Regular Singularities Research Articles

Topological triviality of deformations of regular mappings

We give a sufficient condition for topological triviality of f+tP, t∈[0,1] for C2 mappings f,P:Rn→Rm, m≤n, in terms of the Hörmander-Łojasiewicz inequality ν(df(x))≥C(dist(x,Z)/(1+|x|2))k, x∈Rn, where Z is the set of critical points of f and ν is the Rabier function. We apply this condition for regular mappings. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated regular mappings' singularities in terms of degrees of polynomials describing the regular mappings.

On the solutions of the universal differential equation with three regular singularities (On solutions of KZ 3 )

On the solutions of the universal differential equation with three regular singularities (On solutions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>K</mml:mi><mml:msub><mml:mi>Z</mml:mi> <mml:mn>3</mml:mn> </mml:msub></mml:mrow></mml:math>)

The regularity transformation equations: An elliptic mechanism for smoothing gravitational metrics in general relativity

A central question in General Relativity (GR) is how to determine whether singularities are geometrical properties of spacetime, or simply anomalies of a coordinate system used to parameterize the spacetime. In particular, it is an open problem whether there always exist coordinate transformations which smooth a gravitational metric to optimal regularity, two full derivatives above the curvature tensor, or whether regularity singularities exist. We resolve this open problem above a threshold level of smoothness by proving in this paper that the existence of such coordinate transformations is equivalent to solving a system of nonlinear elliptic equations in the unknown Jacobian and transformed connection, both viewed as matrix valued differential forms. We name these the Regularity Transformation equations, or RT-equations. In a companion paper we prove existence of solutions to the RT-equations for connections $\Gamma\in W^{m,p},$ curvature ${\rm Riem}(\Gamma) \in W^{m,p}$, assuming $m\geq1$, $p>n$. Taken together, these results imply that there always exist coordinate transformations which smooth arbitrary connections to optimal regularity, (one derivative more regular than the curvature), and there are no regularity singularities, above the threshold $m\geq1$, $p>n$. Authors are currently working on extending these methods to the case of GR shock waves, when gravitational metrics are only Lipschitz continuous, ($m=0$, $p=\infty$), and optimal regularity is required to recover basic properties of spacetime.

A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer

The work is devoted to the construction of the asymptotics of the solution of a singularly perturbed system of equations of parabolic type, in the case when the limit equation has a regular singularity as the small parameter tends to zero. The asymptotics of the solution of such problems contains, along with parabolic boundary layer functions, and power boundary layer functions.

Series Solution of a Ten-Parameter Second-Order Differential Equation with Three Regular Singularities and One Irregular Singularity

We introduce a ten-parameter ordinary linear differential equation of the second order with four singular points. Three of these are finite and regular whereas the fourth is irregular at infinity. We use the tridiagonal representation approach to obtain a solution of the equation as bounded infinite series of square integrable functions that are written in terms of the Jacobi polynomials. The expansion coefficients of the series satisfy three-term recursion relation, which is solved in terms of a modified version of the continuous Hahn orthogonal polynomial.

Confluence of Singularities in Hypergeometric Systems

A system in a Birkhoff normal form with an irregular singularity of Poincare rank 1 at the origin and a regular singularity at infinity is through the Borel-Laplace transform dual to a system in an Okubo form. Schafke has showed that the Birkhoff system can also be obtained from the Okubo system by a simple limiting procedure. The Okubo system comes naturally with two kinds of mixed solution bases, both of which converge under the limit procedure to the canonical solutions of the limit Birkhoff system on sectors near the irregular singularity at the origin. One can then define Stokes matrices of the Okubo system as connection matrices between different branches of the mixed solution bases and use them to relate the monodromy matrices of the Okubo system to the usual Stokes matrices of the limit system at the irregular singularity. This is illustrated on the example of confluence in the generalized hypergeometric equation.

Adversarial Learning for Invertible Steganography

Deep neural networks have revolutionised the research landscape of steganography. However, their potential has not been explored in invertible steganography, a special class of methods that permits the recovery of distorted objects due to steganographic perturbations to their pristine condition. In this paper, we revisit the regular-singular (RS) method and show that this elegant but obsolete invertible steganographic method can be reinvigorated and brought forwards to modern generation via neuralisation. Towards developing a renewed RS method, we introduce adversarial learning to capture the regularity of natural images automatically in contrast to handcrafted discrimination functions based on heuristic image prior. Specifically, we train generative adversarial networks (GANs) to predict bit-planes that have been used to carry hidden information. We then form a synthetic image and use it as a reference to provide guidance on data embedding and image recovery. Experimental results showed a significant improvement over the prior implementation of the RS method based on large-scale statistical evaluations.

Exterior multiplication with singularities: a Saito theorem in vector bundles

Let $E$ be a vector bundle over a suitable differential manifold $M$ and let $\wedge^p E$ denote $p$-exterior product of $E$. Given sections $\omega_1,\dots,\omega_k$ of $E$ and a section $\eta$ of $\wedge^p E$, we consider the problem if $\eta$ can be written in the form $$\eta=\sum \omega_i\wedge\gamma_i,$$ where $\gamma_i$ are sections of $\wedge^{p-1}E$. An obvious necessary condition $\Omega\wedge\eta=0$, where $\Omega=\omega_1\wedge\cdots\wedge\omega_k$, has to be supplemented with a condition that the form $\Omega$ has sufficiently regular singularities at points where $\Omega(x)=0$. Such a local condition is suggested by an algebraic theorem of K. Saito and is given in terms of the depth of the ideal defined by coefficients of $\Omega$. Working in the smooth, real analytic and holomorphic (with $M$ Stein manifold) categories, we show that the condition is sufficient for the above property to hold. Moreover, in the smooth category it is sufficient for existence of a continuous right inverse to the operator defined by $(\gamma_1,\dots,\gamma_k)\mapsto\sum \omega_i\wedge\gamma_i$. All these results are also proven in the case where $E$ is a bundle over a suitable closed subset of $M$.

Shock Wave Interactions and the Riemann-Flat Condition: The Geometry Behind Metric Smoothing and the Existence of Locally Inertial Frames in General Relativity

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether {\it regularity singularities} (points where the curvature is in $L^\infty$ but the essential smoothness of the gravitational metric is only Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to $C^{1,1}$ by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to $C^{1,1}$ locally. This latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme.

Global spacetime structure of compactified inflationary universe

We investigate the global spacetime structure of torus de Sitter universe, which is exact de Sitter space with torus identification based on the flat chart. We show that past incomplete null geodesics in torus de Sitter universe are locally extendible. Then we give an extension of torus de Sitter universe so that at least one of the past incomplete null geodesics in the original spacetime becomes complete. However, we find that extended torus de Sitter universe has two ill behaviors. The first one is a closed causal curve. The second one is so called quasi regular singularity, which means that there is no global, consistent extension of spacetime where all curves become complete, nevertheless each curve is locally extensible.

Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes four restrictions imposed on the parameters: two of the singularities should have non-zero integer characteristic exponents and the accessory parameters should obey polynomial equations.

A linear system for pipe flow stability analysis allowing for boundary condition modifications

An accurate system to study the stability of pipe flow that ensures regularity is presented. The system produces a spectrum that is as accurate as Meseguer and Trefethen (2000), while providing flexibility to amend the boundary conditions without a need to modify the formulation. The accuracy is achieved by formulating the state variables to behave as analytic functions. We show that the resulting system retains the regular singularity at the pipe centre with a multiplicity of poles such that the wall boundary conditions are complemented with precisely the needed number of regularity conditions for obtaining unique solutions. In the case of axisymmetric and axially constant perturbations the computed eigenvalues match, to double precision accuracy, the values predicted by the analytical characteristic relations. The derived system is used to obtain the optimal inviscid disturbance pattern, which is found to hold similar structure as in plane shear flows.

Indicator-based lightweight steganography on 32-bit RISC architectures for IoT security

Embedded devices with highly constrained resources are emerging in numerous application areas which include wireless sensor networks, Radio-Frequency IDentification (RFID) tags, and Internet of Things (IoT). These devices need to typically communicate small payload in the form of text/image/audio for which security is exceptionally essential. Considering the resource limitation on constrained devices, many crypto algorithms and a few stego algorithms have been designed with lightweight properties. Majority of these algorithms have been tested for lightweight property only based on their algorithmic attributes. Conversely, ensuring such lightweight characteristics by analysing their feasibility to reside and run in a constrained environment based on the device’s architectural attribute is inevitable for IoT applications. This paper aims to contribute by proposing an indicator based lightweight Least Significant Bit (LSB) steganography algorithm and to compare it’s algorithmic and device dependent implementation aspects with similar algorithms on popular 32-bit Reduced Instruction Set Computer (RISC) microcontrollers used in IoT platforms. The proposed variable embedding algorithm achieves a Peak Signal to Noise Ratio (PSNR) of over 46 dB with Normalised Cross Correlation (NCC) & Structural Similarity Index Measure (SSIM) being 0.9999 and 0.9998 respectively for an average embedding capacity of 1.5 bits per pixel. In addition to the above mentioned benchmarking parameter results, the Regular & Singular (RS) group and Sample Pair (SP) steganalysis, were also carried out to validate the security level of the proposed algorithm. On analysing the suitability of the proposed algorithm in terms of timing performance and memory requirements by implementing on different IoT hardware, the microcontroller with PIC32 core achieves a higher embedding throughput of over 2.7 Mega bits per second with a smaller memory footprint of less than 2 KB. Finally, the results obtained from the proposed work outperform the microcontroller implementation of stego algorithms reported in the literature.

Analytical study of Bragg resonance by singly periodic sinusoidal ripples based on the modified mild-slope equation

The wave propagation over singly periodic sinusoidal ripples is studied analytically based on the modified mild-slope equation (MMSE). Firstly, each ripple region is divided into four monotonic subintervals such that there is only one regular singular point of the MMSE within each subinterval which is located at one of the two endpoints. Secondly, in each subinterval, due to the monotonicity, the implicit MMSE can be transformed into an explicit MMSE (EMMSE). Thirdly, in each subinterval, due to the unique regular singularity, a general solution in terms of Frobenius series to the EMMSE can be constructed and the convergence condition of the series solution is analyzed and graphically demonstrated. Finally, by using the mass-conserving matching conditions at each common boundary between any two adjacent subintervals, an analytical formula of the reflection coefficient is established. The present solution is validated against various existing solutions, especially, it is identical to a numerical solution to the MMSE. It is shown that, as the number of ripples increases, the peak amplitudes of both the primary and subharmonic Bragg resonances increase while the resonance bandwidths decrease, and the zero reflection occurs more frequently. When the number becomes very large, the peak amplitudes of the primary and subharmonic Bragg resonances achieve unity and the resonance bandwidths approach to their limit resonance bands. As the height of ripples increases, the peak amplitudes of both the primary and subharmonic Bragg resonances and the bandwidth of the primary resonance increase accordingly, and the positions of both the primary and subharmonic Bragg resonances downward shift to low frequency more significantly, but the number of the zero reflection keeps the same.

On a degenerate boundary value problem to the two-dimensional self-similar nonlinear wave system

This paper focuses on a degenerate boundary value problem arising from the study of the two-dimensional Riemann problem to the nonlinear wave system. In order to deal with the parabolic degeneracy, we introduce a partial hodograph transformation to transform the nonlinear wave system into a new system, which displays a clear regularity–singularity structure. The local existence of classical solutions for the new system is established in a weighted metric space. Returning the solution to the original variables, we obtain the existence of classical solutions to the degenerate boundary value problem for the nonlinear wave system.

Poisson metrics on flat vector bundles over non-compact curves

Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian-Yang-Mills equation for holomorphic vector bundles on K3 surfaces in the large complex structure limit. We define a notion of slope stability, and show that if the flat connection D has regular singularities, and the Riemannian metric g has finite volume then E admits a Poisson metric with asymptotics determined by the parabolic structure if and only if (E,D,P) is slope polystable.

On the Regular Integral Solutions of a Generalized Bessel Differential Equation

The original Bessel differential equation that describes, among many others, cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalized Bessel differential equation that describes coupled acoustic-vortical waves. The solutions of the generalized Bessel differential equation are obtained for all possible combinations of the two complex parameters, order and degree, and finite complex variable, as Frobenius-Fuchs series around the regular singularity at the origin; the series converge in the whole complex plane of the variable, except for the point-at-infinity, that is, the only other singularity of the differential equation. The regular integral solutions of the first and second kinds lead, respectively, to the generalized Bessel and Neumann functions; these reduce to the original Bessel and Neumann functions for zero degree and have alternative expressions for nonzero degree.

A Proof of the Gaudin Bethe Ansatz Conjecture

Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe Ansatz conjecture in the Feigin-Frenkel form which states that the joint eigenvalues of the higher Gaudin Hamiltonians on the tensor product of irreducible finite-dimensional $\mathfrak{g}$-modules are in 1-1 correspondence with monodromy-free ${}^LG$-opers on the projective line with regular singularities at the points $z_1,\ldots,z_N,\infty$ and the prescribed residues at the singular points.

Generalized confluent hypergeometric solutions of the Heun confluent equation

We show that the Heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions. For each of these solutions a characteristic exponent of a regular singularity of the Heun confluent equation is a non-zero integer and the accessory parameter obeys a polynomial equation. Each of the solutions can be written as a linear combination with constant coefficients of a finite number of either the Kummer confluent hypergeometric functions or the Bessel functions.

On q-Deformations of the Heun Equation

The $q$-Heun equation and its variants arise as degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the $q$-Heun equation by using analysis of regular singularities. We also consider the quasi-exact solvability of the $q$-Heun equation and its variants. Namely we investigate finite-dimensional subspaces which are invariant under the action of the $q$-Heun operator or variants of the $q$-Heun operator.