Events
Department of Mathematics and Statistics
Texas Tech University
We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\alpha$ for some $\alpha>0$. We prove that the asymptotic behavior of a solution $y(t)$ near a finite blow-up time $T_*$ is $(T_*-t)^{-1/\alpha}\xi_*$ for some nonzero vector $\xi_*$. Specific error estimates for $|(T_*-t)^{1/\alpha}y(t)-\xi_*|$ are provided. In some typical cases, they can be a positive power of $(T_*-t)$ or $1/|\ln(T_*-t)|$. This depends on whether the decaying rate of the lower order term, relative to the size of the dominant term, is of a power or logarithmic form. Similar results are obtained for a class of nonlinear differential inequalities with finite time blow-up solutions. An application to a model of inhomogeneous populations will be given.
A class of space-like rotational hypersurfaces, represented by the parametrization $\mathbf{x}(u, v, w)$, is investigated within the four-dimensional pseudo-Euclidean space $\mathbb{E}^4_2$. The curvatures of the hypersurface are derived. In addition, the associated Laplace--Beltrami operator is computed, and it is shown that the hypersurface satisfies the eigenvalue equation $\Delta \mathbf{x} = \mathcal{A} \mathbf{x}$, where $\mathcal{A}$ is a $4 \times 4$ matrix.
 | Wednesday
Sep. 17
|
| Algebra and Number Theory
No Seminar
|
Abstract: Rooted in Approximation Theory, Optimal Recovery can be viewed as a trustworthy learning theory focusing on the worst case. Regrettably, compared to more popular Machine Learning alternatives, the classical theory of Optimal Recovery overlooked the computational aspect, with a few exceptions, e.g., the development of spline functions. Nowadays, modern optimization techniques facilitate advances — even theoretical ones — on the minimax problems that abound in the field. I will illustrate this point by selecting a few snippets from my recent work.
KEY WORDS/RELATED TOPICS: Prediction of (multivalued) functions based on merely convex models; Estimation of linear functionals in the space of continuous functions; Full Recovery from deterministically inaccurate data in Hilbert spaces; Estimation of linear functionals from stochastically inaccurate data; Prediction of the maxima of Lipschitz functions from inaccurate point values
When: 3:00 pm (Lubbock's local time is GMT -5)
Where: room Math 011 (Math Basement)
ZOOM details:
- Choice #1: use this
Direct Link that embeds meeting ID and passcode
or
- Choice #2: Log into zoom, then join by manually inputting the meeting ID and passcode ...
* Meeting ID: 915 2866 2672
* Passcode: applied
Abstract pdf
The Biomath seminar may be attended virtually Friday at 11:00 AM CDT (UT-5) via this Zoom link.
Meeting ID: 938 8653 3169
Passcode: 883472
abstract 2 PM CDT (UT-5)
Zoom link available from Dr. Brent Lindquist upon request.