Schur Q-functions and degeneracy locus formulas for morphisms with symmetries

We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle, and is based on a certain push-forward formula for these polynomials in a Grassmann bundle.Comment: 22 pages, AMSTEX, misprints corrected, exposition improved. to appear in the Proceedings of Intersection Theory Conference in Bologna, "Progress in Mathematics", Birkhause

Resolution of null fiber and conormal bundles on the Lagrangian Grassmannian

We study the null fiber of a moment map related to dual pairs. We construct an equivariant resolution of singularities of the null fiber, and get conormal bundles of closed $K_C$-orbits in the Lagrangian Grassmannian as the categorical quotient. The conormal bundles thus obtained turn out to be a resolution of singularities of the closure of nilpotent $K_C$-orbits, which is a "quotient" of the resolution of the null fiber.Comment: 17 pages; completely revised and add reference

Containment and equivalence of weighted automata: Probabilistic and max-plus cases

This paper surveys some results regarding decision problems for probabilistic and max-plus automata, such as containment and equivalence. Probabilistic and max-plus automata are part of the general family of weighted automata, whose semantics are maps from words to real values. Given two weighted automata, the equivalence problem asks whether their semantics are the same, and the containment problem whether one is point-wise smaller than the other one. These problems have been studied intensively and this paper will review some techniques used to show (un)decidability and state a list of open questions that still remain

Class and rank of differential modules

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a substitute for the length of a free complex--and on the rank of a differential module in terms of invariants of its homology. These results specialize to basic theorems in commutative algebra and algebraic topology. One instance is a common generalization of the equicharacteristic case of the New Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning complexes over noetherian commutative rings, and of a theorem of G. Carlsson on differential graded modules over graded polynomial rings.Comment: 27 pages. Minor changes; mainly stylistic. To appear in Inventiones Mathematica

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

Joint analysis of stressors and ecosystem services to enhance restoration effectiveness

With increasing pressure placed on natural systems by growing human populations, both scientists and resource managers need a better understanding of the relationships between cumulative stress from human activities and valued ecosystem services. Societies often seek to mitigate threats to these services through large-scale, costly restoration projects, such as the over one billion dollar Great Lakes Restoration Initiative currently underway. To help inform these efforts, we merged high-resolution spatial analyses of environmental stressors with mapping of ecosystem services for all five Great Lakes. Cumulative ecosystem stress is highest in near-shore habitats, but also extends offshore in Lakes Erie, Ontario, and Michigan. Variation in cumulative stress is driven largely by spatial concordance among multiple stressors, indicating the importance of considering all stressors when planning restoration activities. In addition, highly stressed areas reflect numerous different combinations of stressors rather than a single suite of problems, suggesting that a detailed understanding of the stressors needing alleviation could improve restoration planning. We also find that many important areas for fisheries and recreation are subject to high stress, indicating that ecosystem degradation could be threatening key services. Current restoration efforts have targeted high-stress sites almost exclusively, but generally without knowledge of the full range of stressors affecting these locations or differences among sites in service provisioning. Our results demonstrate that joint spatial analysis of stressors and ecosystem services can provide a critical foundation for maximizing social and ecological benefits from restoration investments. www.pnas.org/lookup/suppl/doi:10.1073/pnas.1213841110/-/DCSupplementa

Diffusion tensor imaging of frontal lobe white matter tracts in schizophrenia

We acquired diffusion tensor and structural MRI images on 103 patients with schizophrenia and 41 age-matched normal controls. The vector data was used to trace tracts from a region of interest in the anterior limb of the internal capsule to the prefrontal cortex. Patients with schizophrenia had tract paths that were significantly shorter in length from the center of internal capsule to prefrontal white matter. These tracts, the anterior thalamic radiations, are important in frontal-striatal-thalamic pathways. These results are consistent with findings of smaller size of the anterior limb of the internal capsule in patients with schizophrenia, diffusion tensor anisotropy decreases in frontal white matter in schizophrenia and hypothesized disruption of the frontal-striatal-thalamic pathway system

Up-regulation of NMDA receptor subunit and post-synaptic density protein expression in the thalamus of elderly patients with schizophrenia

Numerous studies have described structural and functional abnormalities of the thalamus in schizophrenia, but surprisingly few studies have examined neurochemical abnormalities that accompany these pathological changes. We previously identified abnormalities of multiple molecules associated with glutamatergic neurotransmission, including changes in NMDA receptor subunit transcripts and binding sites and NMDA receptor-associated post-synaptic density (PSD) protein transcripts in the thalamus of elderly patients with schizophrenia. In the present study, we performed western blot analysis to determine whether protein levels of NMDA receptor subunits (NR1, NR2A, NR2B) and associated PSD proteins (NF-L, PSD95, SAP102) are altered in schizophrenia. Thalamic tissue from each subject was grossly dissected into two regions: a dorsomedial region containing limbic-associated dorsomedial, anterior and central medial thalamic nuclei; and a ventral thalamus region that primarily consisted of the ventral lateral nucleus. We observed increased protein expression of the NR2B NMDA receptor subunit and its associated intracellular protein, PSD95, in the dorsomedial thalamus of patients with schizophrenia, but the other molecules were unchanged, and we found no changes in the ventral thalamus. These data provide additional evidence of thalamic neurochemical abnormalities, particularly in thalamic nuclei which project to limbic regions of the brain. Further, these findings provide additional evidence of NMDA receptor alterations in schizophrenia, which may play an important role in the neurobiology of the illness.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/65970/1/j.1471-4159.2006.03954.x.pd

Brains studying brains: look before you think in vision

Using our own brains to study our brains is extraordinary. For example, in vision this makes us naturally blind to our own blindness, since our impression of seeing our world clearly is consistent with our ignorance of what we do not see. Our brain employs its 'conscious' part to reason and make logical deductions using familiar rules and past experience. However, human vision employs many 'subconscious' brain parts that follow rules alien to our intuition. Our blindness to our unknown unknowns and our presumptive intuitions easily lead us astray in asking and formulating theoretical questions, as witnessed in many unexpected and counter-intuitive difficulties and failures encountered by generations of scientists. We should therefore pay a more than usual amount of attention and respect to experimental data when studying our brain. I show that this can be productive by reviewing two vision theories that have provided testable predictions and surprising insights