Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schr\"odinger operators in dimension three

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,\Delta^2)$ associated with fourth-order Schr\"odinger operators $H=\Delta^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-\mu}$ for some $\mu>0$. A recent work by Goldberg and Green \cite{GoGr21} has demonstrated that wave operators $W_\pm(H,\Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1<p<\infty$ under the condition that $\mu>9$, and zero is a regular point of $H$. In this paper, we aim to further establish endpoint estimates for $W_\pm(H,\Delta^2)$ in two significant ways. First, we provide counterexamples that illustrate the unboundedness of $W_\pm(H,\Delta^2)$ on the endpoint spaces $L^1(\mathbb{R}^3)$ and $L^\infty(\mathbb{R}^3)$, even for non-zero compactly supported potentials $V$. Second, we establish weak (1,1) estimates for the wave operators $W_\pm(H,\Delta^2)$ and their dual operators $W_\pm(H,\Delta^2)^*$ in the case where zero is a regular point and $\mu>11$. These estimates depend critically on the singular integral theory of Calder\'on-Zygmund on a homogeneous space $(X,d\omega)$ with a doubling measure $d\omega$.Comment: 29 pages. Any comments are welcome

LpL^p-boundedness of wave operators for fourth order Schr\"odinger operators with zero resonances on R3\mathbb{R}^3

Let $H = \Delta^2 + V$ be the fourth-order Schr\"odinger operator on $\mathbb{R}^3$ with a real-valued fast-decaying potential $V$. If zero is neither a resonance nor an eigenvalue of $H$, it has recently been shown by Goldberg and Green \cite{GoGr21} that the wave operators $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for $1 < p < \infty$. Additionally, it has been proved by the authors in \cite{MWY23} that these wave operators are unbounded at the endpoints, i.e., for $p=1$ and $p=\infty$. In this paper, our primary focus is to further establish the $L^p$-bounds of $W_\pm(H, \Delta^2)$ that exhibit all types of singularities at the zero energy threshold. We first prove that $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for $1 < p < \infty$ in the first kind resonance case. We then proceed to establish for the second kind resonance case that they are bounded on $L^p(\mathbb{R}^3)$ for $1 < p < 3$, but not for $3 \le p < \infty$. In the third kind resonance case, we show that $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for $1<p<3$. Remarkably, we can also prove that $W_\pm(H, \Delta^2)$ remain bounded on $L^p(\mathbb{R}^3)$ for $3\le p<\infty$ if in addition $H$ has a zero eigenvalue, but no $p$-wave zero resonance and all zero eigenfunctions are orthogonal to $x_ix_jx_kV$ in $L^2(\mathbb{R}^3)$ for all $i,j,k=1,2,3$ with $x=(x_1,x_2,x_3)$ and that, without such additional conditions, they are unbounded on $L^p(\mathbb{R}^3)$ for $3\le p<\infty$. These results precisely delineate all $L^p$-bounds of the wave operators $W_\pm(H, \Delta^2)$ in $\mathbb{R}^3$ with the exception of the endpoints $p=1$ and $p=\infty$. As a result, $L^p$-$L^q$ decay estimates are derived for solutions of fourth-order Schr\"odinger equations and beam equations with zero resonance singularities.Comment: 70 pages! Any comments are welcome

EPiK-a Workflow for Electron Tomography in Kepler.

Scientific workflows integrate data and computing interfaces as configurable, semi-automatic graphs to solve a scientific problem. Kepler is such a software system for designing, executing, reusing, evolving, archiving and sharing scientific workflows. Electron tomography (ET) enables high-resolution views of complex cellular structures, such as cytoskeletons, organelles, viruses and chromosomes. Imaging investigations produce large datasets. For instance, in Electron Tomography, the size of a 16 fold image tilt series is about 65 Gigabytes with each projection image including 4096 by 4096 pixels. When we use serial sections or montage technique for large field ET, the dataset will be even larger. For higher resolution images with multiple tilt series, the data size may be in terabyte range. Demands of mass data processing and complex algorithms require the integration of diverse codes into flexible software structures. This paper describes a workflow for Electron Tomography Programs in Kepler (EPiK). This EPiK workflow embeds the tracking process of IMOD, and realizes the main algorithms including filtered backprojection (FBP) from TxBR and iterative reconstruction methods. We have tested the three dimensional (3D) reconstruction process using EPiK on ET data. EPiK can be a potential toolkit for biology researchers with the advantage of logical viewing, easy handling, convenient sharing and future extensibility

High-performance blob-based iterative three-dimensional reconstruction in electron tomography using multi-GPUs

<p>Abstract</p> <p>Background</p> <p>Three-dimensional (3D) reconstruction in electron tomography (ET) has emerged as a leading technique to elucidate the molecular structures of complex biological specimens. Blob-based iterative methods are advantageous reconstruction methods for 3D reconstruction in ET, but demand huge computational costs. Multiple graphic processing units (multi-GPUs) offer an affordable platform to meet these demands. However, a synchronous communication scheme between multi-GPUs leads to idle GPU time, and a weighted matrix involved in iterative methods cannot be loaded into GPUs especially for large images due to the limited available memory of GPUs.</p> <p>Results</p> <p>In this paper we propose a multilevel parallel strategy combined with an asynchronous communication scheme and a blob-ELLR data structure to efficiently perform blob-based iterative reconstructions on multi-GPUs. The asynchronous communication scheme is used to minimize the idle GPU time so as to asynchronously overlap communications with computations. The blob-ELLR data structure only needs nearly 1/16 of the storage space in comparison with ELLPACK-R (ELLR) data structure and yields significant acceleration.</p> <p>Conclusions</p> <p>Experimental results indicate that the multilevel parallel scheme combined with the asynchronous communication scheme and the blob-ELLR data structure allows efficient implementations of 3D reconstruction in ET on multi-GPUs.</p

Face hallucination under an image decomposition perspective

In this paper we propose to convert the task of face hallucination into an image decomposition problem, and thenuse the morphological component analysis (MCA) for hallucinating a single face image, based on a novel three-stepframework. Firstly, a low-resolution input image is up-sampled by interpolation. Then, the MCA is employed to decompose the interpolated image into a high-resolution image and an unsharp masking, as MCA can properly decompose a signal into special parts according to typical dictionaries. Finally, a residue compensation, which is based on the neighbour reconstruction of patches, is performed to enhance the facial details. The proposed method can effectively exploit the facial properties for face hallucination under the image decomposition perspective. Experimental results demonstrate the effectiveness of our method, in terms of the visual quality of the hallucinated face images

Wireless coverage with disparate ranges

Coverage has been one of the most fundamental yet chal-lenging issues in wireless networks. Given a set of nodes and a set of disks of disparate radii, the problem Minimum Disk Cover seeks a disk cover of all nodes with minimum cardi-nality. We present the first polynomial time approximation scheme. We also consider a classical generalization where each input disk is associated with a positive cost, the prob-lem Min-Cost Disk Cover seeks a disk cover of all nodes with minimum total cost. We present a randomized algo-rithm that can achieve an approximation ratio of 2O(log ∗ n) with high probability, where n is the number of input disks. Another line of this work is exploring the relations be-tween disk cover and an important practical problem which seeks a wireless covering schedule of maximum life sub-ject to an energy budget function. We present two algo-rithms: Ellipsoid Algorithm (EA) and Price-Directive Al-gorithm (PDA), and prove that by applying our algorithmic results on disk cover, the approximation ratios for EA and PDA are 2O(log ∗ n) and (1 + ǫ) 2O(log ∗ n) respectively

3D reconstruction of biological structures: automated procedures for alignment and reconstruction of multiple tilt series in electron tomography

Transmission electron microscopy allows the collection of multiple views of specimens and their computerized three-dimensional reconstruction and analysis with electron tomography. Here we describe development of methods for automated multi-tilt data acquisition, tilt-series processing, and alignment which allow assembly of electron tomographic data from a greater number of tilt series, yielding enhanced data quality and increasing contrast associated with weakly stained structures. This scheme facilitates visualization of nanometer scale details of fine structure in volumes taken from plastic-embedded samples of biological specimens in all dimensions. As heavy metal-contrasted plastic-embedded samples are less sensitive to the overall dose rather than the electron dose rate, an optimal resampling of the reconstruction space can be achieved by accumulating lower dose electron micrographs of the same area over a wider range of specimen orientations. The computerized multiple tilt series collection scheme is implemented together with automated advanced procedures making collection, image alignment, and processing of multi-tilt tomography data a seamless process. We demonstrate high-quality reconstructions from samples of well-described biological structures. These include the giant Mimivirus and clathrin-coated vesicles, imaged in situ in their normal intracellular contexts. Examples are provided from samples of cultured cells prepared by high-pressure freezing and freeze-substitution as well as by chemical fixation before epoxy resin embedding

The role of basic health insurance on depression: an epidemiological cohort study of a randomized community sample in Northwest China

BACKGROUND: Little research has focused on the relationship between health insurance and mental health in the community. The objective of this study is to determine how the basic health insurance system influences depression in Northwest China. METHODS: Participants were selected from 32 communities in two northwestern Chinese cities through a three-stage random sampling. Three waves of interviews were completed in April 2006, December 2006, and January 2008. The baseline survey was completed by 4,079 participants. Subsequently, 2,220 participants completed the first follow-up, and 1,888 completed the second follow-up. Depression symptoms were measured by the Center for Epidemiologic Studies Depression Scale (CES-D). RESULTS: A total of 40.0% of participants had at least one form of health insurance. The percentages of participants with severe depressive symptoms in the three waves were 21.7%, 22.0%, and 17.6%. Depressive symptoms were found to be more severe among participants without health insurance in the follow-up surveys. After adjusting for confounders, participants without health insurance were found to experience a higher risk of developing severe depressive symptoms than participants with health insurance (7 months: OR, 1.40; 95% CI, 1.09-1.82; p = 0.01; 20 months: OR, 1.89; 95% CI, 1.37-2.61; p < 0.001). CONCLUSION: A lack of basic health insurance can dramatically increase the risk of depression based on northwestern Chinese community samples