arXiv subjects

Abhishek Saha

Publications and source records attributed to Abhishek Saha.

At least 19 recordsLinked to original sources

arXiv2026

An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms

We compute the local integrals appearing in the refined Gan--Gross--Prasad conjecture for Fourier--Jacobi periods of $\mathrm{Sp}_4$ in new ramified cases and use this to formulate an explicit conjectural identity relating Petersson norms of degree 2 Siegel cusp forms and associated half-integral weight forms. We note consequences of our identity for the growth of Petersson norms, the size of Fourier coefficients, and non-vanishing of central $L$-values.

math.NT↗

arXiv2026

Simple supercuspidal representations of $\mathrm{GSp}_4$ and test vectors

We consider simple supercuspidal representations of $\mathrm{GSp}_4$ over a $p$-adic field and show that they have conductor exponent 5. We study (paramodular) newvectors and minimal vectors in these representations, obtain formulas for their matrix coefficients, and compute key local integrals involving these as test vectors. Our local computations lead to several explicit global period formulas involving automorphic representations $π$ of $\mathrm{GSp}_4(\mathbb{A})$ whose local components (at ramified primes) are simple supercuspidal representations, and where the global test vectors are chosen to be (diagonal shifts of) newforms or automorphic forms of minimal type. As an analytic application of our work to the sup-norm problem, we show the existence of paramodular newforms on $\mathrm{GSp}_4(\mathbb{A})$ of conductor $p^5$ that take ``large values" on a fixed compact set as $p\rightarrow \infty$.

math.NT↗

arXiv2026

DUALFloodGNN: Physics-informed Graph Neural Network for Operational Flood Modeling

Flood models inform strategic disaster management by simulating the spatiotemporal hydrodynamics of flooding. While physics-based numerical flood models are accurate, their substantial computational cost limits their use in operational settings where rapid predictions are essential. Models designed with graph neural networks (GNNs) provide both speed and accuracy while having the ability to process unstructured spatial domains. Given its flexible input and architecture, GNNs can be leveraged alongside physics-informed techniques with ease, significantly improving interpretability and generalizability. We introduce a novel flood GNN architecture, DUALFloodGNN, which embeds physical constraints at both global and local scales through explicit loss terms. The model jointly predicts water volume at nodes and flow along edges through a shared message-passing framework. To improve performance for autoregressive inference, model training is conducted with a multi-step loss enhanced with dynamic curriculum learning. Compared with standard GNN architectures and state-of-the-art GNN flood models, DUALFloodGNN achieves substantial improvements in predicting multiple hydrologic variables (e.g., water volume, flow, and depth) while maintaining high computational efficiency. The model is open sourced at https://github.com/acostacos/dual_flood_gnn. The dataset is open sourced at https://hdl.handle.net/2123/35293 with the DOI 10.25910/9xav-0s86.

cs.LG↗

arXiv2025

Flood-LDM: Generalizable Latent Diffusion Models for rapid and accurate zero-shot High-Resolution Flood Mapping

Flood prediction is critical for emergency planning and response to mitigate human and economic losses. Traditional physics-based hydrodynamic models generate high-resolution flood maps using numerical methods requiring fine-grid discretization; which are computationally intensive and impractical for real-time large-scale applications. While recent studies have applied convolutional neural networks for flood map super-resolution with good accuracy and speed, they suffer from limited generalizability to unseen areas. In this paper, we propose a novel approach that leverages latent diffusion models to perform super-resolution on coarse-grid flood maps, with the objective of achieving the accuracy of fine-grid flood maps while significantly reducing inference time. Experimental results demonstrate that latent diffusion models substantially decrease the computational time required to produce high-fidelity flood maps without compromising on accuracy, enabling their use in real-time flood risk management. Moreover, diffusion models exhibit superior generalizability across different physical locations, with transfer learning further accelerating adaptation to new geographic regions. Our approach also incorporates physics-informed inputs, addressing the common limitation of black-box behavior in machine learning, thereby enhancing interpretability. Code is available at https://github.com/neosunhan/flood-diff.

cs.CV↗

arXiv2025

Shear-layer effects on the dynamics of unsteady premixed laminar counterflow flames

The influence of flow non-uniformity and unsteadiness on premixed flames is of considerable interest due to its direct relevance to practical combustion systems. The steady counterflow flame has long served as a canonical configuration for investigating flame dynamics under controlled, spatially non-uniform conditions. A commonly studied variation, referred to as the unsteady counterflow, introduces a controlled temporal perturbation to the otherwise steady flow from the nozzles, thereby enabling the systematic examination of the coupled effects of unsteadiness and non-uniformity. Prior investigations have focused on flame dynamics along the line of symmetry, where the reduced dimensionality of the problem facilitates analysis. In the present study, we extend this perspective by experimentally examining flame behavior at off-center locations, where multi-dimensional effects of non-uniformity and unsteadiness are more pronounced. Results reveal markedly different dynamics away from the centerline, characterized by a dominant contribution from higher harmonic responses. Further analysis of the associated vortex dynamics in the shear layer demonstrates that the intensity of these vortical structures directly governs the strength of the observed higher harmonics, and thereby the altered flame behavior.

physics.flu-dyn↗

arXiv2025

Sub-Pixel Scale Structured Illumination for Lateral Resolution Enhancement of Non-Diffraction-Limited Flow Imaging

In fluid flow imaging, intensity gradients are a good measure of spatial variations in scalar properties, which play an important role in controlling transport processes. However, current flow imaging techniques exhibit system-limited spatial resolutions, thus inhibiting the ability to accurately detect intensity gradients. To address this challenge, we present a method and system, inspired by Structured Illumination Microscopy (SIM), which can be implemented in dynamic flow imaging to enhance pixel resolution and, thereby, the estimation of scalar gradients. We utilize sub-pixel-scale patterned light matching the system pixel scale and multi-frame imaging that creates quasi-static images over four frames, with scalability for high-speed imaging. These multi-frame images are then processed using a bespoke recombination algorithm that produces a new image with twice the pixel resolution compared to the original images. The sub-pixel spatial-resolution enhancement capabilities are shown with static images and dynamic fluid flow, for which enhancement in the flow gradient is demonstrated.

physics.optics↗

arXiv2025

Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2

Let $F$ be an $L^2$-normalized Siegel cusp form for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Hecke eigenform and not a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients satisfy the bound $|a(F,S)| \ll_ε\frac{k^{1/4+ε} (4π)^k}{Γ(k)} c(S)^{-\frac12} \det(S)^{\frac{k-1}2+ε}$ where $c(S)$ denotes the gcd of the entries of $S$, and that its global sup-norm satisfies the bound $\|(\det Y)^{\frac{k}2}F\|_\infty \ll_εk^{\frac54+ε}.$ The former result depends on new bounds that we establish for the relevant local integrals appearing in the refined global Gan-Gross-Prasad conjecture (which is now a theorem due to Furusawa and Morimoto) for Bessel periods.

math.NT↗

arXiv2024

Mass equidistribution for Saito-Kurokawa lifts

Let $F$ be a holomorphic cuspidal Hecke eigenform for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis (GRH), we prove that the mass of $F$ equidistributes on the Siegel modular variety as $k\longrightarrow \infty$. As a corollary, we show under GRH that the zero divisors of Saito--Kurokawa lifts equidistribute as their weights tend to infinity.

math.NT↗

arXiv2023

Spreading dynamics of droplets impacting on oscillating hydrophobic substrates

Droplet impact on oscillating substrates is important for both natural and industrial processes. Recognizing the importance of the dynamics that arise from the interplay between droplet transport and substrate motion, in this work, we present an experimental investigation of the spreading of a droplet impacting a sinusoidally oscillating hydrophobic substrate. We particularly focus on the maximum spread of droplets as a function of various parameters of substrate oscillation. We first quantify the maximum spreading diameter attained by the droplets as a function of frequency, amplitude of vibration, and phase at the impact for various impact velocities. We highlight that there can be two stages of spreading. Stage-I, which is observed at all impact conditions, is controlled by the droplet inertia and affected by the substrate oscillation. For certain conditions, a Stage-II spreading is also observed, which occurs during the retraction process of Stage-I due to additional energies imparted by the substrate oscillation. Subsequently, we derive scaling analyses to predict the maximum spreading diameters and the time for this maximum spread for both Stage-I and Stage-II. Furthermore, we identify the necessary condition for Stage-II spreading to be greater than Stage-I. The results will enable optimization of the parameters in applications where substrate oscillation is used to control the droplet spread and, thus, heat and mass transfer between the droplet and the substrate.

physics.flu-dyn↗

arXiv2023

Role of Ambient Pressure on Colliding Jets

In this letter, the merging-vs-bouncing response of obliquely-oriented colliding jets under elevated and reduced gaseous environment pressures was experimentally examined. Experiments with water and n-tetradecane confirmed that the collision outcome transitions from merging to bouncing, and then to merging again, when the impact velocity was increased. This behavior which was previously reported for atmospheric pressure, has now also been observed at elevated and reduced pressures. New results also show that there exists a critical pressure (0.9 bar for tetradecane and 5 bar for water) below which increasing pressure promotes bouncing (expands the bouncing regime), while beyond this, merging is promoted (reduces the bouncing regime) instead. This leads to a non-monotonic influence of pressure on the non-coalescence outcomes of collisional jets, which was not previously reported. The study provides evidence of new behaviors in colliding jets at reduced and elevated pressures, which differs from well-studied droplet-droplet collisions.

physics.flu-dyn↗

arXiv2022

The Manin constant and the modular degree

The Manin constant $c$ of an elliptic curve $E$ over $\mathbb{Q}$ is the nonzero integer that scales the differential $ω_f$ determined by the normalized newform $f$ associated to $E$ into the pullback of a Néron differential under a minimal parametrization $ϕ\colon X_0(N)_{\mathbb{Q}} \twoheadrightarrow E$. Manin conjectured that $c = \pm 1$ for optimal parametrizations, and we prove that in general $c \mid \mathrm{deg}(ϕ)$ under a minor assumption at $2$ and $3$ that is not needed for cube-free $N$ or for parametrizations by $X_1(N)_{\mathbb{Q}}$. Since $c$ is supported at the additive reduction primes, which need not divide $\mathrm{deg}(ϕ)$, this improves the status of the Manin conjecture for many $E$. Our core result that gives this divisibility is the containment $ω_f \in H^0(X_0(N), Ω)$, which we establish by combining automorphic methods with techniques from arithmetic geometry; here the modular curve $X_0(N)$ is considered over $\mathbb{Z}$ and $Ω$ is its relative dualizing sheaf over $\mathbb{Z}$. We reduce this containment to $p$-adic bounds on denominators of the Fourier expansions of $f$ at all the cusps of $X_0(N)_{\mathbb{C}}$ and then use the recent basic identity for the $p$-adic Whittaker newform to establish stronger bounds in the more general setup of newforms of weight $k$ on $X_0(N)$. To overcome obstacles at $2$ and $3$, we analyze nondihedral supercuspidal representations of $\mathrm{GL}_2(\mathbb{Q}_2)$ and exhibit new cases in which $X_0(N)_{\mathbb{Z}}$ has rational singularities.

math.NT↗

arXiv2022

On Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree 2

We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of Saito-Kurokawa type, we prove an improved bound in the $k$-aspect for the smallest prime at which its Hecke eigenvalue is negative. Secondly, we show that there are infinitely many sign changes among the Hecke eigenvalues of $F$ at primes lying in an arithmetic progression. Third, we show that there are infinitely many positive as well as infinitely many negative Fourier coefficients in any ``radial" sequence comprising of prime multiples of a fixed fundamental matrix. Finally we consider the case when $F$ is of Saito--Kurokawa type, and in this case we prove the (essentially sharp) bound $| a(T) | ~\ll_{F, ε}~ \big( \det T \big)^{\frac{k-1}{2}+ε}$ for the Fourier coefficients of $F$ whenever $\gcd(4 \det(T), N)$ is squarefree, confirming a conjecture made (in the case $N=1$) by Das and Kohnen.

math.NT↗

arXiv2022

Synchronization based model for turbulent thermoacoustic systems

We present a phenomenological reduced-order model to capture the transition to thermoacoustic instability in turbulent combustors. The model is based on the framework of synchronization and considers the acoustic field and the unsteady heat release rate from turbulent reactive flow as two nonlinearly coupled sub-systems. Previous experimental studies have reported a route from low amplitude chaotic oscillation (i. e. combustion noise) to periodic oscillations through intermittency in turbulent combustors. By varying the coupling strength, our proposed model can replicate the route that is observed in experiments. Instead of assessing combustion noise as background noise to the system, the model considered a coupled oscillator system that produces multifractal chaotic oscillations to represent the combustion noise. This set of coupled oscillators is then nonlinearly coupled to a linear oscillator representing the acoustic field. As the coupling strength increases, the system loses its multifractality and exhibits large-amplitude periodic oscillations in a manner consistent with the dynamics observed in experimental studies.

nlin.AO↗

arXiv2021

The special values of the standard $L$-functions for $\mathrm{GSp}_{2n} \times \mathrm{GL}_1$

We prove the expected algebraicity property for the critical values of character twists of the standard $L$-function associated to vector-valued holomorphic Siegel cusp forms of archimedean type $(k_1, k_2, \ldots, k_n)$, where $k_n \geq n+1$ and all $k_i$ are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach.

math.NT↗

arXiv2021

Penetration and Secondary Atomization of Droplets Impacted on Wet Facemasks

Face covering, commonly known as facemask, is considered to be one of the most effective Personal Protective Equipments (PPEs) to reduce transmissions of pathogens through respiratory droplets - both large drops and liquid aerosol particles. Face masks, not only inhibit the expulsion of such respiratory droplets from the user, but also protects the user from inhaling pathogen-laden potentially harmful droplets or their dried nuclei. While the efficacies of various dry face masks have been explored in the recent past, a comprehensive investigation of a wet mask is lacking. Yet, users wear masks for a long period of time and during this period, owing to respiratory droplets released through multiple respiratory events, the mask matrix becomes wet. We, herein, present an experimental study on the dynamics of sequential impacts of droplets on masks to understand how wetness affects possible penetration and secondary atomization of the impacted droplet. Two different types of masks, hydrophobic and hydrophilic, were used in this study to evaluate the underlying physical mechanism that controls the penetration in each of them.

physics.flu-dyn↗

arXiv2021

On fundamental Fourier coefficients of Siegel cusp forms of degree 2

Let $F$ be a Siegel cusp form of degree 2, even weight $k \geq 2$ and odd squarefree level $N$. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of $F$ at fundamental matrices $S$ (i.e., with $-4 det(S)$ equal to a fundamental discriminant). We prove that as $S$ varies along the equivalence classes of fundamental matrices with $det(S) \asymp X$, the sequence $a(F,S)$ has at least $X^{1-ε}$ sign changes, and takes at least $X^{1-ε}$ "large values". Furthermore, assuming the Generalized Riemann Hypothesis as well as the refined Gan--Gross--Prasad conjecture, we prove the bound $|a(F,S)| \ll_{F, ε} \frac{\det(S)^{\frac{k}2 - \frac{1}{2}}}{ (\log |\det(S)|)^{\frac18 - ε}}$ for fundamental matrices $S$.

math.NT↗

arXiv2021

Integrality and cuspidality of pullbacks of nearly holomorphic Siegel Eisenstein series

We study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions.

math.NT↗

arXiv2021

Neural ODE to model and prognose thermoacoustic instability

In reacting flow systems, thermoacoustic instability characterized by high amplitude pressure fluctuations, is driven by a positive coupling between the unsteady heat release rate and the acoustic field of the combustor. When the underlying flow is turbulent, as a control parameter of the system is varied and the system approach thermoacoustic instability, the acoustic pressure oscillations synchronize with heat release rate oscillations. Consequently, during the onset of thermoacoustic instability in turbulent combustors, the system dynamics transition from chaotic oscillations to periodic oscillations via a state of intermittency. Thermoacoustic systems are traditionally modeled by coupling the model for the unsteady heat source and the acoustic subsystem, each estimated independently. The response of the unsteady heat source, the flame, to acoustic fluctuations are characterized by introducing external unsteady forcing. This necessitates a powerful excitation module to obtain the nonlinear response of the flame to acoustic perturbations. Instead of characterizing individual subsystems, we introduce a neural ordinary differential equation (neural ODE) framework to model the thermoacoustic system as a whole. The neural ODE model for the thermoacoustic system uses time series of the heat release rate and the pressure fluctuations, measured simultaneously without introducing any external perturbations, to model their coupled interaction. Further, we use the parameters of neural ODE to define an anomaly measure that represents the proximity of system dynamics to limit cycle oscillations and thus provide an early warning signal for the onset of thermoacoustic instability.

physics.flu-dyn↗