Super approximation for SL2(Z/qZ)×SL2(Z/qZ)\text{SL}_2(\mathbb Z/q\mathbb Z)\times \text{SL}_2(\mathbb Z/q\mathbb Z)

Let $S\subset \text{SL}_2(\mathbb Z)\times \text{SL}_2(\mathbb Z)$ be finite symmetric and assume $S$ generates a group $G$ which is Zariski-dense in $\text{SL}_2\times \text{SL}_2$. We prove that the Cayley graphs $$\{\mathcal Cay(G (\text{mod }q), S (\text{mod }q))\}_{q\in \mathbb Z_+}$$ form a family of expanders

Sum-product phenomenon in quotients of rings of algebraic integers

We obtain a bounded generation theorem over $\mathcal O/\mathfrak a$, where $\mathcal O$ is the ring of integers of a number field and $\mathfrak a$ a general ideal of $\mathcal O$. This addresses a conjecture of Salehi-Golsefidy. Along the way, we obtain nontrivial bounds for additive character sums over $\mathcal O/\mathcal P^n$ for a prime ideal $\mathcal P$ with the aid of certain sum-product estimates

Fast Diffusion GAN Model for Symbolic Music Generation Controlled by Emotions

Diffusion models have shown promising results for a wide range of generative tasks with continuous data, such as image and audio synthesis. However, little progress has been made on using diffusion models to generate discrete symbolic music because this new class of generative models are not well suited for discrete data while its iterative sampling process is computationally expensive. In this work, we propose a diffusion model combined with a Generative Adversarial Network, aiming to (i) alleviate one of the remaining challenges in algorithmic music generation which is the control of generation towards a target emotion, and (ii) mitigate the slow sampling drawback of diffusion models applied to symbolic music generation. We first used a trained Variational Autoencoder to obtain embeddings of a symbolic music dataset with emotion labels and then used those to train a diffusion model. Our results demonstrate the successful control of our diffusion model to generate symbolic music with a desired emotion. Our model achieves several orders of magnitude improvement in computational cost, requiring merely four time steps to denoise while the steps required by current state-of-the-art diffusion models for symbolic music generation is in the order of thousands

Data-driven structural control of monopile wind turbine towers based on machine learning

This paper studies the data-driven structural control of monopile wind turbine towers based on machine learning approach, by using an active tuned mass damper (TMD) located in the nacelle. The adaptive dynamic programming (ADP) approach is employed to obtain the optimal controller which is derived on the modern large-scale machine learning platform Tensorflow. The proposed network structure includes three simple three-layer neural networks (NNs), i.e. a plant network, a critic network, and an action network. The plant network is used to capture the fully nonlinear dynamics of the structural system while the action network is used to approximate the optimal controller. Their training requires the gradient information flowing through the whole network. The automatic differentiation is used in this paper for all the gradient derivations, which greatly improves the employed ADP algorithm’s ability in solving complex practical problems. The simulation results of structural control of monopile turbine towers show that on average the active TMD achieves 15% performance improvement on tower fatigue load reduction over a passive TMD, with small active power consumption (less than 0.24% of the turbine’s nominal power production). Besides, the controller design considers the trade-off between control performance and power consumption

Long-distance and high-impact wind farm wake effects revealed by SAR: a global-scale study

Wind, as a clean and sustainable source of energy, has witnessed significant growth in recent years. However, with a growing number of wind farms authorised, constructed and commissioned, the wake effect (the reduced wind speed caused by upstream wind farms) is emerging as a pressing concern for both farm owners and policymakers. Here, to systematically and comprehensively investigate the wake effects in real-world wind farms, we analyse the wind speed retrieved from 7122 Sentinel 1A/B SAR images spanning over three years, encompassing more than 60 large-scale wind farms across Europe and Asia. Our study reveals that long-distance wakes can propagate more than 100 km. Additionally, we identify that wake effects lead to, on average, a 1.204 m/s (or 12.4%) speed reduction for downstream wake-affected areas. We envisage that our quantitative findings can provide vital support to wake-related planning and legislation for future wind energy projects where wind power plants are expected to be in close proximity

Reinforcement learning-based structural control of floating wind turbines

The structural control of floating wind turbines using active tuned mass damper is investigated in this article. To our knowledge, this is for the first time that reinforcement learning-based control approach is employed to this type of application. Specifically, an adaptive dynamic programming (ADP) algorithm is used to derive the optimal control law based on the nonlinear structural dynamics, and the large-scale machine learning platform Tensorflow is employed for the design and implementation of the neural network (NN) structure. Three fully connected NNs, i.e., a plant network, a critic network, and an action network, are included in the proposed NN structure. Their training requires the gradient information flowing through the whole network, which is tackled by automatic differentiation, a popular technique for deriving the gradients of complex networks automatically. While to our knowledge, the network structures in the existing literature are rather simple and the training of the hidden layer is usually ignored. This allows their gradients to be derived analytically, which is infeasible with complex network structures. Thus, automatic differentiation greatly improves the employed ADP algorithm's ability in solving complex problems. The simulation results of structural control of floating wind turbines show that ADP controller performs very well in both normal and extreme conditions, with the standard deviation of the platform pitch displacement being reduced by around 40%. A clear advantage of ADP controllers over the H∞ controller is observed, especially in extreme conditions. Moreover, our design considers the tradeoff between the control performance and power consumption