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In this paper we use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the $L^2$ sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a multiply-covered circle. Moreover, we show that curves in any homotopy class with initially small $L^3lVert k_srVert_2^2$ enjoy a uniform length bound under the flow, yielding the convergence result in these cases.
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In this paper we consider the steepest descent $L^2$-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally convex of class $C^2$ or embedded of class $W^{2,2}$ bounding a convex domain), the flow converges smoothly to a round expanding multiply-covered circle.
The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between topological spaces endowed with continuous real-valued functions. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance.
In this paper, we consider the $L^2$-gradient flow for the modified $p$-elastic energy defined on planar closed curves. We formulate a notion of weak solution for the flow and prove the existence of global-in-time weak solutions with $p ge 2$ for initial curves in the energy space via minimizing movements. Moreover, we prove the existence of unique global-in-time solutions to the flow with $p=2$ and obtain their subconvergence to an elastica as $t to infty$.
A compact complex manifold $V$ is called Vaisman if it admits an Hermitian metric which is conformal to a Kahler one, and a non-isometric conformal action by $mathbb C$. It is called quasi-regular if the $mathbb C$-action has closed orbits. In this case the corresponding leaf space is a projective orbifold, called the quasi-regular quotient of $V$. It is known that the set of all quasi-regular Vaisman complex structures is dense in the appropriate deformation space. We count the number of closed elliptic curves on a Vaisman manifold, proving that their number is either infinite or equal to the sum of all Betti numbers of a Kahler orbifold obtained as a quasi-regular quotient of $V$. We also give a new proof of a result by Rukimbira showing that the number of Reeb orbits on a Sasakian manifold $M$ is either infinite or equal to the sum of all Betti numbers of a Kahler orbifold obtained as an $S^1$-quotient of $M$.
Closed timelike curves are among the most controversial features of modern physics. As legitimate solutions to Einsteins field equations, they allow for time travel, which instinctively seems paradoxical. However, in the quantum regime these paradoxes can be resolved leaving closed timelike curves consistent with relativity. The study of these systems therefore provides valuable insight into non-linearities and the emergence of causal structures in quantum mechanics-essential for any formulation of a quantum theory of gravity. Here we experimentally simulate the non-linear behaviour of a qubit interacting unitarily with an older version of itself, addressing some of the fascinating effects that arise in systems traversing a closed timelike curve. These include perfect discrimination of non-orthogonal states and, most intriguingly, the ability to distinguish nominally equivalent ways of preparing pure quantum states. Finally, we examine the dependence of these effects on the initial qubit state, the form of the unitary interaction, and the influence of decoherence.
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