Geometrical Spaces Research Articles

Role of the Cerebellum in the Construction of Functional and Geometrical Spaces.

The perceptual and motor systems appear to have a set of movement primitives that exhibit certain geometric and kinematic invariances. Complex patterns and mental representations can be produced by (re)combining some simple motor elements in various ways using basic operations, transformations, and respecting a set of laws referred to as kinematic laws of motion. For example, point-to-point hand movements are characterized by straight hand paths with single-peaked-bell-shaped velocity profiles, whereas hand speed profiles for curved trajectories are often irregular and more variable, with speed valleys and inflections extrema occurring at the peak curvature. Curvature and speed are generically related by the 2/3 power law. Mathematically, such laws can be deduced from a combination of Euclidean, affine, and equi-affine geometries, whose neural correlates have been partially detected in various brain areas including the cerebellum and the basal ganglia. The cerebellum has been found to play an important role in the control of coordination, balance, posture, and timing over the past years. It is also assumed that the cerebellum computes forward internal models in relationship with specific cortical and subcortical brain regions but its motor relationship with the perceptual space is unclear. A renewed interest in the geometrical and spatial role of the cerebellum may enable a better understanding of its specific contribution to the action-perception loop and behavior's adaptation. In this sense, we complete this overview with an innovative theoretical framework that describes a possible implementation and selection by the cerebellum of geometries adhering to different mathematical laws.

Difficulties and Coping Strategies in Understanding Mathematical Concepts in a Private Higher Education in Tagum City, Davao del Norte, Philippines

Understanding mathematical concepts involve developing an intuitive grasp of the underlying ideas and principles that govern mathematical operations and relationships. The primary purpose of this study is to investigate the difficulties and coping strategies in understanding mathematical concepts. This study used an explanatory sequential research design through the collection and analysis of quantitative data followed by a qualitative inquiry. To collect data, the researchers developed questionnaires for the quantitative and qualitative inquiries, which were validated by external experts. The sample of the study for the quantitative part included conveniently selected 297 students from a private Higher Education Institution (HEI) in Tagum City, Davao del Norte, Philippines, in the academic year 2017-2018. In addition, seven (7) individuals among the respondents participated in the Key Informant Interviews (KIIs). This was followed by one session of Focus Group Discussion (FGD) which was also participated by seven first-year students enrolled during the second semester of S.Y. 2017-2018. Five skills areas were examined: number fact, arithmetic, language, informational, and visual. Based on the results, students find difficulties in understanding mathematical concepts, specifically in visualizing mathematical concepts and manipulating geometrical shapes and spaces meaningfully. The students also demonstrated difficulties in analyzing the problem, understanding the concept, identifying the correct solution, establishing the equation, and simplifying the expression. In addition, the students cope with the difficulties through perseverance, expert consultation, reading and studying, constant practice, and confidence building. Mathematics teachers are recommended to design need-based teaching strategies to optimize student learning.

The Geometry of Misinformation: Embedding Twitter Networks of Users Who Spread Fake News in Geometrical Opinion Spaces

To understand why internet users spread fake news online, many studies have focused on individual drivers, such as cognitive skills, media literacy, or demographics. Recent findings have also shown the role of complex socio-political dynamics, highlighting that political polarization and ideologies are closely linked to a propensity to participate in the dissemination of fake news. Most of the existing empirical studies have focused on the US example by exploiting the self-reported or solicited positioning of users on a dichotomous scale opposing liberals with conservatives. Yet, left-right polarization alone is insufficient to study socio-political dynamics when considering non binary and multi-dimensional party systems, in which relevant ideological stances must be characterized in additional dimensions, relating for example to opposition to elites, government, political parties or mainstream media. In this article we leverage ideological embeddings of Twitter networks in France in multi-dimensional opinions spaces, where dimensions stand for attitudes towards different issues, and we trace the positions of users who shared articles that were rated as misinformation by fact-checkers. In multi-dimensional settings, and in contrast with the US, opinion dimensions capturing attitudes towards elites are more predictive of whether a user shares misinformation. Most users sharing misinformation hold salient anti-elite sentiments and, among them, more so those with radical left- and right-leaning stances. Our results reinforce the importance of enriching one-dimensional left-right analyses, showing that other ideological dimensions, such as anti-elite sentiment, are critical when characterizing users who spread fake news. This lends support to emerging accounts of social drivers of misinformation through political polarization, but also stresses the role of the entanglement between fake news, anti-elite polarization, and the role of scientific authorities in public debate.

Collective migration of cells in geometric spaces: Intrinsic correlation length racing against extrinsic confinement size

Many experiments showed that cell groups can exhibit various impressive motion modes (e.g., global rotation and local swirling) in different-sized geometrical spaces, with the underlying mechanisms still unclear. However, recent experiments showed that different types of cells perform distinct motion modes (i.e., global rotation and coherent oscillation) in the same constraint conditions, which brings further puzzles on the movements of collective cells. Here, by comprehensively considering the social interactions between adjacent cells and internal mechanical processes of the cell itself, we propose an active vertex model to bring physical insights into collective cell migration in geometrical constraints. Using this model, we can reproduce all reported motion modes of cells in circular constraints observed in existing experiments. Specifically, as the confinement size decreases, local swirling and global rotation modes successively appear in cell groups with finite correlation length, while eccentric rotation and coherent oscillation modes successively emerge in cell groups with scale-free correlations. We demonstrate that these distinct modes are subtly coded by the intrinsic correlation length of cell groups and the extrinsic confinement size. Unexpectedly, we discover that, in small-size confinements, cell groups with scale-free correlations can spontaneously transition from eccentric rotation into coherent oscillation modes, because the strong social interactions reconcile the movements of all cells. In addition, we propose a velocity indicator that can directly distinguish migration modes emerging in small-scale confinements. These findings are in broad agreement with many experiments, and shed light on the spatiotemporal dynamics of active matter.

Data enhanced iterative few-sample learning algorithm-based inverse design of 2D programmable chiral metamaterials

Abstract A data enhanced iterative few-sample (DEIFS) algorithm is proposed to achieve the accurate and efficient inverse design of multi-shaped 2D chiral metamaterials. Specifically, three categories of 2D diffractive chiral structures with different geometrical parameters, including widths, separation spaces, bridge lengths, and gold lengths are studied utilising both the conventional rigorous coupled wave analysis (RCWA) approach and DEIFS algorithm, with the former approach assisting the training process for the latter. The DEIFS algorithm can be divided into two main stages, namely data enhancement and iterations. Firstly, some “pseudo data” are generated by a forward prediction network that can efficiently predict the circular dichroism (CD) response of 2D diffractive chiral metamaterials to reinforce the dataset after necessary denoising. Then, the algorithm uses the CD spectra and the predictions of parameters with smaller errors iteratively to achieve accurate values of the remaining parameters. Meanwhile, according to the impact of geometric parameters on the chiroptical response, a new functionality is added to interpret the experimental results of DEIFS algorithm from the perspective of data, improving the interpretability of the DEIFS. In this way, the DEIFS algorithm replaces the time-consuming iterative optimization process with a faster and simpler approach that achieves accurate inverse design with dataset whose amount is at least one to two orders of magnitude less than most previous deep learning methods, reducing the dependence on simulated spectra. Furthermore, the fast inverse design of multiple shaped metamaterials allows for different light manipulation, demonstrating excellent potentials in applications of optical coding and information processing. This work belongs to one of the first attempts to thoroughly characterize the flexibility, interpretability, and generalization ability of DEIFS algorithm in studying various chiroptical effects in metamaterials and accelerating the inverse design of hypersensitive photonic devices.

Cultural divisions and time: Mapping diachronic homologies using class-specific MCA (CSA)

In this article, we present an alternative approach to study dimensional stability and change in cultural divisions across time. Drawing on recent developments in Geometric Data Analysis (GDA), we combine the use of two distinct statistical techniques: Multiple Correspondence Analysis (MCA) and Class Specific MCA (CSA). Specifically, the approach allows for the systematic investigation of three key aspects: (i) whether and how the dimensionality and the structure of primary axes of geometrical spaces have been stable or subject to change; (ii) how the oppositions depicted by the axes of such spaces have become weaker or stronger; and (iii), how the connection between cultural divisions and class divisions has developed over time. Our exemplary case is Norway and the data stem from five different rounds of The Culture and Media Survey, conducted by Statistics Norway between 2000 and 2016.

Geometrical analysis of motion schemes on fencing experts from competition videos.

Geometrical fencing is a scientific approach to fencing pioneered by Camillo Agrippa in the XVIth century which consists of characterizing the geometrical structure of fencing movements. Many geometrical spaces are involved in a duel, which evolve over time according to the skills of the fencers and the game rules. In this article, the concept of motion scheme is introduced as a flexible geometrical structure to represent fencing spaces evolving over time. The method is applied to the video of a duel of the Olympic games 2016. Five main results are presented. First, decisive actions of the duel are deduced from the distance between fencers. Second, footwork is reconstructed from horizontal movements of the feet. Third, a kinematic model is developed and compared with data in the literature. Fourth, the lunge attack is characterized and compared with data in the literature. Fifth, the role of the free hand is studied in the case of protective and balancing gestures. These findings provide rich information on the geometrical structure of fencing movements as well as on the tactical-strategic choices made by the fencers in real competition conditions. Finally, four applications illustrate the scientific value of motion schemes in fencing and other sports.

Unique water H-bonding types on metal surfaces: from the bonding nature to cooperativity rules

Understanding the nature of H-bonding interactions is essential to modern sciences, such as biology, chemistry, and physics. Using density functional theory calculations, herein, we have identified two unique H-bonding types existing in a single sheet of a mixed water–hydroxyl phase on close-packed metal surfaces, in sharp contrast to conventional H-bonds in liquid water and water ices. Interestingly, the shallow H-bonds show reduced electrostatic and Pauli repulsion interactions, with an electrostatic polar character resulted from complete σ resonances, whereas the deep H-bonds exhibit enhanced electrostatic and Pauli repulsion interactions, with an electrostatic dipolar feature originated from hybrid orbital interactions. A trade-off-like cooperativity law of the two types of H-bonds was discovered, that is, strengthening in the internal bonds (dO–H) leads to weakening in the external bonds (dO:H) or vice versa. However, the shallow H-bonds exhibit a non-linear cooperativity, whereas the deep H-bonds show a linear cooperativity. We also identified an oxygen backbone cooperativity rule that strengthening the adsorbate–metal interactions has a net effect in analogy to reducing the O–O repulsion within the adlayer. Furthermore, we have discovered several universality classes in geometrical, vibrational, and electronic spaces for the two H-bonding types. Although shared by electronic universality classes, the two contrasting H-bonding types are featured by divergent trends with significant overlapping, where competitive variations in the electrostatic and Pauli repulsion strengths are basic rules for the cooperative H-bonding types. The knowledge of the unconventional H-bonding types expands our current understanding of H-bonding interactions in liquid water and water ices and points to the importance of H-bonding manipulation at electronic levels. These findings not only shed new light on probing the fundamental nature of H-bonds in general but also have insightful implications for resolving the cooperative H-bonding nature of interfacial water, liquid water, water ices, and aqueous solutions.

Topographically guided hierarchical mineralization.

Material platforms based on interaction between organic and inorganic phases offer enormous potential to develop materials that can recreate the structural and functional properties of biological systems. However, the capability of organic-mediated mineralizing strategies to guide mineralization with spatial control remains a major limitation. Here, we report on the integration of a protein-based mineralizing matrix with surface topographies to grow spatially guided mineralized structures. We reveal how well-defined geometrical spaces defined within the organic matrix by the surface topographies can trigger subtle changes in single nanocrystal co-alignment, which are then translated to drastic changes in mineralization at the microscale and macroscale. Furthermore, through systematic modifications of the surface topographies, we demonstrate the possibility of selectively guiding the growth of hierarchically mineralized structures. We foresee that the capacity to direct the anisotropic growth of such structures would have important implications in the design of biomineralizing synthetic materials to repair or regenerate hard tissues.

Topological surfaces as gridded surfaces in geometrical spaces

In this paper, we study topological surfaces as discrete models by means of gridded surfaces in the two-dimensional scaffolding of cubic honeycombs in Euclidean and hyperbolic spaces.

Increasing the Sampling Efficiency of Protein Conformational Change by Combining a Modified Replica Exchange Molecular Dynamics and Normal Mode Analysis.

Understanding conformational change at an atomic level is significant when determining a protein functional mechanism. Replica exchange molecular dynamics (REMD) is a widely used enhanced sampling method to explore protein conformational space. However, REMD with an explicit solvent model requires huge computational resources, immensely limiting its application. In this study, a variation of parallel tempering metadynamics (PTMetaD) with the omission of solvent-solvent interactions in exchange attempts and the use of low-frequency modes calculated by normal-mode analysis (NMA) as collective variables (CVs), namely ossPTMetaD, is proposed with the aim to accelerate MD simulations simultaneously in temperature and geometrical spaces. For testing the performance of ossPTMetaD, five protein systems with diverse biological functions and motion patterns were selected, including large-scale domain motion (AdK), flap movement (HIV-1 protease and BACE1), and DFG-motif flip in kinases (p38α and c-Abl). The simulation results showed that ossPTMetaD requires much fewer numbers of replicas than temperature REMD (T-REMD) with a reduction of ∼70% to achieve a similar exchange ratio. Although it does not obey the detailed balance condition, ossPTMetaD provides consistent results with T-REMD and experimental data. The high accessibility of the large conformational change of protein systems by ossPTMetaD, especially in simulating the very challenging DFG-motif flip of protein kinases, demonstrated its high efficiency and robustness in the characterization of the large-scale protein conformational change pathway and associated free energy profile.

METASET: Exploring Shape and Property Spaces for Data-Driven Metamaterials Design

Abstract Data-driven design of mechanical metamaterials is an increasingly popular method to combat costly physical simulations and immense, often intractable, geometrical design spaces. Using a precomputed dataset of unit cells, a multiscale structure can be quickly filled via combinatorial search algorithms, and machine learning models can be trained to accelerate the process. However, the dependence on data induces a unique challenge: an imbalanced dataset containing more of certain shapes or physical properties can be detrimental to the efficacy of data-driven approaches. In answer, we posit that a smaller yet diverse set of unit cells leads to scalable search and unbiased learning. To select such subsets, we propose METASET, a methodology that (1) uses similarity metrics and positive semi-definite kernels to jointly measure the closeness of unit cells in both shape and property spaces and (2) incorporates Determinantal Point Processes for efficient subset selection. Moreover, METASET allows the trade-off between shape and property diversity so that subsets can be tuned for various applications. Through the design of 2D metamaterials with target displacement profiles, we demonstrate that smaller, diverse subsets can indeed improve the search process as well as structural performance. By eliminating inherent overlaps in a dataset of 3D unit cells created with symmetry rules, we also illustrate that our flexible method can distill unique subsets regardless of the metric employed. Our diverse subsets are provided publicly for use by any designer.

A New Geometric Metric in the Shape and Size Space of Curves in R n

Shape analysis of curves in Rn is an active research topic in computer vision. While shape itself is important in many applications, there is also a need to study shape in conjunction with other features, such as scale and orientation. The combination of these features, shape, orientation and scale (size), gives different geometrical spaces. In this work, we define a new metric in the shape and size space, S2, which allows us to decompose S2 into a product space consisting of two components: S4×R, where S4 is the shape space. This new metric will be associated with a distance function, which will clearly distinguish the contribution that the difference in shape and the difference in size of the elements considered makes to the distance in S2, unlike the previous proposals. The performance of this metric is checked on a simulated data set, where our proposal performs better than other alternatives and shows its advantages, such as its invariance to changes of scale. Finally, we propose a procedure to detect outlier contours in S2 considering the square-root velocity function (SRVF) representation. For the first time, this problem has been addressed with nearest-neighbor techniques. Our proposal is applied to a novel data set of foot contours. Foot outliers can help shoe designers improve their designs.

Structure of passive states and its implication in charging quantum batteries.

In this article, in addition to the characterization of geometrical state spaces for the passive states, an operational approach has been introduced to distinguish them on their charging capabilities of a quantum battery. Unlike the thermal states, the structural instability of passive states assures the existence of a natural number n, for which n+1 copies of the state can charge a quantum battery while n copies cannot. This phenomenon can be presented in an n copy resource-theoretic approach, for which the free states are unable to charge the battery in n copies. Here we have exhibited the single copy scenario explicitly. We also show that general ordering of the passive states on the basis of their charging capabilities is not possible and even the macroscopic entities (viz. energy and entropy) are unable to order them precisely. Interestingly, for some of the passive states, the majorization criterion gives sufficient order to the charging and discharging capabilities. However, the charging capacity for the set of thermal states (for which charging is possible) is directly proportional to their temperature.

Geometrical structure of perceptual color space: Mental representations and adaptation invariance.

Similarity between percepts and concepts is used to accomplish many everyday tasks, e.g., object identification; so this similarity is widely used to construct geometrical spaces that represent stimulus qualities, but the intrinsic validity of the geometry, i.e., whether similarity operations support a particular geometry, is almost never tested critically. We introduce an experimental approach for equating relative similarities by setting perceived midpoints between pairs of stimuli. Midpoint settings are used with Varignon's Theorem to test the intrinsic geometry of a representation space, and its mapping to a physical space of stimuli. For perceptual color space, we demonstrate that geometrical structure depends on the mental representation used in judging similarity: An affine geometry was valid when observers used an opponent-color mental representation. Similarities based on a conceptual space of complementary colors thus power a geometric coordinate system. An affine geometry implies that similarity can be judged within straight lines and across parallel lines, and its neural coding could involve ratios of responses. We show that this perceptual space is invariant to changes in illumination color, providing a formal justification to generalize color constancy results measured for color categories, to all of color space. The midpoint measurements deviate significantly from midpoints in the extensively used “uniform” color spaces CIELAB and CIELUV, showing that these spaces do not provide adequate metric representation of perceived colors. Our paradigm can thus test for intrinsic geometrical assumptions underlying the representation space for many perceptual modalities, and for the extrinsic perceptual geometry of the space of physical stimuli.

Multi-lead ECG signal analysis for myocardial infarction detection and localization through the mapping of Grassmannian and Euclidean features into a common Hilbert space

Background and objectiveElectrocardiogram is commonly used as a diagnostic tool for the monitoring of cardiac health and the detection of possible heart diseases. However, the procedure followed for the diagnosis of heart abnormalities is time consuming and prone to human errors. Thus, the development of computer-aided techniques for the automatic analysis of electrocardiogram signals is of vital importance for the diagnosis and prevention of heart diseases. The most serious outcome of coronary heart disease is the myocardial infarction, i.e., the rapid and irreversible damage of cardiac muscles, which, if not diagnosed and treated in time, continues to damage further the myocardial structure and function. In this paper we propose a novel approach for the automatic detection and localization of myocardial infarction from multi-lead electrocardiogram signals. MethodsThe proposed method initially reshapes the multidimensional signal into a third-order tensor structure and subsequently extracts feature representations in both Euclidean and Grassmannian space. In addition, two different methods are proposed for the mapping of the two different feature representations into a common Hilbert space before the final classification of signals. The first approach is based on the mapping of both Grassmannian and Euclidean features in a Reproducing Kernel Hilbert Space (RKHS), while the second one attempts to initially apply Vector of Locally Aggregated Descriptors (VLAD) encoding directly to Grassmann manifold and then concatenate the two VLAD representations. ResultsFor the evaluation of the proposed method, we have conducted extensive tests using a publicly available dataset, namely PTB Diagnostic ECG database, containing 549 multi-lead ECG data recordings from 290 subjects and from different diagnostic classes. The method provides an excellent detection rate of 100%, and localization rate, i.e., 100% with the first fusion method and 99.7% with the second one. ConclusionsThe Experimental results presented in this paper show the superiority of the proposed methodology against a number of state-of-the-art approaches. The main advantage of the proposed approach is that it exploits better the intercorrelations between signals of different ECG leads, by extracting feature representations that lie in different geometrical spaces and contain complementary information with regard to the dynamics of signals.

UNIFIED TOPOLOGICAL FRAMEWORK FOR RETRIEVING 2D AND 3D MULTI-SCALE SPATIAL INFORMATION

Abstract. Different applications or users need different model (differ requirement on level of details; geometry and information), less focuses in topological element. Improper manage geometry, attribute and topology elements in multi-scape environment will cause data repetition/redundancy, storage capacity and performance issues. This research focuses on utilizing topological graph which is important for various Geographical Information Science applications such as 3D indoor, network analysis, and sharing information/attribute from multiple level of details (LoD). A conceptual framework towards a unified data structure/model for representing several levels of details (LoDs) which integrates 2D and 3D topological and geometrical spaces in a single view is presented. The proposed framework has a potential of adopting into the CityGML/IndoorGML standards with representing different CityGML LoDs in a single view representation. Potential applications of the proposed conceptual framework including solar energy and utility mapping are discussed.

Sensory studies, or when physics was psychophysics: Ernst Mach and physics between physiology and psychology, 1860-71.

This paper highlights the significance of sensory studies and psychophysical investigations of the relations between psychic and physical phenomena for our understanding of the development of the physics discipline, by examining aspects of research on sense perception, physiology, esthetics, and psychology in the work of Gustav Theodor Fechner, Hermann von Helmholtz, Wilhelm Wundt, and Ernst Mach between 1860 and 1871. It complements previous approaches oriented around research on vision, Fechner's psychophysics, or the founding of experimental psychology, by charting Mach's engagement with psychophysical experiments in particular. Examining Mach's study of the senses and esthetics, his changing attitudes toward the mechanical worldview and atomism, and his articulation of comparative understandings of sensual, geometrical, and physical spaces helps set Mach's emerging epistemological views in the context of his teaching and research. Mach complemented an analytic strategy focused on parallel psychic and physical dimensions of sensation, with a synthetic comparative approach - building analogies between the retina, the individual, and social life, and moving between abstract and sensual spaces. An examination of the broadly based critique that Mach articulated in his 1871 lecture on the conservation of work shows how his historical approach helped Mach cast what he now saw as a narrowly limiting emphasis on mechanics as a phase yet to be overcome.

The creation and projection of ambiophonic and geometrical sonic spaces with reference to Denis Smalley's 'Base Metals'.

The creation and projection of ambiophonic and geometrical sonic spaces with reference to Denis Smalley's 'Base Metals'.

The creation and projection of ambiophonic and geometrical sonic spaces with reference to Denis Smalley's 'Base Metals'.

The creation and projection of ambiophonic and geometrical sonic spaces with reference to Denis Smalley's 'Base Metals'.