Configuration Space Research Articles

Homology and Euler characteristic of generalized anchored configuration spaces of graphs

AbstractIn this paper we consider the generalized anchored configuration spaces on n labeled points on a graph. These are the spaces of all configurations of n points on a fixed graph G, subject to the condition that at least q vertices in some pre-determined set K of vertices of G are included in each configuration. We give a non-alternating formula for the Euler characteristic of such spaces for arbitrary connected graphs, which are not trees. Furthermore, we completely determine the homology groups of the generalized anchored configuration spaces of n points on a circle graph.

A realist interpretation of unitarity in quantum gravity

Unitarity is a difficult concept to implement in canonical quantum gravity because of state non-normalisability and the problem of time. We take a realist approach based on pilot-wave theory to address this issue in the Ashtekar formulation of the Wheeler–DeWitt equation. We use the postulate of a definite configuration in the theory to define a global time for the gravitational-fermionic system recently discussed in Alexander et al (2022 Phys. Rev. D 106 106012), by parameterising a variation of a Weyl-spinor that depends on the Kodama state. The total Hamiltonian constraint yields a time-dependent Schrodinger equation, without semi-classical approximations, which we use to derive a local continuity equation over the configuration space. We implement the reality conditions at the level of the guidance equation, and obtain a real spin-connection, extrinsic curvature and triad along the system trajectory. We obtain quantum corrections to deSitter spacetime from the guidance equation. The non-normalisable Kodama state is naturally factored out of the full quantum state in the conserved current density, opening the possibility for quantum-mechanical unitarity. We also give a pilot-wave generalisation of the notion of unitarity applicable to non-normalisable states, and show the existence of equilibrium density for our system. Lastly, we find unitary states in mini-superspace by finding an approximate solution to the Hamiltonian constraint.

Effects of general relativistic post-Newtonian approximations on the configuration space properties of elliptical orbital trajectories

Post-Newtonian approximations have been successful in describing various astronomical phenomena that classical Newtonian gravity could not. One such phenomenon is the precession of the periapsis of elliptical orbits. Einstein originally solved the problem in 1916 and since then others have been applying the technique to numerous complex systems. However, not much attention has been given to studying the changes in the configuration space when switching from Newtonian to post-Newtonian mechanics. Here, we compare the sizes and topologies of the configuration spaces of both these approaches and further explore the likelihood of planetary collisions in other star systems when configuration space expansion due to general relativity is taken into account.

Unified invariant of knots from homological braid action on Verma modules

AbstractWe re‐build the quantum unified invariant of knots from braid groups' action on tensors of Verma modules. It is a two variables series having the particularity of interpolating both families of colored Jones polynomials and ADO polynomials, that is, semisimple and non‐semisimple invariants of knots constructed from quantum . We prove this last fact in our context that re‐proves (a generalization of) the famous Melvin–Morton–Rozansky conjecture first proved by Bar‐Natan and Garoufalidis. We find a symmetry of nicely generalizing the well‐known one of the Alexander polynomial, ADO polynomials also inherit this symmetry. It implies that quantum non‐semisimple invariants are not detecting knots' orientation. Using the homological definition of Verma modules we express as a generating sum of intersection pairing between fixed Lagrangians of configuration spaces of disks. Finally, we give a formula for using a generalized notion of determinant, that provides one for the ADO family. It generalizes that for the Alexander invariant.

Uncertainty-biased molecular dynamics for learning uniformly accurate interatomic potentials

Efficiently creating a concise but comprehensive data set for training machine-learned interatomic potentials (MLIPs) is an under-explored problem. Active learning, which uses biased or unbiased molecular dynamics (MD) to generate candidate pools, aims to address this objective. Existing biased and unbiased MD-simulation methods, however, are prone to miss either rare events or extrapolative regions—areas of the configurational space where unreliable predictions are made. This work demonstrates that MD, when biased by the MLIP’s energy uncertainty, simultaneously captures extrapolative regions and rare events, which is crucial for developing uniformly accurate MLIPs. Furthermore, exploiting automatic differentiation, we enhance bias-forces-driven MD with the concept of bias stress. We employ calibrated gradient-based uncertainties to yield MLIPs with similar or, sometimes, better accuracy than ensemble-based methods at a lower computational cost. Finally, we apply uncertainty-biased MD to alanine dipeptide and MIL-53(Al), generating MLIPs that represent both configurational spaces more accurately than models trained with conventional MD.

Topological actions of Temperley–Lieb monoids and representation stability

In this paper, we consider the Temperley–Lieb algebras [Formula: see text] at [Formula: see text]. Since [Formula: see text], we can consider the multiplicative monoid structure and ask how this monoid acts on topological spaces. Given a monoid action on a topological space, we get an algebra action on each homology group. The main theorem of this paper explicitly deduces the representation structure of the homology groups in terms of a natural filtration associated with our [Formula: see text]-space. As a corollary of this result, we are able to study stability phenomena. There is a natural way to define representation stability in the context of [Formula: see text], and the presence of filtrations enables us to define a notion of topological stability. We are able to deduce that a filtration-stable sequence of [Formula: see text]-spaces results in representation-stable sequence of homology groups. This can be thought of as the analogue of the statement that the homology of configuration spaces forms a finitely generated FI-module.

K-core attack, equilibrium K-core, and kinetically constrained spin system

Abstract Kinetically constrained spin systems are toy models of supercooled liquids and amorphous solids. In this Perspective, we revisit the prototypical Fredrickson-Andersen (FA) kinetically constrained model from the viewpoint of K-core combinatorial optimization. Each kinetic cluster of the FA system, containing all the mutually visitable microscopic occupation configurations, is exactly the solution space of a specific instance of the K-core attack problem. The whole set of different jammed occupation patterns of the FA system is the configuration space of an equilibrium K-core problem. Based on recent theoretical results achieved on the K-core attack and equilibrium K-core problems, we discuss the thermodynamic spin glass phase transitions and the maximum occupation density of the fully unfrozen FA kinetic cluster, and the minimum occupation density and extreme vulnerability of the partially frozen (jammed) kinetic clusters. The equivalence between K-core attack and the fully unfrozen FA kinetic cluster also implies a new way of sampling K-core attack solutions.

Automatic Determination of the Non-Covalent Stable Conformations of the NO2-Pyrene Cluster in Full Dimensionality (81D) Using the vdW-TSSCDS Approach.

We present the detailed topographical characterisation (stationary points and minimum energy paths connecting them) of the full dimensional (81D) intermolecular potential energy surface associated with the non-covalent interactions between the NO2 radical and the pyrene (C16H10) molecule. The whole procedure is (quasi) fully automated. We have used our recent algorithm vdW-TSSCDS as implemented on the freely-available AutoMekin software package. To this end, a series of inexpensive classical trajectories using forces from a low-level (semi-empirical) theory are used to sample the configuration space of the system in the search for candidates to first order saddle points. These guess structures are determined by means of a graph-theory based algorithm using the concept of adjacency matrix. Low-level optimizations are followed by re-optimizations at a final high-level of theory (DFT and CCSD(T)-F12 in our case.). The resulting set of stationary points and paths connecting them constitutes the so-called reaction network. In the case of NO2-pyrene, this network exhibits four major basins which can be characterized by their point-group symmetry. A central one, of global C2 symmetry, comprises the global minimum (as well as all other permutationally related conformers) together with the corresponding C2v saddle points connecting them. This central basin is connected to three others of lower C1 symmetry. The latter can be distinguished by the projection of the position of the NO2 nitrogen atom on the pyrene plane in combination with the relative orientation of the oxygen pair pointing either inwards, outwards, upwards or downwards.

Chaotic Phenomena for Generalised N-centre Problems

We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision-less periodic solutions, we encode them in bi-infinite sequences of symbols. Solutions are obtained as minimisers of the Maupertuis functional in suitable free homotopy classes of the punctured surface, without any collision regularisation. For any sufficiently large value of the energy, we prove that the generalised N-centre problem admits a symbolic dynamics. Moreover, when the Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant subsets.

The unequal connectivity of parks and tree canopy in 10 U.S. cities

The connectivity of urban green space has a strong influence on the benefits that green space provides—more connected green space has been linked to greater biodiversity, increased recreational opportunities, improved air quality, and increased noise attenuation. Previous urban green space equity research largely focuses on the total amount or accessibility of green space, ignoring connectivity and other measures of configuration that also impact green space equity. Taking both a demographic and spatial approach, we examine whether green space (i.e., park and tree canopy) area, patch size, and connectivity are distributed equally in ten U.S. cities. We find that not only are wealthy, college-educated, and predominately White census tracts more likely to have greater total green space, the green space patches are larger and more connected. This inequality was stronger for tree canopy than for parks, pointing to the need for policy interventions to improve tree canopy connectivity in areas of systematic disinvestment.

Quantum geometrodynamics revived II. Hilbert space of positive definite metrics

Abstract This paper represents the second in a series of works aimed at reinvigorating the quantum geometrodynamics program. Our approach introduces a lattice regularization of the hypersurface deformation algebra, such that each lattice site carries a set of canonical variables given by the components of the spatial metric and the corresponding conjugate momenta. In order to quantize this theory, we describe a representation of the canonical commutation relations that enforces the positivity of the operators q_{ab} s^a s^b for all choices of s. Moreover, symmetry of q_{ab} and p^{ab} is ensured. This reflects the physical requirement that the spatial metric should be a positive definite, symmetric tensor. To achieve this end, we resort to the Cholesky decomposition of the spatial metric into upper triangular matrices with positive diagonal entries. Moreover, our Hilbert space also carries a representation of the vielbein fields and naturally separates the physical and gauge degrees of freedom. Finally, we introduce a generalization of the Weyl quantization for our representation. We want to emphasize that our proposed methodology is amenable to applications in other fields of physics, particularly
in scenarios where the configuration space is restricted by complicated relationships among the degrees of freedom.

Effect of Plant Spacing and Nutrient Levels on Growth and Yield of Red Cabbage (Brassica oleracea var. capitata f. rubra)

An experiment was conducted to investigate the impact of plant spacing and nutrient levels on the growth and yield of red cabbage. This experiment involved three distinct spacing configurations (45cmx30cm, 45cmx45cm, and 45cmx60cm) and nutrient levels (75% NPK, 100% NPK, and 125% NPK). The experiment was designed as a Factorial Randomized Block Design (FRCBD) and carried out at the Department of Horticulture, University of Agricultural Sciences, GKVK, Bengaluru, during the 2021-22 period. This study infers that spacing and nutrients promote the vegetative growth of plants. The spacing configuration of 45cm X 60cm with 125% NPK resulted in the most significant outcomes, including the highest number of leaves (28.93), maximum plant height (36.71 cm), maximum fresh weight of the head (1421.80 g) and highest head volume (1360.19 cc) which could be because of wider spacing and higher nutrient doses. Conversely, the spacing configuration of 45cm X 30cm with 75% NPK demonstrated the early head initiation (45.60 days) and the most compact head formation (0.38). The highest yield per hectare (74.43 t) was achieved with the spacing configuration of 45cmx30cm using 125% NPK which improved greater availability of nutrients and, as a result, boosted the vegetative development.

The rule of four: anomalous distributions in the stoichiometries of inorganic compounds

Why are materials with specific characteristics more abundant than others? This is a fundamental question in materials science and one that is traditionally difficult to tackle, given the vastness of compositional and configurational space. We highlight here the anomalous abundance of inorganic compounds whose primitive unit cell contains a number of atoms that is a multiple of four. This occurrence—named here the rule of four—has to our knowledge not previously been reported or studied. Here, we first highlight the rule’s existence, especially notable when restricting oneself to experimentally known compounds, and explore its possible relationship with established descriptors of crystal structures, from symmetries to energies. We then investigate this relative abundance by looking at structural descriptors, both of global (packing configurations) and local (the smooth overlap of atomic positions) nature. Contrary to intuition, the overabundance does not correlate with low-energy or high-symmetry structures; in fact, structures which obey the rule of four are characterized by low symmetries and loosely packed arrangements maximizing the free volume. We are able to correlate this abundance with local structural symmetries, and visualize the results using a hybrid supervised-unsupervised machine learning method.

A new approach to the evaluation and solution of the relativistic kinetic dispersion relation and verification with continuum kinetic simulation

The present work describes a new approach to evaluation and root finding for the kinetic dispersion relation of Langmuir waves, which is central to the analytical understanding of collisionless damping in plasmas. The plasma dispersion function is solved to machine precision using direct integration in the complex plane in combination with an analytic evaluation of the residue to account for the deformation along the Landau contour. To efficiently attain machine precision, the contour is displaced in the complex plane prior to integration, and numerical subtleties related to the placement of the contour are discussed. The approach is generic in that it applies to arbitrary distribution functions, with the present manuscript focused on relativistic cases. Detailed verification of results via direct kinetic simulation in a variety of configuration space dimensions is also presented. Finally, the technique is applied to the challenging case of highly relativistic (i.e. extremely hot) plasmas. Here we show both qualitative agreement with prior work, as well as the disappearance of the Landau root which would have significant implication for real-life observation or experiment.

Visibility, Precarity and Public Spaces: Reading Matthew Arnold’s “West London” in an Indian Classroom

This essay is an account of my experience of teaching Matthew Arnold's "West London" in an Indian classroom through the conceptual prism of vulnerability. The essay argues that the configuration of public space in Arnold's poem in terms of differential class relations in Victorian society offers a literary mode of visualizing vulnerablity and prompts an ethical recognition of the humanness of marginalized characters. It further demonstrates how the peadgogic exercise of 'reading' such poetry in the contemporary urban Indian context can alert students to the politics of occupying and negotiating public spaces in terms of (the lack of) class position and privilege across diverse cultural and historical contexts. 

Redundant Non-Serial Implicit Manipulator Kinematics and Dynamics

Abstract Redundant non-serial manipulators that encompass a broad spectrum of parallel and non-parallel heavy load bearing construction and material handling equipment are treated, using foundations of differential geometry. Kinematics of this category of manipulator are defined in configuration space by algebraic equations in input and output coordinates that cannot be explicitly solved for either set of coordinates as a function of the other. New sets called assembly components of manipulator configuration space are defined that partition the space into disjoint topological components whose assembled configurations cannot be connected by continuous trajectories. Forward and inverse kinematically singular configurations are characterized by criteria that partition each assembly component into singularity free assembly components in which equations of kinematics and dynamics are well behaved. Three typical applications and associated simplified model problems are studied throughout the paper to illustrate methods and results presented. Singularity free differentiable configuration space manifolds are parameterized by both input and operational coordinates, leading to well posed ordinary differential equations of manipulator dynamics in both input and operational coordinates. It is shown that generalized inverse velocity kinematic equations that are problematic for serial manipulators are likewise plagued with problems for non-serial implicit manipulators that can be avoided using the methods presented.

Optimizing Nurse Workflow Efficiency: An Examination of Nurse Walking Behavior and Space Accessibility in Medical Surgical Units.

This study aimed to investigate the effect of spatial adjacencies on nurses' walking patterns and the subsequent impact on staff satisfaction with perceived accessibility and adjacency-related issues. Recognizing the crucial importance of spatial adjacencies in healthcare facilities is essential, as they significantly affect staff morale, fatigue management, operational efficiency, error reduction, and overall patient care excellence, highlighting the need for objective assessments to evaluate the impact of facility layout and space configuration on workflow patterns and staff satisfaction in patient care units. Integrating on-site observations with survey data, we explored how spatial adjacencies affect staff walking behavior and satisfaction in two med-surgical unit floors. The findings highlighted a significant frequency of movements between nurse stations, patient rooms, and medication areas. Regression analysis identified several contributing factors to staff satisfaction, including the proximity of supplies, team visibility, ease of access across departments, and the location of equipment rooms. Specifically, satisfaction with the proximity of supplies was positively associated with increased provider satisfaction with workflow, quality of care, and workplace. Additionally, valuable feedback from staff revealed concerns regarding break room placement, medication area functionality, and disparities in the availability of supplies. This study highlighted the critical need for carefully planned spatial adjacency strategies to enhance workflow efficiency and raise clinical staff satisfaction within healthcare facilities. The actionable insights gleaned from this research offer valuable direction to architects, healthcare administrators, and design professionals, enabling the creation of environments that positively resonate with healthcare providers and improve healthcare operations.

Identifying the charge states of carbon vacancies in 4H-SiC by ab initio metadynamics

4H Silicon carbide (4H-SiC) is widely recognized as a highly promising material for high-voltage and high-power electronic applications due to its exceptional properties. The performance of devices based on 4H-SiC is often weakened by the presence of carbon-related point defects, particularly carbon vacancies (VC). The defects of VC introduce deep-level traps (e.g., Z1/2 and EH6/7) that deteriorate device functionality. Experimental and theoretical studies on VC have led to some conflicting results about the charge states of VC, especially for the charge state ordering of EH6/7. We now employ ab initio metadynamics (META) to systematically investigate configuration space including the direction and magnitude of bond distortion and identify the most stable structures of VC. Eventually, the charge states of VC in 4H-SiC are identified. The Z1 (EH6) and Z2 (EH7) indicate transitions from acceptor (donor) levels of VC, located on the h and k sublattice sites, respectively. Z1 and Z2 demonstrate negative-U ordering, characterized by U values of −0.16 and −0.37 eV, respectively. Conversely, EH6 and EH7 display positive-U ordering, with U values of 0.16 and 0.08 eV, respectively. The current results provide insights into the properties of VC in 4H-SiC, highlighting the effectiveness of META in the exploration of complex potential energy surfaces associated with point defects in solids.

Reconfigurable piezoelectric metamaterial for selective noise directivity

This work explores strategies to change the directivity of sound radiated from an oscillating metamaterial plate. To this end, the vibro-acoustic behavior of a plate that hosts an array of piezoelectric unit cells whose inductive shunt resonance frequencies can assume different predefined configurations in space is numerically investigated. The effects of such configurations on (i) the structural normal velocity of the plate, (ii) the hemispherical acoustic field estimated based on Rayleigh’s integral, and (iii) the non-negative acoustic intensity are discussed. The numerical results show that simple changes in the unit cell’s tuning configuration in space can reduce, totally attenuate, or enhance the sound radiated in specific directions, findings that may pave the way to the use of reconfigurable metamaterials as effective systems for selective radiated noise directivity.

Low Fidelity Data Driven Machine Learning Based Optimisation Method for Box-Wing Configuration

Wing design optimization traditionally involves computationally expensive high-fidelity simulations, limiting the exploration of design spaces. In this study, we propose a methodology that combines low-fidelity numerical models with machine learning algorithms to efficiently navigate the complex parameter space of box-wing configurations. Through the utilisation of a surrogate model trained on a limited dataset derived from low-fidelity simulations, our method strives to predict results within an acceptable range, significantly curtailing computational costs and time. The effectiveness of this methodology is demonstrated through a series of case studies, involving the Onera M6 and NASA CRM wing as test cases and Bionica box-wing optimization as an application case. The initial application of the proposed methodology to the box-wing case successfully achieved an almost 9.82% increase in overall aerodynamic efficiency. Its competitive performance compared to conventional optimization methods, along with its substantial reduction in computational time and resource requirements, is evident. This efficient methodology holds promise for enhancing the design optimization process for aviation start-ups by efficiently exploring complex design spaces with reduced computational burden.