Marginal Covariance of Parameters in New Observations

We have observed a common problem of solving for the marginal covariance of parameters introduced in new observations. This problem arises in several situations, including augmenting parameters to a Kalman filter, and computing weight for relative pose constraints. To handle this problem, we derive a solution in a least squares sense. The solution is applied to the above two instance situations and verified by independently reported results.Comment: 3 pages, short technical repor

Recent Progress on Ricci Solitons

Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton's Ricci flow and play important roles in the singularity study of the Ricci flow. In this paper, we survey some of the recent progress on Ricci solitons.Comment: 32 pages; to appear in Proceedings of International Conference on Geometric Analysis (Taipei, July 2007

Derivation of Gell-Mann-Nishijima formula from the electromagnetic field modes of a hadron

When an electron probes another elementary particle Q, the wave function of the electron can be separated into two independent parts, the first part represents the electronic motion, the second part represents the electromagnetic field mode around the particle Q. In analogy with optical modes $TEM_{nlm}$ for a laser resonator, when the electromagnetic field around the particle Q forms into a mode, the quantum numbers of the mode satisfy the Gell-Mann-Nishijima formula, these quantum numbers are recognized as the charge number, baryon number and strangeness number. The modes are used as a visual model to understand the abstract baryon number and strangeness number of hadrons.Comment: LaTex, 10 pages, 5 figure

On The Eigenvalues of Some Vectorial Sturm-Liouville Eigenvalue Problems

The author tries to derive the asymptotic expression of the large eigevalues of some vectorial Sturm-Liouville differential equations. A precise description for the formula of the square root of the large eiegnvalues up to the $O(1/n)$-term is obtained.Comment: 12 pages, latex, no figure

Geometry of Complete Gradient Shrinking Ricci Solitons

We survey some of the recent progress on complete gradient shrinking Ricci solitons, including the classifications in dimension three and asymptotic behavior of potential functions as well as volume growths of geodesic balls in higher dimensions. This article is written for the conference proceedings dedicated to Yau's 60th birthday.Comment: 16 pages; updated versio

The K\"ahler-Ricci flow on Fano manifolds

In this lecture notes, we aim at giving an introduction to the K\"ahler-Ricci flow (KRF) on Fano manifolds. It covers some of the developments of the KRF in its first twenty years (1984-2003), especially an essentially self-contained exposition of Perelman's uniform estimates on the scalar curvature, the diameter, and the Ricci potential function for the normalized K\"ahler-Ricci flow (NKRF), including the monotonicity of Perelman's \mu-entropy and \kappa-noncollapsing theorems for the Ricci flow on compact manifolds. The Notes is based on a mini-course on KRF delivered at University of Toulouse III in February 2010, a talk on Perelman's uniform estimates for NKRF at Columbia University's Geometry and Analysis Seminar in Fall 2005, and several conference talks, including "Einstein Manifolds and Beyond" at CIRM (Marseille - Luminy, fall 2007), "Program on Extremal K\"ahler Metrics and K\"ahler-Ricci Flow" at the De Giorgi Center (Pisa, spring 2008), and "Analytic Aspects of Algebraic and Complex Geometry" at CIRM (Marseille - Luminy, spring 2011).Comment: v.2: corrected a number of typos and added the proof of Theorem 2.3 on preserving positive orthogonal bisectional curvature. To appear as a book chapter in An Introduction to the K\"ahler-Ricci Flow, Lecture Notes in Mathematics, vol. 2086, Springer, 201

Derived Kodaira Spencer map, Cosection lemma, and semiregularity

The cosection lemma proved by J. Li and Y.H. Kiem said the intrinsic normal cone lies inside the kernel of any cosection of the obstruction sheaf when the moduli has a perfect obstruction theory. With a definition of higher tangent vectors of a scheme at a point, and a construction of the derived Kodaira Spencer map by K. Behrend and B. Fantechi, we prove a derived version of cosection lemma without perfect obstruction theory condition. As an application we give a short proof of the Kodaira's Principle \textit{ambient cohomology annihilates obstruction} (semiregularity), assuming the existence of locall universal family.Comment: 10 page

On dimension reduction in the K\"ahler-Ricci flow

We consider dimension reduction for solutions of the K\"ahler-Ricci flow with nonegative bisectional curvature. When the complex dimension $n=2$, we prove an optimal dimension reduction theorem for complete translating K\"ahler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reduction theorem for complete ancient solutions of the K\"ahler-Ricci flow with nonnegative bisectional curvature on noncompact complex manifolds under a finiteness assumption on the Chern number $c^n_1$.Comment: 15 pages, Late

On the Construction of Isospectral Vectorial Sturm-Liouville Differential Equations

The author extends the idea of Jodeit and Levitan for constructing isospectral problems of the classical scalar Sturm-Liouville differential equations to the vectorial Sturm-Liouville differential equations. Some interesting relations are found.Comment: 15 pages, latex, no figures, revised versio

Laser agitates probability flow in atoms to form alternating current and its peak-dip phenomenon

By using trajectory-based approaches to quantum transition, it is found that laser can agitate the probability flow in atoms to form alternating current with the frequency of the laser. The detailed physical process of quantum transition is investigated, during which the alternating current in atomic probability flow becomes a key role connecting the external electromagnetic wave with the evolution of the quantum states in atoms. Computer was employed to simulate the physical process. The atomic alternating current may have the peak-dip phenomenon.Comment: 8 pages, 8 figure