Singular Points Research Articles

Functions of a Complex Variable with a Large Parameter and Construction of Regions

The study of the asymptotic behavior of solutions of singularly perturbed equations in complex domains comes down to the study of integrals of exponential functions containing a large parameter. Such integrals differ significantly from the integrals to which the saddle point method is applicable. The saddle point method is not applicable to study such integrals. Thus, the problem arises of constructing domains and choosing integration paths for studying such integrals. In this work, specific examples of integrals show the construction of regions in the complex plane and the choice of integration paths. The chosen integration paths ensure that the integrals are bounded over a large parameter as this parameter tends to infinity. When constructing the domain and choosing integration paths, level lines of some harmonic functions that have zeros and singular points were used. The principle of symmetry is also used. In early works, cases were considered when the eigenvalues ​​of the first approximation matrix of a singularly perturbed equation had only zeros or only poles. Cases where the eigenvalues ​​have zeros and poles were not considered.

Image formation near hyperbolic umbilic in strong gravitational lensing

Hyperbolic umbilic (HU) is a point singularity of the gravitational lens equation, giving rise to a ring-shaped image formation made of four highly magnified images, off-centred from the lens centre. Recent observations have revealed new strongly lensed image formations near HU singularities, and many more are expected in ongoing and future observations. Like fold/cusp, image formations near HU also satisfy magnification relation (), i.e., the signed magnification sum of the four images equals zero. Here, we study how deviates from zero as a function of area () covered by the image formation near HU and the distance ( d) of the central maxima image (which is part of the HU image formation) from the lens centre for ideal single- and double-component cluster-scale lenses. For lens ellipticity values ≥0.3, the central maxima image will form sufficiently far from the lens centre ( d≳5″), similar to the observed HU image formations with . We also find that, in some cases, double-component and actual cluster-scale lenses can lead to large cross-sections for HU image formations for sources at z≳5, effectively increasing the chances to observe HU image formation at high redshifts. Finally, we study the time delay distribution in the observed HU image formation, finding that not only are these images highly magnified, but the relative time delay between various pairs of HU characteristic image formation has a typical value of ∼100 days, an order of magnitude smaller than generic five-image formations in cluster lenses, making such image formations optimal targets for time delay cosmography studies.

Rectifiability, finite Hausdorff measure, and compactness for non‐minimizing Bernoulli free boundaries

AbstractWhile there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less is known about critical points of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time‐dependent problem occur naturally in applied problems such as water waves and combustion theory. For such critical points —which can be obtained as limits of classical solutions or limits of a singular perturbation problem—it has been open since (Weiss, 2003) whether the singular set can be large and what equation the measure satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a frequency formula for the Bernoulli problem as well as the celebrated Naber–Valtorta procedure to answer this more than 20 year old question in an affirmative way: For a closed class we call variational solutions of the Bernoulli problem, we show that the topological free boundary (including degenerate singular points , at which as ) is countably ‐rectifiable and has locally finite ‐measure, and we identify the measure completely. This gives a more precise characterization of the free boundary of in arbitrary dimension than was previously available even in dimension two. We also show that limits of (not necessarily minimizing) classical solutions as well as limits of critical points of a singularly perturbed energy are variational solutions, so that the result above applies directly to all of them.

Dynamic light scattering in aqueous solutions of γ-picoline

line-water solutions was established and their dynamics were studied depending on the temperature of the solution and the concentration of γ-picoline in the solution. Analysis of the research results confirms the assumption of the microheterogeneous structure of aqueous solutions in the vicinity of a singular point on the phase diagram of the state in temperatureconcentration coordinates. The microheterogeneity of the structure of an aqueous solution of γ-picoline determines the maximum intensity of light scattering and the negative dispersion of the speed of sound in the hypersonic frequency range.

Black hole singularity from OPE

Eternal asymptotically AdS black holes are dual to thermofield double states in the boundary CFT. It has long been known that black hole singularities have certain signatures in boundary thermal two-point functions related to null geodesics bouncing off the singularities (bouncing geodesics). In this paper we shed light on the manifestations of black hole singularities in the dual CFT. We decompose the boundary CFT correlator of scalar operators using the Operator Product Expansion (OPE) and focus on the contributions from the identity, the stress tensor, and its products. We show that this part of the correlator develops singularities precisely at the points that are connected by bulk bouncing geodesics. Black hole singularities are thus encoded in the analytic behavior of the boundary correlators determined by multiple stress tensor exchanges. Furthermore, we show that in the limit where the conformal dimension of the operators is large, the sum of multi-stress-tensor contributions develops a branch point singularity as predicted by the geodesic analysis. We also argue that the appearance of complexified geodesics, which play an important role in computing the full correlator, is related to the contributions of the double-trace operators in the boundary CFT.

Shrinking rates of horizontal gaps for generic translation surfaces

A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\\rightarrow \\infty $$\\end{document} for the difference in angle between two almost horizontal saddle connections.

Smooth connectivity in real algebraic varieties

AbstractA standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When considering the complement of the set of singular points of a real algebraic variety, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate the approach.

On prescribing the number of singular points in a Cosserat-elastic solid

Abstract In a geometrically non-linear Cosserat model for micro-polar elastic solids, we prove that critical points of the Cosserat energy functional with an arbitrary large (finite) number of singularities do exist, whereas Cosserat energy minimizers are known to be locally Hölder continuous. To reach that goal, we first develop a technique to insert dipole pairs of singularities into smooth maps while controlling the amount of Cosserat energy needed to do so. We then use this method to force an arbitrary number of singular points into (weak) Cosserat-elastic solids by prescribing smooth boundary data. The boundary data themselves are given in such a way, that they contain no topological obstruction to regularity. Throughout this paper, we often exploit connections between harmonic maps and Cosserat-elastic solids, so that we are able to adapt and incorporate ideas of R. Hardt and F.-H. Lin for harmonic maps with singularities, as well as of F. Béthuel for dipole pairs of singularities.

On bifurcation of non-isolated singular points arising in a problem of two-phase fluid motion in a pipe

The dynamics of gas suspension in a pipe is studied. It is described by a system of three autonomous differential equations, whose equilibrium (singular) points are not isolated, but form several continuous curves. One of these curves belongs to the transonic plane. The dynamics in a neighborhood of this plane is interesting from the physical viewpoint. In this work, the dynamics in a neighborhood of the curves of singular points is investigated. The main tools of this work are the center manifold theorem and the reduction principle. A peculiarity of the system is that the curves of singular points intersect each other. Moreover, at the intersection point a bifurcation occurs. Results of the work are the description of the dynamics in a neighborhood of singular points, the proof that a curve of singular points belongs to a center manifold and the description of the bifurcation, which occurs at the intersection point of the curves of singular points.

Quantum entanglement assisted via Duffing nonlinearity

We propose a scheme to enhance quantum entanglement in an optomechanical system by exploiting the so-called Duffing nonlinearity. Our model system consists of two mechanically coupled mechanical resonators, both driven by an optical field. One resonator supports Duffing nonlinearity, while the other does not. The resonators are coupled to each other via the so-called phonon-hopping mechanism. The hopping rate is θ-phase dependent that induces exceptional point singularities in the system. Interestingly, while the resonator with Duffing nonlinearity exhibits vanishing entanglement with light, we observe an increase in entanglement between light and the other mechanical resonator. This enhanced entanglement persists longer against thermal fluctuations compared to the one without the nonlinearity. Additionally, this entanglement features a sudden death and revival phenomenon, where the peaks happen at multiples of θ=π2. This work opens another avenue for exploiting nonlinear resources to generate strong quantum entanglement, paving the way for advancements in quantum information processing, quantum sensing, and quantum computing within complex systems. Published by the American Physical Society 2024

Health-related quality of life of Canadian patients with human epidermal growth factor receptor 2 (HER2)-positive metastatic breast cancer.

397 Background: The EuroQol 5 Dimension 5 Level (EQ-5D-5L) is a validated tool that assesses health utility (HU) as a self-reported measure of health-related quality of life (HRQoL). This study aims to generate Canadian-specific HU data for patients with stable, HER2-positive (HER2+) metastatic breast cancer (mBC) receiving standard therapies. Methods: A cross-sectional study was conducted at McGill University Health Centre in Quebec, Canada. Eligibility criteria included: age ≥18 years, de novo or recurrent mBC, stable disease, and active treatment with a Health Canada-approved systemic therapy regimen (any line/time point on treatment). The EQ-5D-5L questionnaire was administered at every visit and each record was treated as a unique singular data point. The HU values were derived using the Canadian algorithm developed by Xie et al., 2016. The EQ-5D-5L measures HRQoL across five domains: mobility, self-care, usual activities, pain/discomfort, and anxiety/depression, and is scored on 5 levels ranging from one “no problems” to five “extreme problems”. A second part of the questionnaire includes a visual analogue scale (VAS) where respondents rank their overall health on scale of 0-100. The primary outcome was mean HU and standard deviation (SD) for HER2+ patients overall. Secondary outcomes included mean HU by hormone receptor (HR) status and describing patient responses to the questionnaire. Results: Between December 2022 and July 2023, a total of 81 HER2+ HU data points were included for analysis, representing 30 patients. For these patients, the mean age was 54.2 years (SD: 10.6). Most patients (58.3%) were on first line and had an Eastern Cooperative Oncology Group Performance Status of 0 (65.5%). The mean HU value was 0.83 (SD: 0.14, n=81). For patients with HR-positive mBC, the mean HU was 0.81 (SD: 0.14, n=44), while for patients with HR-negative mBC, it was 0.85 (SD: 0.13, n=37). The majority of patients reported “no problems” across all domains: mobility (69.1%), self-care (81.5%), usual activities (46.9%), pain/discomfort (35.8%), and anxiety/depression (58%). The mean VAS score was 77.3 (SD: 17.0). Conclusions: This study provides the first Canadian data on contemporary HU values for patients with HER2+ mBC, which reflects current HRQoL amidst a changing treatment landscape.

Deep sediment heterogeneity inferred using very low-frequency features from merchant shipsa).

The very low-frequency noise from merchant ships provides a good broadband sound source to study the deep layers of the seabed. The nested striations that characterize ship time-frequency spectrograms contain unique acoustic features corresponding to where the waveguide invariant β becomes infinite. In this dataset, these features occur at frequencies between 20 and 80 Hz, where pairs of modal group velocities become equal. The goal of this study is to identify these β = ∞ frequencies in ship noise spectrograms and use them to perform statistical inference for the deep layer sound speeds and thicknesses in the New England Mudpatch for a larger number of ships and acoustic arrays over a larger geographical region than previously studied. Marginal probability distributions of the data indicate that using singular points for a feature-based inversion yields an estimate of the sound speed and a limiting value for the thickness of the first deep layer. Heterogeneity is examined by correlating spatial variability of the deep layer sound speeds with ship tracks.

Застосування методу двобічних наближень до знаходження додатних аксіально-симетричних розв’язків крайових задач з монотонними нелінійностями

This paper considers the application of the method of two-sided approximations for finding positive solutions to boundary value problems for nonlinear elliptic differential equations with axial symmetry. The case of Dirichlet boundary conditions (first kind) is considered, with a power-type monotone nonlinearity, where the exponent ranges from 0 to 1. By transitioning to polar coordinates in the boundary value problem for the elliptic equation, due to the axial symmetry of the solution, the problem is reduced to a boundary value problem for an ordinary differential equation on an interval with respect to a function that depends only on the polar radius, thus eliminating the dependence on the angle. The pole of the polar coordinate system becomes a singular point, where a boundedness condition must be imposed. For the boundary value problem, a Green's function is constructed to further reduce the problem to a Hammerstein integral equation. The integral equation is treated as a nonlinear operator equation in a Banach space of continuous functions on the interval, partially ordered by a cone of non-negative continuous functions on this interval. The operator is examined for properties such as monotonicity (isotonicity), positivity, boundedness, and concavity. Next, the initial approximation is found as the endpoints of the invariant conical segment for the isotone operator in such a way as to ensure the highest convergence rate of the iterative process. The following iterative sequences of two-sided approximations are constructed: the first sequence, which is non-decreasing with respect to the cone, and the second sequence, which is non-increasing with respect to the cone. At each iteration, the arithmetic mean of the upper and lower approximations is taken as the next approximation. The iterative process is terminated when the error estimate of the solution satisfies the specified accuracy. The theoretical results obtained in this work were validated through a computational experiment. The dependence of the solution on the parameters in the right-hand side was analyzed and illustrated with corresponding graphs

Cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures

We consider the existence of cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures on the following classes of torsion-free G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds: the Euclidean R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} with its standard G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structure, metric cylinders over Calabi–Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant–Salamon G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.

A contour integral-based method for nonlinear eigenvalue problems for semi-infinite photonic crystals

In this study, we introduce an efficient method for determining isolated singular points of two-dimensional semi-infinite and bi-infinite photonic crystals, equipped with perfect electric conductor and quasi-periodic mixed boundary conditions. This specific problem can be modeled by a Helmholtz equation and is recast as a generalized eigenvalue problem involving an infinite-dimensional block quasi-Toeplitz matrix. Through an intelligent implementation of cyclic structure-preserving matrix transformations, the contour integral method is elegantly employed to calculate the isolated eigenvalue and to extract a component of the associated eigenvector. Moreover, a propagation formula for electromagnetic fields is derived. This formulation enables rapid computation of field distributions across the expansive semi-infinite and bi-infinite domains, thus highlighting the attributes of edge states. The preliminary MATLAB implementation is available at https://github.com/FAME-GPU/2D_Semi-infinite_PhC.

Discretization of Thin-Walled Sections with Variable Wall Thickness

At the stage of designing thin-walled aircraft structures, to simplify calculations, their cross sections are idealized. To do this, the section with the skin and the longitudinal elements reinforcing it is replaced (when determining normal stresses) by a discrete one, consisting of concentrated areas at characteristic points. In this case, the equality of the moments of inertia of the initial and discrete sections is preserved. Such idealization is used in the calculation of thin-walled rods for normal and shear stresses (Wagner model). For sections consisting of a system of rectangular strips of constant thickness, discretization allows to set approximate values of normal and shear stresses and accurately find the locations of the singular points of the bending center (in an open contour) and the center of rigidity (in a closed one). The discrete model of a strip consists of three lumped areas: two at the edges and one in the center. The paper proposes to extend the discrete model to sections in which the skin thickness changes according to a linear law. In addition to the rectangular strip, it is possible to use elongated triangles and trapezoids, replaced by three and four concentrated areas, respectively. The possibility of using a discrete model for calculating some thin-walled sections of open and closed contours is considered. The section of an open contour is studied - the problem of transverse bending without torsion of a channel having flanges with a linearly variable thickness. Differences in the flows of tangential forces calculated from the exact and discrete models are shown. The coincidence of the results in determining the position of the bending center according to two models was established. When studying the application of a discrete model to a closed contour, its simplified option is proposed. The problem of transverse bending without torsion and finding the center of rigidity in a section with a contour line in the form of a trapezoid with front and rear walls of constant thickness and upper and lower skins and a similar section with a contour in the form of a rectangle was considered. Differences in the flows of tangential forces calculated by exact and discrete models are established. For a closed section in the form of a rectangle, the decrease in the moment of inertia for torsion due to the redistribution of material in the cross section was studied separately. It has been established that when finding the position of the center of rigidity, the discrepancy between the results of the exact and discrete models in sections with geometric parameters close to real ones was less than 1% for a rectangular contour, and 4% for a trapezoidal contour. The results indicate the possibility of extending the application of the discrete model of thin-walled cross section to the design calculations of thin-walled rods with variable skin thickness, representing practical structures.

Описание физических эффектов акустического поля в волноводе мелкого моря

The field of the complex vector of acoustic intensity of a moving tonal low-frequency source in a regular waveguide of a shallow sea is investigated. The following phenomena are considered: phase front dislocations, singular points, an acoustic intensity vortex and the mechanism of its stability, angular momentum of the vortex, vortex rotor. The value of the signal-to-noise ratio has been established at which the appearance of a phase front dislocation and a vortex in the far field of the signal source is possible in a coherent field.

Double Laplace Transform Method for Solving Fractional Fourth-Order Partial Integro-Differential Equations with Weakly Singular Kernel

Objectives: To investigates the solutions of fourth order partial integro-differential equations with high-order non-integer derivatives and weakly singular kernels. Methods: Weakly singular kernels present challenges in both analytical and numerical treatments due to their intricate behaviour near singular points. In this article, we introduce a novel approach utilizing the double Laplace transform method to effectively address these challenges. Findings: By solving a series of precise and understandable examples, the double Laplace transform clearly transforms the fractional partial integro-differential equation into an algebraic equation that can be solved easily. Novelty: The double Laplace transform is an effective instrument for solving fractional partial integro-differential equations, which are commonly used in fields such as physics, fluid mechanics, gas dynamics, and signal processing. Mathematics Subject Classifications: 35R09, 35R11, 44A10, 44A30. Keywords: Fractional PIDE, Weakly singular kernel, Double Laplace transform, Inverse double Laplace transform, Exact solution

Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the original problem to a problem in convex geometry. Proving either of the two conjectural statements will in particular resolve a long-standing conjecture by Briançon and Iarrobino back in the ’70s for the case of the powers of the maximal ideal. Furthermore, for specific classes of lengths, we conjecturally classify points satisfying the conjectural sufficient conditions. This in particular (conjecturally) provides many new explicit families of examples of maximum dimension tangent space at a point of the Hilbert schemes of points of lengths strictly between two consecutive tetrahedral numbers ( 3 + k 3 ) .

Dulac maps of real saddle-nodes

Abstract Consider a germ of a holomorphic vector field at the origin on the coordinate complex plane. This germ is called a saddle-node if the origin is its singular point, one of its eigenvalues at zero is zero, and the other is not. A saddle-node germ is real if its restriction to the real plane is real. The monodromy transformation for this germ has a multiplier at zero equal to 1. The germ of this map is parabolic and admits a ‘normalizing cochain’. In this note we express the Dulac map of any real saddle-node up to a left composition with a real germ ( C , 0 ) → ( C , 0 ) through one component of the cochain normalizing the monodromy transformation.