Varieties of Cost Functions.

Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring

Quantifying the ki-67 heterogeneity profile in prostate cancer.

BackgroundKi-67 is a robust predictive/prognostic marker in prostate cancer; however, tumor heterogeneity in prostate biopsy samples is not well studied.MethodsUsing an MRI/US fusion device, biopsy cores were obtained systematically and by targeting when indicated by MRI. Prostate cores containing cancer from 77 consecutive men were analyzed. The highest Ki-67 was used to determine interprostatic variation. Ki-67 range (highest minus lowest) was used to determine intraprostatic and intralesion variation. Apparent diffusion coefficient (ADC) values were evaluated in relation to Ki-67.ResultsInterprostatic Ki-67 mean ± standard deviation (SD) values for NCCN low (L), intermediate (I), and high (H) risk patients were 5.1 ± 3.8%, 7.4 ± 6.8%, and 12.0 ± 12.4% (ANOVA P = 0.013). Intraprostatic mean ± SD Ki-67 ranges in L, I, and H risk patients were 2.6 ± 3.6%, 5.3 ± 6.8%, and 10.9 ± 12.3% (ANOVA P = 0.027). Intralesion mean ± SD Ki-67 ranges in L, I, and H risk patients were 1.1 ± 0.9%, 5.2 ± 7.9%, and 8.1 ± 10.8% (ANOVA P = 0.22). ADC values at Ki-67 > and <7.1% were 860 ± 203 and 1036 ± 217, respectively (P = 0.0029).ConclusionsHigh risk patients have significantly higher inter- and intraprostatic Ki-67 heterogeneity. This needs to be considered when utilizing Ki-67 clinically

Molecular Signatures in the Near Infrared Dayside Spectrum of HD 189733b

We have measured the dayside spectrum of HD 189733b between 1.5 and 2.5 microns using the NICMOS instrument on the Hubble Space Telescope. The emergent spectrum contains significant modulation, which we attribute to the presence of molecular bands seen in absorption. We find that water (H2O), carbon monoxide (CO), and carbon dioxide (CO2) are needed to explain the observations, and we are able to estimate the mixing ratios for these molecules. We also find temperature decreases with altitude in the ~0.01 < P < ~1 bar region of the dayside near-infrared photosphere and set an upper limit to the dayside abundance of methane (CH4) at these pressures.Comment: 13 pages, 3 figures. accepted in Astrophysical Journal Letter

Deriving relativistic momentum and energy. II. Three-dimensional case

We generalise a recent derivation of the relativistic expressions for momentum and kinetic energy from the one-dimensional to the three-dimensional case.Comment: 7 page