This paper is concerned with minimal-length representatives of equivalence classes of words in $F_2$ under Aut $F_2$. We give a simple inequality characterizing words of minimal length in their equivalence class. We consider an operation that “grows” words from other words, increasing the length, and we study root words—minimal words that cannot be grown from other minimal words. Root words are “as minimal as possible” in the sense that their characterization is the boundary case of the minimality inequality. The property of being a root word is respected by equivalence classes, and the length of each root word is divisible by 4.

A natural notion for the signature $C_{\pm}({\mathcal V}(p))$ of the curvature of the zero set ${\mathcal V}(p)$ of a non-commutative polynomial $p$ is introduced. The main result of this paper is the bound \[ \operatorname{deg} p \leq2 C_\pm \bigl({\mathcal V}(p) \bigr) + 2. \] It is obtained under some irreducibility and nonsingularity conditions, and shows that the signature of the curvature of the zero set of $p$ dominates its degree.

The condition $C_+({\mathcal V}(p))=0$ means that the non-commutative variety ${\mathcal V}(p)$ has positive curvature. In this case, the preceding inequality implies that the degree of $p$ is at most two. Non-commutative varieties ${\mathcal V}(p)$ with positive curvature were introduced in Indiana Univ. Math. J. 56 (2007) 1189-1231). There a slightly weaker irreducibility hypothesis plus a number of additional hypotheses yielded a weaker result on $p$. The approach here is quite different; it is cleaner, and allows for the treatment of arbitrary signatures.

In J. Anal. Math. 108 (2009) 19-59), the degree of a non-commutative polynomial $p$ was bounded by twice the signature of its Hessian plus two. In this paper, we introduce a modified version of this non-commutative Hessian of $p$ which turns out to be very appropriate for analyzing the variety ${\mathcal V}(p)$.

We prove that the depth formula holds for Tor-independent modules in certain cases over a Cohen–Macaulay local ring, provided one of the modules has reducible complexity.

The real locus in the moduli space of complex algebraic curves of given genus consists of curves having real forms, that is, Riemann surfaces admitting a symmetry (anticonformal involution). The real locus is covered by subsets, each formed by curves having a given topological type determined by the number of connected components and the separability type of the real models of the curves. In this paper, we study the structure of the nerve corresponding to this covering, called the real nerve of complex algebraic curves, for even genera. We find its geometrical and homological dimension, and we obtain some results concerning its global geometrical properties. In the proofs, we use the equivalent language of Riemann surfaces and their symmetries.

This paper extends the theory of scalar-valued integrable functions with respect to vector measures defined on $δ$-rings to the case of vector-valued tensor integrable functions with respect to vector measures defined on $δ$-rings. This paper also generalizes some results of G. F. Stefánsson for tensor integration theory of vector-valued functions with respect to vector measures defined on $σ$-algebras.

We construct the Kuranishi spaces, or in other words, the versal deformations, for the following classes of connections with fixed divisor of poles $D$: all such connections, as well as for its subclasses of integrable, integrable logarithmic and integrable logarithmic connections with a parabolic structure over $D$. The tangent and obstruction spaces of deformation theory are defined as the hypercohomology of an appropriate complex of sheaves, and the Kuranishi space is a fiber of the formal obstruction map.

Our aim in this note is to estimate the Orlicz–Sobolev capacity of balls.

We study linear extremal problems in the Bergman space $A^p$ of the unit disc for $p$ an even integer. Given a functional on the dual space of $A^p$ with representing kernel $k \in A^q$, where $1/p + 1/q = 1$, we show that if the Taylor coefficients of $k$ are sufficiently small, then the extremal function $F \in H^{\infty}$. We also show that if $q \le q_1 < \infty$, then $F \in H^{(p-1)q_1}$ if and only if $k \in H^{q_1}$.

We prove a weighted version of the Hardy–Littlewood–Sobolev inequality for radially symmetric functions, and show that the range of admissible power weights appearing in the classical inequality due to Stein and Weiss can be improved in this particular case.

We show the $L^2$-convergence of continuous time ergodic averages of a product of functions evaluated at return times along polynomials. These averages are the continuous time version of the averages appearing in Furstenberg's proof of Szemerédi’s Theorem. For each average, we show that it is sufficient to prove convergence on special factors, the Host-Kra factors, which have the structure of a nilmanifold. We also give a description of the limit. In particular, if the polynomials are independent over the real numbers then the limit is the product of the integrals. We further show that if the collection of polynomials has “low complexity”, then for every set $E$ of real numbers with positive density and for every $\delta>0$, the set of polynomial return times for the “$\delta$-thickened” set $E_{\delta}$ has bounded gaps. We give bounds for the flow average complexity and show that in some cases the flow average complexity is strictly less than the discrete average complexity.

Let $G$ be a finite abelian group. A number field $K$ is called a Hilbert-Speiser field of type $G$ if, for every tame $G$-Galois extension $L/K$, the ring of integers $\mathcal{O}_L$ is free as an $\mathcal{O}_K[G]$-module. If $\mathcal{O}_L$ is free over the associated order $\mathcal{A}_{L/K}$ for every $G$-Galois extension $L/K$, then $K$ is called a Leopoldt field of type $G$. It is well known (and easy to see) that if $K$ is Leopoldt of type $G$, then $K$ is Hilbert–Speiser of type $G$. We show that the converse does not hold in general, but that a modified version does hold for many number fields $K$ (in particular, for $K/\mathbb{Q}$ Galois) when $G=C_{p}$ has prime order. We give examples with $G=C_5$ to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

We generalize Abel's classical theorem on linear\break equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$-manifold, or more generally for any $d$-cycle $Z$ relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$-gerbe $\Lambda_{Z}$, the Abel gerbe determined by $Z$, whose vanishing as a Deligne cohomology class generalizes the notion of ‘linear equivalence to zero’. In this setting, Abel's theorem remains valid. Moreover, we generalize the classical Inversion theorem for the Abel–Jacobi map, thereby proving that the moduli space of Abel gerbes is isomorphic to the harmonic Deligne cohomology; that is, gerbes with harmonic curvature.

The inhomogeneous self-similar measure $\mu$ is defined by the relation \[ \mu= \sum_{j=1}^N p_j \mu oS_j^{-1} + p \nu, \] where $(p_1, \ldots, p_N, p)$ is a probability vector, $ S_j : \mathbb R^n \rightarrow\mathbb R^n, j = 1, \ldots, N$ are contracting similarities and $\nu$ is a probability measure on $\mathbb R^n$ with compact support. The existence of such measures is well known, see Math. Proc. Cambridge Philos. Soc. 144 (2008) 465-493) and the references therein. In Math. Proc. Cambridge Philos. Soc. 144 (2008) 465-493), the authors have studied the Fourier transforms of inhomogeneous self-similar measures and they give relations about the asymptotic behavior of the Fourier transform of $\nu$ and $\mu$. Some constructions which are given with precise asymptotic behavior arise from a discrete measure $\nu$. Here we will see that these constructions can be extended with purely continuous measures $\nu$. In order to prove this, we will construct suitable symmetric Bernoulli convolution measures Essays in Commutative Harmonic Analysis (1979) Springer) and will use the results of J. Math. Anal. Appl. 299 (2004) 550–562).

For given positive integers $m$ and $n$, we consider the frequency of representations of $m/n$ as a sum of unit fractions.

The complete classification of three-dimensional homogeneous paracontact metric manifolds is obtained. In the symmetric case, such a manifold is either flat or of constant sectional curvature −1. In the non-symmetric case, it is a Lie group equipped with a left-invariant paracontact metric structure.