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Series: SIAM Student Seminar

In this talk, we provide a deterministic algorithm for robotic path finding in unknown environment and an associated graph generator use only potential information. Also we will generalize the algorithm into a path planning algorithm for certain type of optimal control problems under some assumptions and will state some approximation methods if certain assumption no longer holds in some cases. And we hope to prove more theoretical results for those algorithms to guarantee the success.

Series: SIAM Student Seminar

A local Hausdorff dimension is defined on a metric space. We study its properties and use it to define a local measure. We show that in many circumstances we can recover the global Hausdorff dimension from the local one. We give an example of a compact metric space with a continuum of local dimension values. We define the dimension of a measure and connect the definition to that of local Hausdorff dimension and measure for a class of spaces called (variable) Ahlfors Q-regular. Very little background knowledge, aside from basic familiarity with metric spaces, will be assumed.

Series: SIAM Student Seminar

This is a summary of result of LUC HILLAIRET AND CHRIS JUDGE.

Series: SIAM Student Seminar

In this paper, we consider selection of a sliding vector fieldof Filippov type on a discontinuity manifold $\Sigma$ of co-dimension 3(intersection of three co-dimension 1 manifolds). We propose an extension of the “moments vector field”to this case, and - under the assumption that $\Sigma$ is nodally attractive -we prove that our extension delivers a uniquely definedFilippov vector field. As it turns out, the justification of our proposed extension requiresestablishing invertibility of certain sign matrices. Finally,we also propose the extension of the moments vector field todiscontinuity manifolds of co-dimension 4 and higher.

Series: SIAM Student Seminar

Periodic eigendecomposition algorithm for calculating eigenvectors
of a periodic
product of a sequence of matrices, an extension of the periodic
Schur decomposition, is formulated
and compared with the recently proposed covariant vectors
algorithms. In contrast to those, periodic
eigendecomposition requires no power iteration and is capable of
determining not only the real
eigenvectors, but also the complex eigenvector pairs. Its
effectiveness, and in particular its ability
to resolve eigenvalues whose magnitude differs by hundreds of
orders, is demonstrated by applying
the algorithm to computation of the full linear stability spectrum
of periodic solutions of Kuramoto-Sivashinsky system.

Series: SIAM Student Seminar

By showing a duality relation between the Sobolev and
Hardy-Littlewood-Sobolev inequalities, I discuss a proof of the sharp
Sobolev inequality. The duality relation between these two inequalities is
known since 1983 and has led to interesting recent work on the inequalities
(which may be the topic of future talks).

Series: SIAM Student Seminar

In 1956 Mark Kac introduced an equation governing the evolution of the velocity distribution
of n particles. In his derivation, he assumed a stochastic
model based on binary collisions which preserves energy but not momentum. In this talk I will
describe Kac's model and the main theorem of Kac's paper :
that solutions with chaotic initial data can be related to the solutions Boltzmann type
equation.

Series: SIAM Student Seminar

We introduce a new model for cell phone signal problem, which is stochastic van der Pol oscillator with condition that ensures global boundedness in phase space and keeps unboundedness for frequency. Also we give a new definition for stochastic Poincare map and find a new approximation to return time and point. The new definition is based on the numerical observation. Also we develop a new approach by using dynamic tools, such as method of averaging
and relaxation method, to estimate the return time and return point. Thus we can show that the return time is always not Gaussian and return point's distribution is
not symmetric under certain section.

Series: SIAM Student Seminar

Erdos and Szemeredi conjectured that if one has a set of n numbers, one must have
either the sumset or product set be of nearly maximal size, cn^2/log(n). In this
talk, he will introduce the sum-product problem in the reals, show previous,
beautiful geometric proofs by Solymosi and Elekes, and discuss some recent progress
by Amirkhanyan, Croot, Pryby and Bush.

Series: SIAM Student Seminar

Fix k vertices in a graph G, say a_1,...,a_k, if there exists
a cycle that visits these vertices with this specified order,
we say such a cycle is (a_1,a_2,...,a_k)-ordered. It is shown
by Thomas and Wollan that any 10k-connected graph is k-linked,
therefore any 10k-connected graph has an (a_1,a_2,...,a_k)-ordered
for any a_1,...,a_k. However, it is possible that we can improve
this bound when k is small. It is shown by W. Goddard that any
4-connected maximal planar graph has an (a_1,...,a_4)-ordered
cycle for any choice of 4 vertices. We will present a complete
characterization of 4-ordered cycle in planar graphs. Namely,
for any four vertices a,b,c,d in planar graph G, if there is no
(a,b,c,d)-ordered cycle in G, then one of the follows holds:
(1) there is a cut S separating {a,c} from {b,d} with |S|\leq 3;
(2) roughly speaking, a,b,d,c "stay" in a face of G with this order.