Approximate Set Union Via Approximate Randomization

We develop an randomized approximation algorithm for the size of set union problem \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert, which given a list of sets $A_1,...,A_m$ with approximate set size $m_i$ for $A_i$ with $m_i\in \left((1-\beta_L)|A_i|, (1+\beta_R)|A_i|\right)$, and biased random generators with Prob(x=\randomElm(A_i))\in \left[{1-\alpha_L\over |A_i|},{1+\alpha_R\over |A_i|}\right] for each input set $A_i$ and element $x\in A_i,$ where $i=1, 2, ..., m$. The approximation ratio for \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert is in the range $[(1-\epsilon)(1-\alpha_L)(1-\beta_L), (1+\epsilon)(1+\alpha_R)(1+\beta_R)]$ for any $\epsilon\in (0,1)$, where $\alpha_L, \alpha_R, \beta_L,\beta_R\in (0,1)$. The complexity of the algorithm is measured by both time complexity, and round complexity. The algorithm is allowed to make multiple membership queries and get random elements from the input sets in one round. Our algorithm makes adaptive accesses to input sets with multiple rounds. Our algorithm gives an approximation scheme with O(\setCount\cdot(\log \setCount)^{O(1)}) running time and $O(\log m)$ rounds, where $m$ is the number of sets. Our algorithm can handle input sets that can generate random elements with bias, and its approximation ratio depends on the bias. Our algorithm gives a flexible tradeoff with time complexity O\left(\setCount^{1+\xi}\right) and round complexity $O\left({1\over \xi}\right)$ for any $\xi\in(0,1)$

A Deep Learning Reconstruction Framework for Differential Phase-Contrast Computed Tomography with Incomplete Data

Differential phase-contrast computed tomography (DPC-CT) is a powerful analysis tool for soft-tissue and low-atomic-number samples. Limited by the implementation conditions, DPC-CT with incomplete projections happens quite often. Conventional reconstruction algorithms are not easy to deal with incomplete data. They are usually involved with complicated parameter selection operations, also sensitive to noise and time-consuming. In this paper, we reported a new deep learning reconstruction framework for incomplete data DPC-CT. It is the tight coupling of the deep learning neural network and DPC-CT reconstruction algorithm in the phase-contrast projection sinogram domain. The estimated result is the complete phase-contrast projection sinogram not the artifacts caused by the incomplete data. After training, this framework is determined and can reconstruct the final DPC-CT images for a given incomplete phase-contrast projection sinogram. Taking the sparse-view DPC-CT as an example, this framework has been validated and demonstrated with synthetic and experimental data sets. Embedded with DPC-CT reconstruction, this framework naturally encapsulates the physical imaging model of DPC-CT systems and is easy to be extended to deal with other challengs. This work is helpful to push the application of the state-of-the-art deep learning theory in the field of DPC-CT

Transition magnetic moment of Majorana neutrinos in the μν\mu\nuSSM

The nonzero vacuum expectative values of sneutrinos induce spontaneously R-parity and lepton number violation, and generate three tiny Majorana neutrino masses through the seesaw mechanism in the $\mu\nu$SSM, which is one of Supersymmetric extensions beyond Standard Model. Applying effective Lagrangian method, we study the transition magnetic moment of Majorana neutrinos in the model here. Under the constraints from neutrino oscillations, we consider the two possibilities on the neutrino mass spectrum with normal or inverted ordering.Comment: 20 pages, 2 figures, accepted for publication in JHEP. arXiv admin note: text overlap with arXiv:1305.4352, arXiv:1304.624