Mellin Transform Research Articles

Integrating the full four-loop negative geometries and all-loop ladder-type negative geometries in ABJM theory

The decomposition of the four-point ABJM amplituhedron into negative geometries produces compact integrands of logarithmic of amplitudes such that the infrared divergence only comes from the last loop integration, from which we can compute the cusp anomalous dimension of the ABJM theory. In this note, we integrate L – 1 loop momenta of the L-loop negative geometries for all four-loop negative geometries and a special class of all-loop ladder-type negative geometries by a method based on Mellin transformation, and from these finite quantities we extract the corresponding contribution to the cusp anomalous dimension. We find that the infrared divergence of a box-type negative geometry at L = 4 is weaker than other negative geometries, then only tree-type negative geometries contribute to the cusp anomalous dimension at L = 4. For the all-loop ladder-type negative geometries, we prove and conjecture some recursive structures as integral equations in Mellin space and find that they cannot contribute zeta values like ζ3, ζ5 to the cusp anomalous dimension.

Holography in flat spacetimes: the case for Carroll

We compare and contrast the two approaches of holography in asymptotically flat spacetimes, viz. the co-dimension two Celestial approach based on the Mellin transformation and the co-dimension one Carrollian approach based on the modified Mellin and elucidate how some of the problems of the Celestial approach can be rectified by the Carrollian one. Considering flat holography as a limit from AdS/CFT makes a co-dimension one dual more plausible, and our previous construction of Carrollian correlations from AdS Witten diagrams is testimony to this. In this paper, we show how to generalize our earlier analysis for operators with spin. We work out a large number of explicit non-trivial examples (twelve) and show matching between the limit of AdS4 Witten diagrams and 3d boundary symmetry considerations, thus making the case for the Carrollian dual even stronger.

Celestial leaf amplitudes

Celestial amplitudes may be decomposed as weighted integrals of AdS3-Witten diagrams associated to each leaf of a hyperbolic foliation of spacetime. We show, for the Kleinian three-point MHV amplitude, that each leaf subamplitude is smooth except for the expected light-cone singularities. Moreover, we find that the full translationally-invariant celestial amplitude is simply the residue of the pole in the leaf amplitude at the point where the total conformal weights of the gluons equals three. This full celestial amplitude vanishes up to light-cone contact terms, as required by spacetime translation invariance, and reduces to the expression previously derived by Mellin transformation of the Parke-Taylor formula.

Solution Strategy and Associated Results for Fuzzy Mellin Transformation

The advancement of the Mellin transformation is that it exhibits a scale-invariant nature, and thus, it is widely used in computer science. Fuzzy Mellin transformation can provide the purposes of such problems in uncertain phenomena. The existing studies on fuzzy Mellin transformation consider the fuzzy level cut approach, which makes the fuzzy valued functions into their crisp couplets regarding lower and uppercuts. This paper reconstructs a distinct mathematical frame for the Mellin transformation in a completely fuzzy environment in the sense of a modified Hukuhara derivative. In this paper, we mean the complete fuzzy environment to describe fuzzy phenomena where the conversion of the fuzzy function to its crisp counterpart is annulled. The proposed fuzzy Mellin transformation’s superiority over existing approaches is that it can be used directly to fuzzy valued functions without converting them to their crisp counter version. The challenges to dealing with fuzzy valued mappings and inputs without its level cut representation have been tackled in this manuscript. Furthermore, numerical examples, possible applications, and further extensions of this work are hinted at in this paper.

Convolutional double copy in (anti) de Sitter space

The double copy is a remarkable relationship between gauge theory and gravity that has been explored in a number of contexts, most notably scattering amplitudes and classical solutions. The convolutional double copy provides a straightforward method to bridge the two theories via a precise map for the fields and symmetries at the linearised level. This method has been thoroughly investigated in flat space, offering a comprehensive dictionary both with and without fixing the gauge degrees of freedom. In this paper, we extend this to curved space with an (anti) de Sitter background metric. We work in the temporal gauge, and employ a modified convolution that involves the Mellin transformation in the time direction. As an example, we show that the point-like charge in gauge theory double copies to the (dS-) Schwarzschild black hole solution.

Generalized Limit Theorem for Mellin Transform of the Riemann Zeta-Function

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z(s), s=σ+it, with fixed 1/2<σ<1, of the square |ζ(1/2+it)|2 of the Riemann zeta-function. We consider probability measures defined by means of Z(σ+iφ(t)), where φ(t), t⩾t0>0, is an increasing to +∞ differentiable function with monotonically decreasing derivative φ′(t) satisfying a certain normalizing estimate related to the mean square of the function Z(σ+iφ(t)). This allows us to extend the distribution laws for Z(s).

Exact Solutions for a Class of Variable Coefficients Fractional Differential Equations Using Mellin Transform and the Invariant Subspace Method

AbstractIn this paper, we propose a class of variable coefficients fractional ordinary differential equations (FODEs). Using Mellin transform (MT), we have transformed this class into a functional equation which can’t be solved in general. So, we have selected many special cases of this functional equation that can be solved exactly. After solving these special cases of the functional equation and using the inverse MT, we obtained some exact solutions for the proposed class. The obtained solutions are given in the form of $$H$$ H -function and the Wright function. The results, as special cases, contain some special forms given in the literature. Also, the invariant subspace method (ISM) is utilized for solving a class of nonlinear fractional diffusion equations with variable coefficients. The solutions of this class of nonlinear fractional diffusion equations depend upon the solutions of the proposed class of FODEs.

Goldstone bosons on celestial sphere and conformal soft theorems

In this paper, we study celestial amplitudes of Goldstone bosons and conformal soft theorems. Motivated by the success of soft bootstrap in momentum space and the important role of the soft limit behavior of tree-level amplitudes, our goal is to extend some of the methods to the celestial sphere. The crucial ingredient of the calculation is the Mellin transformation, which transforms four-dimensional scattering amplitudes to correlation functions of primary operators in the celestial CFT. The soft behavior of the amplitude is then translated to the singularities of the correlator. Only for amplitudes in “UV completed theories” (with sufficiently good high energy behavior) the Mellin integration can be properly performed. In all other cases, the celestial amplitude is only defined in a distributional sense with delta functions. We provide many examples of celestial amplitudes in UV-completed models, including linear sigma models and Z-theory, which is a certain completion of the SU(N) non-linear sigma model. We also comment on the BCFW-like and soft recursion relations for celestial amplitudes and the extension of soft bootstrap ideas.

A KLT-like construction for multi-Regge amplitudes

Inspired by the calculational steps originally performed by Kawai, Lewellen and Tye, we decompose scattering amplitudes with single-valued coefficients obtained in the multi-Regge-limit of N=4 super-Yang–Mills theory into products of scattering amplitudes with multi-valued coefficients. We consider the simplest non-trivial situation: the six-point remainder function complementing the Bern–Dixon–Smirnov ansatz for multi-loop amplitudes.Utilizing inverse Mellin transformations, all single-valued amplitude components can indeed be decomposed into multi-valued amplitude components. Although the final expression is very similar in structure to the Kawai–Lewellen–Tye construction, moving away from the highly symmetric string scenario comes with several imponderabilities, some of which become more pronounced when considering more than six external legs in the remainder function.

The Fractional Dunkl Laplacian: Definition and Harmonization via the Mellin Transform

In this paper, we extend the scope of the Tate and Ormerod Lemmas to the Dunkl setting, revealing a profound interconnection that intricately links the Dunkl transform and the Mellin transform. This illumination underscores the pivotal significance of the Mellin integral transform in the realm of fractional calculus associated with differential-difference operators. Our primary focus centers on the Dunkl–Laplace operator, which serves as a prototype of a differential-difference second-order operator within an unbounded domain. Following influential research by Pagnini and Runfola, we embark on an innovative exploration employing Bochner subordination approaches tailored for the fractional Dunkl Laplacian (FDL). Notably, the Mellin transform emerges as a robust and enlightening tool, particularly in its application to the FDL.

Classification of Skin Lesion Images Using Artificial Intelligence Methodologies through Radial Fourier–Mellin and Hilbert Transform Signatures

This manuscript proposes the possibility of concatenated signatures (instead of images) obtained from different integral transforms, such as Fourier, Mellin, and Hilbert, to classify skin lesions. Eight lesions were analyzed using some algorithms of artificial intelligence: basal cell carcinoma (BCC), squamous cell carcinoma (SCC), melanoma (MEL), actinic keratosis (AK), benign keratosis (BKL), dermatofibromas (DF), melanocytic nevi (NV), and vascular lesions (VASCs). Eleven artificial intelligence models were applied so that eight skin lesions could be classified by analyzing the signatures of each lesion. The database was randomly divided into 80% and 20% for the training and test dataset images, respectively. The metrics that are reported are accuracy, sensitivity, specificity, and precision. Each process was repeated 30 times to avoid bias, according to the central limit theorem in this work, and the averages and ± standard deviations were reported for each metric. Although all the results were very satisfactory, the highest average score for the eight lesions analyzed was obtained using the subspace k-NN model, where the test metrics were 99.98% accuracy, 99.96% sensitivity, 99.99% specificity, and 99.95% precision.

Robust watermarking method based on the Analytical Clifford Fourier Mellin Transform

Robust watermarking method based on the Analytical Clifford Fourier Mellin Transform

On Approximation by an Absolutely Convergent Integral Related to the Mellin Transform

In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical zero in the strip {s∈C:1/2<σ<1}.

Fractional $$(p,q)$$-Mellin Transform and Its Applications

Fractional $$(p,q)$$-Mellin Transform and Its Applications

A modified semi-weight function method for SIF calculation of arbitrary inclined interface crack with both single complex-root and double real-root singularities

An arbitrary inclined crack terminated at the interface usually has singularity of single complex-root or double real-root at crack tip. Currently, most methods of the stress intensity factor (SIF) calculation (e.g., Mellin transformation method, enrichment strategy method) only consider the single complex-root singularity, with difficulty in solving auxiliary functions. Although the scaled boundary method takes the double real-root singularity into account, it needs to set special finite elements at crack tip. In this study, based on the sub-matrix method and the reciprocal theorem of work, a modified semi-weight function method is proposed to calculate SIF of the arbitrary interface crack with consideration of both single complex-root and double real-root singularity, and also analyze the effect of bimaterial parameters (Gi), crack inclination angle (w) and relative length (a/L) on the SIFs. This modified method has the advantages of uniform auxiliary function (independent of crack size and external load) and simple finite element calculation (without special singularity element). Moreover, its validity is verified by good agreement of our calculation results with the available literature results and finite element results. The research results can provide a theoretical basis for anti-crack design of multi-layer caverns of the compressed air energy storage (CAES).

Construction of highly non linear component of block cipher based on mclaurin series and mellin transformation with application in image encryption

Construction of highly non linear component of block cipher based on mclaurin series and mellin transformation with application in image encryption

New generalized Mellin transform and applications to partial and fractional differential equations

Abstract In this paper, we introduce a generalized Mellin transformation in a general form that encompasses the generalized Mellin transformations found in the literature. Then, we give the fundamental properties of this new integral transformation and apply it to some elementary functions. Furthermore, we obtain the solutions of partial and fractional differential equations by means of this new integral transformation. Finally, we examine the relations between generalized Mellin transformations in the literature and the new generalized Mellin transformation and present a table showing the new integral transformations of functions commonly used in mathematics, physics and engineering applications.

On the Approximation by Mellin Transform of the Riemann Zeta-Function

This paper is devoted to the approximation of a certain class of analytic functions by shifts Z(s+iτ), τ∈R, of the modified Mellin transform Z(s) of the square of the Riemann zeta-function ζ(1/2+it). More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z(s+iτ), which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure.

On the Discrete Approximation by the Mellin Transform of the Riemann Zeta-Function

In the paper, it is obtained that there are infinite discrete shifts Ξ(s+ikh), h>0, k∈N0 of the Mellin transform Ξ(s) of the square of the Riemann zeta-function, approximating a certain class of analytic functions. For the proof, a probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied.

CRYPTOGRAPHIC METHOD TO ENHANCE DATA SECURITY USING RSA ALGORITHM AND MELLIN TRANSFORM

Cryptography is the technique of using mathematical algorithms to encrypt and decrypt the information. The process of converting plaintext to ciphertext is known as encryption, whereas the process of converting ciphertext to plaintext is known as decryption. Encryption and decryption methods based on Mellin Transform are unable to provide more security while transmitting the information. RSA algorithm is an Asymmetric key cryptography algorithm. The purpose of this study is to present a cryptographic method that uses the RSA algorithm and Mellin Transform to improve communication security.