Geometric Insight Research Articles

Introducing Novel Geometric Insights and Three-Dimensional Depictions of the Pythagorean Theorem for any Triangles

Introducing Novel Geometric Insights and Three-Dimensional Depictions of the Pythagorean Theorem for any Triangles

Geometric Landscape Annealing as an Optimization Principle Underlying the Coherent Ising Machine

Given the fundamental importance of combinatorial optimization across many diverse domains, there has been widespread interest in the development of unconventional physical computing architectures that can deliver better solutions with lower resource costs. However, a theoretical understanding of their performance remains elusive. We develop such understanding for the case of the coherent Ising machine (CIM), a network of optical parametric oscillators that can be applied to any quadratic unconstrained binary optimization problem. We focus on how the CIM finds low-energy solutions of the Sherrington-Kirkpatrick spin glass. As the laser gain of this system is annealed, the CIM interpolates between gradient descent on coupled soft spins to descent on coupled binary spins. By combining the Kac-Rice formula, the replica method, and supersymmetry breaking, we develop a detailed understanding of the evolving geometry of the high-dimensional energy landscape of the CIM as the laser gain increases, finding several phase transitions in the landscape, from flat to rough to rigid. Additionally, we develop a novel cavity method that provides a geometric interpretation of supersymmetry breaking in terms of the reactivity of a rough landscape to specific external perturbations. Our energy landscape theory successfully matches numerical experiments, provides geometric insights into the principles of CIM operation, and yields optimal annealing schedules. Published by the American Physical Society 2024

Nonintegrability of 3 DoF Hamiltonian Resonant Systems

We review a number of methods to prove nonintegrability of Hamiltonian systems and focus on 3 Degrees-of-Freedom (DoF) systems listing the known results for the prominent resonances. Associated with the Hamiltonian systems are the averaged-normal forms that provide us with geometric insight, approximations of orbits and measures of chaos. Symmetries do change the qualitative and quantitative pictures; we illustrate this for the 1:2:1 resonance with discrete symmetry in the 1st and 3rd DoF. In this case, the averaged-normal form is still nonintegrable, but it becomes integrable when adding discrete symmetry in all DoF. Apart from the short-periodic solutions obtained by averaging, we find many periodic solutions. There is numerical evidence of the presence of Šilnikov bifurcation which clarifies the presence of nonintegrability phenomena qualitatively and quantitatively.

Universal phase-field mixture representation of thermodynamics and shock-wave mechanics in porous soft biologic continua.

A continuum mixture theory is formulated for large deformations, thermal effects, phase interactions, and degradation of soft biologic tissues suitable at high pressures and low to very high strain rates. Tissues consist of one or more solid and fluid phases and can demonstrate nonlinear anisotropic elastic, viscoelastic, thermoelastic, and poroelastic physics. Under extreme deformations or shock loading, tissues may fracture, tear, or rupture. Existing models do not account for all physics simultaneously, and most poromechanics and soft-tissue models assume incompressibility of some or all constituents, generally inappropriate for modeling shock waves or extreme compressions. Motivated by these prior limitations, a thermodynamically consistent formulation that combines a continuum theory of mixtures, compressible nonlinear anisotropic thermoelasticity, viscoelasticity, and phase-field mechanics of fracture is constructed to resolve the pertinent physics. A metric tensor of generalized Finsler space supplies geometric insight on effects of rearrangements of microstructure, for example degradation, growth, and remodeling. Shocks are modeled as singular surfaces. Hugoniot states and shock decay are analyzed: Solutions account for concurrent viscoelasticity, fracture, and interphase momentum and energy exchange not all contained in previous analyses. Suitability of the framework for representing blood, skeletal muscle, and liver is demonstrated by agreement with experimental data and observations across a range of loading rates and pressures. Insight into previously unresolved physics is obtained, for example importance of rate sensitivity of damage and quantification of effects of dissipation from viscoelasticity and phase interactions on shock decay.

Geometric Amplitude Accompanying Local Responses: Spinor Phase Information from the Amplitudes of Spin-Polarized STM Measurements.

Solving the Hamiltonian of a system yields the energy dispersion and eigenstates. The geometric phase of the eigenstates generates many novel effects and potential applications. However, the geometric properties of the energy dispersion go unheeded. Here, we provide geometric insight into energy dispersion and introduce a geometric amplitude, namely, the geometric density of states (GDOS) determined by the Riemann curvature of the constant-energy contour. The geometric amplitude should accompany various local responses, which are generally formulated by the real-space Green's function. Under the stationary phase approximation, the GDOS simplifies the Green's function into its ultimate form. In particular, the amplitude factor embodies the spinor phase information of the eigenstates, favoring the extraction of the spin texture for topological surface states under an in-plane magnetic field through spin-polarized STM measurements. This work opens a new avenue for exploring the geometric properties of electronic structures and excavates the unexplored potential of spin-polarized STM measurements to probe the spinor phase information of eigenstates from their amplitudes.

Rao's Theorem for forcibly planar sequences revisited

We consider the graph degree sequences such that every realisation is a polyhedron. It turns out that there are exactly eight of them. All of these are unigraphic, in the sense that each is realised by exactly one polyhedron. This is a revisitation of a Theorem of Rao about sequences that are realised by only planar graphs.Our proof yields additional geometrical insight on this problem. Moreover, our proof is constructive: for each graph degree sequence that is not forcibly polyhedral, we construct a non-polyhedral realisation.

Word Measures on Unitary Groups: Improved Bounds for Small Representations

Abstract Let $F$ be a free group of rank $r$ and fix some $w\in F$. For any compact group $G$ we can define a measure $\mu _{w,G}$ on $G$ by (Haar-)uniformly sampling $g_{1},...,g_{r}\in G$ and evaluating $w(g_{1},...,g_{r})$. In [23], Magee and Puder study the behavior of the moments of $\mu _{w,U(n)}$ as a function of $n$, establishing a connection between their asymptotic behavior and certain algebraic invariants of $w$, such as its commutator length. We employ geometric insights to refine their analysis, and show that the asymptotic behavior of the moments is also governed by the primitivity rank of $w$. Additionally, we also apply our methods to prove a special case of a conjecture of Hanany and Puder [13, Conjecture 1.13] regarding the asymptotic behavior of expected values of irreducible characters of $U(n)$ under $\mu _{w,U(n)}$.

MeshPointNet: 3D Surface Classification Using Graph Neural Networks and Conformal Predictions on Mesh-Based Representations

Abstract In many design automation applications, accurate segmentation and classification of 3D surfaces and extraction of geometric insight from 3D models can be pivotal. This paper primarily introduces a machine learning-based scheme that leverages graph neural networks for handling 3D geometries, specifically for surface classification. Our model demonstrates superior performance against two state-of-the-art models, PointNet + + and PointMLP, in terms of surface classification accuracy, beating both models. Central to our contribution is the novel incorporation of conformal predictions, a method that offers robust uncertainty quantification and handling with marginal statistical guarantees. Unlike traditional approaches, conformal predictions enable our model to ensure precision, especially in challenging scenarios where mistakes can be highly costly. This robustness proves invaluable in design applications, and as a case in point, we showcase its utility in automating the computational fluid dynamics meshing process for aircraft models based on expert guidance. Our results reveal that our automatically generated mesh, guided by the proposed rules by experts enabled through the segmentation model, is not only efficient but matches the quality of expert-generated meshes, leading to accurate simulations.

The Synge G-Method: cosmology, wormholes, firewalls, geometry

Unphysical equations of state result from the unrestricted use of the Synge G-trick of running the Einstein field equations backwards; in particular often this results in ρ+p<0 which implies negative inertial mass density, which does not occur in reality. This is the basis of some unphysical spacetime models including phantom energy in cosmology and traversable wormholes. The slogan ‘ER = EPR’ appears to have no basis in physics and is merely the result of vague and unbridled speculation. Wormholes (the ‘ER’ of the slogan) are a mathematical curiosity of general relativity that have little to no application to a description of our Universe. In contrast quantum correlations (the ‘EPR’ of the slogan) are a fundamental property of quantum mechanics that follows from the principle of superposition and is true regardless of the properties of gravity. The speculative line of thought that led to ‘ER = EPR’ is part of a current vogue for anti-geometrical thinking that runs counter to (and threatens to erase) the great geometrical insights of the global structure program of general relativity.

A refinement of Morse–Novikov cohomology on manifolds with boundary and the cohomology of the space of solutions of kerΔ𝜃

In this paper, we consider the Morse–Novikov coboundary operator [Formula: see text] where [Formula: see text], and which is given by [Formula: see text], and for which the associated Morse–Novikov cohomology groups are denoted by [Formula: see text]. The ultimate objective of this paper is to uncover more geometric and topological insights about [Formula: see text] when the boundary of [Formula: see text] is nonvanishing. To begin, we employ a different approach to prove that the concrete realizations of the Morse–Novikov cohomology groups of [Formula: see text] intersect at the origin, and then we use the long exact sequence and cohomological algebra as powerful tools to decompose the absolute and relative Morse–Novikov cohomologies into interior and boundary portions, respectively. On the other hand, we use a novel reasoning based on cohomological algebra to determine the cohomology of [Formula: see text], and as a result the absolute Morse–Novikov cohomology of [Formula: see text] can be characterized in terms of two successive degrees in just one degree of the cohomology of [Formula: see text]. Consequently, the cohomology of [Formula: see text] can be recovered from the interior and boundary sections.

Curvature-enhanced graph convolutional network for biomolecular interaction prediction

Geometric deep learning has demonstrated a great potential in non-Euclidean data analysis. The incorporation of geometric insights into learning architecture is vital to its success. Here we propose a curvature-enhanced graph convolutional network (CGCN) for biomolecular interaction prediction. Our CGCN employs Ollivier-Ricci curvature (ORC) to characterize network local geometric properties and enhance the learning capability of GCNs. More specifically, ORCs are evaluated based on the local topology from node neighborhoods, and further incorporated into the weight function for the feature aggregation in message-passing procedure. Our CGCN model is extensively validated on fourteen real-world bimolecular interaction networks and analyzed in details using a series of well-designed simulated data. It has been found that our CGCN can achieve the state-of-the-art results. It outperforms all existing models, as far as we know, in thirteen out of the fourteen real-world datasets and ranks as the second in the rest one. The results from the simulated data show that our CGCN model is superior to the traditional GCN models regardless of the positive-to-negative-curvature ratios, network densities, and network sizes (when larger than 500).

Phase response to arbitrary perturbations: Geometric insights and resetting surfaces

Phase response to arbitrary perturbations: Geometric insights and resetting surfaces

Insights into the dosimetric and geometric characteristics of stereotactic radiosurgery for multiple brain metastases: A systematic review.

GammaKnife (GK) and CyberKnife (CK) have been the mainstay stereotactic radiosurgery (SRS) solution for multiple brain metastases (MBM) for several years. Recent technological advancement has seen an increase in single-isocentre C-arm linac-based SRS. This systematic review focuses on dosimetric and geometric insights into contemporary MBM SRS and thereby establish if linac-based SRS has matured to match the mainstay SRS delivery systems. The PubMed, Web of Science and Scopus databases were interrogated which yielded 891 relevant articles that narrowed to 20 articles after removing duplicates and applying the inclusion and exclusion criteria. Primary studies which reported the use of SRS for treatment of MBM SRS and reported the technical aspects including dosimetry were included. The review was limited to English language publications from January 2015 to August 2023. Only full-length papers were included in the final analysis. Opinion papers, commentary pieces, letters to the editor, abstracts, conference proceedings and editorials were excluded. The Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines were followed. The reporting of conformity indices (CI) and gradient indices, V12Gy, monitor units and the impact of translational and rotational shifts were extracted and analysed. The single-isocentre technique for MBM dominated recent SRS studies and the most studied delivery platforms were Varian. The C-arm linac-based SRS plan quality and normal brain tissue sparing was comparable to GK and CK and in some cases better. The most used nominal beam energy was 6FFF, and optimised couch and collimator angles could reduce mean normal brain dose by 11.3%. Reduction in volume of the healthy brain receiving a certain dose was dependent on the number and size of the metastases and the relative geometric location. GK and CK required 4.5-8.4 times treatment time compared with linac-based SRS. Rotational shifts caused larger changes in CI in C-arm linac-based single-isocentre SRS. C-arm linac-based SRS produced comparable MBM plan quality and the delivery is notably shorter compared to GK and CK SRS.

Spectroscopic definition of ferrous active sites in non-heme iron enzymes.

Non-heme iron enzymes play key roles in antibiotic, neurotransmitter, and natural product biosynthesis, DNA repair, hypoxia regulation, and disease states. These enzymes had been refractory to traditional bioinorganic spectroscopic methods. Thus, we developed variable-temperature variable-field magnetic circular dichroism (VTVH MCD) spectroscopy to experimentally define the excited and ground ligand field states of non-heme ferrous enzymes (Solomon et al., 1995). This method provides detailed geometric and electronic structure insight and thus enables a molecular level understanding of catalytic mechanisms. Application of this method across the five classes of non-heme ferrous enzymes has defined that a general mechanistic strategy is utilized where O2 activation is controlled to occur only in the presence of all cosubstrates.

Spin labels for 19F ENDOR distance determination: resolution, sensitivity and distance predictability.

19F electron-nuclear double resonance (ENDOR) has emerged as an attractive method for determining distance distributions in biomolecules in the range of 0.7-2 nm, which is not easily accessible by pulsed electron dipolar spectroscopy. The 19F ENDOR approach relies on spin labeling, and in this work, we compare various labels' performance. Four protein variants of GB1 and ubiquitin bearing fluorinated residues were labeled at the same site with nitroxide and trityl radicals and a Gd(III) chelate. Additionally, a double-histidine variant of GB1 was labeled with a Cu(II) nitrilotriacetic acid chelate. ENDOR measurements were carried out at W-band (95 GHz) where 19F signals are well separated from 1H signals. Differences in sensitivity were observed, with Gd(III) chelates providing the highest signal-to-noise ratio. The new trityl label, OXMA, devoid of methyl groups, exhibited a sufficiently long phase memory time to provide an acceptable sensitivity. However, the longer tether of this label effectively reduces the maximum accessible distance between the 19F and the Cα of the spin-labeling site. The nitroxide and Cu(II) labels provide valuable additional geometric insights via orientation selection. Prediction of electron-nuclear distances based on the known structures of the proteins were the closest to the experimental values for Gd(III) labels, and distances obtained for Cu(II) labeled GB1 are in good agreement with previously published NMR results. Overall, our results offer valuable guidance for selecting optimal spin labels for 19F ENDOR distance measurement in proteins.

Local stability of spheres via the convex hull and the radical Voronoi diagram.

Jamming is an emergent phenomenon wherein the local stability of individual particles percolates to form a globally rigid structure. However, the onset of rigidity does not imply that every particle becomes rigid, and indeed some remain locally unstable. These particles, if they become unmoored from their neighbors, are called rattlers, and their identification is critical to understanding the rigid backbone of a packing, as these particles cannot bear stress. The accurate identification of rattlers, however, can be a time-consuming process, and the currently accepted method lacks a simple geometric interpretation. In this manuscript, we propose two simpler classifications of rattlers in hard sphere systems based on the convex hull of contacting neighbors and the maximum inscribed sphere of the radical Voronoi cell, each of which provides geometric insight into the source of their instability. Furthermore, the convex hull formulation can be generalized to explore stability in hyperstatic soft sphere packings, spring networks, nonspherical packings, and mean-field non-central-force potentials.

Analytical Optimal Solution for the Reachable Domain of Low-Thrust Spacecraft

Low-thrust electric propulsion system has drawn increasing attention from researchers because of its high propellant efficiency. The reachable domain of low-thrust spacecraft can provide a geometric insight for space mission planning. In this paper, analytical solutions of the envelope of reachable domain are obtained by employing the Pontryagin’s maximum principle in the well-known linearized model of relative motion, the Tschauner–Hempel equations. Specifically, this study focuses on two contributions. First, an analytical solution of the envelope of reachable domain is obtained, and the associated optimal control profile is derived. Second, an ellipsoid approximation of the reachable domain is proposed to represent the envelope directly based on the analytical solution. Numerical simulations are conducted to demonstrate the accuracy of the proposed solutions. The results show that the reachable domain obtained analytically coincides well with that solved by the numerical indirect method based on the two-body model with low thrust.

Astrodynamics-Informed Kinodynamic Sampling-Based Motion Planning for Relative Spacecraft Motion

This paper presents a sampling-based spacecraft relative motion planning algorithm to reconfigure a spacecraft from a given initial state to a desired final state, subject to kinodynamic (simultaneous kinematic and dynamics) constraints. We leverage theoretical advancement in the field of sampling-based motion planning in robotics and extend their application to achieve motion planning in astrodynamics. We present an astrodynamics-informed kinodynamic motion planning (AIKMP) algorithm that takes advantage of a spacecraft’s natural motion to compute a safe, fuel-efficient motion plan for spacecraft reconfiguration in a cluttered environment. This algorithm is developed in the relative orbital element space of the deputy spacecraft around a chief spacecraft that provides very useful geometric insight to relative spacecraft motion, as well as helps in generating an overall smooth final transfer trajectory without demanding additional postprocessing of the transfer solution. Through several example scenarios, we demonstrate that the AIKMP algorithm is regularly able to compute safe and fuel-efficient solutions to relative transfer problems. We also show that this algorithm iteratively improves on its computed solutions and thus holds the capability of finding near-optimal transfer solutions given a sufficient number of algorithm iterations—without requiring an initial guess of the solution.

The invariant manifold approach applied to global long-time dynamics of FitzHugh-Nagumo systems

We consider the FitzHugh-Nagumo system equipped with boundary condition of Dirichlet type on some two/three-dimensional domains. This system describes the signal transmission across the axonal membrane in neurophysiology. It is a semilinear parabolic PDE for the voltage variable coupled with a first-order ODE of space-time type for the recovery variable. We prove that there exists a finite-dimensional global manifold in the case of the fast recovery variable. Since the manifold is uniformly attracting, it gives geometric insight into the global long-time dynamics of the solutions. The proof is based on an abstract invariant manifold theorem for dynamical systems on a Banach space.

Geometric Insights into the Multivariate Gaussian Distribution and Its Entropy and Mutual Information.

In this paper, we provide geometric insights with visualization into the multivariate Gaussian distribution and its entropy and mutual information. In order to develop the multivariate Gaussian distribution with entropy and mutual information, several significant methodologies are presented through the discussion, supported by illustrations, both technically and statistically. The paper examines broad measurements of structure for the Gaussian distributions, which show that they can be described in terms of the information theory between the given covariance matrix and correlated random variables (in terms of relative entropy). The content obtained allows readers to better perceive concepts, comprehend techniques, and properly execute software programs for future study on the topic's science and implementations. It also helps readers grasp the themes' fundamental concepts to study the application of multivariate sets of data in Gaussian distribution. The simulation results also convey the behavior of different elliptical interpretations based on the multivariate Gaussian distribution with entropy for real-world applications in our daily lives, including information coding, nonlinear signal detection, etc. Involving the relative entropy and mutual information as well as the potential correlated covariance analysis, a wide range of information is addressed, including basic application concerns as well as clinical diagnostics to detect the multi-disease effects.