Mathematics
http://hdl.handle.net/2022/13431
2015-05-25T05:36:11ZConvergence of Hill's method for nonselfadjoint operators
http://hdl.handle.net/2022/19108
Convergence of Hill's method for nonselfadjoint operators
Johnson, M.A.; Zumbrun, K.
By the introduction of a generalized Evans function defined by an appropriate 2- modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of non-self-adjoint operators which previously had not been treated in a simple way. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrices
2012-01-01T00:00:00ZA khintchine decomposition for free probability
http://hdl.handle.net/2022/19106
A khintchine decomposition for free probability
Williams, J.D.
Let μ be a probability measure on the real line. In this paper we prove that there exists a decomposition $\mu = \mu_{0}\boxplus \mu_{1}\dots\boxplus \mu_{n}$such that $\mu_{0}$ is infinitely divisible, and $\mu_{i}$ is indecomposable for $i \geq 1$. Additionally, we prove that the family of all $\boxplus$-divisors of a measure $\mu$ is compact up to translation. Analogous results are also proven in the case of multiplicative convolution
2012-01-01T00:00:00ZHandle addition for doubly-periodic Scherk surfaces
http://hdl.handle.net/2022/19104
Handle addition for doubly-periodic Scherk surfaces
Weber, M.; Wolf, M.
We prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus g with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The proof of the family of immersed surfaces is by induction on genus, while the proof of embeddedness is by the conjugate Plateau method.
2012-01-01T00:00:00ZTetrahedral forms in monoidal categories and 3-manifold invariants
http://hdl.handle.net/2022/19102
Tetrahedral forms in monoidal categories and 3-manifold invariants
Geer, N.; Kashaev, R.; Turaev, V.
We introduce systems of objects and operators in linear monoidal categories called $\hat{\Psi}$-systems. A $\hat{\Psi}$-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold $M$, a principal bundle over $M$, a link in $M$). This construction generalizes the quantum dilogarithmic invariant of links appearing in the original formulation of the volume conjecture. We conjecture that all quantum groups at odd roots of unity give rise to $\hat{\Psi}$-systems and we verify this conjecture in the case of the Borel subalgebra of quantum sl$_{2}$.
2012-01-01T00:00:00Z