Rooted Maximum Agreement Supertrees

Given a set $\T$ of rooted, unordered trees, where each $T_i \in \T$ is distinctly leaf-labeled by a set $\Lambda(T_i)$ and where the sets $\Lambda(T_i)$ may overlap, the maximum agreement supertree problem~(MASP) is to construct a distinctly leaf-labeled tree $Q$ with leaf set $\Lambda(Q) \subseteq $\cup$_{T_i \in \T} \Lambda(T_i)$ such that $|\Lambda(Q)|$ is maximized and for each $T_i \in \T$, the topological restriction of $T_i$ to $\Lambda(Q)$ is isomorphic to the topological restriction of $Q$ to $\Lambda(T_i)$. Let $n = \left| $\cup$_{T_i \in \T} \Lambda(T_i)\right|$, $k = |\T|$, and $D = \max_{T_i \in \T}\{\deg(T_i)\}$. We first show that MASP with $k = 2$ can be solved in $O(\sqrt{D} n \log (2n/D))$ time, which is $O(n \log n)$ when $D = O(1)$ and $O(n^{1.5})$ when $D$ is unrestricted. We then present an algorithm for MASP with $D = 2$ whose running time is polynomial if $k = O(1)$. On the other hand, we prove that MASP is NP-hard for any fixed $k \geq 3$ when $D$ is unrestricted, and also NP-hard for any fixed $D \geq 2$ when $k$ is unrestricted even if each input tree is required to contain at most three leaves. Finally, we describe a polynomial-time $(n/\!\log n)$-approximation algorithm for MASP.     

On the relevance of non-standard theories of uncertainty in modeling and pooling expert opinions

Wu, Apostolakis and Okrent¹ have recently analyzed the current status of emerging alternatives to classical probabilistic methods in the modeling and pooling of expert opinions in safety analysis of engineering systems. They have pointed out some difficulties faced by these theories, due to their relative lack of maturity. This paper pursues the investigation so as to clarify some aspects of belief functions and possibility theories, and also to point out the need for further research. A comparison between the mathematical models of expert opinion pooling offered by Bayesian probabilities, belief functions and possibility theory is carried out. It is proved that the Bayesian approach that these authors advocate suffers from the same numerical stability problems as possibilistic and evidential rules of combination in the presence of strongly conflicting information due to their strong structural similarities. The problem of dependence between experts is briefly addressed. The other main point of this paper is that a single combination rule cannot reasonably address all situations where expert opinions must be pooled. It is suggested that the framework of possibility and evidence theories offers a more flexible framework for representing and combining subjective uncertain judgments than the one of subjective probability alone although some progress is required to reach the maturity of the Bayesian theory.     

Nonlinear state affine identification methods: Applications to electrical power plants

A nonlinear identification method by simple state-space models is proposed in the discrete-time case. This approach is suggested by theoretical results similar to the approximation property known for a long time for Volterra series. The proposed technique uses extensions with polynomial inputs of bilinear systems called state-affine systems. This modelling requires a small computing time and enables us to obtain nonlinear models which can be easily implemented on microprocessors and remain valid over a wide range of operating conditions. Several physical examples concerning electric power plants are also reported.     

Periodic-polynomial interpretation for structural properties of linear periodic discrete-time systems

The structural properties of linear periodic discrete-time systems are analyzed in the periodic polynomial representation. It is shown that the classical polynomial approach for linear time-invariant systems can be extended to periodic systems. New definitions and properties are given in terms of skew polynomial rings and periodic difference algebra. Necessary and sufficient conditions for the characterization of reachability, controllability, observability and reconstructibility are given in this framework.     

Numerical techniques for the calculation of binary phase diagrams

Numerical techniques are presented for the calculation of binary phase diagrams. Most compositions are obtained directly through a Taylor series expansion of the composition with respect to temperature. The series is limited to the second order term and analytic expressions of the slopes and curvatures of phase boundaries have been developed for its use. A Newton-Raphson iteration technique tests the convergence of the results. The method is efficient and accommodates various formalisms for the free energy functions. The series is not applicable to very dilute solutions of a component, near the critical point of a miscibility gap or in the vicinity of the congruent transformation point of a compound; in these cases, alternate equations have been derived. Formerly Graduate Student, Department of Metallurgy and Materials Science, Carnegie-Mellon University,     

Sampling frequency and accuracy of SPM flux estimates in two contrasted drainage basins

The present paper is based on discharges and suspended particulate matter concentrations from a 9-years high-resolution database for the Garonne River (large plain river) covering contrasted hydrologic years, and a 12-months high frequency sampling for the Nivelle River (small mountainous river). Annual SPM fluxes in the Garonne River range from 0.6 x 10(6) t year(-1) (1997) to 3.9 x 10(6) t year(-1) (1996). In contrast, the Nivelle River transported 11 x 10(3) t year(-1) from December 1995 to December 1996. From the long-term observation of the Garonne River an empirical relation between SPM* (discharge-weighted mean annual SPM concentrations) and annual discharge was established. This relation allows estimating annual SPM fluxes for the Garonne River with less than 30% deviation from reference values for the whole range of mean annual discharge observed during the past decade. Specific (=area-normalized) annual SPM fluxes (YSPM) range from 11 to 74 t km(-2) year(-1) for the Garonne River. Comparison of these results with YSPM of the Nivelle River (69 t km(-2) year(-1) in 1996) suggests that interannual hydrological variations may have a greater impact on fluvial SPM transport than basin-specific parameters. By extracting individual SPM concentrations and corresponding discharge values from the database, different sampling frequencies were simulated and resulting SPM fluxes were then compared to reference fluxes derived from the complete database. If a deviation of simulated flux estimates from reference fluxes lower than +/-20% is accepted, the Garonne River (large plain river) must be sampled at least every 3 days (10 samples per month) and the Nivelle River every 7 h (approx. 100 samples per month). For the Garonne River this minimum sampling frequency is valid for all contrasted hydrologic years of the observation period. Below these minimum sampling frequencies, annual SPM flux estimates may greatly differ from reference fluxes (up to 200%) and there is high probability of systematic underestimation. Consequently, annual SPM flux estimates for the Garonne River derived from the empirical relation (SPM*-annual discharge) are likely to be more satisfactory (errors <30%) than estimates based on sampling frequencies lower than the minimum frequency. These findings underline the need of adapted sampling strategies for erosion assessment, reliable chemical (e.g. nutrients and pollutants) mass balances and characterisation of fluvial transport mechanisms in the world's contrasted watersheds.