Global Generation of Adjoint Line Bundles on Projective 55-folds

Let $X$ be a smooth projective variety of dimension $5$ and $L$ be an ample line bundle on $X$ such that $L^5>7^5$ and $L^d\cdot Z\geq 7^d$ for any subvariety $Z$ of dimension $1\leq d\leq 4$. We show that $\mathcal{O}_X(K_X+L)$ is globally generated.Comment: Final version to appear in manuscripta mathematica. We notice that mistakes were introduced by the journal to some fractions in the form "expression/expression" which should be read as "(expression)/(expression)

Single soft gluon emission at two loops

We study the single soft-gluon current at two loops with two energetic partons in massless perturbative QCD, which describes, for example, the soft limit of the two-loop amplitude for $gg\to Hg$. The results are presented as Laurent expansions in $\epsilon$ in $D=4-2\epsilon$ spacetime dimension. We calculate the expansion to order $\epsilon^2$ analytically, which is a necessary ingredient for Higgs production at hadron colliders at next-to-next-to-next-to-leading order in the soft-virtual approximation. We also give two-loop results of the single soft-gluon current in ${\cal N}=4$ Super-Yang-Mills theory, and find that it has uniform transcendentality. By iteration relation of splitting amplitudes, our calculations can determine the three-loop single soft-gluon current to order $\epsilon^0$ in ${\cal N}=4$ Super-Yang-Mills theory in the limit of large $N_c$.Comment: typos corrected; journal versio

Fast Estimation of True Bounds on Bermudan Option Prices under Jump-diffusion Processes

Fast pricing of American-style options has been a difficult problem since it was first introduced to financial markets in 1970s, especially when the underlying stocks' prices follow some jump-diffusion processes. In this paper, we propose a new algorithm to generate tight upper bounds on the Bermudan option price without nested simulation, under the jump-diffusion setting. By exploiting the martingale representation theorem for jump processes on the dual martingale, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal-dual algorithm, therefore significantly improves the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our proposed algorithm

Prognostic value of routine laboratory variables in prediction of breast cancer recurrence.

The prognostic value of routine laboratory variables in breast cancer has been largely overlooked. Based on laboratory tests commonly performed in clinical practice, we aimed to develop a new model to predict disease free survival (DFS) after surgical removal of primary breast cancer. In a cohort of 1,596 breast cancer patients, we analyzed the associations of 33 laboratory variables with patient DFS. Based on 3 significant laboratory variables (hemoglobin, alkaline phosphatase, and international normalized ratio), together with important demographic and clinical variables, we developed a prognostic model, achieving the area under the curve of 0.79. We categorized patients into 3 risk groups according to the prognostic index developed from the final model. Compared with the patients in the low-risk group, those in the medium- and high-risk group had a significantly increased risk of recurrence with a hazard ratio (HR) of 1.75 (95% confidence interval [CI] 1.30-2.38) and 4.66 (95% CI 3.54-6.14), respectively. The results from the training set were validated in the testing set. Overall, our prognostic model incorporating readily available routine laboratory tests is powerful in identifying breast cancer patients who are at high risk of recurrence. Further study is warranted to validate its clinical application