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Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which not only gives, as a special case, a famous modular relation between the Rogers-Ramanujan functions $G(q)$ and $H(q)$ but also a relation between two fifth order mock theta functions and $G(q)$ and $H(q)$. We generalize Ramanujan's relation with the help of a parameter $s$ to get an infinite family of such identities. Our result shows that a theta function can always be ``dissected'' as a finite sum of products of generalized Rogers-Ramanujan functions. Several well-known results are shown to be consequences of our theorem, for example, a generalization of the Jacobi triple product identity and Andrews' relation between two of his generalized third order mock theta functions. We give enough evidence, through asymptotic analysis as well as by other means, to show that the identities we get from our main result for $s>2$ transcend the modular world and hence look difficult to be written in the form of a modular relation. Using asymptotic analysis, we also offer a clinching evidence that explains how Ramanujan may have arrived at his generalized modular relation., Comment: 23 pages, submitted for publication. Comments are welcome

This work introduces a novel adaptive mesh refinement (AMR) method that utilizes dominant balance analysis (DBA) for efficient and accurate grid adaptation in computational fluid dynamics (CFD) simulations. The proposed method leverages a Gaussian mixture model (GMM) to classify grid cells into active and passive regions based on the dominant physical interactions in the equation space. Unlike traditional AMR strategies, this approach does not rely on heuristic-based sensors or user-defined parameters, providing a fully automated and problem-independent framework for AMR. Applied to the incompressible Navier-Stokes equations for unsteady flow past a cylinder, the DBA-based AMR method achieves comparable accuracy to high-resolution grids while reducing computational costs by up to 70%. The validation highlights the method's effectiveness in capturing complex flow features while minimizing grid cells, directing computational resources toward regions with the most critical dynamics. This modular and scalable strategy is adaptable to a range of applications, presenting a promising tool for efficient high-fidelity simulations in CFD and other multiphysics domains., Comment: 22 pages, 12 figures

Recent experimental studies reveal that the near-wake region of a circular cylinder at hypersonic Mach numbers exhibits self-sustained flow oscillations. The oscillation frequency was found to have a universal behavior. Experimental observations suggest an aeroacoustic feedback loop to be the driving mechanism of oscillations. An analytical aeroacoustic model which predicts the experimentally observed frequencies and explains the universal behavior is presented here. The model provides physical insights and informs of flow regimes where deviations from universal behavior are to be expected., Comment: 6 pages, 6 figures

Transformer-based models have revolutionized the field of image super-resolution (SR) by harnessing their inherent ability to capture complex contextual features. The overlapping rectangular shifted window technique used in transformer architecture nowadays is a common practice in super-resolution models to improve the quality and robustness of image upscaling. However, it suffers from distortion at the boundaries and has limited unique shifting modes. To overcome these weaknesses, we propose a non-overlapping triangular window technique that synchronously works with the rectangular one to mitigate boundary-level distortion and allows the model to access more unique sifting modes. In this paper, we propose a Composite Fusion Attention Transformer (CFAT) that incorporates triangular-rectangular window-based local attention with a channel-based global attention technique in image super-resolution. As a result, CFAT enables attention mechanisms to be activated on more image pixels and captures long-range, multi-scale features to improve SR performance. The extensive experimental results and ablation study demonstrate the effectiveness of CFAT in the SR domain. Our proposed model shows a significant 0.7 dB performance improvement over other state-of-the-art SR architectures., Comment: Accepted to CVPR 2024

Savonius turbines, prominent in small-scale wind turbine applications operating under low-speed conditions, encounter limitations due to opposing torque on the returning blade, impeding high efficiency. A viable solution involves mitigating this retarding torque by directing incoming airflow through a cylindrical deflector. However, such flow control is highly contingent upon the location and size of the cylindrical deflector, and its angular velocity. This study introduces a novel design optimization framework tailored for enhancing the turbine-deflector system's performance. Leveraging surrogate models for computational efficiency, six different models were assessed, with Kriging selected for subsequent analysis based on its superior performance at approximating the relation between design parameters and objective function. The training data for the surrogate model and the flow field data around the system were obtained through Unsteady Reynolds-Averaged Navier Stokes (URANS) simulations using a sliding mesh technique. An in-house code for the Quantum-based Salp Swarm Optimization (QSSO) algorithm was then employed to obtain design parameters corresponding to the peak power coefficient (Cp) for the stationary deflector-turbine system. Additionally, the QSSO algorithm was quantitatively compared with nine other competing algorithms. The optimized stationary deflector-turbine system showed an improvement of 26.94% in Cp at Tip Speed Ratio (TSR) of 0.9 compared to the baseline case. Further investigation into the effect of deflector rotational velocity ($\omega_d$) revealed significant improvements: 40.98% and 11.33% enhancement at $\omega_d$ = 3 rad/s, and 51.23% and 19.42% at $\omega_d$ = 40 rad/s, compared to configurations without a deflector and with the optimized stationary deflector, respectively at a TSR of 0.9., Comment: 42 pages, 16 figures

Jointly learning multiple tasks with a unified model can improve accuracy and data efficiency, but it faces the challenge of task interference, where optimizing one task objective may inadvertently compromise the performance of another. A solution to mitigate this issue is to allocate task-specific parameters, free from interference, on top of shared features. However, manually designing such architectures is cumbersome, as practitioners need to balance between the overall performance across all tasks and the higher computational cost induced by the newly added parameters. In this work, we propose \textit{InterroGate}, a novel multi-task learning (MTL) architecture designed to mitigate task interference while optimizing inference computational efficiency. We employ a learnable gating mechanism to automatically balance the shared and task-specific representations while preserving the performance of all tasks. Crucially, the patterns of parameter sharing and specialization dynamically learned during training, become fixed at inference, resulting in a static, optimized MTL architecture. Through extensive empirical evaluations, we demonstrate SoTA results on three MTL benchmarks using convolutional as well as transformer-based backbones on CelebA, NYUD-v2, and PASCAL-Context., Comment: Under review

Coherent small-amplitude unsteadiness of the shock wave and the separation region over a canonical double cone flow, termed in literature as oscillation-type unsteadiness, is experimentally studied at Mach 6. The double cone model is defined by three non-dimensional geometric parameters: fore- and aft-cone angles ($\theta_1$ and $\theta_2$), and ratio of the conical slant lengths ($\Lambda$). Previous studies of oscillations have been qualitative in nature, and mostly restricted to a special case of the cone model with fixed $\theta_1 = 0^\circ$ and $\theta_2 = 90^\circ$ (referred to as the spike-cylinder model), where $\Lambda$ becomes the sole governing parameter. In the present effort we investigate the self-sustained flow oscillations in the $\theta_1$-$\Lambda$ parameter space for fixed $\theta_2 = 90^\circ$ using time-resolved schlieren visualization. The experiments reveal two distinct sub-types of oscillations, characterized by the motion (or lack thereof) of the separation point on the fore-cone surface. The global time scale associated with flow oscillation is extracted using spectral proper orthogonal decomposition. The non-dimensional frequency (Strouhal number) of oscillation is seen to exhibit distinct scaling for the two oscillation sub-types. The relationship observed between the local flow properties, instability of the shear layer, and geometric constraints on the flow suggests that an aeroacoustic feedback mechanism sustains the oscillations. Based on this insight, a simple model with no empiricism is developed for the Strouhal number. The model predictions are found to match well with experimental measurements. The model provides helpful physical insight into the nature of the self-sustained flow oscillations over a double cone at high-speeds., Comment: 26 pages, 19 figures