1,965 research outputs found
O-minimality and certain atypical intersections
We show that the strategy of point counting in o-minimal structures can be
applied to various problems on unlikely intersections that go beyond the
conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called
Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound
for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In
the context of abelian varieties we obtain the Zilber-Pink Conjecture for
curves unconditionally when everything is defined over a number field. For
higher dimensional subvarieties of abelian varieties we obtain some weaker
results and some conditional results
Multiarray Signal Processing: Tensor decomposition meets compressed sensing
We discuss how recently discovered techniques and tools from compressed
sensing can be used in tensor decompositions, with a view towards modeling
signals from multiple arrays of multiple sensors. We show that with appropriate
bounds on a measure of separation between radiating sources called coherence,
one could always guarantee the existence and uniqueness of a best rank-r
approximation of the tensor representing the signal. We also deduce a
computationally feasible variant of Kruskal's uniqueness condition, where the
coherence appears as a proxy for k-rank. Problems of sparsest recovery with an
infinite continuous dictionary, lowest-rank tensor representation, and blind
source separation are treated in a uniform fashion. The decomposition of the
measurement tensor leads to simultaneous localization and extraction of
radiating sources, in an entirely deterministic manner.Comment: 10 pages, 1 figur
Coalition structure generation over graphs
We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph
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