Geometric Measure Research Articles

Entropy-based geometric measures of correlations for bell diagonal states via measurement

Abstract The classification and quantification of quantum correlations are the important issues in quantum information theory. Quantum relative entropy and Rényi relative entropy are both non-negative, and can therefore be regarded as generalized distances between quantum states. We first introduced the method proposed by Brodutch and Modi for constructing measurements of classical and quantum correlations. Subsequently, we use this method to construct measurement-induced geometric classical and quantum correlations based on quantum relative entropy and Rényi− 1 2 relative entropy. For two-qubit Bell diagonal states, the analytic expressions for these geometric classical correlations and quantum correlations are obtained.

Mathematical skills classification through primary education

By the end of primary school, children are expected to acquire a range of mathematical skills that progressively develop. This study aimed to gain insight into how a large number of numerical and geometrical measures are grouped and whether the structures shift or remain invariant along child’s development based on the data obtained from a sample of 1346 s to sixth grade children. On the basis of correlation analyses and exploratory factor analyses, we came up with an invariant four-factor structure for each grade. The four factors obtained were the following: (a) early and retrieval skills (subitizing, enumeration, number facts) (b) transcoding and ordinality skills, (c) numerical advanced skills and (d) visual-spatial advanced skills. Confirmatory factor analyses showed that the four-factor model fits well to the data (RMSEA < 0.06, CFI > 0.95, GFI > 0.93, SRMR < 0.05) at all grades. The associations between the mathematical measures were captured in each factor and the revealed invariant structure is discussed and compared with previous classification models of mathematical skills. Given that the difficulties in learning mathematics is currently being viewed as a continuum of academic abilities instead of a distinct problem, the underlying invariant four-factor structure can facilitate experts and educators to better understand how a broad area of mathematical skills are related across the primary education, in order to carry our comprehensive assessment of both the mathematical strengths and weaknesses of their students and to apply the appropriate customized teaching strategies.

Fractal dimension and lacunarity measures of glioma subcomponents are discriminative of the grade of gliomas and IDH status.

Since the overall glioma mass and its subcomponents-enhancing region (malignant part of the tumor), non-enhancing (less aggressive tumor cells), necrotic core (dead cells), and edema (water deposition)-are complex and irregular structures, non-Euclidean geometric measures such as fractal dimension (FD or "fractality") and lacunarity are needed to quantify their structural complexity. Fractality measures the extent of structural irregularity, while lacunarity measures the spatial distribution or gaps. The complex geometric patterns of the glioma subcomponents may be closely associated with the grade and molecular landscape. Therefore, we measured FD and lacunarity in the glioma subcomponents and developed machine learning models to discriminate between tumor grades and isocitrate dehydrogenase (IDH) gene status. 3D fractal dimension (FD3D) and lacunarity (Lac3D) were measured for the enhancing, non-enhancing plus necrotic core, and edema-subcomponents using preoperative structural-MRI obtained from the The Cancer Genome Atlas (TCGA) and University of California San Francisco Preoperative Diffuse Glioma MRI (UCSF-PDGM) glioma cohorts. The FD3D and Lac3D measures of the tumor-subcomponents were then compared across glioma grades (HGGs: high-grade gliomas vs. LGGs: low-grade gliomas) and IDH status (mutant vs. wild type). Using these measures, machine learning platforms discriminative of glioma grade and IDH status were developed. Kaplan-Meier survival analysis was used to assess the prognostic significance of FD3D and Lac3D measurements. HGG exhibited significantly higher fractality and lower lacunarity in the enhancing subcomponent, along with lower fractality in the non-enhancing subcomponent compared to LGG. This suggests that a highly irregular and complex geometry in the enhancing-subcomponent is a characteristic feature of HGGs. A comparison of FD3D and Lac3D between IDH-wild type and IDH-mutant gliomas revealed that mutant gliomas had ~2.5-fold lower FD3D in the enhancing-subcomponent and higher FD3D with lower Lac3D in the non-enhancing subcomponent, indicating a less complex and smooth enhancing subcomponent, and a more continuous non-enhancing subcomponent as features of IDH-mutant gliomas. Supervised ML models using FD3D from both the enhancing and non-enhancing subcomponents together demonstrated high-sensitivity in discriminating glioma grades (~97.9%) and IDH status (~94.4%). A combined fractal estimation of the enhancing and non-enhancing subcomponents using MR images could serve as a non-invasive, precise, and quantitative measure for discriminating glioma grade and IDH status. The combination of 2-hydroxyglutarate-magnetic resonance spectroscopy (2HG-MRS) with FD3D and Lac3D quantification may be established as a robust imaging signature for glioma subtyping.

Percolation Statistics in the MillenniumTNG Simulations

The statistical analysis of cosmic large-scale structure is most often based on simple two-point summary statistics, like the power spectrum or the two-point correlation function of a sample of galaxies or other types of tracers. In contrast, topological measures of clustering are also sensitive to higher-order correlations and thus offer the prospect to access additional information that may harbor important constraining power. We here revisit one such geometric measure of the cosmic web in the form of the so-called percolation analysis, using the recent MillenniumTNG simulation suite of the ΛCDM paradigm. We analyze continuum percolation statistics both for high-resolution dark matter particle distributions and for galaxy mock catalogs from a semianalytic galaxy formation model within a periodic simulation volume of 3000 Mpc on a side. For comparison, we also investigate the percolation statistics of random particle sets and neutrino distributions with two different summed particle masses. We find that the percolation statistics of the dark matter distribution evolves strongly with redshift and thus clustering strength, yielding a progressively lower percolation threshold toward later times. However, there is a sizable residual dependence on numerical resolution, which we interpret as a residual influence of different levels of shot noise. This is corroborated by our analysis of galaxy mock catalogs, whose results depend on sampling density more strongly than on galaxy selection criteria. While this limits the discriminative power of percolation statistics, our results suggest that it still remains useful as a complementary cosmological test when controlled for sampling density.

Can imaginarity be broadcast via real operations?

Imaginarity has proven to be a valuable resource in various quantum information processing tasks. A natural question arises: can the imaginarity of quantum states be broadcast via real operations? In this work, we present explicit structures for nonreal states whose imaginarity can be broadcast and cloned. That is, for a nonreal state, its imaginarity can be cloned if and only if it is a direct sum of several maximally imaginary states under orthogonal transformation, and its imaginarity can be broadcast if and only if it is a direct sum of a real state and some nonreal qubit states which are mixtures of two orthogonal maximally imaginary states under orthogonal transformation. In particular, we show that for a nonreal pure state, its imaginarity cannot be broadcast unless it is a maximally imaginary state. Furthermore, we derive a trade-off relation on the imaginarity broadcasting of pure states in terms of the measure of irreversibility of quantum states concerning real operations and the geometric measure of imaginarity. In addition, we demonstrate that any faithful measure of imaginarity is not superadditive.

A generalised entropic measure of steering using Tsallis entropies and the relationship with existent steering criteria

Abstract EPR steering is an asymmetric quantum correlation, as such, it is an important resource for quantum information protocols, like those used in quantum key distribution that are of tremendous relevance nowadays. The security of the aforementioned protocols can be shown by using entropic uncertainty measures, which are based on quantum information quantities, among which the entropies are found. In this work, we propose an entropic measure of steering, starting from the detection of steering given in a previous entropic uncertainty relation-based criterion, as well as using a generalised entropy, the Tsallis entropies. We define the entropic measure of steering in terms of an indicator, a quantity depending on the entropies of the measurements of the system, which assigns a value to the steering content of quantum states, while also comprising the asymmetry property of steering. We investigate the properties of the proposed quantity as a measure of steering and its asymmetry and show how it can be used for specific examples. For instance, a class of noisy two-qubit states that includes the Werner states is used to demonstrate the potential of such a proposal. Furthermore, a comparison is also made with the detection of steering in these states using existing steering criteria and the amount of steering a geometric measure for the Werner state gives.

Convergence to the planar interface for a nonlocal free‐boundary evolution

AbstractWe capture optimal decay for the Mullins–Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the planar profile in the physically relevant case of ambient space dimension three. Far from assuming small or well‐prepared initial data, we allow for initial interfaces that do not have graph structure and are not connected, hence explicitly including the regime of Ostwald ripening. In terms only of initially finite (not small) excess mass and excess surface energy, we establish that the surface becomes a Lipschitz graph within a fixed timescale (quantitatively estimated) and remains trapped within this setting. To obtain the graph structure, we leverage regularity results from geometric measure theory. At the same time, we extend a duality method previously employed for one‐dimensional PDE problems to higher dimensional, nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy, dissipation, and graph height are obtained.

Entanglement of multi-qubit states representing directed networks and its detection with quantum computing

We consider quantum graph states that can be mapped to directed weighted graphs, also known as directed networks. The geometric measure of entanglement of the states is calculated for the quantum graph states corresponding to arbitrary graphs. We find relationships between the entanglement and the properties of the corresponding graphs. Namely, we obtain that the geometric measure of entanglement of a qubit with other qubits in the graph state is related to the weights of ingoing and outgoing arcs with respect to the vertex representing the qubit, outdegree and indegree of the corresponding vertex in the graph. For unweighted and undirected graphs, the entanglement depends on the degree of the corresponding vertex. Quantum protocol for quantifying of the entanglement of the quantum graph states is constructed. As an example, a quantum graph state corresponding to a chain is examined, and the entanglement of the state is calculated on AerSimulator.

Quantifying subspace entanglement with geometric measures

Quantifying subspace entanglement with geometric measures

Fully nonlinear dead-core systems

We study fully nonlinear dead-core systems coupled with strong absorption terms. We discover a chain reaction, exploiting properties of an equation along the system and obtain higher sharp regularity across the free boundary. Additionally, we prove geometric measure estimates and obtain coincidence of the free boundaries. We also derive Liouville type theorems for entire solutions. These results are new even for linear systems.

A stabilised Total Lagrangian Element-Free Galerkin method for transient nonlinear solid dynamics

AbstractThis paper presents a new stabilised Element-Free Galerkin (EFG) method tailored for large strain transient solid dynamics. The method employs a mixed formulation that combines the Total Lagrangian conservation laws for linear momentum with an additional set of geometric strain measures. The main aim of this paper is to adapt the well-established Streamline Upwind Petrov–Galerkin (SUPG) stabilisation methodology to the context of EFG, presenting three key contributions. Firstly, a variational consistent EFG computational framework is introduced, emphasising behaviours associated with nearly incompressible materials. Secondly, the suppression of non-physical numerical artefacts, such as zero-energy modes and locking, through a well-established stabilisation procedure. Thirdly, the stability of the SUPG formulation is demonstrated using the time rate of Hamiltonian of the system, ensuring non-negative entropy production throughout the entire simulation. To assess the stability, robustness and performance of the proposed algorithm, several benchmark examples in the context of isothermal hyperelasticity and large strain plasticity are examined. Results show that the proposed algorithm effectively addresses spurious modes, including hour-glassing and spurious pressure fluctuations commonly observed in classical displacement-based EFG frameworks.

Temporal evolution of ascending aortic aneurysm wall stress predicts all-cause mortality.

Diameter-based risk stratification for elective repair of ascending aortic aneurysm fails to prevent type A dissection in many patients. Aneurysm wall stresses may contribute to risk prediction; however, rates of wall stress change over time are poorly understood. Our objective was to examine aneurysm wall stress changes over 3-5 years and subsequent all-cause mortality. Male veterans with <5.5 cm ascending aortic aneurysms and computed tomography at baseline and 3- to 5-year follow-up underwent three-dimensional aneurysm model construction. Peak circumferential and longitudinal wall stresses at systole were calculated using finite element analysis. Temporal trends were assessed by mixed-effects modelling. Changes in aortic wall stresses, diameter and length over time were evaluated as predictors of subsequent 3-year all-cause mortality by Cox proportional hazards modelling. Sixty-two male veterans were included in the study. Yearly changes in geometric and biomechanical measures were 0.12 mm/year (95% confidence interval, 0.04-0.20) for aortic diameter, 0.41 mm/year (0.12-0.71) for aortic length, 1.19 kPa/year -5.94 to 8.33) for peak circumferential stress, and 0.48 kPa/year (-3.89 to 4.84) for peak longitudinal stress. Yearly change in peak circumferential stress was significantly associated with hazard of death-hazard ratio for peak circumferential stress growth per 10 kPa/year, 1.27 (95% CI, 1.02-1.60; P = 0.037); hazard ratio for peak circumferential stress growth ≥ 32 kPa/year, 8.47 (95% CI, 2.42-30; P < 0.001). In this population of nonsurgical aneurysm patients, large temporal changes in peak circumferential stress, but not aortic diameter or length, was associated with all-cause mortality. Biomechanical stress and stress changes over time may be beneficial as additional risk factors for elective surgery in small aneurysms.

Holographic entanglement renormalisation for fermionic quantum matter

We demonstrate the emergence of a holographic dimension in a system of 2D non-interacting Dirac fermions placed on a torus, by studying the scaling of multipartite entanglement measures under a sequence of renormalisation group (RG) transformations applied in momentum space. Geometric measures defined in this emergent space can be related to the RG beta function of the spectral gap, hence establishing a holographic connection between the spatial geometry of the emergent spatial dimension and the entanglement properties of the boundary quantum theory. We prove, analytically, that changing the boundedness of the holographic space involves a topological transition accompanied by a critical Fermi surface in the boundary theory. We go on to show that this results in the formation of a quantum wormhole geometry that connects the UV and the IR of the emergent dimension. The additional conformal symmetry at the transition also supports a relation between the emergent metric and the stress-energy tensor. In the presence of an Aharonov–Bohm flux, the entanglement gains a geometry-independent piece which is shown to be topological, sensitive to changes in boundary conditions, and related to the Luttinger volume of the system. Upon the insertion of a strong transverse magnetic field, we show that the Luttinger volume is linked to the Chern number of the occupied single-particle Landau levels.

Multipartite generalization of geometric measure of discord

Radhakrishnan et al. (2020) proposed a generalization of quantum discord to multipartite systems, which is consistent with the conventional definition of discord in bipartite systems, and derived explicit formulae for any multipartite state. These results are significant in capturing quantum correlations for multi-qubit systems. We propose a generalization of the geometric measure of bipartite quantum discord to the multipartite case in the same manner. We find generic forms of the generalization of the geometric measure of quantum discord in a general N-partite quantum state. Further, we obtain computable exact formulas for the generalization of the geometric measure of quantum discord in an N-qubit quantum state, which includes the results obtained in [arXiv:2104.12344].

Horizontal Gaze Assessment: An Extensive Narrative Review

Bipedalism was a significant milestone in the evolutionary development of Homo sapiens sapiens, influencing neocortical evolution and subsequent behavioral changes. Coordinated visual and sensory inputs are crucial for posture, environmental interaction, and surgical planning, with horizontal gaze being a pivotal parameter. This narrative review aims to explore various geometric measures used to assess horizontal gaze in patients, highlighting their applications in surgical planning. A literature review was conducted in indexed databases using Mesh terms like "Cervical Vertebrae" and "Visual Fields" along with keywords such as "horizontal gaze" and "sagittal spine parameters." Among 477 initially identified articles, 41 were selected for inclusion after rigorous filtering. The most recognized method for assessing horizontal gaze is the Chin Brow Vertical Angle (CBVA), initially described in patients with ankylosing spondylitis. Clinical photography is employed as a tool for CBVA calculation, while other measures like McGregor slope and Slope of the Line of Sight have been considered as alternatives to CBVA. Each method presents its unique advantages and limitations. This review highlights the need for further research into horizontal gaze measurement methods. Developing novel approaches to determine horizontal gaze can significantly enhance surgical planning and, consequently, improve patient outcomes. The ongoing exploration of these geometric measures offers promising prospects for advancing the field and optimizing patient care.

Dosimetric impact of contour editing on CT and MRI deep-learning autosegmentation for brain OARs.

To establish the clinical applicability of deep-learning organ-at-risk autocontouring models (DL-AC) for brain radiotherapy. The dosimetric impact of contour editing, prior to model training, on performance was evaluated for both CT and MRI-based models. The correlation between geometric and dosimetric measures was also investigated to establish whether dosimetric assessment is required for clinical validation. CT and MRI-based deep learning autosegmentation models were trained using edited and unedited clinical contours. Autosegmentations were dosimetrically compared to gold standard contours for a test cohort. D1%, D5%, D50%, and maximum dose were used as clinically relevant dosimetric measures. The statistical significance of dosimetric differences between the gold standard and autocontours was established using paired Student's t-tests. Clinically significant cases were identified via dosimetric headroom to the OAR tolerance. Pearson's Correlations were used to investigate the relationship between geometric measures and absolute percentage dose changes for each autosegmentation model. Except for the right orbit, when delineated using MRI models, the dosimetric statistical analysis revealed no superior model in terms of the dosimetric accuracy between the CT DL-AC models or between the MRI DL-AC for any investigated brain OARs. The number of patients where the clinical significance threshold was exceeded was higher for the optic chiasm D1% than other OARs, for all autosegmentation models. A weak correlation was consistently observed between the outcomes of dosimetric and geometric evaluations. Editing contours before training the DL-AC model had no significant impact on dosimetry. The geometric test metrics were inadequate to estimate the impact of contour inaccuracies on dose. Accordingly, dosimetric analysis is needed to evaluate the clinical applicability of DL-AC models in the brain.

Boundary dynamics for holomorphic sequences, non-autonomous dynamical systems and wandering domains

There are many classical results, related to the Denjoy–Wolff theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, we address such questions in the very general setting of sequences (Fn) of holomorphic maps between simply connected domains. We show that, while some classical results can be generalised, with an interesting dependence on the geometry of the domains, a much richer variety of behaviours is possible. One of our main results is new even in the classical setting.Our methods apply in particular to non-autonomous dynamical systems, when (Fn) are forward compositions of holomorphic maps, and to the study of wandering domains in holomorphic dynamics.The proofs use techniques from geometric function theory, measure theory and ergodic theory, and the construction of examples involves a ‘weak independence’ version of the second Borel–Cantelli lemma and the concept from ergodic theory of ‘shrinking targets’.

Signed eigenvalue/vector distribution of complex order-three random tensor

Abstract We compute the signed distribution of the eigenvalues/vectors of the complex order-three random tensor by computing a partition function of a four-fermi theory, where signs are from a Hessian determinant associated to each eigenvector. The issue of the presence of a continuous degeneracy of the eigenvectors is properly treated by a gauge-fixing. The final expression is compactly represented by a generating function, which has an expansion whose powers are the dimensions of the tensor index spaces. A crosscheck is performed by Monte Carlo simulations. By taking the large-N limit we obtain a critical point where the behavior of the signed distribution qualitatively changes, and also the end of the signed distribution. The expected agreement of the end of the signed distribution with that of the genuine distribution provides a few applications, such as the largest eigenvalue, the geometric measure of entanglement, and the best rank-one approximation in the large-N limit.

Exotic self-assembly of hard spheres in a morphometric solvent

The self-assembly of spheres into geometric structures, under various theoretical conditions, offers valuable insights into complex self-assembly processes in soft systems. Previous studies have utilized pair potentials between spheres to assemble maximum contact clusters in simulations and experiments. The morphometric approach to solvation free energy that we utilize here goes beyond pair potentials; it is a geometry-based theory that incorporates a weighted combination of geometric measures over the solvent accessible surface for solute configurations in a solvent. In this paper, we demonstrate that employing the morphometric model of solvation free energy in simulating the self-assembly of sphere clusters results, under most conditions, in the previously observed maximum contact clusters. Under other conditions, it unveils an assortment of extraordinary sphere configurations, such as double helices and rhombohedra. These exotic structures arise specifically under conditions where the interactions take multibody potentials into account. This investigation establishes a foundation for comprehending the diverse range of geometric forms in self-assembled structures, emphasizing the significance of the morphometric approach in this context.

GIPC: Fast and Stable Gauss-Newton Optimization of IPC Barrier Energy

Barrier functions are crucial for maintaining an intersection- and inversion-free simulation trajectory but existing methods, which directly use distance can restrict implementation design and performance. We present an approach to rewriting the barrier function for arriving at an efficient and robust approximation of its Hessian. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, from which analytic eigensystems are derived and enhanced with filtering and stiffening terms that ensure robustness with respect to the convergence of a Project-Newton solver. A further advantage of our rewriting of the barrier function is that it naturally caters to the notorious case of nearly parallel edge-edge contacts for which we also present a novel analytic eigensystem. Our approach is thus well suited for standard second-order unconstrained optimization strategies for resolving contacts, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The efficiency of our eigensystems alone yields a 3× speedup over the standard Incremental Potential Contact (IPC) barrier formulation. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration.