Singular Locus Research Articles

Segre-driven radicality testing

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

The Ricci flow and the early universe

A framework of quantum spacetime reference frame is proposed and reviewed, in which the quantum spacetime at the Gaussian approximation is deformed by the Ricci flow. At sufficient large scale, the Ricci flow not only smooths out local small irregularities making the universe a homogeneous and isotropic Friedmann–Robertson–Walker metric, but also develops a local singularity at the physical-time origin. Due to the phenomenological suppression of the non-Gaussian primordial perturbations, we assume the validity of the Ricci flow applying to the high curvature region near the local singularity of the early universe. The no-local-collapsing theorem of Perelman ensures the existence of a canonical neighborhood around the large curvature pinching point, which resembles a gradient shrinking Ricci soliton (GSRS) solution of the Ricci flow. Without any inflaton field, the GSRS naturally reproduces an exact inflationary deSitter universe near the singularity at the leading order. Without any rolling-down behavior of inflaton, the deviation from exact deSitter described by the “slow roll parameters” can be calculated by a small deviation from the singular flow-time via the Ricci flow, and the primordial perturbations can also be studied on the GSRS background, the power spectrum of the scalar perturbation agrees with present observations, and the one of the tensor perturbation is predicted too small to be detectable than the standard inflation. The previous treatment of the cosmological constant and the effective gravity are also briefly reviewed in the framework. So we argue that the Ricci flow provides us a possible unified view and treatment of the late epoch accelerating expansion and early epoch inflation of the universe without introducing dark energy or inflaton (dark energy of the second kind).

Binary forms of suprageneric rank and the multiple root loci

We state the relation between the variety of binary forms of given rank and the dual of the multiple root loci. This is a new result for the suprageneric rank that appears as a continuation of the cited work by Buczyński, Han, Mella and Teitler. We describe the strata of these varieties and explore their singular loci.

METHODS OF CONTROLLING BOUNDARY TRANSPARENCY IN NON-HOMOGENOUS ELASTIC WAVEGUIDES

The work is devoted to the development of methods for controlling the efficiency of energy transfer in a composite elastic waveguide. Based on the analysis of the scattered field at the boundary of the stepped waveguide, formed by the rigid contact of two half-layers with the same mechanical characteristics, but with the different thicknesses, the main factors that affect the transparency of the interface were established. Harmonic oscillations generated by the first normal wave propagating from infinity in the narrower half-layer were considered. Mathematical difficulties of the posed boundary problem are due to the presence of a local singularity in the stresses at the point of change of the boundary conditions at the boundary of the two hemispheres. The solution is built by the superposition method, which allows taking into account the local singularity due to the asymptotic features of the unknowns. The quality criterion of the obtained solution was the control of the accuracy of the fulfillment of the conjugation conditions at the boundary of the two half-layers. The main attention in the work is focused on establishing the conditions for changing the transparency of the boundary depending on the frequencies, the symmetry of the oscillations, and the ratio of the half-layer widths. It was shown in the work that for both symmetric and antisymmetric oscillations of a stepped waveguide, there are frequency ranges in which the transparency of the boundary changes significantly. For both types of symmetry, in the frequency range up to the critical frequency for the third propagating normal wave, there are two frequency ranges in which the transparency of the boundary increases rather sharply. The frequencies at which local energy maxima are observed in the reflected field are different for symmetric and anti-symmetric oscillations. For symmetric oscillations, the first energy maximum in the reflected field is observed at the frequency when only one wave can propagate in both half-layers. This effect is due to the increase in the role of inhomogeneous waves in the transmitted field. The second energy maximum in the reflected field is due to the transformation of the energy of the incident wave into propagating waves of higher orders. In the case of antisymmetric oscillations, both maxima are due to the energy features of propagating waves of higher orders. The quality of energy resonance in the reflected field depends significantly on the symmetry of the oscillations. The established features of the scattered field make it possible to develop recommendations for controlling the transparency of the boundary in a stepped waveguide.

One-dimensional convolutional neural network for mapping mineral prospectivity: A case study in Changba ore concentration area, Gansu Province

Predictive modeling of mineral prospects is a critical but challenging procedure for delineating undiscovered prospective targets in mineral exploration. In the present study, the one-dimensional convolutional neural network (1D CNN) was utilized for mineral prospectivity mapping (MPM). Local singularity analysis and deposit buffer analysis were used to determine the unfavorable metallogenic regions, and random points were taken as negative samples. Synthetic minority over-sampling technique (SMOTE) was used to generate the training data to enhance the generalization ability of the model. The effects of different hyperparameters on classification performance in 1D CNN were studied and the optimal combination of hyperparameters was determined. The average classification accuracy of this model was 96.2%, and the standard deviation was 2.14%, indicating that the model constructed by this hyperparameter set was robust. On this basis, the geochemical data of 19 elements were used as the input characteristic variable, and the prospecting prospect of lead–zinc deposits in Changba ore concentration area was predicted based on 1D CNN. The results showed that the prospectivity map of lead–zinc deposits generated by the 1D CNN model can effectively link the multivariate geochemical data with the known positions of lead–zinc deposits and greatly increase the precision of the potential exploration areas for lead–zinc deposits.

Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals

Gauss–Legendre quadrature, Clenshaw–Curtis quadrature and the trapezoid rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals that arise in three-dimensional viscous fluid flow.

A New Anisotropic Singularity Algorithm to Characterize Geo-Chemical Anomalies in the Duolong Mineral District, Tibet, China

With the increasing exploitation of mineral resources by humans, exploring non-traditional areas for hidden resources such as deep earth and sediment-covered regions has become a significant challenge in the field of mineral exploration. Geochemical data, as a crucial information carrier of geological bodies, serves as one of the direct and effective sources for quantitative analysis of regional geological evolution and mineralization prediction studies. It plays an indispensable role in geographic information system (GIS)-based mineral exploration. Due to the neglect of spatial distribution characteristics and the variability of statistical features with spatial metrics in traditional statistical methods, this paper employs fractal/multifractal and the local singularity analysis to identify geochemical anomalies from background and characterize geochemical distributions associated with porphyry Cu-Au mineralization in the Duolong mineral district, Tibet, China. A novel algorithm for estimating the singularity index, which takes anisotropy into consideration, is proposed and practically applied to the Duolong district. By comparing with the isotropic singularity index, this new method objectively identifies anisotropic geochemical signatures and investigates non-linear behaviors of ore-forming elements, making it more practical and effective in geo-anomaly extraction. Furthermore, the current method is capable of indicating variations in geochemical distributions at different scales through directional arrows marking analytical windows. The summed-up direction of these multi-scale vectors effectively demonstrates migration trends of ore materials at each location within the study area. The new method can pinpoint the location of ore-forming element accumulation and migration directions, unlocking valuable insights from complex datasets. This promises to revolutionize our understanding of how minerals are formed and distributed within the Earth’s crust.

Almost-Riemannian structures on nonnilpotent, solvable 3D Lie groups

In this paper we study Almost-Riemannian Structures (ARS) on the class of nonnilpotent, solvable, connected 3D Lie groups. The nice structures present in such groups allow us to show that the singular locus of ARSs on such groups are always embedded submanifolds.

A generic instability in clustering dark energy?

In this paper, we study the effective field theory (EFT) of dark energy (DE) for the k-essence model beyond linear order. Using particle-mesh N-body simulations that consistently solve the DE evolution on a grid, we find that the next-to-leading order in the EFT expansion, which comprises the terms of the equations of motion that are quadratic in the field variables, gives rise to a generic instability in the regime of low speed of sound (high Mach number). We rule out the possibility of a numerical artefact by considering simplified cases in spherically and plane symmetric situations analytically. If the speed of sound vanishes exactly, the non-linear instability makes the evolution singular in finite time, signalling a breakdown of the EFT framework. The case of finite (but small) speed of sound is subtle, and the local singularity could be replaced by some other type of behaviour with strong non-linearities. While an ultraviolet completion may cure the problem in principle, there is no reason why this should be the case in general. As a result, for a large range of the effective speed of sound cs , a linear treatment is not adequate.

Hybrid Visual-Ranging Servoing for Positioning Based on Image and Measurement Features.

In this article, a hybrid visual-ranging servoing method is proposed to realize high-precision positioning tasks with a 6-degree of freedom (DOF) manipulator. This method utilizes the image and measurement features directly in the control loop. Without the need of complex image feature design and attitude estimation, this method realizes the 6-DOF control of a robot. A vital challenge in traditional vision-based systems is avoiding local minima and singularity problems. To tackle this issue, a full-rank interaction matrix hybrid visual servo (FRHVS) design criterion is proposed, which guarantees that the hybrid interaction matrix and its pseudoinverse matrix are both full rank. Moreover, the interaction matrix for these hybrid strategies, which combines image features with other sensors features, is derived in an analytical form. Experiments on a 6-DOF manipulator show that the proposed method is effective and has global asymptotic stability and high precision.

Numerical analytic continuation

Let f be an analytic function on a simply-connected compact continuum E of the complex z-plane. This might be an interval of the real line, where f might be real analytic. How can we calculate good estimates of the analytic continuation of f to other points zin {mathbb C}? How can we estimate the locations of real or complex singularities of f? We review both the theory and the practice of some existing methods for these problems and propose that excellent results can be obtained from the computation of rational approximations of f by the AAA algorithm. In the case of analytic functions of two or more variables, the rational approximations are applied along line segments or other analytic arcs.

Rigorous Bounds on Transport from Causality.

We use causality to derive a number of simple and universal constraints on dispersion relations, which describe the location of singularities of retarded two-point functions in relativistic quantum field theories. We prove that all causal dissipative dispersion relations have a finite radius of convergence in cases where stochastic fluctuations are negligible. We then give two-sided bounds on all transport coefficients in units of this radius, including an upper bound on diffusivity.

Universal location of Yang-Lee edge singularity in classic O(N) universality classes

Employing the functional renormalization group approach at next-to-leading order of the derivative expansion, we refine our earlier findings for the location of the Yang-Lee edge singularity in classic O(N) universality classes. For the universality classes of interest to QCD, in three dimensions, we found $|z_c|/R_\chi^{1/\gamma} = 1.612(9),\ 1.597(3)$ for $N=2$, $4$ correspondingly. We also established $|z_c| = 2.04(8),\ 1.69(3)$ for $N=2$, $4$ albeit with greater systematic error.

Generalized solutions to degenerate dynamical systems

The solutions to degenerate dynamical systems of the form A(x)ẋ=f(x) are studied by considering the equation as a differential inclusion. The set Z={det(A(x))=0}, called the singular set, is assumed to have an empty interior. The reasons leading us to the definition of the sets used for differential inclusion are exposed in detail. This definition is then applied on the one hand to generic cases and on the other hand to the particular cases resulting from physics, which can be found in Saavedra, Troncoso, and Zanelli [J. Math. Phys. 42, 4383 (2001)]. It is shown that generalized solutions may enter, leave, or remain in the singular locus.

Free-Space Optical Communication with an Optimized Lipschitz Exponent for Biosignal Telemetry

Abstract Healthcare monitoring is a rapidly developing network in the field of advanced medical treatment. The network combines the ideology of wireless communication, signal processing, medical information and real-time processing units to support the medical monitoring system. The proposed work focuses on the development of a Free-Space Optical (FSO) system to transmit the biosignals from a remote distance to the physician. Generally, the data transmitted over the FSO system is affected by various atmospheric conditions such as air medium, O2, and H2O molecules. To tackle these problems, the Biosignals Electrocardiogram (ECG) and Electroencephalogram (EEG) are processed in the Optimized Lipschitz Exponent (OLE) function before transmission over the FSO medium. In this novel technique, the OLE function measures the informative data from the biosignals by calculating the local regularities and singularity. This collects the most informative signals and transmits them in the signal over the FSO medium. This particular hybridization helps to transmit the required data without distortion. The Bit Error Rate (BER) of 10−9 is obtained, which satisfies the healthcare monitoring condition. The result section shows that the proposed model has minimum losses compared to the original signal.

A relative version of the Turaev–Viro invariants and the volume of hyperbolic polyhedral 3‐manifolds

AbstractWe define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3‐manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus of the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the volume conjecture for the Turaev–Viro invariants proposed by Chen–Yang [8] for hyperbolic 3‐manifolds with totally geodesic boundary.

Evaluation of an open-source pipeline to create patient-specific left atrial models: A reproducibility study

This work presents an open-source software pipeline to create patient-specific left atrial models with fibre orientations and a fibrDEFAULTosis map, suitable for electrophysiology simulations, and quantifies the intra and inter observer reproducibility of the model creation. The semi-automatic pipeline takes as input a contrast enhanced magnetic resonance angiogram, and a late gadolinium enhanced (LGE) contrast magnetic resonance (CMR). Five operators were allocated 20 cases each from a set of 50 CMR datasets to create a total of 100 models to evaluate inter and intra-operator variability. Each output model consisted of: (1) a labelled surface mesh open at the pulmonary veins and mitral valve, (2) fibre orientations mapped from a diffusion tensor MRI (DTMRI) human atlas, (3) fibrosis map extracted from the LGE-CMR scan, and (4) simulation of local activation time (LAT) and phase singularity (PS) mapping. Reproducibility in our pipeline was evaluated by comparing agreement in shape of the output meshes, fibrosis distribution in the left atrial body, and fibre orientations. Reproducibility in simulations outputs was evaluated in the LAT maps by comparing the total activation times, and the mean conduction velocity (CV). PS maps were compared with the structural similarity index measure (SSIM). The users processed in total 60 cases for inter and 40 cases for intra-operator variability. Our workflow allows a single model to be created in 16.72 ± 12.25 min. Similarity was measured with shape, percentage of fibres oriented in the same direction, and intra-class correlation coefficient (ICC) for the fibrosis calculation. Shape differed noticeably only with users’ selection of the mitral valve and the length of the pulmonary veins from the ostia to the distal end; fibrosis agreement was high, with ICC of 0.909 (inter) and 0.999 (intra); fibre orientation agreement was high with 60.63% (inter) and 71.77% (intra). The LAT showed good agreement, where the median ± IQR of the absolute difference of the total activation times was 2.02 ± 2.45 ms for inter, and 1.37 ± 2.45 ms for intra. Also, the average ± sd of the mean CV difference was -0.00404 ± 0.0155 m/s for inter, and 0.0021 ± 0.0115 m/s for intra. Finally, the PS maps showed a moderately good agreement in SSIM for inter and intra, where the mean ± sd SSIM for inter and intra were 0.648 ± 0.21 and 0.608 ± 0.15, respectively. Although we found notable differences in the models, as a consequence of user input, our tests show that the uncertainty caused by both inter and intra-operator variability is comparable with uncertainty due to estimated fibres, and image resolution accuracy of segmentation tools.

Remarks on 2d unframed quiver gauge theories

We study 2d mathcal{N} = (2, 2) quiver gauge theories without flavor nodes. There is a special class of quivers whose gauge group ranks stay positive in any duality frame. We illustrate this with the Abelian Kronecker quiver and the Abelian Markov quiver as the simplest examples. In the geometric phase, they engineer an infinite sequence of projective spaces and hypersurfaces in Calabi-Yau spaces, respectively. We show that the Markov quiver provides an Abelianization of SU(3) SQCD. Turning on the FI parameters and the θ angles for the Abelian quiver effectively deform SQCD by such parameters. For an Abelian necklace quiver corresponding to SU(k) SQCD, we find evidence for singular loci supporting non-compact Coulomb branches in the Kähler moduli space.

Iterated and mixed discriminants

Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schäfli for the study of hyperdeterminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hypersurfaces was given by Ottaviani. We propose and study a generalization of this question for systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant . We define a related polynomial called the multivariate iterated discriminant , generalizing the classical Schäfli method for hyperdeterminants. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre–Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.

On the structure of connected components of the moduli spaces of compact Riemann surfaces

Using the Broughton’s equisymmetric stratification of the moduli space of compact Riemann surfaces of genus g ge 2, Bartolini, Costa and Izquierdo have shown that its singular locus S_{g} is disconnected except for g = 3, 4, 7, 13, 17, 19, 59. One of a lot of problems motivated by this result concerns the study of its connected components, both from the quantitative and qualitative points of view. In 2012, Bartolini and Izquierdo have shown that all strata corresponding to the actions of Z_{2} and Z_{3} are contained in a single connected component C^{2,3}_g. The other components which were known to these authors were those composed by single strata and they asked whether there are components different from C^{2,3}_ g being the union at least two equisymmetric strata. The aim of this paper is to give an affirmative answer to this question, by showing that there are connected components composed of precisely two strata in S_{g} for g = rp(p -1)/2 where r ge 6 and p > 5r + 3 is a prime for which pr + 2 is also a prime.