lca_algebraic: a library bringing symbolic calculus to LCA for comprehensive sensitivity analysis

Purpose In this paper, we present new tools to ease the analysis of the effect of variability and uncertainty on life cycle assessment (LCA) results. Methods The tools consist of a standard protocol and an open-source library: lca_algebraic. This library, written in Python and based on the framework Brightway2 (Mutel in J Open Source Softw 2(12):236, 2017) provides functions to support sensitivity analysis by bringing symbolic calculus to LCA. The use of symbolic calculus eases the definition of parametric inventories and enables a very fast evaluation of impacts by factorizing the background activities. Thanks to this processing speed, a large number of Monte Carlo simulations can be generated to evaluate the variation of the impacts and apply advanced statistic tools such as Sobol indices to quantify the contribution of each parameter to the final variance (Sobol in Math Comput Simul 55(1–3):271–280, 2001). An additional algorithm uses the key parameters, identified from their high Sobol indices, to generate simplified arithmetic models for fast estimates of LCA results. Results and discussion The protocol and library were validated through their application to the assessment of impacts of mono crystalline photovoltaic (PV) systems. A comprehensive sensitivity analysis was performed based on the protocol and the complementary functions provided by lca_algebraic. The proposed tools helped building a detailed parametric reference LCA model of the PV system to identify the range of variation of multi-criterion LCA results and the key foreground-related parameters explaining these variations. Based on these key parameters, we generated simplified arithmetic models for quick and simple multi-criteria environmental assessments to be used by non-expert LCA users. The resulting models are both compact and aligned with the reference parametric LCA model of crystalline silicon PV systems. Conclusion This work brings powerful and practical tools to the LCA community to better understand, identify, and quantify the sources of variation of environmental impacts and produce simplified models to spread the use of LCA among non-experts. The library mainly explores the uncertainties of the foreground activities. Further work could also integrate the uncertainty of background activities, described, for example, by pedigree matrices.

Symbolic Computation to Solving an Irrational Equation on Based Symmetric Polynomials Method

In this article, we examine the use of symmetry groups for modeling applied problems through computer symbolic calculus. We consider the problem of solving radical equations symbolically using computer mathematical packages. We propose some methods to obtain a correct analytical solution for this class of equations by means of the Mathcad package. The application of symmetric polynomials is proposed to ensure a correct approach to the solution. Issues on the solvability based on the physical sense of a problem are discussed. Common errors in solving radical equations related to the specificity of the computer usage are analyzed. Provable electrical and geometrical problems are illustrated as example.

Operator Symbols and Operator Indices

We suggest a certain variant of symbolic calculus for special classes of linear bounded operators acting in Banach spaces. According to the calculus we formulate an index theorem and give applications to elliptic pseudo-differential operators on smooth manifolds with non-smooth boundaries.

Asymptotics of fluctuations in Crump‒Mode‒Jagers processes: the lattice case

Consider a supercritical Crump‒Jagers process in which all births are at integer times (the lattice case). Let μ̂(z) be the generating function of the intensity of the offspring process, and consider the complex roots of μ̂(z)=1. The root of smallest absolute value is e-α=1∕m, where α>0 is the Malthusian parameter; let γ* be the root of second smallest absolute value. Subject to some technical conditions, the second-order fluctuations of the age distribution exhibit one of three types of behaviour: (i) when γ*>e-α∕2=m-1∕2, they are asymptotically normal; (ii) when γ*=e-α∕2, they are still asymptotically normal, but with a larger variance; and (iii) when γ*<e-α∕2, the fluctuations are in general oscillatory and (degenerate cases excluded) do not converge in distribution. This trichotomy is similar to what has been observed in related situations, such as some other branching processes and for Pólya urns. The results lead to a symbolic calculus describing the limits. The asymptotic results also apply to the total of other (random) characteristics of the population.

THE LOGIC OF LEIBNIZ’S GENERALES INQUISITIONES DE ANALYSI NOTIONUM ET VERITATUM

The Generales Inquisitiones de Analysi Notionum et Veritatum is Leibniz’s most substantive work in the area of logic. Leibniz’s central aim in this treatise is to develop a symbolic calculus of terms that is capable of underwriting all valid modes of syllogistic and propositional reasoning. The present paper provides a systematic reconstruction of the calculus developed by Leibniz in the Generales Inquisitiones. We investigate the most significant logical features of this calculus and prove that it is both sound and complete with respect to a simple class of enriched Boolean algebras which we call auto-Boolean algebras. Moreover, we show that Leibniz’s calculus can reproduce all the laws of classical propositional logic, thus allowing Leibniz to achieve his goal of reducing propositional reasoning to algebraic reasoning about terms.