The Department of Mathematics, SRM University-AP, is pleased to announce that Assistant Professor Dr Subha Sandeep Repaka has published a significant research paper titled “On Reducibility of Induced Representations of Odd Unitary Groups: The Depth Zero Case.” This accomplishment reflects Dr Repaka’s expertise and dedication to advancing mathematical research, further enriching the academic contributions of the department and the university.

Abstract:

We study a problem concerning parabolic induction in certain $p$-adic unitary groups. More precisely, for $E/F$ a quadratic extension of $p$-adic fields the associated unitary group $G=\mathrm{U}(n,n+1)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $\mathrm{GL}_n(E) \times \mathrm{U}_1(E)$. Let $\pi$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $\iota_P^G \pi$ is reducible.

Future Research Plans:

We would like to solve the same problem which I had solved in this paper for the groups U(n,n) and U(n,n+1) over p-adic fields using L-Functions which is a very novel approach.

The link to the article:

http://nyjm.albany.edu/j/2024/30-50.html

Dr Shekhar Singh Negi from the Department of Mathematics has published a research paper titled “A note on Sugeno exponential function with respect to distortion.” Dr Negi’s research investigates the Sugeno exponential function. This research develops new mathematical tools and rules to work with a different way of measuring things, which can be useful in various fields like economics, biology, or any area where traditional measurements don’t quite fit the problem at hand.

Abstract:

This study explores the Sugeno exponential function, which is the solution to a first order differential equation with respect to nonadditive measures, specifically distorted Lebesgue measures. We define k-distorted semigroup property of the Sugeno exponential function, introduce a new addition operation on a set of distortion functions, and discuss some related results. Furthermore, m-Bernoulli inequality, a more general inequality than the well-known Bernoulli inequality on the real line, is established for the Sugeno exponential function. Additionally, the above concept is extended to a system of differential equations with respect to the distorted Lebesgue measure which gives rise to the study of a matrix m-exponential function.

Finally, we present an appropriate m-distorted logarithm function and describe its behaviour when applied to various functions, such as the sum, product, quotient, etc., while maintaining basic algebraic structures. The results are illustrated throughout the paper with a variety of examples.

Collaborations:

Prof. Vicenc Torra, Professor at the Department of Computing Science at Umea University. His area of research include artificial intelligence, data privacy, approximate reasoning, and decision making.

Future Research Plans:

To explore the aforementioned derivative and investigate results with applications in real life.

The link to the article

Dr Koyel Chakravarty, Assistant Professor in the Department of Mathematics, has made a significant contribution to the field of cancer research with her paper “Analysis and Regulation of Chaos Dynamics in a Cancer Model through Chemotherapeutic Intervention and Immune System Augmentation,” which was recently published in the International Journal of Dynamics and Control. In her paper, Dr Chakravarty delves into the intricate world of chaos dynamics within a cancer model and explores the potential for regulating these dynamics through the combined approach of chemotherapeutic intervention and immune system augmentation.

Her research offers insights into understanding the complex behaviour of cancer cells and how such insights can be leveraged to develop more effective treatment strategies. Dr Chakravarty’s work marks a crucial step forward in the ongoing efforts to combat cancer, shedding light on the dynamic interplay between therapeutic interventions and the body’s immune response.

The publication of this paper not only underscores Dr Koyel’s expertise in the field of mathematical analysis in cancer research but also signifies a promising advancement in the collective pursuit of understanding and addressing the challenges posed by cancer.

Abstract

The focus of the current investigation lies in the formulation and analysis of a dynamic model depicting cancer growth, incorporating the joint influences of chemotherapy and immune system augmentation. The primary emphasis of this study revolves around the analysis of the dynamic behaviour within a living-cell closed carcinoma system, specifically one devoid of external vitamin support, with a particular exploration of chaos dynamics. Subsequently, the authors aim to scrutinise the pivotal impact of infused vitamins in attaining stable system dynamics through the application of chaos control techniques.

The formulated model exhibits fundamental mathematical properties, revealing a spectrum of co-dimension one and co-dimension two bifurcations. The identification of specific bifurcation types relies on algebraic criteria techniques, where conditions necessary and sufficient for bifurcation types are developed. Notably, these criteria are distinct from traditional approaches based on the characteristics of the eigenvalues of the Jacobian matrix, instead relying on coefficients derived from characteristic equations. The accuracy of the analytical conclusions is validated through numerical findings, elucidating diverse bifurcation structures. The article enriches its contribution by delving into the control of chaos through the reinforcement of the internal immune system and the maintenance of the biological system’s stability. This work culminates in proposing future directions aimed at advancing a more realistic approach to eradicating cancer.

Research in Layperson’s Terms

This study focuses on developing and analysing a model that simulates how cancer grows, considering both chemotherapy and the immune system’s response. The main goal is to understand how cancer behaves over time in a system that doesn’t have external vitamin support, especially looking at how chaotic or unpredictable the growth can become. The researchers also investigate how adding vitamins might help stabilise this chaotic system using specific control techniques. The model they created has certain mathematical features that show different types of changes, called bifurcations, which can occur under specific conditions.
Additionally, the study explores how strengthening the immune system might help control this chaos and stabilise the biological system. The paper concludes by suggesting future research directions that could lead to more effective cancer treatment strategies.

Practical implementation

The practical implementation and social implications of analysing and regulating chaos dynamics in a cancer model through chemotherapeutic intervention and immune system augmentation can be profound. Insights gained from this research could be applied to optimize cancer treatment protocols, potentially leading to more effective therapies with reduced side effects. By understanding and controlling the chaotic behaviour in cancer systems, patient outcomes could be improved through personalized treatment strategies.
Socially, the adoption of these findings may lead to increased public confidence in advanced cancer treatments, as well as a broader acceptance of integrating immune system support with traditional therapies. The potential for more stable and predictable treatment outcomes may also reduce the emotional and financial burden on patients and healthcare systems. Additionally, this approach may encourage further interdisciplinary research, bridging gaps between Mathematics, Biology, and Medicine, thus fostering innovation in cancer therapy development.

Collaborations
Dr Lakshmi Narayan Guin, Associate Professor, Department of Mathematics, Siksha Bhavana, Visva-Bharati

Future research plans
Potential areas for further exploration include:

• Personalised Medicine: Developing patient-specific models that consider individual biological variations could lead to more tailored and effective cancer treatments, minimising side effects and improving outcomes.
• Integration with Advanced Therapies: Combining the insights from chaos dynamics with emerging therapies such as immunotherapy, targeted therapy, and gene editing could enhance the precision and efficacy of cancer treatments.

In a significant stride towards understanding cholesterol homeostasis, Dr Koyel Chakravarty, Assistant Professor and Mr Sukdeb Manna, a PhD Scholar in the Department of Mathematics has co-authored a paper titled “Unlocking Cholesterol Homeostasis: A Mathematical Modelling Perspective.” The paper has been published in the esteemed journal, The European Physical Journal Plus, with an impact factor of 3.4. This collaborative effort showcases the innovative application of mathematical modelling in unravelling the complexities of cholesterol regulation within the body.

The research not only contributes to the existing body of knowledge in this field but also sheds light on potential avenues for further exploration and understanding. Dr Chakravarty and Mr. Manna’s work underscores the importance of interdisciplinary approaches in scientific research and highlights SRM University-AP’s commitment to fostering cutting-edge research and innovation.

Their collaborative efforts serve as an inspiration to aspiring researchers and underscore the university’s dedication to pushing the boundaries of knowledge and discovery. Congratulations to Dr Chakravarty and Mr Manna on this remarkable achievement, and we look forward to more groundbreaking contributions from them in the future.

Abstract:
Limited progress in the mathematical modelling of cholesterol transport systems is hampering novel therapeutic interventions. This issue is addressed by the present study through precise design, employing four compartmental models to elucidate cholesterol dynamics in the comprehensive bloodstream. Disparities in medical advancements, particularly in cholesterol-related pathophysiology, are aimed to be bridged, advancing medical science and patient care outcomes.

Therapeutic strategies for reducing blood cholesterol are explored by the model, with parameter influences on equilibrium stability revealed through sensitivity analysis. System parameters are effectively manipulated by imposing sensitivity analysis, and pinpointing areas for model refinement. Stability analysis contributes to diverse realistic models, confirming local asymptotic stability. Model efficacy in studying cholesterol transport dynamics is supported by analytical and numerical assessments. The study concludes with the present model validation to substantiate it by comparing the present outcomes with the existing ones.

Explanation of The Research in Layperson’s Terms:

Basically, scientists are having trouble figuring out how to model how cholesterol moves around in the body, which is important for developing new treatments. This study tries to solve that problem by creating detailed models that show how cholesterol behaves in different parts of the bloodstream. The goal is to bridge the gap in medical knowledge about cholesterol-related problems and improve how we treat patients. The models help us understand how different treatments might affect cholesterol levels, and by analyzing them closely, we can figure out which factors are most important. This lets us tweak the models to make them more accurate. The study shows that the models are reliable by testing them both analytically and numerically, and comparing the results to what we already know.

Practical Implementation or the Social Implications of the Research

1. Personalized Medicine: The mathematical models developed in this research could help in designing personalized treatment plans for individuals with high cholesterol levels. By understanding how different factors affect cholesterol dynamics, doctors can tailor therapies to each patient’s specific needs, leading to more effective and targeted treatments.
2. Drug Development: Pharmaceutical companies could use these models to screen potential drugs for lowering cholesterol. By simulating how different compounds interact with cholesterol transport systems, researchers can identify promising candidates for further testing, potentially speeding up the drug development process.
3. Healthcare Cost Reduction: Better understanding of cholesterol dynamics could lead to more efficient use of healthcare resources. By optimizing treatment strategies and preventing complications related to high cholesterol, healthcare costs associated with conditions like heart disease could be reduced, benefiting both individuals and society.
4. Public Health Initiatives: Insights from the research could inform public health initiatives aimed at reducing cholesterol-related diseases. For example, policymakers could use the models to design targeted interventions such as education campaigns promoting healthy lifestyle choices or policies to improve access to cholesterol-lowering medications.
5. Improved Patient Outcomes: Ultimately, the goal of this research is to improve patient outcomes by better understanding and managing cholesterol homeostasis. By developing more accurate models of cholesterol transport dynamics, healthcare providers can make more informed decisions, leading to better control of cholesterol levels and reduced risk of cardiovascular disease and other related conditions.

Future Research Plans:

1. Current models may focus on a limited set of factors influencing cholesterol transport. Future research could explore the integration of additional biological factors such as genetic variations, hormonal influences, and dietary components into the models to create a more comprehensive understanding of cholesterol homeostasis.
2. Most existing models of cholesterol transport assume static conditions. Future research could develop dynamic models that capture the time-dependent changes in cholesterol levels in response to various stimuli, such as meals, physical activity, and medication intake. Dynamic models would provide a more accurate representation of real-world cholesterol dynamics and enable the evaluation of time-sensitive interventions.

Picture Related to the Research

Link to the article