Department of Mathematics

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    Theoretical Investigations of Some Blood Flow Problems in Biomathematics
    (1998) Maiti, Arun Kumar
    Biomathematics (also called mathematical biosciences) is an interdisciplinary subject with a vast and exponentially growing literature belonging to different disciplines. Contributions to it have been made by mathematicians, physicists, statisticians, computer scientists, botanists, zoologists, ecologists, agricultural scientists, demographers, physiologists, medical scientists, mechanical, electrical and chemical engineers and many others. A large number of mathematical models have been developed to get an insight into complex biological, ecological and physiological situations. A great variety of mathematical techniques have been employed to solve these models. These include techniques for solutions of differential, difference, integral, delay-differential and integro-differential equations as well as techniques of linear, nonlinear, dynamic and stochastic programming, calculus of variations, maximum principle and so on. Some of the latest results of algebraic topology, fuzzy sets and catastrophe theory have been successfully used to probe deeply into the problems of life sciences.
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    WEAVING FRAMES IN HILBERT SPACES
    (University of Delhi, 2017-10) Deepshikha
    The work in the thesis entitled “Weaving Frames in Hilbert Spaces” deals with the study of weaving properties of different types of frames in separable Hilbert spaces. This is motivated by a new concept of “weaving frames” in separable Hilbert spaces; introduced by Bemrose, Casazza, Gr¨ochenig, Lammers and Lynch [13] with an eye on various applications of frames in distributed signal processing. Let I be a countable set. Two discrete frames {fi}i∈I and {gi}i∈I for a separable Hilbert space H are said to be woven if there are universal positive constants A and B such that for every subset σ ⊂ I, the family {fi}i∈σ ∪ {gi}i∈σc is a frame for H with lower and upper frame bounds A and B, respectively. We say that {fi}i∈I and {gi}i∈I are weakly woven if for every subset σ ⊂ I, the family {fi}i∈σ ∪ {gi}i∈σc is a frame for H. Weaving frames has potential applications in wireless sensor networks that require distributed processing under different frames, as well as pre-processing of signals using Gabor frames. Casazza and Lynch in [30] reviewed fundamental properties of weaving frames. They also discussed weaving equivalent of an unconditional basis. Fundamental properties of weaving fusion frames can be found in [56]. Casazza, Freeman and Lynch [31] extended the concept of weaving Hilbert space frames to the Banach space setting. We study weaving properties of ordinary frames, generalized frames and continuous frames in separable Hilbert spaces. The contents of the thesis have been published/accepted/communicated in [55, 57, 58, 101, 102]. The thesis is apportioned into five chapters. Chapter 1 provides a brief review of frames and Riesz bases in separable Hilbert spaces. Chapters 2 to 5 are the embodiment of our study in the thesis.