Dynamics and the Godbillon-Vey Class of C^1 Foliations

Let F be a codimension-one, C^2-foliation on a manifold M without boundary. In this work we show that if the Godbillon--Vey class GV(F) \in H^3(M) is non-zero, then F has a hyperbolic resilient leaf. Our approach is based on methods of C^1-dynamical systems, and does not use the classification theory of C^2-foliations. We first prove that for a codimension--one C^1-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F) has positive Lebesgue measure. We then prove that if E(F) has positive measure for a C^1-foliation F, then F must have a hyperbolic resilient leaf, and hence its geometric entropy must be positive. The proof of this uses a pseudogroup version of the Pliss Lemma. The theorem then follows, as a C^2-foliation with non-zero Godbillon-Vey class has non-trivial Godbillon measure. These results apply for both the case when M is compact, and when M is an open manifold.Comment: This manuscript is a revision of the section 3 material from the previous version, and includes edits to the pictures in the tex

On a conjecture of Helleseth

We are concern about a conjecture proposed in the middle of the seventies by Hellesseth in the framework of maximal sequences and theirs cross-correlations. The conjecture claims the existence of a zero outphase Fourier coefficient. We give some divisibility properties in this direction

Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials

We consider Weil sums of binomials of the form $W_{F,d}(a)=\sum_{x \in F} \psi(x^d-a x)$, where $F$ is a finite field, $\psi\colon F\to {\mathbb C}$ is the canonical additive character, $\gcd(d,|F^\times|)=1$, and $a \in F^\times$. If we fix $F$ and $d$ and examine the values of $W_{F,d}(a)$ as $a$ runs through $F^\times$, we always obtain at least three distinct values unless $d$ is degenerate (a power of the characteristic of $F$ modulo $|F^\times|$). Choices of $F$ and $d$ for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if $F$ is a field of order $3^n$ with $n$ odd, and $d=3^r+2$ with $4 r \equiv 1 \pmod{n}$, then $W_{F,d}(a)$ assumes only the three values $0$ and $\pm 3^{(n+1)/2}$. This proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The proof employs diverse methods involving trilinear forms, counting points on curves via multiplicative character sums, divisibility properties of Gauss sums, and graph theory.Comment: 19 page

Dynamic Surface Tension of Aqueous Solutions of Ionic Surfactants: Role of Electrostatics

The adsorption kinetics of the cationic surfactant dodecyltrimethylammonium bromide at the air-water interface has been studied by the maximum bubble pressure method at concentrations below the critical micellar concentration. At short times, the adsorption is diffusion-limited. At longer times, the surface tension shows an intermediate plateau and can no longer be accounted for by a diffusion limited process. Instead, adsorption appears kinetically controlled and slowed down by an adsorption barrier. A Poisson-Boltzmann theory for the electrostatic repulsion from the surface does not fully account for the observed potential barrier. The possibility of a surface phase transition is expected from the fitted isotherms but has not been observed by Brewster angle microscopy.Comment: 13 pages, 5 figure

Foamed emulsion drainage: flow and trapping of drops

Foamed emulsions are ubiquitous in our daily life but the ageing of such systems is still poorly understood. In this study we investigate foam drainage and measure the evolution of the gas, liquid and oil volume fractions inside the foam. We evidence three regimes of ageing. During an initial period of fast drainage, both bubbles and drops are very mobile. As the foam stabilises drainage proceeds leading to a gradual decrease of the liquid fraction and slowing down of drainage. Clusters of oil drops are less sheared, their dynamic viscosity increases and drainage slows down even further, until the drops become blocked. At this point the oil fraction starts to increase in the continuous phase. The foam ageing leads to an increase of the capillary pressure until the oil acts as an antifoaming agent and the foam collapses.Comment: Soft Matter 201