Algebraic cycles on a generalized Kummer variety

We compute explicitly the Chow motive of any generalized Kummer variety associated to any abelian surface. In fact, it lies in the rigid tensor subcategory of the category of Chow motives generated by the Chow motive of the underlying abelian surface. One application of this calculation is to show that the Hodge conjecture holds for arbitrary products of generalized Kummer varieties. As another application, all numerically trivial 1-cycles on arbitrary products of generalized Kummer varieties are smash-nipotent.Comment: 11 pages. Comments are welcom

Bose-Einstein condensation of photons in the matter-dominated universe

In 1914, Planck introduced the concept of a white body. In nature, no true white bodies are known. We assume that the universe after last-scattering is an ideal white body that contains a tremendously large number of thermal photons and is at an extremely high temperature. Bose-Einstein condensation of photons in an ideal white body is investigated within the framework of quantum statistical mechanism. The computation shows that the transition temperature $T_c$ is a monotonically increasing function of the number density $n$ of photons. At finite temperature, we find that the condensate fraction $N_0(T)/N$ decreases continuously from unity to zero as the temperature increases from zero to the transition temperature $T_c$. Further, we study the radiation properties of an ideal white body. It is found that in the condensation region of $T<T_c$, the spectral intensity $I(\omega,T)$ of white body radiation is identical with Planck's law for blackbody radiation

Endpoint Strichartz estimates for magnetic wave equations on two dimensional hyperbolic spaces

In this paper, we prove that Kato smoothing effects for magnetic Schr\"odinger operators can yield the endpoint Strichartz estimates for linear wave equation with magnetic potential on two dimensional hyperbolic spaces. This result serves as a cornerstone for the author's work \cite{Lize} and collaborative work \cite{LMZ} in the study of asymptotic stability of harmonic maps for wave maps from $\Bbb R\times\Bbb H^2$ to $\Bbb H^2$.Comment: revised and enlarge

MRF-ZOOM: A Fast Dictionary Searching Algorithm for Magnetic Resonance Fingerprinting

Magnetic resonance fingerprinting (MRF) is a new technique for simultaneously quantifying multiple MR parameters using one temporally resolved MR scan. But its brute-force dictionary generating and searching (DGS) process causes a huge disk space demand and computational burden, prohibiting it from a practical multiple slice high-definition imaging. The purpose of this paper was to provide a fast and space efficient DGS algorithm for MRF. Based on an empirical analysis of properties of the distance function of the acquired MRF signal and the pre-defined MRF dictionary entries, we proposed a parameter separable MRF DGS method, which breaks the multiplicative computation complexity into an additive one and enabling a resolution scalable multi-resolution DGS process, which was dubbed as MRF ZOOM. The evaluation results showed that MRF ZOOM was hundreds or thousands of times faster than the original brute-force DGS method. The acceleration was even higher when considering the time difference for generating the dictionary. Using a high precision quantification, MRF can find the right parameter values for a 64x64 imaging slice in 117 secs. Our data also showed that spatial constraints can be used to further speed up MRF ZOOM.Comment: 7 figure

Circle patterns with obtuse exterior intersection angles

Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the case of obtuse exterior intersection angles.Comment: 24 pages, 1 figur

Asymptotic stability of solitons to 1D Nonlinear Schrodinger Equations in subcritical case

In this paper, we prove the asymptotic stability of solitary waves to 1D nonlinear Schr\"odinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by Cuccagna, Georgiev, Visciglia to overcome the weak decay with respect to $t$ of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schr\"odinger equations obtained by Staffilani to control the high moments of the solutions emerging from the vector fields method

Asymptotic stability of large energy harmonic maps under the wave map from 2D hyperbolic spaces to 2D hyperbolic spaces

In this paper, we prove that the large energy harmonic maps from $\Bbb H^2$ to $\Bbb H^2$ are asymptotically stable under the wave map equation.Comment: We improve the expressions, some improper notations are removed, grammer errors and typos are corrected, more background materials are involve

Global Schr\"odinger map flows to K\"ahler manifolds with small data in critical Sobolev spaces: High dimensions

In this paper, we prove that the Schr\"odinger map flows from $\Bbb R^d$ with $d\ge 3$ to compact K\"ahler manifolds with small initial data in critical Sobolev spaces are global. This is a companion work of our previous paper [23] where the energy critical case $d=2$ was solved. In the first part of this paper, for heat flows from $\Bbb R^d$ ($d\ge 3$) to Riemannian manifolds with small data in critical Sobolev spaces, we prove the decay estimates of moving frame dependent quantities in the caloric gauge setting, which is of independent interest and may be applied to other problems. In the second part, with a key bootstrap-iteration scheme in our previous work [23], we apply these decay estimates to the study of Schr\"odinger map flows by choosing caloric gauge. This work with our previous work solves the open problem raised by Tataru.Comment: slightly enlarged, submitte

Energy-momentum non-conservation on noncommutative spacetime and the existence of infinite spacetime dimension

From the constructions of the quantum spacetime, a four dimensional quantized spacetime can be embedded in a five dimensional continuous spacetime. Thus to observe from the five dimensional continuous spacetime where the four dimensional quantized spacetime is embedded, there exist the energy-momentum flows between the five dimensional continuous spacetime and the four dimensional quantized spacetime. This makes the energy-momentum not locally conserved generally on the four dimensional quantized spacetime. We propose that energy-momentum tensors of noncommutative field theories constructed from the Noether approach are just the correct forms for the energy-momentum tensors of noncommutative field theories. The non-vanishing of the total divergences of the energy-momentum tensors of noncommutative field theories just reflect that energy-momentum are not locally conserved on noncommutative spacetime. At the same time, from the constructions of the quantum spacetime, we propose that the total spacetime dimension of the quantum spacetime is infinite.Comment: 18 pages Late

Anticommutators and propagators of Moyal star-products for Dirac field on noncommutative spacetime

We study the Moyal anticommutators and their expectation values between vacuum states and non-vacuum states for Dirac fields on noncommutative spacetime. Then we construct the propagators of Moyal star-products for Dirac fields on noncommutative spacetime. We find that the propagators of Moyal star-products for Dirac fields are equal to the propagators of Dirac fields on ordinary commutative spacetime.Comment: 9 pages, Latex, some references adde