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This paper focuses on the existence and multiplicity of normalized solutions for the coupled Schrodinger system with Sobolev critical coupling term. We present several existence and multiplicity results under some explicit conditions. Furthermore, we present a non-existence result for the defocusing case. This paper, together with the paper [T. Bartsch, H. W. Li and W. M. Zou. Calc. Var. Partial Differential Equations 62 (2023) ], provides a more comprehensive understanding of normalized solutions for Sobolev critical systems. We believe our methods can also address the open problem of the multiplicity of normalized solutions for Schrodinger systems with Sobolev critical growth, with potential for future development and broader applicability., Comment: Any comments are welcome

Due to the recurrent structure of RNN, the long information propagation path poses limitations in capturing long-term dependencies, gradient explosion/vanishing issues, and inefficient sequential execution. Based on this, we propose a novel paradigm called Parallel Gated Network (PGN) as the new successor to RNN. PGN directly captures information from previous time steps through the designed Historical Information Extraction (HIE) layer and leverages gated mechanisms to select and fuse it with the current time step information. This reduces the information propagation path to $\mathcal{O}(1)$, effectively addressing the limitations of RNN. To enhance PGN's performance in long-range time series forecasting tasks, we propose a novel temporal modeling framework called Temporal PGN (TPGN). TPGN incorporates two branches to comprehensively capture the semantic information of time series. One branch utilizes PGN to capture long-term periodic patterns while preserving their local characteristics. The other branch employs patches to capture short-term information and aggregate the global representation of the series. TPGN achieves a theoretical complexity of $\mathcal{O}(\sqrt{L})$, ensuring efficiency in its operations. Experimental results on five benchmark datasets demonstrate the state-of-the-art (SOTA) performance and high efficiency of TPGN, further confirming the effectiveness of PGN as the new successor to RNN in long-range time series forecasting. The code is available in this repository: \url{https://github.com/Water2sea/TPGN}.

Motivated by large-scale applications, there is a recent trend of research on using first-order methods for solving LP. Among them, PDLP, which is based on a primal-dual hybrid gradient (PDHG) algorithm, may be the most promising one. In this paper, we present a geometric viewpoint on the behavior of PDHG for LP. We demonstrate that PDHG iterates exhibit a spiral pattern with a closed-form solution when the variable basis remains unchanged. This spiral pattern consists of two orthogonal components: rotation and forward movement, where rotation improves primal and dual feasibility, while forward movement advances the duality gap. We also characterize the different situations in which basis change events occur. Inspired by the spiral behavior of PDHG, we design a new crossover algorithm to obtain a vertex solution from any optimal LP solution. This approach differs from traditional simplex-based crossover methods. Our numerical experiments demonstrate the effectiveness of the proposed algorithm, showcasing its potential as an alternative option for crossover.

Accurate reconstruction of plant models for phenotyping analysis is critical for optimising sustainable agricultural practices in precision agriculture. Traditional laboratory-based phenotyping, while valuable, falls short of understanding how plants grow under uncontrolled conditions. Robotic technologies offer a promising avenue for large-scale, direct phenotyping in real-world environments. This study explores the deployment of emerging robotics and digital technology in plant phenotyping to improve performance and efficiency. Three critical functional modules: environmental understanding, robotic motion planning, and in-situ phenotyping, are introduced to automate the entire process. Experimental results demonstrate the effectiveness of the system in agricultural environments. The pheno-robot system autonomously collects high-quality data by navigating around plants. In addition, the in-situ modelling model reconstructs high-quality plant models from the data collected by the robot. The developed robotic system shows high efficiency and robustness, demonstrating its potential to advance plant science in real-world agricultural environments.

Zinc electrolysis is one of the key processes in zinc smelting, and maintaining stable operation of zinc electrolysis is an important factor in ensuring production efficiency and product quality. However, poor contact between the zinc electrolysis cathode and the anode is a common problem that leads to reduced production efficiency and damage to the electrolysis cell. Therefore, online monitoring of the contact status of the plates is crucial for ensuring production quality and efficiency. To address this issue, we propose an end-to-end network, the Frequency-masked Multimodal Autoencoder (FM-AE). This method takes the cell voltage signal and infrared image information as input, and through automatic encoding, fuses the two features together and predicts the poor contact status of the plates through a cascaded detector. Experimental results show that the proposed method maintains high accuracy (86.2%) while having good robustness and generalization ability, effectively detecting poor contact status of the zinc electrolysis cell, providing strong support for production practice., Comment: 2023 The 34th Chinese Process Control Conference (CPCC 2023)

A recent GPU implementation of the Restarted Primal-Dual Hybrid Gradient Method for Linear Programming was proposed in Lu and Yang (2023). Its computational results demonstrate the significant computational advantages of the GPU-based first-order algorithm on certain large-scale problems. The average performance also achieves a level close to commercial solvers for the first time in history. However, due to limitations in experimental hardware and the disadvantage of implementing the algorithm in Julia compared to C language, neither the commercial solver nor cuPDLP reached their maximum efficiency. Therefore, in this report, we have re-implemented and optimized cuPDLP in C language. Utilizing state-of-the-art CPU and GPU hardware, we extensively compare cuPDLP with the best commercial solvers. The experiments further highlight its substantial computational advantages and potential for solving large-scale linear programming problems. We also discuss the profound impact this breakthrough may have on mathematical programming research and the entire operations research community., Comment: fix typos, update numerical results

We propose and demonstrate a general method to calibrate the frequency-dependent response of self-compensating noble-gas-alkali-metal comagnetometers to arbitrary spin perturbations. This includes magnetic and nonmagnetic perturbations like rotations and exotic spin interactions. The method is based on a fit of the magnetic field response to an analytical model. The frequency-dependent response of the comagnetometer to arbitrary spin perturbations can be inferred using the fit parameters. We demonstrate the effectiveness of this method by comparing the inferred rotation response to an experimental measurement of the rotation response. Our results show that experiments relying on zero-frequency calibration of the comagnetometer response can over- or under-estimate the comagnetometer sensitivity by orders of magnitude over a wide frequency range. Moreover, this discrepancy accumulates over time as operational parameters tend to drift during comagnetometer operation. The demonstrated calibration protocol enables accurate prediction and control of comagnetometer sensitivity to, for example, ultralight bosonic dark-matter fields coupling to electron or nuclear spins as well as accurate monitoring and control of the relevant system parameters.

In this paper, we consider the logarithmic elliptic equations with critical exponent \begin{equation} \begin{cases} -\Delta u=\lambda u+ |u|^{2^*-2}u+\theta u\log u^2, \\ u \in H_0^1(\Omega), \quad \Omega \subset \R^N. \end{cases} \end{equation} Here, the parameters $N\geq 6$, $\lambda\in \R$, $\theta>0$ and $ 2^*=\frac{2N}{N-2} $ is the Sobolev critical exponent. We prove the existence of sign-changing solution with exactly two nodal domain for an arbitrary smooth bounded domain $\Omega\subset \mathbb{R}^{N}$. When $\Omega=B_R(0)$ is a ball, we also construct infinitely many radial sign-changing solutions with alternating signs and prescribed nodal characteristic.

Linear programming has been practically solved mainly by simplex and interior point methods. Compared with the weakly polynomial complexity obtained by the interior point methods, the existence of strongly polynomial bounds for the length of the pivot path generated by the simplex methods remains a mystery. In this paper, we propose two novel pivot experts that leverage both global and local information of the linear programming instances for the primal simplex method and show their excellent performance numerically. The experts can be regarded as a benchmark to evaluate the performance of classical pivot rules, although they are hard to directly implement. To tackle this challenge, we employ a graph convolutional neural network model, trained via imitation learning, to mimic the behavior of the pivot expert. Our pivot rule, learned empirically, displays a significant advantage over conventional methods in various linear programming problems, as demonstrated through a series of rigorous experiments.

In this paper, we study the existence and nonexistence of positive solutions for a coupled elliptic system with critical exponent and logarithmic terms. The presence of the the logarithmic terms brings major challenges and makes it difficult to use the previous results established in the work of Chen and Zou without new ideas and innovative techniques., Comment: The authors welcome any comments

The development of SQUID current sensors with a thermally actuated input current limiter (TCL) integrated into the input circuit of the sensor is presented. The TCL is based on an unshunted Josephson Junction (JJ) series array, around which meanders a galvanically isolated, but tightly, thermally coupled, resistive heater element. By applying a current to the heater, the JJ critical currents can be reduced or completely suppressed, while the other on-chip SQUID circuit elements remain unaffected. The functional parameters of the TCL are determined by direct transport measurements and by static and dynamic flux coupling tests via the input circuit. In liquid Helium, a heater power, which reliably suppresses the critical current of the JJ array to zero, of $\sim$1.5 mW was needed, with the TCL normal state resistance being at 500 $\Omega$. In this configuration, typical TCL switching times below 10 $\mu$s were observed in the direct transport measurements. The TCL can be used to temporarily or even permanently disable malfunctioning channels in multichannel SQUID magnetometer systems, when feedback into their otherwise superconducting input circuits is not possible. In doing so, significant signal distortions in neighbouring channels from screening currents in these input circuits are avoided. Furthermore, by completely suppressing and restoring the TCL critical current, dc offset currents in the superconducting input circuits can be prevented. This is relevant, for instance, in SQUID-based spin precession experiments on hyperpolarized noble gases, in which even dc currents of a few $\mu$A in the input circuit can lead to significant magnetic field distortions in the adjacent sample region contributing to the transverse spin-spin relaxation rate $1/T_{2}$, Comment: 7 pages, 6 figures, ASC 2022, submitted to Superconducting Science and Technology

Self-compensated comagnetometers, employing overlapping samples of spin-polarized alkali and noble gases (for example K-$^3$He) are promising sensors for exotic beyond-the-standard-model fields and high-precision metrology such as rotation sensing. When the comagnetometer operates in the so-called self-compensated regime, the effective field, originating from contact interactions between the alkali valence electrons and the noble-gas nuclei, is compensated with an applied magnetic field. When the comagnetometer begins operation in a given magnetic field, spin-exchange optical pumping establishes equilibrium between the alkali electron-spin polarization and the nuclear-spin polarization. Subsequently, when the magnetic field is tuned to the compensation point, the spin polarization is brought out of the equilibrium conditions. This causes a practical issue for long measurement times. We report on a novel method for closed-loop control of the compensation field. This method allows optimization of the operating parameters, especially magnetic field gradients, in spite of the inherently slow (hours to days) dynamics of the system. With the optimization, higher stable nuclear polarization, longer relaxation times and stronger electron-nuclear coupling are achieved which is useful for nuclear-spin-based quantum memory, spin amplifiers and gyroscopes. The optimized sensor demonstrates a sensitivity comparable to the best previous comagnetometer but with four times lower noble gas density. This paves the way for applications in both fundamental and applied science.

SOLNP+ is a derivative-free solver for constrained nonlinear optimization. It starts from SOLNP proposed in 1989 by Ye Ye with the main idea that uses finite difference to approximate the gradient. We incorporate the techniques of implicit filtering, new restart mechanism and modern quadratic programming solver into this new version with an ANSI C implementation. The algorithm exhibits a great advantage in running time and robustness under noise compared with the last version by MATLAB. SOLNP+ is free to download at https://github.com/COPT-Public/SOLNP_plus.

The ADMM-based interior point (ABIP, Lin et al. 2021) method is a hybrid algorithm that effectively combines interior point method (IPM) and first-order methods to achieve a performance boost in large-scale linear optimization. Different from traditional IPM that relies on computationally intensive Newton steps, the ABIP method applies the alternating direction method of multipliers (ADMM) to approximately solve the barrier penalized problem. However, similar to other first-order methods, this technique remains sensitive to condition number and inverse precision. In this paper, we provide an enhanced ABIP method with multiple improvements. Firstly, we develop an ABIP method to solve the general linear conic optimization and establish the associated iteration complexity. Secondly, inspired by some existing methods, we develop different implementation strategies for ABIP method, which substantially improve its performance in linear optimization. Finally, we conduct extensive numerical experiments in both synthetic and real-world datasets to demonstrate the empirical advantage of our developments. In particular, the enhanced ABIP method achieves a 5.8x reduction in the geometric mean of run time on $105$ selected LP instances from Netlib, and it exhibits advantages in certain structured problems such as SVM and PageRank. However, the enhanced ABIP method still falls behind commercial solvers in many benchmarks, especially when high accuracy is desired. We posit that it can serve as a complementary tool alongside well-established solvers.

In the present paper, we consider the coupled Schr\"{o}dinger systems with critical exponent: \begin{equation*} \begin{cases} -\Delta u_i+\lambda_{i}u_i=\sum\limits_{j=1}^{d} \beta_{ij}|u_j|^{3}|u_i|u_i \quad ~\text{ in } \Omega,\\ u_i \in H_0^1(\Omega) ,\quad i= 1,2,...,d. \end{cases} \end{equation*} Here, $\Omega\subset \mathbb{R}^{3}$ is a smooth bounded domain, $d \geq 2$, $\beta_{ii}>0$ for every $i$, and $\beta_{ij}=\beta_{ji}$ for $i \neq j$. We study a Br\'{e}zis-Nirenberg type problem: $-\lambda_{1}(\Omega)<\lambda_{1},\cdots,\lambda_{d}<-\lambda^*(\Omega)$, where $\lambda_{1}(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary conditions and $\lambda^*(\Omega)\in (0, \lambda_1(\Omega))$. We acquire the existence of least energy positive solutions to this system for weakly cooperative case ($\beta_{ij}>0$ small) and for purely competitive case ($\beta_{ij}\leq 0$) by variational arguments. The proof is performed by mathematical induction on the number of equations, and requires more refined energy estimates for this system. Besides, we present a new nonexistence result, revealing some different phenomena comparing with the higher-dimensional case $N\geq 5$. It seems that this is the first paper to give a rather complete picture for the existence of least energy positive solutions to critical Schr\"{o}dinger system in dimension three.

We argue that information from countries who had earlier COVID-19 surges can be used to inform another country's current model, then generating what we call back-to-the-future (BTF) projections. We show that these projections can be used to accurately predict future COVID-19 surges prior to an inflection point of the daily infection curve. We show, across 12 different countries from all populated continents around the world, that our method can often predict future surges in scenarios where the traditional approaches would always predict no future surges. However, as expected, BTF projections cannot accurately predict a surge due to the emergence of a new variant. To generate BTF projections, we make use of a matching scheme for asynchronous time series combined with a response coaching SIR model., Comment: 21 pages, 7 figures