Singularity Index Research Articles

Singularity indices of hyperelliptic fibrations

Singularity indices of hyperelliptic fibrations

Sharp focusing of beams with V-point polarization singularities

It is theoretically and numerically shown that when tightly focusing an n-th order vector light field that has the central V-point (at which the linear polarization direction is undetermined), the polarization singularity index n, and a "flower"-shaped intensity pattern with 2(n-1) lobes it forms a transverse intensity distribution with 2(n-1) local maxima. At the same time, a vector light field with the polarization singularity index -n, which has the form of a "web" with 2(n+1) cells generates at the sharp focus a transverse intensity distribution with 2(n+1) local maxima. In the focal spot, either 2(n-1) or 2(n+1) V-point polarization singularities with alternating indices +1 or -1 are formed at the intensity zero.

Tightly focusing vector beams containing V-point polarization singularities

We show theoretically and numerically that an n-th order vector light field containing the central V-point singularity of indefinite linear polarization with polarization singularity index n, with a 'flower' 2(n – 1)-petal polarization pattern centered on it, produces an intensity pattern with 2(n – 1) local maxima at the tight focus. Meanwhile, a vector light field with polarization singularity index –n, leading to a 'web' of polarization singularities composed of 2(n + 1) cells, is tightly focused into an intensity pattern with 2(n + 1) intensity maxima. At the intensity nulls at the focus, either 2(n – 1) or 2(n + 1) V-points with alternating + 1 or –1 indices are produced. In addition, we study more general vector fields of the order (n, m) and analytically derive their Poincare-Hopf indices for many values of n and m. Application areas of such light fields with polarization singularities are laser information technologies, laser material processing, microscopy and optical trapping.

A Riemannian metric for geometry-aware singularity avoidance by articulated robots

Articulated robots such as manipulators increasingly must operate in uncertain and dynamic environments where interaction (with human coworkers, for example) is necessary. In these situations, the capacity to quickly adapt to unexpected changes in operational space constraints is essential. At certain points in a manipulator’s configuration space, termed singularities, the robot loses one or more degrees of freedom (DoF) and is unable to move in specific operational space directions. The inability to move in arbitrary directions in operational space compromises adaptivity and, potentially, safety. We introduce a geometry-aware singularity index, defined using a Riemannian metric on the manifold of symmetric positive definite matrices, to provide a measure of proximity to singular configurations. We demonstrate that our index avoids some of the failure modes and difficulties inherent to other common indices. Further, we show that our index can be differentiated easily, making it compatible with local optimization approaches used for operational space control. Our experimental results establish that, for reaching and path following tasks, optimization based on our index outperforms a common manipulability maximization technique and ensures singularity-robust motions.

Application of multivariate canonical harmonic trend analysis, singularity analysis with radius-areal metal amount and improved adaptive fuzzy self-organizing mapping to identify geochemical anomaly related to iron polymetallic mineralization in Hunjiang district, Northeastern China

How to more effectively perform anomaly detection of combination information has always been an important issue for the scholars in various fields. In order to identify and extract the geochemical anomaly information related to polymetallic mineralization in the Hunjiang area, this article uses the hybrid method that combines multivariate canonical harmonic trend analysis (MCHTA), singularity analysis with radius-areal metal amount and improved adaptive fuzzy self-organizing map (IAFSOM). First, multiple sets of combination feature information with multi-dimensional variables will be obtained through the MCHTA method, which information is considered as the initial information for the subsequent analysis. Next, the singularity analysis method is used to process the combination concentration value to calculate the singularity indexes. Finally, the singularity indexes are classified by the IAFSOM method, and nine groups of sample data are obtained. The analysis results found that the samples information in fourth group covered most of the low α-values. The main conclusions in this study are as follows: (1) The MCHTA method can effectively detect the combination information related to geochemical anomaly; (2) The application of singularity analysis method with radius-areal metal amount can reveal the significant characteristics of mineralization combination elements; (3) IAFSOM can be used as an effective tool for the classification and identification of geochemical anomaly with combination information; (4) the hybrid method that combines MCHTA method, singularity analysis and IAFSOM model has a good indication significance in the prospecting of geochemical anomalies, and could provide a good method for geochemical prospecting.

Multifractal analysis on spatial variability of soil particles and nutrients of Benggang in granite hilly region, China

Benggang erosion is a large-scale, rapidly developing phenomenon that commonly occurs in the granite hilly region of southern China and severely endangers local land resources and the ecological environment. A developing Benggang in southeast China was selected to explore the non-uniformity of soil particle size distribution (PSD) and the nutrient heterogeneity during Benggang erosion using multifractal theory. The results showed significant differences in the soil particle composition and nutrient components of different parts of Benggang. The generalized dimension spectra exhibited an inverse “S” decreasing pattern and the singularity spectra showed a left hook shape, indicating that the soil PSD of Benggang features high non-uniformity and can be described by the multifractal theory. The information dimension (D1) and the correlation dimension (D2) were high in the uppercatchment, the lowerportion ofthe collapsingwall, and the top ofthe alluvialfan. D1, D2, and the spectral width (Δα) were negatively correlated with clay and silt content but positively correlated with sand content. In contrast, D1–D2 and the degree of symmetry (Δf) showed opposite relationships. The multifractal singularity indexes α0 and Δα were significantly negatively correlated with soil organic matter but positively correlated with bulk density; opposite correlations were observed for Δf. The findings revealed that the multifractal theory can quantitatively characterize the characteristics of soil PSD and laterally reflect soil nutrient loss. This research introduces a scientific method for quantitatively describing the heterogeneity of soil PSD during Benggang development. In addition, it also provides a theoretical basis for controlling Benggang erosion to protect the environment of the southern granite region.

H-plane horn antenna with enhanced directivity using conformal transformation optics

Conformal transformation optics is employed to enhance an H-plane horn’s directivity by designing a graded-index all-dielectric lens. The transformation is applied so that the phase error at the aperture is gradually eliminated inside the lens, leading to a low-profile high-gain lens antenna. The physical space shape is modified such that singular index values are avoided, and the optical path inside the lens is rescaled to eliminate superluminal regions. A prototype of the lens is fabricated using three-dimensional printing. The measurement results show that the realized gain of an H-plane horn antenna can be improved by 1.5–2.4 dB compared to a reference H-plane horn.

Flood characteristic description using the singularity method

An advanced understanding of the flood mechanism will provide scientific evidence for decision makers for assessing water resources and development planning. Mathematical models can be used to develop new indices for flood characteristics description and forecasting. In this study, a singularity index (fractal model) was developed by using peak flow and flow discharge data with time to describe flooding events in the Oak Ridges Moraine (ORM) area, especially for characterizing flow change properties. It was found that the singularity index can be used to describe flood characteristics, which describes the changing behaviour around the peak of the flood flow. The higher the singularity index, the more quickly the flood process changes, and vice versa. It was also observed that the singularity index is higher when more watersheds have urban land cover, and it is lower when fewer watersheds have urban land cover and more have agricultural land cover.

Efficiency of singularity and PCA mapping of mineralization-related geochemical anomalies: a comparative study using BLEG and <180μm stream sediment geochemical data

In gold (Au) exploration, the analysis of both bulk leach extractable gold (BLEG) and acid-extractable Au in the <180μm stream sediment fraction are the two most common approaches. The Eskişehir- Sivrihisar region in Western Türkiye hosts several orogenic type mineral deposits. The purpose of this study is to delineate geochemical anomalies of ore-related elements and track their dispersion, which may lead to discovery of unknown ore deposits. This research also compares the capability of conventional statistical and principal component analysis (PCA), with concentration area (CA) and number-size/concentration (N-S/C) fractal methods as well as singularity index method to differentiate anomalous and background Au distributions. Known Au mineralization in the region of interest is strongly reflected in stream sediment BLEG Au patterns, which have robust singularity indices with C-A and N-S multifractal modeling and PCA. A hundred % of the Au deposits were detected using either BLEG Au and Ag singularity index mapping with C-A fractal analysis whereas the factor analysis of which revealed 85% efficiency. Several strong Au-Ag anomalies defined by the singularity index and factor analysis in this study requires further follow up for the discovery of new deposits.

Sharp Focusing of a Hybrid Vector Beam with a Polarization Singularity

The key result of this work is the use of the global characteristics of the polarization singularities of the entire beam as a whole, rather than the analysis of local polarization, Stokes and Poincare–Hopf indices. We extend Berry’s concept of the topological charge of scalar beams to hybrid vector beams. We discuss tightly focusing a new type of nth-order hybrid vector light field comprising n C-lines (circular polarization lines). Using a complex Stokes field, it is shown that the field polarization singularity index equals n/2 and does not preserve in the focal plane. The intensity and Stokes vector components in the focal plane are expressed analytically. It is theoretically and numerically demonstrated that at an even n, the intensity pattern at the focus is symmetrical, and instead of C-lines, there occur C-points around which axes of polarization ellipses are rotated. At n = 4, C-points characterized by singularity indices 1/2 and ‘lemon’-type topology are found at the focus. For an odd source field order n, the intensity pattern at the focus has no symmetry, and the field becomes purely vectorial (with no elliptical polarization) and has n V-points, around which linear polarization vectors are rotating.

Three-dimensional singularity index at interface corner in prismatic butt joint under arbitrary material combination

In this paper, the three-dimensional (3D) singularity index at the interface corner λcorner is newly analyzed for the prismatic butt joint in comparison with the two-dimensional (2D) singularity index along the interface edge λ2D previously studied. The finite element eigenanalysis reveals that the ratio of the singularity index is in the range of 0.85≤λcorner/λ2D≤1.15 for the all range of material combination. Under 2D bad pair condition expressed by Dundurs’ parameters α,β as αα-2β>0, the ratio is in the range 0.85≤λcorner/λ2D<1. The 3D bad pair condition when the singular stress appears at the corner is clarified on the α,β map in comparison with 2D well-known expression αα-2β>0. Simple and convenient discriminant formula at the 3D interface corner is also indicated for the appearance of the stress singularity.

A new approach on quantification of flexibility index in multi-carrier energy systems towards optimally energy hub management

Multi-carrier energy systems (MCESs) are identified as intermediate frameworks among different energy carriers aiming to optimally meet diverse energy demands. In this study, a novel robust-based flexibility evaluation method is proposed for MCESs to quantify the maximum potentiality of the energy hub to compensate for the maximum uncertainty. For this purpose, a singular flexibility index is first calculated via the definition of apposite corrective actions, including purchasing the supplementary input carriers and implementing the demand response programs (DRPs). Then, given various individual uncertainty sets, a new bi-level optimization framework is introduced to achieve the optimal and deterministic scheduling of MCESs in the upper-level and determining the largest uncertainty radiuses in the lower-level. The demonstrated method is formulated as a second-order cone programming and is solved through an iteration-based search algorithm. The presented structure is implemented for a sample energy hub to confirm its extensive capability to apply various flexible options as compared to the information gap decision theory (IGDT). The maximum uncertainty margin in the electrical load is averagely equal to 79% of the predicted demand. The proposed method could be also fruitful for the robust scheduling of the energy hub in a worst-case scenario while considering the maximum uncertainty.

Polarization singularity index determination by using a tilted lens.

The superposition of spin and orbital angular momentum states of light generates polarization singularities. By perturbing and disintegrating their component orbital angular momentum (OAM) states, the polarization singularity indices can be determined. The spatially varying polarization distribution of these beams possesses information about the helical wavefront structures of the component OAM states, although they have plane wavefronts. The polarization singular beam (PSB) is focused using a tilted lens, and the intensity distribution at a predicted position in the direction of propagation is used to determine the component OAM content in the beam. Astigmatism introduced by the tilt of the lens modulates the vortex beam to introduce intensity nulls in the propagated beam. We demonstrate by simulations and experiments the index determination of the V points and C points using a tilted lens. This method is effective in the index determination of V points and C points formed by the superposition of component scalar vortices having opposite-sign topological charges. The degeneracy of C points with the same Stokes indices can be lifted through this technique.

Singular Lenses for Flexural Waves on Elastic Thin Curved Plates

Transformation optics, which is generically applicable to other classical waves such as acoustic and elastic waves, provides an emerging design paradigm to manipulate waves. However, some lenses and optical-transformation devices require a singular refractive index; meeting this requirement is a significant challenge. A method called transmutation can relax some types of index singularity into finite anisotropy around the singularity. Here, we show that such lenses with a singularity for flexural waves can be obtained by approaching a near-zero thickness of the plate precisely at the location of the singularity. As examples, we demonstrate a series of Eaton lenses theoretically and experimentally by projecting the refractive index in space onto the thickness in plates and by working in a broad frequency range in which impedance mismatch is negligible. This framework offers an insight into feasible methods that can be used to develop singular devices such as cloaking devices on thin flexible curved plates and can be further extended to a general methodology for shaping elastic waves. We hope that this elastic platform can also be a test bed to indirectly study unprecedented phenomena enabled by gravitational and quantum fields in terms of analog models.

Mineral prospectivity modelling using singularity mapping and multifractal analysis of stream sediment geochemical data from the auriferous Hutti-Maski schist belt, S. India

The regional stream sediment geochemical survey has become an all the more important exploration technique for delineating potential mineral zones to target exploration. However, the effective appraisal of pathfinder elements has remained a fundamental challenge in exploration geochemistry. In this regard, the singularity mapping technique based on modern nonlinear theory and fractal-multifractal geometry is a new efficient method to distinguish thresholds of anomalies based on stream sediment geochemical data. The auriferous Hutti-Maski schist belt in Eastern Dharwar Craton (EDC), S. India has been taken up for delineation future targets using local singularity analysis (LSA) of stream sediment geochemical data. Ordinary Singularity Mapping (OSM) and Spatially Weighted Singularity Mapping (SWSM) are employed for mapping singularity of gold pathfinder elements (Au, As, Ag, Hg, Sb, Se, Cu, and Zn). In SWSM, a distance-based fuzzy weighting method is used to estimate the spatial weighting factor with respect to ore controlling shear zones. The local isotropic singularity index and the local anisotropic singularity index are estimated for each of the eight elements. The Student’s t-statistics based on Weight of Evidence (WOE) model is used to determine the threshold singularity index value to separate geochemical anomaly from background. Principal Component Analysis (PCA) is applied to singularity maps of multiple elements derived using OSM and SWSM techniques. The first principal component (PC-1) scores estimated by the OSM-PCA and SWSM-PCA lead to comprehensive geochemical anomaly maps. Results of two different singularity mapping techniques are compared using Geochemical Mineralization Probability Index (GMPI), Concentration-Area (C-A) fractal analysis and Prediction-Area (P-A) plots. It is observed that the SWSM method results in a smaller target with higher predictive ability compared to OSM method that delineates a larger target area.

Detection of partially coherent polarization singular vector beams using Stokes polarimetry

We have demonstrated, theoretically and experimentally, a method based on Stokes polarimetry to detect the polarization singularity index (PSI) and the state-of-polarization (SOP) distribution of a partially coherent polarization singular vector beam. It is observed that on reducing the spatial correlation in polarization singular vector beams, the information of the phase vortex and polarization vortex is lost, yielding identical intensity profiles tending toward Gaussian distribution. However, the residual correlated field that comprises the polarized part of irradiance distribution still preserves these vortex structures. The PSI of a partially coherent vector beam can be readily obtained from the flower patterned image of the Stokes parameter's distribution. A uniform deterioration in the magnitude of Stokes intensity all across the beam cross section is observed as a signature of reduced two-point correlation. The respective Stokes phase maps are also invariant and can be used to determine the SOP distribution. The present study foresees potential application in free-space optical communication, optical trapping, and imaging, where the coherence properties of the polarization singular vector beams are of considerable importance.

Renormalized Energy Between Vortices in Some Ginzburg–Landau Models on 2-Dimensional Riemannian Manifolds

We study a variational Ginzburg–Landau type model depending on a small parameter \(\varepsilon >0\) for (tangent) vector fields on a 2-dimensional Riemannian manifold S. As \(\varepsilon \rightarrow 0\), these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus \({\mathfrak {g}}\) of S is different than 1. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorticity-dependent lattice in the \(2{\mathfrak {g}}\)-dimensional space of harmonic 1-forms on S if \({\mathfrak {g}}\geqq 1\). Our second main result determines the interaction energy (called renormalized energy) between vortex points as a \(\Gamma \)-limit (at the second order) as \(\varepsilon \rightarrow 0\). The renormalized energy governing the optimal location of vortices depends on the Gauss curvature of S as well as on the quantized flux. The coupling between flux quantization constraints and vorticity, and its impact on the renormalized energy, are new phenomena in the theory of Ginzburg–Landau type models. We also extend this study to two other (extrinsic) models for embedded hypersurfaces \(S\subset {{\mathbb {R}}}^3\), in particular, to a physical model for non-tangent maps to S coming from micromagnetics.

Research on spatial feature differentiation of basin river network geomorphology based on multifractal algorithm

Using multi-fractal algorithm to scientifically quantify watershed morphology is one of the hot spots of geomorphology research. In this paper, based on the DEM data of the Northwest China Basin (Daxiahe), the remote sensing images of the Daxia River Basin were processed. The multi-fractal model was used to extract and analyze the multi-fractal characteristics and quantified characteristic parameters of the Daxia River and 7 sub-basins of the river system, and discussed the geomorphic significance of the spatial characteristics of the watershed. The results showed that the multi-fractal spectrum of the watershed was in the shape of a right hook (Δf < 0). The span of the singular index Δa was between 0.860 and 1.160, and the value was relatively small. This indicated that the Daxia River watershed was evenly distributed in most areas, and the water system was not fully developed. The average values of the multi-fractal spectral height difference (Δf) and the singular index distribution range (Δa) from downstream to upstream were increasing (1.184∼1.570) and decreasing (1.160∼0.933) in sequence, which indicated that the proportion of the geomorphic unit with the smallest slope in the watershed increased sequentially from downstream to upstream, and the change of geomorphic shape from upstream to downstream tended to be complicated. This study is of great significance to effectively solve the complex problems in geomorphology and to master the characteristics and laws of various geomorphological phenomena.

A robustness measure for singular point and index estimation in discretized orientation and vector fields

AbstractThe identification of singular points or topological defects in discretized vector fields occurs in diverse areas ranging from the polarization of the cosmic microwave background to liquid crystals to fingerprint recognition and bio‐medical imaging. Due to their discrete nature, defects and their topological charge cannot depend continuously on each single vector, but they discontinuously change as soon as a vector changes by more than a threshold. Considering this threshold of admissible change at the level of vectors, we develop a robustness measure for discrete defect estimators. Here, we compare different template paths for defect estimation in discretized vector or orientation fields. Sampling prototypical vector field patterns around defects shows that the robustness increases with the length of template path, but less so in the presence of noise on the vectors. We therefore find an optimal trade‐off between resolution and robustness against noise for relatively small templates, except for the “single pixel” defect analysis, which cannot exclude zero robustness. The presented robustness measure paves the way for uncertainty quantification of defects in discretized vector fields.

Comparison of concentration-area (C-A) fractal method and singularity index in separation of geochemical anomalies of Cu element in Malayer-Aligudarz-Esfahan metallogenic zone

کمربند فلززایی ملایر-الیگودرز-اصفهان با طولی افزون بر 400 کیلومتر و عرض 90 کیلومتر بزرگترین کمربند فلززایی سرب و روی در ایران است. با توجه به اینکه یکی از مهم­ترین فلزات همراه با زون­های سرب و روی، مس می­باشد، در این پژوهش، به منظور جدایش جوامع آنومالی ژئوشیمیایی از زمینه این فلز از روش­های فرکتال عیار- مساحت (C-A) و شاخص سینگولاریتی (SI) استفاده شده است. لذا تجزیه و تحلیل 19974 نمونه از رسوبات آبراهه­ای با استفاده از روش­های نام برده در زون فلززایی ملایر- الیگودرز- اصفهان، منجر به تفکیک مناطق آنومال عنصر مس گردید. در نقشه­های حاصل از مدل فرکتال عیار- مساحت (C-A)، قسمت­های شمال­غرب و جنوب­شرق زون بیشترین میزان آنومالی را از خود نشان می­دهد. در نقشه­های حاصل از روش شاخص سینگولاریتی (SI)، آنومالی­های پنهان بهتر نمایش داده شده و انطباق خوبی بین آنومالی­ها و موقعیت فعلی معادن مس در زون مورد نظر وجود دارد. با تطبیق آنومالی­های به دست آمده از هر دو روش با نقشه زمین­شناسی محدوده مورد نظر، مشخص گردید که آنومالی­های به دست آمده هم­پوشانی بالایی با واحد آهک کرتاسه در منطقه نشان می­دهد و لذا این واحد می­تواند راهنمای اکتشافی مناسبی برای پی­جویی عناصری نظیر سرب، روی و مس در این منطقه محسوب شود.