Singular Points Research Articles

The total absolute curvature of closed curves with singularities

Abstract In this paper, we give a generalization of Fenchel’s theorem for closed curves as frontals in Euclidean space ℝ n . We prove that, for a non-co-orientable closed frontal in ℝ n , its total absolute curvature is greater than or equal to π. It is equal to π if and only if the curve is a planar locally L-convex closed frontal whose rotation index is 1/2 or –1/2. Furthermore, if the equality holds and if every singular point is a cusp, then the number N of cusps is an odd integer greater than or equal to 3, and N = 3 holds if and only if the curve is simple.

Algebraic Montgomery-Yang problem and smooth obstructions

Let S S be a rational homology complex projective plane with quotient singularities. The algebraic Montgomery-Yang problem conjectures that the number of singular points of S S is at most three if its smooth locus is simply-connected. In this paper, we leverage results from the study of smooth 4-manifolds, including the Donaldson diagonalization theorem and Heegaard Floer correction terms, to establish additional conditions on S S . As a result, we eliminate the possibility of a rational homology complex projective plane of specific types with four singularities. Moreover, we identify large families encompassing infinitely many types of singularities that satisfy the orbifold BMY inequality, a key property in algebraic geometry, yet are obstructed from being a rational homology complex projective plane due to smooth conditions. Additionally, we discuss computational results related to this problem, offering new insights into the algebraic Montgomery-Yang problem.

A comparative analysis of alpha olefin sulfonate and dodecyl sulfate in aphronic fluid containing xanthan gum in a wide range of temperatures

Geothermal energy, oil industry, and underground gas storage technology require deep drilling. Although oil-based drilling fluids have been widely used, they cause environmental issues. Environmentally friendly Aphronic fluid has emerged as an alternative to oil-based drilling fluid. Despite various investigations on Aphron’s performance, the effect of temperature has not yet been comprehensively studied. In this study, Aphronic fluid was developed with xanthan gum (XG) as a polymer and alpha-olefin sulfonate (AOS) as a surfactant. Furthermore, another Aphronic fluid was developed with xanthan gum (XG) as a polymer and sodium dodecyl sulfate (SDS) as a surfactant. This study evaluates their rheological properties, including shear rate (ranging from 0.1 to 10 s− 1), across a range of temperatures (25 ℃, 49 ℃, 71 ℃, and 93 ℃). Stability testing was conducted at ambient temperature, and rheology tests were performed at various temperatures. It was found that the rheological properties decrease as the temperature increases. An increase in the temperature and shear rate decreases the viscosity of the drilling fluid. AOS demonstrated higher stability and less reduction in rheological properties at elevated temperatures than SDS. This difference can be attributed to SDS’s tendency to denature the XG polymer, leading to a notable decrease in viscosity. Maintaining viscosity is crucial for fluid stability and enhancement of rheological properties. Moreover, the best stability was achieved at 1.5 times the critical micelle concentration (CMC). This behavior is consistent with the Gibbs elasticity theory, indicating that the optimal concentration for both AOS and SDS surfactants is 1.5 times the CMC. Deviation from this optimal concentration resulted in decreased system stability. The determination of the CMC value showed a broader region rather than a singular point, potentially confirming the Gibbs-Marangoni theory. The implications of this research underscore the importance of selecting the appropriate Aphronic fluid formula for drilling operations to ensure stability and rheological properties conducive to overcoming reservoir pressure and accurately facilitating the cutting transfer process. This research highlights the need to choose the right Aphronic fluid formula for drilling. Using the right formula helps in drilling to ensure stability and rheological properties conducive to overcoming reservoir pressure and accurately facilitating the cutting transfer process.

Modeling of flows under dams using the method of singular points

Today, the relevance of issues related to regular and safe water supply for human beings, society and the economy is beyond doubt. One of the main ways to store and man age water resources is the construction and operation of dams. The most important task of sustainable existence and development of reservoir projects is dam safety. To ensure this, theoretical research, creation of mathematical and computer models of the physical phe nomena and engineering structures under study are necessary. The paper proposes methods for constructing flows in the middle sections of a dam with a linear filtration law by ap propriately placing special flow points and applying conformal transformations. Filtration f lows with different outlines of impermeable floodplain, flows under a dam with filters in homogeneously isotropic and heterogeneously anisotropic soils are considered.

Gaussian Curvature of the Slowness Surface in Vicinity of Singularity Point in Anisotropic Media

Gaussian Curvature of the Slowness Surface in Vicinity of Singularity Point in Anisotropic Media

Asymptotic Properties of Functionals of the Squared Periodograms for Stationary Random Fields

The paper presents conditions for the asymptotic normality of functionals of squared periodograms based on tapered data. Stationary Gaussian random fields are considered. Two limit theorems are stated: for the first one the certain condition of integrability of the spectral density of the field is assumed, and the second result is for spectral densities with the prescribed behavior near the points of singularities.

Inverse energy cascade in turbulent Taylor–Couette flows

The inverse energy cascade in turbulent Taylor–Couette flow is studied in line with the results of the large eddy simulation. The simulation results show that the inverse energy cascade first occurs within the core region of the flow channel of the Taylor–Couette flow at higher Reynolds numbers. It is uncovered that this phenomenon is induced by the pulsed zero shear stress resulting from the singularities of the Navier–Stokes equation. In the core area between the two cylinders, the shear stress is nearly zero at higher Reynolds numbers. The turbulence generated there has high turbulent energy due to discontinuity of the tangential velocity. Since the energy transfer between the fluid layers is inhibited due to the low shear stress, the turbulent energy cannot be transferred along the radial direction, and small-scale vortices with high turbulent energy are produced. These small-scale vortices are located with the large-scale vortices and cannot be dissipated owing to low shear stress. A peak in the energy spectrum at the middle frequency (or wave number) is formed due to the concentration of the small-scale vortices. As the number of the singular points of the Navier–Stokes equation increases with the increasing Reynolds number, the region with zero shear stress expands along the radial direction, intensifying nonlinear instability and energy accumulation. This, in turn, leads to more prominent peaks in the energy spectrum, resulting in a more pronounced inverse energy cascade.

Study on Shear Postbuckling and Failure of Perforated Plates: Modeling and Experiments

Based on the Kirchhoff–Karman nonlinear theory, the postbuckling bearing capacity (PBC) of perforated plates under shearing load was studied and compared using a self-developed quasi-conforming element program with singular point identification and switching approach. The influences of geometric parameters on the PBC were analyzed. During calculation, the edge beams were introduced to simulate effects of boundary on PBC. The perforated plates were manufactured and tested to verify the accuracy of the numerical results. The results show that the influence of in-plane constraints along the normal of edges on the PBC is more significant. The PBC increases with the increase of plate thickness but is not sensitive to the change of the hole’s location. The reinforcement can reduce the stress concentration and increase the PBC by 7–133%. The PBC is greatly affected by the aperture D, and it is found that the optimal aperture D* exists when the side length a is constant. The PBC under the optimal aperture is higher than that under other apertures. To facilitate the optimization design, a more refined geometric parameter library is provided by the least-squares method combined with the linear interpolation method. The interpolation results are verified by the existing numerical results.

Monodromic singularities without curves of zero angular speed

<p>We consider planar analytic vector fields $ \mathcal{X} $ having a monodromic singular point with Poincaré map $ \Pi $. We use the fact that there always exists a real analytic invariant curve $ F = 0 $ of $ \mathcal{X} $ in a neighborhood of that singularity. We find some relations between $ \Pi $ and $ F $ that can be used to determine new conditions that guarantee the analyticity of $ \Pi $ at the singularity. In the special case that $ F $ becomes an inverse integrating factor of $ \mathcal{X} $, we rediscover formulas obtained previously by other methods. Applications to the center-focus problem and also to vector fields with degenerate infinity are given.</p>

Dirichlet Functions Generated by Blaschke Products

The continuation of general Dirichlet series to meromorphic functions in the complex plane remains an outstanding problem. It has been completely solved only for Dirichlet L-series. A sufficient condition for the general case exists, however it is impossible to verify that it is fulfilled. We solve this problem here for another class of general Dirichlet series, namely those series which are obtained from infinite Blaschke products by a particular change of variable. This is a source of examples of general Dirichlet series with infinitely many poles. An interesting new case is now revealed, in which the singular points of the extended function form a continuum. We take a closer look at the case of Dirichlet series with natural boundary and give examples of such series. Some figures illustrate the theory.

Some features satellite attitude control using quaternions

Algorithms using the kinematics equations in quaternions are widely used in the attitude control systems of modern spacecraft, primarily microsatellites. They ensure large-angle rotations and avoidance of singularity points characteristic of Euler-Krylov angles. However, the specificity of single quaternions as attitude representation parameters can create an "unwinding effect" in the control system, when turning the spacecraft to a desirable position is not performed by the shortest path and increases the electricity consumption, that is limited on board. This work, summarizing known publications, illustrates the causes of this effect and demonstrates a simple algorithm for its prevention. The effectiveness of this approach is shown on the MATLAB platform numerical simulation examples of the microsatellite control system with reaction wheels and a PD controller.

ТОЧКИ РАВНОВЕСИЯ МЕХАНИЧЕСКИХ СИСТЕМ

This paper deals with the differences between regular and singular points of equilibrium states of elastic systems. The author provides a classification of singular points based on mathematical, mechanical and geometrical approaches. Aspects concerning nondegenerate (regular) critical points in which the first differential is zero and the second differential (quadratic form) is not degenerate are disclosed. The conditions of stability and instability of isolated equilibria are discussed, as well as the influence of the primary nonzero terms of the potential energy expansion on the stability of degenerate equilibria. The conditions for bifurcation loss of stability in the context of incomplete equilibrium of elastic systems are considered. It is analyzed that a necessary condition for the appearance of bifurcation points is the presence of an energetically orthogonal complement. The influence of initial geometrical imperfections on the character of stability loss is also considered.

Unveiling van Hove Singularity-Boosted Photothermoelectric Response for Wearable Human-Radiation Detection.

Van Hove singularity (vHs), the singularity point of density of states (DOS) in crystalline solids, is a research hotspot in emerging phenomena such as light-matter interaction, superconducting, and quantum anomalous Hall effect. Although the significance of vHs in photothermoelectric (PTE) effect has been recognized, its integral role in electron excitation and thermoelectric effect is still unclear, particularly in the mid-infrared band that suffers from Pauli blockade in semimetals. Here, we unveil the Fermi-level-modulated PTE behavior in the vicinity of vHs in carbon nanotubes, employing ionic-liquid gating. The concurrent enhancement of optical absorption and thermoelectric effect effectively improves the overall photoresponse by tens of folds at the vHs point. Generally applicable to strongly correlated systems such as metallic 1D nanomaterials and 2D Moiré systems, a quantitative correlation between PTE photodetectivity and electronic DOS is derived in the vicinity of the vHs point. Finally, chemically doped PTE mid-infrared detectors with graded doping levels are demonstrated to exhibit human-radiation sensitivity, high flexibility, and high transparency, paving the way for wearable sensor networks in healthcare systems and the Internet of Things.

“Vill ‘Hem’, Men Vet ej var Hemmet Ligger”: Migration and the Aesthetics of Estrangement in Verner von Heidenstam’s Formative Art and Prose

This essay argues that Nobel laureate Verner von Heidenstam’s campaign against naturalist aesthetics in late nineteenth-century Swedish literature was motivated, in part, by the sense of estrangement he developed from Swedish cultural life during his adolescent years as a migrant. It also contends that the aesthetic discontent he experienced in his early career foreshadowed a wider sense of alienation from place and nation that would accompany the rise of globalization and normalized migration in the twentieth century. While recent scholarship on Heidenstam’s early oeuvre situates the writer’s bibliography within the fin de siècle, this project refocuses the discussion on the contemporaneous artistic debates Heidenstam addresses in his polemic Renässans, as well as the migratory themes he explores in his 1892 novel, Hans Alienus. This approach illuminates how Heidenstam’s youthful quest for aesthetic reinvention upended the notion that artists and writers can be tethered to singular points of origin, offering new pathways for understanding the emergence of a distinct migrant literature and visual art in Sweden. Although Heidenstam’s later works took a sharp nationalistic turn and have receded from popular consciousness in contemporary times, reexamining his earliest paintings and prose as products of a migrant imagination can help scholars more firmly affix his legacy to modern and Modernist traditions, inviting fresh perspectives on his paradigm-shifting aesthetic of estrangement.

Colossal Shift Currents in Band-Inverted Bi Nanotubes Driven by the Interplay of Curvature and Spin-Orbit Coupling.

The concept of non-trivial electronic structure combined with reduced dimensionality presents a promising strategy for advancing optical applications and energy harvesting technologies. Symmetry breaking in low dimensional system enables the emergence of non-linear optical responses, which are greatly amplified by the singular points of band inversion. Here, using first-principles calculations, the significant enhancement of the shift current in Bi nanotubes is investigated, driven by the combined effects of 1D geometry and non-trivial band order. By rolling a 2D Bi monolayer, which exhibits a quantum spin Hall phase, into a 1D nanotube, an extraordinarily large shift current of 4.94 A·Å2V-2 is generated, two orders of magnitude larger than the experimental values in WS2 nanotubes. The enhancement of the shift current in Bi nanotubes is demonstrated and attributed to the state mixing induced by the interplay of strong spin-orbit coupling, the curvature of the tube, and the non-trivial band order of Bi nanotubes. The robustness of this enhanced shift current against various perturbations is also discussed, such as oxidation and doping. The novel approach integrating non-trivial band order with reduced dimensionality sheds new light on advanced photovoltaic applications by harnessing both geometric advantages and exotic electronic structures for energy conversion.

Desingularization of finitely smooth vector fields on the plane

The paper is devoted to desingularization of finitely smooth vector fields on the plane. Let the difference of germs of a finitely smooth vector field and an analytic vector field be sufficiently flat at a common singular point. Then these vector fields share similar behavior under desingularization. In particular, the nice desingularizations of both vector fields are achieved in the same number of steps. Moreover, for any analytic vector field with a singular point of multiplicity μ 0 \mu _0 , nice desingularization of the finitely smooth vector field is achieved after μ 0 + 2 \mu _0+2 steps. In addition, a description is presented for topologically sufficient jets of finitely smooth vector fields at nonmonodromic singular points.

Method for Investigation of Convergence of Formal Series Involved in Asymptotics of Solutions of Second-Order Differential Equations in the Neighborhood of Irregular Singular Points

The aim of the article is to create a method for studying the asymptotics of solutions to second-order differential equations with irregular singularities. The method allows us to prove the convergence of formal series included in the asymptotics of solutions for a wide class of second-order differential equations in the neighborhoods of their irregular singular points, including the point at infinity, which is generally irregular. The article provides a number of applications of the method for studying the asymptotics of solutions to both ordinary differential equations and partial differential equations.

Neurospectral Computation for the Resonant Characteristics of an Equilateral Triangular Patch Antenna on Suspended Substrates

Modeling and design of an equilateral triangular patch antenna on suspended and single substrate are accomplished in this paper. The spectral domain approach is important due to its accuracy, but has a high computational cost. On the other hand, the Artificial Neural Networks (ANNs) have recently become a fast and flexible vehicle for modeling and designing microwave antennas. This paper introduces electromagnetic knowledge combined with ANNs to compute the resonant frequency of the fundamental and higher order modes and to eliminate the difficulties of handling the singularity points encountered in the numerical evaluation of integrals. The resonant frequency results obtained from the neural model are in very good agreement with the experimental and theoretical results available in the literature.

In Vivo Mapping of Human Ventricular Fibrillation in Brugada Syndrome: The Role of Repolarization Heterogeneity.

Brugada syndrome (BrS) is associated with ventricular fibrillation (VF). Different VF mechanisms have been described, and repolarization gradients were associated with VF in a BrS model. The aim of this study is to map VF in BrS with ECG imaging. Furthermore, spatial correlation between sinus rhythm maps and VF maps was evaluated. Inclusion criteria were (1) BrS diagnosis and (2) VF mapped with ECG imaging during right ventricle outflow tract ablation. VF mechanism was classified into (1) rotational, (2) focal, and (3) irregular. For comparison, 6 controls were enrolled. The following sinus rhythm maps were performed: activation, recovery time, and activation-recovery interval time. Spatial overlap between steep repolarization gradients (cliffs) at recovery time and activation-recovery interval time maps and initiating VF rotational activity was evaluated with photogrammetry. A total of 28 VF maps in 21 patients with BrS were analyzed. In the first ≈7 seconds of VF, rotational, focal, and irregular mechanisms were found. In 19 patients with BrS (90.5%) and none of the controls, a right ventricle outflow tract repolarization cliff only was found. In all these patients, the singularity point of the first initiating rotational VF activity spatially overlapped with the right ventricle outflow tract cliff. Abolition of right ventricle outflow tract repolarization cliffs was confirmed in all but 2 patients (94.3%). In one patient with recurrence, VF was mapped on the anterior right ventricle over a cliff that was not targeted at the first ablation procedure. In patients with BrS, repolarization heterogeneity has a critical role in VF. Repolarization cliffs might be a therapeutic target in VF ablation.

Fabry-Perot Resonances in Bilayer Metasurfaces.

In this study, we construct Fabry-Perot cavities in which nanostructured, thin resonant metasurfaces act as mirrors. We develop a temporal coupled-mode theory and provide an accurate description of the resonances supported by these cavities, deriving analytically their transmission characteristics. The presence of metasurface mirrors introduces a substantial group delay, causing the field concentration to shift from the bulk of the cavity towards the regions close to the two metasurfaces. This shift is accompanied by a significant increase in the quality factor of the resonances that is associated with the emergence of singular points in frequency space. These singular points exist even when the cavity separations are smaller than the cavity length of the fundamental mode in standard cavities, enabling meta-mirror Fabry-Perot cavities to outperform traditional cavities by achieving much higher quality factors despite displaying shorter cavity lengths.