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By combining the philosophical literature on statistical evidence and the interdisciplinary literature on algorithmic fairness, we revisit recent objections against classification parity in light of causal analyses of algorithmic fairness and the distinction between predictive and diagnostic evidence. We focus on trial proceedings as a black-box classification algorithm in which defendants are sorted into two groups by convicting or acquitting them. We defend a novel principle, causal equal protection, that combines classification parity with the causal approach. In the do-calculus, causal equal protection requires that individuals should not be subject to uneven risks of classification error because of their protected or socially salient characteristics. The explicit use of protected characteristics, however, may be required if it equalizes these risks., Comment: 16 pages, 5 figures

We shortly review the historical path of Morrey-Campanato functional spaces and the fundamentals of Carnot groups. Then, we merge these two topics, by recovering several classical results concerning regularity of Morrey-Campanato spaces in the framework of Carnot groups.

In this paper we prove existence of solutions to Schr\"odinger-Maxwell type systems involving mixed local-nonlocal operators. Two different models are considered: classical Schr\"odinger-Maxwell equations and Schr\"odinger-Maxwell equations with a coercive potential, and the main novelty is that the nonlocal part of the operator is allowed to be nonpositive definite according to a real parameter. We then provide a range of parameter values to ensure the existence of solitary standing waves, obtained as Mountain Pass critical points for the associated energy functionals., Comment: arXiv admin note: text overlap with arXiv:2303.11663

Let $\mathcal{P}$ be the set of points of a finite-dimensional projective space over a local field $F$, endowed with the topology $\tau$ naturally induced from the canonical topology of $F$. Intuitively, continuous incidence abelian group structures on $\mathcal{P}$ are abelian group structures on $\mathcal{P}$ preserving both the topology $\tau$ and the incidence of lines with points. We show that the real projective line is the only finite-dimensional projective space over an Archimedean local field which admits a continuous incidence abelian group structure. The latter is unique up to isomorphism of topological groups. In contrast, in the non-Archimedean case we construct continuous incidence abelian group structures in any dimension $n \in \mathbb{N}$. We show that if $n>1$ and the characteristic of $F$ does not divide $n+1$, then there are finitely many possibilities up to topological isomorphism and, in any case, countably many., Comment: 22 pages

RNA folding prediction remains challenging, but can be also studied using a topological mathematical approach. In the present paper, the mathematical method to compute the topological classification of RNA structures and based on matrix field theory is shortly reviewed, as well as a computational software, McGenus, used for topological and folding predictions. Additionally, two types of analysis are performed: the prediction results from McGenus are compared with topological information extracted from experimentally-determined RNA structures, and the topology of RNA structures is investigated for biological significance, in both evolutionary and functional terms. Lastly, we advocate for more research efforts to be performed at intersection of physics-mathematics and biology, and in particular about the possible contributions that topology can provide to the study of RNA folding and structure., Comment: 11 pages, 9 figures

Classical results concerning Klein-Gordon-Maxwell type systems are shortly reviewed and generalized to the setting of mixed local-nonlocal operators, where the nonlocal one is allowed to be nonpositive definite according to a real parameter. In this paper, we provide a range of parameter values to ensure the existence of solitary (standing) waves, obtained as Mountain Pass critical points for the associated energy functionals in two different settings, by considering two different classes of potentials: constant potentials and continuous, bounded from below, and coercive potentials., Comment: 26 pages, 2 figures

In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, together with a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey-predator or rumor spreading., Comment: 18 pages, 4 figures

Contra Dardashti, Th\'ebault, and Winsberg (2017), this paper defends an analysis of arguments from analogue simulations as instances of a familiar kind of inductive inference in science: arguments from material analogy (Hesse, Models and Analogies in Science, Univ Notre Dame Press, 1963). When understood in this way, the capacity of analogue simulations to confirm hypotheses about black holes can be deduced from a general account - fully consistent with a Bayesian standpoint - of how ordinary arguments from material analogy confirm. The proposed analysis makes recommendations about what analogue experiments are worth pursuing that are more credible than Dardashti, Hartmann, Th\'ebault, and Winsberg's (2019). It also offers a more solid basis for addressing the concerns by Crowther, Linneman, and W\"utrich (2019), according to which analogue simulations are incapable of sustaining hypotheses concerning black hole radiation., Comment: 26 pages, 1 figure

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., Comment: 25 pages, 2 figures

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., Comment: 20 pages, 2 figures

The testimony and practice of notable mathematicians indicate that there is an important phenomenological and epistemological difference between superficial and deep analogies in mathematics. In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of 'suggesting' a conjecture in the psychological sense). The proposed conditions generalize the criteria put forward by Hesse (1963) in her influential work on analogical reasoning in the empirical sciences. By reference to several case-studies, we argue that the account proposed in this paper does a better job in vindicating the use of analogical inference in mathematics than the prominent alternative defended by Bartha (2009). Moreover, our proposal offers novel insights into the practice of extending to the infinite case mathematical properties known to hold in finite domains., Comment: 35 pages, 4 figures. Forthcoming in Philosophia Mathematica

The classical Lanchester's model is shortly reviewed and analysed, with particular attention to the critical issues that intrinsically arise from the mathematical formalization of the problem. We then generalize a particular version of such a model describing the dynamics of warfare when three or more armies are involved in the conflict. Several numerical simulations are provided.

A new method to compute Schauder Estimates for multidimensional fourth order heat-type equations is proposed. In particular, we show how knowing Schauder or Sobolev estimates for the one-dimensional fourth order heat equation allows to derive their multidimensional analogs for equations with time inhomogeneous coefficients with the same constants as in the case of the one-dimensional heat equation. Our method relies on a merger between (Krylov and Priola, 2017), where they actually showed the same result for the classical second order heat equation and (Funaki, 1979), where a probabilistic construction of solutions for the fourth order heat equation is presented., Comment: We have found an error in the argument, which require a more deep analysis

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., Comment: 15 pages, 1 figure

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., Comment: 10 pages, 4 figures

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., Comment: 8 pages, 5 figures

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 V\^ilcu and collaborators. A similar characterization is given for a more general production function, namely the Kadiyala production function, in the case of developable surfaces.

A Feynman path integral formula for the Schr\"odinger 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 Schr\"odinger and heat equation with magnetic field., Comment: 43 pages

The Ogawa stochastic integral is shortly reviewed and formulated in the framework of abstract Wiener spaces. The condition of universal Ogawa integrability in the multidimensional case is investigated, proving that it cannot hold in general without the introduction of a "renormalization term". Explicit examples are provided.