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Computational tomography (CT) provides high-resolution medical imaging, but it can expose patients to high radiation. X-ray scanners have low radiation exposure, but their resolutions are low. This paper proposes a new conditional diffusion model, DX2CT, that reconstructs three-dimensional (3D) CT volumes from bi or mono-planar X-ray image(s). Proposed DX2CT consists of two key components: 1) modulating feature maps extracted from two-dimensional (2D) X-ray(s) with 3D positions of CT volume using a new transformer and 2) effectively using the modulated 3D position-aware feature maps as conditions of DX2CT. In particular, the proposed transformer can provide conditions with rich information of a target CT slice to the conditional diffusion model, enabling high-quality CT reconstruction. Our experiments with the bi or mono-planar X-ray(s) benchmark datasets show that proposed DX2CT outperforms several state-of-the-art methods. Our codes and model will be available at: https://www.github.com/intyeger/DX2CT.

A physical model of Geant4-based simulation of galactic cosmic ray (GCR) particles interaction with the lunar surface matter has been developed to investigate the production rate of cosmogenic nuclei. In this model the GCRs, mainly very high energy protons and $\alpha$ particles, bombard the surface of the Moon and produce many secondary particles such as protons and neutrons. The energies of proton and neutron at different depths are recorded and saved into ROOT files, and the analytical expressions for the differential proton and neutron fluxes are obtained through the best-fit procedure under the ROOT software. To test the validity of this model, we calculate the production rates of long-lived nuclei $^{10}$Be and $^{26}$Al in the Apollo 15 long drill core by combining the above differential fluxes and the newly evaluated spallation reaction cross sections. Numerical results show that the theoretical production rates agree quite well with the measured data. It means that this model works well. Therefore, it can be expected that this model can be used to investigate the cosmogenic nuclei in lunar samples returned by Chinese lunar exploration program and can be extended to study other objects, such as the meteorites and the Earth's atmosphere.

In [11] the authors investigated a family of quotient Hilbert modules in the Cowen-Douglas class over the unit disk constructed from classical Hilbert modules such as the Hardy and Bergman modules. In this paper we extend the results to the multivariable case of higher multiplicity. Moreover, similarity as well as isomorphism results are obtained., Comment: Revised. To appear in Journal of Functional Analysis

One can view contraction operators given by a canonical model of Sz.-Nagy and Foias as being defined by a quotient module where the basic building blocks are Hardy spaces. In this note we generalize this framework to allow the Bergman and weighted Bergman spaces as building blocks, but restricting attention to the case in which the operator obtained is in the Cowen-Douglas class and requiring the multiplicity to be one. We view the classification of such operators in the context of complex geometry and obtain a complete classification up to unitary equivalence of them in terms of their associated vector bundles and their curvatures., Comment: 11 pages

This paper has been withdrawn by the author, since the author does not have enough time to answer every questions on this result., Comment: This paper has been withdrawn

We prove that if a completely non-unitary contraction T in L(H) has a non-trivial algebraic element h, then T has a non-trivial invariant subspace., Comment: 6 pages

We study reducing subspaces for an analytic multiplication operator M_{z^{n}} on the Bergman space L_{a}^{2}(A_{r}) of the annulus A_{r}, and we prove that M_{z^{n}} has exactly 2^n reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context of complex geometry., Comment: 13 pages

In this paper, to solve the invariant subspace problem, contraction operators are classified into three classes ; (Case 1) completely non-unitary contractions with a non-trivial algebraic element, (Case 2) completely non-unitary contractions without a non-trivial algebraic element, or (Case 3) contractions which are not completely non-unitary. We know that every operator of (Case 3) has a non-trivial invariant subspace. In this paper, we answer to the invariant subspace problem for the operators of (Case 2). Since (Case 1) is simpler than (Case 2), we leave as a question.

We introduce the notions of L(H)-valued norms and Banach spaces with respect to L(H)-valued norms. In particular, we introduce Hilbert spaces with respect to L(H)-valued inner products. In addition, we provide several fundamental examples of Hilbert spaces with respect to L(H)-valued inner products., Comment: 13 page

If $R$ and $M$ are Hilbert modules (in the sense of R. G. Douglas and V. I. Paulsen), we study the relationship between invertible module maps $X:R\to{M}$ and $X_{z}:R/R_{z}\to{M/M_{z}}$. In particular, for quasi-free Hilbert modules $R$ and $M$, we provide a condition of a module map $X:R\to{M}$, such that if $X_{z}:R/R_{z}\to{M/M_{z}}$ is invertible for every $z$ in a domain $\Omega$ in the complex plane, then $X$ is also invertible., Comment: 7 pages

We introduce a new norm, called $N^{p}$-norm $(1\leq{p}<\infty)$ on a space $N^{p}(V,W)$ where $V$ and $W$ are abstract operator spaces. By proving some fundamental properties of the space $N^{p}(V,W)$, we also obtain that if $W$ is complete, then the space $N^{p}(V,W)$ is also a Banach space with respect to this norm for $1\leq{p}<\infty$., Comment: 9 pages

We provide the definition and fundamental properties of algebraic elements with respect to an operator satisfying hypothesis (h). Furthermore, we analyze Hilbert modules using C_0-operators relative to a bounded finitely connected region Omega in the complex plane., Comment: 14 pages

We introduce two kinds of operator-valued norms. One of them is an $L(H)$-valued norm. The other one is an $L(C(K))$-valued norm. We characterize the completeness with respect to a bounded $L(H)$-valued norm. Furthermore, for a given Banach space $\textbf{B}$, we provide an $L(C(K))$-valued norm on $\textbf{B}$. and we introduce an $L(C(K))$-valued norm on a Banach space satisfying special properties., Comment: 8 pages

We introduce two kinds of quasi-inner functions. Since every rationally invariant subspace for a shift operator $S_K$ on a vector-valued Hardy space $H^{2}(\Omega,K)$ is generated by a quasi-inner function, we also provide relationships of quasi-inner functions by comparing rationally invariant subspaces generated by them. Furthermore, we discuss fundamental properties of quasi-inner functions, and quasi-inner divisors.

We give definitions and some properties of the shift operator S_{L(H^2)} and multiplication operator on L(H^2). In addition, we obtain some properties of the commutant of the shift operator S_{L(H^2)} and characterize S_{L(H^2)}-invariant subspaces., Comment: 9 pages

We study modularity of the lattice Lat $(T)$ of closed invariant subspaces for a $C_0$-operator $T$ and find a condition such that Lat $(T)$ is a modular. Furthermore, we provide a quasiaffinity preserving modularity., Comment: 12 pages

We study $C_{0}$-operators relative to a multiply connected domain using a substitute of the characteristic function. This method allows us to prove certain relations between the Jordan model of an operator and that of its restriction to an invariant subspace., Comment: 12pages, To appear in JOT