arXiv subjects

Abhishek Saha

Publications and source records attributed to Abhishek Saha.

At least 37 records · Page 2Linked to original sources

arXiv2020

On Secondary Atomization and blockage of surrogate cough droplets in single and multi-layer face masks

By now it is well-understood that the usage of facemasks provides protection from transmission of viral loads through exhalation and inhalation of respiratory droplets. Therefore, during the current Covid-19 pandemic the usage of face masks is strongly recommended by health officials. Although three-layer masks are generally advised for usage, many commonly available or homemade masks contain only single and double layers. In this study, we show through detailed physics based analyses and high speed imaging that high momentum cough droplets on impingement on single- and double-layer masks can lead to significant partial penetration and more importantly atomization into numerous much smaller daughter droplets, thereby increasing the total population of the aerosol, which can remain suspended for a longer time. The possibility of secondary atomization of high momentum cough droplets due to impingement, hydrodynamic focusing and extrusion through the microscale pores in the fibrous network of the mask has not been explored before. However, this unique mode of aerosol generation poses a finite risk of infection as shown in this work. We also demonstrate that in single layer masks close to 70 % of a given droplet volume is atomized and only 30 % is trapped within the fibers. The entrapped volume is close to 90 % for double layer masks which still allows some atomization into smaller droplets. We however found that a triple-layer surgical mask permits negligible penetration and hence should be effective in preventing disease transmission.

physics.app-ph↗

arXiv2020

Analyzing the dominant SARS-CoV-2 transmission routes towards an ab initio SEIR model

Identifying the relative importance of the different transmission routes of the SARS-CoV-2 virus is an urgent research priority. To that end, the different transmission routes, and their role in determining the evolution of the Covid-19 pandemic are analyzed in this work. Probability of infection caused by inhaling virus-laden droplets (initial, ejection diameters between $0.5-750μm$) and the corresponding desiccated nuclei that mostly encapsulate the virions post droplet evaporation, are individually calculated. At typical, air-conditioned yet quiescent indoor space, for average viral loading, cough droplets of initial diameter between $10-50 μm$ have the highest infection probability. However, by the time they are inhaled, the diameters reduce to about $1/6^{th}$ of their initial diameters. While the initially near unity infection probability due to droplets rapidly decays within the first $25s$, the small yet persistent infection probability of desiccated nuclei decays appreciably only by $\mathcal{O} (1000s)$, assuming the virus sustains equally well within the dried droplet nuclei as in the droplets. Combined with molecular collision theory adapted to calculate frequency of contact between the susceptible population and the droplet/nuclei cloud, infection rate constants are derived ab-initio, leading to a SEIR model applicable for any respiratory event - vector combination. Viral load, minimum infectious dose, sensitivity of the virus half-life to the phase of its vector and dilution of the respiratory jet/puff by the entraining air are shown to mechanistically determine specific physical modes of transmission and variation in the basic reproduction number $\mathcal{R}_0$, from first principle calculations.

q-bio.PE↗

arXiv2020

Insights on drying and precipitation dynamics of respiratory droplets in the perspective of Covid-19

We isolate a nano-colloidal droplet of surrogate mucosalivary fluid to gain fundamental insights into the infectivity of air borne nuclei during the Covid-19 pandemic. Evaporation experiments are performed with salt-water solutions seeded with a viral load of inactive nanoparticles in an acoustic levitator. We seek to emulate the drying, flow and precipitation dynamics of such air borne mucosalivary droplets. Observations with the surrogate fluid are validated by similar experiments with actual samples from a healthy subject. A unique feature emerges with regards to the final crystallite dimension; it is always 20-30% of the initial droplet diameter for different sizes and ambient conditions. The preserved precipitates from levitated droplets show that 15% of the total virion population remain dispersed on the outer surface of air-desiccated air borne nuclei. This fraction increases to ~90% if the respiratory droplets (of larger initial size) settle on a surface and then evaporate in the sessile mode.

cond-mat.soft↗

arXiv2020

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$ where $D$ is an indefinite quaternion division algebra over $\mathbb{Q}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times θ_χ)$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $θ_χ$ is an (essentially fixed) automorphic form on $\mathrm{GL}_2$ obtained as the theta lift of a Hecke character $χ$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to \emph{any} family whose associated matrix coefficients have such a decay property.

math.NT↗

arXiv2020

Synchronization framework for modeling transition to thermoacoustic instability in laminar combustors

We, herein, present a new model based on the framework of synchronization to describe a thermoacoustic system and capture the multiple bifurcations that such a system undergoes. Instead of applying flame describing function to depict the unsteady heat release rate as the flame's response to acoustic perturbation, the new model considers the acoustic field and the unsteady heat release rate as a pair of nonlinearly coupled damped oscillators. By varying the coupling strength, multiple dynamical behaviors, including limit cycle oscillation, quasi-periodic oscillation, strange nonchaos, and chaos can be captured. Furthermore, the model was able to qualitatively replicate the different behaviors of a laminar thermoacoustic system observed in experiments by Kabiraj et al.~[Chaos 22, 023129 (2012)]. By analyzing the temporal variation of the phase difference between heat release rate oscillations and pressure oscillations under different dynamical states, we show that the characteristics of the dynamical states depend on the nature of synchronization between the two signals, which is consistent with previous experimental findings.

nlin.AO↗

arXiv2020

Modeling ambient temperature and relative humidity sensitivity of respiratory droplets and their role in Covid-19 outbreaks

One of the many unresolved questions that revolves around the Covid-19 pandemic is whether local outbreaks can depend on ambient conditions like temperature and relative humidity. In this paper, we develop a model that tries to explain and describe the temperature and relative humidity sensitivity of respiratory droplets and their possible connection in determining viral outbreaks. The model has two parts. First, we model the growth rate of the infected population based on a reaction mechanism - the final equations of which are similar to the well-known SIR model. The advantage of modeling the pandemic using the reaction mechanism is that the rate constants have sound physical interpretation. The infection rate constant is derived using collision rate theory and shown to be a function of the respiratory droplet lifetime. In the second part, we have emulated the respiratory droplets responsible for disease transmission as salt solution droplets and computed their evaporation time accounting for droplet cooling, heat and mass transfer and finally crystallization of the salt. The model output favourably compares with the experimentally obtained evaporation characteristics of levitated droplets of pure water and salt solution, respectively, ensuring fidelity of the model. Droplet evaporation/desiccation time is indeed dependent on ambient temperature and relative humidity, considered at both outdoor and indoor conditions. Since the droplet evaporation time determines the infection rate constant, ambient temperature and relative humidity are shown to impact the outbreak growth rates.

physics.flu-dyn↗

arXiv2020

Universality in spectral condensation

Self-organization is the spontaneous formation of spatial, temporal, or spatiotemporal patterns in complex systems far from equilibrium. During such self-organization, energy distributed in a broadband of frequencies gets condensed into a dominant mode, analogous to a condensation phenomena. We call this phenomenon spectral condensation and study its occurrence in fluid mechanical, optical and electronic systems. We define a set of spectral measures to quantify this condensation spanning several dynamical systems. Further, we uncover an inverse power law behaviour of spectral measures with the power corresponding to the dominant peak in the power spectrum in all the aforementioned systems.

nlin.AO↗

arXiv2020

On the standard $L$-function for $GSp_{2n} \times GL_1$ and algebraicity of symmetric fourth $L$-values for $GL_2$

We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to $L$-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case $n=2$ we are able to prove a reciprocity law -- predicted by Deligne's conjecture -- for the critical special values of the twisted standard $L$-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. By specializing further to the case of Siegel cusp forms obtained via the Ramakrishnan--Shahidi lift, we obtain a reciprocity law for the critical special values of the symmetric fourth $L$-function of a classical newform.

math.NT↗

arXiv2019

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces

Let $D$ be an indefinite quaternion division algebra over $\mathbb{Q}$. We approach the problem of bounding the sup-norms of automorphic forms $ϕ$ on $D^\times(\mathbb{A})$ that belong to irreducible automorphic representations and transform via characters of unit groups of orders of $D$. We obtain a non-trivial upper bound for $\|ϕ\|_\infty$ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for $\|ϕ\|_\infty$ in the setting of newforms for Eichler orders. In the special case when the index of the order in a maximal order is a squarefull integer $N$, our result specializes to $\|ϕ\|_\infty \ll_{π_\infty, ε} N^{1/3 + ε} \|ϕ\|_2$. A key application of our result is to automorphic forms $ϕ$ which correspond at the ramified primes to either minimal vectors (in the sense of Hu-Nelson-Saha), or $p$-adic microlocal lifts (in the sense of Nelson). For such forms, our bound specializes to $\| ϕ\|_\infty \ll_ε C^{\frac16 + ε}\|ϕ\|_2$ where $C$ is the conductor of the representation $π$ generated by $ϕ$. This improves upon the previously known local bound $\|ϕ\|_\infty \ll_{λ, ε} C^{\frac14 + ε}\|ϕ\|_2$ in these cases.

math.NT↗

arXiv2019

Universality in the emergence of oscillatory instabilities in turbulent flows

Spontaneous emergence of periodic oscillations due to self-organization is ubiquitous in turbulent flows. The emergence of such oscillatory instabilities in turbulent fluid mechanical systems is often studied in different system-specific frameworks. We uncover the existence of a universal scaling behaviour during self-organization in turbulent flows leading to oscillatory instability. Our experiments show that the spectral amplitude of the dominant mode of oscillations scales inversely with the Hurst exponent of a fluctuating state variable following an inverse power law relation. Interestingly, we observe the same power law behaviour with a constant exponent near -2 across various turbulent systems such as aeroacoustic, thermoacoustic and aeroelastic systems.

nlin.AO↗

arXiv2019

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms.

math.NT↗

arXiv2019

Lowest weight modules of Sp_4(R) and nearly holomorphic Siegel modular forms

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest-weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2. Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight $det^\ell$ sym^m with respect to an arbitrary congruence subgroup of Sp_4(Q). We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight $det^{\ell'} sym^{m'}$ with $(\ell', m')$ varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight $det^{3}sym^{m'}$ that cannot be obtained from holomorphic forms. As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction.

math.NT↗

arXiv2018

On the order of vanishing of newforms at cusps

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$. We obtain an explicit formula, as a product of local terms, for the ramification index at each cusp of a modular parametrization of $E$ by $X_0(N)$. Our formula shows that the ramification index always divides 24, a fact that had been previously conjectured by Brunault as a result of numerical computations. In fact, we prove a more general result which gives the order of vanishing at each cusp of a holomorphic newform of arbitary level, weight and character, provided its field of rationality satisfies a certain condition. The above result relies on a purely $p$-adic computation of possibly independent interest. Let $F$ be a non-archimedean local field and $π$ an irreducible, admissible, generic representation of $\mathrm{GL}_2(F)$. We introduce a new integral invariant, which we call the \emph{vanishing index} and denote $e_π(l)$, that measures the degree of "extra vanishing" at matrices of level $l$ of the Whittaker function associated to the newvector of $π$. Our main local result writes down the value of $e_π(l)$ in every case.

math.NT↗

arXiv2018

Some analytic aspects of automorphic forms on GL(2) of minimal type

Let $π$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $ϕ$ be an $L^2$-normalized automorphic form in the space of $π$. The sup-norm problem asks for bounds on $\| ϕ\|_\infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|ϕ|^2 (g) \, d g$ of $ϕ$. All previous work on these problems in the conductor-aspect has focused on the case that $ϕ$ is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor $p$ of $C$, the local component $π_p$ is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms $ϕ$ for which the local components $ϕ_p \in π_p$ are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of $PGL_2(\mathbb{R})$. For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if $π_\infty$ is a holomorphic discrete series of lowest weight $k$, we obtain the optimal bound $C^{1/8 -ε} k^{1/4 - ε} \ll_ε |ϕ|_\infty \ll_ε C^{1/8 + ε} k^{1/4+ε}$. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence for newforms of powerful level.

math.NT↗

arXiv2018

Integral representation and critical $L$-values for holomorphic forms on $GSp_{2n} \times GL_1$

We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to $L$-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case $n=2$, we are able to prove a reciprocity law -- predicted by Deligne's conjecture -- for the critical special values of the twisted standard $L$-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. The proof of this application uses our recent structure theorem [arXiv:1501.00524] for the space of nearly holomorphic Siegel modular forms of degree 2 and arbitrary level.

math.NT↗

arXiv2017

Hybrid sup-norm bounds for Maass newforms of powerful level

Let $f$ be an $L^2$-normalized Hecke--Maass cuspidal newform of level $N$, character $χ$ and Laplace eigenvalue $λ$. Let $N_1$ denote the smallest integer such that $N|N_1^2$ and $N_0$ denote the largest integer such that $N_0^2 |N$. Let $M$ denote the conductor of $χ$ and define $M_1= M/\gcd(M,N_1)$. In this paper, we prove the bound $|f|_\infty$ $\ll_ε$ $N_0^{1/6 + ε} N_1^{1/3+ε} M_1^{1/2} λ^{5/24+ε}$, which generalizes and strengthens previously known upper bounds for $|f|_\infty$. This is the first time a hybrid bound (i.e., involving both $N$ and $λ$) has been established for $|f|_\infty$ in the case of non-squarefree $N$. The only previously known bound in the non-squarefree case was in the N-aspect; it had been shown by the author that $|f|_\infty \ll_{λ, ε} N^{5/12+ε}$ provided $M=1$. The present result significantly improves the exponent of $N$ in the above case. If $N$ is a squarefree integer, our bound reduces to $|f|_\infty \ll_εN^{1/3 + ε}λ^{5/24 + ε}$, which was previously proved by Templier. The key new feature of the present work is a systematic use of p-adic representation theoretic techniques and in particular a detailed study of Whittaker newforms and matrix coefficients for $GL_2(F)$ where $F$ is a local field.

math.NT↗

arXiv2016

A note on the growth of nearly holomorphic vector-valued Siegel modular forms

Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $ρ$ with respect to some congruence subgroup of $\mathrm{Sp}_{2n}(\mathbb Q)$. In this note, we prove that the function on $\mathrm{Sp}_{2n}(\mathbb R)$ obtained by lifting $F$ has the moderate growth (or "slowly increasing") property. This is a consequence of the following bound that we prove: $\|ρ(Y^{1/2})F(Z) \| \ll \prod_{i=1}^n (μ_i(Y)^{λ_1/2} + μ_i(Y)^{-λ_1/2})$ where $ λ_1 \ge \ldots \ge λ_n$ is the highest weight of $ρ$ and $μ_i(Y)$ are the eigenvalues of the matrix $Y$.

math.NT↗

arXiv2015

On sup-norms of cusp forms of powerful level

Let f be an L^2-normalized Hecke--Maass cuspidal newform of level N and Laplace eigenvalue λ. It is shown that |f|_\infty <<_{λ, ε} N^{-1/12 + ε} for any ε>0. The exponent is further improved in the case when N is not divisible by "small squares". Our work extends and generalizes previously known results in the special case of N squarefree.

math.NT↗