Department of Mathematics

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This paper deals with the renormalization of symmetric bimodal maps with low smoothness. We prove the existence of the renormalization fixed point in the space C1+Lip symmetric bimodal maps. Moreover, we show that the topological entropy of the renormalization operator defined on the space of C1+Lip symmetric bimodal maps is infinite. Further we prove the existence of a continuum of fixed points of renormalization. Consequently, this proves the non-rigidity of the renormalization of symmetric bimodal maps. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

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Journal of Statistical Physics

Renormalization of Symmetric Bimodal Maps with Low Smoothness

The role of a floating elastic plate on the hydrodynamic instability of a gravity-driven flow down an inclined plane is examined in the linear and nonlinear regimes. Linear instability of the system with respect to infinitesimal disturbances is captured using normal-mode analysis. The critical conditions for instability are obtained analytically utilizing the small film aspect ratio. The bifurcation of the nonlinear evolution equation is analyzed using weakly nonlinear stability analysis. The time evolution of the surface elevation is analyzed using the nonlinear analysis. The Orr–Sommerfeld boundary value problem corresponding to the perturbed flow is derived, and it is solved numerically using the spectral collocation method. The behavior of the marginal stability curves and temporal growth of the unstable waves are portrayed for a range of dimensionless flow parameters. Moreover, the pressure acting on the surface are calculated and analyzed for various structural parameters, applicable to different flow configurations. The study reveals that the structural parameters such as rigidity and mass per unit length play a crucial role in suppressing and facilitating the unstable surface waves of the flow. However, the compressive force acting on the plate results in the flexural destabilization. Thus, the plate parameters are more efficient in damping the shorter wave disturbances. Numerical observations imply that the floating elastic plate assists in stabilizing the free-falling flow and dampens the high amplitude waves. © 2021

Wave Motion

Hydroelastic response of a floating plate on the falling film: A stability analysis

The authentication and content protection of multimedia data is a challenging task in the present scenario. One solution is to generate the perceptual hash which essentially authenticates the multimedia data and can also be dealt with image database search problems. In this paper, a novel system for generating an image hash is presented. The proposed system utilizes the global and local features in the hash generation process. The local features are obtained from non-linear scale-space based KAZE features. These KAZE features have the ability to capture the most stable point under several content preserving distortions. In contrast, first the input image is converted to a normalized image, it is transformed to the log-polar coordinate system, and then a reference image using the local contrast in the wavelet domain is obtained to extrcat the global features. An intermediate hash sequence is achieved by extracting the significant information from the reference image using the singular value decomposition. The final hash sequence combines both the vectors followed by the randomization process. Extensive experiments are conducted to demonstrate the feasibility and the robustness of the proposed hashing system against a wide range of intentional/unintentional distortions. Further, the comparative analysis with some state-of-the-art techniques validates the better discrimination of the proposed work. IEEE

IEEE Transactions on Dependable and Secure Computing

A New Robust Reference Image Hashing System

In this article, a blockwise regression-based image inpainting framework is proposed. The core idea is to fill the unknown region in two stages: Extrapolate the edges to the unknown region and then fill the unknown pixels values in each sub-region demarcated by the extended edges. Canny edge detection and linear edge extension are used to respectively identify and extend edges to the unknown region followed by regression within each sub-region to predict the unknown pixel values. Two different regression models based on K-nearest neighbours and support vectors machine are used to predict the unknown pixel values. The proposed framework has the advantage of inpainting without requiring prior training on any image dataset. The extensive experiments on different images with contrasting distortions demonstrate the robustness of the proposed framework and a detailed comparative analysis shows that the proposed technique outperforms existing state-of-the-art image inpainting methods. Finally, the proposed techniques are applied to MRI images suffering from susceptibility artifacts to illustrate the practical usage of the proposed work. © 2021 Association for Computing Machinery.

ACM Transactions on Internet Technology

A Novel Image Inpainting Framework Using Regression

In this paper, a prey-predator model in polluted environment with disease in prey has been proposed and studied. It is assumed that only prey population is prone to disease whereas, both the populations are affected by the pollutant. Boundedness of the solution of the system is discussed. Existence of all possible equilibrium points has been established. Using Routh Hurwitz criterion, local stability of all the possible equilibrium points has been obtained. Also, interior equilibrium point has been proved to be globally asymptotically stable using Lyapunov function. Then time delay has been introduced in the system making the model more realistic. Existence and direction of Hopf bifurcation in the delay model has been established using normal form theory and center manifold theorem. By taking a set of hypothetical and biologically feasible parameters, model has been studied numerically using MATLAB and the effect of pollutant on the system has been deduced. © 2021 the author(s).

Communications in Mathematical Biology and Neuroscience

Dynamical analysis of polluted prey-predator system with infected prey

In this article, we discuss the piecewise polynomial based Galerkin method to approximate the solutions of second kind Volterra-Hammerstein integral equations. We discuss the convergence of the approximate solutions to the exact solutions and obtain the orders of convergence O(hr) and O(h2r), respectively, for Galerkin and its iterated Galerkin methods in uniform norm, where h, r denotes the norm of the partition and smoothness of the kernel, respectively. We also obtain the superconvergence results for multi-Galerkin and iterated multi-Galerkin methods. We show that iterated multi-Galerkin method has the order of convergence O(h3r) in the uniform norm. Numerical results are provided to demonstrate the theoretical results. © 2021, Semnan University, Center of Excellence in Nonlinear Analysis and Applications. All rights reserved.

International Journal of Nonlinear Analysis and Applications

Projection and multi-projection methods for second kind volterra-hammerstein integral equation

In this article, the concept of the single-objective sequential quadratically constrained quadratic programming method is extended to the multiobjective case and a new line search technique is developed for nonlinear multiobjective optimization problems. The proposed method ensures global convergence as well as spreading of the Pareto front. A descent direction is obtained by solving a quadratically constrained quadratic programming subproblem. A nondifferentiable penalty function is used to restrict the constraint violations. Convergence of the descent sequence is established under the Mangasarian-Fromovitz constraint qualification and some mild assumptions. In addition to this, a new technique is designed for selecting initial points to ensure the spreading of the Pareto front. The method is compared with existing methods using a set of test problems. © 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.

SIAM Journal on Optimization

A globally convergent sqcqp method for multiobjective optimization problems

In this article, we deal with the existence of non-negative solutions of the class of following non local problem {−M(∫Rn∫Rn|u(x)−u(y)|ns|x−y|2ndxdy)(−Δ)n/ssu=(∫ΩG(y,u)|x−y|μdy)g(x,u)inΩ,u=0inRn∖Ω, where (−Δ)n/ss is the n/ s-fractional Laplace operator, n≥ 1 , s∈ (0 , 1) such that n/ s≥ 2 , Ω ⊂ Rn is a bounded domain with Lipschitz boundary, M: R+→ R+ and g: Ω × R→ R are continuous functions, where g behaves like exp(|u|nn−s) as | u| → ∞. The key feature of this article is the presence of Kirchhoff model along with convolution type nonlinearity having exponential growth which appears in several physical and biological models. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.}, author_keywords={Choquard nonlinearity; Doubly non local problems; Kirchhoff equation; Trudinger-Moser nonlinearity}, keywords={Applications; Mathematical techniques, Biological models; Continuous functions; Exponential growth; Fractional Laplacian; Kirchhoff equation; Lipschitz boundary; Non-negative solutions; Nonlocal problems, Laplace transforms}, funding_details={Department of Science and Technology, Ministry of Science and Technology, IndiaDepartment of Science and Technology, Ministry of Science and Technology, India, डीएसटी, ECR/2017/002651}, funding_details={Science and Engineering Research BoardScience and Engineering Research Board, SERB}, funding_text_1={This research is supported by Science and Engineering Research Board, Department of Science and Technology, Government of India, Grant number: ECR/2017/002651. The second author wants to thank Bennett University for its hospitality during her visit there.}, funding_text_2={This research is supported by Science and Engineering Research Board, Department of Science and Technology, Government of India, Grant number: ECR/2017/002651. The second author wants to thank Bennett University for its hospitality during her visit there.}, references={Alves, C.O., Yang, M., Existence of solutions for a nonlocal variational problem in R2 with exponential critical growth (2017) J. Convex Anal., 24 (4), pp. 1197-1215; Alves, C.O., Cassani, D., Tarsi, C., Yang, M., Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in R2 (2016) J. Differ. Equ., 261 (3), pp. 1933-1972; Applebaum, D., Lévy processes—from probability to finance and quantum groups (2004) Not. Am. Math. Soc., 51 (11), pp. 1336-1347; Arora, R., Giacomoni, J., Mukherjee, T., Sreenadh, K., n -Kirchhoff-Choquard equations with exponential nonlinearity (2019) Nonlinear Anal., 108, pp. 113-144; Brasco, L., Lindgren, E., Parini, E., The fractional Cheeger problems (2014) Interfaces Free Bound., 16, pp. 419-458; Brasco, L., Parini, E., Squassina, M., Stability of variational eigenvalues for the fractional p -Laplacian (2016) Discrete Contin. Dyn. Syst., 36, pp. 439-455; Brezis, H., (2011) Functional Analysis, Sobolev Spaces and Partial Differential Equations, , Universitext, Springer, New York; Caffarelli, L., Non-local diffusions, drifts and games (2012) Nonlinear Partial Differential Equations, 7, pp. 37-52. , Abel Symp., Springer, Heidelberg; Caffarelli, L., Silvestre, L., An extension problem related to the fractional Laplacian (2007) Commun. Partial Differ. Equ., 32 (7-9), pp. 1245-1260; Di Nezza, E., Palatucci, G., Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces (2012) Bull. Sci. Math., 136 (5), pp. 521-573; Giacomoni, J., Mishra, P.K., Sreenadh, K., Fractional elliptic equations with critical exponential nonlinearity (2016) Adv. Nonlinear Anal., 5 (1), pp. 57-74; Giacomoni, J., Mishra, P.K., Sreenadh, K., Fractional Kirchhoff equation with critical exponential nonlinearity (2016) Complex Var. Elliptic Equ., 61 (9), pp. 1241-1266; Goyal, S., Sreenadh, K., Nehari manifold for non-local elliptic operator with concave-convex nonlinearities and sign-changing weight functions (2015) Proc. Indian Acad. Sci. Math. Sci., 125 (4), pp. 545-558; Kirchhoff, G., (1876) Mechanik. Vorlesungen über mathematische Physik, , Teubner, Leipzig; Li, F., Gao, C., Zhu, X., Existence and concentration of sign-changing solutions to Kirchhoff-type system with Hartree-type nonlinearity (2017) J. Math. Anal. Appl., 448, pp. 60-80; Lieb, E.H., Existence and uniqueness of the minimizing solution of Choquard nonlinear equation (1976) Stud. Appl. Math, 57, pp. 93-105; Lieb, E.H., Loss, M., (2001) Analysis, 14. , 2, Graduate Studies Mathematics, Am. Math. Soc., Providence; Lü, D., A note on Kirchhoff-type equations with Hartree-type nonlinearities (2014) Nonlinear Anal., 99, pp. 35-48; Martinazzi, L., Fractional Adams-Moser-Trudinger type inequalities (2015) Nonlinear Anal., 127, pp. 263-278; Moroz, V., Schaftingen, J.V., A guide to the Choquard equation (2017) J. Fixed Point Theory Appl., 19 (1), pp. 773-813; Parini, E., Ruf, B., On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces (2018) Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl., 29 (2), pp. 315-319; Pekar, S., (1954) Untersuchung über die Elektronentheorie der Kristalle, , Akademie Verlag, Berlin; Perera, K., Squassina, M., Yang, Y., Bifurcation and multiplicity results for critical fractional p -Laplacian problems (2016) Math. Nachr., 289 (2-3), pp. 332-342; Pucci, P., Xiang, M., Zhang, B., Existence results for Schödinger-Choquard-Kirchhoff equations involving the fractional p -Laplacian (2019) Adv. Calc. Var., 12 (3), pp. 253-275; Servadei, R., Valdinoci, E., Mountain pass solutions for non-local elliptic operators (2012) J. Math. Anal. Appl., 389 (2), pp. 887-898; Xiang, M., Zhang, B., Repovs, D., Existence and multiplicity of solutions for fractional Schrödinger-Kirchhoff equations with Trudinger-Moser nonlinearity (2018) Nonlinear Anal., 186, pp. 74-98; Xiang, M., Rădulescu, V.D., Zhang, B., Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity (2019) Calc. Var. Partial Differ. Equ., 58 (2)}, correspondence_address1={Mukherjee, T.; Department of Mathematics, India; email: tulimukh@gmail.com}, publisher={Springer Science and Business Media B.V.}, issn={01678019}, coden={AAMAD}, language={English}, abbrev_source_title={Acta Appl Math}, document_type={Article}, source={Scopus},

Acta Applicandae Mathematicae

Kirchhoff Equations with Choquard Exponential Type Nonlinearity Involving the Fractional Laplacian

In this article, we study the discrete version of Legendre spectral and iterated Legendre spectral techniques to solve the second kind Hammerstein type weakly singular integral equations. To obtain the convergence analysis, we use the appropriate numerical quadrature rule and obtain the order (Formula presented.) in discrete Legendre spectral method. If the quadrature rule is minimal, i.e. the number of quadrature nodes and the dimension of the approximating subspace are same, then the optimal rate is obtained (Formula presented.) in iterated form of discrete Legendre spectral collocation method in (Formula presented.) norm and uniform norm. Numerical aspects are given to verify the hypothetical results. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

International Journal of Computer Mathematics

Discrete Legendre spectral methods for Hammerstein type weakly singular nonlinear Fredholm integral equations

The protection of sensitive and confidential data become a challenging task in the present scenario as more and more digital data is stored and transmitted between the end users. The privacy is vitally necessary in case of medical data, which contains the important information of the patients. In this article, a novel biometric inspired medical encryption technique is proposed based on newly introduced parameterized all phase orthogonal transformation (PR-APBST), singular value, and QR decomposition. The proposed technique utilizes the biometrics of the patient/owner to generate a key management system to obtain the parameters involved in the proposed technique. The medical image is then encrypted employing PR-APBST, QR and singular value decomposition and is ready for secure transmission or storage. Finally, a reliable decryption process is employed to reconstruct the original medical image from the encrypted image. The validity and feasibility of the proposed framework have been demonstrated using an extensive experiments on various medical images and security analysis. © 1989-2012 IEEE.