C. Lin - Finnish MAGIC Consortium, Tuorla Observatory, University of Turku and Department of Physics, University of Oulu, Finland

C. Lin
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C. Lin
Finnish MAGIC Consortium, Tuorla Observatory, University of Turku and Department of Physics, University of Oulu, Finland

Pubs By Year

Pub Categories

High Energy Astrophysical Phenomena (6)
Nuclear Experiment (4)
High Energy Physics - Phenomenology (4)
Cosmology and Nongalactic Astrophysics (4)
Computer Science - Learning (4)
Instrumentation and Methods for Astrophysics (4)
Astrophysics of Galaxies (3)
Computer Science - Human-Computer Interaction (3)
Physics - Mesoscopic Systems and Quantum Hall Effect (3)
High Energy Physics - Lattice (2)
Physics - Chemical Physics (2)
Solar and Stellar Astrophysics (2)
Physics - Strongly Correlated Electrons (2)
High Energy Physics - Theory (2)
Mathematics - Analysis of PDEs (2)
Computer Science - Computer Vision and Pattern Recognition (2)
Statistics - Methodology (2)
Physics - Instrumentation and Detectors (2)
Earth and Planetary Astrophysics (1)
Mathematics - Mathematical Physics (1)
Mathematics - Functional Analysis (1)
Mathematical Physics (1)
Physics - Superconductivity (1)
Statistics - Machine Learning (1)
Physics - Physics and Society (1)
Physics - Data Analysis; Statistics and Probability (1)
Nuclear Theory (1)
General Relativity and Quantum Cosmology (1)
Computer Science - Cryptography and Security (1)
Physics - Materials Science (1)
Physics - Optics (1)
Mathematics - Complex Variables (1)
Physics - Accelerator Physics (1)
Physics - Plasma Physics (1)
Mathematics - Information Theory (1)
High Energy Physics - Experiment (1)
Computer Science - Information Theory (1)

Publications Authored By C. Lin

The use of geometrical constraints opens many new perspectives in photonics and in fundamental studies of nonlinear waves. By implementing surface structures in vertical cavity surface emitting lasers as manifolds for curved space, we experimentally study the impacts of geometrical constraints on nonlinear wave localization. We observe localized waves pinned to the maximal curvature in an elliptical-ring, and confirm the reduction in the localization length of waves by measuring near and far field patterns, as well as the corresponding dispersion relation. Read More

We compare the existent methods including the minimum spanning tree based method and the local stellar density based method in measuring mass segregation of star clusters. We find that the minimum spanning tree method reflects more the compactness, which represents the global spatial distribution of massive stars, while the local stellar density method reflects more the crowdedness, which provides the local gravitational potential information. It is suggested to measure the local and the global mass segregation simultaneously. Read More

The design of energy and spectrally efficient Wireless Sensor Networks (WSN) is crucial to support the upcoming expansion of IoT/M2M mobile data traffic. In this work, we consider an energy harvesting WSN where sensor data are periodically reported to a Fusion Center (FC) by a sparse set of active sensors. Unlike most existing works, the transmit power levels of each sensor are assumed to be unknown at the FC in this distributed setting. Read More

High temperature superconductivity appears in the cuprates when a spin order is destroyed, while the role of charge is less known. Recent experimental progress suggests that the spin and charge degrees of freedom are highly entangled, which may lead to the enigmatic normal-state pseudogap that spans from the parent antiferromagnetic (AF) insulating phase to the overdoped superconducting regime. For example, short-range charge density wave (CDW) sets in right at the pseudogap temperature $T^{*}$ in single-layered Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$, below which spin fluctuations are suppressed. Read More

Fragmentation cross section of $^{28}$Si + $^{9}$Be reaction at 75.8 MeV/u was analyzed for studying the decay mode of single-proton emission in $^{21}$Al (the proton-rich nucleus with neutron closed-shell of $N = 8$ and $T_z = -5/2$). With the comparison between the measured fragmentation cross section and the theoretical cross section produced by EPAX3. Read More

Authors: F. P. An, A. B. Balantekin, H. R. Band, M. Bishai, S. Blyth, D. Cao, G. F. Cao, J. Cao, Y. L. Chan, J. F. Chang, Y. Chang, H. S. Chen, Q. Y. Chen, S. M. Chen, Y. X. Chen, Y. Chen, J. Cheng, Z. K. Cheng, J. J. Cherwinka, M. C. Chu, A. Chukanov, J. P. Cummings, Y. Y. Ding, M. V. Diwan, M. Dolgareva, J. Dove, D. A. Dwyer, W. R. Edwards, R. Gill, M. Gonchar, G. H. Gong, H. Gong, M. Grassi, W. Q. Gu, L. Guo, X. H. Guo, Y. H. Guo, Z. Guo, R. W. Hackenburg, S. Hans, M. He, K. M. Heeger, Y. K. Heng, A. Higuera, Y. B. Hsiung, B. Z. Hu, T. Hu, E. C. Huang, H. X. Huang, X. T. Huang, Y. B. Huang, P. Huber, W. Huo, G. Hussain, D. E. Jaffe, K. L. Jen, X. P. Ji, X. L. Ji, J. B. Jiao, R. A. Johnson, D. Jones, L. Kang, S. H. Kettell, A. Khan, S. Kohn, M. Kramer, K. K. Kwan, M. W. Kwok, T. J. Langford, K. Lau, L. Lebanowski, J. Lee, J. H. C. Lee, R. T. Lei, R. Leitner, J. K. C. Leung, C. Li, D. J. Li, F. Li, G. S. Li, Q. J. Li, S. Li, S. C. Li, W. D. Li, X. N. Li, X. Q. Li, Y. F. Li, Z. B. Li, H. Liang, C. J. Lin, G. L. Lin, S. Lin, S. K. Lin, Y. -C. Lin, J. J. Ling, J. M. Link, L. Littenberg, B. R. Littlejohn, J. L. Liu, J. C. Liu, C. W. Loh, C. Lu, H. Q. Lu, J. S. Lu, K. B. Luk, X. Y. Ma, X. B. Ma, Y. Q. Ma, Y. Malyshkin, D. A. Martinez Caicedo, K. T. McDonald, R. D. McKeown, I. Mitchell, Y. Nakajima, J. Napolitano, D. Naumov, E. Naumova, H. Y. Ngai, J. P. Ochoa-Ricoux, A. Olshevskiy, H. -R. Pan, J. Park, S. Patton, V. Pec, J. C. Peng, L. Pinsky, C. S. J. Pun, F. Z. Qi, M. Qi, X. Qian, R. M. Qiu, N. Raper, J. Ren, R. Rosero, B. Roskovec, X. C. Ruan, H. Steiner, P. Stoler, J. L. Sun, W. Tang, D. Taychenachev, K. Treskov, K. V. Tsang, C. E. Tull, N. Viaux, B. Viren, V. Vorobel, C. H. Wang, M. Wang, N. Y. Wang, R. G. Wang, W. Wang, X. Wang, Y. F. Wang, Z. Wang, Z. Wang, Z. M. Wang, H. Y. Wei, L. J. Wen, K. Whisnant, C. G. White, L. Whitehead, T. Wise, H. L. H. Wong, S. C. F. Wong, E. Worcester, C. -H. Wu, Q. Wu, W. J. Wu, D. M. Xia, J. K. Xia, Z. Z. Xing, J. L. Xu, Y. Xu, T. Xue, C. G. Yang, H. Yang, L. Yang, M. S. Yang, M. T. Yang, Y. Z. Yang, M. Ye, Z. Ye, M. Yeh, B. L. Young, Z. Y. Yu, S. Zeng, L. Zhan, C. Zhang, C. C. Zhang, H. H. Zhang, J. W. Zhang, Q. M. Zhang, R. Zhang, X. T. Zhang, Y. M. Zhang, Y. X. Zhang, Y. M. Zhang, Z. J. Zhang, Z. Y. Zhang, Z. P. Zhang, J. Zhao, L. Zhou, H. L. Zhuang, J. H. Zou

The Daya Bay experiment has observed correlations between reactor core fuel evolution and changes in the reactor antineutrino flux and energy spectrum. Four antineutrino detectors in two experimental halls were used to identify 2.2 million inverse beta decays (IBDs) over 1230 days spanning multiple fuel cycles for each of six 2. Read More

Weak-lensing (WL) peak counts provide a straightforward way to constrain cosmology, and results have been shown promising. However, the importance of understanding and dealing with systematics increases as data quality reaches an unprecedented level. One of the sources of systematics is the convergence-shear inversion. Read More

Beaming effect is important for the observational properties of blazars. In this work, we collect 91 $Fermi$ blazars with available radio Doppler factors. $\gamma$-ray Doppler factors are estimated and compared with radio Doppler factors for some sources. Read More

The various types of communication technologies and mobility features in Internet of Things (IoT) on the one hand enable fruitful and attractive applications, but on the other hand facilitates malware propagation, thereby raising new challenges on handling IoT-empowered malware for cyber security. Comparing with the malware propagation control scheme in traditional wireless networks where nodes can be directly repaired and secured, in IoT, compromised end devices are difficult to be patched. Alternatively, blocking malware via patching intermediate nodes turns out to be a more feasible and practical solution. Read More

In a two-component jet model, the emissions are the sum of the core and extended emissions: $S^{\rm ob}=S_{\rm core}^{\rm ob}+S_{\rm ext}^{\rm ob}$, with the core emissions, $S_{\rm core}^{\rm ob}= f S_{\rm ext}^{\rm ob}\delta^{q}$, being a function of the Doppler factor, $\delta$, the extended emission, $S_{\rm ext}^{\rm ob}$, jet type dependent factor, $q$, and the ratio of the core to the extended emissions in the comoving frame, $f$. The $f$ is an unobservable but important parameter. Following our previous work, we collect 65 blazars with available Doppler factor, $\delta$, superluminal velocity, $\beta_{app}$, and core-dominance parameter, $R$, calculate the ratio, $f$, and peform statistical analyses. Read More

We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter $\tau$, we introduce a conceptual proof by applying Painlev\'{e} VI\ equation. As an application of this simple zero property, we obtain the smoothness of all the degeneracy curves of trivial critical points for some multiple Green function. Read More

As virtual reality (VR) emerges as a mainstream platform, designers have started to experiment new interaction techniques to enhance the user experience. This is a challenging task because designers not only strive to provide designs with good performance but also carefully ensure not to disrupt users' immersive experience. There is a dire need for a new evaluation tool that extends beyond traditional quantitative measurements to assist designers in the design process. Read More

There is only a handful of scanning techniques that can provide surface topography at nanometre resolution. At the same time, there are no methods that are capable of non-invasive imaging of the three-dimensional surface topography of a thin free-standing crystalline material. Here we propose a new technique - the divergent beam electron diffraction (DBED) and show that it can directly image the inhomogeneity in the atomic positions in a crystal. Read More

The properties of a bound electron system immersed in a plasma environment are strongly modified by the surrounding plasma. The modification of a basic quantity, the ionization energy, is described by the electronic self-energy and by dynamical screening within the framework of the quantum statistical theory. Introducing the ionic dynamical structure factor as the indicator for the ionic micro-field, we demonstrate that ionic correlations and fluctuations play a critical role in determining the ionization potential depression. Read More

In many phase II trials in solid tumours, patients are assessed using endpoints based on the Response Evaluation Criteria in Solid Tumours (RECIST) scale. Often, analyses are based on the response rate. This is the proportion of patients who have an observed tumour shrinkage above a pre-defined level and no new tumour lesions. Read More

The pioneering work of Brezis-Merle [7], Li-Shafrir [27], Li [26] and Bartolucci-Tarantello [4] showed that any sequence of blow up solutions for (singular) mean field equations of Liouville type must exhibit a "mass concentration" property. A typical situation of blow-up occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. Read More

Motivated by the absence of signals of new physics at the LHC, which seems to imply the presence of large mass hierarchies, we investigate the theoretical possibility that these could arise dynamically in new strongly-coupled gauge theories extending the standard model of particle physics. To this purpose, we study lattice data on non-Abelian gauge theories in the (near-)conformal regime---specifically, $\mathrm{SU}(2)$ with $N_{\mathrm{f}}=1$ and $2$ dynamical fermion flavours in the adjoint representation. We focus our attention on the ratio $R$ between the masses of the lightest spin-2 and spin-0 resonances, and draw comparisons with a simple toy model in the context of gauge/gravity dualities. Read More

Variability is one of the extreme observational properties of BL Lacertae objects. AO 0235+164 is a well studied BL Lac through the whole electro-magnetic wavebands. In the present work, we show its optical R band photometric observations carried out during the period of Nov, 2006 to Dec. Read More

Blazars are an extreme subclass of active galactic nuclei. Their rapid variability, luminous brightness, superluminal motion, and high and variable polarization are probably due to a beaming effect. However, this beaming factor (or Doppler factor) is very difficult to measure. Read More

Here we propose a linear microbolometric array based on VOx thin films. The linear microbolometric array is fabricated by using micromachining technology, and its thermo-sensitive VOx thin film has excellent infrared response spectrum and TCR characteristics. Nano-scale VOx thin films deposited on SiO2/Si substrates were obtained by e-beam vapor deposition. Read More

One big challenge that hinders the transition of brain-computer interfaces (BCIs) from laboratory settings to real-life applications is the availability of high-performance and robust learning algorithms that can effectively handle individual differences, i.e., algorithms that can be applied to a new subject with zero or very little subject-specific calibration data. Read More

Several active areas of research in novel energy storage technologies, including three-dimensional solid state batteries and passivation coatings for reactive battery electrode components, require conformal solid state electrolytes. We describe an atomic layer deposition (ALD) process for a member of the lithium phosphorus oxynitride (LiPON) family, which is employed as a thin film lithium-conducting solid electrolyte. The reaction between lithium tert-butoxide (LiO$^t$Bu) and diethyl phosphoramidate (DEPA) produces conformal, ionically conductive thin films with a stoichiometry close to Li$_2$PO$_2$N between 250 and 300$^\circ$C. Read More

Eclipsing binaries provide a unique opportunity to determine fundamental stellar properties. In the era of wide-field cameras and all-sky imaging surveys, thousands of eclipsing binaries have been reported through light curve classification, yet their basic properties remain unexplored due to the extensive efforts needed to follow them up spectroscopically. In this paper we investigate three M2-M3 type double-lined eclipsing binaries discovered by cross-matching eclipsing binaries from the Catalina Sky Survey wtih spectroscopically classified M dwarfs from the Large Sky Area Multi-Object Fiber Spectroscopic Telescope survey data release one and two. Read More

Comparing with conventional accelerator, laser plasma accelerator can generate high energy ions in a greatly reduced scale due to its TV/m acceleration gradient. A compact laser plasma accelerator (CLAPA) has been built at Institute of Heavy Ion physics in Peking University, based on which application researches like biological irradiation, astrophysics simulation etc. will be carried on. Read More

Recent advent in graph signal processing (GSP) has led to the development of new graph-based transforms and wavelets for image / video coding, where the underlying graph describes inter-pixel correlations. In this paper, we develop a new transform called signed graph Fourier transform (SGFT), where the underlying graph G contains negative edges that describe anti-correlations between pixel pairs. Specifically, we first construct a one-state Markov process that models both inter-pixel correlations and anti-correlations. Read More

Electroencephalogram (EEG) signals are frequently used in brain-computer interfaces (BCIs), but they are easily contaminated by artifacts and noises, so preprocessing must be done before they are fed into a machine learning algorithm for classification or regression. Spatial filters have been widely used to increase the signal-to-noise ratio of EEG for BCI classification problems, but their applications in BCI regression problems have been very limited. This paper proposes two common spatial pattern (CSP) filters for EEG-based regression problems in BCI, which are extended from the CSP filter for classification, by making use of fuzzy sets. Read More

This work presents a systematic review of the feature-rich essential properties in graphene-related systems using the first-principles method. The geometric and electronic properties are greatly diversified by the number of layers, the stacking configurations, the sliding-created configuration transformation, the rippled structures, and the distinct adatom adsorptions. The top-site adsorptions can induce the significantly buckled structures, especially for hydrogen and fluorine adatoms. Read More

A cluster-transfer experiment $^9$Be($^9$Be,$^{14}$C$^*\rightarrow\alpha$+$^{10}$Be)$\alpha$ was carried out using an incident beam energy of 45 MeV. This reaction channel has a large $Q$-value that favors populating the high-lying states in $^{14}$C and separating various reaction channels. A number of resonant states are reconstructed from the forward emitting $^{10}$Be + $\alpha$ fragments with respect to three sets of well discriminated final states in $^{10}$Be, most of which agree with the previous observations. Read More

We perform a Hamiltonian analysis and prove that a large class of scalar-tensor theories whose Lagrangian is given by a generic function $F\left(K_{ab},N,R_{ab},h^{ab},\nabla_a,t\right)$ is free from the Ostrogradsky ghost, and thus these theories contain at most 3 degrees of freedom at classical level. Read More

Two of the main problems encountered in the development and accurate validation of photometric redshift (photo-z) techniques are the lack of spectroscopic coverage in feature space (e.g. colours and magnitudes) and the mismatch between photometric error distributions associated with the spectroscopic and photometric samples. Read More

The structures, point defects and impacts of fission products for U3Si (\b{eta}-U3Si and {\gamma}-U3Si) and USi2 ({\alpha}-USi2 and \b{eta}-USi2) are studied by first-principles calculations. The lattice parameters of U3Si and USi2 are calculated and the stability of different types of point defects is predicted by their formation energies. The results show that silicon vacancies are more prone to be produced than uranium vacancies in \b{eta}-USi2 matrix, while uranium vacancies are the most stable defects of other three types of crystallographic structures. Read More

The Compton Spectrometer and Imager (COSI) is a balloon-borne, soft-gamma ray imager, spectrometer, and polarimeter with sensitivity from 0.2 to 5 MeV. Utilizing a compact Compton telescope design with twelve cross-strip, high-purity germanium detectors, COSI has three main science goals: study the 511 keV positron annihilation line from the Galactic plane, image diffuse emission from stellar nuclear lines, and perform polarization studies of gamma-ray bursts and other extreme astrophysical environments. Read More

The Compton Spectrometer and Imager (COSI) is a balloon-borne gamma-ray (0.2-5 MeV) telescope with inherent sensitivity to polarization. COSI's main goal is to study astrophysical sources such as $\gamma$-ray bursts, positron annihilation, Galactic nucleosynthesis, and compact objects. Read More

We consider a cosmological model in which the tensor mode becomes massive during inflation, and study the Cosmic Microwave Background (CMB) temperature and polarization bispectra arising from the mixing between the scalar mode and the massive tensor mode during inflation. The model assumes the existence of a preferred spatial frame during inflation. The local Lorentz invariance is already broken in cosmology due to the existence of a preferred rest frame. Read More

The localized low-energy interfacial excitations, or Nambu-Goldstone modes, of phase-segregated binary mixtures of Bose-Einstein condensates are investigated analytically by means of a double-parabola approximation (DPA) to the Lagrangian density in Gross-Pitaevskii theory for a system in a uniform potential. Within this model analytic expressions are obtained for the excitations underlying capillary waves or "ripplons" for arbitrary strength $K\,(>1)$ of the phase segregation. The dispersion relation $\omega \propto k^{3/2}$ is derived directly from the Bogoliubov-de Gennes equations in limit that the wavelength $2\pi/k$ is much larger than the healing length $\xi $. Read More

Weak gravitational lensing (WL) causes distortions of galaxy images and probes massive structures on large scales, allowing us to understand the late-time evolution of the Universe. One way to extract the cosmological information from WL is to use peak statistics. Peaks are tracers of massive halos and therefore probe the mass function. Read More

In this paper, we establish a theoretical connection between the classical Lucas & Kanade (LK) algorithm and the emerging topic of Spatial Transformer Networks (STNs). STNs are of interest to the vision and learning communities due to their natural ability to combine alignment and classification within the same theoretical framework. Inspired by the Inverse Compositional (IC) variant of the LK algorithm, we present Inverse Compositional Spatial Transformer Networks (IC-STNs). Read More

We consider ground states of two-dimensional Bose-Einstein condensates in a trap with attractive interactions, which can be described equivalently by positive minimizers of the $L^2-$critical constraint Gross-Pitaevskii energy functional. It is known that ground states exist if and only if $a< a^*:= \|w\|_2^2$, where $a$ denotes the interaction strength and $w$ is the unique positive solution of $\Delta w-w+w^3=0$ in $R^2$. In this paper, we prove the local uniqueness and refined spike profiles of ground states as $a\nearrow a^*$, provided that the trapping potential $h(x)$ is homogeneous and $H(y)=\int_{R^2} h(x+y)w^2(x)dx$ admits a unique and non-degenerate critical point. Read More

In this manuscript, the long ignored Type-III Natural-Flavor-Conserving 2HDM is reviewed and the general pattern for the fermion mass matrices in such a model is derived. With such a model, one do not have to assume extremely small tree-level FCN interactions or loop corrections as large as the tree-level interactions so as to cancel each other and keep the FCN interactions small enough to fit the empirical values. As a side effect, this model also provide a mechanism to break CP symmetry with complex elements in the CKM matrix. Read More

Peak statistics in weak lensing maps access the non-Gaussian information contained in the large-scale distribution of matter in the Universe. They are therefore a promising complement to two-point and higher-order statistics to constrain our cosmological models. To prepare for the high-precision data of next-generation surveys, we assess the constraining power of peak counts in a simulated Euclid-like survey on the cosmological parameters $\Omega_\mathrm{m}$, $\sigma_8$, and $w_0^\mathrm{de}$. Read More

The recent accelerated growth in the computing power has generated popularization of experimentation with dynamic computer models in various physical and engineering applications. Despite the extensive statistical research in computer experiments, most of the focus had been on the theoretical and algorithmic innovations for the design and analysis of computer models with scalar responses. In this paper, we propose a computationally efficient statistical emulator for a large-scale dynamic computer simulator (i. Read More

We study formation and long-term evolution of a circumstellar disk in a collapsing molecular cloud core using a resistive magnetohydrodynamic simulation. While the formed circumstellar disk is initially small, it grows as accretion continues and its radius becomes as large as 200 AUs toward the end of the Class-I phase. A pair of grand-design spiral arms form due to gravitational instability in the disk, and they transfer angular momentum in the highly resistive disk. Read More

We generalize the string-net construction to multiple flavors of strings, each of which is labeled by the elements of an abelian group $G_i$. The same flavor of strings can branch while different flavors of strings can cross one another and thus they form intersecting string-nets. We systematically construct the exactly soluble lattice Hamiltonians and the ground state wave functions for the intersecting string-net condensed phases. Read More

In a semi-supervised learning scenario, (possibly noisy) partially observed labels are used as input to train a classifier, in order to assign labels to unclassified samples. In this paper, we study this classifier learning problem from a graph signal processing (GSP) perspective. Specifically, by viewing a binary classifier as a piecewise constant graph-signal in a high-dimensional feature space, we cast classifier learning as a signal restoration problem via a classical maximum a posteriori (MAP) formulation. Read More

We extensively study the rotational group structure inside the patch space by introducing the fiber bundle structure. The rotational group structure leads to a new image denoising algorithm called the \textit{vector non-local Euclidean median} (VNLEM). The theoretical aspect of VNLEM is studied, which explains why the VNLEM and traditional non-local mean/non-local Euclidean median (NLEM) algorithm work. Read More

Neural node embeddings have recently emerged as a powerful representation for supervised learning tasks involving graph-structured data. We leverage this recent advance to develop a novel algorithm for unsupervised community discovery in graphs. Through extensive experimental studies on simulated and real-world data, we demonstrate that the proposed approach consistently improves over the current state-of-the-art. Read More

We study the scaling properties of Higgs-Yukawa models. Using the technique of Finite-Size Scaling, we are able to derive scaling functions that describe the observables of the model in the vicinity of a Gaussian fixed point. A feasibility study of our strategy is performed for the pure scalar theory in the weak-coupling regime. Read More

The decay of the proton-rich nucleus $^{22}$Si was studied by a silicon array coupled with germanium clover detectors. Nine charged-particle groups are observed and most of them are recognized as $\beta$-delayed proton emission. A charged-particle group at 5600 keV is identified experimentally as $\beta$-delayed two-proton emission from the isobaric analog state of $^{22}$Al. Read More

We classify and construct models for two-dimensional (2D) interacting fermionic symmetry-protected topological (FSPT) phases with general finite Abelian unitary symmetry $G_f$. To obtain the classification, we couple the FSPT system to a dynamical discrete gauge field with gauge group $G_f$ and study braiding statistics in the resulting gauge theory. Under reasonable assumptions, the braiding statistics data allows us to infer a potentially complete classification of 2D FSPT phases with Abelian symmetry. Read More