arXiv subjects

Zhijie Dong

Publications and source records attributed to Zhijie Dong.

7 recordsLinked to original sources

arXiv2024

Loop Grassmannian of quivers and Compactified Coulomb branch of quiver gauge theory with no framing

Mirković introduced the notion of loop Grassmannian for symmetric integer matrix $κ$. It is a two-step limit of the local projective space $Z_κ^α$, which generalizes the usual Zastava for a simply laced group $G$. The usual loop Grassmannian of $G$ is recovered when the matrix $κ$ is the Cartan matrix of $G$. On the other hand, Braverman, Finkelberg, and Nakajima showed that the Compactified Coulomb branch $\mathbf{M}_{Q}^α$ for the quiver gauge theory with no framing also generalizes the usual Zastava. We show that in the case when $κ$ is the associated matrix of the quiver $Q$, these two generalizations of Zastava coincide, i.e $\mathbf{M}_{Q}^α\cong Z_{κ(Q)}^α$.

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arXiv2023

Non-invasive Deep-Brain Imaging with 3D Integrated Photoacoustic Tomography and Ultrasound Localization Microscopy (3D-PAULM)

Photoacoustic computed tomography (PACT) is a proven technology for imaging hemodynamics in deep brain of small animal models. PACT is inherently compatible with ultrasound (US) imaging, providing complementary contrast mechanisms. While PACT can quantify the brain's oxygen saturation of hemoglobin (sO$_2$), US imaging can probe the blood flow based on the Doppler effect. Further, by tracking gas-filled microbubbles, ultrasound localization microscopy (ULM) can map the blood flow velocity with sub-diffraction spatial resolution. In this work, we present a 3D deep-brain imaging system that seamlessly integrates PACT and ULM into a single device, 3D-PAULM. Using a low ultrasound frequency of 4 MHz, 3D-PAULM is capable of imaging the whole-brain hemodynamic functions with intact scalp and skull in a totally non-invasive manner. Using 3D-PAULM, we studied the mouse brain functions with ischemic stroke. Multi-spectral PACT, US B-mode imaging, microbubble-enhanced power Doppler (PD), and ULM were performed on the same mouse brain with intrinsic image co-registration. From the multi-modality measurements, we future quantified blood perfusion, sO$_2$, vessel density, and flow velocity of the mouse brain, showing stroke-induced ischemia, hypoxia, and reduced blood flow. We expect that 3D-PAULM can find broad applications in studying deep brain functions on small animal models.

physics.ins-det↗

arXiv2023

High-level synthesis design of scalable ultrafast ultrasound beamformer with single FPGA

Ultrafast ultrasound imaging is essential for advanced ultrasound imaging techniques such as ultrasound localization microscopy (ULM) and functional ultrasound (fUS). Current ultrafast ultrasound imaging is challenged by the ultrahigh data bandwidth associated with the radio frequency (RF) signal, and by the latency of the computationally expensive beamforming process. As such, continuous ultrafast data acquisition and beamforming remain elusive with existing software beamformers based on CPUs or GPUs. To address these challenges, the proposed work introduces a novel method of implementing an ultrafast ultrasound beamformer specifically for ultrafast plane wave imaging (PWI) on a field programmable gate array (FPGA) by using high-level synthesis. A parallelized implementation of the beamformer on a single FPGA was proposed by 1) utilizing a delay compression technique to reduce the delay profile size, which enables both run-time pre-calculated delay profile loading from external memory and delay reuse 2) vectorizing channel data fetching which is enabled by delay reuse, and 3) using fixed summing networks to reduce consumption of logic resources. Our proposed method presents two unique advantages over current FPGA beamformers: 1) high scalability that allows fast adaptation to different FPGA resources and beamforming speed demands by using Xilinx High-Level Synthesis as the development tool, and 2) allow a compact form factor design by using a single FPGA to complete the beamforming instead of multiple FPGAs. With the proposed method, a sustainable average beamforming rate of 4.83 G samples/second in terms of input raw RF sample was achieved. The resulting image quality of the proposed beamformer was compared with the software beamformer on the Verasonics Vantage system for both phantom imaging and in vivo imaging of a mouse brain.

eess.SP↗

arXiv2021

Application of Schur-Weyl duality to Springer theory

In \cite{FMX19}, it is proved that the convolution algebra of top Borel-Moore homology on Steinberg variety of type $B/C$ realizes $U(sl_n^θ)$, where $sl_{n}^θ$ is the fixed point subalgebra of involution on $sl_n$. So top Borel-Moore homology of the partial Springer's fibers gives the representations of $U(sl_n^θ)$. In this paper, we study these representations using the Schur-Weyl duality and Springer theory.

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arXiv2019

Convolution algebra of diagram automorphism fixed quiver variety

We study the convolution algebra $H_{*}(Z^θ_{W})$ of homology on diagram automorphism fixed point quiver variety and prove that there exists an algebra homomorphism from the universal enveloping algebra of the diagram automorphism fixed algebra of the split quiver to $H_{*}(Z^θ_{W})$.

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arXiv2019

Equivariant homology theory and twisted Yangian

We study the convolution algebra $H^{G\times \CC^{*}}_{*}(Z)$ of $G$-equivariant homology group on the Steinberg variety of type B/C and define an algebra $\widetilde{Y}$ that maps to $H^{G\times \CC^{*}}_{*}(Z)$. The Drinfeld new realization of the twisted Yangian associated to symmetric pairs is a quotient of $\widetilde{Y}$. We also study the $G$-equivariant case and prove that the twisted Yangian is the deformation of the twisted current algebra.

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arXiv2018

A relation between Mirković-Vilonen cycles and modules over preprojective algebra of Dynkin quiver of type ADE

The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the reduceness conjecture.

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