Geometric Notions Research Articles

Smooth and proper maps with respect to a fibration

Abstract This paper explain how the geometric notions of local contractibility and properness are related to the $\Sigma$ -types and $\Pi$ -types constructors of dependent type theory. We shall see how every Grothendieck fibration comes canonically with such a pair of notions—called smooth and proper maps—and how this recovers the previous examples and many more. This paper uses category theory to reveal a common structure between geometry and logic, with the hope that the parallel will be beneficial to both fields. The style is mostly expository, and the main results are proved in external references.

Measuring the industry golem location and posture repetition exactitude, according to the improved sparrow search arithmetic-improved Gaussian curve matching-improved hough transform arithmetic

Industry golems have been extensively used in the domain of factory automation fabrication. The location and posture repetition exactitude often sharply decreases with the long-term operation and wear of the golem. Therefore, the location and posture repetition exactitude greatly limits the further application of industry golems in high-precision manufacturing scenarios. In the field of industry golem detection, the development of precise evaluation machinery for golem location and posture repetition exactitude has become an urgent problem to be solved in the field of precision determination. However, due to the abstract geometric concept of post-repetition exactitude, it is difficult to measure the location and posture repetition exactitude directly. The primary difficulty in attitude determination lies in how to reflect the post-repetition exactitude. Taking the insufficient of available methods into account, a location and posture repetition exactitude measured model is presented according to the orientation-cosine to study a highly efficient method for measuring the repetition location and posture deviation of industry golems. According to the model, an industry golem location and posture repetition exactitude determination machinery is cultivated and then the industry golem location and posture repetition exactitude is envisioned. Through using the determination machinery, a novel method according to an ISSA-IGCM-IHT arithmetic is presented to determine the Industry golem’s location and posture repetition exactitude. The location and angular determination exactitude of ±1.5 μm and ±2″ are demonstrated by the experimentations, respectively. The dedication of this paper is that the nonobjective geometrical notion of post-repetition exactitude is presented by a novel location and posture guideline subassembly of optic and consanguinity pictorial information detection accessories according to the directional-cosine. By acquiring the location and posture of the stauro-laser graphic, multi-dot determination of the industry golem location and posture repetition exactitude has been realized, according to the ISSA-IGCM-IHT arithmetic.

Implementation of energy interaction between laser beam and keyhole wall based on algebraically reconstructing free surface: An experimentally validated numerical framework

The interaction of the laser beam with the material is critical for determining laser processing quality, with ray tracing methods mostly used in energy transfer research. These techniques generally rely on only the volume of fluid (VOF) method or level set (LS) function to track the molten pool free surface, resulting in either accuracy loss or difficulty in ensuring mass conservation of the interface. The few coupled algorithms are built on geometric notions that are difficult to understand while being limited to structural grids. In this paper, we have proposed a revised ray tracing framework, so-called ARCLS, to derive the exact intersections of the rays with the keyhole wall which is represented by a continuous LS surface algebraically rebuilt on top of the VOF interface. It is easy to code and convenient for unstructured meshes. This framework is tested through a V-groove case and a set of Ti6Al4V titanium alloy laser welding examples. The findings reveal that the reconstructed LS surface perfectly matches the VOF interface, and the predicted molten pool profile agrees with the actual data well. The keyhole morphology and molten flow patterns are also consistent with the physical truths. Spattering, porosity, necking, and swelling are observed in the developing keyhole, which are the possible reasons for the potential welding defects. Further investigation suggested that the ARCLS-based algorithm collected more laser rays at the keyhole bottom compared to the standard technique, achieving a deeper keyhole profile and a reflection pattern that was more optically logical. The work presents a reliable approach for estimating laser energy distribution, which can be effectively applied to the heat-fluid coupling simulation for laser manufacturing processes in order to forecast the dynamic behaviors of keyhole and molten pool.

Beyond the Quantum Membrane Paradigm: A Philosophical Analysis of the Structure of Black Holes in Full QG

This paper presents a philosophical analysis of the structure of black holes, focusing on the event horizon and its fundamental status. While black holes have been at the centre of countless paradoxes arising from the attempt to merge quantum mechanics and general relativity, recent experimental discoveries have emphasised their importance as objects for the development of Quantum Gravity. In particular, the statistical mechanical underpinning of black hole thermodynamics has been a central research topic. The Quantum Membrane Paradigm, proposed by Wallace (Stud Hist Philos Sci Part B 66:103-117, 2019), posits a real membrane made of black hole microstates at the black hole horizon to provide a statistical mechanical understanding of black hole thermodynamics from an exterior observer’s point of view. However, we argue that the Quantum Membrane Paradigm is limited to low-energy Quantum Gravity and needs to be modified to avoid reference to geometric notions, such as the event horizon, which presumably do not make sense in the non-spatiotemporal context of full Quantum Gravity. Our proposal relies on the central dogma of black hole physics. It considers recent developments, such as replica wormholes and entanglement wedge reconstruction, to provide a new framework for understanding the nature of black hole horizons in full Quantum Gravity.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Advancements in Continuum Mechanics and Electrodynamics by a spacetime geometric approach

Results achieved by the authors in the course of research activity on continuum mechanics and electrodynamics (CME) during the past twenty years are illustrated, revised and discussed. Adoption of a geometric approach leads to renewal of concepts and methods of classical CME and to formulation in Euclid (3+1)D ambient spacetime wherein innovation, clarity and depth of a geometric treatment naturally emerge. The dissemination of novel concepts and methods in CME is not delayable, with critical revisitation of problematic notions, analyses and results still currently on the scene. Material frame indifference, equilibrium in a reference configuration, extremality principles in Dynamics, finite elasticity, chain decomposition of finite strain in elasto-thermo-plasticity, variance of electro-magnetic induction laws under frame changes, action on electric charges moving in a magnetic field, are under the spotlight of innovation and advancement. Fostering basic knowledge of Differential Geometry and application of geometric notions and methods contribute effective tools in formulating meaningful rules, amending misstatements and dimming debates based on vague affirmations.

Port-Hamiltonian discontinuous Galerkin finite element methods

Abstract A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a nonzero energy flow through the boundaries. In this paper, we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving semidiscrete finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes–Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.

K\"{a}hler-Norden structures on Hom-Lie groups and Hom-Lie algebras

The aim of this paper is to describe two geometric notions, holomorphic Norden structures and K\"{a}hler-Norden structures on Hom-Lie groups, and study their relationships in the left invariant setting. We study K\"{a}hler-Norden structures with abelian complex structures and give the curvature properties of holomorphic Norden structures on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie group is a flat (holomorphic Norden Hom-Lie algebra carries a Hom-Left-symmetric algebra) if its left-invariant complex structure (complex structure) is abelian.

A simpler proof of Sternfeld’s Theorem

In Sternfeld’s work on Kolmogorov’s Superposition Theorem appeared the combinatorial–geometric notion of a basic set and a certain kind of arrays. A subset [Formula: see text] is basic if any continuous function [Formula: see text] could be represented as the sum of compositions of continuous functions [Formula: see text] and projections to the coordinate axes. The definition of a Sternfeld array is presented in this paper. Sternfeld’s Arrays Theorem. If a closed bounded subset[Formula: see text] contains Sternfeld arrays of arbitrary large size then[Formula: see text] is not basic. The paper provides a simpler proof of this theorem.

On classes of ζ-uniformly q-analogue of analytic functions with some subordination results

This article presents new classes of ζ-uniformly q-starlike and q-convex analytic functions of order υ using the principle of q-calculus. The renowned classes of starlike and convex functions are utilized to establish these classes. The investigation of these uniformly classes leads to the study of some geometric notions, including coefficient estimates, sharpness, distortion and growth theorems, and convex linear combinations. Furthermore, some subordination properties involving integral means inequalities and subordinate sequences are examined.

How do pre-service teachers view Galois theory? A questionnaire study

Abstract algebra is an important part of mathematics teacher education as it provides the rigorous mathematical foundations for many mathematics topics covered in school classrooms. Throughout their academic career, many secondary mathematics teachers even enroll in more advanced algebra courses, which often culminate in Galois theory. However, very little is known about educational aspects of Galois theory and respective mathematics education research is scarce. We contribute to closing this gap by reporting on the results of an exploratory questionnaire study with a sample of n=39 pre-service mathematics teachers, inquiring about the raison d’être of incorporating Galois theory into teacher education: Is Galois theory viewed as useful for their later profession and which connections are drawn to the secondary mathematics classroom? On the one hand, the results of our study indicate that a vast majority of pre-service teachers do not perceive studying Galois theory as meaningful and struggle to exemplify connections between Galois theory and secondary school mathematics. On the other hand, a small share of the participants experienced Galois theory as an important part of mathematics that elegantly connects a variety of algebraic and geometric notions.

On a Geometric Notion of Duality in Nonlinear Control Systems

On a Geometric Notion of Duality in Nonlinear Control Systems

On the consequences of Raychaudhuri equation in Kantowski-Sachs space-time

The paper aims to study the geometry and physics of the Raychaudhuri equation (RE) in the background of a homogeneous and anisotropic space–time described by Kantowski–Sachs (KS) metric. Role of anisotropy/shear in the context of convergence and possible avoidance of singularity has been analyzed subject to a physically motivated constraint. Moreover, using a suitable transformation the first order RE has been converted to a second order differential equation analogous to a Harmonic Oscillator and criterion for convergence has been shown to be associated with the time varying frequency of the Oscillator. Finally, the paper points out a geometric and physical notion of anisotropy along with their corresponding behavior towards convergence.

T-Duality and Lie bialgebroid structures

We show that the geometric notion of duality behind T-duality, between two string theories on different manifolds E, Ê in the sense of [3][4], is precisely that of Lie bialgebroids due to Mackenzie and Xu [11].

Segre-driven radicality testing

We present a probabilistic algorithm to test if a homogeneous polynomial ideal I defining a scheme X in Pn is radical using Segre classes and other geometric notions from intersection theory which is applicable for certain classes of ideals. If all isolated primary components of the scheme X are reduced and it has no embedded components outside of the singular locus of Xred=V(I), then the algorithm is not applicable and will return that it is unable to decide radically; in all the other cases it will terminate successfully and in either case its complexity is singly exponential in n. The realm of the ideals for which our radical testing procedure is applicable and for which it requires only single exponential time includes examples which are often considered pathological, such as the ones drawn from the famous Mayr-Meyer set of ideals which exhibit doubly exponential complexity for the ideal membership problem.

Cooperative learning for geometric notions in kids of 5-years-old in kindergarten

Cooperative learning has become a fundamental teaching-learning strategy for the social construction of knowledge. This methodology is ideal for students at kindergarten in regions such as Latin America and the Caribbean, where the effects of learning poverty condition socioeconomic development. Therefore, the objective was to evaluate the incidence of cooperative learning in the teaching of geometry in 5-year-old students belonging to educational institutions of the kindergarten in the three natural regions of Peru. This was qualitative, descriptive-correlational research. Two data collection instruments were applied to 78 students who served as a sample. To analyze the relationship between the variables, Spearman's Rank-Order was used, which confirmed the positive relationship between them; therefore, the influence of incorporating this type of learning does have an impact on the logical development of children. Quality education with optimal learning strategies will lead these students in the future to a good level of performance in academic tests such as PISA.

Finite-time bounds on the probabilistic violation of the second law of thermodynamics

Jarzynski’s equality sets a strong bound on the probability of violating the second law of thermodynamics by extracting work beyond the free energy difference. We derive finite-time refinements to this bound for driven systems in contact with a thermal Markovian environment, which can be expressed in terms of the geometric notion of thermodynamic length. We show that finite-time protocols converge to Jarzynski’s bound at a rate slower than 1/\sqrt{\tau}1/τ, where \tauτ is the total time of the work-extraction protocol. Our result highlights a new application of minimal dissipation processes and demonstrates a connection between thermodynamic geometry and the higher order statistical properties of work.

Exchange graphs for mutation-finite non-integer quivers

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).

Large structures within the class of summing operators

We investigate lineability/spaceability problems within the setting of multilinear summing operators on quasi-Banach sequence spaces. Furthermore, we deal with the contemporary geometric notions of pointwise-lineability and (α,β)-lineability. Among other results, we prove that the class of multilinear operators taking values on ℓq(0<q≤∞) that are absolutely but not multiple summing is pointwise c-spaceable when non-empty. Spaceability in other classes, such as Dunford-Pettis operators and multilinear operators on general summable scalar families, is also studied.

SMEFTs living on the edge: determining the UV theories from positivity and extremality

We study the “inverse problem” in the context of the Standard Model Effective Field Theory (SMEFT): how and to what extend can one reconstruct the UV theory, given the measured values of the operator coefficients in the IR? The main obstacle of this problem is the degeneracies in the space of coefficients: a given SMEFT truncated at a finite dimension can be mapped to infinitely many UV theories. We discuss these degeneracies at the dimension-8 level, and show that positivity bounds play a crucial role in the inverse problem. In particular, the degeneracies either vanish or become significantly limited for SMEFTs that live on or close to the positivity bounds. The UV particles of these SMEFTs, and their properties such as spin, charge, other quantum numbers, and interactions with the SM particles, can often be uniquely determined, assuming dimension-8 coefficients are measured. The allowed region for SMEFTs, which forms a convex cone, can be systematically constructed by enumerating its generators. We show that a geometric notion, extremality, conveniently connects the positivity problem with the inverse problem. We discuss the implications of a SMEFT living on an extremal ray, on a k-face, and on the vertex of the positive cone. We also show that the information of the dimension-8 coefficients can be used to set exclusion limits on all individual UV states that interact with the SM, independent of specific model assumptions. Our results indicate that the dimension-8 operators encode much more information about the UV than one would naively expect, which can be used to reverse engineer the UV physics from the SMEFT.