lowest order in $\alpha'$ 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 $P, Q$ 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 $P$ by using the Reeb graph derived from a certain horizontal arc in the ambient space $[0,1]\times[0,1]$ of the Rubinstein-Scharlemann graphic derived from $P$ and $Q$. 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 $p$ and a free profinite group $S$, let $S^{(n,p)}$ be the $n$th term of its lower $p$-central filtration, and $S^{[n,p]}$ the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical basis of the cohomology group $H^2(S^{[n,p]},\mathbb{Z}/p)$, 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 $S^{(n,p)}/S^{(n+1,p)}$. We prove that the cohomology group satisfies shuffle relations, which for small values of $n$ 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 $F \tilde F$ term, the sourced GW spectrum is exponentially enhanced and parametrically the square of the vacuum fluctuations spectrum, ${\cal P}^s_T\sim ({\cal P}^v_T)^2$, giving an even bluer spectrum than the standard vacuum one. When the scalar field is also coupled to $F^2$ term, the amplitude is still exponentially enhanced, but the spectrum can be arbitrarily close to scale invariant (still slightly blue), $n_T\gtrsim 0$, 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