Every filter is homeomorphic to its square

We show that every filter $\mathcal{F}$ on $\omega$, viewed as a subspace of $2^\omega$, is homeomorphic to $\mathcal{F}^2$. This generalizes a theorem of van Engelen, who proved that this holds for Borel filters.Comment: 4 page

Products and countable dense homogeneity

Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we construct a zero-dimensional separable metrizable space $X$ such that $X$ is countable dense homogeneous while $X^2$ is not. It follows from results of Hru\v{s}\'ak and Zamora Avil\'es that such a space $X$ cannot be Borel. Furthermore, $X$ can be made homogeneous and completely Baire as well.Comment: 7 page

Countable dense homogeneity in powers of zero-dimensional definable spaces

We show that, for a coanalytic subspace $X$ of $2^\omega$, the countable dense homogeneity of $X^\omega$ is equivalent to $X$ being Polish. This strengthens a result of Hru\v{s}\'ak and Zamora Avil\'es. Then, inspired by results of Hern\'andez-Guti\'errez, Hru\v{s}\'ak and van Mill, using a technique of Medvedev, we construct a non-Polish subspace $X$ of $2^\omega$ such that $X^\omega$ is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer to a question of Hru\v{s}\'ak and Zamora Avil\'es. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a question of Medini and Milovich. Finally, we show that if every countable subset of a zero-dimensional separable metrizable space $X$ is included in a Polish subspace of $X$ then $X^\omega$ is countable dense homogeneous.Comment: 14 page

Atomic Scale Fractal Dimensionality in Proteins

The soft condensed matter of biological organisms exhibits atomic motions whose properties depend strongly on temperature and hydration conditions. Due to the superposition of rapidly fluctuating alternative motions at both very low temperatures (quantum effects) and very high temperatures (classical Brownian motion regime), the dimension of an atomic ``path'' is in reality different from unity. In the intermediate temperature regime and under environmental conditions which sustain active biological functions, the fractal dimension of the sets upon which atoms reside is an open question. Measured values of the fractal dimension of the sets on which the Hydrogen atoms reside within the Azurin protein macromolecule are reported. The distribution of proton positions was measured employing thermal neutron elastic scattering from Azurin protein targets. As the temperature was raised from low to intermediate values, a previously known and biologically relevant dynamical transition was verified for the Azurin protein only under hydrated conditions. The measured fractal dimension of the geometrical sets on which protons reside in the biologically relevant temperature regime is given by $D=0.65 \pm 0.1$. The relationship between fractal dimensionality and biological function is qualitatively discussed.Comment: ReVTeX4 format with 5 *.eps figure

Contrast between spin and valley degrees of freedom

We measure the renormalized effective mass (m*) of interacting two-dimensional electrons confined to an AlAs quantum well while we control their distribution between two spin and two valley subbands. We observe a marked contrast between the spin and valley degrees of freedom: When electrons occupy two spin subbands, m* strongly depends on the valley occupation, but not vice versa. Combining our m* data with the measured spin and valley susceptibilities, we find that the renormalized effective Lande g-factor strongly depends on valley occupation, but the renormalized conduction-band deformation potential is nearly independent of the spin occupation.Comment: 4+ pages, 2 figure

Transference of Transport Anisotropy to Composite Fermions

When interacting two-dimensional electrons are placed in a large perpendicular magnetic field, to minimize their energy, they capture an even number of flux quanta and create new particles called composite fermions (CFs). These complex electron-flux-bound states offer an elegant explanation for the fractional quantum Hall effect. Furthermore, thanks to the flux attachment, the effective field vanishes at a half-filled Landau level and CFs exhibit Fermi-liquid-like properties, similar to their zero-field electron counterparts. However, being solely influenced by interactions, CFs should possess no memory whatever of the electron parameters. Here we address a fundamental question: Does an anisotropy of the electron effective mass and Fermi surface (FS) survive composite fermionization? We measure the resistance of CFs in AlAs quantum wells where electrons occupy an elliptical FS with large eccentricity and anisotropic effective mass. Similar to their electron counterparts, CFs also exhibit anisotropic transport, suggesting an anisotropy of CF effective mass and FS.Comment: 5 pages, 5 figure

A homogeneous space whose complement is rigid

We construct a homogeneous subspace of $2^\omega$ whose complement is dense in $2^\omega$ and rigid. Using the same method, assuming Martin's Axiom, we also construct a countable dense homogeneous subspace of $2^\omega$ whose complement is dense in $2^\omega$ and rigid.Comment: 9 page