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Axiom
Published in Paul W. Ross, The Handbook of Software for Engineers and Scientists, 2018
AXIOM,1 a language, compiler, library, interactive shell, and hypertext help system for graphics and symbolic and numeric computation, is an integrated suite of applications for performing numeric and symbolic computations and creating two- and three-dimensional graphs. AXIOM is distinguished from the other major computer algebra systems by having an advanced object-oriented language (with compiler) and a library structured for maximum reuse and natural expression of algorithms. The interactive shell supports a subset of the compiled AXIOM language. The shell interprets user input and calls the appropriate library functions. The system documentation is available on-line in a hypertext help system. With one mouse click, the user can have an AXIOM expression (and all the expressions on which it depends) appearing in the documentation entered into an AXIOM workspace where it will be evaluated. Similarly, if a user clicks on a graph then the full graphics application becomes active and the graph can be interactively transformed. A library browser is fully integrated with the hypertext system.
Sand dune risk assessment in Sabha region, Libya using Landsat 8, MODIS, and Google Earth Engine images
Published in Geomatics, Natural Hazards and Risk, 2018
Biswajeet Pradhan, Ahmed Ali Alazhari Moneir, Ratiranjan Jena
Table 2 presents the weights of the conditioning factors for each expert and the geometric mean of those weights. The experts provided the importance degree in the pairwise table for the factors, and the AHP method was used to calculate the weights in the table. The main reason for choosing geometric mean as opposed to arithmetic mean is the only aggregation form that makes a valid and approximate value by keeping the reciprocity. Geometric mean holds the axiom while the arithmetic mean do not. However, the geometric mean is convenient and easy to implement in AHP method (Althuwaynee et al. 2014a). The geometric mean is exactly a concave aggregator that provides good performance against arithmetic means. By analyzing the similarities and differences among the expert opinions, the correlation analysis confirms that the lowest R2 is 0.29 between the second and third experts. By contrast, others opinions are nearly similar to R2 >0.90. According to the estimated geometric means, land use, and soil seem to be the most influential factors that affect the sand dunes in the study area, whereas rainfall is the least significant factor.
A family of genuine and non-algebraisable C-systems
Published in Journal of Applied Non-Classical Logics, 2021
Mauricio Osorio, Aldo Figallo-Orellano, Miguel Pérez-Gaspar
The proof is done by induction on the length of the theorem proof. First, one can check that all axioms of are tautologies by means of truth-tables. Now, let α be a theorem and let be its proof; and, assume α is not an axiom, otherwise we are done, then there are two previous steps such that , by induction hypothesis and are tautologies, then α is a tautology.
Nearness as context-dependent expression: an integrative review of modeling, measurement and contextual properties
Published in Spatial Cognition & Computation, 2020
Marc Novel, Rolf Grütter, Harold Boley, Abraham Bernstein
Crangle and Suppes (1994) and Suppes (1991) discusses which geometry is appropriate for different prepositions. To model “near” they use absolute geometry, which is a weakened Euclidean geometry with the axiom of parallels removed. Since only congruence is required (for instance qualitative equidistance or ordering of qualitative distance, as investigated by Suppes, Krantz, Luce and Tversky (1989)), such a geometry allows for strict-positiveness and non-symmetry (including weak symmetry). Unfortunately, this approach does not solve the problem of the triangle inequality, which still holds true in absolute geometry.