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Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n-$th Bernoulli numbers. In this paper, we will generalize this problem by using congruent theory and combinatorial methods, and we get some curious congruences.

Visualizing high-dimensional data is an important routine for understanding biomedical data and interpreting deep learning models. Neighbor embedding methods, such as t-SNE, UMAP, and LargeVis, among others, are a family of popular visualization methods which reduce high-dimensional data to two dimensions. However, recent studies suggest that these methods often produce visual artifacts, potentially leading to incorrect scientific conclusions. Recognizing that the current limitation stems from a lack of data-independent notions of embedding maps, we introduce a novel conceptual and computational framework, LOO-map, that learns the embedding maps based on a classical statistical idea known as the leave-one-out. LOO-map extends the embedding over a discrete set of input points to the entire input space, enabling a systematic assessment of map continuity, and thus the reliability of the visualizations. We find for many neighbor embedding methods, their embedding maps can be intrinsically discontinuous. The discontinuity induces two types of observed map distortion: ``overconfidence-inducing discontinuity," which exaggerates cluster separation, and ``fracture-inducing discontinuity," which creates spurious local structures. Building upon LOO-map, we propose two diagnostic point-wise scores -- perturbation score and singularity score -- to address these limitations. These scores can help identify unreliable embedding points, detect out-of-distribution data, and guide hyperparameter selection. Our approach is flexible and works as a wrapper around many neighbor embedding algorithms. We test our methods across multiple real-world datasets from computer vision and single-cell omics to demonstrate their effectiveness in enhancing the interpretability and accuracy of visualizations., Comment: 43 pages, 15 figures

Deep supervised models possess significant capability to assimilate extensive training data, thereby presenting an opportunity to enhance model performance through training on multiple datasets. However, conflicts arising from different label spaces among datasets may adversely affect model performance. In this paper, we propose a novel approach to automatically construct a unified label space across multiple datasets using graph neural networks. This enables semantic segmentation models to be trained simultaneously on multiple datasets, resulting in performance improvements. Unlike existing methods, our approach facilitates seamless training without the need for additional manual reannotation or taxonomy reconciliation. This significantly enhances the efficiency and effectiveness of multi-dataset segmentation model training. The results demonstrate that our method significantly outperforms other multi-dataset training methods when trained on seven datasets simultaneously, and achieves state-of-the-art performance on the WildDash 2 benchmark.

Embedding high-dimensional data into a low-dimensional space is an indispensable component of data analysis. In numerous applications, it is necessary to align and jointly embed multiple datasets from different studies or experimental conditions. Such datasets may share underlying structures of interest but exhibit individual distortions, resulting in misaligned embeddings using traditional techniques. In this work, we propose \textit{Entropic Optimal Transport (EOT) eigenmaps}, a principled approach for aligning and jointly embedding a pair of datasets with theoretical guarantees. Our approach leverages the leading singular vectors of the EOT plan matrix between two datasets to extract their shared underlying structure and align the datasets accordingly in a common embedding space. We interpret our approach as an inter-data variant of the classical Laplacian eigenmaps and diffusion maps embeddings, showing that it enjoys many favorable analogous properties. We then analyze a data-generative model where two observed high-dimensional datasets share latent variables on a common low-dimensional manifold, but each dataset is subject to data-specific translation, scaling, nuisance structures, and noise. We show that in a high-dimensional asymptotic regime, the EOT plan recovers the shared manifold structure by approximating a kernel function evaluated at the locations of the latent variables. Subsequently, we provide a geometric interpretation of our embedding by relating it to the eigenfunctions of population-level operators encoding the density and geometry of the shared manifold. Finally, we showcase the performance of our approach for data integration and embedding through simulations and analyses of real-world biological data, demonstrating its advantages over alternative methods in challenging scenarios.

Low-dimensional embeddings (LDEs) of high-dimensional data are ubiquitous in science and engineering. They allow us to quickly understand the main properties of the data, identify outliers and processing errors, and inform the next steps of data analysis. As such, LDEs have to be faithful to the original high-dimensional data, i.e., they should represent the relationships that are encoded in the data, both at a local as well as global scale. The current generation of LDE approaches focus on reconstructing local distances between any pair of samples correctly, often out-performing traditional approaches aiming at all distances. For these approaches, global relationships are, however, usually strongly distorted, often argued to be an inherent trade-off between local and global structure learning for embeddings. We suggest a new perspective on LDE learning, reconstructing angles between data points. We show that this approach, Mercat, yields good reconstruction across a diverse set of experiments and metrics, and preserve structures well across all scales. Compared to existing work, our approach also has a simple formulation, facilitating future theoretical analysis and algorithmic improvements.

Sulfur chemistry in the formation process of low-mass stars and planets remains poorly understood. The protoplanetary disks (PPDs) are the birthplace of planets and its distinctive environment provides an intriguing platform for investigating models of sulfur chemistry. We analyzed the ALMA observations of CS 7-6 transitions in the HD 163296 disk and perform astrochemical modeling to explore its sulfur chemistry. We simulated the distribution of sulfur-containing molecules and compared it with observationally deduced fractional column densities. We have found that the simulated column density of CS is consistent with the observationally deduced fractional column densities, while the simulated column density of C$_2$S is lower than the observationally deduced upper limits on column densities. This results indicate that we have a good understanding of the chemical properties of CS and C$_2$S in the disk. We also investigated the influence of the C/O ratio on sulfur-containing molecules and found that the column densities of SO, SO$_2$, and H$_2$S near the central star are dependent on the C/O ratio. Additionally, we found that the $N$[CS]/$N$[SO] ratio can serve as a promising indicator of the disk's C/O ratio in the HD 163296. Overall, the disk of HD 163296 provides a favorable environment for the detection of sulfur-containing molecules., Comment: 16 pages, 7 figures

We introduce a wall model (WM) for large-eddy simulation (LES) applicable to rough surfaces with Gaussian and non-Gaussian distributions for both transitionally and fully rough regimes. The model is applicable to arbitrary complex geometries where roughness elements are assumed to be underresolved. The wall model is implemented using a feedforward neural network, with the geometric properties of the roughness topology and near-wall flow quantities serving as input. The optimal set of non-dimensional input features is identified using information theory, selecting variables that maximize information about the output while minimizing redundancy among inputs. The model incorporates a confidence score based on Gaussian process modeling, enabling the detection of low model performance for unseen rough surfaces. The model is trained using a direct numerical simulation roughness database comprising approximately 200 cases. The roughness geometries for the database are selected from a large repository through active learning. This approach ensures that the rough surfaces incorporated into the database are the most informative. The model performance is evaluated both a-priori and a-posteriori in WMLES of turbulent channel flows with rough walls. Over 120 channel flow cases are considered, including untrained roughness geometries, roughness Reynolds numbers, and grid resolutions for both transitionally- and fully-rough regimes. The results show that the rough-wall model predicts the wall shear stress within 15% accuracy. The model is also assessed on a high-pressure turbine blade with two different rough surfaces. The WM predicts the skin friction and the mean velocity deficit within 10% accuracy except the region with shock waves.

Integrative analysis of multiple heterogeneous datasets has become standard practice in many research fields, especially in single-cell genomics and medical informatics. Existing approaches oftentimes suffer from limited power in capturing nonlinear structures, insufficient account of noisiness and effects of high-dimensionality, lack of adaptivity to signals and sample sizes imbalance, and their results are sometimes difficult to interpret. To address these limitations, we propose a novel kernel spectral method that achieves joint embeddings of two independently observed high-dimensional noisy datasets. The proposed method automatically captures and leverages possibly shared low-dimensional structures across datasets to enhance embedding quality. The obtained low-dimensional embeddings can be utilized for many downstream tasks such as simultaneous clustering, data visualization, and denoising. The proposed method is justified by rigorous theoretical analysis. Specifically, we show the consistency of our method in recovering the low-dimensional noiseless signals, and characterize the effects of the signal-to-noise ratios on the rates of convergence. Under a joint manifolds model framework, we establish the convergence of ultimate embeddings to the eigenfunctions of some newly introduced integral operators. These operators, referred to as duo-landmark integral operators, are defined by the convolutional kernel maps of some reproducing kernel Hilbert spaces (RKHSs). These RKHSs capture the either partially or entirely shared underlying low-dimensional nonlinear signal structures of the two datasets. Our numerical experiments and analyses of two single-cell omics datasets demonstrate the empirical advantages of the proposed method over existing methods in both embeddings and several downstream tasks., Comment: 32 pages, 5 figures; comments are welcome

To realize large-scale quantum information processes, an ideal scheme for two-qubit operations should enable diverse operations with given hardware and physical interaction. However, for spin qubits in semiconductor quantum dots, the common two-qubit operations, including CPhase gates, SWAP gates, and CROT gates, are realized with distinct parameter regions and control waveforms, posing challenges for their simultaneous implementation. Here, taking advantage of the inherent Heisenberg interaction between spin qubits, we propose and verify a fast composite two-qubit gate scheme to extend the available two-qubit gate types as well as reduce the requirements for device properties. Apart from the formerly proposed CPhase (controlled-phase) gates and SWAP gates, theoretical results indicate that the iSWAP-family gate and Fermionic simulation (fSim) gate set are additionally available for spin qubits. Meanwhile, our gate scheme limits the parameter requirements of all essential two-qubit gates to a common J~{\Delta}E_Z region, facilitate the simultaneous realization of them. Furthermore, we present the preliminary experimental demonstration of the composite gate scheme, observing excellent match between the measured and simulated results. With this versatile composite gate scheme, broad-spectrum two-qubit operations allow us to efficiently utilize the hardware and the underlying physics resources, helping accelerate and broaden the scope of the upcoming noise intermediate-scale quantum (NISQ) computing., Comment: 23 pages, 6 figures

We present an unbiased $\lambda$ 3 mm spectral line survey (between 84.5 and 115.8 GHz), conducted by the Purple Mountain Observatory 13.7 meter radio telescope, together with updated modeling results, towards the carbon-rich Asymptotic Giant Branch star, IRC+10216 (CW Leo). A total of 75 spectral lines (96 transitions) are detected, and identified to arise from 19 molecules: C$_2$H, $l$-C$_3$H, C$_4$H, CN, C$_3$N, HC$_3$N, HC$_5$N, HCN, HNC, CH$_3$CN, MgNC, CO, $c$-C$_3$H$_2$, SiC$_2$, SiO, SiS, CS, C$_2$S, C$_3$S, and their isotopologues. Among them, one molecular emission line (H$^{13}$CCCN $J=13-12$) is discovered in IRC+10216 for the first time. The excitation temperature, column density, and fractional abundance of the detected species are deduced by assuming they are in local thermodynamic equilibrium. In addition, the isotopic ratios of [$^{12}$C]/[$^{13}$C], [$^{32}$S]/[$^{34}$S], [$^{28}$Si]/[$^{29}$Si], and [$^{12}$C$^{34}$S]/[$^{13}$C$^{32}$S] are obtained and found to be consistent with previous studies. Finally, we summarize all of the 106 species detected in IRC+10216 to date with their observed and modeled column densities for the convenience of future studies., Comment: 71 pages, 39 figures, 10 tables. Accepted for publication in ApJS

Nonreciprocal transport in uniform systems has attracted great research interest recently and the existing theories mainly focus on the diffusive regime. In this study, we uncover a novel scenario for nonreciprocal charge transport in the ballistic regime enabled by asymmetric band structures of the system. The asymmetry of the bands induces unequal Coulomb potentials within the system as the bias voltage imposed by the electrodes inverts its sign. As a result, the bands undergo different energy shifts as the current flows in opposite directions, giving rise to the nonreciprocity. Utilizing the gauge-invariant nonlinear transport theory, we show that the nonreciprocal transport predominantly originates from the second-order conductance, which violates the Onsager reciprocal relation but fulfills a generalized reciprocal relation similar to that of unidirectional magnetoresistance. The ballistic nonreciprocal transport phenomena differ from the diffusive ones by considering the internal asymmetric Coulomb potential, a factor not accounted for in diffusive cases but undeniably crucial in ballistic scenarios. Our work opens a avenue for implementing nonreciprocal transport in the ballistic regime and provides an alternative perspective for further experimental explorations for nonreciprocal transport., Comment: 7 pages, 4 figures

With one- and two-qubit gate fidelities approaching the fault-tolerance threshold for spin qubits in silicon, how to scale up the architecture and make large arrays of spin qubits become the more pressing challenges. In a scaled-up structure, qubit-to-qubit connectivity has crucial impact on gate counts of quantum error correction and general quantum algorithms. In our toolbox of quantum gates for spin qubits, SWAP gate is quite versatile: it can help solve the connectivity problem by realizing both short- and long-range spin state transfer, and act as a basic two-qubit gate, which can reduce quantum circuit depth when combined with other two-qubit gates. However, for spin qubits in silicon quantum dots, high fidelity SWAP gates have not been demonstrated due to the requirements of large circuit bandwidth and a highly adjustable ratio between the strength of the exchange coupling J and the Zeeman energy difference Delta E_z. Here we demonstrate a fast SWAP gate with a duration of ~25 ns based on quantum dots in isotopically enriched silicon, with a highly adjustable ratio between J and Delta E_z, for over two orders of magnitude in our device. We are also able to calibrate the single-qubit local phases during the SWAP gate by incorporating single-qubit gates in our circuit. By independently reading out the qubits, we probe the anti-correlations between the two spins, estimate the operation fidelity and analyze the dominant error sources for our SWAP gate. These results pave the way for high fidelity SWAP gates, and processes based on them, such as quantum communication on chip and quantum simulation by engineering the Heisenberg Hamiltonian in silicon., Comment: 25 pages, 5 figures

Preserving qubit coherence and maintaining high-fidelity qubit control under complex noise environment is an enduring challenge for scalable quantum computing. Here we demonstrate an addressable fault-tolerant single spin qubit with an average control fidelity of 99.12% via randomized benchmarking on a silicon quantum dot device with an integrated micromagnet. Its dephasing time T2* is 1.025 us and can be enlarged to 264 us by using the Hahn echo technique, reflecting strong low-frequency noise in our system. To break through the noise limitation, we introduce geometric quantum computing to obtain high control fidelity by exploiting its noise-resilient feature. However, the control fidelities of the geometric quantum gates are lower than 99%. According to our simulation, the noise-resilient feature of geometric quantum gates is masked by the heating effect. With further optimization to alleviate the heating effect, geometric quantum computing can be a potential approach to reproducibly achieving high-fidelity qubit control in a complex noise environment., Comment: 10 pages, 8 figures

High-fidelity singlet-triplet state readout is essential for large-scale quantum computing. However, the widely used threshold method of comparing a mean value with the fixed threshold will limit the judgment accuracy, especially for the relaxed triplet state, under the restriction of relaxation time and signal-to-noise ratio. Here, we achieve an enhanced latching readout based on Pauli spin blockade in a Si-MOS double quantum dot device and demonstrate an average singlet-triplet state readout fidelity of 97.59% by the threshold method. We reveal the inherent deficiency of the threshold method for the relaxed triplet state classification and introduce machine learning as a relaxation-independent readout method to reduce the misjudgment. The readout fidelity for classifying the simulated single-shot traces can be improved to 99.67% by machine learning method, better than the threshold method of 97.54% which is consistent with the experimental result. This work indicates that machine learning method can be a strong potential candidate for alleviating the restrictions of stably achieving high-fidelity and high-accuracy singlet-triplet state readout in large-scale quantum computing., Comment: 11 pages,11 figures

Pulse distortion, as one of the coherent error sources, hinders the characterization and control of qubits. In the semiconductor quantum dot system, the distortions on measurement pulses and control pulses disturb the experimental results, while no effective calibration procedure has yet been reported. Here, we demonstrate two different calibration methods to calibrate and correct the distortion using the two-qubit system as a detector. The two calibration methods have different correction accuracy and complexity. One is the coarse predistortion (CPD) method, with which the distortion is partly relieved. The other method is the all predistortion (APD) method, with which we measure the transfer function and significantly improve the exchange oscillation homogeneity. The two methods use the exchange oscillation homogeneity as the metric and are appropriate for any qubit that oscillates with a diabatic pulse. With the APD procedure, an arbitrary control waveform can be accurately delivered to the device, which is essential for characterizing qubits and improving gate fidelity., Comment: 19 pages, 5 figures

Single-cell data integration can provide a comprehensive molecular view of cells, and many algorithms have been developed to remove unwanted technical or biological variations and integrate heterogeneous single-cell datasets. Despite their wide usage, existing methods suffer from several fundamental limitations. In particular, we lack a rigorous statistical test for whether two high-dimensional single-cell datasets are alignable (and therefore should even be aligned). Moreover, popular methods can substantially distort the data during alignment, making the aligned data and downstream analysis difficult to interpret. To overcome these limitations, we present a spectral manifold alignment and inference (SMAI) framework, which enables principled and interpretable alignability testing and structure-preserving integration of single-cell data with the same type of features. SMAI provides a statistical test to robustly assess the alignability between datasets to avoid misleading inference, and is justified by high-dimensional statistical theory. On a diverse range of real and simulated benchmark datasets, it outperforms commonly used alignment methods. Moreover, we show that SMAI improves various downstream analyses such as identification of differentially expressed genes and imputation of single-cell spatial transcriptomics, providing further biological insights. SMAI's interpretability also enables quantification and a deeper understanding of the sources of technical confounders in single-cell data.

Whether SiC$_2$ is a parent species, that is formed in the photosphere or as a by-product of high-temperature dust formation, or a daughter species, formed in a chemistry driven by the photodestruction of parent species in the outer envelope, has been debated for a long time. Here, we analyze the ALMA observations of four SiC$_2$ transitions in the CSEs of three C-rich AGB stars (AI Vol, II Lup, and RAFGL 4211), and found that SiC$_2$ exhibits an annular, shell-like distribution in these targets, suggesting that SiC$_2$ can be a daughter species in the CSEs of carbon-rich AGB stars. The results can provide important references for future chemical models., Comment: Accepted in Frontiers in Astronomy and Space Sciences

It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum_{\substack{i+j+k=p \\ i, j, k>0}} \frac{1}{i j k} \equiv-2 B_{p-3}\pmod p$$ has been generalized along different directions, where $B_n$ denote the $n$th Bernoulli number. In this paper, we obtain several new generalizations of the above congruence by applying congruences involving multiple harmonic sums. For example, we have $$\sum_{\substack{k_1+k_2+\cdots+k_n=p \\ k_i> 0, 1 \le i \le n}} \dfrac{(-1)^{k_1}\left(\dfrac{k_1}{3}\right)}{k_1 \cdots k_n} \equiv \dfrac{(n-1)!}{n}\dfrac{2^{n-1}+1}{3\cdot6^{n-1}}B_{p-n}\left(\dfrac{1}{3}\right)\pmod p,$$ where $n$ is even, $B_n(x)$ denote the Bernoulli polynomials., Comment: 12 pages

The influence of roughness spacing on boundary layer transition over distributed roughness elements is studied using direct numerical simulation (DNS) and global stability analysis, and compared to isolated roughness elements at the same Reh. Small spanwise spacing ($\lambda_z = 2.5h$) inhibits the formation of counter-rotating vortices (CVP) and as a result, hairpin vortices are not generated and the downstream shear layer is steady. For $\lambda_z = 5h$, the CVP and hairpin vortices are induced by the first row of roughness, perturbing the downstream shear layer and causing transition. The temporal periodicity of the primary hairpin vortices appears to be independent of the streamwise spacing. Distributed roughness leads to a lower critical Reh for instability to occur and a more significant breakdown of the boundary layer compared to isolated roughness. When the streamwise spacing is comparable to the region of flow separation ($\lambda_x = 5h$), the high-momentum fluid barely moves downward into the cavities and the wake flow has little impact on the following roughness elements. The leading unstable varicose mode is associated with the central low-speed streaks along the aligned roughness elements, and its frequency is close to the shedding frequency of hairpin vortices in the isolated case. For larger streamwise spacing ($\lambda_x = 10h$), two distinct modes are obtained from global stability analysis. The first mode shows varicose symmetry, corresponding to the primary hairpin vortex shedding induced by the first-row roughness. The high-speed streaks formed in the longitudinal grooves are also found to be unstable and interacting with the varicose mode. The second mode is a sinuous mode with lower frequency, induced as the wake flow of the first-row roughness runs into the second row. It extracts most energy from the spanwise shear between the high- and low-speed streaks.

Let $[x]$ be the integral part of $x$, $n>1$ be a positive integer and $\chi_n$ denote the trivial Dirichlet character modulo $n$. In this paper, we use an identity established by Z. H. Sun to get congruences of $T_{m,k}(n)=\sum_{x=1}^{[n/m]}\frac{\chi_n(x)}{x^k}\left(\bmod n^{r+1}\right)$ for $r\in \{1,2\}$, any positive integer $m $ with $n \equiv \pm 1 \left(\bmod m \right)$ in terms of Bernoulli polynomials. As its an application, we also obtain some new congruences involving binomial coefficients modulo $n^4$ in terms of generalized Bernoulli numbers., Comment: 21pages

Dimension reduction and data visualization aim to project a high-dimensional dataset to a low-dimensional space while capturing the intrinsic structures in the data. It is an indispensable part of modern data science, and many dimensional reduction and visualization algorithms have been developed. However, different algorithms have their own strengths and weaknesses, making it critically important to evaluate their relative performance for a given dataset, and to leverage and combine their individual strengths. In this paper, we propose an efficient spectral method for assessing and combining multiple visualizations of a given dataset produced by diverse algorithms. The proposed method provides a quantitative measure -- the visualization eigenscore -- of the relative performance of the visualizations for preserving the structure around each data point. Then it leverages the eigenscores to obtain a consensus visualization, which has much improved { quality over the individual visualizations in capturing the underlying true data structure.} Our approach is flexible and works as a wrapper around any visualizations. We analyze multiple simulated and real-world datasets from diverse applications to demonstrate the effectiveness of the eigenscores for evaluating visualizations and the superiority of the proposed consensus visualization. Furthermore, we establish rigorous theoretical justification of our method based on a general statistical framework, yielding fundamental principles behind the empirical success of consensus visualization along with practical guidance., Comment: Under revision of Nature Communications

Spin qubits based on silicon metal-oxide semiconductor (Si-MOS) quantum dots (QDs) are promising platforms for large-scale quantum computers. To control spin qubits in QDs, electric dipole spin resonance (EDSR) has been most commonly used in recent years. By delocalizing an electron across a double quantum dots charge state, flopping-mode EDSR has been realized in Si/SiGe QDs. Here, we demonstrate a flopping-mode spin qubit in a Si-MOS QD via Elzerman single-shot readout. When changing the detuning with a fixed drive power, we achieve s-shape spin resonance frequencies, an order of magnitude improvement in the spin Rabi frequencies, and virtually constant spin dephasing times. Our results offer a route to large-scale spin qubit systems with higher control fidelity in Si-MOS QDs., Comment: 5 pages, 4 figures