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Series: Dissertation Defense

This thesis addresses four topics in the area of applied harmonic analysis. First, we show that the affine densities of separable wavelet frames affect the frame properties. In particular, we describe a new relationship between the affine densities, frame bounds and weighted admissibility constants of the mother wavelets of pairs of separable wavelet frames. This result is also extended to wavelet frame sequences. Second, we consider affine pseudodifferential operators, generalizations of pseudodifferential operators that model wideband wireless communication channels. We find two classes of Banach spaces, characterized by wavelet and ridgelet transforms, so that inclusion of the kernel and symbol in appropriate spaces ensures the operator if Schatten p-class. Third, we examine the Schatten class properties of pseudodifferential operators. Using Gabor frame techniques, we show that if the kernel of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. This result improves existing theorems and is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class. The implications of this result for the Kohn-Nirenberg symbol of a pseudodifferential operator are also described. Lastly, Fourier integral operators are analyzed with Gabor frame techniques. We show that, given a certain smoothness in the phase function of a Fourier integral operator, the inclusion of the symbol in appropriate mixed modulation spaces is sufficient to guarantee that the operator is Schatten p-class.

Series: Other Talks

Join math club for Dr. Baker's mathematical magic show.

Series: Mathematical Biology Seminar

I will consider a class of mathematical models of decision

making. These models are based on dynamics in the neighborhood of

unstable equilibria and involve random perturbations due to small

noise. I will report results on the vanishing noise limit for these

systems, providing precise predictions about the statistics of

decision making times and sequences of unstable equilibria visited by

the process. Mathematically, the results are based on the analysis of

random Poincare maps in the neighborhood of each equilibrium point. I

will also discuss some experimental data.

making. These models are based on dynamics in the neighborhood of

unstable equilibria and involve random perturbations due to small

noise. I will report results on the vanishing noise limit for these

systems, providing precise predictions about the statistics of

decision making times and sequences of unstable equilibria visited by

the process. Mathematically, the results are based on the analysis of

random Poincare maps in the neighborhood of each equilibrium point. I

will also discuss some experimental data.

Series: PDE Seminar

A classic story of nonlinear science started with the

particle-like

water wave that Russell famously chased on horseback in 1834. I will

recount progress regarding the robustness of solitary waves in

nonintegrable model systems such as FPU lattices, and discuss progress

toward a proof (with Shu-Ming Sun) of spectral stability of small

solitary waves for the 2D Euler equations for water of finite depth

without surface tension.

particle-like

water wave that Russell famously chased on horseback in 1834. I will

recount progress regarding the robustness of solitary waves in

nonintegrable model systems such as FPU lattices, and discuss progress

toward a proof (with Shu-Ming Sun) of spectral stability of small

solitary waves for the 2D Euler equations for water of finite depth

without surface tension.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

The Scenery Reconstruction Problem consists in trying to reconstruct

a coloring of the integers given only the observations made by

a random walk. For this we consider a random walk S and

a coloring of the integers X. At time $t$ we observe

the color $X(S(t))$. The coloring is i.i.d. and we show that

given only the sequence of colors

$$X(S(0)),X(S(1)),X(S(2)),...$$

it is possible to reconstruct $X$ up to translation

and reflection. The solution depends on the property of the

random walk and the distribution of the coloring.

Longest Common Subsequences (LCS) are widely used in genetics.

If we consider two sequences X and Y, then a common subsequence

of X and Y is a string which is a subsequence of X and of Y at the same

time. A Longest Common Subsequence of X and Y is a common

subsequence of X and Y of maximum length. The problem of the asymptotic

order of the flucutation for the LCS of independent random

strings has been open for decades. We have now been able to

make progress on this problem for several important cases.

We will also show the connection to the Scenery Reconstruction

Problem.

a coloring of the integers given only the observations made by

a random walk. For this we consider a random walk S and

a coloring of the integers X. At time $t$ we observe

the color $X(S(t))$. The coloring is i.i.d. and we show that

given only the sequence of colors

$$X(S(0)),X(S(1)),X(S(2)),...$$

it is possible to reconstruct $X$ up to translation

and reflection. The solution depends on the property of the

random walk and the distribution of the coloring.

Longest Common Subsequences (LCS) are widely used in genetics.

If we consider two sequences X and Y, then a common subsequence

of X and Y is a string which is a subsequence of X and of Y at the same

time. A Longest Common Subsequence of X and Y is a common

subsequence of X and Y of maximum length. The problem of the asymptotic

order of the flucutation for the LCS of independent random

strings has been open for decades. We have now been able to

make progress on this problem for several important cases.

We will also show the connection to the Scenery Reconstruction

Problem.

Series: Algebra Seminar

We discuss some arithmetic properties of modular varieties

of D-elliptic sheaves, such as the existence of rational points or

the structure of their "fundamental domains" in the Bruhat-Tits

building. The notion of D-elliptic sheaf is a generalization of the

notion of Drinfeld module. D-elliptic sheaves and their moduli

schemes were introduced by Laumon, Rapoport and Stuhler in their

proof of certain cases of the Langlands conjecture over function

fields.

of D-elliptic sheaves, such as the existence of rational points or

the structure of their "fundamental domains" in the Bruhat-Tits

building. The notion of D-elliptic sheaf is a generalization of the

notion of Drinfeld module. D-elliptic sheaves and their moduli

schemes were introduced by Laumon, Rapoport and Stuhler in their

proof of certain cases of the Langlands conjecture over function

fields.

Monday, March 8, 2010 - 13:00 ,
Location: Skiles 255 ,
Chun Liu ,
Penn State/IMA ,
Organizer:

Almost all models for complex fluids can be fitted into the energetic variational framework. The advantage of the approach is the revealing/focus of the competition between the kinetic energy and the internal "elastic" energies. In this talk, I will discuss two very different engineering problems: free interface motion in Newtonian fluids and viscoelastic materials. We will illustrate the underlying connections between the problems and their distinct properties. Moreover, I will present the analytical results concerning the existence of near equilibrium solutions of these problems.

Series: Other Talks

For more information, see the <A href="/~rohrs/FranklinColloquium.pdf">flyer</a>.

Statistics is now a part of the K-12 curriculum (including elementary school) and there is much activity in the area of statistics education. This colloquium is intended for any and all faculty, staff, and students, who are interested in, have taught, or have children in k-12 schools.

Series: Other Talks

The speaker has combined his two loves to create a dynamic presentation called "Mathemagics," suitable for all audiences, where he demonstrates and explains his secrets for performing rapid mental calculations faster than a calculator. Reader's Digest calls him "America's Best Math Whiz". He has presented his high energy talk for thousands of groups throughout the world. This event is free but reservations are required. The signup form will be available before 5pm on February 25.

See details about the speaker.

See details about the speaker.

Series: Other Talks

The SIAM Student Chapter at Georgia Tech will be hosting this conference. It is an extension of the ACES Workshop which has been held yearly by the universities of Auburn, Clemson, Emory, and South Carolina since 2006. As with the ACES Workshop, this conference is an opportunity for graduate students to present their research in applied mathematics and related fields as well as to meet with other graduate students from different universities and departments. See the conference site for more details.