Speakers

Professor, Sciences and Mathematics, 

Department of Computer Science , University of Niš, Serbia

Title: Improvements of Gradient Neural Networks for Solving Ill Conditioned Problems

Abstract: Gradient neural networks (GNNs) are dynamical systems that evolve continuously over time and are becoming increasingly prevalent in both academic research and engineering applications. This work focuses on using GNNs to solve the matrix equation $AXB=D$, where $X$ is the unknown matrix, with a special emphasis on handling problems involving poorly conditioned data. The standard GNN dynamics for solving this equation rely on an error matrix (EM) defined as $E_t = AV_tB - D$, where $V_t$ represents the state variables matrix that changes over time. The state trajectories follow the gradient-descent trajectory of the Frobenius norm $||E_t||^2_F$ to solve the equation $E_t=0$. The authors suggest two specific improvements to address ill-conditioned problems. The first improvement, known as the best approximate GNN (BGNN), uses a modified error function based on two error matrices: $E_{1t} = AV_tB - D$ and $E_{2t} = V_t$. These are combined using a regularization factor $\alpha$ into the expression $E_t = E_{1t} + \alpha E_{2t}$. In this model, the term $E_{1t}$ approximates least squares solutions, while $E_{2t}$ ensures the minimum-norm solution among them. Convergence for the BGNN is related to the exact solution of a generalized Sylvester equation. The second proposed approach incorporates both the current value $E_t$ and the value from the previous time step $E_{tp}$. The error matrix for this model is expressed as $E_t = E_{1t} + \beta_t E_{2t}$, where $\beta_t$ is defined by analogy to conjugate gradient methods. These methods are validated through simulation examples conducted on ill-conditioned test matrices in Matlab.

Professor, University of Johannesburg

Associate Professor of Mathematics and Statistical Sciences, School of Pure and Applied Sciences, Botswana International University of Science and Technology (BIUST), Southern Africa Technology (BIUST), Southern Africa

Title: Hybrid nanofluid Flow with Thermal Performance and Applications: Future Scope and challenges 

Abstract: The suspension of nanoparticles in the base fluid has significant attention from researchers and industries with an aspect of improving the thermo-physical properties of the fluid. Hybrid nanofluid is a new kind of nanofluid synthesized by impregnating two nanosized particles (of metal or metal oxide or combination of both particles) in base fluid. Hybrid nanofluid supports a better enhancement in thermophysical properties especially in thermal conductivity as compared the single or mono nanoparticle in the base fluid. It is observed that hybrid nanofluid will be a better replacement of mono-nanofluid since it provides more heat transfer enhancement especially in the areas of automobile, electro-mechanical, manufacturing process, HVAC and solar energy. In this study shows importance of different phenomena as interfacial nanolayer thickness, flow geometry, different flow and temperature control parameters and their significant contribution. Further, discuss the future scope and challenges.

Professor, Mathematics, NIT Rourkela, India 

Title: SQUARE IN ARITHMETIC PROGRESSIONS 

Abstract: There are infinitely many arithmetic progression of three integer squares. Each Pythagorean triple produces three squares in an arithmetic progression. However, there is no nontrivial arithmetic progression of four squares. In addition, there is no nontrivial arithmetic progression of three squares with a square common difference.  Alternatively, the Diophantine equation 1 + x^4  = 2y^2 has no solution in positive integers and the Diophantine equation 8x^4+ 1 = y^2 has no positive integral solution other than x = 1  and y=3. A positive integer u is a balancing number if 8u^2+1 is a square.  The solution of the last Diophantine equation suggests that the only balancing number, which is also a square, is 1.

Professor, Mathematics, IIT Kharagpur, India 

Title: On convexity of Chebyshev Sets 

Abstract: The research systematically develops best approximation theory, focusing on existence, uniqueness, characterization, continuity, and computational elements of best approximations. Concepts such as suns, proximal sets, approximate compactness, and metric projections are employed to examine the geometric characteristics of Chebyshev sets. The results provide partial progress toward resolving the long-standing Chebyshev set problem and offer useful insights for applications in convex optimization and projection theory.

Associate Professor and Head of the Department, Mathematics, Central University Jharkhand, India 

Title: Riemannian Manifold and Convergence on Mann’s Method under it 

Abstract: In this talk we will concentrate about the basic introduction of Riemannian manifold. Here we will give some important definitions and their applications. Here convergence analysis of Mann’s iteration method using Kantorovich’s theorem in the context of connected and complete Riemannian manifolds has been examined. We also provide an algorithm for Mann’s method to find a singularity in a two dimensional sphere S2. Finally, we provide an example that shows the better convergence result of Mann’s method in comparison to that of Newton’s method.

Associate Professor, BITS Pilani, Hyderabad, India

Title: Fractional Fourier Transform 

Abstract: The Fractional Fourier Transform (FrFT) is a natural generalization of the classical Fourier transform that provides a flexible framework for time–frequency analysis and operator theory. This talk introduces the basic concepts and properties of the FrFT and extends the transform to the setting of tempered distributions. Within this framework, generalized pseudo-differential operators associated with the FrFT are defined and their boundedness properties on suitable function spaces are discussed. Finally, the application of the Fractional Fourier Transform to generalized heat equations is presented, demonstrating how the FrFT approach facilitates the analysis of fractional diffusion and solution behavior. The results highlight the relevance of the FrFT in harmonic analysis, partial differential equations, and applied mathematical modeling.  

Professor, School of Mathematics, Gangadhar Meher University, Sambalpur, 768001, India

Title: Wavelet Galerkin Method for Eigenvalue Problem of a Compact Integral Operator 

Abstract: Many practical problems in science and engineering are formulated as eigenvalue problems of compact linear integral operators. For many years, numerical solution of eigenvalue prob lems have attracted much attention. The Galerkin, petrove-Galerkin, collocation, Nystr¨om and degenerate kernel methods are the commonly used methods for the approximation of eigenfunctions of the compact integral operator. The analysis for the convergence of Galerkin, petrove-Galerkin, collocation, Nystr¨om and degenerate kernel methods are well documented in literature. Standard numerical treatment in the Galerkin method for the eigenvalue problem is normally to discretize the compact integral operator into a matrix and then solve the eigenvalue problem of the resulting matrix. The computed eigenfunctions of the matrix are considered as approximations of eigenfunctions of the compact linear integral operator. It is well known that matrix resulting from a discretization of a compact integral operator is a full matrix. Solving the eigenvalue problem of a full matrix requires significant amount of computational effort. Hence, fast algorithms for such a problem is highly desirable. We discussed a fast wavelet Galerkin method for solving the eigenvalue problem of a com pact linear integral operator with a smooth kernel. The approximation of eigenfunctions of a compact integral operator with a smooth kernel by Galerkin method using wavelet bases are considered.The essence of these methods is to approximate the matrix by a sparse matrix and solve the eigenvalue problem of sparse matrix. Such algorithms are called matrix truncation or matrix compression. This matrix compression technique leads to fast algorithms for solving the eigenvalue problems. Truncated wavelet Galerkin method are computationally economic in comparison to the wavelet Galerkin method for the eigenvalue problem. The convergence rates for the eigenvalues and eigen vectors, and computational complexity of the truncated eigen value problem have been discussed. The numerical results are solved to validate the theoretical estimates. 

Professor, Department of Mathematics, 

Sambalpur University, Jyoti Vihar, Burla, Sambalpur, India  

Title: Balancing numbers based statistical convergence and a Korovkin type theorem 

Abstract: Statistical convergence is a concept in mathematical analysis that provides a way to measure the convergence of a sequence of numbers based on their sta tistical properties. Unlike traditional notions of convergence, which focus on the limit of individual terms in the sequence, statistical convergence considers the behavior of the sequence as a whole. More formally, a sequence of real numbers (xn) converges statistically to a real number L if, for every ϵ > 0, limn→∞ 1 n{k ≤ n : |xk −L| ≥ ϵ} = 0. This means that the proportion of terms in the sequence that are far away from L is small in some precise sense. In this work, we introduce the concept of balancing numbers based statisti cal convergence, a new form of convergence defined via transformations involving balancing numbers. We establish fundamental properties of balancing numbers based statistical convergence and study its relation to ordinary statistical convergence. Furthermore, we extend this framework by defining balancing number based statistical convergence via modulus functions, which we denote by Bf-statistical convergence. We show that Bf-statistical convergence implies balancing number based statistical convergence. The study also develops the corresponding notions of Cauchy sequences within these settings. Finally, we prove Korovkin-type approx imation theorems in the space C∗[0,∞) by employing Bf-statistical convergence, using the classical test functions {1,e−x,e−2x}.