Geometric Language Research Articles

Primitive-Based 3D Human-Object Interaction Modelling and Programming

Embedding Human and Articulated Object Interaction (HAOI) in 3D is an important direction for a deeper human activity understanding. Different from previous works that use parametric and CAD models to represent humans and objects, in this work, we propose a novel 3D geometric primitive-based language to encode both humans and objects. Given our new paradigm, humans and objects are all compositions of primitives instead of heterogeneous entities. Thus, mutual information learning may be achieved between the limited 3D data of humans and different object categories. Moreover, considering the simplicity of the expression and the richness of the information it contains, we choose the superquadric as the primitive representation. To explore an effective embedding of HAOI for the machine, we build a new benchmark on 3D HAOI consisting of primitives together with their images and propose a task requiring machines to recover 3D HAOI using primitives from images. Moreover, we propose a baseline of single-view 3D reconstruction on HAOI. We believe this primitive-based 3D HAOI representation would pave the way for 3D HAOI studies. Our code and data are available at https://mvig-rhos.com/p3haoi.

Interpretable Geometry Problem Solving Using Improved RetinaNet and Graph Convolutional Network

This paper proposes an interpretable geometry solution based on the formal language set of text and diagram. Geometry problems are solved using machines; however, machines encounter challenges in natural language processing and computer vision. Significant progress has improved existing methods in the extraction of geometric formal languages. However, the neglect of the graph structure information in the formal language and the lack of further refinement of the extracted language set can lead to poor theorem prediction and poor accuracy in problem solving. In this paper, a formal language graph is constructed using the extracted formal language set and applied to theorem prediction using a graph convolutional network. To better extract the relationship set of diagram elements, an improved diagram parser is proposed. The test results indicate that the improved method has good results when solving interpretable geometry problems.

Protein–protein contact prediction by geometric triangle-aware protein language models

Protein–protein contact prediction by geometric triangle-aware protein language models

SHAPER: can you hear the shape of a jet?

The identification of interesting substructures within jets is an important tool for searching for new physics and probing the Standard Model at colliders. Many of these substructure tools have previously been shown to take the form of optimal transport problems, in particular the Energy Mover’s Distance (EMD). In this work, we show that the EMD is in fact the natural structure for comparing collider events, which accounts for its recent success in understanding event and jet substructure. We then present a Shape Hunting Algorithm using Parameterized Energy Reconstruction (Shaper), which is a general framework for defining and computing shape-based observables. Shaper generalizes N-jettiness from point clusters to any extended, parametrizable shape. This is accomplished by efficiently minimizing the EMD between events and parameterized manifolds of energy flows representing idealized shapes, implemented using the dual-potential Sinkhorn approximation of the Wasserstein metric. We show how the geometric language of observables as manifolds can be used to define novel observables with built-in infrared-and-collinear safety. We demonstrate the efficacy of the Shaper framework by performing empirical jet substructure studies using several examples of new shape-based observables.

Quantum integrability: Lagrangian 1-form case

A new notion of integrability called the multi-dimensional consistency for the integrable systems with the Lagrangian 1-form structure is captured in the geometrical language for quantum level. A zero-curvature condition, which implies the multi-dimensional consistency, will be a key relation, e.g. Hamiltonian operators. Therefore, the existence of the zero-curvature condition directly leads to the path-independent feature of the mapping, e.g. multi-time evolution in the Schrödinger picture. Another important result is the formulation of the continuous multi-time propagator. With this new type of the propagator, a new perspective on summing all possible paths unavoidably arises as not only all possible paths in the space of dependent variables but also in the space of independent variables must be taken into account. The semi-classical approximation is applied to the multi-time propagator expressing in terms of the classical action and the fluctuation around it. Therefore, the extremum propagator, resulting in path independent feature on the space of independent variables, would guarantee the integrability of the system.

La precision dans la mesure du temps et la Nouvelle Philosophie: Étude de cas (Alexandre Koyré & Richard Westfall)

The 17th century is considered to be a century of great upheaval, of a real transformation in the way scholars apprehended the world. This change, for the majority of historical studies, goes hand in hand with the birth of the so-called New Natural Philosophy. It is argued that before the 17th century, in order to describe, understand and explain the world, scholars used a philosophictheological language that was later replaced by a mathematical (geometrical) language. This new way of contemplating and trying to understand the world is often associated, rightly or wrongly, with the birth of the quest for accuracy, and more precisely for accuracy in time measurement. By means of a case study, this article seeks to question this link highlighted by historians of the second half of the twentieth century. The present article is a study of two key articles that share the same object, accuracy in time measurement, written by two authors who have influenced, and still do, a large part of scholarly thinking, Alexander Koyre and Richard Westfall.

Analyzing the Architectural Characteristics of Tadao Ando’s Museum Projects

The museum is one of the building types that lead the development of architecture, transcending physical borders and regionalism and revealing the experimental nature of modern architecture. Japanese architect Tadao Ando is a representative architect of contemporary architecture but also designed the most significant number of museums and sublimated traditional Japanese architecture into a modern interpretation. This paper explores Ando’s philosophical background, which seeks to inform the relationship between architecture, humans, and the environment via unique Japanese culture. As a methodology, an existing literature study collected and analyzed various materials necessary to understand Tadao Ando's architectural theory and museum. His workspace's components and details of expression were observed and explored through field visits. Regarding the characteristics of his work, the architectural background and history, and the overall spatial composition and features of the pieces were analyzed with references and drawings. The result showed that Tadao Ando has been constructing architecture using the geometric formative language formed through his experiences in various fields, influenced by Western classical architecture and abstract art. Tadao Ando embodies his attitude towards nature, which is entrenched within him. As a result of a holistic analysis of Tadao Ando's architecture after 1985, It was proven that it is a phenomenon that appears throughout the work, and as the plane of geometry is repeated, the logic of form composition is consistently shown.

The isospectral problem for flat tori from three perspectives

Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed. In 1964, Milnor used a construction of Witt to find an example of isospectral nonisometric Riemannian manifolds, a striking and concise result that occupied one page in the Proceedings of the National Academy of Science of the USA. Milnor’s example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues are identical, in spite of the fact that these tori are not isometric. A natural question is, What is the lowest dimension in which such isospectral nonisometric pairs exist? This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We explore this question from all three perspectives and describe its resolution by Schiemann in the 1990s. Moreover, we share a number of open problems.

ŚULBA-SŪTRAS and PYTHAGORAS’ Theorem

Śulba-sūtras, composed in the later part of the Vedic period (c.800 – 500 BCE), are the texts of early geometry in India. The most notable feature of this text, in terms of geometric language, is the profound statement of the theorem of square on the diagonal. This famous theorem is now universally associated with the name of the Greek philosopher and mathematician Pythagoras (c.570 – c.495 BCE). The objective of this paper is to present a few geometrical propositions discussed in the Śulba-sūtras, which highlight themselves as applications of the theorem stated by Baudhāyana, a distinguished Indian R. s. i (Rishi) as well as a geometrician. In this paper, an attempt has been made to establish that this general theorem was not only known to the Indians long before the date of Pythagoras but also gave an explicit statement of it.

Elasto-plastic evolution of single crystals driven by dislocation flow

This work introduces a model for large-strain, geometrically nonlinear elasto-plastic dynamics in single crystals. The key feature of our model is that the plastic dynamics are entirely driven by the movement of dislocations, that is, 1-dimensional topological defects in the crystal lattice. It is well known that glide motion of dislocations is the dominant microscopic mechanism for plastic deformation in many crystalline materials, most notably in metals. We propose a novel geometric language, built on the concepts of space-time “slip trajectories” and the “crystal scaffold” to describe the movement of (discrete) dislocations and to couple this movement to plastic flow. The energetics and dissipation relationships in our model are derived from first principles drawing on the theories of crystal modeling, elasticity, and thermodynamics. The resulting force balances involve a new configurational stress tensor describing the forces acting against slip. In order to place our model into context, we further show that it recovers several laws that were known in special cases before, most notably the equation for the Peach–Koehler force (linearized configurational force) and the fact that the combination of all dislocations yields the curl of the plastic distortion field. Finally, we also include a brief discussion on how a number of other effects, such as hardening, softening, dislocation climb, and coarse-graining, could be incorporated into our model.

Partial Smoothness and Constant Rank

The idea of partial smoothness in optimization blends certain smooth and nonsmooth properties of feasible regions and objective functions. As a consequence, the standard first-order conditions guarantee that diverse iterative algorithms (and post-optimality analyses) identify active structure or constraints. However, by instead focusing directly on the first-order conditions, the formal concept of partial smoothness simplifies dramatically: in basic differential geometric language, it is just a constant-rank condition. In this view, partial smoothness extends to more general mappings, such as saddlepoint operators underlying primal-dual splitting algorithms.

Relativistic dynamics of the motion of heat

The motion of heat can be described by a thermal energy–momentum tensor. A general description of heat conduction in the geometrical language is developed based on a two-phase continuum model within the framework of relativistic continuum dynamics, and the momentum balance equation of heat, or the general heat conduction equation (GHCE), is derived by the balance equation of the thermal energy–momentum tensor. We demonstrate that the low-speed limit of the GHCE coincides the former version of GHCE of the thermomass theory that is based on the energy–mass equivalence and Newtonian dynamics. Numerical solutions of the GHCE are also provided to demonstrate that the GHCE not only overcomes the paradox of instantaneous heat propagation of the Fourier’s law, but also resolves the defect of the CV model that temperatures may drop below absolute zero under some conditions. The present work formulates a general description of the thermal transport processes, and can provide a deeper insight into the heat transfer discipline from the relativistic point of view.

The evolution of geometric Robertson–Schrödinger uncertainty principle for spin 1 system

Geometric Quantum Mechanics is a mathematical framework that shows how quantum theory may be expressed in terms of Hamiltonian phase-space dynamics. The states are points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is specified by the Schrödinger equation in this framework. The quest to express the uncertainty principle in geometrical language has recently become the focus of significant research in geometric quantum mechanics. One has demonstrated that the Robertson–Schrödinger uncertainty principle, which is a stronger version of the uncertainty relation, can be defined in terms of symplectic form and Riemannian metric. On the basis of this formulation, we study the dynamical behavior of the uncertainty relation for the spin 1 system in this work. We show that under Hamiltonian flow, the Robertson–Schrödinger uncertainty principles are not invariant. This is because, unlike the symplectic area, the Riemannian metric is not invariant under Hamiltonian flow throughout the evolution process.

THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM

Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow.

Scalar and fermion on-shell amplitudes in generalized Higgs effective field theory

Beyond-the-standard-model (BSM) particles should be included in effective field theory in order to compute the scattering amplitudes involving these extra particles. We formulate an extension of Higgs effective field theory which contains an arbitrary number of scalar and fermion fields with arbitrary electric and chromoelectric charges. The BSM Higgs sector is described by using the nonlinear sigma model in a manner consistent with the spontaneous electroweak symmetry breaking. The chiral-order counting rule is arranged consistently with the loop expansion. The leading-order Lagrangian is organized in accord with the chiral-order counting rule. We use a geometrical language to describe the particle interactions. The parametrization redundancy in the effective Lagrangian is resolved by describing the on-shell scattering amplitudes only with the covariant quantities in the scalar/fermion field space. We introduce a useful coordinate (normal coordinate), which simplifies the computations of the on-shell amplitudes significantly. We show that the high-energy behaviors of the scattering amplitudes determine the ``curvature tensors'' in the scalar/fermion field space. The massive spinor--wave function formalism is shown to be useful in the computations of on-shell helicity amplitudes.

An implementation of sign language alphabet hand posture recognition using geometrical features through artificial neural network (part 2)

In the sign language alphabet, several hand signs are in use. Automatic recognition of performed hand signs can facilitate the communication between hearing and none hearing people. This framework proposes hand posture recognition of the American Sign Language alphabet based on a neural network (NN) which works on geometrical feature extraction of the hand. The user’s hand is captured by a 3D depth-based sensor camera. Consequently, the hand is segmented according to the depth features. The proposed system is called ‘Depth-based Geometrical Sign Language Recognition’ (DGSLR). The DGSLR adopted an easier hand segmentation approach, which is further used in other segmentation applications. The proposed geometrical feature extraction framework improves the accuracy of recognition due to unchangeable features against hand orientation or rotation compared to Discrete Cosine Transform (DCT) and Moment Invariant. As a support vector machine (SVM) is a type of artificial neural network (ANN), it is used to drive desired outcomes. Since there are 26 different signs in the Sign Language alphabet, a multi-class SVM versus a single SVM classifier with 26 classes by an RBF kernel was used to validate each class. The proposed framework is proficient to hand posture recognition and provides an accuracy of up to 96.78%. The findings of the iterations demonstrated that the combination of the extracted features resulted in a better accuracy rate in the recognition process in the classification step.

Is SMEFT enough?

There are two canonical approaches to treating the Standard Model as an Effective Field Theory (EFT): Standard Model EFT (SMEFT), expressed in the electroweak symmetric phase utilizing the Higgs doublet, and Higgs EFT (HEFT), expressed in the broken phase utilizing the physical Higgs boson and an independent set of Goldstone bosons. HEFT encompasses SMEFT, so understanding whether SMEFT is sufficient motivates identifying UV theories that require HEFT as their low energy limit. This distinction is complicated by field redefinitions that obscure the naive differences between the two EFTs. By reformulating the question in a geometric language, we derive concrete criteria that can be used to distinguish SMEFT from HEFT independent of the chosen field basis. We highlight two cases where perturbative new physics must be matched onto HEFT: (i) the new particles derive all of their mass from electroweak symmetry breaking, and (ii) there are additional sources of electroweak symmetry breaking. Additionally, HEFT has a broader practical application: it can provide a more convergent parametrization when new physics lies near the weak scale. The ubiquity of models requiring HEFT suggests that SMEFT is not enough.

Model of the essential attributes of visual culture as the criteria for studying artifacts and design of new visual images

The object of this research is visual culture in the context of studying artifacts and design of new visual images. The subject of this research is the theoretical reflection on the essential attributed as the general characteristics of visual culture. The author compares the depiction of the artifacts of visual culture artifacts in postmodernism and constructivism. Postmodernism does not aim to build holistic images and reveal the content through visual form. Constructivism, in turn, structures holistic and aesthetically expressive forms of artifacts in unity with the content. The knowledge and meanings in visual artifacts are constructed via graphic spatial and artistic features. Special attention is given to the constructivist approach as a theoretical-methodological substantiation for studying the essential attributes of visual artifacts. Such approach allows the cultural entities to build semantic models by means of the general cultural geometric language in the dialectical synthesis of various forms, content, and conventional meanings. The artistic interpretations contribute to creation of the unique content. The constructivist approach includes the following methods: deconstructivism of conducting detailed analysis of the artifacts; constructivism of building new aesthetically expressive images. The result of this research lies in systematization of the essential attributes of visual culture for all forms and content. The novelty of consists in correspondence of the constructivist approach to the object and subject of the research, as well as in examination of the actual reality of the modern visual culture of digital technologies and new constructive methods of visualizing form and content in the images.

A Survey on Quaternion Algebra and Geometric Algebra Applications in Engineering and Computer Science 1995–2020

Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020. After a brief introduction of GA, the applications of GA are reviewed across many fields. We discuss the characteristics of the applications of GA to various problems of computer science and engineering. In addition, the challenges and prospects of various applications proposed by many researchers are analyzed. We analyze the developments using GA in image processing, computer vision, neurocomputing, quantum computing, robot modeling, control, and tracking, as well as improvement of computer hardware performance. We believe that up to now GA has proven to be a powerful geometric language for a variety of applications. Furthermore, there is evidence that this is the appropriate geometric language to tackle a variety of existing problems and that consequently, step-by-step GA-based algorithms should continue to be further developed. We also believe that this extensive review will guide and encourage researchers to continue the advancement of geometric computing for intelligent machines.

Ensino da álgebra em uma turma do oitavo ano de uma escola pública de São Mateus - ES

O presente trabalho, visa investigar possíveis contribuições da utilização de dois jogos educativos em aulas de matemática, para ensinar as operações de adição, subtração e multiplicação de polinômios para uma turma do 8º ano do Ensino Fundamental, que tiveram um papel investigativo, mostrando que os alunos possuem dificuldades de interpretação e portanto, dificuldades na passagem da linguagem natural para a algébrica e geométrica, bem como em conceitos importantes como os de perímetro e área. Por este motivo, se torna importante metodologias de ensino que visem enfatizar a interpretação de problemas e a contextualização desse conteúdo a realidade dos alunos, visto que alguns deles não conseguem visualizar a utilidade desses conceitos no dia a dia.