arXiv subjects

Fridolin Melong

Publications and source records attributed to Fridolin Melong.

18 recordsLinked to original sources

arXiv2026

$q$-Deformed Topological Recursion: Quantum Curves and Non-perturbative Analysis

This work investigates the $q$-deformation of $(r,s)$-Airy structures and their realization via $q$-difference operators, providing a bridge between quantum spectral curves and integrable systems. We construct an all-order $q$-WKB solution for the matrix systems associated with the $q$-quantized curve $E_q(x,y)=0$. We demonstrate that the resulting non-perturbative connected $q$-amplitudes satisfy a set of shifted $q$-loop equations, which can be interpreted as the Ward identities of a $q$-deformed $\mathcal{W}(\mathfrak{gl}_r)$ algebra. Our main result provides a rigorous classification of admissible $(r,s,q)$ pairs and $q$-Casimir configurations that satisfy the $q$-topological type property. This ensures that the semi-classical expansion is uniquely governed by the $q$-topological recursion, offering new insights into the $q$-quantization of mirror curves and their underlying algebraic structures.

math-ph↗

arXiv2026

$q$-Deformed Topological Recursion, Weight Vectors and Algebraic Structures

We investigate a $q$-deformation of the shifted topological recursion, extending the construction of Belliard-Bouchard-Kramer-Nelson to the context of quantum algebras. Through the study of highest weight vectors in $q$-deformed of $\mathcal{W}$-algebra representations, we derive a $q$-analogue of the topological recursion and show it yields $q$-deformed quantum curves. This framework unifies various approaches to quantum integrability and provides new insights into the geometry of $q$-deformed moduli spaces.

math-ph↗

arXiv2026

Confinement-Tunable Synthetic Gauge Fields and Floquet Topological Phenomena in a Driven Quantum Wire Qubit

Theoretical analysis demonstrates that a spin qubit in a parabolic quantum wire, when driven by a bichromatic field, exhibits a confinement-tunable synthetic gauge field leading to novel Floquet topological phenomena. The underlying mechanism for topological protection of qubit states against time-periodic perturbations is presented. The analysis reveals a confinement-induced topological Landau-Zener transition, characterized by a shift from preserved symmetries to chiral interference patterns in Landau-Zener-St$\ddot{u}$ckelberg-Majorana interferometry. The emergence of non-Abelian geometric phases under cyclic evolution in curved confinement and phase-parameter space is identified, enabling holonomic quantum computation. Furthermore, the prediction of unconventional Floquet-Bloch oscillations in the quasi-energy and resonance transition probability spectra as a function of the biharmonic phase indicates exotic properties, such as fractal spectra and fractional Floquet tunnelling. These phenomena provide direct evidence of coherent transport in the synthetic dimension. Concrete experimental pathways for realizing these effects in semiconductor heterostructures are proposed, and the framework is extended to multi-qubit entanglement generation with a quantitative analysis of its inherent resilience to decoherence. Collectively, these findings position quantum wire materials as a versatile and scalable platform for Floquet engineering, topological quantum control, and fault-tolerant quantum information processing.

cond-mat.mes-hall↗

arXiv2025

Characterization of the $W_{1+\infty}$-n-algebra and applications

In this paper, we construct the $W_{1+\infty}$-n-algebras in the framework of the generalized quantum algebra. We characterize the $\mathcal{R}(p,q)$-multi-variable $W_{1+\infty}$-algebra and derive its $n$-algebra which is the generalized Lie algebra for $n$ even. Furthermore, we investigate the $\mathcal{R}(p,q)$-elliptic hermitian matrix model and determine a toy model for the generalized quantum $W_{\infty}$ constraints. Also, we deduce particular cases of our results.

math-ph↗

arXiv2025

Generalized super-$W_{1+\infty}$-$n$-algebra and Landau Problem

We investigate the $\mathcal{R}(p,q)$-super $n$-bracket and study their properties such that the generalized super Jacobi identity (GJSI). Furthermore, from the $\mathcal{R}(p,q)$-operators in a Supersymmetric Landau problem, we furnish the $\mathcal{R}(p,q)$-super $W_{1+\infty}$ $n$-algebra which obey the generalized super Jacobi identity (GSJI) for $n$ even. Also, we derive the $\mathcal{R}(p,q)$-super $W_{1+\infty}$ sub-$2n$-algebra and deduce particular cases induced by quantum algebras existing in the literature.

math-ph↗

arXiv2025

Asymptotic normality and strong consistency of kernel regression estimation in q-calculus

We construct a family of estimators for a regression function based on a sample following a qdistribution. Our approach is nonparametric, using kernel methods built from operations that leverage the properties of q-calculus. Furthermore, under appropriate assumptions, we establish the weak convergence and strong consistency of this family of estimators.

math.ST↗

arXiv2023

Multi-parameter Fermi-Dirac and Bose-Einstein Stochastic Distributions

In this paper, we characterize the multivariate uniform probability distribution of the first and second kinds in the framework of the $\mathcal{R}(p,q)$-deformed quantum algebras. Their bivariate distributions and related properties, namely ($\mathcal{R}(p,q)$-mean, $\mathcal{R}(p,q)$-variance and $\mathcal{R}(p,q)$-covariance) are computed and discussed. Particular cases corresponding to quantum algebras existing in literature are deduced.

math-ph↗

arXiv2023

Conformal super Virasoro algebra: matrix model and quantum deformed algebra

In this paper, we construct the super Virasoro algebra with an arbitrary conformal dimension $Δ$ from the generalized $\mathcal{R}(p,q)$-deformed quantum algebra and investigate the $\mathcal{R}(p,q)$-deformed super Virasoro algebra with the particular conformal dimension $Δ=1$. Furthermore, we perform the R(p,q)-conformal Virasoro n-algebra, the $\mathcal{R}(p,q)$-conformal super Virasoro n-algebra ($n$ even) and discuss a toy model for the $\mathcal{R}(p,q)$-conformal Virasoro constraints and R(p,q)-conformal super Virasoro constraints. Besides, we generalized the notion of the $\mathcal{R}(p,q)$-elliptic hermitian matrix model with an arbitrary conformal dimension $Δ$. Finally, we deduce relevant particular cases generated by quantum algebras known in the literature.

math-ph↗

arXiv2023

Generalized Heisenberg-Virasoro algebra and matrix models from quantum algebra

In this paper, we construct the Heisenberg-Virasoro algebra in the framework of the $\mathcal{R}(p,q)$-deformed quantum algebras. Moreover, the $\mathcal{R}(p,q)$-Heisenberg-Witt $n$-algebras is also investigated. Furthermore, we generalize the notion of the elliptic hermitian matrix models. We use the constraints to evaluate the $\mathcal{R}(p,q)$-differential operators of the Virasoro algebra and generalize it to higher order differential operators. Particular cases corresponding to quantum algebras existing in literature are deduced.

math.QA↗

arXiv2022

$\mathcal{R}(p,q)$-multivariate discrete probability distributions

We construct the multivariate probability distributions (Pólya, inverse Pólya, hypergeometric and negative hypergeometric) from the generalized quantum algebra. Moreover, we derive the bivariate probability distributions and determine their properties($\mathcal{R}(p,q)$-factorial moments and covariance). Besides, we deduce particular cases of probability distributions from the quantum algebras known in the literature.

math.QA↗

arXiv2022

Multinomial probability distribution and quantum deformed algebras

The multinomial coefficient and their recurrence relations from the generalized quantum deformed algebras are examined. Moreover, the $\mathcal{R}(p,q)-$ deformed multinomial probability distribution and the negative $\mathcal{R}(p,q)-$ deformed multinomial probability distribution are constructed. The recurrence relations are also determined. Particular cases of our results corresponding to the quantum algebras in the literature are deduced from the general formalism.

math-ph↗

arXiv2022

$\mathcal{R}(p,q)$-trinomial probability distribution: properties and particular cases

In this paper, we investigate the trinomial probability distribution of the first and second kind from the $\mathcal{R}(p,q)$-quantum algebras. Moreover, we compute their $\mathcal{R}(p,q)$-factorial moments and derive the corresponding covariance. Particular cases of trinomial probability distribution are deduced from the formalism developed.

math.QA↗

arXiv2022

$\mathcal{R}(p,q)$-deformed super Virasoro $n$-algebra

In this paper, we construct the super Witt algebra and super Virasoro algebra in the framework of the $\mathcal{R}(p,q)$-deformed quantum algebras. Moreover, we perform the super $\mathcal{R}(p,q)$-deformed Witt $n$-algebra, the $\mathcal{R}(p,q)$-deformed Virasoro $n$-algebra and discuss the super $\mathcal{R}(p,q)$-Virasoro $n$-algebra ($n$ even). Besides, we define and construct another super $\mathcal{R}(p,q)$-deformed Witt $n$-algebra and study a toy model for the super $\mathcal{R}(p,q)$-Virasoro constraints. Relevant particular cases induced from the quantum algebras known in the literature are deduced from the formalism developped.

math-ph↗

arXiv2020

Generalized Witt and Witt n-algebras, Virasoro algebras and constraints, and KdV equations from R(p,q)-deformed quantum algebras

We perform generalizations of Witt and Virasoro algebras, and derive the corresponding Korteweg-de Vries equations from known R(p,q)-deformed quantum algebras previously introduced in J. Math. Phys. 51, 063518, (2010). Related relevant properties are investigated and discussed. Besides, we construct the R(p,q)-deformed Witt n- algebra, and determine the Virasoro constraints for a toy model, which play an important role in the study of matrix models. Finally, as matter of illustration, explicit results are provided for main particular deformed quantum algebras known in the literature.

math-ph↗

arXiv2019

R(p,q)- analogs of discrete distributions: general formalism and application

In this paper, we define and discuss $\mathcal{R}(p,q)$- deformations of basic univariate discrete distributions of the probability theory. We mainly focus on binomial, Euler, Pólya and inverse Pólya distributions. We discuss relevant $\mathcal{R}(p,q)-$ deformed factorial moments of a random variable, and establish associated expressions of mean and variance. Futhermore, we derive a recursion relation for the probability distributions. Then, we apply the same approach to build main distributional properties characterizing the generalized $q-$ Quesne quantum algebra, used in physics. Other known results in the literature are also recovered as particular cases.

math.PR↗

arXiv2019

R(p,q)-deformed combinatorics: full characterization and illustration

This paper addresses a theory of R(p,q)-deformed combinatorics in discrete probability. It mainly focuses on R(p,q)-deformed factorials, binomial coefficients, Vandermonde's formula, Cauchy's formula, binomial and negative binomial formulae, factorial and binomial moments, and Stirling numbers. Moreover, the R(p,q)-Stirling numbers of the second kind and the R(p,q)-Bell numbers for graphs are also derived. Related relevant properties are investigated and discussed. Finally, as a concrete illustration, the developed formalism is displayed for the well known generalized q-Quesne deformed quantum algebra to construct the corresponding deformed combinatorics, as a particular case.

math.GM↗

arXiv2018

Geometry and probability on the noncommutative 2-torus in a magnetic field

In this work, we describe the geometric and probabilistic properties of a noncommutative 2- torus in a magnetic field. We study the volume invariance, integrated scalar curvature and volume form by using the method of perturbation by inner derivation of the magnetic Laplacian in the noncommutative 2-torus. Then, we analyze the magnetic stochastic process describing the motion of a particle subject to a uniform magnetic field on the noncommutative 2-torus, derive and discuss the related main properties.

math-ph↗

arXiv2018

R(p,q)-deformed conformal Virasoro algebra

This paper addresses an R(p,q)-deformed conformal Virasoro algebra with an arbitrary conformal dimension Delta. Wellknown deformations constructed in the literature are deduced as particular cases. Then, the special case of the conformal dimension Delta=1 is elucidated for its interesting properties. The R(p,q)-KdV equation, associated with the deformed Virasoro algebra, is also derived and discussed. Finally, the (p,q)-deformed energy-momentum tensor, consistent with the central extension term, is computed and analyzed.

math-ph↗