Minimal kernels of Dirac operators along maps

Let $M$ be a closed spin manifold and let $N$ be a closed manifold. For maps $f\colon M\to N$ and Riemannian metrics $g$ on $M$ and $h$ on $N$, we consider the Dirac operator $D^f_{g,h}$ of the twisted Dirac bundle $\Sigma M\otimes_{\mathbb{R}} f^*TN$. To this Dirac operator one can associate an index in $KO^{-dim(M)}(pt)$. If $M$ is $2$-dimensional, one gets a lower bound for the dimension of the kernel of $D^f_{g,h}$ out of this index. We investigate the question whether this lower bound is obtained for generic tupels $(f,g,h)$

Spectral and Dynamical Properties in Classes of Sparse Networks with Mesoscopic Inhomogeneities

We study structure, eigenvalue spectra and diffusion dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their internal structure as scale-free and correlated subgraphs, and the topology of connecting network. Within the exhaustive spectral analysis for both the adjacency matrix and the normalized Laplacian matrix we identify the spectral properties which characterize the mesoscopic structure of sparse cyclic graphs and trees. The minimally connected nodes, clustering, and the average connectivity affect the central part of the spectrum. The number of distinct modules leads to an extra peak at the lower part of the Laplacian spectrum in cyclic graphs. Such a peak does not occur in the case of topologically distinct tree-subgraphs connected on a tree. Whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs. We also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue equal 2 of the Laplacian. We corroborate the results with simulations of the random walk on several types of networks. Our results for the distribution of return-time of the walk to the origin (autocorrelator) agree well with recent analytical solution for trees, and it appear to be independent on their mesoscopic and global structure. For the cyclic graphs we find new results with twice larger stretching exponent of the tail of the distribution, which is virtually independent on the size of cycles. The modularity and clustering contribute to a power-law decay at short return times

Qudit surface codes and gauge theory with finite cyclic groups

Surface codes describe quantum memory stored as a global property of interacting spins on a surface. The state space is fixed by a complete set of quasi-local stabilizer operators and the code dimension depends on the first homology group of the surface complex. These code states can be actively stabilized by measurements or, alternatively, can be prepared by cooling to the ground subspace of a quasi-local spin Hamiltonian. In the case of spin-1/2 (qubit) lattices, such ground states have been proposed as topologically protected memory for qubits. We extend these constructions to lattices or more generally cell complexes with qudits, either of prime level or of level $d^\ell$ for $d$ prime and $\ell \geq 0$, and therefore under tensor decomposition, to arbitrary finite levels. The Hamiltonian describes an exact $\mathbb{Z}_d\cong\mathbb{Z}/d\mathbb{Z}$ gauge theory whose excitations correspond to abelian anyons. We provide protocols for qudit storage and retrieval and propose an interferometric verification of topological order by measuring quasi-particle statistics.Comment: 26 pages, 5 figure

Homotopy types of stabilizers and orbits of Morse functions on surfaces

Let $M$ be a smooth compact surface, orientable or not, with boundary or without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$ the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule $h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in C^{\infty}(M,P)$. Let $f:M \to P$ be a Morse function and $O(f)$ be the orbit of $f$ under this action. We prove that $\pi_k O(f)=\pi_k M$ for $k\geq 3$, and $\pi_2 O(f)=0$ except for few cases. In particular, $O(f)$ is aspherical, provided so is $M$. Moreover, $\pi_1 O(f)$ is an extension of a finitely generated free abelian group with a (finite) subgroup of the group of automorphisms of the Reeb graph of $f$. We also give a complete proof of the fact that the orbit $O(f)$ is tame Frechet submanifold of $C^{\infty}(M,P)$ of finite codimension, and that the projection $Diff(M) \to O(f)$ is a principal locally trivial $S(f)$-fibration.Comment: 49 pages, 8 figures. This version includes the proof of the fact that the orbits of a finite codimension of tame action of tame Lie group on tame Frechet manifold is a tame Frechet manifold itsel

Insights into the composition of humin from an irish grassland soil.

Humin is defined as the fraction of the humic substances (HS) that is insoluble in aqueous solution at any pH value (1). Because of its insolubility it has been the least studied of all the humic fractions (2). Humin typically represents more than 50% of the organic carbon (C) in a soil, and is regarded as the residual organic C that remains after extraction of the HAs and FAs (3). Based on data from solid-state 13C Nuclear Magnetic Resonance (NMR) spectroscopy, it has been concluded that a repeating aliphatic structural unit, possibly attributable to branched and cross-linked algal or microbial lipids, are common to both sediment and soil humin samples (4). Humin becomes enriched in these aliphatic substances because of their selective preservation during organic diagenesis (2). To investigate the composition of humin, The Clonakilty soil, a well drained Brown Podzolic soil in grassland lysimeter studies at the Teagasc Soils Research Centre, Johnstown Castle, Co. Wexford, Ireland and was used in the present study

Tverberg-type theorems for intersecting by rays

In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the center point theorem, and Tverberg's theorem on partitions of a point set

Aggregation and Degradation of Dispersants and Oil by Microbial Exopolymers (ADDOMEx): Toward a Synthesis of Processes and Pathways of Marine Oil Snow Formation in Determining the Fate of Hydrocarbons

Microbes (bacteria, phytoplankton) in the ocean are responsible for the copious production of exopolymeric substances (EPS) that include transparent exopolymeric particles. These materials act as a matrix to form marine snow. After the Deepwater Horizon oil spill, marine oil snow (MOS) formed in massive quantities and influenced the fate and transport of oil in the ocean. The processes and pathways of MOS formation require further elucidation to be better understood, in particular we need to better understand how dispersants affect aggregation and degradation of oil. Toward that end, recent work has characterized EPS as a function of microbial community and environmental conditions. We present a conceptual model that incorporates recent findings in our understanding of the driving forces of MOS sedimentation and flocculent accumulation (MOSSFA) including factors that influence the scavenging of oil into MOS and the routes that promote decomposition of the oil post MOS formation. In particular, the model incorporates advances in our understanding of processes that control interactions between oil, dispersant, and EPS in producing either MOS that can sink or dispersed gels promoting microbial degradation of oil compounds. A critical element is the role of protein to carbohydrate ratios (P/C ratios) of EPS in the aggregation process of colloid and particle formation. The P/C ratio of EPS provides a chemical basis for the stickiness ; factor that is used in analytical or numerical simulations of the aggregation process. This factor also provides a relative measure for the strength of attachment of EPS to particle surfaces. Results from recent laboratory experiments demonstrate (i) the rapid formation of microbial assemblages, including their EPS, on oil droplets that is enhanced in the presence of Corexit-dispersed oil, and (ii) the subsequent rapid oil oxidation and microbial degradation in water. These findings, combined with the conceptual model, further improve our understanding of the fate of the sinking MOS (e.g., subsequent sedimentation and preservation/degradation) and expand our ability to predict the behavior and transport of spilled oil in the ocean, and the potential effects of Corexit application, specifically with respect to MOS processes (i.e., formation, fate, and half-lives) and Marine Oil Snow Sedimentation and Flocculent Accumulation