Improved Bounds for rr-Identifying Codes of the Hex Grid

For any positive integer $r$, an $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and pairwise distinct. For a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We find a code of density less than $5/(6r)$, which is sparser than the prior best construction which has density approximately $8/(9r)$.Comment: 12p

Bottom-up retinotopic organization supports top-down mental imagery

Finding a path between locations is a routine task in daily life. Mental navigation is often used to plan a route to a destination that is not visible from the current location. We first used functional magnetic resonance imaging (fMRI) and surface-based averaging methods to find high-level brain regions involved in imagined navigation between locations in a building very familiar to each participant. This revealed a mental navigation network that includes the precuneus, retrosplenial cortex (RSC), parahippocampal place area (PPA), occipital place area (OPA), supplementary motor area (SMA), premotor cortex, and areas along the medial and anterior intraparietal sulcus. We then visualized retinotopic maps in the entire cortex using wide-field, natural scene stimuli in a separate set of fMRI experiments. This revealed five distinct visual streams or ‘fingers’ that extend anteriorly into middle temporal, superior parietal, medial parietal, retrosplenial and ventral occipitotemporal cortex. By using spherical morphing to overlap these two data sets, we showed that the mental navigation network primarily occupies areas that also contain retinotopic maps. Specifically, scene-selective regions RSC, PPA and OPA have a common emphasis on the far periphery of the upper visual field. These results suggest that bottom-up retinotopic organization may help to efficiently encode scene and location information in an eye-centered reference frame for top-down, internally generated mental navigation. This study pushes the border of visual cortex further anterior than was initially expected

BRST Formulation of 4-Monopoles

A supersymmetric gauge invariant action is constructed over any 4-dimensional Riemannian manifold describing Witten's theory of 4-monopoles. The topological supersymmetric algebra closes off-shell. The multiplets include the auxiliary fields and the Wess-Zumino fields in an unusual way, arising naturally from BRST gauge fixing. A new canonical approach over Riemann manifolds is followed, using a Morse function as an euclidean time and taking into account the BRST boundary conditions that come from the BFV formulation. This allows a construction of the effective action starting from gauge principles.Comment: 18 pages, Amste

Relativistic Coulomb Green's function in dd-dimensions

Using the operator method, the Green's functions of the Dirac and Klein-Gordon equations in the Coulomb potential $-Z\alpha/r$ are derived for the arbitrary space dimensionality $d$. Nonrelativistic and quasiclassical asymptotics of these Green's functions are considered in detail.Comment: 9 page

Two scenarios for quantum multifractality breakdown

We expose two scenarios for the breakdown of quantum multifractality under the effect of perturbations. In the first scenario, multifractality survives below a certain scale of the quantum fluctuations. In the other one, the fluctuations of the wave functions are changed at every scale and each multifractal dimension smoothly goes to the ergodic value. We use as generic examples a one-dimensional dynamical system and the three-dimensional Anderson model at the metal-insulator transition. Based on our results, we conjecture that the sensitivity of quantum multifractality to perturbation is universal in the sense that it follows one of these two scenarios depending on the perturbation. We also discuss the experimental implications.Comment: 5 pages, 4 figures, minor modifications, published versio

On the Stability of Compactified D=11 Supermembranes

We prove D=11 supermembrane theories wrapping around in an irreducible way over $S^{1} \times S^{1}\times M^{9}$ on the target manifold, have a hamiltonian with strict minima and without infinite dimensional valleys at the minima for the bosonic sector. The minima occur at monopole connections of an associated U(1) bundle over topologically non trivial Riemann surfaces of arbitrary genus. Explicit expressions for the minimal connections in terms of membrane maps are presented. The minimal maps and corresponding connections satisfy the BPS condition with half SUSY.Comment: 15 pages, latex. Added comments in conclusions and more reference