Geometric Context Research Articles

Cohomological rank functions on abelian varieties

Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of $\mathbb Q$-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincare transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GV-subschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.

Spontaneous object-location memory based on environmental geometry is impaired by both hippocampal and dorsolateral striatal lesions.

We examined the role of the hippocampus and the dorsolateral striatum in the representation of environmental geometry using a spontaneous object recognition procedure. Rats were placed in a kite-shaped arena and allowed to explore two distinctive objects in each of the right-angled corners. In a different room, rats were then placed into a rectangular arena with two identical copies of one of the two objects from the exploration phase, one in each of the two adjacent right-angled corners that were separated by a long wall. Time spent exploring these two objects was recorded as a measure of recognition memory. Since both objects were in different locations with respect to the room (different between exploration and test phases) and the global geometry (also different between exploration and test phases), differential exploration of the objects must be a result of initial habituation to the object relative to its local geometric context. The results indicated an impairment in processing the local geometric features of the environment for both hippocampus and dorsolateral striatum lesioned rats compared with sham-operated controls, though a control experiment showed these rats were unimpaired in a standard object recognition task. The dorsolateral striatum has previously been implicated in egocentric route-learning, but the results indicate an unexpected role for the dorsolateral striatum in processing the spatial layout of the environment. The results provide the first evidence that lesions to the hippocampus and dorsolateral striatum impair spontaneous encoding of local environmental geometric features.

Catadioptric images compression using an adapted neighborhood and the shape-adaptive DCT

In the catadioptric images, the neighborhood topology is usually modified by the mirror shape which consists of significant radial distortions and low non-uniform resolution because of its convexity geometry. Consequently, conventional compression approaches (i.e. H.264 intra-frame coding or JPEG standard) based on rectangular block partitioning cannot be applied directly to catadioptric images because they inevitably produce significant block effects. In this investigation, we propose a new coding scheme based on an adaptive block partitioning. Particularly, to reduce the blocking artifacts, we propose the shape-adaptive discrete cosine transform (SA-DCT) and adapted neighborhood. The adapted neighborhood is obtained from the equivalence between the catadioptric image and the scene point’s projection on the unit sphere. The adapted blocks shape correctly follows the geometric context and varies according to the resolution of the catadioptric images. Such changes improve considerably the compression efficiency using the block partitioning in catadioptric images. Experimental results figured out that our proposal scheme outperforms the JPEG standard and the Intra-frame coder H264. Moreover the proposed coding scheme based on SA-DCT transform and the suitable partitioning for catadioptric images provides high PSNR and a considerable reduction of block effects compared to conventional approaches, especially in the case of low bit rate.

Nori Diagrams and Persistent Homology

Recently, it was found that there is a remarkable intuitive similarity between studies in theoretical computer science dealing with large data sets on the one hand, and categorical methods of topology and geometry in pure mathematics, on the other. In this article, we treat the key notion of persistency from computer science in the algebraic geometric context involving Nori motivic constructions and related methods. We also discuss model structures for persistent topology.

On the geometry of φ-curvatures

The aim of this paper is to continue the geometric-analytic study of φ-curvatures initiated in [2]. These curvatures arise naturally in many geometric contexts, notably in Ricci-harmonic solitons theory. In the present paper we prove two rigidity results related to harmonic-Einstein manifolds, a generalization of the notion of Einstein manifolds to the present situation. We observe that, when we restrict our theorems to the classical case of Einstein manifolds, we obtain some new results even in this setting.

Razonamiento configural y desarrollo del discurso en la resolución de problemas empíricos en contexto geométrico

El objetivo de este estudio es identificar relaciones entre los procesos cognitivos involucrados en la resolución de problemas geométricos empíricos, desde la perspectiva del razonamiento configural, y el modo como los estudiantes desarrollan sus discursos escritos para comunicar la solución. Analizamos las respuestas escritas de 33 estudiantes de primero de Bachillerato a dos problemas empíricos. Los resultados muestran la tendencia de los estudiantes a ignorar la configuración geométrica presentada una vez establecidas en registro algebraico las relaciones necesarias y suficientes que permiten resolver el problema. Además, se pone de manifiesto, en el desarrollo del discurso, la transición desde el modo de acumulación al modo de sustitución en los razonamientos que permiten establecer una solución válida, así como la influencia de la configuración inicial en el desenlace del razonamiento configural.

A look to differential

This paper reveals results of a research related to the presentation of the concept of differential for functions of a variable, from some calculus textbooks. Given that the concept of differential is closely linked to the concepts of derivative, integral and differential equation, the objective is to analyze the different ways of presenting the concept from those instances, in order to determine whether the presentation facilitates its understanding. The research follows a qualitative methodology based on a bibliographic review and conceptual didactic analysis, proceeding from content analysis corresponding to the notion of differential. It was found that the concept of differential in textbooks is presented in a geometric context, as an increment of a variable and as a function, dispossessed of its arithmetic meaning, being impossible to suppress the concept in the formal construction of the calculus.

Individual variation of the masticatory system dominates 3D skull shape in the herbivory-adapted marsupial wombats

BackgroundWithin-species skull shape variation of marsupial mammals is widely considered low and strongly size-dependent (allometric), possibly due to developmental constraints arising from the altricial birth of marsupials. However, species whose skulls are impacted by strong muscular stresses – particularly those produced through mastication of tough food items – may not display such intrinsic patterns very clearly because of the known plastic response of bone to muscle activity of the individual. In such cases, allometry may not dominate within-species shape variation, even if it is a driver of evolutionary shape divergence; ordination of shape in a geometric morphometric context through principal component analysis (PCA) should reveal main variation in areas under masticatory stress (incisor region/zygomatic arches/mandibular ramus); but this main variation should emerge from high individual variability and thus have low eigenvalues.ResultsWe assessed the evidence for high individual variation through 3D geometric morphometric shape analysis of crania and mandibles of three species of grazing-specialized wombats, whose diet of tough grasses puts considerable strain on their masticatory system. As expected, we found little allometry and low Principal Component 1 (PC1) eigenvalues within crania and mandibles of all three species. Also as expected, the main variation was in the muzzle, zygomatic arches, and masticatory muscle attachments of the mandibular ramus. We then implemented a new test to ask if the landmark variation reflected on PC1 was reflected in individuals with opposite PC1 scores and with opposite shapes in Procrustes space. This showed that correspondence between individual and ordinated shape variation was limited, indicating high levels of individual variability in the masticatory apparatus.DiscussionOur results are inconsistent with hypotheses that skull shape variation within marsupial species reflects a constraint pattern. Rather, they support suggestions that individual plasticity can be an important determinant of within-species shape variation in marsupials (and possibly other mammals) with high masticatory stresses, making it difficult to understand the degree to which intrinsic constraints act on shape variation at the within-species level. We conclude that studies that link micro- and macroevolutionary patterns of shape variation might benefit from a focus on species with low-impact mastication, such as carnivorous or frugivorous species.

MUGGLE: MUlti-Stream Group Gaze Learning and Estimation

Being able to accurately predict the common gaze point of a group of persons is of particular interest to precise marketing and automatic group attention assessment. Group gaze estimation faces challenges including small face/head size and outlier observers. To address these challenges, we proposed a novel framework called Multi-stream Group Gaze Learning and Estimation (MUGGLE). The MUGGLE infrastructure includes two inference streams: 1) a holistic stream which utilizes fused attention map as input to a global deep convolutional structure to explore the global geometric configurations and contexts of interesting persons in the scene; and 2) an aggregative stream which robustly aggregates individual gazes via a recurrent structure (e.g., LSTM) to obtain outlier-tolerant estimation. Both streams are seamlessly integrated via a fusion network. Extensive experiments are performed on a fully annotated group gaze image dataset with 8,000+ images and 100,000+ faces (which is publicly releasable). The results demonstrate the effectiveness of the proposed MUGGLE framework in group gaze estimation.

Formality and Kontsevich–Duflo type theorems for Lie pairs

Kontsevich's formality theorem states that there exists an L∞ quasi-isomorphism from the dgla Tpoly•(M) of polyvector fields on a smooth manifold M to the dgla Dpoly•(M) of polydifferential operators on M, which extends the classical Hochschild–Kostant–Rosenberg map. In this paper, we extend Kontsevich's formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and g-manifolds (that is, manifolds endowed with an action of a Lie algebra g). The spaces tot(Γ(Λ•A∨)⊗RTpoly•) and tot(Γ(Λ•A∨)⊗RDpoly•) associated with a Lie pair (L,A) each carry an L∞ algebra structure canonical up to L∞ isomorphism. These two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators, respectively. Their corresponding cohomology groups HCE•(A,Tpoly•) and HCE•(A,Dpoly•) admit canonical Gerstenhaber algebra structures. We establish the following formality theorem for Lie pairs: there exists an L∞ quasi-isomorphism from tot(Γ(Λ•A∨)⊗RTpoly•) to tot(Γ(Λ•A∨)⊗RDpoly•) whose first Taylor coefficient is equal to hkr∘(tdL/A∇)12. Here the cocycle (tdL/A∇)12 acts on tot(Γ(Λ•A∨)⊗RTpoly•) by contraction. Furthermore, we prove a Kontsevich–Duflo type theorem for Lie pairs: the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the Lie pair (L,A) is an isomorphism of Gerstenhaber algebras from HCE•(A,Tpoly•) to HCE•(A,Dpoly•). As applications, we establish formality and Kontsevich–Duflo type theorems for complex manifolds, foliations, and g-manifolds. In the case of complex manifolds, we recover the Kontsevich–Duflo theorem of complex geometry.

Resonant Mirković–Vilonen polytopes and formulas for highest-weight characters

Formulas for the product of an irreducible character $$\chi _\lambda $$ of a complex Lie group and a deformation of the Weyl denominator as a sum over the crystal $${\mathcal {B}}(\lambda +\rho )$$ go back to Tokuyama. We study the geometry underlying such formulas using the expansion of spherical Whittaker functions of p-adic groups as a sum over the canonical basis $${\mathcal {B}}(-\infty )$$ , which we show may be understood as arising from tropicalization of certain toric charts that appear in the theory of total positivity and cluster algebras. We use this to express the terms of the expansion in terms of the corresponding Mirkovic–Vilonen polytope. In this non-archimedean setting, we identify resonance as the appropriate analogue of total positivity, and introduce resonant Mirkovic–Vilonen polytopes as the corresponding geometric context. Focusing on the exceptional group $$G_2$$ , we show that these polytopes carry new crystal graph structures which we use to compute a new Tokuyama-type formula as a sum over $${\mathcal {B}}(\lambda +\rho )$$ plus a geometric error term coming from finitely many crystals of resonant polytopes.

Étale stacks as prolongations

We give a complete and categorical characterization of étale stacks (generalized orbifolds) in various geometric contexts, including differentiable stacks and topological stacks. This characterization states that étale stacks are precisely those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms, and makes no mention of an atlas or groupoid presentation. Moreover, we show that the bicategory of étale differentiable stacks and local diffeomorphisms is equivalent to the bicategory of stacks on the site of smooth manifolds and local diffeomorphisms. An analogous statement holds for other flavors of manifolds (topological, Ck, complex, super...), and topological spaces locally homeomorphic to a given space X. A slight modification of this result also holds in an even more general context, including all étale topological stacks, and Zariski étale stacks, which are analogues of Deligne-Mumford stacks for the Zariski topology, and we also sketch a proof of an analogous characterization of classical Deligne-Mumford algebraic stacks (which we prove in full in [7] by different methods). We go on to characterize effective étale stacks (e.g. reduced orbifolds) as precisely those stacks arising as the prolongations of sheaves of sets (rather than stacks of groupoids). It follows that étale stacks (and in particular orbifolds) induce a small gerbe over their effective part, and all gerbes over effective étale stacks arise in this way. We show that well known Lie groupoids from foliation theory literature can be reproduced from this framework as natural presentations for moduli stacks of certain geometric structures. For example, there exists a classifying stack for Riemannian metrics, presented by Haefliger's groupoid RΓ [13] and submersions into this stack classify Riemannian foliations, and similarly for symplectic structures, with the role of RΓ replaced with ΓSp. We also prove some unexpected results, for example: the category of smooth n-manifolds and local diffeomorphisms has binary products.

An Original and Additional Mathematical Model Characterizing a Bayesian Approach to Decision Theory

We propose an original mathematical model according to a Bayesian approach explaining uncertainty from a point of view connected with vector spaces. A parameter space can be represented by means of random quantities by accepting the principles of the theory of concordance into the domain of subjective probability. We observe that metric properties of the notion of $\alpha$-product mathematically fulfill the ones of a coherent prevision of a bivariate random quantity. We introduce fundamental metric expressions connected with transformed random quantities representing changes of origin. We obtain a posterior probability law by applying the Bayes' theorem into a geometric context connected with a two-dimensional parameter space.

On the Poisson boundary of the relativistic Brownian motion

Dans cet article, nous determinons la frontiere de Poisson du mouvement brownien relativiste dans deux classes de varietes lorentziennes, les espaces modeles de courbure constante et les espaces de Robertson–Walker qui constituent une vaste famille d’espace-temps courbes. Notre objectif est double : d’une part, il s’agit de comprendre les relations entre la geometrie a l’infini de ces varietes et le comportement asymptotique des trajectoires browniennes, avec comme objectif de comparer la compactification stochastique formee par les points de sortie du processus aux frontieres purement geometriques, conformes ou causales. D’autre part, nous souhaitons illustrer la pertinence de la methode de devissage introduite par les auteurs (Seminaire de Probabilites XLVIII (2016) 199–229 Springer), methode qui s’avere particulierement bien adaptee aux differents contextes geometriques consideres.

Construcción cognitiva del conjunto solución de un sistema de ecuaciones lineales con dos incógnitas

En esta investigación proponemos una descomposición genética para el conjunto solución de un sistema de ecuaciones lineales con dos incógnitas, mediante un tránsito desde sistemas homogéneos a no homogéneos, en un contexto geométrico-cartesiano. Para validar nuestra descomposición genética diseñamos instrumentos que aplicamos a estudiantes de formación inicial del profesorado de matemáticas para educación secundaria. Con ello, y haciendo uso de la estadística implicativa, logramos confirmar las estructuras mentales dispuestas en nuestra descomposición genética. Los resultados evidencian cierta incomprensión de lo que es una solución para un sistema, dificultades para articular los aspectos geométricos con los algebraicos y la conveniencia de utilizar una estrategia alternativa para el caso de un sistema de tres o más ecuaciones lineales.

Risk and Reliability Analysis of Geometric Design Criteria: A Critical Synthesis

Highway and street designers deal with the challenge of designing for a broad range of driver, vehicle, and roadway characteristics and conditions. There is significant variability in design inputs and design controls that influence design criteria and design decisions. This variability has traditionally been considered implicitly through selected values for geometric design parameters and criteria. Average values are used when the variability in the input design parameters is small. Conservative values are used if the variability is larger, often the case within the highway geometric design context. Previous research has demonstrated that addressing this variability and uncertainty more explicitly as part of design decisions can lead to better-informed and more cost-effective design decisions. Probabilistic design approaches that quantify both risk and reliability have been successfully incorporated into other design disciplines for those reasons. These approaches have also been explored in the highway geometric design literature and have shown promise. They are likely to be central to future performance-based design initiatives, as outlined in a recently published framework on conducting performance-based geometric design analysis. Given the emerging importance of performance-based design and the need to address challenges regarding the current method of handling variability and uncertainty in the input design parameters, this paper presents a collective review and assessment of methodological alternatives for quantifying risk and reliability associated with geometric design criteria and decisions.

Are all identically conserved geometric tensors metric variations of actions? A status report

Noether's theorem, that local gauge variations of gauge invariant actions are identically conserved (more tautologically, that gauge variations of gauge invariants vanish) was established a century ago. Its converse, in the geometric context: are all identically conserved local symmetric tensors variations of some coordinate invariant action? remains unsolved to this day. We survey its present state and discuss some of our concrete attempts at a solution, including a significant improvement. For notational simplicity, details are primarily given in D=2, but we discuss generic D as well.

Geometria alla scuola dell’infanzia: una pseudo-definizione di circonferenza

This paper deals with a teaching experiment concerning the notion of circumference in a geometrical context, conducted in kindergarten with children aged 4-5. Through a classroom-based intervention, designed according to the Semiotic Mediation theoretical framework and developed on a multimodal approach, children produce a “pseudo-definition of circumference” (Robotti, 2018), which still refers to perceptual elements linked to the shape, but where it is possible to identify the dynamic nature of the curve as a trace generated by the movement of a point. The analysis of the teaching experiment highlights the specific roles of the teacher and of artifacts in supporting the process of semiotic mediation through which the children and the teacher transform the signs linked to artifacts into mathematical signs.

3D floor plan recovery from overlapping spherical images

We present a novel approach to automatically recover, from a small set of partially overlapping spherical images, an indoor structure representation in terms of a 3D floor plan registered with a set of 3D environment maps. We introduce several improvements over previous approaches based on color and spatial reasoning exploiting Manhattan world priors. In particular, we introduce a new method for geometric context extraction based on a 3D facet representation, which combines color distribution analysis of individual images with sparse multi-view clues. We also introduce an efficient method to combine the facets from different viewpoints in a single consistent model, taking into the reliability of the facet information. The resulting capture and reconstruction pipeline automatically generates 3D multi-room environments in cases where most previous approaches fail, e.g., in the presence of hidden corners and large clutter, without the need for additional dense 3D data or tools. We demonstrate the effectiveness and performance of our approach on different real-world indoor scenes. Our test data is available to allow further studies and comparisons.

Integration of point cloud data and hyperspectral imaging as a data gathering methodology for refurbishment projects using building information modelling (BIM)

PurposeBuilding information modelling (BIM) is a digital representation of the physical and functional characteristics of a building. Its use offers a range of benefits in terms of achieving the efficient design, construction, operation and maintenance of buildings. Applying BIM at the outset of a new build project should be relatively easy. However, it is often problematic to apply BIM techniques to an existing building, for example, as part of a refurbishment project or as a tool supporting the facilities management strategy, because of inadequacies in the previous management of the dataset that characterises the facility in question. These inadequacies may include information on as built geometry and materials of construction. By the application of automated retrospective data gathering for use in BIM, such problems should be largely overcome and significant benefits in terms of efficiency gains and cost savings should be achieved.Design/methodology/approachLaser scanning can be used to collect geometrical and spatial information in the form of a 3D point cloud, and this technique is already used. However, as a point cloud representation does not contain any semantic information or geometrical context, such point cloud data must refer to external sources of data, such as building specification and construction materials, to be in used in BIM.FindingsHyperspectral imaging techniques can be applied to provide both spectral and spatial information of scenes as a set of high-resolution images. Integrating of a 3D point cloud into hyperspectral images would enable accurate identification and classification of surface materials and would also convert the 3D representation to BIM.Originality/valueThis integrated approach has been applied in other areas, for example, in crop management. The transfer of this approach to facilities management and construction would improve the efficiency and automation of the data transition from building pathology to BIM. In this study, the technological feasibility and advantages of the integration of laser scanning and hyperspectral imaging (the latter not having previously been used in the construction context in its own right) is discussed, and an example of the use of a new integration technique is presented, applied for the first time in the context of buildings.