61 research outputs found
A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes
The Shi arrangement is the arrangement of affine hyperplanes
in of the form or , for . It dissects into regions, as was first proved
by Shi. We give a simple bijective proof of this result. Our bijection
generalizes easily to any subarrangement of containing the
hyperplanes and to the extended Shi arrangements
A simple bijection for the regions of the Shi arrangement of hyperplanes
AbstractThe Shi arrangement Ln is the arrangement of affine hyperplanes in Rn of the form xi − xj = 0 or 1, for 1 ⩽ i < j ⩽ n. It dissects Rn into (n + 1)n−1 regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Ln containing the hyperplanes xi − xj = 0 and to the extended Shi arrangements. It also implies the fact that the number of regions of Ln which are relatively bounded is (n − 1)n−1
Thomassen's Choosability Argument Revisited
Thomassen (1994) proved that every planar graph is 5-choosable. This result
was generalised by {\v{S}}krekovski (1998) and He et al. (2008), who proved
that every -minor-free graph is 5-choosable. Both proofs rely on the
characterisation of -minor-free graphs due to Wagner (1937). This paper
proves the same result without using Wagner's structure theorem or even planar
embeddings. Given that there is no structure theorem for graphs with no
-minor, we argue that this proof suggests a possible approach for
attacking the Hadwiger Conjecture
Matrix product solution to an inhomogeneous multi-species TASEP
We study a multi-species exclusion process with inhomogeneous hopping rates.
This model is equivalent to a Markov chain on the symmetric group that
corresponds to a random walk in the affine braid arrangement. We find a matrix
product representation for the stationary state of this model. We also show
that it is equivalent to a graphical construction proposed by Ayyer and
Linusson, which generalizes Ferrari and Martin's construction
On the number of -cycles in the assignment problem for random matrices
We continue the study of the assignment problem for a random cost matrix. We
analyse the number of -cycles for the solution and their dependence on the
symmetry of the random matrix. We observe that for a symmetric matrix one and
two-cycles are dominant in the optimal solution. In the antisymmetric case the
situation is the opposite and the one and two-cycles are suppressed. We solve
the model for a pure random matrix (without correlations between its entries)
and give analytic arguments to explain the numerical results in the symmetric
and antisymmetric case. We show that the results can be explained to great
accuracy by a simple ansatz that connects the expected number of -cycles to
that of one and two cycles.Comment: To appear in Journal of Statistical Mechanic
Random multi-index matching problems
The multi-index matching problem (MIMP) generalizes the well known matching
problem by going from pairs to d-uplets. We use the cavity method from
statistical physics to analyze its properties when the costs of the d-uplets
are random. At low temperatures we find for d>2 a frozen glassy phase with
vanishing entropy. We also investigate some properties of small samples by
enumerating the lowest cost matchings to compare with our theoretical
predictions.Comment: 22 pages, 16 figure
Kinome-wide interaction modelling using alignment-based and alignment-independent approaches for kinase description and linear and non-linear data analysis techniques
<p>Abstract</p> <p>Background</p> <p>Protein kinases play crucial roles in cell growth, differentiation, and apoptosis. Abnormal function of protein kinases can lead to many serious diseases, such as cancer. Kinase inhibitors have potential for treatment of these diseases. However, current inhibitors interact with a broad variety of kinases and interfere with multiple vital cellular processes, which causes toxic effects. Bioinformatics approaches that can predict inhibitor-kinase interactions from the chemical properties of the inhibitors and the kinase macromolecules might aid in design of more selective therapeutic agents, that show better efficacy and lower toxicity.</p> <p>Results</p> <p>We applied proteochemometric modelling to correlate the properties of 317 wild-type and mutated kinases and 38 inhibitors (12,046 inhibitor-kinase combinations) to the respective combination's interaction dissociation constant (K<sub>d</sub>). We compared six approaches for description of protein kinases and several linear and non-linear correlation methods. The best performing models encoded kinase sequences with amino acid physico-chemical z-scale descriptors and used support vector machines or partial least- squares projections to latent structures for the correlations. Modelling performance was estimated by double cross-validation. The best models showed high predictive ability; the squared correlation coefficient for new kinase-inhibitor pairs ranging P<sup>2 </sup>= 0.67-0.73; for new kinases it ranged P<sup>2</sup><sub>kin </sub>= 0.65-0.70. Models could also separate interacting from non-interacting inhibitor-kinase pairs with high sensitivity and specificity; the areas under the ROC curves ranging AUC = 0.92-0.93. We also investigated the relationship between the number of protein kinases in the dataset and the modelling results. Using only 10% of all data still a valid model was obtained with P<sup>2 </sup>= 0.47, P<sup>2</sup><sub>kin </sub>= 0.42 and AUC = 0.83.</p> <p>Conclusions</p> <p>Our results strongly support the applicability of proteochemometrics for kinome-wide interaction modelling. Proteochemometrics might be used to speed-up identification and optimization of protein kinase targeted and multi-targeted inhibitors.</p
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