Non-perturbative Quantum Dynamics of the Order Parameter in the Pairing Model

We consider quantum dynamics of the order parameter in the discrete pairing model (Richardson model) in thermodynamic equilibrium. The integrable Richardson Hamiltonian is represented as a direct sum of Hamiltonians acting in different Hilbert spaces of single-particle and paired/empty states. This allows us to factorize the full thermodynamic partition function into a combination of simple terms associated with real spins on singly-occupied states and the partition function of the quantum XY-model for Anderson pseudospins associated with the paired/empty states. Using coherent-state path-integral, we calculate the effects of superconducting phase fluctuations exactly. The contribution of superconducting amplitude fluctuations to the partition function in the broken-symmetry phase is shown to follow from the Bogoliubov-de Gennes equations in imaginary time. These equations in turn allow several interesting mappings, e.g., they are shown to be in a one-to-one correspondence with the one-dimensional Schr\"odinger equation in supersymmetric Quantum Mechanics. However, the most practically useful approach to calculate functional determinants is found to be via an analytical continuation of the quantum order parameter to real time, \Delta(\tau -> it), such that the problem maps onto that of a driven two-level system. The contribution of a particular dynamic order parameter to the partition function is shown to correspond to the sum of the Berry phase and dynamic phase accumulated by the pseudospin. We also examine a family of exact solutions for two-level-system dynamics on a class of elliptic functions and suggest a compact expression to estimate the functional determinants on such trajectories. The possibility of having quantum soliton solutions co-existing with classical BCS mean-field is discussed.Comment: 34 pages (v2: Typos corrected, references added

On-line blind unmixing for hyperspectral pushbroom imaging systems

International audienceIn this paper, the on-line hyperspectral image blind unmixing is addressed. Inspired by the Incremental Non-negative Matrix Factorization (INMF) method, we propose an on-line NMF which is adapted to the acquisition scheme of a pushbroom imager. Because of the non-uniqueness of the NMF model, a minimum volume constraint on the endmembers is added allowing to reduce the set of admissible solutions. This results in a stable algorithm yielding results similar to those of standard off-line NMF methods, but drastically reducing the computation time. The algorithm is applied to wood hyperspectral images showing that such a technique is effective for the on-line prediction of wood piece rendering after finishing. Index Terms— Hyperspectral imaging, Pushbroom imager, On-line Non-negative Matrix Factorization, Minimum volume constraint

Exact Covers via Determinants

Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such that every hyperedge includes one vertex from each part, the k-dimensional matching problem asks whether there is a disjoint collection of the hyperedges which covers all vertices. We show it can be solved by a randomized polynomial space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors polynomial in n and k. When we drop the partition constraint and permit arbitrary hyperedges of cardinality k, we obtain the exact cover by k-sets problem. We show it can be solved by a randomized polynomial space algorithm in time O*(c_k^n), where c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k. Both results substantially improve on the previous best algorithms for these problems, especially for small k, and follow from the new observation that Lovasz' perfect matching detection via determinants (1979) admits an embedding in the recently proposed inclusion-exclusion counting scheme for set covers, despite its inability to count the perfect matchings