Singularity Analysis Research Articles

Fractal Derivatives and Singularity Analysis of Frequency—Depth Clusters of Earthquakes along Converging Plate Boundaries

Fractional calculus (FC) has recently received increasing attention due to its applications in many fields involving complex and nonlinear systems. However, one of the key challenges in using FC to deal with fractal or multifractal phenomena is how to relate functions to geometries with fractal dimensions. The current paper demonstrates how fractal calculus can be used to represent physical properties such as density defined on fractal geometries that no longer have the Lebesgue additive properties required for ordinary calculus. First, it introduces the recently proposed concept of fractal density, that is, densities defined on fractals and multifractals, and then shows how fractal calculus can be used to describe fractal densities. Finally, the singularity analysis based on fractal density calculation is used to analyze the depth clustering distribution of seismic frequencies around the Moho transition zone in the subduction zone of the Pacific plates and the Tethys collision zones. The results show that three solutions (linear, log-linear, and double log-linear) of a unified differential equation can describe the decay rate of the fractal density of depth clusters at the number (frequencies) of earthquakes. The spatial distribution of the three groups of earthquakes is further divided according to the three attenuation relationships. From north latitude to south latitude, from the North Pacific subduction zone to the Tethys collision zone, and then to the South Pacific subduction zone, the attenuation relationships of the earthquake depth distribution are generally from a linear, to log-linear, to double log-linear pattern. This provides insight into the nonlinearity of the depth distribution of earthquake swarms.

On an injectivity theorem for log-canonical pairs with analytic adjoint ideal sheaves

As an application of the residue functions corresponding to the lc-measures developed by the authors, the proof of the injectivity theorem on compact Kähler manifolds for plt pairs by Matsumura is improved in this article to allow multiplier ideal sheaves of plurisubharmonic functions with neat analytic singularities in the coefficients of the relevant cohomology groups. With the use of a refined version of analytic adjoint ideal sheaves, a plan towards a solution to the generalised version of Fujino’s conjecture (i.e. an injectivity theorem on compact Kähler manifolds for lc pairs with multiplier ideal sheaves) is laid down and, in addition to the result for plt pairs, a proof for lc pairs in dimension 2 2 , which is also an improvement of Matsumura’s result, is given.

Organization scale of photospheric magnetic imbalance in coronal holes

To investigate the statistical properties of the photospheric magnetic fields underlying coronal holes (CHs) and “normal” coronal regions a classical technique, the signed measure, is used. This technique allows to characterize the scaling behavior and the topology of sign-oscillating magnetic structures in selected regions of line of sigth (LoS) magnetograms recorded by the Heliosismic Magnetic Imager on board of the Solar Dynamic Observatory (SDO/HMI). To this end we have compared the properties of the photospheric magnetic field underlying 60 CHs and 60 non-coronal holes (NCHs). In particular, in addition to having studied distributions and momenta of photospheric magnetic fields associated to the selected regions, we have performed the sign singularity analysis computing the cancellation functions of the highly fluctuating photospheric magnetic fields. We have found that photospheric magnetic fields associated to CHs are imbalanced in the sign and that this imbalance emerges mainly at the supergranular scales.

Determination of singular and higher order non-singular stress for angularly heterogeneous material notch

The complete stress distribution around the tip of angularly heterogeneous material V-notch includes singular stress term and higher order non-singular stress term. The latter has important role on the prediction of the fracture, while it is usually neglected due to the evaluation difficulty. The idea of combing the singularity analysis near the notch vertex with the finite element analysis is established to deal with this difficulty. Firstly, a set of variable coefficient differential equations for the singular analysis of angularly heterogeneous material notch are built. The interpolating matrix method is introduced to solve the singular characteristic equations to prepare the singularity order and characteristic angular function. The V-notched structure is then analyzed by the coarse finite element meshes. The displacements from the finite element analysis are substituted into the singularity asymptotic expression to form a set of over-determined equation. The least square method is applied to yield the amplitude coefficients in the series asymptotic expansion. The complete stress field near the angularly heterogeneous material V-notch tip is then constructed according to the obtained amplitude coefficients, singularity orders, characteristic angular functions, together with their first order derivatives. The singular and non-singular stress term can be respectively calculated according to the truncated series terms. The reconstructed stresses have the same precision of the displacement from the view point of the finite element analysis. The role of the non-singular stress on the stress intensity factors of angularly heterogeneous material notch is then investigated.

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).

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.

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.

Singularity analysis of spatial single-DOF mechanisms based on the locations of the instantaneous screw axes

The instantaneous kinematics of a mechanism can be fully described through the instantaneous screw axes (ISAs) associated to the relative motions between mechanism's links. Characterizing the instantaneous kinematics of a mechanism includes its singularity analysis, which is the identification of mechanism configurations (singularities) where the instantaneous degrees-of-freedom (DOFs) of the mechanism “somehow” change. In single-DOF mechanisms with time-independent either holonomic or first-order non-holonomic constraints, the locations of the instantaneous screw axes only depend on the mechanism configuration. This property is fully exploited in planar single-DOF mechanisms where all the singularity conditions have been associated to the coincidence of particular instant centers (ICs), thus making the singularity identification possible with a simple analysis of the mechanism configuration. Differently, for single-DOF spatial mechanisms, the geometric conditions on ISAs that identify a singular configuration have not been enunciated for the general case, yet. This paper fills this lack by providing the explicit expressions of all the possible instantaneous input-output relationships, the exhaustive enumeration of the geometric conditions on the ISAs that identify singular configurations and a geometric and analytic technique for the singularity analysis of spatial single-DOF mechanisms, which is the natural extension of the IC-based ones conceived for planar single-DOF mechanisms.

Design of 4-DOF hybrid parallel robots with an integrated three-fingered robot end effector

Grippers are essential components of robotic automation, allowing general purpose robots to perform specialized tasks. This paper aims to design a novel family of 4-DOF hybrid parallel robots, which has an integrated three-fingered robot end effector. As the proposed hybrid parallel robot features a parallel structure, it holds the potential to offer advantages synonymous with parallel mechanisms. The three-fingered robot end effector can perform stable grasping of objects with different sizes and shapes. First, the condition for stable grasping is derived and the synthesis method is proposed. Second, a family of novel 4-DOF hybrid parallel robots is designed through integrating a newly invented three-fingered robot end effector. Finally, a comparative study including the workspace analysis, singularity analysis and performance evaluation of an example hybrid parallel robot with two different topologies is carried out. Then the hybrid parallel robot with a better performance can be selected. The results of performance analysis demonstrate the correctness and rationality of the synthesis method and the high application potential of the obtained hybrid parallel robots.

The classical and quantum implications of the Raychaudhuri equation in f(T)-gravity

The present work deals with the classical and quantum aspects of the Raychaudhuri equation (RE) in the framework of f(T)-gravity theory. In the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker space-time, the RE has been formulated and used to examine the focusing theorem and convergence condition for different choices of f(T). Finally in quantum cosmology, the wave function of the Universe has been shown to be the energy eigen function of the time-independent Schrödinger equation of a particle. Also probability measure on the minisuperspace has been examined at zero volume for singularity analysis in the quantum regime. Lastly, the Bohmian trajectory for the present quantum system has been explicitly determined for some particular choices.

Raychaudhuri equation from Lagrangian and Hamiltonian formulation: A quantum aspect

The paper deals with a suitable transformation related to the metric scalar of the hyper-surface so that the Raychaudhuri Equation (RE) can be written as a second order nonlinear differential equation. A first integral of this second order differential equation gives a possible analytic solution of the RE. Also, it is shown that construction of a Lagrangian (and hence a Hamiltonian) is possible, from which the RE can be derived. Wheeler-Dewitt equation has been formulated in canonical quantization scheme and norm of its solution (wave function of the universe) is shown to affect the singularity analysis in the quantum regime for any spatially homogeneous and isotropic cosmology. Finally Bohmian trajectories are formulated with causal interpretation and these quantum trajectories unlike classical geodesics obliterate the initial big-bang singularity when the quantum potential is included.

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.

Mixed-Mode Oscillations in Coupled FitzHugh–Nagumo Oscillators: Blow-Up Analysis of Cusped Singularities

Mixed-Mode Oscillations in Coupled FitzHugh–Nagumo Oscillators: Blow-Up Analysis of Cusped Singularities

Performance analysis of single-focus phase singularity based on elliptical reflective annulus quadrangle-element coded spiral zone plates

Optical vortices generated by the conventional vortex lens are usually disturbed by the undesired higher-order foci, which may lead to additional artifacts and thus degrade the contrast sensitivity. In this work, we propose an efficient methodology to combine the merit of elliptical reflective zone plates (ERZPs) and the advantage of spiral zone plates (SZPs) in establishing a specific single optical element, termed elliptical reflective annulus quadrangle-element coded spiral zone plates (ERAQSZPs) to generate single-focus phase singularity. Differing from the abrupt reflectance of the ERZPs, a series of randomly distributed nanometer apertures are adopted to realize the sinusoidal reflectance. Typically, according to our physical design, the ERAQSZPs are fabricated on a bulk substrate; therefore, the new idea can significantly reduce the difficulty in the fabrication process. Based on the Kirchhoff diffraction theory and convolution theorem, the focusing performance of ERAQSZPs is calculated. The results reveal that apart from the capability of generating optical vortices, ERAQSZPs can also integrate the function of focusing, energy selection, higher-order foci elimination, as well as high spectral resolution together. In addition, the focusing properties can be further improved by appropriately adjusting the parameters, such as zone number and the size of the consisted primitives. These findings are expected to direct a new direction toward improving the performance of optical capture, x-ray fluorescence spectra, and forbidden transition.

Natural frequency prediction of the 3-RPS parallel manipulator using the substructure synthesis technology

AbstractThis paper proposed an elastodynamic modeling method combined with independent displacement coordinates and substructure synthesis technology. Firstly, the connecting rod was discretized, and the elastodynamic control equation for each element was established. The multipoint constraint element theory, linear algebra, and singularity analysis were used to identify the globally independent displacement coordinates of the manipulator. On this basis, the elastodynamic model using the substructure synthesis for the 3-PRS parallel manipulator (PM) was developed, with its natural frequencies distribution in the regular workspace discussed. The comparison with the finite-element results showed that the maximum error of the first three-order natural frequencies was within 1.03%, which verified the correctness of the analytical model. The proposed elastodynamic model included all the kinematic constraints of the manipulator without increasing the Lagrangian multiplier. The method is computationally efficient and assesses the dynamic behaviors of the mechanism at the predesign phase.

Generalised Ordinary vs Fully Simple Duality for n-Point Functions and a Proof of the Borot–Garcia-Failde Conjecture

We study a duality for the n-point functions in VEV formalism that we call the ordinary vs fully simple duality. It provides an ultimate generalisation and a proper context for the duality between maps and fully simple maps observed by Borot and Garcia-Failde. Our approach allows to transfer the algebraicity properties between the systems of n-point functions related by this duality, and gives direct tools for the analysis of singularities. As an application, we give a proof of a recent conjecture of Borot and Garcia-Failde on topological recursion for fully simple maps.

A 2R1T Redundantly Actuated Parallel Manipulator With an Offset Moving Platform and Fixed Linear Actuators

Abstract Redundantly actuated parallel manipulators with two rotations and one translation (2R1T RAPMs) have the potential for machining complex surfaces, where a large orientation workspace and high stiffness are required. Considering the advantages of an offset moving platform, such as an enlarged orientation workspace and improved stiffness, a novel 2R1T (2PRR)R-PRS-PSS RAPM with an offset moving platform is proposed in this paper, called M2. Compared with the existing 2R1T RAPM with an offset moving platform, the main advantage of the proposed RAPM is that the heavy motors of four limbs are mounted on the base to reduce the movable mass and improve dynamic response. The kinematical analysis is investigated, including mobility, inverse, forward kinematics, and singularity analysis. Meanwhile, comprehensive evaluations of the properties of the offset moving platform and actuation redundancy are carried out. Compared with M2 RAPM’s form without an offset in moving platform, i.e., no auxiliary platform, and M2 RAPM’s nonredundantly actuated form, the proposed M2 RAPM can achieve a larger orientation workspace and higher stiffness. Particularly, the maximum stiffness of the proposed M2 is 68.8% larger than its form without an auxiliary platform. Finally, the dimensional parameters of the proposed M2 are optimized to obtain an improved satisfactory workspace.

Singularity analysis and workspace optimization of a novel symmetrical 4-limb 5-DOF hybrid robot

Singularity analysis and workspace optimization of a novel symmetrical 4-limb 5-DOF hybrid robot

PO-01-205 COMPUTATIONAL MODELING UNCOVERS A UNIFYING MECHANISM LINKING CONDUCTION BLOCK TO VT LOCALIZATION AND STABILITY IN ARRHYTHMOGENIC CARDIOMYOPATHY

We recently showed in a murine model of arrhythmogenic cardiomyopathy (ACM) that VT was driven by discrete rotors deep within the RV. Mechanisms underlying the evolution of these RV-pinned rotors, however, remained unknown considering that the initial wavebreaks that preceded VT were at the LV/RV interface. We hypothesized that the unique spatial action potential duration (APD) profile (RV>LV) in ACM during rapid pacing and adrenergic stimulation, independent of other factors such as structural remodeling, is sufficient to explain the rapid evolution of wavebreaks into discrete rotors that efficiently translocate and stabilize in the RV to maintain VT. We developed 2D virtual sheets of mouse ventricular myocardium using isotropic cellular automaton models with realistic restitution properties that mimic experimentally measured LV to RV APD gradients in wildtype (WT) and ACM hearts (Fig A). VT rotors were induced by burst stimulus trains at increasing frequencies. VT threshold was defined as the duration of the burst stimulus train required to initiate stable VT. Rotors were localized in space and time using phase singularity (PS) analysis. To determine the relationship between spatial APD profile and rotor dynamics, we varied the steepness of the LV-to-RV APD gradient. Once induced, we examined rotor dynamics, speed of rotor drift, meander, stability and dominant frequency (DF). VT threshold was markedly decreased in ACM (2.9s) vs WT (14.5s) reflecting increased vulnerability. VT rotors were more readily induced when stimuli were applied from the zone of short versus long APD. While VT rotors were randomly distributed across homogeneous tissues, PS density analysis revealed exclusive localization of VT rotors within areas of increased APD in heterogeneous tissues (Fig B). Varying the slope of the APD gradient revealed divergent effects on forward drift and DF of VT rotors (Fig C). As the slope of the APD gradient steepened, the velocity of rotor forward drift to the RV increased exponentially while rotor DF decreased in a linear manner consistent with increased stability. Simulation of experimentally measured LV-to-RV APD gradient in ACM caused forward drift of multiple rotors at 2.18cm/sec before pinning within narrow RV loci. The unique spatial APD profile at early stages of ACM is sufficient to explain not only the initiation but also the translocation, dynamics and stability of VT rotors within the RV.

Inverse kinematics and singularity analysis of a redundant parallel robot

The 3C (computer, communications, consumer electronics) industry belongs to labor-intensive and the requirements of the 3C industry are precisely high. The parallel robot is a good choice to replace human labor to manufacture these products. This paper presents a new 3- redundant parallel mechanism for 3C manufacturing. Firstly, some advantages of redundant parallel robots in the current industrial demand are discussed. Then, the inverse kinematics of the redundant parallel manipulator is analyzed according to the closed-loop equation, the inverse displacement problem is solved. The machine is simple and low cost, and as a less freedom robot, it has a higher precision than the traditional robot and has a broad prospect for 3C manufacturing. Finally, the Jacobian matrix of the redundant parallel manipulator was calculated and established by the inverse kinematics formula, and three kinds of the singularity of the mechanism were analyzed by the Jacobian matrix, which are the forward singularity, reverse singularity, mixed singularity respectively.