We study an efficient numerical method for solving difficult `saddle point' linear systems that arise at every time step in the discretization of incompressible flow problems, including those modeled by the Navier-Stokes equations (e.g. water, oil, air under 220 mph) and magnetohydrodynamics (flows on conducting fluids). By combining an algebraic splitting of the block saddle point matrix, a particular approximation of the Schur complement system, and an incremental version of the associated time stepping algorithm, we are able to decompose the linear systems into smaller pieces that are easier to solve. We prove that the approximations made in the solve process are third (or fourth) order, and so are appropriate for use with second order time stepping methods. Numerical tests are performed which verify excellent performance of the methods on a variety of test problems.
Ramsey theory dates back to the 1930's and computing Ramsey numbers is a notoriously difficult problem in combinatorics. We study Ramsey numbers of graphs under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph such that no triangle has all its edges colored differently. Given a graph H and apositive integer k, the Gallai-Ramsey number of H is the least positive integer N such that every Gallai coloring of the complete graph K_N using k colors contains a monochromatic copy of H. Gallai-Ramsey numbers of graphs are more well-behaved, though computing them is far from trivial. In this talk, I will present our recent results on Gallai-Ramsey numbers of cycles.
Quantum information theory has emerged at the junction of multiple
disciplines, blending concepts and techniques from physics,
mathematics, and computer science. At the heart of this new field is
the effort to understand and answer the following questions: How is
information stored and manipulated in a quantum system, and how well
is this information preserved under physical processes? In this talk,
I will introduce some key notions in quantum information using
mathematical formalism, including quantum states, quantum channels,
and quantum entanglement. Some of the mathematical tools needed to
understand the problems of detecting and manipulating entanglement
will be presented.
In finite dimensional quantum information, transformations
between systems are represented by quantum channels: completely
positive and trace preserving linear maps between matrix spaces.
Single-shot quantum channel discrimination is the task of determining
which of two known channels is acting on a system, given only a single
use. We will review how entanglement can be used in this task to gain
an advantage, and how this phenomenon is directly connected to
properties of norms measuring the distance between the channels. In
particular, the advantage provided by entanglement is quantified by
the gap between the completely bounded trace norm and the induced
trace norm. We will discuss recent results related to these norms and
single-shot quantum channel discrimination, as well as open problems.
A beautiful example of spontaneous pattern formation appears in the distribution of vegetation in some dry-land environments.
Examples from Africa, Australia and the Americas reveal that vegetation, at a community scale, may spontaneously form into stripe-like bands, alternating with striking regularity with bands of bare soil, in response to aridity stress. A typical length scale for such patterns is
100 m; they are readily surveyed by modern satellites (and explored from your armchair in Google maps). These ecosystems represent some of Earth’s most vulnerable under threats of desertification, and some ecologists have suggested that the patterns, so easily monitored by satellites, may have potential as early warning signs of ecosystem collapse. I will describe efforts based in simple mathematical models, inspired by decades of physics research on pattern formation, to understand the morphology of the patterns. I will also describe efforts at analyzing the patterns via the satellite images, which, in some cases, we can accurately align with the aerial survey photographs from the 1950s to investigate details of the pattern evolution.
Combining statistical parametric maps (SPM) from individual subjects is the goal in some types of group-level analyses of functional magnetic resonance imaging (fMRI) data. Brain maps are usually combined using a simple average across subjects, making them susceptible to subjects with outlying values. Furthermore, t tests are prone to false positives and false negatives when outlying values are observed. We propose a regularized unsupervised aggregation method for SPMs to find an optimal weight for aggregation, which aids in detecting and mitigating the effect of outlying subjects. We also present a bootstrap-based weighted t test using the optimal weights to construct an activation map robust to outlying subjects. We validate the performance of the proposed aggregation method and test using simulated and real data examples. Results show that the regularized aggregation approach can effectively detect outlying subjects, lower their weights, and produce robust SPMs.
In the Hilbert space formulation, quantum states are density matrices, i.e., positive
semidefinite matrices with trace one, and quantum channels are trace preserving completely positive linear maps on matrices. In this talk, we will present some results on the existence of quantum channels that send certain quantum states to other quantum states. Additional conditioons on the quantum channels may be imposed to satisfy certain
We consider bootstrap percolation in tilings of the plane by regular polygons. First, we determine the percolation threshold for each of the infinite Archimedean lattices.
More generally, let T denote the set of plane tilings t by regular polygons such that if t contains one instance of a vertex type, then t contains infinitely many instances of that type. We show that no tiling in T has threshold 4 or more.
This material is self-contained, and requires no particular background. We'll share many open problems, as well as the intuition behind these results.
This will be an introduction to how the graph (of a real symmetric matrix, or a general matrix) constrains the multiplicities of its eigenvalues. The case of trees is most interesting and this will be described in some detail, including the maximum multiplicity, the minimum number of distinct eigenvalues, the possible lists of multiplicities and how they come about. This talk will be an overview of the subject and a second talk in the GAG seminar will continue the description. The subject has just been covered in a new book from Cambridge University Press (same title), and REU students have helped to make some very important contributions over the years. There is still plenty of work to be done in the area, which has been of interest to algebraic graph theorists and numerical analysts, as well as matrix theorists.