Department of Mathematics

It is known that the smaller the values of Gowers U2 and U3 norms of a Boolean function, the greater its strength against linear and quadratic approximations. This article gives recursive relation between the Gowers U2 and U3 norms of Boolean functions and their restriction to hyperplanes. These results will help analyze the second and third-order Gowers norms of Boolean functions in a higher number of variables.

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian (Pλ) {(−Δ)su=λu−q+u2s⁎−1,u>0in Ω,A(u)=0on∂Ω=∑D∪∑N, where Ω⊂RN is a bounded domain with smooth boundary ∂Ω, 1/20 is a real parameter, 02s, 2s⁎=2N/(N−2s) and [Formula presented] Here ∑D, ∑N are smooth (N−1) dimensional submanifolds of ∂Ω such that ∑D∪∑N=∂Ω, ∑D∩∑N=∅ and ∑D∩∑N‾=τ′ is a smooth (N−2) dimensional submanifold of ∂Ω. Within a suitable range of λ, we establish existence of at least two opposite energy solutions for (Pλ) using the standard Nehari manifold technique.

Let X be a multi-dimensional subshift of finite type generated by a finite set of finite forbidden blocks. We give an algorithm for generating the elements of the shift space using a sequence of finite matrices (of increasing order). We prove that the sequence generated yields precisely the elements of the shift space X and hence characterizes the elements of the shift space X. In the process, we prove that elements of d-dimensional shift of finite type can be characterized by a sequence of finite matrices.

This paper deals with the time delayed SIR model with saturated treatment function under the effect of pollution. As the pollution rate has hiked in this era and the presence of toxicants are found in various resources, therefore it is important to study their effects on disease dynamic of population. Hence, we consider the populace affected by the pollution. As the effect of treatment is not immediate, rather involves time lag to show its effect, therefore we consider time that is taken by infective to recover as the time delay. Existence of equilibrium points and the boundedness of the system has been obtained. Stability analysis using reproduction number R0 has been done. Global stability of the endemic equilibrium point is established using Lyapunov Lasalles theorem. Considering time delay as the critical parameter, existence of hopf bifurcation has been proved. Also,the direction and stability of Hopf Bifurcation has been obtained. Numerical simulations are done in support of results obtained analytically. It is observed that as pollution increases, infective take more time to recover and we also note that recovered individuals decrease with increase in the pollution. Therefore, it important to reduce pollution and provide timely treatment to the patients. Thus, we prove that, pollution and time delay play a critical role in shaping the dynamics of the system.

For detecting skin cancer, a silicon nitride based ring resonator refractive index sensor is modelled using coupled mode theory. This device has a high sensitivity of 249nm/RIU and Q factor of 1310.

The Markovian arrival process (MAP) is a versatile counting process with dependent and non-identically distributed inter-arrival times following the phase-type distribution. In this article, we study the classical (Formula presented.) -shock model and a mixed (Formula presented.) -shock model by assuming the MAP of shocks. We derive explicit expressions for the reliability and the mean lifetime of the system. Further, we study an optimal replacement policy based on the MAP. We illustrate the developed results through several numerical examples.

We investigated the effect of Au nanoparticles in Zinc o xide semiconductor matrix for enhanced absorption. Apart from ZnO optical absorption at 360 n m, the matrix also shows plasmonic absorption at 560 nm.

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.

Instrumented versions of functional geriatric screening tests have been developed to improve clinical precision. Several different instrumented versions of the Sit-to-Stand (iSTS) test have been developed using a range of sensors such as accelerometers and cameras. An instrumented chair equipped with load cells and an ultrasound sensor was developed to detect phases of the STS (Sit to Stand). The chair was designed to be able to detect all the phases of the STS, including when the person was not in contact with the chair. Performance of the iSTS chair was compared between an RGB camera approach, and a data-fusion approach using the load-cell and ultrasound equipped chair. Ten adult subjects were tested performing the 5STS at two self-selected speeds. The accuracy of the load cell equipped chair was 70%, while the RGB camera achieved 76% accuracy. The ultrasound version of the chair and the fusion of the RGB and load cells technique both achieved significantly better accuracy at 86% and 89%, respectively. The new version of the instrumented chair obtained a high degree of accuracy in detecting the different phases of the STS and is suitable to detect STS phases without requiring additional sensors. Future work will test older subjects and aim to develop new parameters based on the phases of the STS as indicators of physical performance.

The work presents an inventive, simple and implementable design approach to enhance the bandwidth of conventional Salisbury Screen Microwave Absorber (SSMA). Theoretically, the maximum fractional bandwidth of SSMA for FR4 substrate with an optimum sheet resistivity of 308 Ω/sq for –10 dB reflection is nearly 42.1%. In comparison, the bandwidth for square patch based SSMA is 59.7% with the same thickness. The design comprises square patches of SSMA placed periodically on a metal sheet. The square patches consist of an FR4 substrate and a 200 Ω/sq resistive sheet. A single reflection null is observed in the SSMA due to λ/4 resonance whereas in the proposed absorber an additional resonant mode is introduced due to coupling between the nearby patches. The simultaneous overlapping of the λ/4 mode and the additional coupling mode results in bandwidth extension. The close agreement between the simulation and measurement data is observed.