13,870 research outputs found
The Algorithmic Numbers in Non-Archimedean Numerical Computing Environments
No full textThere are many natural phenomena that can best be described by the
use of infinitesimal and infinite numbers (see e.g. [1, 5, 13, 23]. However,
until now, the Non-standard techniques have been applied to theoretical
models. In this paper we investigate the possibility to implement such
models in numerical simulations. First we define the field of Euclidean
numbers which is a particular eld of hyperreal numbers. Then, we introduce
a set of families of Euclidean numbers, that we have called altogether
algorithmic numbers, some of which are inspired by the IEEE 754 standard
for floating point numbers. In particular, we suggest three formats which
are relevant from the hardware implementation point of view:
the Polynomial Algorithmic Numbers, the Bounded Algorithmic Numbers and the
Truncated Algorithmic Numbers. In the second part of the paper, we show
a few applications of such numbers
Non-Standard Analysis Revisited: an Easy Axiomatic Presentation Oriented Towards the Applications
No full textAlpha-Theory has been introduced in 1995 to provide a simplified version of Robinson’s Non-Standard Analysis which
overcomes the technicalities of symbolic logic. The theory has been improved during the years, and recently it has been
used also to solve practical problems in a pure numerical way, thanks to the introduction of the algorithmic numbers. In this
paper, we introduce Alpha-Theory using a novel axiomatic approach oriented towards real-world applications, to avoid the
need to master mathematical logic and model theory. To corroborate the strong link of this Alpha-Theory axiomatization and
scientific computations, we report numerical illustrative applications never carried out by means of non-standard numbers
within a computer, i.e., the computation of the eigenvalues of a non-Archimedean matrix, some computations related to
non-Archimedean Markov Chains, and the Cholesky factorization of a non-Archimedean matrix. We also highlight the
differences between our numerical routines and pure symbolic approaches: as expected, the former scales better when the
dimension of the problem increases
An Aristotelian notion of size
No full textThe naıve idea of “size” for collections seems to obey both to Aristotle’s Principle: “the whole is greater than its parts” and to Cantor’s Principle: “1-to-1 correspondences preserve size”. Notoriously, Aristotle’s and Cantor’s principles are incompatible for infinite collections.
Cantor’s theory of cardinalities weakens the former principle to “the
part is not greater than the whole”, but the outcoming cardinal arith-
metic is very unusual. It does not allow for inverse operations, and so there is no direct way of introducing infinitesimal numbers. (Sizes are added by means of disjoint unions and multiplied by means of disjoint unions of equinumerous collections.)
Here we maintain Aristotle’s principle, halving instead Cantor’s principle to “equinumerous collections are in 1-1 correspondence”. In
this way we obtain a very nice arithmetic: in fact, our “numerosities” may be taken to be nonstandard integers. These numerosities appear
naturally suited to sets of ordinals, but they depend, for generic sets, on a “labelling” of the universe by ordinals. The problem of finding a canonical way of attaching numerosities to all sets seems to be worth of further investigation
An Euclidean measure of size for mathematical universes
No full textWe show that a measure of size satisfying the ve common notions
of Euclid's Elements can be consistently assumed for all sets in the
universe of ?classical" mathematics. In particular, such a universal
Euclidean measure maintains the ancient principle that ?the whole
is greater than the part". Values are taken in the positive part of a
discretely ordered ring (actually, into a set of hypernatural numbers
of nonstandard analysis) in such a way that measures of disjoint
sums and Cartesian products correspond to sums and products, respectively.
Moreover, universal Euclidean measures can be taken in
such a way that they satisfy a natural continuity property for suitable
(normal) approximations
Environmental life cycle assessment of rice production in northern Italy: a case study from Vercelli
No full textPURPOSE: The study’s objective is to assess the environmental performance of rice production in Northern Italy, in particular in Piedmont, the first Italian and European district for the rice-growing area, and thus identify the most critical hotspots and agricultural processes. In particular, as a case study, a farm located in Vercelli (VC) has been chosen. Subsequently, the study results were compared with other different cultivation practices to evaluate the most sustainable choice. METHODS: The application of the LCA has been performed, highlighting the phases of rice production that have the most significant impact. Then, uncertainty and sensitivity analyses have been made to estimate the robustness of the results and assess the influence of changing some input variables on emission reduction. Finally, multivariate statistical, specifically a principal component analysis (PCA), was conducted to aid the interpretation of the output dataset of this case study. LCA, uncertainty analysis, and sensitivity analysis were performed with SimaPro 9.2.0, using ReCiPe 2016 Midpoint (H) methodology, and PCA with R software. RESULTS AND DISCUSSIONS: The hotspot with the highest environmental load is irrigation, which compared to the other phases impacts more in 15 out of 18 categories, including 12 with impacts greater than + 75%. This is because irrigation causes direct impacts, related to the methanogenesis in rice fields, but also indirect impacts related mainly to the production of the energy mix required to move the large masses of irrigation water. Therefore, different water management systems were compared and results show that the irrigation systems based on intermittent paddy submergence (DSI) could result in − 40% lower impacts, resulting to be the preferable technique over the other irrigation systems analyzed, including the traditional one used in this study. CONCLUSIONS: In order to reduce the environmental impacts related to the irrigation process, a water management system characterized by intermittent flooding of the paddy field (DSI) could be used as it reduces the environmental impacts the most (− 40%), while the least suitable system is one characterized by continuous flooding without drought periods, as it causes the highest impacts. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11367-022-02109-x
The nonlinear Schroedinger equation: Soliton dynamics
No full textIn this paper we investigate the dynamics of solitons occurring in the nonlinear Schroedinger equation when a parameter h → 0.
We prove that under suitable assumptions, the soliton approximately follows the dynamics of a point particle, namely, the motion of its barycenter q(t) satisfies the equation
q'' ( t ) + ∇ V (q(t))=H(t)
where sup_t |H(t)| goes to 0 while h tends to 0
L'architettura militante. Colloquio con Vieri Quilici
No full textNell'articolo in forma di intervista Vieri Quilici ricorda il sodalizio culturale e personale con Manfredo Tafuri. Il pensiero di Tafuri “sulla città sovietica” si evolve parallelamente al lavoro di Quilici sulle avanguardie russe. Un parallelismo che all’inizio corrisponde alle date, del ’65 e del ’68, rispettivamente di "Architettura sovietica contemporanea" e di "Teorie e Storia dell’architettura" e che poi porta a momenti di confronto diretto nel 1976 e 1981, rispettivamente con "Città russa e Città sovietica" e "La sfera e il labirinto". Punto d’incontro diventa allora la rivista “Rassegna Sovietica” dove Quilici pubblica e commenta periodicamente materiali storici d’epoca e dove anche Tafuri nel genn-febbr.’74 pubblica il saggio "Le prime ipotesi di pianificazione urbanistica nella Russia sovietica: Mosca 1918-1924" (poi diventato un importante capitolo de La Sfera e il Labirinto).
Nello studio dell'AUA di piazza Caprettari 70 i due giovani architetti si trovavano spesso a discutere insieme sugli esiti delle rispettive ricerche. Sicuramente decisiva fu per entrambi la riflessione sulle implicazioni critiche della Scuola formalista (di Slutsky in particolare), sull’autonomia del linguaggio rispetto alla propaganda e all’ideologia, e, per quanto riguarda le sorti della Città sovietica, sul controverso drammatico destino del cosiddetto Socialismo in un solo Paese, ecc
MABS validation through repeated execution and data mining analysis
No full textAgent Based Modelling is the most interesting and advanced approach for simulating a complex system: in a social context, the single parts and the whole are often very hard to describe in detail. Besides, there are agent based formalisms which allow to study the emergency of social behaviour with the creation and study of models, known as artificial societies. Thanks to the ever increasing computational power, it's been possible to use such models to create software, based on intelligent agents, which aggregate behaviour is complex and difficult to predict, and can be used in open and distributed systems. Data mining is born in the last decades in order to help users in finding useful knowledge from the otherwise overwhelming amount of data available nowadays from the web and the data collected every day by companies. Data Mining techniques can therefore be the keystone to reveal non-trivial knowledge expressed by the initial assumption used to build the micro-level of the model and the structure of the society of agents that emerged from the simulation
An elementary approach to stochastic differential equations using the infinitesimals.
No full textThe aim of this paper is to evidence two points relative to Nonstandard analysis (NSA): 1. In most applications of NSA to analysis, only elementary facts and techniques of nonstandard calculus seems to be necessary. 2. The advantages of a theory which includes infinitesimals rely more on the possibility of making new models rather than in the proving techniques.
These two points will be illustrated in the theory of Brownian motion which can be considered as a classical model to test the power of the infinitesimal approach. Starting from a naive idea of Brownian motion, we deduce the Fokker-Plank equation in a simple and rigorous way. It is possible to keep every things to a simple level since all the theory of stochastic differential equations is treated as a hyperfinite theory and it is not translated in a "standard model". The only standard object is the final one: the Fokker-Plank equation
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