Variational formulas of higher order mean curvatures

In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total $2p$-th mean curvature functional $\mathcal {M}_{2p}$ of a submanifold $M^n$ in a general Riemannian manifold $N^{n+m}$ for $p=0,1,...,[\frac{n}{2}]$. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional $\mathcal {M}_{2p}$, called relatively $2p$-minimal submanifolds, for all $p$. At last, we discuss the relations between relatively $2p$-minimal submanifolds and austere submanifolds in real space forms, as well as a special variational problem.Comment: 13 pages, to appear in SCIENCE CHINA Mathematics 201

Yield Spread Selection in Predicting Recession Probabilities: A Machine Learning Approach

The literature on using yield curves to forecast recessions customarily uses 10-year--three-month Treasury yield spread without verification on the pair selection. This study investigates whether the predictive ability of spread can be improved by letting a machine learning algorithm identify the best maturity pair and coefficients. Our comprehensive analysis shows that, despite the likelihood gain, the machine learning approach does not significantly improve prediction, owing to the estimation error. This is robust to the forecasting horizon, control variable, sample period, and oversampling of the recession observations. Our finding supports the use of the 10-year--three-month spread

Normal scalar curvature conjecture and its applications

In this paper, we proved the Normal Scalar Curvature Conjecture and the Bottcher-Wenzel Conjecture. We also established some new pinching theorems for minimal submanifolds in spheres.Comment: minor and final typo correction

Card Games Unveiled: Exploring the Underlying Linear Algebra

We discuss four famous card games that can help learn linear algebra. The games are: SET, Socks, Spot it!, and EvenQuads. We describe the game in the language of vector, affine, and projective spaces. We also show how these games are connected to each other. A separate section is devoted to playing Socks with the EvenQuads deck and vice versa.Comment: 21 pages, 13 figure

Quad Squares

We study 4-by-4 squares formed by cards from the EvenQuads deck. EvenQuads is a card game with 64 cards where cards have 3 attributes with 4 values in each attribute. A quad is four cards with all attributes the same, all different, or half and half. We define Latin quad squares as squares where the cards in each row and column have different values for each attribute. We define semimagic quad squares as squares where each row and column form a quad. For magic quad squares, we add a requirement that the diagonals have to form a quad. We also define strongly magic quad squares. We analyze types of semimagic and strongly magic quad squares. We also calculate the number of semimagic, magic, and strongly magic quad squares for quad decks of any size. These squares can be described in terms of integers. Four integers form a quad when their bitwise XOR is zero.Comment: 24 pages, 10 figure

Light-Front Model of Transition Form-Factors in Heavy Meson Decay

Electroweak transition form factors of heavy meson decays are important ingredients in the extraction of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements from experimental data. In this work, within a light-front framework, we calculate electroweak transition form factor for the semileptonic decay of $D$ mesons into a pion or a kaon. The model results underestimate in both cases the new data of CLEO for the larger momentum transfers accessible in the experiment. We discuss possible reasons for that in order to improve the model.Comment: Paper with 5 pages and 2 eps figures. To appear to Nuclear Physics B. Talk at Light Cone 2009: Relativistic Hadronic and Particle Physics (LC 2009), Sao Jose dos Campos, S.P, Brazil, 8-13 Jul 2009