arXiv subjects

Souvik Mandal

Publications and source records attributed to Souvik Mandal.

8 recordsLinked to original sources

arXiv2026

A Coprimality Topology on the Gaussian Integers: Kolmogorov Quotient and Gaussian Prime Density

This article investigates the coprimality topology on the set of non-zero Gaussian integers, $\mathbb{Z}[i]\setminus\{0\}$, generated by the arithmetic basis $\sigma_\alpha=\{\beta\neq 0 : \gcd(\alpha,\beta)\sim 1\}$. By analyzing prime supports up to associates, we establish that two points are topologically indistinguishable if and only if their supports coincide. We demonstrate that the space is both hyperconnected and ultraconnected, and we explicitly characterize its Kolmogorov quotient $X$ as the space of finite subsets of associate classes of Gaussian primes, endowed with the basis $\mathcal{O}_F=\{S\in X:S\cap F=\emptyset\}$ for $F\in X$. Finally, we demonstrate that the set of Gaussian primes is dense in $\mathbb{Z}[i]\setminus\{0\}$ with respect to the coprimality topology, thereby establishing a topological proof of the infinitude of Gaussian primes.

math.GN↗

arXiv2026

A Precise Treatment of Soft Quotient Topology and Soft Covering Maps

We develop a foundational theory of soft quotient topology, providing a systematic approach to quotient constructions in soft topological spaces. We establish the universal property of soft quotient topology and investigate the relationship between global soft topology and its parametric slices, noting that slice-wise quotient behaviour is not sufficient to characterise soft quotients. The usefulness of the framework is illustrated through the construction of soft quotient spaces, together with a study of soft group actions and their orbit spaces. We propose a precise definition of soft covering maps that resolves inconsistencies found in the existing literature. Finally, to illustrate the applicability of our framework, we discuss a potential application in multi-agent motion planning, showing how it can significantly reduce combinatorial complexity under parametric uncertainty.

math.GN↗

arXiv2026

Relative Smooth Surgery Structure Sets of Thickenings of the Cayley Projective Plane and Applications

We compute the relative smooth surgery structure sets of the thickenings $\mathbb{OP}^{2}\times\mathbb{D}^{k}$ of the Cayley projective plane $\mathbb{OP}^{2}$ for every $k\geq 1$ with $k\equiv 0\pmod 4$, by determining the corresponding normal invariants and surgery obstruction map. We show that the latter is not surjective and determine the $2$-adic valuation of the generator of its image. As applications, we construct infinitely many pairwise non-homeomorphic closed smooth manifolds of dimension $16+k$, homotopy equivalent to $\mathbb{OP}^{2}\times\mathbb{S}^{k}$ and distinguished by their Pontryagin numbers; we compute the rational homotopy groups of the block diffeomorphism group $\widetilde{\operatorname{Diff}}(\mathbb{OP}^{2})$ in every degree congruent to $3$ modulo $4$; and we construct smooth $\mathbb{OP}^{2}$-bundles over $\mathbb{S}^{4}$, $\mathbb{S}^{8}$, and $\mathbb{S}^{12}$ whose total spaces have non-vanishing $\widehat{\mathfrak{A}}$-genus. These bundles yield elements of infinite order in the homotopy groups of the spaces of metrics of positive sectional, Ricci, and scalar curvature on $\mathbb{OP}^{2}$ in degrees $3$, $7$, and $11$.

math.AT↗

arXiv2026

Countable Fan Tightness and Selection Games in Group-Valued Function Spaces

Game-theoretic characterizations of selection principles provide a powerful framework for analyzing covering properties through strategic interactions. For a Tychonoff space $X$ and a non-trivial metrizable arc-connected topological group $G$, we prove that Player~II has a winning strategy in the $\Omega$-Menger game on $X$ if and only if Player~II has a winning strategy in the countable fan tightness game on $C_p(X, G)$ at the identity function. The analogous equivalence is established between the $\Omega$-Rothberger game on $X$ and the countable strong fan tightness game on $C_p(X, G)$ at the identity function. These results extend the game-theoretic characterizations of Clontz from $G = \mathbb{R}$ to arbitrary metrizable arc-connected groups, and lift the selection-principle equivalences of Ko\v{c}inac to the game-theoretic setting. As consequences, we establish that the game-theoretic tightness properties of $C_p(X,G)$ are independent of $G$, preserved under $G$-equivalence, and remain valid for Markov strategies.

math.GN↗

arXiv2026

Prime Density and Classification of Mac\'ias Spaces over Principal Ideal Domains

Recently, the Mac\'ias topology has been generalized over integral domains that are not fields, to furnish a topological proof of the infinitude of prime elements under the assumption that the set of units is finite or not open. In this article, we remove this cardinality assumption completely by using the Jacobson radical. We prove that in any semiprimitive integral domain, the group of units is not open in the Mac\'ias topology. Consequently, for a principal ideal domain, this gives an equivalence between the triviality of the Jacobson radical, the density of the set of prime elements, and the group of units not being open in the Mac\'ias topology. Furthermore, we completely characterize when Mac\'ias spaces over different infinite principal ideal domains are homeomorphic in terms of cardinalities of certain subsets of the domains. As an application we resolve an open problem concerning homeomorphism of Mac\'ias spaces over countably infinite semiprimitive principal ideal domains.

math.GN↗

arXiv2024

A quantum mechanical evaluation of the intermediate scattering function

The intermediate scattering function is interpreted as a correlation function of thermal wave packets of the scattering centers perturbed by the scattering particles at different times. A proof of concept is given at the example of ballistic moving centers. The ensuing numerical method is then illustrated at the example of CO adsorbed on Cu(100).

quant-ph↗

arXiv2022

Stochastic Multi Configuration Time-Dependent Hartree for Dissipative Quantum Dynamics with Strong Intramolecular Coupling

In this article, we explore the dissipation dynamics of a strongly coupled multidimensional system in contact with a Markovian bath following a system-bath approach. We use in this endeavour the recently developed stochastic Multi-Configuration Time-Dependent Hartree approach within the Monte Carlo wave packet formalism [J.Chem.Phys.156, 094109 (2022)]. The method proved to yield thermalized ensembles of wave packets when intramolecular coupling is weak. To treat strongly coupled systems, new Lindblad dissipative operators are constructed as linear combinations of the system coordinates and associated momenta. These are obtained by an unitary transformation to a normal mode representation, which reduces intermode coupling up to second order. Additionally, we use combinations of generalized raising/lowering operators to enforce the Boltzmann distribution in the dissipation operators, which yield perfect thermalization in the harmonic limit. The two ansatz are tested using a model two-dimensional hamiltonian parameterized to disentangle the effects of intramolecular potential coupling, of strong mode mixing observed in Fermi resonances, and of anharmonicity.

physics.chem-ph↗

arXiv2021

Effect of surface temperature on quantum dynamics of H$_2$ on Cu(111) using a chemically accurate potential energy surface

The effect of surface atom vibrations for H$_2$ scattering from a Cu(111) surface at different temperatures is being investigated for hydrogen molecules in their rovibrational ground state ($v$=0, $j$=0). We assume weakly correlated interactions between molecular degrees of freedom and surface modes through a Hartree product type wavefunction. While constructing the six dimensional effective Hamiltonian, we employ: (a) a chemically accurate potential energy surface according to the Static Corrugation Model [Wijzenbroek and Somers, J. Chem. Phys. 137, 054703 (2012)]; (b) normal mode frequencies and displacement vectors calculated with different surface atom interaction potentials within a cluster approximation; (c) initial state distributions for the vibrational modes according to Bose-Einstein probability factors. We carry out 6D quantum dynamics with the so-constructed effective Hamiltonian, and analyze sticking and state-to-state scattering probabilities. The surface atom vibrations affect the chemisorption dynamics. The results show physically meaningful trends both for reaction as well as scattering probabilities compared to experimental and other theoretical results.

cond-mat.other↗