Search results for "Singularity theory"
showing 5 items of 5 documents
The Topology of the Milnor Fibration
2020
The fibration theorem for analytic maps near a critical point published by John Milnor in 1968 is a cornerstone in singularity theory. It has opened several research fields and given rise to a vast literature. We review in this work some of the foundational results about this subject, and give proofs of several basic “folklore theorems” which either are not in the literature, or are difficult to find. Examples of these are that if two holomorphic map-germs are isomorphic, then their Milnor fibrations are equivalent, or that the Milnor number of a complex isolated hypersurface or complete intersection singularity \((X, \underline {0})\) does not depend on the choice of functions that define …
Topological charge selection rule for phase singularities
2009
We present a study of the dynamics and decay pattern of phase singularities due to the action of a system with a discrete rotational symmetry of finite order. A topological charge conservation rule is identified. The role played by the underlying symmetry is emphasized. An effective model describing the short range dynamics of the vortex clusters has been designed. A method to engineer any desired configuration of clusters of phase singularities is proposed. Its flexibility to create and control clusters of vortices is discussed.
Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media
2009
[EN] We determine the functional behavior near the discrete rotational symmetry axis of discrete vortices of the nonlinear Schrodinger equation. We show that these solutions present a central phase singularity whose charge is restricted by symmetry arguments. Consequently, we demonstrate that the existence of high-charged discrete vortices is related to the presence of other off-axis phase singularities, whose positions and charges are also restricted by symmetry arguments. To illustrate our theoretical results, we offer two numerical examples of high-charged discrete vortices in photonic crystal fibers showing hexagonal discrete rotational invariance
Three-qutrit entanglement and simple singularities
2016
In this paper, we use singularity theory to study the entanglement nature of pure three-qutrit systems. We first consider the algebraic variety $X$ of separable three-qutrit states within the projective Hilbert space $\mathbb{P}(\mathcal{H}) = \mathbb{P}^{26}$. Given a quantum pure state $|\varphi\rangle\in \mathbb{P}(\mathcal{H})$ we define the $X_\varphi$-hypersuface by cutting $X$ with a hyperplane $H_\varphi$ defined by the linear form $\langle\varphi|$ (the $X_\varphi$-hypersurface of $X$ is $X\cap H_\varphi \subset X$). We prove that when $|\varphi\rangle$ ranges over the SLOCC entanglement classes, the "worst" possible singular $X_\varphi$-hypersuface with isolated singularities, has…
The Calderón problem for the fractional Schrödinger equation with drift
2020
We investigate the Calder\'on problem for the fractional Schr\"odinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does \emph{not} enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many \emph{generic} measurements is discussed. Here the genericity is obtained through \emph{singularity theory} which might also be interesting in the context of hybrid inverse pro…