preprints

23. Multivariate Dirichlet Moments and a Polychromatic Ewens Sampling Formula

    L. Dello Schiavo, F. Quattrocchi

    We present an elementary non-recursive formula for the multivariate moments of the Dirichlet distribution on the standard simplex, in terms of the pattern inventory of the moments' exponents. We obtain analog formulas for the multivariate moments of the Dirichlet—Ferguson and Gamma measures. We further introduce a polychromatic analogue of Ewens sampling formula on colored integer partitions, discuss its relation with suitable extensions of Hoppe's urn model and of the Chinese restaurant process, and prove that it satisfies an adapted notion of consistency in the sense of Kingman.

24. Persistence of Rademacher-type and Sobolev-to-Lipschitz properties

    L. Dello Schiavo, K. Suzuki

    We consider the Rademacher- and Sobolev-to-Lipschitz-type properties for arbitrary quasi-regular strongly local Dirichlet spaces. We discuss the persistence of these properties under localization, globalization, transfer to weighted spaces, tensorization, and direct integration. As byproducts we obtain: necessary and sufficient conditions to identify a quasi-regular strongly local Dirichlet form on an extended metric topological $\sigma$-finite possibly non-Radon measure space with the Cheeger energy of the space; the tensorization of intrinsic distances; the tensorization of the Varadhan short-time asymptotics.

25. Wasserstein geometry and Ricci curvature bounds for Poisson spaces

    L. Dello Schiavo, R. Herry, K. Suzuki

    Let $\Upsilon$ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure $\pi$. We study the geometry of $\Upsilon$ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on $\mathscr{P}_1(\Upsilon)$, the space of probability measures over $\Upsilon$ with finite first moment, and we construct an extended distance $\mathcal W$ on $\mathscr{P}_1(\Upsilon)$. The distance $\mathcal W$ corresponds, in our setting, to the Benamou—Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with $\mathcal W$. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein—Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has a Ricci curvature, in the entropic sense, bounded below by $1$; (c) the distance $\mathcal W$ satisfies an HWI inequality.

26. Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous

    L. Dello Schiavo, R. Herry, E. Kopfer, K.-T. Sturm

    For an arbitrary dimension $n$, we study: (a) the Polyharmonic Gaussian Field $h_L$ on the discrete torus $\mathbb{T}^n_L = \frac{1}{L} \mathbb{Z}^{n} / \mathbb{Z}^{n}$, that is the random field whose law on $\mathbb{R}^{\mathbb{T}^{n}_{L}}$ given by \begin{equation*} c_n\, e^{-b_n\|(-\Delta_L)^{n/4}h\|^2} dh, \end{equation*} where $dh$ is the Lebesgue measure and $\Delta_{L}$ is the discrete Laplacian; (b) the associated discrete Liouville Quantum Gravity measure associated with it, that is the random measure on $\mathbb{T}^{n}_{L}$ \begin{equation*} \mu_{L}(dz) = \exp \Big( \gamma h_L(z) - \frac{\gamma^{2}}{2} \mathbf{E} h_{L}(z) \Big) dz, \end{equation*} where $\gamma$ is a regularity parameter. As $L\to\infty$, we prove convergence of the fields $h_L$ to the Polyharmonic Gaussian Field $h$ on the continuous torus $\mathbb{T}^n = \mathbb{R}^{n} / \mathbb{Z}^{n}$, as well as convergence of the random measures $\mu_L$ to the LQG measure $\mu$ on $\mathbb{T}^n$, for all $|\gamma| < \sqrt{2n}$.

27. Configuration spaces over singular spaces — II. Curvature

    L. Dello Schiavo, K. Suzuki

    This is the second paper of a series on configuration spaces $\Upsilon$ over singular spaces $X$. Here, we focus on geometric aspects of the extended metric measure space $(\Upsilon, \mathsf{d}_{\Upsilon}, \mu)$ equipped with the $L^2$-transportation distance $\mathsf{d}_{\Upsilon}$, and a mixed Poisson measure $\mu$. Firstly, we establish the essential self-adjointness and the $L^p$-uniqueness for the Laplacian on $\Upsilon$ lifted from $X$. Secondly, we prove the equivalence of Bakry–Émery curvature bounds on $X$ and on $\Upsilon$, without any metric assumption on $X$. We further prove the Evolution Variation Inequality on $\Upsilon$, and introduce the notion of synthetic Ricci-curvature lower bounds for the extended metric measure space $\Upsilon$. As an application, we prove the Sobolev-to-Lipschitz property on $\Upsilon$ over singular spaces $X$, originally conjectured in the case when $X$ is a manifold by M. Röckner and A. Schield. As a further application, we prove the $L^\infty$-to-$\mathsf{d}_{\Upsilon}$-Lipschitz regularization of the heat semigroup on $\Upsilon$ and gives a new characterization of the ergodicity of the corresponding particle systems in terms of optimal transport.

28. Configuration spaces over singular spaces — I. Dirichlet-Form and Metric Measure Geometry

    L. Dello Schiavo, K. Suzuki

    This paper is the first in a series on configuration spaces over singular spaces. Here, we construct a canonical differential structure on the configuration space $\Upsilon$ over a singular base space $X$ and with a general invariant measure $\mu$ on $\Upsilon$.
    We first present an analytic structure on $\Upsilon$, constructing a strongly local Dirichlet form $\mathcal E$ on $L^2(\Upsilon, \mu)$ for $\mu$ in a large class of probability measures. We then investigate the geometric structure of the extended metric measure space $\Upsilon$ endowed with the $L^2$-transportation extended distance $\mathsf{d}_\Upsilon$ and with the measure $\mu$. By establishing various Rademacher- and Sobolev-to-Lipschitz-type properties for $\mathcal E$, we finally provide a complete identification of the analytic and the geometric structure — the canonical differential structure induced on $\Upsilon$ by $X$ and $\mu$ — showing that $\mathcal E$ coincides with the Cheeger energy of $(\Upsilon,\mathsf{d}_\Upsilon,\mu)$ and that the intrinsic distance of $\mathcal E$ coincides with $\mathsf{d}_\Upsilon$.
    The class of base spaces to which our results apply includes sub-Riemannian manifolds, $\mathsf{RCD}$ spaces, locally doubling metric measure spaces satisfying a local Poincaré inequality, and path/loop spaces over Riemannian manifolds; as for $\mu$, our results include Campbell measures and quasi-Gibbs measures, in particular: Poisson measures, canonical Gibbs measures, as well as some determinantal/permanental point processes (sine$_\beta$, Airy$_\beta$, Bessel$_{\alpha, \beta}$, Ginibre).
    A number of applications to interacting particle systems and infinite-dimensional metric measure geometry are also discussed. In particular, we prove the quasi-regularity of $\mathcal E$ and consequently the existence of a Markov diffusion associated to $\mathcal E$ describing a $\mu$-invariant particle system. We further show the universality of the $L^2$-transportation distance $\mathsf{d}_\Upsilon$ for the Varadhan short-time asymptotics for diffusions on $\Upsilon$, regardless of the choice of $\mu$. Our approach does not rely on any relation between $\mu$ and a possible smooth structure on $X$. In particular, we assume no quasi-invariance property of $\mu$ w.r.t. actions of any group of transformations of $X$. Many of our results are new even in the case of standard Euclidean spaces.

29. Conformally Invariant Random Fields, Quantum Liouville Measures, and Random Paneitz Operators on Riemannian Manifolds of Even Dimension

    L. Dello Schiavo, R. Herry, E. Kopfer, K.-T. Sturm

    For large classes of even-dimensional Riemannian manifolds $(M,g)$, we construct and analyze conformally invariant random fields. These centered Gaussian fields $h=h_g$, called co-polyharmonic Gaussian fields, are characterized by their covariance kernels $k$ which exhibit a precise logarithmic divergence: $\big| k(x,y)-\log\frac{1}{d(x,y)} \big| \leq C$. They share the fundamental quasi-invariance property under conformal transformations: if $g'=e^{2\varphi}g$, then $$ h_{g'}\overset{\rm d}= e^{n\varphi} h_g-C\cdot \mathrm{vol}_{g'} $$ with an appropriate random variable $C=C_\varphi$. In terms of the co-polyharmonic Gaussian field $h$, we define the quantum Liouville measure, a random measure on $M$, heuristically given as $$ \mathrm{d}\mu_g^h(x):= e^{\gamma h(x)-\frac{\gamma^2}2k(x,x)}\,\mathrm{dvol}_g(x)\,, $$ and rigorously obtained as almost sure weak limit of the right-hand side with $h$ replaced by suitable regular approximations $h_\ell, \ell\in\mathbb N$. These measures share a crucial quasi-invariance property under conformal transformations: if $g'=e^{2\varphi}g$, then $$ \mathrm{d}\mu^{h'}_{g'}(x)\overset{\rm (d)} = e^{F^h(x)}\,\mathrm{d}\mu_g^h(x) $$ for an explicitly given random variable $F^h(x)$. In terms on the quantum Liouville measure, we define the Liouville Brownian motion on $M$ and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimensions with an ansatz based on Branson's $Q$-curvature: we give a rigorous meaning to the Polyakov—Liouville measure $$ \mathrm{d}\boldsymbol{\nu}^*_g(h) =\frac{1}{Z^*_g} \exp\left(- \int \Theta\,Q_g h + m\, e^{\gamma h} \mathrm{dvol}_g\right) \exp\left(-\frac{a_n}{2} {\mathfrak p}_g(h,h)\right) \mathrm{d}h $$ for suitable positive constants $\Theta$, $m$, $\gamma$ and $a_n$, and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact $2$-dimensional Riemannian manifolds, all compact non-negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even-dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel $k$ — defined as the Green kernel for the GJMS operator on $(M,g)$ — rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary compact manifolds.

publications

1. Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces
   Tran. AMS (2024) online-first

   L. Dello Schiavo, J. Maas, F. Pedrotti

   journal |

   This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka—Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on $\mathbb{R}^n$. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on $\mathbb{R}^n$.

2. A Characterization of Maps of Bounded Compression
   Math. Commun. 29(1):137-142 (2024)

   L. Dello Schiavo

   journal |

   A measurable map between measure spaces is shown to have bounded compression if and only if its image via the measure-algebra functor is Lipschitz-continuous w.r.t. the measure-algebra distances. This provides a natural interpretation of maps of bounded compression/deformation by means of the measure-algebra functor and corroborates the assertion that maps of bounded deformation are the natural class of morphisms for the category of complete and separable metric measure spaces.

3. A Discovery Tour in Random Riemannian Geometry
   Potential Anal. (2024) online-first

   L. Dello Schiavo, E. Kopfer, K.-T. Sturm

   journal |

   We study random perturbations of Riemannian manifolds $(\mathsf{𝖬},\mathsf{𝗀})$ by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields $h^\bullet\colon \omega\mapsto h^\omega$ will act on the manifolds via conformal transformation $\mathsf{g}\mapsto \mathsf{g}^\omega\colon\!= e^{2h^\omega}\mathsf{g}$. Our focus will be on the regular case with Hurst parameter $H>0$, the celebrated Liouville geometry in two dimensions being borderline. We want to understand how basic geometric and functional analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap will change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise. Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.

4. Scaling Limits of Random Walks, Harmonic Profiles, and Stationary Non-Equilibrium States in Lipschitz Domains
   Ann. Appl. Probab. 34(2): 1789-1845 (2024)

   L. Dello Schiavo, L. Portinale, F. Sau

   journal |

   We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain $\Omega$, with both fast and slow boundary. For the random walks on $\Omega$ dual to SEP/SIP we establish: a functional-CLT-type convergence to the Brownian motion on $\Omega$ with either Neumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality) boundary conditions; the discrete-to-continuum convergence of the corresponding harmonic profiles. As a consequence, we rigorously derive the hydrodynamic and hydrostatic limits for SEP/SIP on $\Omega$, and analyze their stationary non-equilibrium fluctuations.

5. A Mecke-type characterization of the Dirichlet—Ferguson measure
   Electron. Commun. Probab. 28: 1-12 (2023)

   L. Dello Schiavo, E. Lytvynov

   journal |

   We prove a characterization of the Dirichlet—Ferguson measure over an arbitrary finite diffuse measure space. We provide an interpretation of this characterization in analogy with the Mecke identity for Poisson point processes.

6. Ergodic decompositions of Dirichlet forms under order isomorphisms
   J. Evol. Equ. 23 (2023)

   L. Dello Schiavo, M. Wirth

   journal |

   We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.

7. Effective Contraction of Skinning Maps
   Proc. Amer. Math. Soc. 9: 445-459 (2022)

   T. Cremaschi, L. Dello Schiavo

   journal |

   Using elementary hyperbolic geometry, we give an explicit formula for the contraction constant of the skinning map over moduli spaces of relatively acylindrical hyperbolic manifolds.

8. The Dirichlet-Ferguson Diffusion on the Space of Probability Measures over a Closed Riemannian Manifold
   Ann. Probab. 50(2): 591-648 (2022)

   L. Dello Schiavo

   journal |

   We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension $d\geq 2$. The process is associated with the Dirichlet form defined by integration of the Wasserstein gradient w.r.t. the Dirichlet-Ferguson measure, and is the counterpart on multi-dimensional base spaces to the Modified Massive Arratia Flow over the unit interval described in V. Konarovskyi, M.-K. von Renesse, Comm. Pure Appl. Math., 72, 0764-0800 (2019). Together with two different constructions of the process, we discuss its ergodicity, invariant sets, finite-dimensional approximations, and Varadhan short-time asymptotics.

9. Sobolev-to-Lipschitz Property on $\mathsf{QCD}$-spaces and Applications
   Math. Ann. 384: 1815–1832 (2022)

   L. Dello Schiavo, K. Suzuki

   journal |

   We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature dimension condition recently introduced in E. Milman, The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifolds, Comm. Pure Appl. Math. (to appear). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These result apply in particular to large classes of (ideal) sub-Riemannian manifolds.

10. Rademacher-type Theorems and Varadhan Short-Time Asymptotics for Local Dirichlet Spaces
    J. Funct. Anal. 281(11): 109234 (2021)

    L. Dello Schiavo, K. Suzuki

    journal |

    We extensively discuss the Rademacher and Sobolev-to-Lipschitz properties for generalized intrinsic distances on strongly local Dirichlet spaces possibly without square field operator. We present many non-smooth and infinite-dimensional examples. As an application, we prove the integral Varadhan short-time asymptotic with respect to a given distance function for a large class of strongly local Dirichlet forms.

11. Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications
    Potential Anal. 58:573-615 (2023) online-first (2021)

    L. Dello Schiavo

    journal |

    We study superpositions and direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin, Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.

12. A Rademacher-type Theorem on $L^2$-Wasserstein Spaces over Closed Riemannian Manifolds
    J. Funct. Anal. 278(6):108397 (2020)

    L. Dello Schiavo

    journal |

    Let $\mathbb P$ be any Borel probability measure on the $L^2$-Wasserstein space $(\mathscr{P}_2(M), W_2)$ over a closed Riemannian manifold $M$. We consider the Dirichlet form $\mathcal E$ induced by $\mathbb P$ and by the Wasserstein gradient on $\mathscr{P}_2(M)$. Under natural assumptions on $\mathbb P$, we show that $W_2$-Lipschitz functions on $\mathscr{P}_2(M)$ are contained in the Dirichlet space $\mathscr{D}(\mathcal E)$ and that $W_2$ is dominated by the intrinsic metric induced by $\mathcal E$. We illustrate our results by giving several detailed examples.

13. Characteristic Functionals of Dirichlet Measures
    Electron. J. Probab. 24(155):1-38 (2019)

    L. Dello Schiavo

    journal |

    We compute the characteristic functional of the Dirichlet–Ferguson measure over a locally compact Polish space and prove continuous dependence of the random measure on the parameter measure. In finite dimension, we identify the dynamical symmetry algebra of the characteristic functional of the Dirichlet distribution with a simple Lie algebra of type $A$. We study the lattice determined by characteristic functionals of categorical Dirichlet posteriors, showing that it has a natural structure of weight Lie algebra module and providing a probabilistic interpretation. A partial generalization to the case of the Dirichlet–Ferguson measure is also obtained.

14. Complete integrability of information processing by biochemical reactions
    Sci. Rep. 6:36314 (2016)

    E. Agliari, A. Barra, L. Dello Schiavo, A. Moro

    journal |

    Statistical mechanics provides an effective framework to investigate information processing in biochemical reactions. Within such framework far-reaching analogies are established among (anti-) cooperative collective behaviors in chemical kinetics, (anti-)ferromagnetic spin models in statistical mechanics and operational amplifiers/flip-flops in cybernetics. The underlying modeling – based on spin systems – has been proved to be accurate for a wide class of systems matching classical (e.g. Michaelis–Menten, Hill, Adair) scenarios in the infinite-size approximation. However, the current research in biochemical information processing has been focusing on systems involving a relatively small number of units, where this approximation is no longer valid. Here we show that the whole statistical mechanical description of reaction kinetics can be re-formulated via a mechanical analogy – based on completely integrable hydrodynamic-type systems of PDEs – which provides explicit finite-size solutions, matching recently investigated phenomena (e.g. noise-induced cooperativity, stochastic bi-stability, quorum sensing). The resulting picture, successfully tested against a broad spectrum of data, constitutes a neat rationale for a numerically effective and theoretically consistent description of collective behaviors in biochemical reactions.

15. Marcinkiewicz continuity estimates for infinite energy solutions of some second order elliptic Dirichlet problems
    Boll. Unione Mat. Ital. 8:97-112 (2015)

    L. Dello Schiavo

    journal |

    Given a non-linear Dirichlet problem, we prove a result of continuous dependence, for both the solution and its gradient, on the zero order term in the case of low Marcinkiewicz summability for the latter, relying on the existence results and estimates recently proved in Boccardo (Ann. Mat. Pura Appl. 188(4):591–601, 2009). As a consequence, solutions obtained as limit of approximations are proved to be unique even in the infinite energy case.

16. Notes on stochastic (bio)-logic gates: computing with allosteric cooperativity
    Sci. Rep. 5:9415 (2015)

    E. Agliari, M. Altavilla, A. Barra, L. Dello Schiavo, E. Katz

    journal |

    Recent experimental breakthroughs have finally allowed to implement in-vitro reaction kinetics (the so called enzyme based logic) which code for two-inputs logic gates and mimic the stochastic AND (and NAND) as well as the stochastic OR (and NOR). This accomplishment, together with the already-known single-input gates (performing as YES and NOT), provides a logic base and paves the way to the development of powerful biotechnological devices. However, as biochemical systems are always affected by the presence of noise (e.g. thermal), standard logic is not the correct theoretical reference framework, rather we show that statistical mechanics can work for this scope: here we formulate a complete statistical mechanical description of the Monod–Wyman–Changeaux allosteric model for both single and double ligand systems, with the purpose of exploring their practical capabilities to express noisy logical operators and/or perform stochastic logical operations. Mixing statistical mechanics with logics and testing quantitatively the resulting findings on the available biochemical data, we successfully revise the concept of cooperativity (and anti-cooperativity) for allosteric systems, with particular emphasis on its computational capabilities, the related ranges and scaling of the involved parameters and its differences with classical cooperativity (and anti-cooperativity).

phd thesis

Diffusions on Wasserstein Spaces

L. Dello Schiavo

ULB

We construct a canonical diffusion process on the space of probability measures over a closed Riemannian manifold, with invariant measure the Dirichlet–Ferguson measure. Together with a brief survey of the relevant literature, we collect several tools from the theory of point processes and of optimal transportation. Firstly, we study the characteristic functional of Dirichlet–Ferguson measures with non-negative finite intensity measure over locally compact Polish spaces. We compute such characteristic functional as a martingale limit of confluent Lauricella hypergeometric functions of type $D$ with diverging arity. Secondly, we study the interplay between the self-conjugate prior property of Dirichlet distributions in Bayesian non-parametrics, the dynamical symmetry algebra of said Lauricella functions and Pólya Enumeration Theory. Further, we provide a new proof of J. Sethuraman’s fixed point characterization of Dirichlet–Ferguson measures, and an understanding of the latter as an integral identity of Mecke- or Georgii–Nguyen–Zessin-type. Thirdly, we prove a Rademacher-type result on the Wasserstein space over a closed Riemannian manifold. Namely, sufficient conditions are given for a probability measure $\mathbb{P}$ on the Wasserstein space, so that real-valued Lipschitz functions be $\mathbb{P}$-a.e. differentiable in a suitable sense. Some examples of measures satisfying such conditions are also provided. Finally, we give two constructions of a Markov diffusion process with values in the said Wasserstein space. The process is associated with the Dirichlet integral induced by the Wasserstein gradient and by the Dirichlet–Ferguson measure with intensity the Riemannian volume measure of the base manifold. We study the properties of the process, including its invariant sets, short-time asymptotics for the heat kernel, and a description by means of a stochastic partial differential equation.

popularization of science

1. La quantità del nulla, Ithaca X (2018, in Italian)
   

   L. Dello Schiavo, A. Baccaglini-Frank

   journal |

   $0$ zero, ovvero la quantità del nulla. Un'idea che — assieme a quella gemella di insieme vuoto — ha segnato la storia della matematica attraverso i secoli e i continenti, ben al di là dell'introduzione dei sistemi numerici posizionali o degli interi relativi.
   $\emptyset$ vuoto, ovvero l'insieme privo di elementi ed unico sottoinsieme di sé, ogni elemento del quale soddisfa ogni proprietà: adynaton per la filosofia antica, paradosso per quella moderna, commodité irrinunciabile per la matematica contemporanea. Dell'uso di questi due concetti, intimamente legati, cercheremo di fornire una storia breve, corredata dalle fonti originali dall'antichità ai giorni nostri ed arricchita da curiosità matematiche, accennando poi a come alcune tappe storiche segnino delicati passaggi cognitivi nel loro apprendimento.

2. Nature of language vs. language of nature — The limits of human languages in describing nature, in A. Altamore, G. Antonini Galileo and the renaissance scientific discourse, Edizioni Nuova Cultura (2010)

   G. Cecchetti, L. Dello Schiavo

   book |

   The composition of scientific and humanistic ambits of experience rises on one hand from a concrete necessity of our culture as in didactical purposes; but if we look at the problem by a more abstract point of view it is possible to follow the same principles that lead Gödel and Wittgenstein to define the limits of mathematical and verbal language in order to highlight the complementarity of their expressive potentialities.

Every Demonstration undergoes a rigorous review process that checks for quality, clarity and accuracy with standards similar to those of traditional academic publications.

from the WDP FAQs

1. Asymmetric Simple Exclusion Processes (2018)

   A. Occelli, L. Dello Schiavo

   WDP

   This Demonstration shows the evolution of a partially asymmetric simple exclusion process (PASEP) on the integer lattice $\mathbb{Z}.$ In PASEP, a particle waits an exponential amount of time, then moves to the right with probability $p$ or to the left with probability $1-p$, unless the site is occupied. In the case $p=1$, the dynamic is totally asymmetric, and the process is known as a totally asymmetric simple exclusion process (TASEP).

   
2. Homogeneous Splines (2014)

   L. Dello Schiavo

   WDP

   This demonstration helps visualizing Runge's phenomenon in spline intepolation.

3. Zonotope Construction via Shephard's Theorem (2014)

   L. Dello Schiavo

   WDP

   Given $X=\{a_1,\dotsc, a_r\}$ a set of generators for $\mathbb{R}^n$, the zonotope $\mathcal{Z}(X)$ generated by $X$ is the convex hull of all vectors of the form $\sum_{a\in X} a$; that is, $\mathcal{Z}(X)$ is the Minkowski sum of all segments $[0,a]$, where $a\in X$. Also, $\mathcal{Z}(X)$ is the shadow of the $r$-dimensional cube $[0,1]^r$ via the projection $(t_1,\dotsc, t_r)\mapsto \sum_i^r t_i a_i$.