On the F.Smarandache LCM Ratio Sequence

In this paper, we use the elementary methods to study the F.Smarandache LCM ratio sequence, and obtain three interesting recurrence relations for it

Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms

Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to $n^{\lfloor p/2 \rfloor}$ for a $p$-th order tensor in $\mathbb{R}^{n^p}$. Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the first quasi-polynomial time algorithm that can decompose a random 3rd order tensor decomposition when the rank is as large as $n^{3/2}/\textrm{polylog} n$. We also give a polynomial time algorithm for certifying the injective norm of random low rank tensors. Our tensor decomposition algorithm exploits the relationship between injective norm and the tensor components. The proof relies on interesting tools for decoupling random variables to prove better matrix concentration bounds, which can be useful in other settings

Nucleon sea in the effective chiral quark model

The asymmetries of both light-flavor antiquark $\bar{d}(x)-\bar{u}(x)$ and strange-antistrange $s(x)-\bar{s}(x)$ distributions of the nucleon sea are considered with more details in the effective chiral quark model. We find that the asymmetric distribution of light-flavor antiquarks $\bar{d}(x)-\bar{u}(x)$ matches the experiment data well and that the asymmetry of strange and antistrange distributions can bring about 60-100% correction to the NuTeV anomaly of $\sin^{2}\theta_{w}$, which are three standard deviations from the world average value measured in other electroweak processes. The results on the correction to the NuTeV anomaly are insensitive to the inputs of the constituent quark distributions and the cut-off parameters. The ratios of $\bar{d}(x)/\bar{u}(x)$ and $s(x)/\bar{s}(x)$ are also discussed, and it is found that the ratio $s(x)/\bar{s}(x)$ is compatible with the available experiments with an additional symmetric sea contribution being considered effectively.Comment: 24 Latex pages, 8 figure