Generic Realization Research Articles

Redundantly globally rigid braced triangulations

By mapping the vertices of a graph G to points in ℝ3, and its edges to the corresponding line segments, we obtain a three-dimensional realization of G. A realization of G is said to be globally rigid if its edge lengths uniquely determine the realization, up to congruence. The graph G is called globally rigid if every generic three-dimensional realization of G is globally rigid. We consider global rigidity properties of braced triangulations, which are graphs obtained from maximal planar graphs by adding extra edges, called bracing edges. We show that for every even integer n ≥ 8 there exist braced triangulations with 3n − 4 edges which remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best possible. This result gives an affirmative answer to a recent conjecture. We also discuss the connections between our results and a related more general conjecture, due to S. Tanigawa and the third author.

A Unified FPGA Realization for Fractional-Order Integrator and Differentiator

This paper proposes a generic FPGA realization of an IP core for fractional-order integration and differentiation based on the Grünwald–Letnikov approximation. All fractional-order dependent terms are approximated to simpler relations using curve fitting to enable an efficient hardware realization. Compared to previous works, the proposed design introduces enhancements in the fractional-order range covering both integration and differentiation. An error analysis between software and hardware results is presented for sine, triangle and sawtooth signals. The proposed generic design is realized on XC7A100T FPGA achieving frequency of 9.328 MHz and validated experimentally for a sine input signal on the oscilloscope. The proposed unified generic design is suitable for biomedical signal processing applications. In addition, it can be employed as a laboratory tool for fractional calculus education.

High Quality QCD Axion at Gravitational Wave Observatories.

The axion solution to the strong CP problem is delicately sensitive to Peccei-Quinn breaking contributions that are misaligned with respect to QCD instantons. Heavy QCD axion models are appealing because they avoid this so-called quality problem. We show that generic realizations of this framework can be probed by the LIGO-Virgo-KAGRA interferometers, through the stochastic gravitational wave (GW) signal sourced by the long-lived axionic string-domain wall network and by upcoming measurements of the neutron and proton electric dipole moments. Additionally, we provide predictions for searches at future GW observatories, which will further explore the parameter space of heavy QCD axion models.

Globally rigid graphs are fully reconstructible

AbstractAd-dimensional framework is a pair$(G,p)$, where$G=(V,E)$is a graph andpis a map fromVto$\mathbb {R}^d$. The length of an edge$uv\in E$in$(G,p)$is the distance between$p(u)$and$p(v)$. The framework is said to be globally rigid in$\mathbb {R}^d$if the graphGand its edge lengths uniquely determine$(G,p)$, up to congruence. A graphGis called globally rigid in$\mathbb {R}^d$if everyd-dimensional generic framework$(G,p)$is globally rigid.In this paper, we consider the problem of reconstructing a graph from the set of edge lengths arising from a generic framework. Roughly speaking, a graphGis strongly reconstructible in$\mathbb {C}^d$if the set of (unlabeled) edge lengths of any generic framework$(G,p)$ind-space, along with the number of vertices ofG, uniquely determine bothGand the association between the edges ofGand the set of edge lengths. It is known that ifGis globally rigid in$\mathbb {R}^d$on at least$d+2$vertices, then it is strongly reconstructible in$\mathbb {C}^d$. We strengthen this result and show that, under the same conditions,Gis in fact fully reconstructible in$\mathbb {C}^d$, which means that the set of edge lengths alone is sufficient to uniquely reconstructG, without any constraint on the number of vertices (although still under the assumption that the edge lengths come from a generic realization).As a key step in our proof, we also prove that ifGis globally rigid in$\mathbb {R}^d$on at least$d+2$vertices, then thed-dimensional generic rigidity matroid ofGis connected. Finally, we provide new families of fully reconstructible graphs and use them to answer some questions regarding unlabeled reconstructibility posed in recent papers.

Associative realizations of κ -deformed extended Snyder model

Usually, the realizations of the noncommutative Snyder model lead to a nonassociative star product. However, it has been shown that this problem can be avoided by adding to the spacetime coordinates new tensorial degrees of freedom. The model so obtained, called extended Snyder model, can be subject to a $\kappa$-deformation, giving rise to a unification of the Snyder and the $\kappa$-Poincar\'e algebras in the formalism of extended spacetime. In this paper we review this construction and consider the generic realizations of the $\kappa$-deformed extended Snyder model, calculating the associated star product, coproduct and twist in a perturbative setting. We also introduce a representation of the Lorentz algebra in the extended space and speculate on possible interpretations of the tensorial degrees of freedom.

On the global rigidity of tensegrity graphs

A tensegrity graph is a graph with edges labeled as bars, cables and struts. A realization of a tensegrity graph T is a pair (T,p), where p maps the vertices of T into Rd for some d≥1. The realization is globally rigid if any realization (T,q) in Rd in which the bars have the same length and the cables and struts are not longer and not shorter, respectively, is an isometric image of (T,p). A tensegrity graph is weakly globally rigid in Rd if it has a generic globally rigid realization in Rd, and strongly globally rigid in Rd if every generic realization in Rd is globally rigid.In this paper we give a necessary condition for weak global rigidity in Rd and prove that in the d=1 case the same condition is also sufficient. In particular, our results imply that a tensegrity graph has a generic globally rigid realization in R1 if and only if it has a generic universally rigid realization in R1. We also show that recognizing strongly globally rigid tensegrity graphs in Rd is co-NP-hard for all d≥1.

Software Components for Smart Industry Based on Microservices: A Case Study in pH Control Process for the Beverage Industry

Modern industries require constant adaptation to new trends. Thus, they seek greater flexibility and agility to cope with disruptions, as well as to solve needs or meet the demand for growth. Therefore, smart industrial applications require a lot of flexibility to be able to react more quickly to continuous market changes, offer more personalized products, increase operational efficiency, and achieve optimum operating points that integrate the entire value chain of a process. This requires the capture of new data that are subsequently processed at different levels of the hierarchy of automation processes, with requirements and technologies according to each level. The result is a new challenge related to the addition of new functionalities in the processes and the interoperability between them. This paper proposes a distributed computational component-based framework that integrates communication, computation, and storage resources and real-time capabilities through container technology, microservices, and the publish/subscribe paradigm, as well as contributing to the development and implementation of industrial automation applications by bridging the gap between generic architectures and physical realizations. The main idea is to enable plug-and-play software components, from predefined components with their interrelationships, to achieve industrial applications without losing or degrading the robustness from previous developments. This paper presents the process of design and implementation with the proposed framework through the implementation of a complex pH control process, ranging from the simulation part to its scaling and implementation to an industrial level, showing the plug-and-play assembly from a definition of components with their relationships to the implementation process with the respective technologies involved. The effectiveness of the proposed framework was experimentally verified in a real production process, showing that the results scaled to an industrial scale comply with the simulated design process. A qualitative comparison with traditional industrial implementations, based on the implementation requirements, was carried out. The implementation was developed in the beverage production plant “Punta Delicia”, located in Colima, Mexico. Finally, the results showed that the platform provided a high-fidelity design, analysis, and testing environment for cyber information flow and their effect on the physical operation of the pH control.

A Middleware Platform for Intelligent Automation: An Industrial Prototype Implementation

The development of dynamic data-based Decision Support Systems (DSSs) along with the increasing availability of data in the industry, makes real-time data acquisition and management a challenge. Intelligent automation appears as a holistic combination of automation with analytics and decisions made by artificial intelligence, delivering smart manufacturing and mass customization while improving resource efficiency. However, challenges towards the development of intelligent automation architectures include the lack of interoperability between systems, complex data preparation steps, and the inability to deal with both high-frequency and high-volume data in a timely fashion. This paper contributes to industrial frameworks focused on the development of standardized system architectures for Industry 4.0, closing the gap between generic architectures and physical realizations. It proposes a platform for intelligent automation relying on a gateway or middleware between field devices, enterprise databases, and DSSs in real-time scenarios. This is achieved by providing the middleware interoperability, determinism, and automatic data structuring over an industrial communication infrastructure such as the OPC UA Standard over Time Sensitive Networks (TSN). Cloud services and database warehousing used to address some of the challenges are handled using fog computing and a multi-workload database. This paper presents an implementation of the platform in the pharmaceutical industry, providing interoperability and real-time reaction capability to changes to an industrial prototype using dynamic scheduling algorithms.

Global rigidity of periodic graphs under fixed-lattice representations

In [9] Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in Rd. Jackson and Jordán [10] confirmed that these conditions are also sufficient in R2, giving a combinatorial characterization of graphs whose generic realizations in R2 are globally rigid. In this paper, we establish analogues of these results for infinite periodic frameworks under fixed lattice representations. Our combinatorial characterization of globally rigid generic periodic frameworks in R2 in particular implies toroidal and cylindrical counterparts of the theorem by Jackson and Jordán.

Ergodic algebras beyond the weak almost periodic functions

Ergodic algebras is a concept introduced in a paper by Zhikov and Krivenko in the 1980’s which generalizes the concept of almost periodic functions and capture the main properties satisfied by generic realizations of a continuous function in a compact topological space endowed with an ergodic dynamical system and an invariant measure. We address the question whether there exist ergodic algebras beyond the weak almost periodic functions, that is, functions whose set of translates is relatively compact in the weak topology of the bounded continuous functions in $${\mathbb {R}}^n$$ . This question is important since, so far, all known ergodic algebras are subalgebras of the weak almost periodic functions. Also, it is known that almost all realizations of a continuous functions in a compact space endowed with a dynamical system and an invariant measure, belongs to an ergodic algebra. In this paper we construct examples of ergodic algebras that are not contained in the weak almost periodic functions. Actually, our examples are not contained in the greater algebra formed by the sum of almost periodic functions with uniformly continuous functions with null mean value in $${\mathbb {R}}^n$$ .

Inflationary magneto-(non)genesis, increasing kinetic couplings, and the strong coupling problem

We study the generation of magnetic fields during inflation making use of a coupling of the inflaton and moduli fields to electromagnetism via the photon kinetic term, and assuming that the coupling is an increasing function of time. We demonstrate that the strong coupling problem of inflationary magnetogenesis can be avoided by incorporating the destabilization of moduli fields after inflation. The magnetic field always dominates over the electric one, and thus the severe constraints on the latter from backreaction, which are the demanding obstacles in the case of a decreasing coupling function, do not apply to the current scenario. However, we show that this loophole to the strong coupling problem comes at a price: the normalization of the amplitude of magnetic fields is determined by this coupling term and is therefore suppressed by a large factor after the moduli destabilization completes. From this we conclude that there is no self-consistent and generic realization of primordial magnetogenesis producing scale-invariant fields in the case of an increasing kinetic coupling.

Configurations of Lines in Space and Combinatorial Rigidity

Let L be a sequence \((\ell _1,\ell _2,\ldots ,\ell _n)\) of n lines in \({\mathbb {C}}^3\). We define the intersection graph \(G_L=([n],E)\) of L, where \([n]:=\{1,\ldots , n\}\), and with \(\{i,j\}\in E\) if and only if \(i\ne j\) and the corresponding lines \(\ell _i\) and \(\ell _j\) intersect, or are parallel (or coincide). For a graph \(G=([n],E)\), we say that a sequence L is a realization of G if \(G\subset G_L\). One of the main results of this paper is to provide a combinatorial characterization of graphs \(G=([n],E)\) that have the following property: For every generic (see Definition 4.1) realization L of G, that consists of n pairwise distinct lines, we have \(G_L=K_n\), in which case the lines of L are either all concurrent or all coplanar. The general statements that we obtain about lines, apart from their independent interest, turns out to be closely related to the notion of graph rigidity. The connection is established due to the so-called Elekes–Sharir framework, which allows us to transform the problem into an incidence problem involving lines in three dimensions. By exploiting the geometry of contacts between lines in 3D, we can obtain alternative, simpler, and more precise characterizations of the rigidity of graphs.

Fingerprints of heavy scales in electroweak effective Lagrangians

The couplings of the electroweak effective theory contain information on the heavy-mass scales which are no-longer present in the low-energy Lagrangian. We build a general effective Lagrangian, implementing the electroweak chiral symmetry breaking SU(2)L ⊗ SU(2)R → SU(2)L+R, which couples the known particle fields to heavier states with bosonic quantum numbers JP = 0± and 1±. We consider colour-singlet heavy fields that are in singlet or triplet representations of the electroweak group. Integrating out these heavy scales, we analyze the pattern of low-energy couplings among the light fields which are generated by the massive states. We adopt a generic non-linear realization of the electroweak symmetry breaking with a singlet Higgs, without making any assumption about its possible doublet structure. Special attention is given to the different possible descriptions of massive spin-1 fields and the differences arising from naive implementations of these formalisms, showing their full equivalence once a proper short-distance behaviour is required.

Disorder-induced topological transitions in multichannel Majorana wires

In this work, we investigate the effect of disorder on the topological properties of multichannel superconductor nanowires. While the standard expectation is that the spectral gap is closed and opened at transitions that change the topological index of the wire, we show that the closing and opening of a transport gap can also cause topological transitions, even in the presence of nonzero density of states across the transition. Such transport gaps induced by disorder can change the topological index, driving a topologically trivial wire into a nontrivial state or vice versa. We focus on the Rashba spin-orbit coupled semiconductor nanowires in proximity to a conventional superconductor, which is an experimentally relevant system, and we obtain analytical formulas for topological transitions in these wires, valid for generic realizations of disorder. Full tight-binding simulations show excellent agreement with our analytical results without any fitting parameters.

Integrating out resonances in strongly-coupled electroweak scenarios

Accepting that there is a mass gap above the electroweak scale, the Electroweak Effective Theory (EWET) is an appropriate tool to describe this situation. Since the EWET couplings contain information on the unknown high-energy dynamics, we consider a generic strongly-coupled scenario of electroweak symmetry breaking, where the known particle fields are coupled to heavier states. Then, and by integrating out these heavy fields, we study the tracks of the lightest resonances into the couplings. The determination of the low-energy couplings (LECs) in terms of resonance parameters can be made more precise by considering a proper short-distance behaviour on the Lagrangian with heavy states, since the number of resonance couplings is then reduced. Notice that we adopt a generic non-linear realization of the electroweak symmetry breaking with a singlet Higgs.

Extension Complexity of Polytopes with Few Vertices or Facets

We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of $d$-polytopes with at most d+4 vertices according to their extension complexity: Out of the super-exponentially-many $d$-polytopes with d+4 vertices, all have extension complexity d+4 except for some families of size $\theta(d^2)$. On the other hand, we show that generic realizations of simplicial/simple $d$-polytopes with $d+1+\alpha$ vertices/facets have extension complexity at least $2 \sqrt{d(d+\alpha)} -d + 1$, which shows that for all $d>(\frac{\alpha-1}{2})^2$ there are $d$-polytopes with $d+1+\alpha$ vertices or facets and extension complexity $d+1+\alpha$.

The unassigned distance geometry problem

Studies of distance geometry problems (DGP) have focused on cases where the vertices at the ends of all or most of the given distances are known or assigned, which we call assigned distance geometry problems (aDGPs). In this contribution we consider the unassigned distance geometry problem (uDGP) where the vertices associated with a given distance are unknown, so the graph structure has to be discovered. uDGPs arises when attempting to find the atomic structure of molecules and nanoparticles using X-ray or neutron diffraction data from non-crystalline materials. Rigidity theory provides a useful foundation for both aDGPs and uDGPs, though it is restricted to generic realizations of graphs, and key results are summarized. Conditions for unique realization are discussed for aDGP and uDGP cases, build-up algorithms for both cases are described and experimental results for uDGP are presented.

The Poincaré algebras p(1,1) and p(1, 2): realizations and deformations

All inequivalent realizations of the Poincaré algebras p(1,1) and p(1, 2) acting in spaces of not more than three variables are constructed. First-order deformations of p(1,1) and p(1,2) are proposed and the generic realizations for the initial and deformed Poincaré algebras are presented.

Strongly rigid tensegrity graphs on the line

Tensegrity frameworks are defined on a set of points in Rd and consist of bars, cables, and struts, which provide upper and/or lower bounds for the distance between their endpoints. The graph of the framework, in which edges are labeled as bars, cables, and struts, is called a tensegrity graph. It is said to be strongly rigid in Rd if every generic realization in Rd as a tensegrity framework is infinitesimally rigid. In this note we show that it is NP-hard to test whether a given tensegrity graph is strongly rigid in R1.

Perpendicular Magnetization and Generic Realization of the Ising Model in Artificial Spin Ice

We have studied frustrated kagome arrays and unfrustrated honeycomb arrays of magnetostatically interacting single-domain ferromagnetic islands with magnetization normal to the plane. The measured pairwise spin correlations of both lattices can be reproduced by models based solely on nearest-neighbor correlations. The kagome array has qualitatively different magnetostatics but identical lattice topology to previously studied artificial spin ice systems composed of in-plane moments. The two systems show striking similarities in the development of moment pair correlations, demonstrating a universality in artificial spin ice behavior independent of specific realization in a particular material system.