Other Methods in Space Discretization 5. Chapter 6. Nonlinear Problems 6. Chapter 7. Domain Decomposition Method 7. In accordance with the developments in computation, theoretical studies on numerical schemes are now fruitful and highly needed. In an article on the finite element method applied to evolutionary problems was published.
Following the method, basically this book studies various schemes from operator theoretical points of view. Many parts are devoted to the finite element method, but other schemes and problems charge simulation method, domain decomposition method, nonlinear problems, and so forth are also discussed, motivated by the observation that practically useful schemes have fine mathematical structures and the converses are also true. Every numerical scheme is thoroughly dissected.
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We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit. If you decide to participate, a new browser tab will open so you can complete the survey after you have completed your visit to this website. Thanks in advance for your time. Skip to content. Search for books, journals or webpages Laprade Professor of Mathematics Keywords in this area Fourier analysis, functional analysis Other research areas.
Jianfeng Lu Associate Professor of Mathematics Keywords in this area electronic structure models, calculus of variations, semiclassical analysis Other research areas. Jonathan Christopher Mattingly James B. James H.
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Nolen Associate Professor of Mathematics Keywords in this area partial differential equations, probability, asymptotic analysis, homogenization Other research areas. William L. Lillian Beatrix Pierce Nicholas J. Leonardy Associate Professor Keywords in this area oscillatory integrals, Carleson operators, discrete operators Other research areas.
Michael C. Reed Professor of Mathematics Other research areas. David G.follow link
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Schaeffer James B. Duke Professor Emeritus of Mathematics Other research areas. Mark A. Stern Professor of Mathematics Keywords in this area geometric analysis, elliptic partial differential equations Other research areas.
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Witelski Professor in the Department of Mathematics Keywords in this area perturbation methods Other research areas. For example, the cohomology of algebraic varieties carries all kinds of "extra" structures that has no counterpart in the cohomology of, say, a manifold or a cell complex. I have been particularly interested in questions concerning moduli spaces in algebraic geometry.
Main supervisor: Sven Raum. They were originally introduced to formalise quantum mechanics on the one hand and to understand the unitary representation theory of groups on the other hand. Keeping this origin of the field in mind, my research merges the rapid development in structure theory of operator algebras that was achieved over the past decade with the needs of an emerging general theory of locally compact groups beyond Lie theory. Fine structure of operator algebras associated with groups acting on trees.
Groups acting on tree provide a rich class of examples, which partially behave like algebraic groups and partially like discrete groups. Superrigidity for locally compact groups. In operator algebras, a superrigidity result recovers a group from one of its group operator algebras.
Such results not only provide a definitve link between the group and its operator algebra, in addition their proofs usual reveal interesting information about the structure of the operator algebra. In the study of Partial Differential Equations and in Harmonic Analysis, an important role is played by the so-called pseudodifferential operators.
For instance, for equations that describe electric potential and steady-state heat flow elliptic equations one can construct explicit solutions using pseudodifferential operators. Roughly speaking, these operators act on functions or signals by filtering attenuating or amplifying specific frequencies of those. For equations that describe wave propagation hyperbolic equations , a similar role is played by Fourier integral operators. These tools allow us to obtain a priori estimates for the solutions, and study their behaviour and properties.
Therefore, being able to estimate these operators in different function spaces is important for measuring the size and regularity of the solutions of PDEs in those spaces. In controlling height and width of a solution, the most important example of such spaces are the Lebesgue spaces Lp. Due to their rearrangement-invariant nature, these spaces are blind to the description of where solutions are concentrated, and thus the consideration of Lebesgue spaces with weights appears naturally. An important role is played by the so-called Muckenhoupt Ap weights.
For nonlinear PDEs , the multilinear counterpart of pseudodifferential and Fourier integral operators play a crucial role. My recent research interests have been dealing with questions regarding both linear and multilinear operators of those described above, and in particular with those of rough type. To get involved in a project in these areas requires a strong background and interest in Harmonic Analysis and PDEs. Some examples of lines of research that one could pursue:. Eigenvalues and, more generally, spectra of differential operators appear naturally in numerous physical models, for instance as frequencies of vibrating strings and membranes, or as energies of quantum systems.
Their investigation is an important and lively field in mathematical physics. Since the spectra of most models cannot be calculated explicitly, there is a strong need for qualitative and quantitative estimates.
Possible research areas for a PhD include:. Main supervisor: Alan Sola. Coordinate shifts acting on Banach spaces of analytic functions represent a concrete and compelling incarnation of operator theory. At first glance, considering the action of a such simple operators on specific function spaces may appear to lose much of the generality that makes operator theory a powerful and flexible tool in mathematics. One can show, however, that many Hilbert space contractions are unitarily equivalent to the shift acting on model subspaces of the Hardy space. The class of analytic 2-isometries has the Dirichlet shift as a natural realization, and there are many other such models.
Thus, understanding the invariant subspaces and cyclic vectors of coordinate shifts is important, and leads to a better understanding of more general operators. Moreover, by working with analytic functions, we are able to connect operator-theoretic questions to deep problems in complex function theory such as boundary behavior, vanishing properties, and so on. For example, answering the question of whether a specific analytic function is cyclic with respect to shifts acting on a function space frequently amounts to analyzing the size and properties of the zero set of the function.
These types of questions have been studied by many mathematicians in the one-variable setting of the unit disk, and many remarkable results have been obtained. However, the higher-dimensional analogs of coordinate shifts acting on function spaces in polydisks have received somewhat less attention, especially for function spaces beyond the Hardy spaces. In recent years, I have been particularly interested in weighted Dirichlet spaces, which can be defined in terms of area-integrability of partial derivatives of an analytic function.
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These spaces are challenging due to the relative smoothness of their elements, yet are rich enough to allow for an interesting subspace structure. In a recent series of papers, my coauthors and I have started making headway on the problem of identifying cyclic vectors in weighted Dirichlet spaces in the bidisk, and we have found techniques for checking membership in such spaces of functions having singularities on the boundary of the bidisk. Last updated: March 5, Source: Department of Mathematics. Department of Mathematics. To submenu. Study Information. First and Second Cycle Courses.
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