Thurston's sphere packings on 3-dimensional manifolds, I

Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is locally determined by combinatorial scalar curvature up to scaling, which generalizes Cooper-Rivin-Glickenstein's local rigidity for tangential sphere packing on 3-dimensional manifolds. We also prove the infinitesimal rigidity that Thurston's Euclidean sphere packing can not be deformed (except by scaling) while keeping the combinatorial Ricci curvature fixed.Comment: Arguments are simplife

Rotated sphere packing designs

We propose a new class of space-filling designs called rotated sphere packing designs for computer experiments. The approach starts from the asymptotically optimal positioning of identical balls that covers the unit cube. Properly scaled, rotated, translated and extracted, such designs are excellent in maximin distance criterion, low in discrepancy, good in projective uniformity and thus useful in both prediction and numerical integration purposes. We provide a fast algorithm to construct such designs for any numbers of dimensions and points with R codes available online. Theoretical and numerical results are also provided

Sliced rotated sphere packing designs

Space-filling designs are popular choices for computer experiments. A sliced design is a design that can be partitioned into several subdesigns. We propose a new type of sliced space-filling design called sliced rotated sphere packing designs. Their full designs and subdesigns are rotated sphere packing designs. They are constructed by rescaling, rotating, translating and extracting the points from a sliced lattice. We provide two fast algorithms to generate such designs. Furthermore, we propose a strategy to use sliced rotated sphere packing designs adaptively. Under this strategy, initial runs are uniformly distributed in the design space, follow-up runs are added by incorporating information gained from initial runs, and the combined design is space-filling for any local region. Examples are given to illustrate its potential application

Rigidity of infinite disk patterns

Let P be a locally finite disk pattern on the complex plane C whose combinatorics are described by the one-skeleton G of a triangulation of the open topological disk and whose dihedral angles are equal to a function \Theta:E\to [0,\pi/2] on the set of edges. Let P^* be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function. We show that P and P^* differ only by a euclidean similarity. In particular, when the dihedral angle function \Theta is identically zero, this yields the rigidity theorems of B. Rodin and D. Sullivan, and of O. Schramm, whose arguments rely essentially on the pairwise disjointness of the interiors of the disks. The approach here is analytical, and uses the maximum principle, the concept of vertex extremal length, and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern. A similar rigidity property holds for locally finite disk patterns in the hyperbolic plane, where the proof follows by a simple use of the maximum principle. Also, we have a uniformization result for disk patterns. In a future paper, the techniques of this paper will be extended to the case when 0 \le \Theta < \pi. In particular, we will show a rigidity property for a class of infinite convex polyhedra in the 3-dimensional hyperbolic space.Comment: 33 pages, published versio

Isospin effect on baryon and charge fluctuations from the pNJL model

We have studied the possible isospin corrections on the skewness and kurtosis of net-baryon and net-charge fluctuations in the isospin asymmetric matter formed in Au+Au collisions at RHIC-BES energies, based on a 3-flavor polyakov-looped Nambu-Jona-Lasinio model. With typical scalar-isovector and vector-isovector couplings leading to the splitting of $u$ and $d$ quark chiral phase transition boundaries and critical points, we have observed dramatic isospin effects on the susceptibilities, especially those of net-charge fluctuations. Reliable experimental measurements at even lower collision energies are encouraged to confirm the observed isospin effects.Comment: 6 pages, 3 figure

Two Particle States in a Box and the ss-Matrix in Multi-Channel Scattering

Using a quantum mechanical model, the exact energy eigenstates for two-particle two-channel scattering are studied in a cubic box with periodic boundary conditions. A relation between the exact energy eigenvalue in the box and the two-channel $S$-matrix elements in the continuum is obtained. This result can be viewed as a generalization of the well-known L\"uscher's formula which establishes a similar relation in elastic scattering.Comment: 4 pages, typeset with ws-ijmpa.cls. Talk presented at International Conference on QCD and Hadronic Physics, June 16-20, 2005, Beijing, China. One reference adde