Floer theory for negative line bundles via Gromov-Witten invariants

Let M be the total space of a negative line bundle over a closed symplectic manifold. We prove that the quotient of quantum cohomology by the kernel of a power of quantum cup product by the first Chern class of the line bundle is isomorphic to symplectic cohomology. We also prove this for negative vector bundles and the top Chern class. We explicitly calculate the symplectic and quantum cohomologies of O(-n) over P^m. For n=1, M is the blow-up of C^{m+1} at the origin and symplectic cohomology has rank m. The symplectic cohomology vanishes if and only if the first Chern class of the line bundle is nilpotent in quantum cohomology. We prove a Kodaira vanishing theorem and a Serre vanishing theorem for symplectic cohomology. In general, we construct a representation of \pi_1(Ham(X,\omega)) on the symplectic cohomology of symplectic manifolds X conical at infinity.Comment: 53 pages; version 3: improved discussion of maximum principle for negative vector bundles. The final version is published in Advances in Mathematic

Circle-actions, quantum cohomology, and the Fukaya category of Fano toric varieties

We define a class of non-compact Fano toric manifolds, called admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed point sets. We prove closed-string mirror symmetry for this class of manifolds: the Jacobian ring of the superpotential is the symplectic cohomology (not the quantum cohomology). Moreover, SH(M) is obtained from QH(M) by localizing at the toric divisors. We give explicit presentations of SH(M) and QH(M), using ideas of Batyrev, McDuff and Tolman. Assuming that the superpotential is Morse (or a milder semisimplicity assumption), we prove that the wrapped Fukaya category for this class of manifolds satisfies the toric generation criterion, i.e. is split-generated by the natural Lagrangian torus fibres of the moment map with suitable holonomies. In particular, the wrapped category is compactly generated and cohomologically finite. The proof uses a deformation argument, via a generic generation theorem and an argument about continuity of eigenspaces. We also prove that for any closed Fano toric manifold, if the superpotential is Morse (or a milder semisimplicity assumption) then the Fukaya category satisfies the toric generation criterion. The key ingredients are non-vanishing results for the open-closed string map, using tools from the paper by Ritter-Smith (we also prove a conjecture from that paper that any monotone toric negative line bundle contains a non-displaceable monotone Lagrangian torus). We also need to extend the class of Hamiltonians for which the maximum principle holds for symplectic manifolds conical at infinity, thus extending the class of Hamiltonian circle actions for which invertible elements can be constructed in SH(M).Comment: 70 pages (51 pages + appendices). Version 2: rewrote the Introduction, fixed a mistake (Remark 1.15), generation theorem generalized to all admissible toric manifolds (Section 1.8

Regional landform thresholds

Remote sensing technology allows us to recognize manifestations of regional thresholds, especially in the spatial characteristics of process agents. For example, a change in river channel pattern over a short distance reflects a threshold alteration in the physical controls of discharge and/or sediment. It is, therefore, a valuable indication of conditions as they exist. However, we probably will have difficulty determining whether the systemic parameters are now close to threshold conditions at which a different change will occur. This, of course, is a temporal and magnitude problem which is difficult to solve from the spatial characteristics

Vector constants of motion for time-dependent Kepler and isotropic harmonic oscillator potentials

A method of obtaining vector constants of motion for time-independent as well as time-dependent central fields is discussed. Some well-established results are rederived in this alternative way and new ones obtained.Comment: 18 pages, no figures, regular Latex article forma

The use of entropy for analysis and control of cognitive models

Measures of entropy are useful for explaining the behaviour of cognitive models. We demonstrate that entropy can not only help to analyse the performance of the model, but it can be used to control model pararmeters and improve the match between the model and data. We present a cognitive model that uses local computations of entropy to moderate its own behaviour and matches the data fairly well

Conservation Laws and the Multiplicity Evolution of Spectra at the Relativistic Heavy Ion Collider

Transverse momentum distributions in ultra-relativistic heavy ion collisions carry considerable information about the dynamics of the hot system produced. Direct comparison with the same spectra from $p+p$ collisions has proven invaluable to identify novel features associated with the larger system, in particular, the "jet quenching" at high momentum and apparently much stronger collective flow dominating the spectral shape at low momentum. We point out possible hazards of ignoring conservation laws in the comparison of high- and low-multiplicity final states. We argue that the effects of energy and momentum conservation actually dominate many of the observed systematics, and that $p+p$ collisions may be much more similar to heavy ion collisions than generally thought.Comment: 15 pages, 14 figures, submitted to PRC; Figures 2,4,5,6,12 updated, Tables 1 and 3 added, typo in Tab.V fixed, appendix B partially rephrased, minor typo in Eq.B1 fixed, minor wording; references adde

Deformations of symplectic cohomology and exact Lagrangians in ALE spaces

We prove that the only exact Lagrangian submanifolds in an ALE space are spheres. ALE spaces are the simply connected hyperkahler manifolds which at infinity look like C^2/G for any finite subgroup G of SL(2,C). They can be realized as the plumbing of copies of the cotangent bundle of a 2-sphere according to ADE Dynkin diagrams. The proof relies on symplectic cohomology.Comment: 35 pages, 3 figures, minor changes and corrected typo

Quantum Channels and Representation Theory

In the study of d-dimensional quantum channels $(d \geq 2)$, an assumption which is not very restrictive, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of SU(n) is a depolarizing channel for all $n$, but for most other representations this is not the case. Since the Bloch sphere is not appropriate here, we develop technology which is a generalization of Bloch's technique. Our method works by representing the density matrix as a polynomial in symmetrized products of Lie algebra generators, with coefficients that are symmetric tensors. Using these tensor methods we prove eleven theorems, derive many explicit formulas and show other interesting properties of quantum channels in various dimensions, with various Lie symmetry algebras. We also derive numerical estimates on the size of a generalized ``Bloch sphere'' for certain channels. There remain many open questions which are indicated at various points through the paper.Comment: 28 pages, 1 figur