Regular Singularities Research Articles

An Improved Methodology for Data Hiding In Images Using Haar Transformed, LSB Replacement Method and Modified PVDMF 1

Steganography is a technique of concealing secret text or image within another image.In this paper, abetterand efficient steganography methodfor hiding imagesand/or text into a color image is proposed. The proposed techniqueis based on two irreversible methods of spatial domain Least Significant Bit (LSB) and Pixel Value Differencing and Modulus Function (PVDMF 1) and one Haar Wavelet Transformation of the frequency domain. The proposed method works in four stages: at the first stage, Haar Wavelet Transformation is applied on the input image and transformed Haar coefficients are obtained. At the second stage, Haar coefficients and secret image are converted into binary form, before applying the LSB replacement method on the green channel by replacing the first two Most Significant Bits (MSB) of the input image withthe last two LSB bits of the secret image. Inthe third stage, modified PVDMF 1 is applied on intermediate stego imagesafter hiding the image by dividing the image into two non-overlapping consecutive pixels of pixels for hiding text. Inthe fourth stage, Inverse Haar Wavelet Transformation is applied to get the stego image. The proposed method offers high hiding capabilitywith goodvisual quality. The visual quality of the stego image is measured by using three widely used objective quality assessment matrices PSNR, RMSE,and SSIM. Furthermore, Regular/Singular (RS) analysis is applied on stego imagesto check theirresistanceagainst RS attack. The hiding capacity and quality assessment results are compared with some existing methods. It is evidentfrom the comparison that the proposed processis superior in termsof visual quality and hiding capacity.

Regular and complex singularities of the generalized thin film equation in two dimensions

Abstract

The Beauville-Narasimhan-Ramanan correspondence for twisted Higgs 𝑉-bundles and components of parabolic 𝑆𝑝(2𝑛,ℝ)-Higgs moduli spaces

We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs V V -bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic S p ( 2 n , R ) \mathrm {Sp}\left ( 2n,\mathbb {R} \right ) -Higgs bundles with fixed parabolic structure.

Multi-directional block based PVD and modulus function image steganography to avoid FOBP and IEP

Since the inception of pixel value differencing (PVD) image steganography, it has drawn considerable interest among the researchers of this field. However, most of the PVD based techniques suffer from either falling-off boundary problem (FOBP) or incorrect extraction problem (IEP). Therefore, to address these two issues, this paper proposes a multi-directional pixel value differencing and modulus function (MDPVDMF) based technique. During the embedding process, the original image (OI) is partitioned into 2 × 2 size pixel blocks. Then, data embedding is performed by exploiting the horizontal, vertical, and diagonal directions for each block. For a 2 × 2 pixel block, two difference values can be obtained in any of the three directions. Next, using the difference values and the remainders of the pixel pairs, the secret bits are embedded. The experiment has been conducted to compute the performance of the proposed technique with regards to the image quality metrics like peak signal-to-noise ratio (PSNR), embedding capacity (EC), and FOBP. Results show that PSNR is optimal for vertical pairs with 39.17 dB whereas the EC is optimal for the diagonal pairs with 3.10 bits per pixel (BPP). Further, the proposed technique has shown exceptional attack resistance ability to regular & singular (RS) attack, salt & pepper (S&P) noise, pixel difference histogram (PDH) analysis, and subtractive pixel adjacency matrix (SPAM) steganalysis.

Chiral algebra of the Argyres-Douglas theory from M5 branes

We study chiral algebras associated with Argyres-Douglas theories engineered from M5 branes. For the theory engineered using 6D (2,0) type $J$ theory on a sphere with a single irregular singularity (without mass parameter), its chiral algebra is the minimal model of W algebra of $J$ type. For the theory engineered using an irregular singularity and a regular full singularity, its chiral algebra is the affine Kac-Moody algebra of $J$ type. We can obtain the Schur index of these theories by computing the vacua character of the corresponding chiral algebra.

Retracing extended sudoku matrix for high-capacity image steganography

Numerous data hiding algorithms have been devised for imperceptibly embedding secret messages into the cover media so as to securely transmit user privacy information over the public communication channels. Among them, the reference matrix-based schemes in spatial domain draw extensive concern on account of the simple but efficient embedding and extraction procedure. The core idea of the planar matrix-based method is let two contiguous cover pixels conceal a base-N secret digit with the guidance of reference matrix, such as Sudoku, Turtle Shell, and so on. A novel image data hiding scheme with great embedding efficiency is presented in this paper, which is based on retracing extended Sudoku (RE-Sudoku) reference matrix. Owing to the number of Sudoku solutions and various extension directions, viz., the multiformity of RE-Sudoku matrix, this scheme is exceedingly more secure than the previous methods. More momentously, the proposed two-dimensional reference matrix can guide two 9-ary notational system secret digits to be embedded into each cover pixel pair simultaneously; thereby resulting in a larger embedding rate which converges to 3.169 bits per pixel. The experimental results reveal that this image steganography outperforms the other related works in terms of hiding capacity while maintaining a desirable image quality around 40 dB. Furthermore, the security of our proposed scheme is verified by demonstrating its resistance to the pixel-value difference histogram (PDH) and regular/singular (RS) steganalysis.

An Efficient Dual-image Reversible Data Hiding Scheme Based on Exploiting Modification Direction

Reversible data hiding (RDH) subtly alter the cover image for embedding the secret data, after extracting the secret data, the cover image can also be completely restored. In this work, we propose a novel RDH scheme based on Exploiting Modification Direction (EMD) with dual images. The characteristics of EMD matrix is exploited, one secret bit and a base-5 digit can be concealed in a cover pixel to generate a pixel pair which will be assigned to two stego images. Without any overhead message, the reversibility can be achieved. In the experiment, the proposed scheme outperforms some state-of-the-art methods in terms of measuring in embedding ratio (ER) or peak-signal-to-noise ratio (PSNR). In addition, the proposed scheme has good performance on resisting static attacks on pixel-value differencing histogram (PDH) and regular singular (RS) analysis.

Supporting data: Regular and complex singularities of the generalised thin film equation in two dimensions

Supporting data: Regular and complex singularities of the generalised thin film equation in two dimensions

Gravitational Measurements in Higher Dimensions

We attempt to study three significant tests of general relativity in higher dimensions, both in commutative and non-commutative spaces. In the context of non-commutative geometry, we will consider a solution of Einstein’s equation in higher dimensions, with a source given by a static, spherically symmetric Gaussian distribution of mass. The resulting metric would describe a regular or curvature singularity free black hole in higher dimensions. The metric should smoothly interpolate between Schwarzschild geometry at large distance, and de-Sitter spacetime at short distance. We will consider gravitational redshift, lensing, and time delay in each sector. It will be shown that, compared to the four-dimensional spacetime, there can be significant modifications due to the presence of extra dimensions and the non-commutative corrected black holes. Finally, we shall attempt to obtain a lower bound on the size of the extra dimensions and on the mass needed to form a black hole in different dimensions.

Analysis of third order linear analytic differential equations with a regular singularity: Bessel and other classical equations

Analysis of third order linear analytic differential equations with a regular singularity: Bessel and other classical equations

Gamma functions, monodromy and Frobenius constants

In their paper on the gamma conjecture in mirror symmetry, Golyshev and Zagier introduce what we refer to as Frobenius constants associated to an ordinary linear differential operator L with a reflection type singularity. These numbers describe the variation around the reflection point of Frobenius solutions to L defined near other singular points. Golyshev and Zagier show that in certain geometric cases Frobenius constants are periods, and they raise the question quite generally how to describe these numbers motivically. In this paper we give a relation between Frobenius constants and Taylor coefficients of generalized gamma functions, from which it follows that Frobenius constants of Picard--Fuchs differential operators are periods. We also study the relation between these constants and periods of limiting Hodge structures. This is a major revision of the previous version of the manuscript. The notion of Frobenius constants and our main result are extended to the general case of regular singularities with any sets of local exponents. In addition, the generating function of Frobenius constants is given explicitly for all hypergeometric connections.

Riemann-Hilbert correspondence and blown up surface defects

The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsian systems has a long history. Recently four-dimensional N = 2 gauge theories joined the party in a multitude of roles. In this paper we study the vacuum expectation values of intersecting half-BPS surface defects in SU(2) theory with Nf = 4 fundamental hypermultiplets. We show they form a horizontal section of a Fuchsian system on a sphere with 5 regular singularities, calculate the monodromy, and define the associated isomonodromic tau-function. Using the blowup formula in the presence of half-BPS surface defects, initiated in the companion paper, we obtain the GIL formula, establishing an unexpected relation of the topological string/free fermion regime of supersymmetric gauge theory to classical integrability.

$$({{\,\mathrm{\mathrm {SL}}\,}}(N),q)$$-Opers, the q-Langlands Correspondence, and Quantum/Classical Duality

A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers - connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for $SL(N)$. We introduce a difference equation version of opers called $q$-opers and prove a $q$-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted $q$-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the $q$-Langlands correspondence. We also describe an application of $q$-opers to the equivariant quantum $K$-theory of the cotangent bundles to partial flag varieties.

A geometric-analytic study of linear differential equations of order two

<p style='text-indent:20px;'>We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational Riccati differential equation and therefore a <i>global holonomy group</i>. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.</p>

Exact WKB and Abelianization for the $$T_3$$ Equation

We describe the exact WKB method from the point of view of abelianization, both for Schr\"odinger operators and for their higher-order analogues (opers). The main new example which we consider is the "$T_3$ equation," an order $3$ equation on the thrice-punctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate system on a moduli space of flat $\mathrm{SL}(3)$-connections. We give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the $T_3$ equation has the expected asymptotic properties. We also briefly revisit the Schr\"odinger equation with cubic potential and the Mathieu equation from the point of view of abelianization.

Regularized Trace of a Sturm–Liouville Operator on a Curve with a Regular Singularity on the Chord

For a Sturm–Liouville operator on a piecewise smooth curve, we study the effect that the spectrum of a nonintegrable singularity of the potential on the segment joining the endpoints of this curve has on the operator asymptotics. It is shown that in the case where the singular point does not give rise to the branching of solutions in its neighborhood (the case of trivial monodromy), the spectral asymptotics and the formula for the regularized trace look the same as for the classical Sturm–Liouville operator on a segment with smooth potential. Further, it is shown that in the case of nontrivial monodromy the spectral asymptotics substantially depends on the commensurability of the parts into which the segment is partitioned by the singular point $$\theta $$ : if $$\theta $$ is rational, then the spectrum is divided into finitely many series, each going to infinity along “its own” parabola. In this case, the regularized trace formula is significantly more complicated and does not show any similarity with the classical formula.

Erratum to: Series solution of a ten-parameter second-order differential equation with three regular singularities and one irregular singularity

Erratum to: Series solution of a ten-parameter second-order differential equation with three regular singularities and one irregular singularity

How to smooth a crinkled map of space-time: Uhlenbeck compactness for L ∞ connections and optimal regularity for general relativistic shock waves by the Reintjes-Temple equations.

We present the authors' new theory of the RT-equations ('regularity transformation' or 'Reintjes-Temple' equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem(Γ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in L p spaces. The theory and results announced in this paper apply to arbitrary L ∞ connections on the tangent bundle of arbitrary manifolds , including Lorentzian manifolds of general relativity.

TBA equations for the Schrödinger equation with a regular singularity

We derive the thermodynamic Bethe ansatz (TBA) equations for the Schrödinger equation with an arbitrary polynomial potential and a regular singular (simple and double pole) term. The TBA equations provide a non-trivial generalization of the ODE/IM correspondence and also give a solution for the Riemann–Hilbert problem in the exact WKB method. We study the TBA equations in detail for the linear and the harmonic oscillator potentials together with inverse and centrifugal terms. As an application, we also compute numerically the Voros spectrum for these potentials using the Bohr–Sommerfeld quantization condition.

Rigid connections on P1 via the Bruhat–Tits building

We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid $G$-connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy-Prasad filtration associated to a point in the Bruhat-Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus $S$, then we have an $S$-toral connection. In this language, the irregular singularity of the Frenkel-Gross connection gives rise to the homogenous toral connection of minimal slope associated to the Coxeter torus $\mathcal{C}$. In the present paper, we consider connections on $\mathbb{G}_m$ which have an irregular homogeneous $\mathcal{C}$-toral singularity at zero of slope $i/h$, where $h$ is the Coxeter number and $i$ is a positive integer coprime to $h$, and a regular singularity at infinity with unipotent monodromy. Our main result is the characterisation of all such connections which are rigid.