arXiv subjects

Pedro Teixeira

Publications and source records attributed to Pedro Teixeira.

16 recordsLinked to original sources

arXiv2022

Aveiro Tech City Living Lab: A Communication, Sensing and Computing Platform for City Environments

This article presents the deployment and experimentation architecture of the Aveiro Tech City Living Lab (ATCLL) in Aveiro, Portugal. This platform comprises a large number of Internet-of-Things devices with communication, sensing and computing capabilities. The communication infrastructure, built on fiber and Millimeter-wave (mmWave) links, integrates a communication network with radio terminals (WiFi, ITS-G5, C-V2X, 5G and LoRa(WAN)), multiprotocol, spread throughout 44 connected points of access in the city. Additionally, public transportation has also been equipped with communication and sensing units. All these points combine and interconnect a set of sensors, such as mobility (Radars, Lidars, video cameras) and environmental sensors. Combining edge computing and cloud management to deploy the services and manage the platform, and a data platform to gather and process the data, the living lab supports a wide range of services and applications: IoT, intelligent transportation systems and assisted driving, environmental monitoring, emergency and safety, among others. This article describes the architecture, implementation and deployment to make the overall platform to work and integrate researchers and citizens. Moreover, it showcases some examples of the performance metrics achieved in the city infrastructure, the data that can be collected, visualized and used to build services and applications to the cities, and, finally, different use cases in the mobility and safety scenarios.

cs.NI↗

arXiv2022

Bernstein's inequality and holonomicity for certain singular rings

In this manuscript we prove the Bernstein inequality and develop the theory of holonomic D-modules for rings of invariants of finite groups in characteristic zero, and for strongly F-regular finitely generated graded algebras with FFRT in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic D-modules, in this context, have finite length. We obtain these results using a more general version of Bernstein filtrations.

math.AC↗

arXiv2021

Bernstein-Sato functional equations, $V$-filtrations, and multiplier ideals of direct summands

This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of $V$-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings.

math.AG↗

arXiv2020

Merkle-CRDTs: Merkle-DAGs meet CRDTs

We study Merkle-DAGs as a transport and persistence layer for Conflict-Free Replicated Data Types (CRDTs), coining the term Merkle-CRDTs and providing an overview of the different concepts, properties, advantages and limitations involved. We show how Merkle-DAGs can act as logical clocks giving Merkle-CRDTs the potential to greatly simplify the design and implementation of convergent data types in systems with weak messaging layer guarantees and a very large number of replicas. Merkle-CRDTs can leverage highly scalable distributed technologies like DHTs and PubSub algorithms running underneath to take advantage of the security and de-duplication properties of content-addressing. Examples of such content-oriented systems could include peer-to-peer content exchange and synchronisation applications between opportunistically connected mobile devices, IoT devices or user applications running in a web browser.

cs.NI↗

arXiv2019

Frobenius powers

This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincides with its test ideals, but appear to be a more refined measure of singularities in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals.

math.AC↗

arXiv2019

The TestIdeals package for Macaulay2

This note describes a \emph{Macaulay2} package for computations in prime characteristic commutative algebra. This includes Frobenius powers and roots, $p^{-e}$-linear and $p^{e}$-linear maps, singularities defined in terms of these maps, different types of test ideals and modules, and ideals compatible with a given $p^{-e}$-linear map.

math.AC↗

arXiv2019

The FrobeniusThresholds package for Macaulay2

This article describes the \emph{Macaulay2} package \emph{FrobeniusThresholds}, designed to estimate and calculate $F$-pure thresholds, more general $F$-thresholds, and related numerical invariants arising in the study of singularities in prime characteristic commutative algebra.

math.AC↗

arXiv2018

On the conservative pasting lemma

Several perturbation tools are established in the volume preserving setting allowing for the pasting, extension, localized smoothing and local linearization of vector fields. The pasting and local linearization hold in all classes of regularity ranging from $C^{1}$ to $C^{\infty}$ (Hölder included). For diffeomorphisms, a conservative linearized version of Franks lemma is proved in the $C^{r,α}$ ($r\in\mathbb{Z}^+$, $0<α<1$) and $C^{\infty}$ settings, the resulting diffeomorphism having the same regularity as the original one.

math.DS↗

arXiv2018

Frobenius powers of some monomial ideals

In this paper, we characterize the (generalized) Frobenius powers and critical exponents of two classes of monomial ideals of a polynomial ring in positive characteristic: powers of the homogeneous maximal ideal, and ideals generated by positive powers of the variables. In doing so, we effectively characterize the test ideals and $F$-jumping exponents of sufficiently general homogeneous polynomials, and of all diagonal polynomials. Our characterizations make these invariants computable, and show that they vary uniformly with the congruence class of the characteristic modulo a fixed integer. Moreover, we confirm that for a diagonal polynomial over a field of characteristic zero, the test ideals of its reduction modulo a prime agree with the reductions of its multiplier ideals for infinitely many primes.

math.AC↗

arXiv2018

Dacorogna-Moser theorem on the Jacobian determinant equation with control of support

The original proof of Dacorogna-Moser theorem on the prescribed Jacobian PDE, $\text{det}\,\nablaφ=f$, can be modified in order to obtain control of support of the solutions from that of the initial data, while keeping optimal regularity. Briefly, under the usual conditions, a solution diffeomorphism $φ$ satisfying \[ \text{supp}(f-1)\subset\varOmega\Longrightarrow\text{supp}(φ-\text{id})\subset\varOmega \] can be found and $φ$ is still of class $C^{r+1,α}$ if $f$ is $C^{r,α}$, the domain of $f$ being a bounded connected open $C^{r+2,α}$ set $\varOmega\subset\mathbb{R}^{n}$.

math.AP↗

arXiv2017

Addendum to: Dacorogna-Moser theorem on the Jacobian determinant equation with control of support

In Dacorogna-Moser theorem on the pullback equation $φ^* (g)=f$ between two prescribed volume forms (with the same total volume), control of support of the solutions can be obtained from that of the initial data, while keeping optimal regularity. This result answers a problem implicitly raised on page 14 of Dacorogna-Moser's original article ("On a partial differential equation involving the Jacobian determinant", Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 1-26), and fully generalizes the solution to the particular case of $g\equiv 1$ (prescribed Jacobian PDE, $\text{det}\,\nablaφ=f$) given in the author's paper "Dacorogna-Moser theorem on the Jacobian determinant equation with control of support", Discrete Cont. Dyn. Syst. 37 (2017), 4071-4089.

math.AP↗

arXiv2016

F-threshold functions: syzygy gap fractals and the two-variable homogeneous case

In this article we study F-pure thresholds (and, more generally, F-thresholds) of homogeneous polynomials in two variables over a field of characteristic p>0. Passing to a field extension, we factor such a polynomial into a product of powers of pairwise prime linear forms, and to this collection of linear forms we associate a special type of function called a syzygy gap fractal. We use this syzygy gap fractal to study, at once, the collection of all F-pure thresholds of all polynomials constructed with the same fixed linear forms. This allows us to describe the structure of the denominator of such an F-pure threshold, showing in particular that whenever the F-pure threshold differs from its expected value its denominator is a multiple of p. This answers a question of Schwede in the two-variable homogeneous case. In addition, our methods give an algorithm to compute F-pure thresholds of homogenous polynomials in two variables.

math.AC↗

arXiv2015

Topological detection of Lyapunov instability

Given an arbitrary continuous flow on a manifold M, let CMin be the set of its compact minimal sets, endowed with the Hausdorff metric, and S the subset of those that are Lyapunov stable. A topological characterization of the interior of S, the set of Lyapunov stable compact minimal sets that are away from Lyapunov unstable ones is given, together with a description of the dynamics around it. In particular, int S is locally a Peano continuum (Peano curve) and each of its countably many connected components admits a complete geodesic metric. This result establishes unexpected connections between the local topology of CMin and the dynamics of the flow, providing criteria for the local detection of Lyapunov instability by merely looking at the topology of CMin. For instance, if CMin is not locally connected at some compact minimal set Q (seen as a "point" of CMin), then every neighbourhood of Q in M contains Lyapunov unstable compact minimal sets (hence, if CMin is nowhere locally connected, then every neighbourhood of each compact minimal set contains infinitely many Lyapunov unstable compact minimal sets).

math.DS↗

arXiv2012

Flows near Compact Invariant Sets - Part I

In this paper it is proved that near a compact, invariant, proper subset of a continuous flow on a compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. This result shows that assuming the connectedness of the phase space implies the existence of a considerably deeper classification of topological flow behaviour, in the vicinity of compact invariant sets, than that described in the classical theorems of Ura-Kimura and Bhatia. The proposed classification brings to light, in a systematic way, the possibility of occurrence of orbits of infinite height arbitrarily near the compact invariant in question, and this under relatively simple conditions. Singularities of smooth vector fields displaying this strange phenomenon occur in every dimension greater than 2 (in this paper, a smooth flow on the 3-dimensional sphere exhibiting such an equilibrium is constructed). Near periodic orbits, the same phenomenon is observable already in dimension 4 (and on every manifold of dimension greater than 4). As a corollary to the main result, an elegant characterization of the topological Hausdorff structure of the set of all compact minimal sets of the flow is obtained (Theorem 2). Keywords: topological behaviour of C0 flows, compact invariant sets, compact minimal sets, topological Hausdorff structure, non-hyperbolic singularities and periodic orbits, orbits of infinite height.

math.DS↗

arXiv2010

Syzygy gap fractals--I. Some structural results and an upper bound

k is a field of characteristic p>0, and l_1,...,l_n are linear forms in k[x,y]. Intending applications to Hilbert--Kunz theory, to each triple C=(F,G,H) of nonzero homogeneous elements of k[x,y] we associate a function delta_C that encodes the "syzygy gaps" of F^q, G^q, and H^q*l_1^{a_1}*...*l_n^{a_n}, for all q=p^e and a_i<= q. These are close relatives of functions introduced in "p-Fractals and power series--I" [P. Monsky, P. Teixeira, p-Fractals and power series--I. Some 2 variable results, J. Algebra 280 (2004) 505--536]. Like their relatives, the delta_C exhibit surprising self-similarity related to "magnification by p," and knowledge of their structure allows the explicit computation of various Hilbert--Kunz functions. We show that these "syzygy gap fractals" are determined by their zeros and have a simple behavior near their local maxima, and derive an upper bound for their local maxima which has long been conjectured by Monsky. Our results will allow us, in a sequel to this paper, to determine the structure of the delta_C by studying the vanishing of certain determinants.

math.AC↗

arXiv2005

p-Fractals and power series--II. Some applications to Hilbert-Kunz theory

We use the results of our paper "p-Fractals and power series--I" (Journal of Algebra 280, 2004, pp. 505--536) to prove the rationality of the Hilbert-Kunz series of a large family of power series, including those of the form \sum_i f_i(x_i,y_i), where the f_i(x_i,y_i) are power series with coefficients in a finite field. The methods are effective, as we illustrate with examples. In the final section, which can be read independently of the others, we obtain more precise results for the Hilbert-Kunz function of the 3 variable power series z^D-h(x,y).

math.AC↗