Deciding first-order properties of nowhere dense graphs

Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.Comment: 30 page

On the number of types in sparse graphs

We prove that for every class of graphs $\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $\phi(\bar x,\bar y)$, whenever one draws a graph $G\in \mathcal{C}$ and a subset of its nodes $A$, the number of subsets of $A^{|\bar y|}$ which are of the form $\{\bar v\in A^{|\bar y|}\, \colon\, G\models\phi(\bar u,\bar v)\}$ for some valuation $\bar u$ of $\bar x$ in $G$ is bounded by $\mathcal{O}(|A|^{|\bar x|+\epsilon})$, for every $\epsilon>0$. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices)

Staining and peeling of the internal limiting membrane using a fluorescent dye (Rhodamine 6 G)

Aim: To assess whether low concentrations of a fluorescent dye such as Rhodamine 6G would help the unaided human eye visualise the vitreous and the internal limiting membrane (ILM) under standard halogen illumination.Material/methods: The UV/Vis absorption (E) and fluorescence (I) spectra of Rhodamine 6G in water were measured and compared with Indocyanine Green (ICG). Surgery was performed in two rhesus monkeys and consisted of standard pars plana vitrectomy with halogen light source used for illumination. Rhodamine 6G was diluted in balanced salt solution (BSS). A few drops of the dye in a concentration of 0.1% (307 mOsm) were applied over the posterior pole in the air-filled globe and washed out by irrigation after 1 min. Immediately after surgery, the globes were enucleated, fixated and prepared for histological evaluation.Results: In contrast to ICG, both the maximum of the absorption and emission of Rhodamin 6G are very much within the spectral sensitivity of the human eye. The Rhodamine 6G--BSS itself appears red in colour. Using a dye concentration of 0.1%, there was no visible red-staining of the ILM as such. As the dye was irrigated out with BSS, a marked green fluorescence of the fluid within the vitreous cavity was noted. With halogen illumination through a standard 20-gauge light pipe, the dye provided a sufficient green fluorescence to identify and safely remove the ILM and to clearly differentiate areas of peeled from non-peeled ILM. During light microscopy, eyes revealed a peeled ILM demarcation with no signs of acute retinal toxicity.Conclusion: The findings indicate that a fluorescent dye can be used for ILM peeling. Assuming that the fluorophore provides a high enough fluorescence quantum yield after adsorption to the ILM, much lower dye concentrations could be used compared with absorbent dyes, thereby minimising toxic effects

On the Parameterized Intractability of Monadic Second-Order Logic

One of Courcelle's celebrated results states that if C is a class of graphs of bounded tree-width, then model-checking for monadic second order logic (MSO_2) is fixed-parameter tractable (fpt) on C by linear time parameterized algorithms, where the parameter is the tree-width plus the size of the formula. An immediate question is whether this is best possible or whether the result can be extended to classes of unbounded tree-width. In this paper we show that in terms of tree-width, the theorem cannot be extended much further. More specifically, we show that if C is a class of graphs which is closed under colourings and satisfies certain constructibility conditions and is such that the tree-width of C is not bounded by \log^{84} n then MSO_2-model checking is not fpt unless SAT can be solved in sub-exponential time. If the tree-width of C is not poly-logarithmically bounded, then MSO_2-model checking is not fpt unless all problems in the polynomial-time hierarchy can be solved in sub-exponential time

Natural-gradient learning for spiking neurons.

In many normative theories of synaptic plasticity, weight updates implicitly depend on the chosen parametrization of the weights. This problem relates, for example, to neuronal morphology: synapses which are functionally equivalent in terms of their impact on somatic firing can differ substantially in spine size due to their different positions along the dendritic tree. Classical theories based on Euclidean-gradient descent can easily lead to inconsistencies due to such parametrization dependence. The issues are solved in the framework of Riemannian geometry, in which we propose that plasticity instead follows natural-gradient descent. Under this hypothesis, we derive a synaptic learning rule for spiking neurons that couples functional efficiency with the explanation of several well-documented biological phenomena such as dendritic democracy, multiplicative scaling, and heterosynaptic plasticity. We therefore suggest that in its search for functional synaptic plasticity, evolution might have come up with its own version of natural-gradient descent

Natural-gradient learning for spiking neurons

In many normative theories of synaptic plasticity, weight updates implicitly depend on the chosen parametrization of the weights. This problem relates, for example, to neuronal morphology: synapses which are functionally equivalent in terms of their impact on somatic firing can differ substantially in spine size due to their different positions along the dendritic tree. Classical theories based on Euclidean gradient descent can easily lead to inconsistencies due to such parametrization dependence. The issues are solved in the framework of Riemannian geometry, in which we propose that plasticity instead follows natural gradient descent. Under this hypothesis, we derive a synaptic learning rule for spiking neurons that couples functional efficiency with the explanation of several well-documented biological phenomena such as dendritic democracy, multiplicative scaling and heterosynaptic plasticity. We therefore suggest that in its search for functional synaptic plasticity, evolution might have come up with its own version of natural gradient descent.Comment: Joint senior authorship: Walter M. Senn and Mihai A. Petrovic

Routing with Congestion in Acyclic Digraphs

We study the version of the $k$-disjoint paths problem where $k$ demand pairs $(s_1,t_1)$, $\dots$, $(s_k,t_k)$ are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than $c$ paths. We show that on directed acyclic graphs the problem is solvable in time $n^{O(d)}$ if we allow congestion $k-d$ for $k$ paths. Furthermore, we show that, under a suitable complexity theoretic assumption, the problem cannot be solved in time $f(k)n^{o(d/\log d)}$ for any computable function $f$

Technical note:On the reliability of laboratory beta-source calibration for luminescence dating

The dose rate of the 90Sr / 90Y beta source used in most luminescence readers is a laboratory key parameter. There is a well-established body of knowledge about parameters controlling accuracy and precision of the calibration value but some hard-to-explain inconsistencies still exist. Here, we have investigated the impact of grain size, aliquot size and irradiation geometry on the resulting calibration value through experiments and simulations. The resulting data indicate that the dose rate of an individual beta source results from the interplay of a number of parameters, most of which are well established by previous studies. Our study provides evidence for the impact of aliquot size on the absorbed dose in particular for grain sizes of 50–200 µm. For this grain-size fraction, the absorbed dose is enhanced by ∼ 10 %–20 % as aliquot size decreases due to the radial increase of dose rate towards the centre of the aliquot. The enhancement is most variable for 50–100 µm grains mounted as aliquots of &lt; 8 mm size. The enhancement is reversed when large grains are mounted as small aliquots due to the edge effect by which the dose induced by backscattered electrons is reduced. While the build-up of charge dictates the increase of absorbed dose with the increase of grain size, this principle becomes more variable with changing irradiation geometry. We conclude that future calibration samples should consist of subsamples composed of small, medium, large and very large quartz grains, each obtaining several gamma doses. The calibration value measured with small, medium and large aliquots is then obtained from the inverse slope of the fitted line, not from a single data point. In this way, all possible irradiation geometries of an individual beta source are covered, and the precision of the calibration is improved.</p

Oxidative DNA damage : mutagenic properties of 5-hydorxycytosine, 5-hydroxyuracil and uracil glycol in Eschericia coli

Thesis (Ph. D.)--Massachusetts Institute of Technology, Division of Toxicology, 1998.Includes bibliographical references (leaves 121-140).by Deborah Anne Kreutzer.Ph.D

Intravitreal bevacizumab for the treatment of macular oedema secondary to branch retinal vein occlusion

Purpose: To evaluate the effect of intravitreal bevacizumab (Avastin) injections on visual acuity (VA) and foveal retinal thickness in patients with macular oedema secondary to branch retinal vein occlusion.Methods: A prospective, non-comparative, consecutive, interventional case series of 34 patients. Patients received repeated intravitreal injections of 1.25 mg bevacizumab. Main outcome measures were VA (Snellen charts and ETDRS) and retinal thickness (optical coherence tomography measurements) in a follow-up period of 6 months.Results: Patients presented at a mean age of 69 years (range 44--86). Mean duration of symptoms was 40 weeks (range 1--300). Mean (SD) VA at baseline was 0.79 (0.39) logMAR, improving to 0.51 (0.34) logMAR at 6 months (p = 0.009). Mean number of letters on the ETDRS chart at baseline was 45.3 (19.0), improving to 60.6 (19.9) at 6 months (p = 0.003). Mean (SD) retinal thickness at baseline was 474 (120) \textgreekmm, declining to 316 (41) \textgreekmm at 6 months.Conclusion: Intravitreal injection of 1.25 mg bevacizumb appears to be an effective treatment option for branch retinal vein occlusion