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Marco Fontana

Publications and source records attributed to Marco Fontana.

At least 19 recordsLinked to original sources
arXiv2025

Poisson multi-Bernoulli mixture filter for trajectory measurements

This paper presents a Poisson multi-Bernoulli mixture (PMBM) filter for multi-target filtering based on sensor measurements that are sets of trajectories in the last two-time step window. The proposed filter, the trajectory measurement PMBM (TM-PMBM) filter, propagates a PMBM density on the set of target states. In prediction, the filter obtains the PMBM density on the set of trajectories over the last two time steps. This density is then updated with the set of trajectory measurements. After the update step, the PMBM posterior on the set of two-step trajectories is marginalised to obtain a PMBM density on the set of target states. The filter provides a closed-form solution for multi-target filtering based on sets of trajectory measurements, estimating the set of target states at the end of each time window. Additionally, the paper proposes computationally lighter alternatives to the TM-PMBM filter by deriving a Poisson multi-Bernoulli (PMB) density through Kullback-Leibler divergence minimisation in an augmented space with auxiliary variables. The performance of the proposed filters are evaluated in a simulation study.

eess.SP↗
arXiv2022

Data-driven clustering and Bernoulli merging for the Poisson multi-Bernoulli mixture filter

This paper proposes a clustering and merging approach for the Poisson multi-Bernoulli mixture (PMBM) filter to lower its computational complexity and make it suitable for multiple target tracking with a high number of targets. We define a measurement-driven clustering algorithm to reduce the data association problem into several subproblems, and we provide the derivation of the resulting clustered PMBM posterior density via Kullback-Leibler divergence minimisation. Furthermore, we investigate different strategies to reduce the number of single target hypotheses by approximating the posterior via merging and inter-track swapping of Bernoulli components. We evaluate the performance of the proposed algorithm on simulated tracking scenarios with more than one thousand targets.

eess.SP↗
arXiv2022

Diaphragm Ankle Actuation for Efficient Series Elastic Legged Robot Hopping

Robots need lightweight legs for agile locomotion, and intrinsic series elastic compliance has proven to be a major ingredient for energy-efficient locomotion and robust locomotion control. Animals' anatomy and locomotion capabilities emphasize the importance of that lightweight legs and integrated, compact, series elastically actuated for distal leg joints. But unlike robots, animals achieve series elastic actuation by their muscle-tendon units. So far no designs are available that feature all characteristics of a perfect distal legged locomotion actuator; a low-weight and low-inertia design, with high mechanical efficiency, no stick and sliding friction, low mechanical complexity, high-power output while being easy to mount. Ideally, such an actuator can be controlled directly and without mechanical cross-coupling, for example remotely. With this goal in mind, we propose a low-friction, lightweight Series ELastic Diaphragm distal Actuator (SELDA) which meets many, although not all, of the above requirements. We develop, implement, and characterize a bioinspired robot leg that features a SELDA-actuated foot segment. We compare two leg configurations controlled by a central pattern generator that both feature agile forward hopping. By tuning SELDA's activation timing, we effectively adjust the robot's hopping height by 11% and its forward velocity by 14%, even with comparatively low power injection to the distal joint.

cs.RO↗
arXiv2018

$t$-local domains and valuation domains

In a valuation domain $(V,M)$ every nonzero finitely generated ideal $J$ is principal and so, in particular, $J=J^t$, hence the maximal ideal $M$ is a $t$-ideal. Therefore, the $t$-local domains (i.e., the local domains, with maximal ideal being a $t$-ideal) are "cousins" of valuation domains, but, as we will see in detail, not so close. Indeed, for instance, a localization of a $t$-local domain is not necessarily $t$-local, but of course a localization of a valuation domain is a valuation domain. So it is natural to ask under what conditions is a $t$-local domain a valuation domain? The main purpose of the present paper is to address this question, surveying in part previous work by various authors containing useful properties for applying them to our goal.

math.AC↗
arXiv2018

Idempotence and divisorialty in Prüfer-like domains

Let $D$ be a Prüfer $\star$-multiplication domain, where $\star$ is a semistar operation on $D$. We show that certain ideal-theoretic properties related to idempotence and divisoriality hold in Prüfer domains, and we use the associated semistar Nagata ring of $D$ to show that the natural counterparts of these properties also hold in $D$.

math.AC↗
arXiv2018

Sharpness and semistar operations in Pruefer-like domains

Let $\star$ be a semistar operation on a domain $D$, $\star_f$ the finite-type semistar operation associated to $\star$, and $D$ a Prüfer $\star$-multiplication domain (P$\star$MD). For the special case of a Prüfer domain (where $\star$ is equal to the identity semistar operation), we show that a nonzero prime $P$ of $D$ is sharp, that is, that $D_P \nsupseteq \bigcap D_M$, where the intersection is taken over the maximal ideals $M$ of $D$ that do not contain $P$, if and only if two closely related spectral semistar operations on $D$ differ. We then give an appropriate definition of $\star_f$-sharpness for an arbitrary P$\star$MD $D$ and show that a nonzero prime $P$ of $D$ is $\star_f$-sharp if and only if its extension to the $\star$-Nagata ring of $D$ is sharp. Calling a P$\star$MD $\star_f$-sharp ($\star_f$-doublesharp) if each maximal (prime) $\star_f$-ideal of $D$ is sharp, we also prove that such a $D$ is $\star_f$-doublesharp if and only if each $(\star, t)$-linked overring of $D$ is $\star_f$-sharp.

math.AC↗
arXiv2018

The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications

Given an arbitrary spectral space $X$, we consider the set ${\boldsymbol{\mathcal{X}}}(X)$ of all nonempty subsets of $X$ that are closed with respect to the inverse topology. We introduce a Zariski-like topology on ${\boldsymbol{\mathcal{X}}}(X)$ and, after observing that it coincides the upper Vietoris topology, we prove that ${\boldsymbol{\mathcal{X}}}(X)$ is itself a spectral space, that this construction is functorial, and that ${\boldsymbol{\mathcal{X}}}(X)$ provides an extension of $X$ in a more `complete' spectral space. Among the applications, we show that, starting from an integral domain $D$, ${\boldsymbol{\mathcal{X}}}(\mathrm{Spec}(D))$ is homeomorphic to the (spectral) space of all the stable semistar operations of finite type on $D$.

math.GN↗
arXiv2017

Topological properties of semigroup primes of a commutative ring

A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space $\scal(R)$ of prime semigroups of $R$. We show that, under a natural topology introduced by B. Olberding in 2010, $\scal(R)$ is a spectral space (after Hochster), spectral extension of $\Spec(R)$, and that the assignment $R\mapsto\scal(R)$ induces a contravariant functor. We then relate -- in the case $R$ is an integral domain -- the topology on $\scal(R)$ with the Zariski topology on the set of overrings of $R$. Furthermore, we investigate the relationship between $\scal(R)$ and the space $\boldsymbol{\mathcal{X}}(R)$ consisting of all nonempty inverse-closed subspaces of $\spec(R)$, which has been introduced and studied in C.A. Finocchiaro, M. Fontana and D. Spirito, "The space of inverse-closed subsets of a spectral space is spectral" (submitted). In this context, we show that $\scal( R)$ is a spectral retract of $\boldsymbol{\mathcal{X}}(R)$ and we characterize when $\scal( R)$ is canonically homeomorphic to $\boldsymbol{\mathcal{X}}(R)$, both in general and when $\spec(R)$ is a Noetherian space. In particular, we obtain that, when $R$ is a Bézout domain, $\scal( R)$ is canonically homeomorphic both to $\boldsymbol{\mathcal{X}}(R)$ and to the space $\overr(R)$ of the overrings of $R$ (endowed with the Zariski topology). Finally, we compare the space $\boldsymbol{\mathcal{X}}(R)$ with the space $\scal(R(T))$ of semigroup primes of the Nagata ring $R(T)$, providing a canonical spectral embedding $\xcal(R)\hookrightarrow\scal(R(T))$ which makes $\xcal(R)$ a spectral retract of $\scal(R(T))$.

math.AC↗
arXiv2016

Algebraic and topological properties of an amalgamated algebra along an ideal

Let $f:A \rightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we study the amalgamation of $A$ with $B$ along $J$ with respect to $f$, a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced by D'Anna and Fontana in 2007, and other classical constructions (such as the $A+ XB[X]$, the $A+ XB[\![X]\!]$ and the $D+M$ constructions). In particular, we completely describe the prime spectrum of the amalgamation and, when it is a local Noetherian ring, we study its embedding dimension and when it turns to be a Cohen-Macaulay ring or a Gorenstein ring.

math.AC↗
arXiv2016

A topological version of Hilbert's Nullstellensatz

We prove that the space of radical ideals of a ring $R$, endowed with the hull-kernel topology, is a spectral space, and that it is canonically homeomorphic to the space of the nonempty Zariski closed subspaces of Spec$(R)$, endowed with a Zariski-like topology.

math.AC↗
arXiv2016

Spectral spaces of semistar operations

We investigate, from a topological point of view, the classes of spectral semistar operations and of eab semistar operations, following methods recently introduced by Finocchiaro and Finocchiaro-Spirito in \cite{Fi, FiSp}. We show that, in both cases, the subspaces of finite type operations are spectral spaces in the sense of Hochster and, moreover, that there is a distinguished class of overrings strictly connected to each of the two types of collections of semistar operations. We also prove that the space of stable semistar operations is homeomorphic to the space of Gabriel-Popescu localizing systems, endowed with a Zariski-like topology, extending to the topological level a result established by Fontana-Huckaba in \cite{fohu}. As a side effect, we obtain that the space of localizing systems of finite type is also a spectral space. Finally, we show that the Zariski topology on the set of semistar operations is the same as the $b$-topology defined recently by B. Olberding \cite{ol, olb_noeth}.

math.AC↗
arXiv2015

New distinguished classes of spectral spaces: a survey

In the present survey paper, we present several new classes of Hochster's spectral spaces "occurring in nature", actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The general setting is the space of the semistar operations (of finite type), endowed with a Zariski-like topology, which turns out to be a natural topological extension of the space of the overrings of an integral domain, endowed with a topology introduced by Zariski. One of the key tool is a recent characterization of spectral spaces, based on the ultrafilter topology, given in a paper by C. Finocchiaro in Comm. Algebra 2014. Several applications are also discussed.

math.AC↗
arXiv2013

Polynomial extensions of semistar operations

We provide a complete solution to the problem of extending arbitrary semistar operations of an integral domain $D$ to semistar operations of the polynomial ring $D[X]$. As an application, we show that one can reobtain the main results of some previous papers concerning the problem in the special cases of stable semistar operations of finite type or semistar operations defined by families of overrings. Finally, we investigate the behavior of the polynomial extensions of the most important and classical operations such as $d_D$, $v_D$, $t_D$, $w_D$ and $b_D$ operations.

math.AC↗
arXiv2012

The constructible topology on spaces of valuation domains

We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an {\sl arbitrary spectral space} and we observe that this topology coincides with the constructible topology. If $K$ is a field and $A$ a subring of $K$, we show that the space Zar$(K|A)$ of all valuation domains, having $K$ as quotient field and containing $A$, (endowed with the Zariski topology) is a spectral space by giving in this general setting the explicit construction of a ring whose Zariski spectrum is homeomorphic to Zar$(K|A)$. We extend results regarding spectral topologies on the spaces of all valuation domains and apply the theory developed to study representations of integrally closed domains as intersections of valuation overrings. As a very particular case, we prove that two collections of valuation domains of $K$ with the same ultrafilter closure represent, as an intersection, the same integrally closed domain.

math.AC↗
arXiv2012

Ultrafilter and Constructible topologies on spaces of valuation domains

Let $K$ be a field and let $A$ be a subring of $K$. We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on the space Zar$(K|A)$ of all valuation domains having $K$ as quotient field and containing $A$. We show that the ultrafilter topology coincides with the constructible topology on the abstract Riemann-Zariski surface Zar$(K|A)$. We extend results regarding distinguished spectral topologies on spaces of valuation domains.

math.AC↗
arXiv2011

On some classes of integral domains defined by Krull's $\boldsymbol{a.b.}$ operations

Let $D$ be an integral domain with quotient field $K$. The $b$-operation that associates to each nonzero $D$-submodule $E$ of $K$, $E^b := \bigcap\{EV \mid V valuation overring of D\}$, is a semistar operation that plays an important role in many questions of ring theory (e.g., if $I$ is a nonzero ideal in $D$, $I^b$ coincides with its integral closure). In a first part of the paper, we study the integral domains that are $b$-Noetherian (i.e., such that, for each nonzero ideal $I$ of $D$, $I^b = J^b$ for some a finitely generated ideal $J$ of $D$). For instance, we prove that a $b$-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily $b$-Noetherian. We also characterize several distinguished classes of $b$-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type $\star_a$ canonically associated to a given semistar operation $\star$ (for instance, the $b$-operation is the e.a.b. semistar operation of finite type canonically associated to the identity operation). These operations, introduced and studied by Krull, Jaffard, Gilmer and Halter-Koch, play a very important role in the recent generalizations of the Kronecker function ring. In particular, in the present paper, we classify several classes of integral domains having some of the fundamental operations $d$, $t$, $w$ and $v$ equal to some of the canonically associated e.a.b. operations $b$, $t_a$, $w_a$ and $v_a$.

math.AC↗
arXiv2010

An overring-theoretic approach to polynomial extensions of star and semistar operations

Call a semistar operation $\ast$ on the polynomial domain $D[X]$ an extension (respectively, a strict extension) of a semistar operation $\star$ defined on an integral domain $D$, with quotient field $K$, if $E^\star = (E[X])^{\ast}\cap K$ (respectively, $E^\star [X]= (E[X])^{\ast}$) for all nonzero $D$-submodules $E$ of $K$. In this paper, we study the general properties of the above defined extensions and link our work with earlier efforts, centered on the stable semistar operation case, at defining semistar operations on $D[X]$ that are "canonical" extensions (or, "canonical" strict extensions) of semistar operations on $D$.

math.AC↗