Geometric Terms Research Articles

Solving the minimum convex partition of point sets with integer programming

The partition of a problem into smaller sub-problems satisfying certain properties is often a key ingredient in the design of divide-and-conquer algorithms. For questions related to location, the partition problem can be modeled, in geometric terms, as finding a subdivision of a planar map – which represents, say, a geographical area – into regions subject to certain conditions while optimizing some objective function. In this paper, we investigate one of these geometric problems known as the Minimum Convex Partition Problem (mcpp ). A convex partition of a point set P in the plane is a subdivision of the convex hull of P whose edges are segments with both endpoints in P and such that all internal faces are empty convex polygons. The mcpp is an NP-hard problem where one seeks to find a convex partition with the least number of faces.We present a novel polygon-based integer programming formulation for the mcpp, which leads to better dual bounds than the previously known edge-based model. Moreover, we introduce a primal heuristic, a branching rule and a pricing algorithm. The combination of these techniques leads to the ability to solve instances with twice as many points as previously possible while constrained to identical computational resources. A comprehensive experimental study is presented to show the impact of our design choices.

The controllability of linear descriptor systems using direct methods

In this paper, the controllability of linear descriptor systems is investigated by using a direct approach. This approach is based on viewing the system as a whole. In general method, the system is first decomposed into two subsystems in the canonical form. Then, these subsystems are dealt with separately. However, in this paper, we developed the system generally not viewed separately. Then, we derived the structure of controllable subspaces for linear descriptor systems. From these results, we can state the definition of a controllable matrix and then we investigated the properties of the linear descriptor systems related to the controllability matrices by using geometric term.

New identification method for computer numerical control geometric errors

To address the problems of the low accuracy of geometric error identification and incomplete identification results of the linear axis detection of computer numerical control (CNC) machine tools, a new 21-item geometric error identification method based on double ball-bar measurement was proposed. The model between the double ball-bar reading and the geometric error term in each plane was obtained according to the three-plane arc trajectory measurement. The mathematical model of geometric error components of CNC machine tools is established, and the error fitting coefficients are solved through the beetle antennae search particle swarm optimization (BAS–PSO) algorithm, in which 21 geometric errors, including roll angle errors, were identified. Experiments were performed to compare the optimization effect of the BAS–PSO and PSO and BAS and genetic particle swarm optimization (GA–PSO) algorithms. Experimental results show that the PSO algorithm is trapped in the local optimum, and the BAS–PSO is superior to the other three algorithms in terms of convergence speed and stability, has higher identification accuracy, has better optimization performance, and is suitable for identifying the geometric error coefficient of CNC machine tools. The accuracy and validity of the identification results are verified by the comparison with the results of the individual geometric errors detected through laser interferometer experiments. The identification accuracy of the double ball-bar is below 2.7 µm. The proposed identification method is inexpensive, has a short processing time, is easy to operate, and possesses a reference value for the identification and compensation of the linear axes of machine tools.

Flutter of geometrical imperfect functionally graded carbon nanotubes doubly curved shells

In this paper, flutter of geometrical imperfect functionally graded carbon nanotubes (FG-CNTs) doubly curved shell subjected to a supersonic flow is investigated. For this purpose, the imperfect doubly curved shell is reinforced by four different carbon nanotubes (CNTs) distributions in the thickness direction. The Hill’s elastic moduli are used to obtain the effective material properties of FG-CNTs doubly curved shell. The piston aerodynamic theory and von Karman geometrical nonlinearity terms are used to study large deflection flutter analysis of FG-CNTs imperfect shell. The effect of different parameters including large deflection, geometrical imperfection and CNTs volume fraction on critical flutter pressure of doubly curved shells are studied. According to the results, distributions with poor CNTs in the middle of thickness are more efficient in improving critical flutter pressure. Additionally, the effect of shell imperfection on flutter pressure is more considerable than CNTs volume fraction and distributions.

Landscape, justice and the quality of life in emblematically embodied nation/states – the case of Denmark

ABSTRACT The modern notion of the landscape of the nation-state, we argue, emerged in part through an ‘emblematic’ fusion of the nation, imagined as a bio-organic body-politic, and the state conceptualised in geo-metric terms as the Euclidean, cartographic framework within which that body operates. The eliding of the geo-metric with the bio-organic has influenced national discourse, law and practice by defining the legal and social right to belong within this landscape in bio-spatial terms. This is exemplified by the international political cause célèbre of the ‘Schleswig-Holstein Border Question’ and its continuing ramifications for the quality of life in Denmark—particularly for those living in the landscapes of state-designated immigrant ‘ghettoes’ scheduled for physical and social eradication because their settlements are perceived as endangering the bio-spatial cohesion of the ‘nation-state’.

Children’s exposure to polycyclic aromatic hydrocarbons in the Valencian Region (Spain): Urinary levels, predictors of exposure and risk assessment

Polycyclic aromatic hydrocarbons (PAHs) are pollutants that are released into the environment during incomplete combustion of organic matter and which can have a negative effect on human health. PAHs enter the human body mostly through ingestion of food or inhalation of tobacco smoke. The purpose of the present study is to evaluate the internal levels of PAHs that children living in the Valencian Region (Spain) are exposed to.In total, we measured eleven biomarkers of exposure to naphthalene, fluorene, phenanthrene, pyrene, and benzo(a)pyrene in the urine of 566 children aged 5–12. The analytical method was based on a liquid-liquid extraction of the PAH metabolites from the urine samples, followed by their determination by liquid chromatography coupled to tandem mass spectrometry. In addition, we used a questionnaire to collect the socio-demographic characteristics and 72 h dietary recall information of the participants in our study.Overall, we detected PAH metabolites in more than 78% of the children, with the exception of 3-hydroxyfluorene and 3-hydroxybenzo(a)pyrene, which were found in less than 37% of the analyzed samples. The most abundant biomarker found was 2-hydroxynaphthalene, with a geometric mean of 10 ng·ml−1. Reference values (RV95) ranging from 0.11 (4-hydroxyphenanthrene) to 53 ng·ml−1 (2-hydroxynaphthalene) in urine of Spanish children were derived from the present study. According to the statistical analysis, the factors that were significantly associated with the internal exposure to PAHs were province of residence, body mass index (BMI), children’s age, consumption of plastic-wrapped food, and dietary habits. The estimated daily intakes in geometric mean terms ranged from 5 (fluorene) to 204 ng·kg-bw−1·day−1 (naphthalene). Risk assessment calculations showed higher hazard quotients and hazard indexes for children aged 5–8 than those aged 9–12, but all were below 1.In conclusion, no potential non-cancer health risk due to PAH exposure was observed in children living in Spain.

Complex hyperk\xe4hler structures defined by Donaldson\u2013Thomas invariants

The notion of a Joyce structure was introduced in Bridgeland (Geometry from Donaldson–Thomas invariants, preprint arXiv:1912.06504) to describe the geometric structure on the space of stability conditions of a hbox {CY}_3 category naturally encoded by the Donaldson-Thomas invariants. In this paper we show that a Joyce structure on a complex manifold defines a complex hyperkähler structure on the total space of its tangent bundle, and give a characterisation of the resulting hyperkähler metrics in geometric terms.

My Love from the Mountain: Contingent bordering processes at Mount Changbai/Baekdu

Critical examinations of the enduring geographical myth of natural borders challenge the technical approach toward borders that tempts us to think about and act in the world in exclusively territorial and geometric terms. Drawing insights from and seeking to contribute to existing literature on the reification, naturalisation, and fetishisation of borders, this two‐part paper presents a unique case study of contingent bordering processes at Mount Changbai/Baekdu in the China–Korea Borderland. The first part investigates the making and remaking of the China–Korea border from the late 17th century to the 20th century, to reveal how the narratives of Mount Changbai/Baekdu as a “holy mountain” and a “natural border” were conceived and imposed to legitimise the Manchu‐Qing Chinese and Chosŏn‐Korean regimes respectively, and how these narratives were challenged and superseded in subsequent social and political circumstances. The second part investigates the complex ways these narratives have been appropriated by examining contemporary bordering processes as revealed in the commodification of mineral water, with a particular focus on the development of two brands of bottled water, one South Korean and one Chinese, both extracted from the Chinese side of Mount Changbai/Baekdu. This paper highlights the distinctiveness of non‐Western border concepts and practices, and the importance of making reference to them when discussing border and territorial issues in specifically Asian contexts with diverse traditions of territorial governance. It also points out the pressing need to investigate the contemporary socio‐economic implications of the complex intertwining of history, memory, and identity in borderlands. This is particularly the case given the surge of domestic and foreign corporate investment flowing into Asia’s borderland regions, transforming 20th‐century economic backwaters into 21st‐century hotbeds of resource exploitation and commodity production, even while the “ghosts of borderland’s past” continue to bedevil relations between cultures that have no contemporary physical borders.

Phase behaviour of a simple fluid confined in a periodic porous material

Using density functional theory for a simple fluid of hard spheres with a square-well attraction, we study the phase behaviour of the fluid confined in triply periodic porous material, where the pore space is bounded by a bicontinuous minimal surface. We combine our numerical results with the morphometric thermodynamics, an ansatz that expresses thermodynamic quantities as a linear combination of four additive geometrical terms, and analyse the dependency of the transition point between a liquid and a gas on the geometrical properties of the confining space. By demonstrating that the morphometric approach can be employed in order to account for the thermodynamic behaviour of a simple fluid in a complex porous material, we make the first step towards a better understanding of experimental observations that can help to characterise the pore space.

The geometry of passivity for quantum systems and a novel elementary derivation of the Gibbs state

Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we reformulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it also provides novel analysis for non-equilibrium quantum systems. We show that, to every passive state, one can associate a simple convex shape in a 2-dimensional plane, and that the area of this shape measures the degree to which the system deviates from the manifold of equilibrium states. This provides a novel geometric measure of athermality with relations to both ergotropy and β--athermality.

Wormhole models in R2-gravity for f(R, T) theory with a hybrid shape function

In the present paper, we explore wormholes in R2-gravity within the f(R, T) formalism by using [Formula: see text] (hybrid shape function). The functional form f(R, T) = R + αR2 + λT is considered where R and T are the Ricci scalar and trace of the energy–momentum tensor, respectively, and α and λ are controlling parameters. With the help of the EoS, ω = pr/ρ, geometric behavior is analyzed and energy conditions are discussed in the anisotropic scenario. In this investigation, significant contribution of the quadratic geometric term (R2) and linear trace term (T) are shown to validate energy conditions for the specific shape function and analyze wormhole solutions in the absence of exotic matter. Also, we observe that the energy conditions are valid in the whole range (r ≥ r0) in the case of constant redshift function with this shape function.

Foundations of population-based SHM, Part IV: The geometry of spaces of structures and their feature spaces

One of the requirements of the population-based approach to Structural Health Monitoring (SHM) proposed in the earlier papers in this sequence, is that structures be represented by points in an abstract space. Furthermore, these spaces should be metric spaces in a loose sense; i.e.there should be some measure of distance applicable to pairs of points; similar structures should then be ‘close’ in the metric. However, this geometrical construction is not enough for the framing of problems in data-based SHM, as it leaves undefined the notion of feature spaces. Interpreting the feature values on a structure-by-structure basis as a type of field over the space of structures, it seems sensible to borrow an idea from modern theoretical physics, and define feature assignments as sections in a vector bundle over the structure space. With this idea in place, one can interpret the effect of environmental and operational variations as gauge degrees of freedom, as in modern gauge field theories. One can then regard data normalisation procedures like cointegration as gauge-fixing operations. This paper will discuss the various geometrical structures required for an abstract theory of feature spaces in SHM, and will draw analogies with how these structures have shown their power in modern physics.Having motivated a number of problems in Population-Based SHM (PBSHM) in geometrical terms, it remains to show how these problems might be solved. In the second part of the paper, the problem of determining the normal condition cross section of a feature bundle is addressed. The solution is provided by the application of Graph Neural Networks (GNN), a versatile non-Euclidean machine learning algorithm which is not restricted to inputs and outputs from vector spaces. In particular, the algorithm is well suited to operating directly on the sort of graph structures which are an important part of the proposed framework for PBSHM. The solution of the normal section problem is demonstrated for a heterogeneous population of truss structures for which the feature of interest is the first natural frequency. The GNN approach is shown to not only solve the normal section problem, but also to accommodate varying temperatures across the population; it thus provides a means of data normalisation.

The geometry of neuronal representations during rule learning reveals complementary roles of cingulate cortex and putamen

Learning new rules and adopting novel behavioral policies is a prominent adaptive behavior of primates. We studied the dynamics of single neurons in the dorsal anterior cingulate cortex and putamen of monkeys while they learned new classification tasks every few days over a fixed set of multi-cue patterns. Representing the rules and the neuronal selectivity as vectors in the space spanned by a set of stimulus features allowed us to characterize neuronal dynamics in geometrical terms. We found that neurons in the cingulate cortex mainly rotated toward the rule, implying a policy search, whereas neurons in the putamen showed a magnitude increase that followed the rotation of cortical neurons, implying strengthening of confidence for the newly acquired rule-based policy. Further, the neural representation at the end of a session predicted next-day behavior, reflecting overnight retention. The novel framework for characterization of neural dynamics suggests complementing roles for the putamen and the anterior cingulate cortex.

The Exact Theory of the Stern–Gerlach Experiment and Why it Does Not Imply that a Fermion Can Only Have Its Spin Up or Down

The Stern–Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern–Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles θ between the spin axis s and the magnetic field B, the fermions can only have two possible energies m0c2±μB. The values ±μB thus do not correspond to the continuum of values −μ·B Einstein and Ehrenfest had conjectured. The energy term V=−μ·B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies ±μB, and as such not valid for individual fermions. The two fermion states with energy ±μB are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.

Rationalizability of square roots

Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a solution in terms of these functions is to rationalize all occurring square roots by a suitable variable change. In this paper, we give a rigorous definition of rationalizability for square roots of ratios of polynomials. We show that the problem of deciding whether a single square root is rationalizable can be reformulated in geometrical terms. Using this approach, we give easy criteria to decide rationalizability in most cases of square roots in one and two variables. We also give partial results and strategies to prove or disprove rationalizability of sets of square roots. We apply the results to many examples from actual computations in high energy particle physics.

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows Past Lipschitz Perturbed Cones

We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry so that the equations contain a singular geometric source term. We first formulate the shock stability problem as an initial-boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and asymptotic behavior of global entropy solutions of bounded variation (BV) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. Then we introduce the Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, as well as the estimates of the center changes of the self-similar solutions. To make sure the decreasing of the Glimm-type functional, we choose appropriate weights by careful asymptotic analysis of the reflection coefficients in the interaction estimates, when the Mach number of the incoming flow is sufficiently large. Finally, we establish the existence of global entropy solutions involving a strong leading conical shock-front, besides weak waves, under the conditions that the Mach number of the incoming flow is sufficiently large and the weighted total variation of the slopes of the generating curve of the Lipschitz perturbed cone is sufficiently small. Furthermore, the entropy solution is shown to approach asymptotically the self-similar solution that is determined by the incoming flow and the asymptotic tangent of the cone boundary at infinity.

A new geometrical method for portfolio optimization

Risk aversion plays a significant and central role in investors’ decisions in the process of developing a portfolio. In this portfolio optimization framework, we determine the portfolio that possesses the minimal risk by using a new geometrical method. For this purpose, we elaborate an algorithm that enables us to compute any Euclidean distance to a standard simplex. With this new approach, we can treat the case of portfolio optimization without short-selling in its entirety, and we also recover in geometrical terms the well-known results on portfolio optimization with allowed short-selling. Then, we apply our results to determine which convex combination of the CAC 40 stocks possesses the lowest risk. Thus, we not only obtain a very low risk compared to the index, but we also get a rate of return that is almost three times better than the one of the index.

The null and force-free electromagnetic field

This paper describes the electrodynamics of a null and force-free field in completely geometric terms. As was previously established in Menon (2020 Class. Quantum Grav. 37 245006), solutions to force-free electrodynamics are governed by the existence of certain special types of foliations of spacetime. Here the nature of the foliations in a coordinate-free formalism in the null case is prescribed. All of the general results are illustrated by constructing a new null, force-free electrodynamic field in a Friedmann–Lemaitre–Robertson–Walker spacetime.

Lyra’s cosmology of homogeneous and isotropic universe in Brans–Dicke theory

In this paper, we investigate a scalar field Brans–Dicke cosmological model in Lyra’s geometry which is based on the modifications in a geometrical term as well as energy term of Einstein’s field equations. We have examined the validity of the proposed cosmological model on the observational scale by performing statistical analysis from the latest [Formula: see text] and SN Ia observational data. We find that the estimated values of Hubble’s constant and matter energy density parameter is in agreement with their corresponding values, obtained from recent observations of Wilkinson Microwave Anisotropy Probe (WMAP) and Plank collaboration. We also derived the deceleration parameter, age of the universe and jerk parameter in terms of red-shift and computed its present values. The dynamics of the deceleration parameter in the derived model of the universe show a signature flipping from positive to a negative value and also indicate that the present universe is in the accelerating phase.

Construcción de ciudadanía y reconciliación a través del acceso a la justicia

La educación es un factor vinculado con la pobreza y con la construcción de ciudadanía plena. Su significado actual dista en términos geométricos del que tuvo en la Revolución Industrial. La doctrina clásica reduce el concepto a la titularidad de los derechos políticos o cívicos, es decir, participación en la soberanía como elector o elegible a través de elecciones, consultas (referéndum o plebiscito) u otra vía. Es en ese contexto donde alcanza su valor e importancia el acceso a la justicia o tutela judicial efectiva, un principio jurídico de alcance general, cuyo significado es la posibilidad igualitaria de ejercer los derechos o resolver los conflictos a través de la justicia. Es así que concluyo que el acceso a la justicia es la vía más expedita para la construcción de ciudadanía plena, reconciliación social y democracia.