Singularity Analysis Method Research Articles

Unsupervised Machine Learning-Based Singularity Models: A Case Study of the Taiwan Strait Basin

The identification of geochemical anomalies in oil and gas indicators is a fundamental task in oil and gas exploration, as the process of oil and gas accumulation is a low probability event. Machine learning algorithms for anomaly detection are applicable to the identification of oil and gas geochemical anomalies related to oil and gas accumulation. However, when using oil and gas indicators for anomaly detection, the diversity of these indicators often leads to the influence of the indicator redundancy on the identification of such features. Therefore, it is particularly important to select appropriate oil and gas indicators for anomaly detection. In this study, a hybrid model combining unsupervised machine learning methods and singularity analysis methods was used to evaluate oil and gas indicator anomalies using geochemical data from the Taiwan Strait Basin. The models used in this study include the singularity index model (LSP), the principal component model combined with the singularity index model (PCA and LSP), and the cluster analysis combined with the principal component model and the singularity index model (CLA-PCA-LSP). PCA models can reduce the dimensions of the data and retain as much information as possible. CLA divides data samples into different groups, so that samples within the same group are more similar and samples between different groups are less similar. LSP is mainly used for measuring the setting and singular degree of local anomalies in multi-scale geochemistry, geophysics, and other types of local anomalies, and it has a unique advantage in extracting low and weak anomalies and nonlinear characteristics. The results of the study show that the results obtained using the CLA-PCA-LSP hybrid model are very similar to those obtained by performing PCA on the entire index and then calculating the singularity index. This also verifies that, for the study areas of the Jiulongjiang Depression and Jinjiang Depression, we can select oil and gas indicators that are favorable for exploration analysis, without including all indicators in the analysis scope, thereby improving the computational efficiency. The application of a singularity analysis method and generalized self-similarity principle in extracting the geochemical information of oil and gas indicators in the Taiwan Strait Basin highlights key technologies such as the identification of weak anomalies, decomposition of composite anomalies, and integration of spatial information. The combination anomalies delineated by the singularity analysis method and S-A method not only reflect the spatial relationship with known oil and gas reservoir distribution, but also show the multiple combination anomalies in unknown areas, providing favorable guidance for the next exploration direction in the Taiwan Strait Basin.

Singularity analysis for V‐notches in a piezoelectric multimaterial and functionally graded piezoelectric materials

AbstractThis study develops a novel singularity analysis method that addresses the limitations posed by piezoelectric material quantity and the variation patterns of property functions in functionally graded piezoelectric materials (FGPMs). Given that piezoelectric composite materials consist of multiple materials, the material properties of FGPMs vary with respect to angle. By introducing the asymptotic assumption that governs the physical fields close to the notch vertex into the static equilibrium equations, a comprehensive set of characteristic equations is formulated. All singularity orders and corresponding characteristic angle functions of the notch are determined by numerically solving the established characteristic equations. The effects of the notch opening angle, piezoelectric material polarization direction, and boundary conditions on the electromechanical field singularity at the notch are assessed. The judicious selection of material variation patterns can alleviate notch singularity in FGPMs.

Unveiling Temporal Cyclicities in Seismic b-Values and Major Earthquake Events in Japan by Local Singularity Analysis and Wavelet Methods

Studying the temporal characteristics of earthquake activity contributes to enhancing earthquake prediction capabilities. The seismic b-value is a key indicator describing the relationship between seismic frequency and magnitude. This study investigates the correlation between the occurrence of major earthquakes and seismic b-values using earthquake activity records in Japan from 1990 to 2023. Local singularity analysis and wavelet analysis of earthquake frequency and b-value time series reveal significant 5-year periodic features in seismic activity in Japan. Furthermore, our research identifies that this periodicity is also prominent in major earthquakes with magnitudes of 7 and above. Additionally, through a detailed analysis of the cross-correlation between seismic b-values and the occurrence time of major earthquakes, we uncover a notable pattern: major earthquakes often occur approximately two years after the peak of seismic b-values. This discovery offers a new perspective on earthquake prediction and may play a crucial role in future earthquake early warning systems.

Spatial sensitivity distribution assessment and Monte Carlo simulations for needle-based bioimpedance imaging during venipuncture using the finite element method.

Despite being among the most common medical procedures, needle insertions suffer from a high error rate. Impedance measurements using electrode-equipped needles offer promise for improved tissue targeting and reduced errors. Impedance visualization usually requires an extensive pre-measured impedance dataset for tissue differentiation and knowledge of the electric fields contributing to the resulting impedances. This work presents two finite element simulation approaches for both problems. The first approach describes the generation of a multitude of impedances with Monte Carlo simulations for both, homogeneous and inhomogeneous tissue to circumvent the need to rely on previously measured data. These datasets could be used for tissue discrimination. The second method describes the simulation of the spatial sensitivity distribution of an electrode layout. Two singularity analysis methods were employed to determine the bulk of the sensitivity within a finite volume, which in turn enables consistent 3D visualization. The modeled electrode layout consists of 12 electrodes radially placed around a hypodermic needle. Electrical excitation was simulated using two neighboring electrodes for current carriage and voltage pickup, which resulted in 12 distinct bipolar excitation states. Both, the impedance simulations and the respective singularity analysis methods were compared with each other. The results show that the statistical spread of impedances is highly dependent on the tissue type and its inhomogeneities. The bounded bulk of sensitivities of both methods are of similar extent and symmetry. Future models should incorporate more detailed tissue properties such as anisotropy or changing material properties due to tissue deformation to gain more accurate predictions.

Compact Design and Task Space Control of a Robotic Transcatheter Delivery System for Mitral Valve Implant.

Mitral regurgitation (MR) is one of the most common valvular abnormalities, and the gold-standard for treatment is surgical mitral valve repair/replacement. Most patients with severe MR are over the age of 75, which makes open-heart surgery challenging. Thus, minimally invasive surgeries using transcatheter approaches are gaining popularity. This paper proposes the next generation of a robotic transcatheter delivery system for the mitral valve implant that focuses on the design of the actuation system, modeling, and task space control. The proposed actuation system is compact while still enabling bidirectional torsion, bending, and prismatic joint motion. A pulley structure is employed to actuate the torsion and bending joints using only one motor per joint in conjunction with an antagonistic passive spring to reduce tendon slack. The robotic transcatheter is also optimized to increase its stability and reduce bending deflection. An inverse kinematics model (with an optimization algorithm), singularity analysis method, and joint hysteresis and compensation model are developed and verified. Finally, a task space controller is also proposed. Experiments, including trajectory tracking and demonstrations of the robot motion in an ex vivo porcine heart and a phantom heart through a tortuous path are presented.

Computing error bounds for asymptotic expansions of regular P-recursive sequences

Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko [SIAM J. Discrete Math. 3 (1990), pp. 216–240], except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes).

Singularity analysis and modular synthesis of over-constrained spatial mechanisms - A case study on Bennett and RPRP linkage

This paper revisits the geometric constraints for a movable spatial RPRP mechanism and further finds that it can be mobile as long as the link lengths and skew angles satisfy specific relationships, regardless of plane-symmetric property. Next, a set of kinematics analysis methods using D-H notations to analyze the modular mechanism are derived. In order to efficiently implement the concatenated over-constrained mechanism, this paper summarizes two singularity analysis methods based on Jacobian matrix form, square matrix or rectangular matrix, obtained by the velocity analysis to find the limited motion range of the mechanism. Then, this paper proposes a modular method to synthesize a family of spatial six-bar over-constrained mechanisms. In addition, by adjusting the geometric relationships between link lengths and skew angles, spatial six-bar over-constrained mechanisms can be further configured to a spatial five-bar or four-bar mechanisms. The modular method can effectively isolate a new concatenated six-bar mechanism synthesized from two different modular mechanisms, Bennett and spatial RPRP mechanisms, by removing overlapped links and joints. Furthermore, it is found that the concatenated mechanism retains the kinematics and singularity characteristics of the source mechanisms.

Symmetries and solutions for the inviscid oceanic Rossby wave equation

The -and -dimensional inviscid Rossby wave equations are analysed using Lie symmetry techniques. The travelling-wave reductions for the equation lead to a third-order ordinary differential equation from which the propagation properties are derived. It is observed that the wave has easterly phase velocity and westerly group velocity. Also, the wave propagates slightly faster in the f-plane than the β-plane. The dispersion relation derived from the third-order equation shows that the Rossby equation transports energy both eastward and westward and its speed is reduced successively and the propagation remains eastward. As for the two-dimensional Rossby wave equation, certain solutions which behave like solitary waves after a certain time are plotted. Interestingly, for certain symmetries the reductions lead to the Riccati's, Abel's, Euler's type and to some linearized equations. Moreover, certain reductions lead to highly nonlinear equations which are analysed by the singularity analysis method.

Application of Singularity Theory to the Distribution of Heavy Metals in Surface Sediments of the Zhongsha Islands

This research aimed to use nonlinear theory and technology to describe the spatial distribution of heavy metals in the surface sediments of the Zhongsha Islands Sea region. The goal of this study is to explore the spatial distribution characteristics of heavy metals in the surface sediments of the Zhongsha Islands. The singularity theory and method were used to delineate heavy metal geochemical anomalies and the generalized self-similarity analysis method was used to decompose heavy metal geochemical anomalies and background concentrations. The results showed that there were abnormally high concentrations of heavy metals in the deep-sea plain area and in the western central sand trough area. The results of this study can inform priority areas for environmental monitoring. The element anomalies extracted by the singularity analysis method in this paper can be a guide to the next step in the investigation and provide the basis for the regional environmental assessment.

On finite time singularities in scalar field dark energy models based in the RS-II Braneworld

The quest of deciphering the true nature of dark energy has proven to be one of the most exciting in recent times in cosmology. Various ideas have been put forward in this regard besides the usual cosmological constant approach, ranging from scalar field based models like Quintessence and Phantom dark energy to various modified gravity approaches as well. A very interesting idea then is to consider scalar field dark energy models in quantum gravitationally corrected cosmologies with the RS-II Braneworld being one of the most well known in this regard. So in this work, we consider RS-II Braneworld based scalar field dark energy models and try to look out for the existence of finite time singularities in these regimes both through a dynamical system perspective, for which we employ the Goriely–Hyde singularity analysis method, and a physical perspective. Our approach is general in the sense that it is not limited to any particular class of potentials or for any constrained parameter region for the brane tension and is valid for both Quintessence and phantom dark energy regimes. We firstly show through Goriely–Hyde procedure that finite time singularities can exist in these models for a limited set of initial conditions and that this result would hold irrespective of any consideration given to the swampland dS conjecture. We then discuss the physical nature of the singularities that can occur in this regime, where we use a well motivated ansatz for the Hubble parameter and show that these models of dark energy can allow for weak singularities like those of Type III and Type IV and can also allow for strong singularities like the Big Rip (Type I).

One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation

Abstract We apply the Lie theory to determine the infinitesimal generators of the one-parameter point transformations which leave invariant the 3 + 1 Kudryashov–Sinelshchikov equation. We solve the classification problem of the one-dimensional optimal system, while we derive all the possible independent Lie invariants; that is, we determine all the independent similarity transformations which lead to different reductions. For an application, the results are applied to prove the existence of travel-wave solutions. Furthermore, the method of singularity analysis is applied where we show that the 3 + 1 Kudryashov–Sinelshchikov equation possess the Painlevé property and its solution can be written by using a Laurent expansion.

Kinematics and Singularity Analysis of a 7-DOF Redundant Manipulator.

A new method of kinematic analysis and singularity analysis is proposed for a 7-DOF redundant manipulator with three consecutive parallel axes. First, the redundancy angle is described according to the self-motion characteristics of the manipulator, the position and orientation of the end-effector are separated, and the inverse kinematics of this manipulator is analyzed by geometric methods with the redundancy angle as a constraint. Then, the Jacobian matrix is established to derive the conditions for the kinematic singularities of the robotic arm by using the primitive matrix method and the block matrix method. Then, the kinematic singularities conditions in the joint space are mapped to the Cartesian space, and the singular configuration is described using the end poses and redundancy angles of the robotic arm, and a singularity avoidance method based on the redundancy angles and end pose is proposed. Finally, the correctness and feasibility of the inverse kinematics algorithm and the singularity avoidance method are verified by simulation examples.

Singularity analysis of igneous zircon U-Pb age and Hf isotopic record in the Zhongdian arc, northwest Yunnan, China: Implications for Indosinian magmatic flare-up and the formation of porphyry copper deposits

Situated in the southern part of Yidun arc in the southeastern Tibetan Plateau, the Zhongdian arc records magmatic activity that ranges from approximately 270 to 190 Ma, which is likely attributable to subduction of the Garze-Litang slab and involved steady low-volume magmatic lulls punctuated by short-lived high-volume flare-ups. Many porphyry copper deposits have been found in these magmatic rocks, but the temporal-spatial relationships among magmatism, porphyry copper deposits, and subduction processes around the Zhongdian arc have long been debated. Frequency distributions calculated from igneous zircon U-Pb ages are commonly utilized to interpret the duration of magmatic events. Recently, a new concept of fractal density and a local singularity analysis (LSA) method have been applied to analyze the geometric properties of zircon U-Pb age peak anomalies, which are linked to deeply rooted avalanches associated with short spurts of convection during the formation of supercontinents and continental crust growth. In this paper, the igneous zircon U-Pb dating data collected from the Zhongdian arc are analyzed by the LSA method from the perspective of fractal density to interpret the causational relationship between age peak and subduction process of the Garze-Litang oceanic crust. The results show that the age density around the age peak of 215 Ma can be well fitted and described by power law functions, and the strong singularity (k > 0.8) indicates the occurrence of extreme geological events. Meanwhile, systematic changes in the hafnium isotopic composition of igneous zircons with time reflect the upwelling of asthenospheric mantle materials, characterized by a rapid increase in εHf(t) values that reach the highest around the 215 Ma peak. Moreover, the log–log plot that shows the cumulative number-age distribution reveals another transitional age of 235 Ma. By comparison of the magmatic intensity between the western and eastern belts of the Zhongdian arc, it is indicated that the eastward magmatic migration might have begun at ca. 235 Ma. Combined with the previous work, we propose that the Indosinian magmatic flare-up in the Zhongdian arc might have been caused by superimposition of slab subduction, steepening of slab dip, and slab breakoff. This long-term magmatic hydrothermal evolution system and extreme singularity events also provide indispensable conditions for the formation of large-scale and high-grade porphyry copper deposits in the region. Our findings provide new insights into the subduction process of the Garze-Litang oceanic crust and Indosinian magmatic framework around the Zhongdian arc. It is believed that the LSA method has potential applications in depicting the variation of magmatic activity and predicting mineral resources.

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.

Cosmological solutions and growth index of matter perturbations in f(Q) gravity

The present work studies one of Einstein's alternative formulations based on the non-metricity scalar $Q$ generalized as $f(Q)$ theory. More specifically, we consider the power-law form of $f(Q)$ gravity i.e. $f(Q)=Q+\alpha\, Q^n$. Here, we analyze the behavior of the cosmological model at the background and perturbation level. At the background level, we find the effective evolution of the model is the same as that of the $\Lambda$CDM for $|n|<1$. Interestingly, the geometric component of the theory solely determined the late-time acceleration of the Universe. We also examine the integrability of the model by employing the method of singularity analysis. In particular, we find the conditions under which field equations pass the Painlev\'{e} test and hence possess the Painlev\'{e} property. While the equations pass the Painlev\'{e} test in the presence of dust for any value of $n$, the test is valid after the addition of radiation fluid only for $n<1$. Finally, at the perturbation level, the behavior of matter growth index signifies a deviation of the model from the $\Lambda$CDM even for $|n|<1$.

Mobility and singularity analyses of a symmetric multi-loop mechanism for space applications

A symmetric, double-tripod multi-loop mechanism (DTMLM), for aerospace applications, is the subject of this paper. Its mobility and singularity are analyzed, while introducing a novel tool, the cell-division method for singularity analysis, applicable to multi-loop mechanisms. The key principle of this method lies in replacing the singularity analysis of the original multi-loop mechanism with: (1) that of an equivalent simpler parallel mechanism; (2) the constraint analysis between loops; and (3) the singularity analysis of simpler kinematic subchains. Then, the mechanism is transformed into a simpler, equivalent parallel mechanism with three identical kinematic subchains. Its mobility and singularity are analyzed based on screw algebra, which leads to a key conclusion about the geometric properties of this mechanism. Results show that: (a) the DTMLM has three degrees of freedom (dof), i.e., two rotational dof around two intersecting axes lying in the middle plane of the mechanism, and one translational dof along the normal to the said plane (2R1T); and (b) the singularities of the 3-RSR parallel mechanism are avoided in the DTMLM by means of prismatic joints, singularities in the DTMLM occurring on the boundary of its workspace. Thus, the DTMLM has a 2R1T mobility everywhere within its workspace. When a set of multi-loop mechanisms of this kind are stacked as modules to assemble a multi-stage manipulator for space applications, the modules can be designed so that, under paradigm operations, all individual loops operate within their workspace, safe from singularities.

Joint string complexity for Markov sources: Small data matters

String complexity is defined as the cardinality of a set of all distinct words (factors) of a given string. For two strings, we introduce the joint string complexity as the cardinality of a set of words that are common to both strings. String complexity finds a number of applications from capturing the richness of a language to finding similarities between two genome sequences. In this paper we analyze the joint string complexity when both strings are generated by Markov sources. We prove that the joint string complexity grows linearly (in terms of the string lengths) when both sources are statistically indistinguishable and sublinearly when sources are statistically distinguishable. Precise analysis of the joint string complexity requires subtle singularity analysis and saddle point method over infinity many saddle points leading to novel oscillatory phenomena with single and double periodicities. To overcome these challenges, we apply analytic techniques such as multivariate generating functions, multivariate depoissonization and Mellin transform, spectral matrix analysis, and complex asymptotic methods.

Random permutations with logarithmic cycle weights

We consider random permutations on Sn with logarithmic growing cycles weights and study the asymptotic behavior as the length n tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables and also compute the total variation distance between both processes. Next, we prove a central limit theorem for the total number of cycles. Furthermore we establish a shape theorem and a functional central limit theorem for the Young diagrams associated to random permutations under this measure. We prove these results using tools from complex analysis and combinatorics. In particular we have to apply the method of singularity analysis to generating functions of the form exp((− log(1 − z))k+1) with k ≥ 1, which have not yet been studied in the literature.

A novel class of reconfigurable parallel kinematic manipulators: Concepts and Fourier-based singularity analysis

In-situ interventions in complex environments require new concepts of Parallel Kinematic Manipulators (PKMs) that present higher versatility for adapting to unstructured working environments without affecting their advantageous characteristics. To address this opportunity a novel class of adaptive PKMs is proposed, the upper joints are repositionable on the moving platform and the lower joints are free from a base. These characteristics give to the proposed PKMs the capability to modify its workspace and performance (i.e. stiffness and singularity avoidance) to work on non-conventional manufacturing/repair environments. Further, the kinematic and workspace models of the proposed PKMs are introduced. More importantly, a unique method for singularity analysis is introduced, as the versatility to modify the workspace comes with the disadvantage of singularities that are not constant. To address this issue a strategy based on the Fourier transform is introduced, the dominant frequencies are identified and configurations with lower dominant frequencies are weighted to stabilise the Jacobian of the parallel mechanism. Finally, a set of validations are presented for proving the proposed method of singularity analysis by FEA with an error smaller than 1%.

A robust geometric method of singularity avoidance for kinematically redundant planar parallel robot manipulators

Jacobian-based methods of singularity analysis are known to be unreliable when applied to kinematically redundant parallel robot manipulators, due to their potential to miss certain singularities and incorrectly identify others in the manipulator’s workspace. In this paper, a geometric method of singularity avoidance for kinematically redundant planar parallel robot manipulators is presented, which firstly determines the manipulator’s proximity to a singularity and then computes how the kinematically redundant degree(s) of freedom should be optimised for the given pose of the end-effector. The singularity analysis is conducted by examining the mechanism in terms of the instantaneous centres of rotation of its corresponding mobility one sub-mechanisms when all but one of the actuators are locked, where the manipulator is in a type-II singularity when these points either are indeterminable or coincide with one another, and an index, rmin, is introduced which describes the minimum normalised distance from such conditions being met. A predictor-corrector method is employed to compute the configuration for which rmin is optimised, and is reachable without crossing a singularity. Finally, the advantages of the geometric method of singularity analysis are shown in comparison to traditional Jacobian-based methods when applied to kinematically redundant parallel robot manipulators.