Geometric Notions Research Articles

Reduced optimal models via cross Gramian for continuous linear time‐invariant systems

In this study, the authors focus on the problem of model order reduction (MOR) suitable for continuous linear time-invariant (LTI) systems. Specifically, for single-input single-output (SISO) LTI systems, a couple of MOR algorithms via the cross Gramian are presented. The authors first give a cross Gramian-based optimal MOR iterative algorithm to reduce the SISO system. Further, the authors explore the optimisation problem on the Stiefel manifold. Based on the geometric notions on this manifold, a conjugate gradient iterative algorithm (CGIAHM) is proposed. The conjugate gradient direction used to search for the minimiser is derived by applying the notion of vector transport. It is worth mentioning that the conjugate gradient used in the CGIAHM algorithm is a decent direction for the cost function due to the ingenious construction of this algorithm. In addition, the authors’ algorithms are extended to solve the optimal MOR problem for general LTI systems. Finally, two numerical examples demonstrate the effectiveness of their algorithms.

Boundary-bulk relation in topological orders

In this paper, we study the relation between an anomaly-free n+1D topological order, which are often called n+1D topological order in physics literature, and its nD gapped boundary phases. We argue that the n+1D bulk anomaly-free topological order for a given nD gapped boundary phase is unique. This uniqueness defines the notion of the “bulk” for a given gapped boundary phase. In this paper, we show that the n+1D “bulk” phase is given by the “center” of the nD boundary phase. In other words, the geometric notion of the “bulk” corresponds precisely to the algebraic notion of the “center”. We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the “bulk” satisfies the same universal property as that of the “center” of an algebra in mathematics, i.e. “bulk = center”. The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions.

Output Regulation by Error Dynamic Feedback in Hybrid Linear Systems with State Jumps

This work deals with a regulation problem for hybrid linear systems which exhibit a continuous-time state motion, ruled by the so-called flow dynamics, except at isolated points of the time axis, where the state has discontinuities governed by a jump behavior. Jump time instants are not a-priori known and may be unequally spaced, the only admissibility constraint being that the set of time intervals between consecutive jumps has a fixed positive lower bound. The considered problem consists in finding a hybrid error feedback compensator that forces the output of a given hybrid plant to asymptotically follow a reference trajectory generated by a hybrid exogenous system, while achieving global asymptotic stability of the closed-loop dynamics, for all the admissible sequences of jump times. The problem is investigated from a structural point of view, using geometric notions and properties. A sufficient condition for the existence of solutions is first stated in an implicit form, by considering the overall compensated system. This result is instrumental in giving a sufficient constructive condition which refers to the output-difference connection of the plant and the exogenous generator. The conditions are expressed in terms of hybrid invariant and controlled invariant subspaces of the systems at issue as well as of their stability and stabilizability properties, respectively. This approach provides a viable algorithmic procedure for the synthesis of solutions, if any exist.

Unknown Input Observers for Hybrid Linear Systems with State Jumps

This paper investigates the problem of constructing an observer that asymptotically estimates a linear function of the state of a linear plant with state jumps in the presence of unknown inputs. Suitable geometric notions are introduced and discussed in order to deal with this problem. A complete characterization of its solvability, together with a procedure for constructing the observer, is given in geometric terms. A sufficient solvability condition that can be practically checked by a simple procedure is also illustrated. Under mildly restrictive hypothesis, the necessary and sufficient solvability conditions can be practically checked and the observer can be practically constructed.

Conditioned invariance and detectability subspaces in the behavioral approach

In this paper we extend the state space geometric notions of conditioned invariance and detectability subspaces to the behavioral framework. This is achieved based on the existing notions of behavioral tracking observer and of behavioral asymptotic observer introduced by Valcher, Willems, Trentelman and Trumph, combined with a notion of behavioral invariance introduced here. Moreover, we provide characterizations for the newly defined behavioral properties.

Topology and convexity in the space of actions modulo weak equivalence

We analyze the structure of the quotient $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We introduce a variant of an abstract construction of Fritz which encapsulates the convex combination operation on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$. This formalism allows us to define the geometric notion of an extreme point. We also discuss a topology on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ due to Abért and Elek in which it is Polish and compact, and show that this topology is equivalent to others defined in the literature. We show that the convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is compatible with the topology, and as a consequence deduce that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is path connected. Using ideas of Tucker-Drob, we are able to give a complete description of the topological and convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ for amenable $\unicode[STIX]{x1D6E4}$ by identifying it with the simplex of invariant random subgroups. In particular, we conclude that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ can be represented as a compact convex subset of a Banach space if and only if $\unicode[STIX]{x1D6E4}$ is amenable. In the case of general $\unicode[STIX]{x1D6E4}$ we prove a Krein–Milman-type theorem asserting that finite convex combinations of the extreme points of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ are dense in this space. We also consider the space $\text{A}_{{\sim}_{s}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of stable weak equivalence classes and show that it can always be represented as a compact convex subset of a Banach space. In the case of a free group $\mathbb{F}_{N}$, we show that if one restricts to the compact convex set $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})\subseteq \text{A}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$ consisting of the stable weak equivalence classes of free actions, then the extreme points are dense in $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$.

Producing or reproducing reasoning? Socratic dialog is very effective, but only for a few.

Successful communication between a teacher and a student is at the core of pedagogy. A well known example of a pedagogical dialog is ‘Meno’, a socratic lesson of geometry in which a student learns (or ‘discovers’) how to double the area of a given square ‘in essence, a demonstration of Pythagoras’ theorem. In previous studies we found that after engaging in the dialog participants can be divided in two kinds: those who can only apply a rule to solve the problem presented in the dialog and those who can go beyond and generalize that knowledge to solve any square problems. Here we study the effectiveness of this socratic dialog in an experimental and a control high-school classrooms, and we explore the boundaries of what is learnt by testing subjects with a set of 9 problems of varying degrees of difficulty. We found that half of the adolescents did not learn anything from the dialog. The other half not only learned to solve the problem, but could abstract something more: the geometric notion that the diagonal can be used to solve diverse area problems. Conceptual knowledge is critical for achievement in geometry, and it is not clear whether geometric concepts emerge spontaneously on the basis of universal experience with space, or reflect intrinsic properties of the human mind. We show that, for half of the learners, an exampled-based Socratic dialog in lecture form can give rise to formal geometric knowledge that can be applied to new, different problems.

Remarks on Minimal Sets for Cyclic Mappings in Uniformly Convex Banach Spaces

ABSTRACTIn this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.

A randomized divide and conquer algorithm for higher-order abstract Voronoi diagrams

Given a set of n sites in the plane, their order-k Voronoi diagram partitions the plane into regions such that all points within one region have the same k nearest sites. The order-k abstract Voronoi diagram offers a unifying framework that represents a wide range of concrete order-k Voronoi diagrams. It is defined in terms of bisecting curves satisfying some simple combinatorial properties, rather than the geometric notions of sites and distance.In this paper we develop a randomized divide-and-conquer algorithm to compute the order-k abstract Voronoi diagram in expected O(kn1+ε) operations. For solving small sub-instances in the divide-and-conquer process, we also give two auxiliary algorithms with expected O(k2nlog⁡n) and O(n22α(n)log⁡n) time, respectively, where α(⋅) is the inverse of the Ackermann function. Our approach directly implies an O(kn1+ε)-time algorithm for several concrete order-k instances such as points in any convex distance, disjoint line segments or convex polygons of constant size in the Lp norms, and others. It also provides basic techniques that can enable the application of well-known random sampling techniques to the construction of Voronoi diagrams in the abstract setting and for non-point sites.

Möbius bands, unstretchable material sheets and developable surfaces.

A Möbius band can be formed by bending a sufficiently long rectangular unstretchable material sheet and joining the two short ends after twisting by 180°. This process can be modelled by an isometric mapping from a rectangular region to a developable surface in three-dimensional Euclidean space. Attempts have been made to determine the equilibrium shape of a Möbius band by minimizing the bending energy in the class of mappings from the rectangular region to the collection of developable surfaces. In this work, we show that, although a surface obtained from an isometric mapping of a prescribed planar region must be developable, a mapping from a prescribed planar region to a developable surface is not necessarily isometric. Based on this, we demonstrate that the notion of a rectifying developable cannot be used to describe a pure bending of a rectangular region into a Möbius band or a generic ribbon, as has been erroneously done in many publications. Specifically, our analysis shows that the mapping from a prescribed planar region to a rectifying developable surface is isometric only if that surface is cylindrical with the midline being the generator. Towards providing solutions to this issue, we discuss several alternative modelling strategies that respect the distinction between the physical constraint of unstretchability and the geometrical notion of developability.

Spacewalks and Amusement Rides: Illustrations of Geometric Phase

SummaryGeometric phase in a dynamical system can be visualized as the interplay between two periodic functions which go in and out of “synch.” Using illustrations of a boy′s walk in space and a dizzying fun park ride, we demonstrate that in certain simple mechanical systems we can compute geometric phase directly from a symmetry—we don′t even need to solve the system of differential equations. We can also use geometric phase to explain how cats (almost) always land on their feet. We conclude with an interpretation of geometric phase in terms of the geometric notions of connection and curvature.

On Visualizing Mixed-Type Data

Survey data are usually of mixed type (quantitative, multistate categorical, and/or binary variables). Multidimensional scaling (MDS) is one of the most extended methodologies to visualize the profile structure of the data. Since the past 60s, MDS methods have been introduced in the literature, initially in publications in the psychometrics area. Nevertheless, sensitivity and robustness of MDS configurations have been topics scarcely addressed in the specialized literature. In this work, we are interested in the construction of robust profiles for mixed-type data using a proper MDS configuration. To this end, we propose to compare different MDS configurations (coming from different metrics) through a combination of sensitivity and robust analysis. In particular, as an alternative to classical Gower’s metric, we propose a robust joint metric combining different distance matrices, avoiding redundant information, via related metric scaling. The search for robustness and identification of outliers is done through a distance-based procedure related to geometric variability notions. In this sense, we propose a statistic for detecting multivariate outliers in the context of mixed-type data and evaluate its performance through a simulation study. Finally, we apply these techniques to a real data set provided by the largest humanitarian organization involved in social programs in Spain, where we are able to find in a robust way the most relevant factors defining the profiles of people that were under risk of being socially excluded in the beginning of the 2008 economic crisis.

Best proximity points, cyclic Kannan maps and geodesic metric spaces

We introduce the concept of cyclic Kannan orbital C-nonexpansive mappings and obtain the existence of a best proximity point on a pair of bounded, closed and convex subsets of a strictly convex metric space by using the geometric notion of seminormal structure. We also study the structure of minimal sets for cyclic Kannan C-nonexpansive mappings and show that results similar to the celebrated Goebel– Karlovitz lemma for nonexpansive self-mappings can be obtained for cyclic Kannan C-nonexpansive mappings.

On cyclic relatively nonexpansive mappings in generalized semimetric spaces

In this article, we prove a fixed point theorem for cyclic relatively nonexpansive mappings in the setting of generalized semimetric spaces by using a geometric notion of seminormal structure and then we conclude some results in uniformly convex Banach spaces. We also discuss on the stability of seminormal structure in generalized semimetric spaces.

The Extension of Distributions on Manifolds, a Microlocal Approach

Let M be a smooth manifold, $${I\subset M}$$ a closed embedded submanifold of M and U an open subset of M. In this paper, we find conditions using a geometric notion of scaling for $${t\in \mathcal{D}^{\prime}(U{\setminus} I)}$$ to admit an extension in $${\mathcal{D}^\prime(U)}$$ . We give microlocal conditions on t which allow to control the wave front set of the extension generalizing a previous result of Brunetti–Fredenhagen. Furthermore, we show that there is a subspace of extendible distributions for which the wave front of the extension is minimal which has applications for the renormalization of quantum field theory on curved spacetimes.

Graph Curvature for Differentiating Cancer Networks

Cellular interactions can be modeled as complex dynamical systems represented by weighted graphs. The functionality of such networks, including measures of robustness, reliability, performance, and efficiency, are intrinsically tied to the topology and geometry of the underlying graph. Utilizing recently proposed geometric notions of curvature on weighted graphs, we investigate the features of gene co-expression networks derived from large-scale genomic studies of cancer. We find that the curvature of these networks reliably distinguishes between cancer and normal samples, with cancer networks exhibiting higher curvature than their normal counterparts. We establish a quantitative relationship between our findings and prior investigations of network entropy. Furthermore, we demonstrate how our approach yields additional, non-trivial pair-wise (i.e. gene-gene) interactions which may be disrupted in cancer samples. The mathematical formulation of our approach yields an exact solution to calculating pair-wise changes in curvature which was computationally infeasible using prior methods. As such, our findings lay the foundation for an analytical approach to studying complex biological networks.

Cook's Local Influence in Generalized Linear Models via the Shape Operator

In this paper we develop an algorithm for assessing the effect of small perturbations of the data on the validity of a postulated generalized linear model. The procedure is based on the geometric notion of shape operator, a single mathematical object that gathers together all the normal curvatures of a given influence graph or hypersurface. In addition to introducing relevant theoretical notions and explaining the foundations of the local influence assessment method in a generalized linear model with a canonical link, we provide a detailed example of application together with the explicit $R$ code implementing the algorithm.

Prática pedagógica de geometria na educação de jovens e adultos: o ensinado e o aprendido

Este artigo apresenta resultados de uma pesquisa que buscou identificar, interpretar e descrever a metodologia adotada por professores da Educação de Jovens e Adultos (ENCCEJA) para ensinar os quadriláteros paralelogramos e verificar as principais ideias assimiladas pelos alunos a respeito desse conteúdo. Como fundamentos teóricos da pesquisa foram utilizados, principalmente, o modelo de desenvolvimento do pensamento geométrico proposto pelos educadores Dina Van Hiele-Geldof e Pierre Van Hiele, o Guia Curricular de Matemática (MINAS GERAIS, 1997), a Proposta Curricular destinada a EJA (BRASIL, 2001) e os estudos de Crowley (2011) sobre o modelo Van Hiele. Foram estudados dois grupos de colaboradores, sendo o primeiro constituído por cinco professores e o segundo por dez alunos. Para coletar os dados necessários ao desenvolvimento da pesquisa pretendida foi realizada uma entrevista gravada com os professores e solicitado aos alunos a resolução de uma atividade prática, envolvendo os saberes inerentes aos quadriláteros paralelogramos. Com o estudo realizado foi constatado que os professores pesquisados desenvolvem sua prática pedagógica de forma expositiva e demonstrativa, predominantemente, com ênfase na transmissão de informações tidas como essenciais, na visualização das imagens das formas geométricas, na resolução de exercícios padronizados, estruturados e elaborados com base nas informações que foram repassadas verbalmente e nos modelos sugeridos em diferentes livros didáticos. Em decorrência dessas estratégias e procedimentos de ensino adotados pelos professores, os alunos da ENCCEJA não conseguem dominar as propriedades fundamentais que caracterizam as figuras geométricas e acabam apresentando dificuldades na identificação correta dessas formas.

Geometric duality theory of cones in dual pairs of vector spaces

This paper will generalize what may be termed the "geometric duality theory" of real pre-ordered Banach spaces which relates geometric properties of a closed cone in a real Banach space, to geometric properties of the dual cone in the dual Banach space. We show that geometric duality theory is not restricted to real pre-ordered Banach spaces, as is done classically, but can be extended to real Banach spaces endowed with arbitrary collections of closed cones. We define geometric notions of normality, conormality, additivity and coadditivity for members of dual pairs of real vector spaces as certain possible interactions between two cones and two convex convex sets containing zero. We show that, thus defined, these notions are dual to each other under certain conditions, i.e., for a dual pair of real vector spaces $(Y,Z)$, the space $Y$ is normal (additive) if and only if its dual $Z$ is conormal (coadditive) and vice versa. These results are set up in a manner so as to provide a framework to prove results in the geometric duality theory of cones in real Banach spaces. As an example of using this framework, we generalize classical duality results for real Banach spaces pre-ordered by a single closed cone, to real Banach spaces endowed with an arbitrary collections of closed cones. As an application, we analyze some of the geometric properties of naturally occurring cones in C*-algebras and their duals.

On the complexity of higher order abstract Voronoi diagrams

Abstract Voronoi diagrams (AVDs) are based on bisecting curves enjoying simple combinatorial properties, rather than on the geometric notions of sites and circles. They serve as a unifying concept. Once the bisector system of any concrete type of Voronoi diagram is shown to fulfill the AVD properties, structural results and efficient algorithms become available without further effort.In a concrete order-k Voronoi diagram, all points are placed into the same region that have the same k nearest neighbors among the given sites. This paper is the first to study abstract Voronoi diagrams of arbitrary order k. We prove that their complexity in the plane is upper bounded by 2k(n−k). So far, an O(k(n−k)) bound has been shown only for point sites in the Euclidean and Lp planes, and, recently, for line segments, in the Lp metric. These proofs made extensive use of the geometry of the sites.Our result on AVDs implies a 2k(n−k) upper bound for a wide range of cases for which only trivial upper complexity bounds were previously known, and a slightly sharper bound for the known cases. Also, our proof shows that the reasons for this bound are combinatorial properties of certain permutation sequences.