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Nicolò Cangiotti

Publications and source records attributed to Nicolò Cangiotti.

26 records · Page 2Linked to original sources
arXiv2022

Confirming Mathematical Conjectures by Analogy

Analogy has received attention as a form of inductive reasoning in the empirical sciences. However, its role in pure mathematics has received less consideration. This paper provides an account of how an analogy with a more familiar mathematical domain can contribute to the confirmation of a mathematical conjecture. By reference to case-studies, we propose a distinction between an incremental and a non-incremental form of confirmation by mathematical analogy. We offer an account of the former within the popular framework of Bayesian confirmation theory. As for the non-incremental notion, we defend its role in rationally informing the prior credences of mathematicians in those circumstances in which no new mathematical evidence is introduced. The resulting 'hybrid' framework captures many important aspects of the use of analogical inference in the realm of pure mathematics.

math.HO↗
arXiv2022

Group Fairness Is Not Derivable From Justice: a Mathematical Proof

We argue that an imperfect criminal law procedure cannot be group-fair, if 'group fairness' involves ensuring the same chances of acquittal or convictions to all innocent defendants independently of their morally arbitrary features. We show mathematically that only a perfect procedure (involving no mistake), a non-deterministic one, or a degenerate one (everyone or no one is convicted) can guarantee group fairness, in the general case. Following a recent proposal, we adopt a definition of group fairness, requiring that individuals who are equal in merit ought to have the same statistical chances of obtaining advantages and disadvantages, in a way that is statistically independent of any of their feature that does not count as merit. We explain by mathematical argument that the only imperfect procedures offering an a-priori guarantee of fairness in relation to all non-merit trait are lotteries or degenerate ones (i.e., everyone or no one is convicted). To provide a more intuitive point of view, we exploit an adjustment of the well-known ROC space, in order to represent all possible procedures in our model by a schematic diagram. The argument seems to be equally valid for all human procedures, provided they are imperfect. This clearly includes algorithmic decision-making, including decisions based on statistical predictions, since in practice all statistical models are error prone.

cs.CY↗
arXiv2022

Normal and pseudonormal numbers

After a short review of the historical milestones on normal numbers, we introduce the Borel numbers as the reals admitting a probability function on their different bases representations. In this setting, we provide two probabilistic characterizations of normality based on the stochastic independence of their digits. Finally, we define the pseudonormality condition, which is satisfied by normal numbers and may be evaluated in a finite number of steps.

math.NT↗
arXiv2020

Exact solutions for a Solow-Swan model with non-constant returns to scale

The Solow-Swan model is shortly reviewed from a mathematical point of view. By considering non-constant returns to scale, we obtain a general solution strategy. We then compute the exact solution for the Cobb-Douglas production function, for both the classical model and the von Bertalanffy model. Numerical simulations are provided.

econ.TH↗
arXiv2020

Notes on a conformal characterization of $2$-dimensional Lorentzian manifolds with constant Ricci scalar curvature

We present a characterization of $2$-dimensional Lorentzian manifolds with constant Ricci scalar curvature. It is well known that every $2$-dimensional Lorentzian manifolds is conformally flat, so we rewrite the Ricci scalar curvature in terms of the conformal factor and we study the solutions of the corresponding differential equations. Several remarkable examples are provided.

math-ph↗
arXiv2020

A geometric characterization of VES and Kadiyala-type production functions

The basic concepts of the differential geometry are shortly reviewed and applied to the study of VES production function in the spirit of the works of Vîlcu and collaborators. A similar characterization is given for a more general production function, namely the Kadiyala production function, in the case of developable surfaces.

econ.TH↗
arXiv2019

A rigorous mathematical construction of Feynman path integrals for the Schrödinger equation with magnetic field

A Feynman path integral formula for the Schrödinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functional. This provides a natural explanation for the appearance of a Stratonovich integral in the path integral formula for both the Schrödinger and heat equation with magnetic field.

math-ph↗