Showing 1–7 of 7 results for author: Rathi, V

Tracking in-silico Lagrangian sensors in a lab-scale stirred tank reactor

Authors: Vamika Rathi, Fatima Sehar, Finn Sommer, Sebastian Goetschel, Eike Steuwe, Alexandra von Kameke, Daniel Ruprecht

Abstract: Lagrangian sensors have shown promise to improve operator awareness of conditions inside a chemical reactor but three-dimensional tracking remains a mostly unsolved challenge. We explore a setup where in-silico sensors, based on a recently proposed real-world design, are tracked using data from an accelerometer and magnetometer available from a built-in inertial measurement unit. Filtering algorit… ▽ More Lagrangian sensors have shown promise to improve operator awareness of conditions inside a chemical reactor but three-dimensional tracking remains a mostly unsolved challenge. We explore a setup where in-silico sensors, based on a recently proposed real-world design, are tracked using data from an accelerometer and magnetometer available from a built-in inertial measurement unit. Filtering algorithms, using a bespoke dynamical model, are used to process these readings into position estimates. We compare tracking performance of an extended Kalman filter, a particle filter and the unscented Kalman filter implemented in the pykalman library. Our numerical experiments track in-silico particles moving in an analytically given three dimensional vortex as well as in the experimentally measured flow-field of a lab-scale stirred tank reactor. Using the Maxey-Riley-Gatignol equations for the movement of inertial particles as ground-truth, we demonstrate that trajectories can be reconstructed from noisy synthetic data with errors below 10%. △ Less

Submitted 11 June, 2026; originally announced June 2026.

Comments: 30 pages, 7 figures

MSC Class: 65C05; 68U07

Approximation of the Basset force in the Maxey-Riley-Gatignol equations via universal differential equations

Authors: Finn Sommer, Vamika Rathi, Sebastian Goetschel, Daniel Ruprecht

Abstract: The Maxey-Riley-Gatignol equations (MaRGE) model the motion of spherical inertial particles in a fluid. They contain the Basset force, an integral term which models history effects due to the formation of wakes and boundary layer effects. This causes the force that acts on a particle to depend on its past trajectory and complicates the numerical solution of MaRGE. Therefore, the Basset force is of… ▽ More The Maxey-Riley-Gatignol equations (MaRGE) model the motion of spherical inertial particles in a fluid. They contain the Basset force, an integral term which models history effects due to the formation of wakes and boundary layer effects. This causes the force that acts on a particle to depend on its past trajectory and complicates the numerical solution of MaRGE. Therefore, the Basset force is often neglected, despite substantial evidence that it has both quantitative and qualitative impact on the movement patterns of modelled particles. Using the concept of universal differential equations, we propose an approximation of the history term via neural networks which approximates MaRGE by a system of ordinary differential equations that can be solved with standard numerical solvers like Runge-Kutta methods. △ Less

Submitted 9 April, 2026; originally announced April 2026.

Comments: 24 pages, 15 figures

MSC Class: 68T07 (Primary) 65L05(Secondary)

ABC implies that Ramanujan's tau function misses almost all primes

Authors: David Kurniadi Angdinata, Evan Chen, Chris Cummins, Ben Eltschig, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Simon Mahns, Aram H. Markosyan, Rithikesh Muddana, Ken Ono, Manooshree Patel, Gaurang Pendharkar, Vedant Rathi, Alex Schneidman, Volker Seeker, Shubho Sengupta, Ishan Sinha, Jimmy Xin, Jujian Zhang

Abstract: Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan's tau-function. Recently, Xiong showed that these prime values form a subset of the primes with density at most $2/11$. Assuming the $abc$ Conjecture, we prove the stronger upper bound \[ S(X):=\#\{\ell\le X:\ \ell\… ▽ More Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan's tau-function. Recently, Xiong showed that these prime values form a subset of the primes with density at most $2/11$. Assuming the $abc$ Conjecture, we prove the stronger upper bound \[ S(X):=\#\{\ell\le X:\ \ell\ \text{prime and } |τ(n)|=\ell \text{ for some } n\ge 1\} = O(X^{13/22}), \] which implies that Ramanujan's tau-function misses a density 1 subset of the primes. We give a heuristic suggesting that $S(X)$ should nevertheless be infinite, with predicted order of magnitude \[ S(X)\asymp \frac{C X^{\frac{1}{11}}}{(\log X)^2}. \] The main engine in this note was formalized and produced automatically in Lean/Mathlib by AxiomProver from a natural-language statement of the problem. △ Less

Submitted 25 April, 2026; v1 submitted 31 March, 2026; originally announced March 2026.

Comments: Minor edits that address two referee reports

MSC Class: 11F11 (Primary); 11F30 (Secondary)

Dead ends in square-free digit walks

Authors: Evan Chen, Chris Cummins, Ben Eltschig, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Markosyan, Ken Ono, Manooshree Patel, Gaurang Pendharkar, Vedant Rathi, Alex Schneidman, Volker Seeker, Shubho Sengupta, Ishan Sinha, Jimmy Xin, Jujian Zhang

Abstract: We study "dead ends" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite square-free walks occur with probability near $1$, corresponding to an asymptotic dead-end density of $\approx 5.218\times 10^{-5}$. We prove that the true asymp… ▽ More We study "dead ends" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite square-free walks occur with probability near $1$, corresponding to an asymptotic dead-end density of $\approx 5.218\times 10^{-5}$. We prove that the true asymptotic dead-end density satisfies \[ c_{\mathrm{dead}} \approx 1.317\times 10^{-9}, \] roughly a factor of $\sim 4\times 10^4$ smaller than the prediction. For every base $b\geq 2$, we prove that dead-end densities exist and are given by a closed-form expression (as a finite alternating sum of Euler products). The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the problem. △ Less

Submitted 6 February, 2026; v1 submitted 4 February, 2026; originally announced February 2026.

Comments: After we posted the first version on arXiv, we learned from K. Soundararajan that the result in this paper was previously obtained by Mirsky in 1947. We mistakenly assumed that the 2024 paper of Miller et al. represented the status of this problem. Their paper makes no reference to Mirsky; his work seems to have been forgotten. This manuscript will not be submitted for journal publication

MSC Class: 11A63; 11B83; 11N05

Almost all primes are partially regular

Authors: Evan Chen, Chris Cummins, Ben Eltschig, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Markosyan, Ken Ono, Manooshree Patel, Gaurang Pendharkar, Vedant Rathi, Alex Schneidman, Volker Seeker, Shubho Sengupta, Ishan Sinha, Jimmy Xin, Jujian Zhang

Abstract: For odd primes $p$, we let $K_p:=\mathbb{Q}(ζ_p)$ be the $p$th cyclotomic field and let $ω$ denote its Teichmuller character. For $α>1/2$, we say that an odd prime $p$ is partially regular if the eigenspaces of the $p$-Sylow subgroup of $\operatorname{Cl}(K_p)$ under the Galois action vanish for all characters $ω^{p-2k}$ with \[ 2\le 2k \le \frac{\sqrt{p}}{(\log p)^α}. \] Equivalently,… ▽ More For odd primes $p$, we let $K_p:=\mathbb{Q}(ζ_p)$ be the $p$th cyclotomic field and let $ω$ denote its Teichmuller character. For $α>1/2$, we say that an odd prime $p$ is partially regular if the eigenspaces of the $p$-Sylow subgroup of $\operatorname{Cl}(K_p)$ under the Galois action vanish for all characters $ω^{p-2k}$ with \[ 2\le 2k \le \frac{\sqrt{p}}{(\log p)^α}. \] Equivalently, $p\nmid \operatorname{num}(B_{2k})$ throughout this range. We prove that a density-one subset of primes is partially regular in this sense. By Leopoldt reflection, this yields a partial Vandiver Theorem: for a density-one set of primes $p$, the even eigenspaces $A_p(ω^{2k})$ vanish for all even $2k$ satisfying the inequality above. This result has consequences for Kubota-Leopoldt $p$-adic $L$-functions, congruences between cusp forms and Eisenstein series, and $p$-torsion in algebraic $K$-groups. The theorem proving partial regularity for almost all $p$ is fully formalized in Lean/Mathlib and was produced automatically by AxiomProver from a natural-language statement of the conjecture. △ Less

Submitted 4 February, 2026; originally announced February 2026.

MSC Class: 11Y40; 11R18

Fel's Conjecture on Syzygies of Numerical Semigroups

Authors: Evan Chen, Chris Cummins, GSM, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Markosyan, Ken Ono, Manooshree Patel, Gaurang Pendharkar, Vedant Rathi, Alex Schneidman, Volker Seeker, Shubho Sengupta, Ishan Sinha, Jimmy Xin, Jujian Zhang

Abstract: Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for $K_p(S)$, for all $p\ge 0$, in terms of the gap power sums $G_r(S)=\sum_{g\notin S} g^r$ and universal symmetric polynomial… ▽ More Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for $K_p(S)$, for all $p\ge 0$, in terms of the gap power sums $G_r(S)=\sum_{g\notin S} g^r$ and universal symmetric polynomials $T_n$ evaluated at the generator power sums $σ_k=\sum_i d_i^k$ (and $δ_k=(σ_k-1)/2^k$). We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for $T_n$ needed for the derivation. The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture. △ Less

Submitted 15 June, 2026; v1 submitted 3 February, 2026; originally announced February 2026.

Comments: Edits that address two referee reports

MSC Class: 20M14; 11P91

Bounds on Threshold of Regular Random $k$-SAT

Authors: Vishwambhar Rathi, Erik Aurell, Lars Rasmussen, Mikael Skoglund

Abstract: We consider the regular model of formula generation in conjunctive normal form (CNF) introduced by Boufkhad et. al. We derive an upper bound on the satisfiability threshold and NAE-satisfiability threshold for regular random $k$-SAT for any $k \geq 3$. We show that these bounds matches with the corresponding bound for the uniform model of formula generation. We derive lower bound on the thresh… ▽ More We consider the regular model of formula generation in conjunctive normal form (CNF) introduced by Boufkhad et. al. We derive an upper bound on the satisfiability threshold and NAE-satisfiability threshold for regular random $k$-SAT for any $k \geq 3$. We show that these bounds matches with the corresponding bound for the uniform model of formula generation. We derive lower bound on the threshold by applying the second moment method to the number of satisfying assignments. For large $k$, we note that the obtained lower bounds on the threshold of a regular random formula converges to the lower bound obtained for the uniform model. Thus, we answer the question posed in \cite{AcM06} regarding the performance of the second moment method for regular random formulas. △ Less

Submitted 23 April, 2010; v1 submitted 5 February, 2010; originally announced February 2010.

Comments: Accepted to SAT 2010