Department of Mathematics

We consider two finite and disjoint sets of homogeneous robots deployed at the nodes of an infinite grid graph. The grid graph also comprises two finite and disjoint sets of prefixed meeting nodes located over the nodes of the grid. The objective of our study is to design a distributed algorithm that gathers all the robots belonging to the first team at one of the meeting nodes belonging to the first type, and all the robots in the second team must gather at one of the meeting nodes belonging to the second type. The robots can distinguish between the two types of meeting nodes. However, a robot cannot identify its team members. This paper assumes the strongest adversarial model, namely the asynchronous scheduler. We have characterized all the initial configurations for which the gathering problem is unsolvable. For the remaining initial configurations, the paper proposes a distributed gathering algorithm. Assuming the robots are capable of global-weak multiplicity detection, the proposed algorithm solves the problem within a finite time period. The algorithm runs in O(dn) moves and O(dn) epochs, where d is the diameter of the minimum enclosing rectangle of all the robots and meeting nodes in the initial configuration, and n is the total number of robots in the system.

Brain tumors are life-threatening and are typically identified by experts using imaging modalities like Magnetic Resonance Imaging (MRI), Computed Tomography (CT), and Positron Emission Tomography (PET). However, any error due to human intervention in brain anomaly detection can have devastating consequences. This study proposes a tumor detection algorithm for brain MRI images. Previous research into tumor detection has drawbacks, paving the way for further investigations. A visual attention-based technique for tumor detection is proposed to overcome these drawbacks. Brain tumors have a wide range of intensity, varying from inner matter-alike intensity to skull-alike intensity, making them difficult to threshold. Thus, a unique approach to threshold using entropy has been utilized. An on-center saliency map accurately captures the biological visual attention-focused tumorous region from the original image. Later, a superpixel-based framework has been proposed and used to capture the true structure of the tumor. Finally, it was experimentally shown that the proposed algorithm outperforms the existing algorithms for brain tumor detection.

Let K=R or C and Rn be the dihedral quandle of order n. In this article, we give decomposition of the quandle ring K[Rn] into indecomposable right K[Rn]-modules for all even n∈N. It follows that the decomposition of K[Rn] given in [2, Prop. 4.18(2)] is valid only in the case when n is not divisible by 4.

The aim of this article is to find a geometric and physical interpretation of fractional order derivatives for a general class of functions defined over a bounded or unbounded domain. We show theoretically and geometrically that the absolute value of the fractional derivative value of a function is inversely proportional to the area of the triangle. Further, we prove geometrically that the fractional derivatives are inversely proportional to the classical integration in some sense. The established results are verified numerically for non-monotonic, trigonometric, and power functions. Further, this article establishes a significant connection between the area of the projected fence and the area of triangles. As the area of triangles decreases, the area of the projected fence increases, and vice versa. We calculate the turning points of the fractional derivative values of different functions with respect to order (Formula presented.), including non-monotonic, trigonometric, and power functions. In particular, we demonstrate that for the power function (Formula presented.), with (Formula presented.) being a positive real number, the value (Formula presented.) is a turning point when (Formula presented.). However, for (Formula presented.), the turning point shifts to the left of point (Formula presented.) and shifts to the right of point (Formula presented.) for (Formula presented.) We discuss the physical interpretation of fractional order derivatives in terms of fractional divergence. We present some applications of fractional tangent lines in the field of numerical analysis.

This study primarily focused on forming pure single-phase FeSb and explored its thermoelectric, mechanical, and electrochemical properties since no reports are available. The FeSb binary alloy has been synthesized through the vacuum melting method, and the phase formation has been confirmed through powder X-ray diffraction (PXRD). The PXRD results show that the synthesized FeSb binary alloy belongs to the hexagonal crystal structure with space group P63/mmc, which coincides with ICSD no. 53971. The pure single phase has been formed by creating a deficiency of 10 % in antimony. The High-Resolution Scanning Electron Microscopy (HR-SEM) and Energy Dispersive X-ray (EDX) analysis have been used to identify the pure single-phase and various elemental components of the hot-pressed pellet of FeSb0.9. The atomic wt.% of iron (Fe) and antimony (Sb) have been identified through EDX spectral analysis. The highest Seebeck coefficient value of −5.4 μV/K is achieved at 497 K, and the lowest electrical conductivity value of 24049 S/m is achieved at 447 K. The hardness of the material is found to be 6.076 GPa, which is much more sufficient for thermoelectric material during industrial handling. The magnetic characteristics of the prepared pure phase FeSb compound have also been measured by Vibrating Scanning Magnetometer (VSM) analysis, which has a weak ferromagnetic nature. Furthermore, three electrodes were employed to study the electrochemical properties, and the alloy has attained the appreciable specific capacitance of 169.5 F/g at 2 A/g.

The hydrodynamic stability behavior of a two-layer falling film is explored with a floating flexible plate on the top layer surface. The stress balance at the surface is modeled using a modified membrane equation. There is an insoluble surfactant at the liquid-liquid interface of the flow system. The linear instability of perturbation waves is captured by numerically solving the generalized Orr-Sommerfeld eigenvalue problem using the Chebyshev spectral collocation technique. Four different types of linear stability modes are identified: surface mode (SM), interface mode (IM), interface surfactant mode (ISM), and shear mode (SHM). The floating flexible plate has an inferior impact on all the existing modes in the longwave zone. However, a significant influence is noticed for the finite wave numbers. The surface and interface wave instabilities can be suppressed in the smaller wavenumber zone by imposing higher structural rigidity and uniform thickness of the flexible plate. The stabilizing nature of the surface mode becomes more powerful when the top layer viscosity is dominant. A new interfacial instability emerges when the top layer is less viscous than the lower one. At moderate Reynolds numbers, the behavior of interface mode is different in two different zones mr<1 and mr>1, where m and r denote the viscosity and density ratio, respectively. Further, the unstable shear modes induced by the top and bottom layers are detected under the low inclination angles with a strong inertia force. The occurrence of the shear modes requires the viscosity and density of the lower layer to be much higher than those of the upper layer. The influence of characteristic parameters of the flexible plate on the lower layer shear mode is not very sensitive. Finally, the competition between the different modes for dominance of stability boundaries is also discussed.

This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and using a simple variable transformation. Taking advantage of orthogonal projection operator and weighted inner product with respect to that same Jacobi weight function, we are able to obtain improved convergence rate for iterated shifted Jacobi spectral Galerkin method (SJSGM) and iterated shifted Jacobi spectral multi-Galerkin method (SJSMGM) in both weighted and infinity norms. Further, we obtain improved superconvergence rate for iterated SJSGM and iterated SJSMGM, by improving the regularity of exact solution, using smoothing transformation. Increasing the value of the smoothing parameter we can improve the regularity of the exact solution upto the desired degree. Numerical results with a comparative study of pre and post smoothing transformation are given to illustrate the theoretical results and efficiency of our proposed methods.

Developed sequential order statistics (DSOS) are very useful in modeling the lifetimes of systems with dependent components, where the failure of one component affects the performance of remaining surviving components. We study some stochastic comparison results for DSOS in both one-sample and two-sample scenarios. Furthermore, we study various ageing properties of DSOS. We state many useful results for generalized order statistics as well as ordinary order statistics with dependent random variables. At the end, some numerical examples are given to illustrate the proposed results.

Non-self adjoint operators describe problems in science and engineering that lack symmetry and unitarity. They have applications in convection–diffusion processes, quantum mechanics, fluid mechanics, optics, wave-guide theory, and other fields of physics. This paper reviews some important aspects of the eigenvalue problems of non-self-adjoint differential operators and discusses the spectral properties of various non-self-adjoint differential operators. Their eigenvalues can be computed for ground and perturbed states by their spectra and pseudospectra. This work also discusses the contemporary results on the finite number of eigenvalues of non-self-adjoint operators and the implications it brings in modeling physical problems.

The rendezvous task calls for two mobile agents, starting from different nodes of a network modeled as a graph to meet at the same node. Agents have different labels which are integers from a set {1,…,L}. They wake up at possibly different times and move in synchronous rounds. In each round, an agent can either stay idle or move to an adjacent node. We consider deterministic rendezvous algorithms. The time of such an algorithm is the number of rounds since the wakeup of the earlier agent till the meeting. In most of the literature concerning rendezvous in graphs, the graph is finite and the time of rendezvous depends on its size. This approach is impractical for very large graphs and impossible for infinite graphs. For such graphs it is natural to design rendezvous algorithms whose time depends on the initial distance D between the agents. In this paper we adopt this approach and consider rendezvous in infinite trees. All our algorithms work in finite trees as well. Our main goal is to study the impact of orientation of a tree on the time of rendezvous. We first design a rendezvous algorithm working for unoriented regular trees, whose time is in O(z(D)log⁡L), where z(D) is the size of the ball of radius D, i.e., the number of nodes at distance at most D from a given node. The algorithm works for arbitrary delay between waking times of agents and does not require any initial information about parameters L or D. Its disadvantage is its complexity: z(D) is exponential in D for any degree d>2 of the tree. We prove that this high complexity is inevitable: Ω(z(D)) turns out to be a lower bound on rendezvous time in unoriented regular trees, even for simultaneous start and even when agents know L and D. Then we turn attention to oriented trees. While for arbitrary delay between waking times of agents the lower bound Ω(z(D)) still holds, for simultaneous start the time of rendezvous can be dramatically shortened. We show that if agents know either a polynomial upper bound on L or a linear upper bound on D, then rendezvous can be accomplished in oriented trees in time O(Dlog⁡L), which is optimal. If no such extra knowledge is available, a significant speedup is still possible: in this case we design an algorithm working in time O(D2+log2⁡L).