Validation of fuzzy logic method for automated mass spectral classification for mineral imaging

Imaging mass spectrometry requires the acquisition and interpretation of hundreds to thousands of individual spectra in order to map the mineral phases within heterogeneous geomatrices. A fuzzy logic inference engine (FLIE) was developed to automate data interpretation. To evaluate the strengths and limitations of FLIE, the chemical images obtained using FLIE were compared with those developed using two chemometric methods: principle component analysis (PCA) and cluster analysis (K-Means). Two heterogeneous geomatrices, a low-grade chalcopyrite ore and basalt, were imaged using a laser-desorption Fourier transform mass spectrometer. Similar mineral distribution patterns in the chalcopyrite ore sample were obtained by the three data analysis methods with most of the differences occurring at the interfaces between mineral phases. PCA missed one minor mineral phase in the chalcopyrite ore sample and did not clearly differentiate among the mineral classes of the basalt. K-Means cluster analysis differentiated among the various mineral phases in both samples, but improperly grouped some spectra in the chalcopyrite sample that only contained unanticipated high mass peaks. Unlike the chemometric methods, FLIE was able to classify spectra as unknowns for those spectra that fell below the confidence level threshold. A nearest neighbor approach, included in FLIE, was used to classify the unknowns to form a visually complete image; however, the unknowns identified by FLIE can be informative because they highlight potential problems or overlooked results. In conclusion, this study validated the fuzzy logic-based approach used in our laboratory and reveald some limitations in the three techniques that were evaluated.     

On deformations of Hamiltonian actions

In this paper we generalize to coisotropic actions of compact Lie groups a theorem of Guillemin on deformations of Hamiltonian structures on compact symplectic manifolds. We show how one can reconstruct from the moment polytope the symplectic form on the manifold. Received: 21 March 2006     

Commutative idempotent residuated lattices

We investigate the variety of residuated lattices with a commutative and idempotent monoid reduct.     

On potentially <Emphasis Type="Italic">H</Emphasis>-graphic sequences

For given a graph H, a graphic sequence π = (d ₁, d ₂,..., d _n) is said to be potentially H-graphic if there is a realization of π containing H as a subgraph. In this paper, we characterize the potentially (K ₅ − e)-positive graphic sequences and give two simple necessary and sufficient conditions for a positive graphic sequence π to be potentially K ₅-graphic, where K _r is a complete graph on r vertices and K _r-e is a graph obtained from K _r by deleting one edge. Moreover, we also give a simple necessary and sufficient condition for a positive graphic sequence π to be potentially K ₆-graphic. Project supported by National Natural Science Foundation of China (No. 10401010).     

Sudoku’s french ancestors

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Multiple Coverings of the Plane with Triangles

We prove that any 43-fold covering of the plane with translates of a triangle can be decomposed into two coverings.