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Annual Progress Seminar
Speaker: Aamir Yousuf (IIT Bombay)
Host: Neela Nataraj
Title: Numerical analysis of a thermoelastic diffusion plate model
Time, day and date: 11:00:00 AM, Monday, February 24
Venue: Online (Link in attached file)
Abstract: We investigate the well-posedness of a coupled
hyperbolic-parabolic system modeling diffusion in thermoelastic plates,
consisting of a fourth-order hyperbolic PDE for plate deflection and
second-order parabolic PDEs for the first moments of temperature and
chemical potential. The unique solvability is established via the
Galerkin approach, and additional regularity of the solution is obtained
under appropriately strengthened given data. For numerical
approximation, we employ the Newmark method for time discretization and
a C 0 -interior penalty (C0-IP) scheme for the spatial discretization of
displacement. For the first moments of temperature and chemical
potential, we use the Crank-Nicolson method for time discretization and
conforming finite elements for spatial discretization. The convergence
of the fully discrete scheme is established, achieving quasi-optimal
rates in space and a quadratic rate in time. Several numerical examples
will be presented to validate the theoretical findings.
Annual Progress Seminar
Speaker: Aratrika Pandey (IIT Bombay)
Host: Ravi Raghunathan
Title: Exact Approximation in the field of formal series
Time, day and date: 4:00:00 PM, Monday, February 24
Venue: Ramanujan Hall
Abstract: We prove a lower bound for the Hausdorff dimension of the set
of exactly $\psi$-approximable matrices with values in a local field of
positive characteristic. This is the analogue of the corresponding
theorem of Bandi and de Saxc\'e over the reals, and is a higher
dimensional version of a theorem.
Seminar
Speaker: Dr. Rajeev Das
Host: Radhendushka Srivastava
Title: Design and Applications of Robust Metaheuristics for Constrained
Optimization Problems
Time, day and date: 4:00:00 PM, Monday, February 24
Venue: Online (https://meet.google.com/dtw-ztpe-jnc)
Abstract: The presentation includes a detailed discussion of my Doctoral
research based on the development and applications of efficient
metaheuristic algorithms for solving constrained optimization problems
(COPs). The primary focus is on the Quadratic Approximation guided Jaya
Algorithm (JaQA) and its enhanced variant, the Self-Adaptive
Multi-Population Quadratic Approximation guided Jaya (SMP-JaQA). These
two algorithms address challenges in solution diversity and balancing
exploration-exploitation, making them highly effective in solving
complex real-world optimization problems. The presentation begins with
an introduction to COPs and the significance of metaheuristics in
overcoming traditional optimization limitations. A comprehensive
discussion on the research motivation, problem statement, and the
identified gaps in existing methodologies follows. The design and
implementation of these two proposed methodologies is elaborated upon,
along with their computational complexity analysis. Experimental
validation is carried out using constrained benchmarks from CEC 2006,
CEC 2010, and CEC 2017. The outcomes demonstrate the superior
performance of the proposed algorithms when compared to state of-the-art
techniques, validated through statistical analysis (say, Friedman Rank
Test and Wilcoxon Signed Rank Test), exploration-exploitation analysis,
convergence analysis, etc. Real-life applications of the research are
also highlighted in areas such as single-stage and two-echelon uncertain
Fixed Charge Transportation Problems (FCTP), dairy feed ration
optimization, mechanical design and electronics design optimization. The
discussion concludes with an overview of research contributions, key
findings, and potential future research directions. Moreover, insights
into the publication status of my research so far is also provided
showcasing peer-reviewed journal and conference publications along with
communicated works.
Analysis seminar
Speaker: Kapil Jaglan (IIT Ropar)
Host: Prachi Mahajan
Title: A study on Planar Harmonic Mappings and Minimal surfaces
Time, day and date: 5:00:00 PM, Monday, February 24
Venue: Online (https://meet.google.com/woo-psdw-wdj)
Abstract: This talk explores the properties of univalent harmonic
functions from a geometric function theoretic perspective and
establishes connections between these functions and minimal surfaces.
The talk consists of four sections followed by future direction, with
the initial section serving as an introduction and containing basic
definitions and results from the existing literature.
In the second section, our main aim is to determine a geometric
condition under which a locally univalent harmonic mapping f defined on
the unit disk D is univalent, and maps D onto a linearly accessible
domain of order β for some β ∈ (0, 1). A linearly accessible domain (a
non-convex domain) is important because, under certain sufficient
conditions stated by Dorff et al., minimal graphs over these domains are
area-minimizing. As a consequence, we derive sufficient conditions for f
to map D onto a linearly accessible domain of order β, in the form of a
convolution result, and a coefficient inequality. By extending the ideas
of Dorff et al., we construct one-parameter families of globally
area-minimizing minimal surfaces over a linearly accessible domain of
order β.
In the third section, we explore the properties of odd univalent
harmonic functions. Our starting point of investigation is to obtain the
sharp coefficient estimates for odd univalent functions exhibiting
convexity in one direction. We then advance our investigation to more
generalized classes, including major geometric subclasses of
sense-preserving univalent harmonic mappings. We examine the growth
pattern of odd univalent harmonic functions and extend the range of 'p'
for which these functions belong to the Hardy space h^p. Our results, in
particular, add to the understanding of the growth pattern between odd
univalent harmonic functions and the harmonic Bieberbach conjecture.
In the final section, we deal with the fundamental problem of
determining the location of the zeros of complex-valued harmonic
polynomials. The best-known results available in this direction are up
to harmonic trinomials only. The exploration of the zeros of a general
harmonic polynomial has been limited due to various challenges. Our
research takes a leap further in identifying the regions encompassing
the zeros of a general harmonic polynomial of arbitrary degree using
various techniques, such as the matrix method and certain other matrix
inequalities. Additionally, we employ the harmonic analog of the
argument principle to examine the distribution of zeros, which we
demonstrate through illustrative examples.
Number theory seminar
Speaker: D. Surya Ramana (Harish-Chandra Research Institute)
Host: Ravi Raghunathan
Title: The Large Sieve with sieving by prime powers.
Time, day and date: 11:30:00 AM, Friday, February 28
Venue: Ramanujan Hall
Abstract The Large Sieve inequality for prime powers provides an upper
bound for the cardinality of a subset of the integers in a real
interval, given the cardinalities of the subsets of
$\mathbb{Z}/p^{\nu}\mathbb{Z}$ in which the reduction of the set modulo
$p^{\nu}$ are known to lie, for various primes powers $p^{\nu}$. We give
a simple proof of this inequality via the principle of the Selberg
sieve.