Department of Mathematics

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IIT Jodhpur

In this paper, we present a method of so-called q-Newton’s type descent direction for solving unconstrained multiobjective optimization problems. The algorithm presented in this paper is implemented by applying an independent parameter q (quantum) in an Armijo-like rule to compute the step length which guarantees that the value of the objective function decreases at every iteration. The search processes gradually shift from global in the beginning to local as the algorithm converges due to q-gradient. The algorithm is experimented on 41 benchmark/test functions which are unimodal and multi-modal with 1, 2, 3, 4, 5, 10 and 50 different dimensions. The performance of the proposed method is confirmed by comparing with three existing schemes. © 2020, Korean Society for Informatics and Computational Applied Mathematics.

World University root node

Journal of Applied Mathematics and Computing

On q-Newton’s method for unconstrained multiobjective optimization problems

The present study deals with the effects of floating flexible membrane on the instability of a gravity-driven flow down an incline. Linear stability of the flow system is explored using normal-mode analysis. Free surface gravity-driven flow is unstable at much lower Reynolds numbers. Instability of such a flow can be controlled either by changing behavioral of the lower wall or by altering the surface waves at the free surface which is done here by including a floating flexible membrane at the top of the liquid layer. Influence of membrane tension is taken into account in terms of stress jump at the free surface. The Orr-Sommerfeld system of the flow is solved numerically using spectral collocation method. The results displays a destabilizing role of membrane tension for a wide range of parameters. The growth rate of the perturbation waves increases with an increase of membrane tension and the critical Reynolds number becomes smaller. Therefore, it is possible to enhance the instability of the flow system with help of membrane properties, which may be useful in Ocean engineering and coating industries. © Springer Nature Singapore Pte Ltd. 2020.

Springer Proceedings in Mathematics and Statistics

Stability analysis of a film flow down an incline in the presence of a floating flexible membrane

This article deals with the study of the following critical exponent problem (Formula presented.) where Hn is the n-dimensional hyperbolic space, n ≥ 4, p = 2n/(n − 2) is the critical exponent, λ,μ &gt; 0 are real parameters, ΔHn denotes the Laplace–Beltrami operator on Hn and g(x) is a real valued potential function on Hn. Using variational methods, we establish the existence of positive ground state solution to (Pλ,μ) and study the convergence of these solutions when μ approaches +∞. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group.

Applicable Analysis

Concentration phenomenon in the critical exponent problems on hyperbolic space

The linear instability of a surfactant-laden two-layer falling film over an inclined slippery wall is analyzed under the influence of external shear, which is imposed on the top surface of the flow. The free surface of the flow and the interface among the fluids are contaminated by insoluble surfactants. Dynamics of the fluid layers are governed by the Navier-Stokes equations, and the surfactant transport equations regulate the motion of the insoluble surfactants at the interface and free surface. Instability mechanisms are compared by imposing the external shear along and opposite to the flow direction. A coupled Orr-Sommerfeld system of equations is derived using the perturbation technique and normal mode analysis. The eigenmodes corresponding to the Orr-Sommerfeld eigenvalue problem are obtained by employing the spectral collocation method. The numerical results imply that the stronger external shear destabilizes the interface mode instability. However, a stabilizing impact of the external shear on the surface mode is noticed if the shear is imposed in the flow direction, which is in contrast to the role of imposed external shear on the surface mode for a surfactant-laden single layer falling film. Furthermore, in the presence of strong imposed shear, the overall stabilization of the surface mode by wall velocity slip for the stratified two-fluid flow is also contrary to that of the single fluid case. The interface mode behaves differently in the two zones at the moderate Reynolds numbers, and higher external shear magnifies the interfacial instability in both zones. An opposite trend is observed in the case of surface instability. Moreover, the impression of shear mode on the primary instability is analyzed in the high Reynolds number regime with sufficiently low inclination angle. Under such configuration, dominance of the shear mode over the surface mode is observed due to the weaker impact of the gravitational force on the surface instability. The shear mode can also be stabilized by applying the external shear in the counter direction of the streamwise flow. Conclusively, the extra imposed shear on the stratified two-layer falling film plays an active role in the control of the attitude of the instabilities. © 2020 Author(s).

Fulltext

Physics of Fluids

Effect of imposed shear on the dynamics of a contaminated two-layer film flow down a slippery incline

The smart healthcare system is an electronic patient records (EPR) sharing system, which significantly helps sharing of EPR data and provides appropriate medical assistance for the patients and a more suitable platform for the potential researchers. However, the security of EPR data is still a major issue in such systems. In this paper, we develop a compression-then-encryption-based dual watermarking to protect the EPR data for the healthcare system, which produces several significant features. Experiments conducted on a large set of medical data indicate the capability of our proposed method for smart healthcare. Finally, when compared with the existing technique, the proposed work offers better performance in terms of robustness and security. © 1994-2012 IEEE.

IEEE Multimedia

Compression-Then-Encryption-Based Secure Watermarking Technique for Smart Healthcare System

We prove that, analogous to the Hilbert–Kunz density function, (used for studying the Hilbert–Kunz multiplicity, the leading coefficient of the Hibert–Kunz function), there exists a β-density function gR , m: [0 , ∞) ⟶ R, where (R, m) is the homogeneous coordinate ring associated with the toric pair (X, D), such that ∫0∞gR,m(x)dx=β(R,m),where β(R, m) is the second coefficient of the Hilbert–Kunz function for (R, m) , as constructed by Huneke–McDermott–Monsky. Moreover, we prove, (1) the function gR , m: [0 , ∞) ⟶ R is compactly supported and is continuous except at finitely many points, (2) the function gR , m is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved. Here we also prove and use a result (which is a refined version of a result by Henk–Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Journal of Algebraic Combinatorics

Density function for the second coefficient of the Hilbert–Kunz function on projective toric varieties

Stochastic precedence order is a natural type of comparison for random variables in numerous engineering applications (e.g., for the stress-strength modeling). In this note, we show that, for a k-out-of-n system, redundancy at the component level is superior to that at the system level with respect to the stochastic precedence order. Cases of active and cold redundancy are considered. Similar results for other stochastic orders were previously discussed in the literature. © 2020 Taylor &amp; Francis Group, LLC.

School of Management & Entrepreneurship

Communications in Statistics - Theory and Methods

A note on the stochastic precedence order between component redundancy and system redundancy for k-out-of-n systems

In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality: for some constant C &gt; 0, where Ω is an open set in ℝN with N ≥ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger. © Copyright Royal Society of Edinburgh 2019.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics

On the generalized Hardy-Rellich inequalities

In this paper, a simple design strategy is presented for designing broadband, polarization insensitive metasurface for radar cross section (RCS) reduction. Two different unit cell structures, circular patch, and loop are chosen to enhance the phase bandwidth. Different phase gradient supercells are designed by positioning these unit cells in a specific phase gradient direction. The eight supercells are then randomly distributed in a checkerboard pattern for further diffusing the incident wave. The proposed metasurface significantly reduce the RCS and demonstrate features such as polarization insensitivity and wideband operational bandwidth. © 2019 IEEE.

IEEE Region 10 Annual International Conference, Proceedings/TENCON

Random Phase Gradient Metasurface for Broadband RCS Reduction

Mixing in numerous medical and chemical applications, involving overly long microchannels, can be enhanced by inducing flow instabilities. The channel length is thus shortened in the inertial microfluidics regime due to the enhanced mixing, thereby making the device compact and portable. Motivated by the emerging applications of a lab on a compact disk based microfluidic devices, we analyze the linear stability of rotationally actuated microchannel flows commonly deployed for biochemical and biomedical applications. The solution of the coupled system of Orr-Sommerfeld and Squire equations yields the growth rate and the neutral curves for the Coriolis force-driven instability. We report on the existence of four different types of unstable modes (Type-I to Type-IV) at low rotation numbers. Furthermore, Types-I and II exhibit competing characteristics, signifying that Type-II can play an important role in the transition to turbulence. Type-III and Type-IV modes have relatively lower growth rates, but the associated normal velocity has an oscillatory nature near the center of the channel. Thus, we infer that Types-III and IV might cause strong mixing locally by virtue of strong velocity perturbation in proximity to the various point depths. Moreover, the situation is reliable if the channel is too short to allow for the amplification of Types-I and II. We quantify the potential of all the unstable modes to induce such localized mixing near an imaginary interface (near a hyphothetical interface) inside the flow using the notion of penetration depth. This study also presents an instability regime diagram obtained from the parametric study over a range of Reynolds numbers, rotation numbers, and streamwise and spanwise wavenumbers to assist the design of efficient microchannels. Further insight into the mechanism of energy transfer, drawn from the evaluation of the kinetic-energy budget, reveals how the Reynolds stress first transfers energy from the mean flow to the streamwise velocity fluctuations. The Coriolis force, thereafter, redistributes the axial momentum into spanwise and wall-normal directions, generating the frequently observed roll-cell structures. A qualitative comparison of our predictions with the reported experiments on roll-cells indicates a good agreement. © 2019 Author(s).