Optimal Flood Control

A mathematical model for optimal control of the water levels in a chain of reservoirs is studied. Some remarks regarding sensitivity with respect to the time horizon, terminal cost and forecast of inflow are made

Subvarieties of generic hypersurfaces in any variety

Let W be a projective variety of dimension n+1, L a free line bundle on W, X in $H^0(L^d)$ a hypersurface of degree d which is generic among those given by sums of monomials from $L$, and let $f : Y \to X$ be a generically finite map from a smooth m-fold Y. We suppose that f is r-filling, i.e. upon deforming X in $H^0(L^d)$, f deforms in a family such that the corresponding deformations of $Y^r$ dominate $W^r$. Under these hypotheses we give a lower bound for the dimension of a certain linear system on the Cartesian product $Y^r$ having certain vanishing order on a diagonal locus as well as on a double point locus. This yields as one application a lower bound on the dimension of the linear system |K_{Y} - (d - n + m)f^*L - f^*K_{W}| which generalizes results of Ein and Xu (and in weaker form, Voisin). As another perhaps more surprising application, we conclude a lower bound on the number of quadrics containing certain projective images of Y.Comment: We made some improvements in the introduction and definitions. In an effort to clarify the arguments we separated the 1-filling case from the r-filling case and we gave a more detailed proof of the key lemma. The article will appear in the Math. Proc. Cambridge Philos. So

FROST -- Fast row-stochastic optimization with uncoordinated step-sizes

In this paper, we discuss distributed optimization over directed graphs, where doubly-stochastic weights cannot be constructed. Most of the existing algorithms overcome this issue by applying push-sum consensus, which utilizes column-stochastic weights. The formulation of column-stochastic weights requires each agent to know (at least) its out-degree, which may be impractical in e.g., broadcast-based communication protocols. In contrast, we describe FROST (Fast Row-stochastic-Optimization with uncoordinated STep-sizes), an optimization algorithm applicable to directed graphs that does not require the knowledge of out-degrees; the implementation of which is straightforward as each agent locally assigns weights to the incoming information and locally chooses a suitable step-size. We show that FROST converges linearly to the optimal solution for smooth and strongly-convex functions given that the largest step-size is positive and sufficiently small.Comment: Submitted for journal publication, currently under revie

Pair density waves and vortices in an elongated two-component Fermi gas

We study the vortex structures of a two-component Fermi gas experiencing a uniform effective magnetic field in an anisotropic trap that interpolates between quasi-one dimensional (1D) and quasi-two dimensional (2D). At a fixed chemical potential, reducing the anisotropy (or equivalently increasing the attractive interactions or increasing the magnetic field) leads to instabilities towards pair density waves, and vortex lattices. Reducing the chemical potential stabilizes the system. We calculate the phase diagram, and explore the density and pair density. The structures are similar to those predicted for superfluid Bose gases. We further calculate the paired fraction, showing how it depends on chemical potential and anisotropy.Comment: 6 pages, 3 figure

Z2 spin liquid in S=1/2 Heisenberg model on Kagome lattice: A projective symmetry group study of Schwinger-fermion mean-field states

With strong geometric frustration and quantum fluctuations, S=1/2 quantum Heisenberg antiferromagnets on the Kagome lattice has long been considered as an ideal platform to realize spin liquid (SL), a novel phase with no symmetry breaking and fractionalized excitations. A recent numerical study of Heisenberg S=1/2 Kagome lattice model (HKLM) show that in contrast to earlier studies, the ground state is a singlet-gapped SL with signatures of Z2 topological order. Motivated by this numerical discovery, we use projective symmetry group to classify all 20 possible Schwinger-fermion mean-field states of Z2 SLs on Kagome lattice. Among them we found only one gapped Z2 SL (which we call Z2[0,\pi]\beta state) in the neighborhood of U(1)-Dirac SL state, whose energy is found to be the lowest among many other candidate SLs including the uniform resonating-valentce-bond states. We thus propose this Z2[0,\pi]\beta state to be the numerically discovered SL ground state of HKLM.Comment: 12 pages, 2 figures, revtex4, published versio