Tootfinder

On $q$-real and $q$-complex numbers
Pavel Etingof
https://arxiv.org/abs/2508.08440 https://arxiv.org/pdf/2508.08440…

On $q$-real and $q$-complex numbers
In arXiv:1812.00170, S. Morier-Genoud and V. Ovsienko introduced the notion of the $q$-rational number $[x]_q$, $x\in \Bbb Q$, a rational function specializing to $x$ at $q=1$, obtained by $q$-deforming the continued fraction expansion of $x$. In arXiv:1908.04365 they introduced $q$-real numbers $[x]_q$, $x\in \Bbb R$ - a Laurent series in $q$ converging to the rational function $[x]_q$ when $x\in \Bbb Q$. In arXiv:2102.00891 it is proved that if $x\in \Bbb Q_{>1}$ then the series $[x]_q$ conve…

Bilateral $q$-ultraspherical functions
Michael J. Schlosser
https://arxiv.org/abs/2508.08908 https://arxiv.org/pdf/2508.08908

Bilateral $q$-ultraspherical functions
We introduce a bilateral extension of the continuous $q$-ultraspherical polynomials which we call bilateral $q$-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ${}_2ψ_2$ series, they are analytic in a variable $x=\cosθ$ and depend on two parameters $β$ and $γ$ and on a base $q$. For these bilateral $q$-ultraspherical functions we derive a bilateral generating function, find a three-term recurrence relation, explain how they behave under the act…

The partition function and elliptic curves
Ken Ono
https://arxiv.org/abs/2508.09608 https://arxiv.org/pdf/2508.09608

The partition function and elliptic curves
We recast congruences for the partition function through the geometry of supersingular elliptic curves. For primes $\ell\ge5$ inert in $K_n=\mathbb{Q}(\sqrt{1-24n})$ (equivalently $\big(\tfrac{1-24n}{\ell}\big)=-1$), we show that $p(n)\pmod{\ell}$ is a "supersingular trace'' on $X_0(6)$: a computable dot product on the level-$6$ Brandt module that is a twisted analogue of the Deuring--Eichler formula for class numbers. In the ramified case $\ell\in\{5,7,11\}$ (i.e. $\ell\mid(1-24n)$), a geometr…

On the proportion of derangements in affine classical groups
Jessica Anzanello
https://arxiv.org/abs/2508.07093 https://arxiv.org/pdf/2508.07093

On the proportion of derangements in affine classical groups
We derive explicit formulas for the proportion of derangements and of derangements of $p$-power order in the affine unitary group $AU_m(q)$ over the finite field $\mathbb{F}_{q^2}$ of characteristic $p$. Both formulas rely on a result of independent interest concerning integer partitions: we obtain a generating function for partitions $λ=(λ_1, \dots, λ_m)$ into $m$ parts, for which $λ_1=1$ or there exists $k \in \{2, \dots,m\}$ such that $λ_{k-1}>λ_k=k$. Moreover, when $p$ is odd, we conj…

Generalized Samorodnitsky noisy function inequalities, with applications to error-correcting codes
Olakunle S. Abawonse, Jan Hazla, Ryan O'Donnell
https://arxiv.org/abs/2508.06940

Generalized Samorodnitsky noisy function inequalities, with applications to error-correcting codes
An inequality by Samorodnitsky states that if $f : \mathbb{F}_2^n \to \mathbb{R}$ is a nonnegative boolean function, and $S \subseteq [n]$ is chosen by randomly including each coordinate with probability a certain $λ= λ(q,ρ) < 1$, then \begin{equation} \log \|T_ρf\|_q \leq \mathbb{E}_{S} \log \|\mathbb{E}(f|S)\|_q\;. \end{equation} Samorodnitsky's inequality has several applications to the theory of error-correcting codes. Perhaps most notably, it can be used to s…

Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications
Minbo Yang, Shuijin Zhang
https://arxiv.org/abs/2508.07719 https:…

Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications
We study the uniqueness and nondegeneracy of positive bubble solutions for the generalized energy-critical Hartree equation on the Heisenberg group $\mathbb{H}^{n}$, \begin{equation}\label{0.1} -Δ_{\mathbb{H}}u=\left(\int_{\mathbb{H}^{n}}\frac{|u(η)|^{Q^{\ast}_μ}}{|η^{-1}ξ|^μ}\mathrm{d}η\right)|u|^{Q^{\ast}_μ-2}u,~~~ξ,η\in\mathbb{H}^{n}, \end{equation} where $Δ_{\mathbb{H}}$ represents the Kohn Laplacian, $u(η)$ is a real-valued function, $Q=2n+2$ is …

Theoretical Mass Function for Secondaries Forming via Gravitational Instability in Circumstellar Disks
Fred C. Adams, Aster G. Taylor, Michael R. Meyer
https://arxiv.org/abs/2508.09015

Theoretical Mass Function for Secondaries Forming via Gravitational Instability in Circumstellar Disks
This paper constructs a theoretical framework for calculating the distribution of masses for secondary bodies forming via gravitational instability in the outer regions of circumstellar disks. We show that several alternate ways to specify the mass scale of forming objects converge to the same result under the constraint that the parental disks are marginally stable with stability parameter $Q=1$. Next we show that the well-known constraint that the formation of secondary bodies requires rapid …

Replaced article(s) found for q-fin.PR. https://arxiv.org/list/q-fin.PR/new
[1/1]:
- A recursive formula for the $n^\text{th}$ survival function and the $n^\text{th}$ first passage t...
Alessio Lapolla

Optimal Thresholds for Monotone Non-Boolean Functions
Saba Lepsveridze, Allen Lin
https://arxiv.org/abs/2509.07246 https://arxiv.org/pdf/2509.07246

Optimal Thresholds for Monotone Non-Boolean Functions
Let $[q] = \{0,1,\ldots,q-1\}$, let $Δ[q]$ denote the simplex of probability measures on $[q]$, and let $λ$ denote the Lebesgue measure. We prove that for any symmetric monotone function $f \colon[q]^n \to [q]$ and any $a \in [q]$ we have \begin{equation*} λ(\{μ\in Δ[q]\;\vert\;\mathbf{Pr}_{x\simμ^{\otimes n}}[f(x)=a] \in (\varepsilon,1-\varepsilon)\}) = O(1/\log n)\textrm{.} \end{equation*} We also show that this bound is tight. This improves Kalai and Mossel's previous bound of $O…

How smooth are restrictions of Besov functions?
Julien Brasseur
https://arxiv.org/abs/2509.07420 https://arxiv.org/pdf/2509.07420

How smooth are restrictions of Besov functions?
In a previous work, we showed that Besov spaces do not enjoy the restriction property unless $q\leq p$. Specifically, we proved that if $p

AILoRA: Function-Aware Asymmetric Initialization for Low-Rank Adaptation of Large Language Models
Xiaoshuang Ji, Zhendong Zhao, Xiaoyan Gu, Xiaojun Chen, Xin Zhao, Zeyao Liu
https://arxiv.org/abs/2510.08034

AILoRA: Function-Aware Asymmetric Initialization for Low-Rank Adaptation of Large Language Models
Parameter-efficient finetuning (PEFT) aims to mitigate the substantial computational and memory overhead involved in adapting large-scale pretrained models to diverse downstream tasks. Among numerous PEFT strategies, Low-Rank Adaptation (LoRA) has emerged as one of the most widely adopted approaches due to its robust empirical performance and low implementation complexity. In practical deployment, LoRA is typically applied to the $W^Q$ and $W^V$ projection matrices of self-attention modules, en…

Rational minimax approximation of matrix-valued functions
Lei-Hong Zhang, Ya-Nan Zhang, Chenkun Zhang, Shanheng Han
https://arxiv.org/abs/2508.06378 https://

Rational minimax approximation of matrix-valued functions
In this paper, we present a rigorous framework for rational minimax approximation of matrix-valued functions that generalizes classical scalar approximation theory. Given sampled data $\{(x_\ell, {F}(x_\ell))\}_{\ell=1}^m$ where ${F}:\mathbb{C} \to \mathbb{C}^{s \times t}$ is a matrix-valued function, we study the problem of finding a matrix-valued rational approximant ${R}(x) = {P}(x)/q(x)$ (with ${P}:\mathbb{C} \to \mathbb{C}^{s \times t}$ a matrix-valued polynomial and $q(x)$ a nonzero scala…

Parking functions and chip-firing on hypergraphs
Timothy Blanton, Anton Dochtermann, Isabelle Hong, SuHo Oh, Zhan Zhan
https://arxiv.org/abs/2508.09720 https://

Parking functions and chip-firing on hypergraphs
For a connected graph $G$ with sink vertex $q$, a $G$-parking function is a vector of nonnegative integers whose entries are determined by cut-sets in $G$. Such objects also arise as the superstable configurations in the context of chip-firing. The set of all $G$-parking functions have various algebraic and combinatorial properties; for instance they relate to evaluations of the Tutte polynomial and in particular are counted by spanning trees of $G$. We extend these constructions to the setting…

Monotonicity in half-spaces for singular quasilinear elliptic problems involving the gradient
Phuong Le
https://arxiv.org/abs/2508.08859 https://arxiv.org/…

Monotonicity in half-spaces for singular quasilinear elliptic problems involving the gradient
We study positive solutions to the problem $-Δ_p u + \vartheta |\nabla u|^q = \frac{1}{u^γ} + f(u)$ in $\mathbb{R}^N_+$ with the zero Dirichlet boundary condition, where $p>1$, $γ>0$, $0

Observational constraints on f(Q,T) gravity from the mass-radius relation and stability of compact stars
S. K. Maurya, Abdul Aziz, Ksh. Newton Singh, G. Mustafa, Y. Sekhmani, Saibal Ray
https://arxiv.org/abs/2510.06003

Observational constraints on f(Q,T) gravity from the mass-radius relation and stability of compact stars
In this investigation we examine the astrophysical consequences of the influence of pressure anisotropy on the physical properties of observed pulsars within the background of $f(Q,T)$ gravity by choosing a specific form $f(Q, T)=ψ_1\, Q + ψ_2 T$, where $ψ_1$ and $ψ_2$ are the model parameters. Initially, we solve the modified field equations for anisotropic stellar configurations by assuming the physically valid metric potential along with anisotropic function for the distribution of the i…

Dynamical Systems with Bounded Condition and $C^{*}$-algebras
Takehiko Mori
https://arxiv.org/abs/2508.05713 https://arxiv.org/pdf/2508.05713

Dynamical Systems with Bounded Condition and $C^{*}$-algebras
By introducing the totally uniqueness condition for maps, we establish a one-to-one correspondence between the family of invariant sets for the $q x{+}d$-function (e.g. the $3 x{+}1$-map, also known as the Collatz map), where $q$ and $d$ are arbitrary positive odd numbers, and the family of reducing subspaces for the associated $C^{*}$-algebra. This extends the connection between the Collatz conjecture (also known as the $3 n{+}1$-problem) and the irreducibility of its associated $C^{*}$-algebr…

Role of non-extensivity $q$-parameter in tectonic seismic forecasting
Oscar Sotolongo-Costa, Miguel Eduardo Mora-Ramos
https://arxiv.org/abs/2508.04932 https://

Role of non-extensivity $q$-parameter in tectonic seismic forecasting
By writing total Tsallis entropy as a function of non-extensivity q-parameter withing the fragment-asperity model for earthquakes, a critical range of values is identified: 1.4

Exploring the nature of the emergent gauge field in composite-fermion metals: A large-scale microscopic study
Amogh Anakru, Mytraya Gattu, Ajit C. Balram, Xiao-Chuan Wu, Prashant Kumar, Zhen Bi, J. K. Jain
https://arxiv.org/abs/2509.07151

Exploring the nature of the emergent gauge field in composite-fermion metals: A large-scale microscopic study
Field theories of the composite-fermion (CF) metal model it as a Fermi sea of composite fermions coupled to an emergent gauge field. Within a random phase approximation, these theories predict that the Landau damping of the gauge field resulting from its coupling to the low-energy, long-wavelength CF particle-hole excitations modifies the electrons' density-density correlation function related to the static structure factor $S(q)$ at wave vector $q$. This produces a non-analytic correction $\pr…

Notes on the Geometry of Electromagnetic Fields and Maxwell's Equations along a non-null curves in non flat-3D space forms $M_{q}^{3}(c)$
Fatma Almaz, Cumali Ekici
https://arxiv.org/abs/2510.00170 …

Notes on the Geometry of Electromagnetic Fields and Maxwell's Equations along a non-null curves in non flat-3D space forms $M_{q}^{3}(c)$
In this paper, the directional derivatives in accordance with the orthonormal frame {T, N, B} are defined in $M_{q}^{3}(c)$, and the extended Serret-Frenet relations by using Frenet formulas are expressed. Furthermore, we express the bending elastic energy function for the same particle in $M_{q}^{3}(c)$ according to curve $α(s,ξ,η)$ and geometrical interpretation of the energy for unit vector fields and we also solve Maxwell's equations for the electric and magnetic field vectors in $M_{q}^…

On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$
Thomas Pawlaschyk
https://arxiv.org/abs/2510.05009 https://arxi…

On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$
In the spirit of Lelong and Bochner, we show that an upper semi-continuous function defined on a open tube set $Ω=ω+ i\mathbb{R}^n$ in $\mathbb{C}^n$, where $ω$ is an open set in $\mathbb{R}^n$, and which is invariant in its imaginary part, is $q$-plurisubharmonic on $Ω$ (in the sense of Hunt and Murray) if and only if it is real $q$-convex on $ω$, i.e., it admits the local maximum property with respect to affine linear functions on real $(q+1)$-dimensional affine subspaces. From this, we …

Holographic F(Q,T) Gravity with Lambert Solution
Houda Filali, Mouhsine Koussou, Mohamed Bennai, Rachid Ahl Laamara
https://arxiv.org/abs/2508.03486 https://

Holographic F(Q,T) Gravity with Lambert Solution
In this work, we study a model of holographic dark energy using FLRW cosmology in the context of modified gravity. An extension of the symmetric teleparallel gravity is obtained by considering the gravitational action L is given by an arbitrary function f of the nonmetricity Q, where the nonmetricity Q is responsible for the gravitational interaction, and the trace of the matter energy momentum tensor T, so that L=f(Q,T). We expand on the features of the derived cosmological model in view of th…

Infinite families of congruences for the second order mock theta function $\mathcal{B}(q)$
Hemjyoti Nath, Hirakjyoti Das
https://arxiv.org/abs/2509.20708 https://

Infinite families of congruences for the second order mock theta function $\mathcal{B}(q)$
The arithmetic properties of the second order mock theta function $\mathcal{B}(q)$, introduced by McIntosh, defined by \begin{equation*} \mathcal{B}(q) := \sum_{n \geq 0} \frac{q^n (-q;q^2)_n}{(q;q^2)_{n+1}} = \sum_{n \geq 0}b(n)q^n, \end{equation*} have been extensively studied. For instance, for all $n\ge0$, Kaur and Rana established congruences such as for all $n\ge0$, \begin{align*} b(12n+10) &\equiv 0 \pmod{36}, \quad b(18n+16) \equiv 0 \pmod{72}, \end{align*} Chen and Mao proved tha…

Boundary estimates for singular elliptic problems involving a gradient term
Phuong Le
https://arxiv.org/abs/2508.07360 https://arxiv.org/pdf/2508.07360

Boundary estimates for singular elliptic problems involving a gradient term
We study the behavior of weak solutions to the singular quasilinear elliptic problem $-Δ_p u + \vartheta |\nabla u|^q = \frac{1}{u^γ} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $γ>0$, $0

Normalized Ensemble-Averaged OAM Spectrum in Disordered Statistical Q-Plates
Netzer Moriya
https://arxiv.org/abs/2509.03920 https://arxiv.org/pdf/2509.0392…

Normalized Ensemble-Averaged OAM Spectrum in Disordered Statistical Q-Plates
This work presents an exact, analytical derivation of the ensemble-averaged orbital angular momentum (OAM) power spectrum for a circularly polarized Gaussian beam traversing a statistical Q-plate with Gaussian spatial disorder. Utilizing the Gaussian moment theorem, a new closed-form expression for the averaged mutual coherence is obtained. This coherence function is then rigorously projected onto OAM modes, yielding an exactly normalized series representation whose absolute convergence is form…

THIS👇👇👇👇👇
https://zirk.us/@ChrisMayLA6/114949958179517536
( And it highlights the fact that we need to rethink utilities, recognizing them as services necessary for a modern equitable society to function)

THIS👇👇👇👇👇
https://zirk.us/@ChrisMayLA6/114949958179517536
( And it highlights the fact that we need to rethink utilities, recognizing them as services necessary for a modern equitable society to function)

The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x,Q^2)$ and $g_1(x,Q^2)$
J. Ablinger, A. Behring, J. Bl\"umlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Sch\"onwald
https://arxiv.org/abs/2509.16124

The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x,Q^2)$ and $g_1(x,Q^2)$
We report quantitative results on the single-mass heavy-flavor contributions up to three-loop order to the unpolarized structure function $F_2(x,Q^2)$ and the polarized structure function $g_1(x,Q^2)$ for the first time. These results are relevant for precision QCD analyses of the World deep-inelastic data and the data taken at future colliders, such as the Electron--Ion Collider, in order to measure the strong coupling constant $α_s(M_Z^2)$, and the twist-2 parton distribution funct…

On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts
Alexey Khartov
https://arxiv.org/abs/2509.07244 https://

On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts
We consider the new class $\boldsymbol{Q}$ of rational-infinitely (or quasi-infinitely) divisible distribution functions on the real line. By definition, $F\in \boldsymbol{Q}$ if there are some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$, where ``$*$'' is the convolution. The characteristic function of such $F$ admits the Lévy--Khintchine-type representation with a ``signed spectral measure''. The class $\boldsymbol{Q}$ is a significant extension of the fa…

Multiplication operators on amalgam of $l^{q), \theta}$ and $L^{p}$
Monika Singh, Jitendra Kumar
https://arxiv.org/abs/2510.07260 https://arxiv.org/pdf/251…

Multiplication operators on amalgam of $l^{q), θ}$ and $L^{p}$
We define the grand amalgam Lebesgue function space $l^{q), θ}(L^p),$ and study the fundamental structural properties of the space, including completeness. Then we define the small Lebesgue sequence space and study its function space properties. Furthermore, we prove a version of the Hölders inequality on the frame work of these spaces. Finally, we study the multiplication operator in the setting of these spaces.

An operator approach to the Madelung-Bohm continuity equation
M. A. Garc\'ia-M\'arquez, H. M. Moya-Cessa, I. Ramos-Prieto, F. Soto-Eguibar
https://arxiv.org/abs/2509.01913

An operator approach to the Madelung-Bohm continuity equation
We solve the probability continuity equation within the Madelung-Bohm framework, assuming a separable phase expressed as $S(x,t) = Q(x)\dotν(t) + μ(t)$. Using operator methods, we reformulate the wave function's amplitude into a form that is more practical for application to any given initial condition. This results in the function $F(x,t)$, which characterizes the wavefunction's amplitude, with $F$ being the transformation of the position operator via a squeeze-like operator. To demonstrate …

Beyond ReLU: Chebyshev-DQN for Enhanced Deep Q-Networks
Saman Yazdannik, Morteza Tayefi, Shamim Sanisales
https://arxiv.org/abs/2508.14536 https://arxiv.or…

Beyond ReLU: Chebyshev-DQN for Enhanced Deep Q-Networks
The performance of Deep Q-Networks (DQN) is critically dependent on the ability of its underlying neural network to accurately approximate the action-value function. Standard function approximators, such as multi-layer perceptrons, may struggle to efficiently represent the complex value landscapes inherent in many reinforcement learning problems. This paper introduces a novel architecture, the Chebyshev-DQN (Ch-DQN), which integrates a Chebyshev polynomial basis into the DQN framework to create…

Spectral analysis of $q$-deformed unitary ensembles with the Al-Salam--Carlitz weight
Sung-Soo Byun, Yeong-Gwang Jung, Jaeseong Oh
https://arxiv.org/abs/2507.18042 https://

Spectral analysis of $q$-deformed unitary ensembles with the Al-Salam--Carlitz weight
We study $q$-deformed random unitary ensembles associated with the weight function of the Al-Salam--Carlitz orthogonal polynomials, indexed by a parameter $a < 0$. In the special case $a = -1$, the model reduces to the $q$-deformed Gaussian unitary ensemble. Employing the Flajolet--Viennot theory together with the combinatorics of matchings, we derive an explicit positive-sum expression for the spectral moments. In the double-scaling regime $q = e^{-λ/N}$, where $N$ denotes the ensemble size a…

Model-Free Dynamic Consensus in Multi-Agent Systems: A Q-Function Perspective
Maryam Babazadeh, Naim Bajcinca
https://arxiv.org/abs/2509.24598 https://arxi…

Model-Free Dynamic Consensus in Multi-Agent Systems: A Q-Function Perspective
This paper presents a new method for achieving dynamic consensus in linear discrete-time homogeneous multi-agent systems (MAS) with marginally stable or unstable dynamics. The guarantee of consensus in this setting involves a set of constraints based on the graph's spectral properties, complicating the design of the coupling gains. This challenge intensifies for large-scale systems with diverse graph Laplacian spectra. The proposed approach reformulates the dynamic consensus problem with a pres…

The universal Harish-Chandra $j$-function
Gil Moss, Justin Trias
https://arxiv.org/abs/2509.20169 https://arxiv.org/pdf/2509.20169

The universal Harish-Chandra $j$-function
Let $F$ be a nonarchimedean local field with residue field of cardinality $q$, let $G$ be the $F$-points of a connected reductive group defined over $F$, let $P$ and $Q$ be two parabolic subgroups with the same Levi factor $M$. We construct intertwining operators $J_{Q|P}$ and the Harish-Chandra $j$-function $j^G$ for finitely generated smooth $A[M]$-modules, where $A$ is any commutative Noetherian algebra over $\mathbb{Z}' :=\mathbb{Z}[\sqrt{q}^{-1}]$. The construction is functorial, comp…

Conjugacy in fibre products, distortion, and the geometry of cyclic subgroups
Martin R Bridson
https://arxiv.org/abs/2507.17598 https://arxiv.org/pdf/2507.…

Conjugacy in fibre products, distortion, and the geometry of cyclic subgroups
We investigate the complexity of the conjugacy problem for fibre products in torsion-free hyperbolic groups. Let $G$ be a torsion-free hyperbolic group and let $P

Approximation by neural network operators of convolution type activated by deformed and parametrized half hyperbolic tangent function
Asiye Arif, Tugba Yurdakadim
https://arxiv.org/abs/2510.05980

Approximation by neural network operators of convolution type activated by deformed and parametrized half hyperbolic tangent function
Here, we introduce three kinds of neural network operators of convolution type which are activated by q-deformed and \b{eta}-parametrized half hyperbolic tangent function. We obtain quantitative convergence results to the identity operator with the use of modulus of continuity. Global smoothness preservation of our operators are also presented and the iterated versions of them are taken into the consideration.

Replaced article(s) found for q-fin.TR. https://arxiv.org/list/q-fin.TR/new
[1/1]:
- Alternative Loss Function in Evaluation of Transformer Models
Jakub Micha\'nk\'ow, Pawe{\l} Sakowski, Robert \'Slepaczuk

Sums of the floor function related to class numbers of imaginary quadratic fields
Marc Chamberland, Karl Dilcher
https://arxiv.org/abs/2510.04387 https://a…

Sums of the floor function related to class numbers of imaginary quadratic fields
A curious identity of Bunyakovsky (1882), made more widely known by Pólya and Szeg{\H o} in their ``Problems and Theorems in Analysis", gives an evaluation of a sum of the floor function of square roots involving primes $p\equiv 1\pmod{4}$. We evaluate this sum also in the case $p\equiv 3\pmod{4}$, obtaining an identity in terms of the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-p})$. We also consider certain cases where the prime $p$ is replaced by a composite integer. C…

Quantum algebra approach to univariate and multivariate rational functions of $q$-Racah type
Wolter Groenevelt, Carel Wagenaar
https://arxiv.org/abs/2507.13483

Quantum algebra approach to univariate and multivariate rational functions of $q$-Racah type
In this paper, we study rational functions of $q$-Racah type and a multivariate extension, using representation theory of $\mathcal U_q(\mathfrak{sl}_2)$. Eigenfunctions of twisted primitive elements in $\mathcal U_q(\mathfrak{su}_2)$ can be expressed in terms of $q^{-1}$-Krawtchouk polynomials. Using this, we show that overlap coefficients of solutions of a generalized eigenvalue problem (GEVP) and an eigenvalue problem (EVP) can be expressed in terms of a rational function of $_4φ_3$-type. W…

Bigraded Polynomials for the Cohomology of Wild Hitchin Systems
Dan Xie
https://arxiv.org/abs/2509.20872 https://arxiv.org/pdf/2509.20872

Bigraded Polynomials for the Cohomology of Wild Hitchin Systems
We introduce a bi-graded polynomial that encodes the cohomology groups of the wild Hitchin system of type~$A_{n-1}$, constructed using an irregular singularity (determined by an integer~$m$) and an arbitrary regular singularity~$f$. When the regular singularity is of the form~$f = [1, \ldots, 1]$, the bi-graded polynomial~$C_{m,n}(q,t)$ coincides with the bigraded rational parking function defined combinatorially, admitting a Schur expansion~$C_{m,n}(q,t) = \sum_λf_λ(q,t) s_λ(x)$. For genera…

Cassini-Catalan Determinants via Ramanujan's Theta Identity
Nagananda K G, Jong Sung Kim
https://arxiv.org/abs/2508.11877 https://arxiv.org/pdf/2508.11…

Cassini-Catalan Determinants via Ramanujan's Theta Identity
In this paper, we show that the classical Cassini and Catalan identities for Fibonacci numbers arise naturally from a single quadratic theta-function identity of Ramanujan. Expanding the identity $ψ(q)ψ(q^{3})=ψ(q^{4})φ(q^{6})+q\,φ(q^{2})ψ(q^{12})$ via the Jacobi triple product and equating coefficients yields the unified $q$-determinant $F_{n+r}(q)F_{n-r}(q)-F_{n}(q)^{2}=(-q)^{\,n-r}F_{r}(q)^{2}$, $n\ge r\ge 1$, where $ψ(q)$ and $φ(q)$ are Ramanujan's theta functions with $q$ a complex…

Replaced article(s) found for q-fin.CP. https://arxiv.org/list/q-fin.CP/new
[1/1]:
- Alternative Loss Function in Evaluation of Transformer Models
Jakub Micha\'nk\'ow, Pawe{\l} Sakowski, Robert \'Slepaczuk

Impact of a Sharpness Based Loss Function for Removing Out-of-Focus Blur
Uditangshu Aurangabadkar, Darren Ramsook, Anil Kokaram
https://arxiv.org/abs/2509.11735 https://

Impact of a Sharpness Based Loss Function for Removing Out-of-Focus Blur
Recent research has explored complex loss functions for deblurring. In this work, we explore the impact of a previously introduced loss function - Q which explicitly addresses sharpness and employ it to fine-tune State-of-the-Art (SOTA) deblurring models. Standard image quality metrics such as PSNR or SSIM do not distinguish sharpness from ringing. Therefore, we propose a novel full-reference image quality metric Omega that combines PSNR with Q. This metric is sensitive to ringing artefacts, bu…

Ruelle's zeta function for non-Archimedean rational maps
Yunping Jiang, Chenxi Wu
https://arxiv.org/abs/2508.19374 https://arxiv.org/pdf/2508.19374

Ruelle's zeta function for non-Archimedean rational maps
We studied the transfer operators defined over $\mathbb{C}_p$-valued analytic functions for subhyperbolic rational maps on $\mathbb{Q}_p$, and showed that the corresponding Ruelle's zeta functions are meromorphic on $\mathbb{C}_p$. We also used $\mathbb{R}$-valued transfer operators to study the shape of the corresponding Julia sets, and proved a Levin-Sodin-Yuditski type identity for general rational maps on $\mathbb{C}_p$. In all the results above, $\mathbb{Q}_p$ can be replaced with any non-…

Logarithmic derivatives of L-functions and small prime quadratic nonresidues
Genheng Zhao
https://arxiv.org/abs/2509.03157 https://arxiv.org/pdf/2509.03157…

Logarithmic derivatives of L-functions and small prime quadratic nonresidues
Let $χ$ be a real non-principal character modulo a prime $q$ and $L(s,χ)$ be the corresponding $L$-function. We prove that for any real number $s\geq 1$ there holds $$ -\frac{L'(s,χ)}{L(s,χ)}\leq c \log q,$$ where $c$ can be taken arbitrarily close to $1/4$ if we assume $q$ is sufficiently large depending upon it. As a consequence, for all large $q$, there are at least $q^{3/50}$ primes $p$ smaller than $q$ such that $χ(p)=-1$.

Well-Posedness for the Euler Equations in Function Spaces of Generalized Smoothness
Nicholas Harrison, Zachary Radke
https://arxiv.org/abs/2510.02626 https://

Well-Posedness for the Euler Equations in Function Spaces of Generalized Smoothness
We consider the question of well-posedness for the incompressible Euler equations in generalized function spaces of the type $B^{s,ψ}_{p,q}(\mathbb{R}^d)$ and $F^{s,ψ}_{p,q}(\mathbb{R}^d)$ where $ψ$ is a slowly varying function in the Karamata sense and $s=d/p+1$. We prove that if $ψ$ grows fast enough, then there is a local in time solution to the Euler equations. We also establish a BKM-type criterion that allows us to obtain global existence in two dimensions.

Crosslisted article(s) found for q-fin.PR. https://arxiv.org/list/q-fin.PR/new
[1/1]:
- A recursive formula for the $n^\text{th}$ survival function and the $n^\text{th}$ first passage t...
Alessio Lapolla

A non-sequential arithmetical theory with pairing
Juvenal Murwanashyaka
https://arxiv.org/abs/2509.15191 https://arxiv.org/pdf/2509.15191

A non-sequential arithmetical theory with pairing
Albert Visser has shown that Robinson's $ \mathsf{Q} $ and Gregorczyk's $ \mathsf{TC} $ are not sequential by showing that these theories are not even poly-pair theories, which, in a strong sense, means these theories lack pairing. In this paper, we use Ehrenfeucht-Fraïssé games to show that the theory $ \mathsf{Q} + Θ$ we obtain by extending Robinson's $ \mathsf{Q} $ with an axiom $ Θ$ which says that the map $ π(x, y ) = (x+y)^2 + x $ is a pairing function is not sequential; in fact, we …

Hyperbinary partitions and q-deformed rationals
Thomas McConville (Kennesaw State University), James Propp (University of Massachusetts Lowell), Bruce E. Sagan (Michigan State University)
https://arxiv.org/abs/2508.20026

Hyperbinary partitions and q-deformed rationals
A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of 2 and every part appears at most twice. We give three applications of the length generating function for such partitions, denoted by h_q(n). Morier-Genoud and Ovsienko defined the q-analogue of a rational number [r/s]_q in various ways, most of which depend directly or indirectly on the continued fraction expansion of r/s. As our first application we show that [r/s]_q = q h_q(n-1)/h_q(n) where r/s…

Replaced article(s) found for q-bio.GN. https://arxiv.org/list/q-bio.GN/new
[1/1]:
- Evaluating DNA function understanding in genomic language models using evolutionarily implausible...
Shiyu Jiang, Xuyin Liu, Zitong Jerry Wang

Bayesian Interpretation of Husimi Function and Wehrl Entropy
Chen Xu, Yiqi Yu, Peng Zhang
https://arxiv.org/abs/2507.08600 https://ar…

Bayesian Interpretation of Husimi Function and Wehrl Entropy
Husimi function (Q-function) of a quantum state is the distribution function of the density operator in the coherent state representation. It is widely used in theoretical research, such as in quantum optics. The Wehrl entropy is the Shannon entropy of the Husimi function, and is non-zero even for pure states. This entropy has been extensively studied in mathematical physics. Recent research also suggests a significant connection between the Wehrl entropy and many-body quantum entanglement in s…

On the quantum algebra $su_q(1,1)$ from a Special Function standpoint
R. Alvarez-Nodarse, A. Arenas-Gomez
https://arxiv.org/abs/2507.14100 https://

On the quantum algebra $su_q(1,1)$ from a Special Function standpoint
In this paper, we study the tensor product of two unitary irreducible representations, as well as the tensor product of a unitary irreducible representation with a finite-dimensional one, and determine the corresponding Clebsch-Gordan coefficients. Using von Neumann's projection operator method, we obtain an explicit representation of these coefficients, which allows us to express them as symmetric q-hypergeometric series. Finally, by leveraging the properties of the q-hypergeometric function, …

Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums
S. Sriram, A. David Christopher
https://arxiv.org/abs/2509.26559

Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums
Let $n$ and $t$ be positive integers with $t\geq 2$. Let $R_t(n)$ be the number of $t$-regular partitions of $n$. A class of functions, denoted $τ_k(n)$, is defined as follows: \[q\prod_{m=1}^{\infty}(1-q^m)^k=\sum_{n=1}^{\infty}τ_k(n)q^n, \] where $k$ is an integer. We express $τ_k(n)$ as a binomial coefficient weighted partition sum. Consequently, we obtain congruence identities that relate $τ_k(n)$, $R_t(n)$ and partition function weighted composition sums.

III. Interacting Dark Energy: Summary of Models, Pathologies, and Constraints
Marcel van der Westhuizen, Amare Abebe, Eleonora Di Valentino
https://arxiv.org/abs/2509.04496 http…

III. Interacting Dark Energy: Summary of Models, Pathologies, and Constraints
We present an overview of the main results from our two companion papers that are relevant for observational constraints on interacting dark energy (IDE) models. We provide analytical solutions for the dark matter and dark energy densities, $ρ_{\rm dm}$ and $ρ_{\rm de}$, as well as the normalized Hubble function $h(z)$, for eight IDE models. These include five linear IDE models, namely $Q=3H(δ_{\rm dm} ρ_{\rm dm} + δ_{\rm de} ρ_{\rm de})$ and four special cases: $Q=3Hδ(ρ_{\rm dm}+ρ_{\r…

The maximal function on spaces of homogeneous type, or adjacent dyadic cubes do good
Alina Shalukhina
https://arxiv.org/abs/2509.02508 https://arxiv.org/pd…

The maximal function on spaces of homogeneous type, or adjacent dyadic cubes do good
We prove that the Hardy--Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(X,d,μ)$ with $1

Polynomial Invariants of q-Matroids and Rank-Metric Codes
Heide Gluesing-Luerssen, Benjamin Jany
https://arxiv.org/abs/2509.21618 https://arxiv.org/pdf/250…

Polynomial Invariants of q-Matroids and Rank-Metric Codes
It is shown that the Whitney function of a representable q-matroid and the collection of all higher weight enumerators of any representing rank-metric code determine each other via a monomial substitution. Moreover, the q-matroid itself and the collection of all higher support enumerators of the code determine each other. Next, it is proven that the Whitney function of a q-matroid and the Whitney function of its projectivization determine each other via a monomial substitution. Finally, q-matro…

Partial regularity for parabolic systems of double phase type
Jihoon Ok, Giovanni Scilla, Bianca Stroffolini
https://arxiv.org/abs/2510.03849 https://arxiv…

Partial regularity for parabolic systems of double phase type
We study partial regularity for nondegenerate parabolic systems of double phase type, where the growth function is given by $H(z,s)=s^p+a(z)s^q$, $z=(x,t)\inΩ_T$, with $\tfrac{2n}{n+2}

Zero-sum turn games using Q-learning: finite computation with security guarantees
Sean Anderson, Chris Darken, Jo\~ao Hespanha
https://arxiv.org/abs/2509.13585 https://

Zero-sum turn games using Q-learning: finite computation with security guarantees
This paper addresses zero-sum ``turn'' games, in which only one player can make decisions at each state. We show that pure saddle-point state-feedback policies for turn games can be constructed from dynamic programming fixed-point equations for a single value function or Q-function. These fixed-points can be constructed using a suitable form of Q-learning. For discounted costs, convergence of this form of Q-learning can be established using classical techniques. For undiscounted costs, we provi…

H\"{o}lder property of the resolvent of a monotone operator in Banach spaces
Changchi Huang, Jigen Peng, Yuchao Tang
https://arxiv.org/abs/2510.03774 https://

Hölder property of the resolvent of a monotone operator in Banach spaces
Let $E$ be a Banach space, and let $J: E \to E^{*}$ denote the normalized duality mapping. In this paper, we establish an upper bound for $\|Jx - Jy\|$ in $q$-uniformly smooth Banach spaces, where the bound is expressed in terms of a relatively simple function of $\|x - y\|$. Subsequently, we derive the Hölder property of mappings of firmly nonexpansive type in 2-uniformly convex and $q$-uniformly smooth Banach spaces ($1

Backbone colouring of chordal graphs
J\'ulio Ara\'ujo, Nicolas Nisse, Lucas Picasarri-Arrieta
https://arxiv.org/abs/2508.02980 https://arxiv.org/pd…

Backbone colouring of chordal graphs
A proper $k$-colouring of a graph $G=(V,E)$ is a function $c: V(G)\to \{1,\ldots,k\}$ such that $c(u)\neq c(v)$ for every edge $uv\in E(G)$. The chromatic number $χ(G)$ is the minimum $k$ such that there exists a proper $k$-colouring of $G$. Given a spanning subgraph $H$ of $G$, a $q$-backbone $k$-colouring of $(G,H)$ is a proper $k$-colouring $c$ of $G$ such that $\lvert c(u)-c(v)\rvert \ge q$ for every edge $uv\in E(H)$. The $q$-backbone chromatic number ${\rm BBC}_q(G,H)$ is the smallest $k…

Reconstruction Codes for Deletions and Insertions: Connection, Distinction, and Construction
Yubo Sun, Gennian Ge
https://arxiv.org/abs/2508.14386 https://…

Reconstruction Codes for Deletions and Insertions: Connection, Distinction, and Construction
Let $\mathcal{B}(\cdot)$ be an error ball function. A set of $q$-ary sequences of length $n$ is referred to as an \emph{$(n,q,N;\mathcal{B})$-reconstruction code} if each sequence $\boldsymbol{x}$ within this set can be uniquely reconstructed from any $N$ distinct elements within its error ball $\mathcal{B}(\boldsymbol{x})$. The main objective in this area is to determine or establish bounds for the minimum redundancy of $(n,q,N;\mathcal{B})$-reconstruction codes, denoted by $ρ(n,q,N;\mathcal{…

Black Hole Mass/Charge Relation and Weak No-hair Theorem Conjecture
Guan-Yi Lu, Meng-Nan Yang, H. Lu
https://arxiv.org/abs/2508.14158 https://arxiv.org/pdf…

Black Hole Mass/Charge Relation and Weak No-hair Theorem Conjecture
We consider Einstein gravity minimally coupled to two Maxwell fields and one dilaton scalar. We study the electrically-charged spherically-symmetric and static solutions that are asymptotic to Minkowski spacetime. General solutions are described by four independent parameters: mass $M$, two electric charges $(Q_1,Q_2)$ and the scalar charge $Σ$. For black holes, the scalar charge is not independent, but a function of $(M,Q_1,Q_2)$. We provide a set of formulae relating the mass to the charges,…

Factorization of multiparticle contributions to amplitudes of B-meson weak decays
Dmitri Melikhov
https://arxiv.org/abs/2507.23441 https://arxiv.org/pdf/25…

Factorization of multiparticle contributions to amplitudes of B-meson weak decays
We show that multiparticle contributions to amplitudes of weak decays of the generic topology (heavy quark hits some intermediate point of the propagator line joining the end-points from which momenta $q$ and $q'$ are emitted) is given in the heavy quark limit by the convolution of (i) hard kernel composed of highly virtual propagators of light degrees of freedom and (ii) the $B$-meson multiparticle wave function, $\langle 0|ϕ(x)ϕ(x_1)\dotsϕ(x_n)ϕ_b(0)ϕ(x'_{n'})\dotsϕ(x'_1)ϕ(x')…

Noncoincidence $f(Q)$-Cosmology with Dark Matter Coupled to Gravity
A. Abebe, P. S. Apostolopoulos, A. Giacomini, G. Leon, F. Moncada, A. Paliathanasis
https://arxiv.org/abs/2510.00535

Noncoincidence $f(Q)$-Cosmology with Dark Matter Coupled to Gravity
We investigate FLRW cosmology in the framework of symmetric teleparallel $f(Q)$ gravity with a nonminimal coupling between dark matter and the gravitational field. In the noncoincidence gauge, the field equations admit an equivalent multi-scalar field representation, which we investigate the phase-space using the Hubble-normalization approach. We classify all stationary points for arbitrary function $f(Q)$ and we discuss the physical properties of the asymptotic solutions. For the power-law the…

Winding quotients for virtual period maps of rank 1
Kyoji Saito
https://arxiv.org/abs/2510.04558 https://arxiv.org/pdf/2510.04558

Winding quotients for virtual period maps of rank 1
We illustrate a rank 1 model of virtual period maps and their associated winding quotient, where the winding quotient is a new phenomenon appeared in a recent study of virtual period maps and it requires a reformulation of the classical inversion problem for the period maps. We answer to the new inversion problem by introducing the q-multiplicatively periodic function, whose pull-back to the winding covering space is the Weierstrass p-function up to a correction by Eisenstein series E2, and whi…

Schwarz-Pick type lemma and Landau type theorem for $\alpha$-harmonic mappings
Vibhuti Arora, Jiaolong Chen, Shankey Kumar, Qianyun Li
https://arxiv.org/abs/2509.06359 https://

Schwarz-Pick type lemma and Landau type theorem for $α$-harmonic mappings
The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the $α$-harmonic mapping $u=P_α[ϕ]$, where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R} )$ and $p\in[1,\infty]$. We get an explicit form of the sharp function $\mathbf{C}_{α, q}(x)$ in the inequality $|\nabla u(x)| \leq \mathbf{C}_{α, q}(x)\|ϕ\|_{L^p(\mathbb{S}^{n-1}, \mathbb{ R} )}$. Second, we prove a Landau type theorem for $u=P_α[ϕ]$, where $ϕ\in L^{\infty}(\mathbb{S}^{n-1},…

An iterative-bijective approach to asymmetric generalizations of Schur's theorem
Laure Velenik
https://arxiv.org/abs/2510.00846 https://arxiv.org/pdf/2…

An iterative-bijective approach to asymmetric generalizations of Schur's theorem
In this paper, we present a new Rogers--Ramanujan type identity for overpartitions by extending the asymmetrical version of Schur's theorem due to Lovejoy to a broader class of infinite products. More precisely, we provide a combinatorial interpretation of the following product, for any positive integer $k$, as a generating function for a class of overpartitions in which parts appear in $2^k - 1$ colors: \[ \frac{(-y_1 q;q)_\infty \cdots (-y_k q;q)_\infty}{(y_1 d q;q)_\infty}. \] Our proof is b…

Hecke $L$-functions Away From The Central Line
Mohammad H. Hamdar
https://arxiv.org/abs/2509.06152 https://arxiv.org/pdf/2509.06152

Hecke $L$-functions Away From The Central Line
We compute the first moment of cubic Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$ evaluated at any $s$ inside the critical strip. The first moment for $s<\frac{1}{2}$ is particularly interesting, and we show there is a phase transition at $s=\frac{1}{3}$. This extends the analogue result of David-Meisner for the first moment over function fields. As in their work, the computation of the moment at $s=\frac{1}{3}$ relies on a cancellation between two terms which are a priori not related: a ma…

Teaching RL Agents to Act Better: VLM as Action Advisor for Online Reinforcement Learning
Xiefeng Wu, Jing Zhao, Shu Zhang, Mingyu Hu
https://arxiv.org/abs/2509.21126 https://…

Teaching RL Agents to Act Better: VLM as Action Advisor for Online Reinforcement Learning
Online reinforcement learning in complex tasks is time-consuming, as massive interaction steps are needed to learn the optimal Q-function.Vision-language action (VLA) policies represent a promising direction for solving diverse tasks; however, their performance on low-level control remains limited, and effective deployment often requires task-specific expert demonstrations for fine-tuning. In this paper, we propose \textbf{VARL} (\textbf{V}LM as \textbf{A}ction advisor for online \textbf{R}einf…

Green's function and volume noncollapsing estimates for the K\"ahler-Ricci flow
Weiqi Zhang, Yashan Zhang
https://arxiv.org/abs/2508.13646 https://

Green's function and volume noncollapsing estimates for the Kähler-Ricci flow
We prove a local volume noncollapsing property for the long-time Kähler-Ricci flow on a minimal Kähler manifold, improving previous results of Guo-Phong-Song-Sturm, Guedj-Tô and Vu. Similar result is also proved for a twisted Kähler-Ricci flow on non-minimal Kähler manifolds. In the course of proof, we develop new Green's function estimates for a general Kähler family under the $L^{p>1}$ and $L^1(\log L)^{q>n}$ volume density conditions.

Lehmer pairs and binomial series
Kapitonets Kirill
https://arxiv.org/abs/2509.00906 https://arxiv.org/pdf/2509.00906

Lehmer pairs and binomial series
The paper defines a generalized Hardy function and simulates the zero of the Zeta function that does not lie on a critical line. $$Z_α=\Reζ(α+it)e^{iθ(t)}$$ where \par $$e^{iθ(t)}=π^{-it/2}\frac{Γ(1/4+it/2)}{|Γ(1/4+it/2)|}$$ \par Using the binomial series and the generalized summation method, the formula for the generalized Hardy function is obtained. $$Z_α=Z_0+\sum^{\infty}_{k=1}(-1)^k\frac{α(α-1)(α-2)...(α-k+1)}{k!}Q_k$$ \par where \par $Z_α$ is a generalized Hardy fun…

Generalized convexity of the Lambert $W$ function
Gendi Wang
https://arxiv.org/abs/2508.17409 https://arxiv.org/pdf/2508.17409

Generalized convexity of the Lambert $W$ function
This paper investigates the generalized convexity properties of the Lambert $W$ function, defined as the solution to $W(z)e^{W(z)}=z$. Focusing on $H_{p,q}$-convexity and concavity with respect to Hölder means, we derive necessary and sufficient conditions for $W$ to exhibit strict $H_{p,q}$-convexity or concavity on the interval $(0,+\infty)$. The main result characterizes these properties in terms of specific parameter regions $(p,q)$-plane. Inequalities involving harmonic, geometric, and ar…

A model free approach for continuous-time optimal tracking control with unknown user-define cost and constrained control input via advantage function
Duc Cuong Nguyen, Quang Huy Dao, Phuong Nam Dao
https://arxiv.org/abs/2509.16821

A model free approach for continuous-time optimal tracking control with unknown user-define cost and constrained control input via advantage function
This paper presents a pioneering approach to solving the linear quadratic regulation (LQR) and linear quadratic tracking (LQT) problems with constrained inputs using a novel off-policy continuous-time Q-learning framework. The proposed methodology leverages a novel concept of the Advantage function for linear continuous systems, enabling solutions to be obtained without the need for prior knowledge of the reward matrix weights, state resetting, or assuming the existence of a predefined admissib…

Algorithm for computing the partition function of the Potts model for SP-graphs
Sofya Mukhamedzhanova, Bulat Sabirov, Amir Mukhamedzhanov
https://arxiv.org/abs/2507.22579 https:…

Algorithm for computing the partition function of the Potts model for SP-graphs
The q-state Potts model is a fundamental framework in statistical physics and graph theory, with its partition function encoding rich information about spin configurations. The multivariate Tutte polynomial (known as the partition function of the Potts model) can be defined on an arbitrary finite graph $G$ and encodes a lot of important combinatorial information about the graph. As a special case, it contains the familiar Tutte polynomial with two variables and, consequently, its specialization…

Transcendental formulas for the coefficients of Ramanujan's mock theta functions
Nickolas Andersen, Gradin Anderson
https://arxiv.org/abs/2509.02777 https://

Transcendental formulas for the coefficients of Ramanujan's mock theta functions
Ramanujan's 1920 last letter to Hardy contains seventeen examples of mock theta functions which he organized into three "orders." The most famous of these is the third-order function $f(q)$ which has received the most attention of any individual mock theta function in the intervening century. In 1964, Andrews--improving on a result of Dragonette--gave an asymptotic formula for the coefficients of $f(q)$ and conjectured an exact formula for the coefficients: a conditionally convergent series tha…

Fokker-Planck equation for stochastic heat equations
Qingyan Meng, Jinqiao Duan, Jinlong Wei, Peter E. Kloeden
https://arxiv.org/abs/2509.01283 https://arx…

Fokker-Planck equation for stochastic heat equations
This work is devoted to the study of the Fokker--Planck equation for a stochastic heat equation with an additive $Q$-Wiener noise and non-homogeneous boundary conditions. We explicitly construct the probability density function and establish the associated Fokker--Planck equation by applying the eigenfunction expansion technique. Moreover, the Feynman--Kac formula is used to obtain the probabilistic representation of the solution. The analysis is further extended to cases with multiplicative no…

Analysis of quantities determining the critical inverse temperature in the annealed Potts model with Pareto vertex weights
A. J. E. M. Janssen
https://arxiv.org/abs/2508.21409 h…

Analysis of quantities determining the critical inverse temperature in the annealed Potts model with Pareto vertex weights
We consider in this work the crucial quantity $t_c$ that determines the critical inverse temperature $β_c$ in the $q$-state Potts model on sparse rank-1 random graphs where the vertices are equipped with a Pareto weight density $(τ-1)\,w^{-τ}\,{\cal X}_{[1,\infty)}(w)$. It is shown in \cite{ref1} that this $t_c$ is the unique positive zero of a function ${\cal K}$ that is obtained by an appropriate combination of the stationarity condition and the criticality condition for the case the exter…

A new parametric observational study of $f(Q,B)$ gravity with modified chaplygin gas
Amit Samaddar, S. Surendra Singh
https://arxiv.org/abs/2507.14251 http…

A new parametric observational study of $f(Q,B)$ gravity with modified chaplygin gas
In this work, we explore the cosmological dynamics of a modified gravity framework based on the function $f(Q,B)=δQ^{2}+βB$, where $Q$ denotes the nonmetricity scalar and $B$ is the boundary term that relates $Q$ to the Ricci scalar. The matter sector is modeled using the Modified Chaplygin Gas (MCG) with the equation of state $p=Aρ-\frac{B}{ρ^α}$, allowing the model to interpolate between early-time matter behavior and late-time cosmic acceleration. By deriving an analytical expression fo…

On the Fermat-type partial differential-difference equations on $\mathbb{C}^n$
Tingbin Cao, Jun Wang, Zhuan Ye
https://arxiv.org/abs/2509.01862 https://arx…

On the Fermat-type partial differential-difference equations on $\mathbb{C}^n$
Assume that $n$ is a positive integer, $p_{j}$ ($j=1,2, \cdots, 6)$ are polynomials, $p$ is an irreducible polynomial, and $f$ is an entire function on $\mathbb{C}^{n}.$ Let $ L(f)=\sum_{j=1}^s q_{t_j}f_{z_{t_j}}$ and $\overline{f}(z)=f(z_{1}+c_{1}, \ldots, z_{n}+c_{n})$, where $q_{t_j}$ ($j=1,2, \cdots, s\le n$) are non-zero polynomials on $\mathbb{C}^{n}$ and $c=(c_{1}, \ldots, c_{n})\in \mathbb{C}^{n}\setminus\{0\}$. We show the structures of all entire solutions to the non-linear partial di…

Eigenvalue Distribution of Large Weighted Random Sparse Uniform $q$-Hypergraphs
Valentin Vengerovsky
https://arxiv.org/abs/2508.13297 https://arxiv.org/pdf…

Eigenvalue Distribution of Large Weighted Random Sparse Uniform $q$-Hypergraphs
We study eigenvalue distribution of the adjacency matrix $A^{(N,p,q)}$ of weighted random uniform $q$-hypergraphs $Γ= Γ_{N,p,q}$. We assume that the graphs have $N$ vertices and the average number of hyperedges attached to one vertex is $(q-1)!\cdot p$. To each edge of the graph $e_{ij}$ we assign a weight given by a random variable $a_{ij}$ with all moments finite. We consider the moments of normalized eigenvalue counting function $σ_{N,p,q}$ of $A^{(N,p,q)}$. Assuming all moments of $a$ fi…

Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes
Thomas Wright
https://arxiv.org/abs/2507.10780 https://arxiv.org/…

Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes
Let $χ$ be a Dirichlet character mod $D$ with $L(s,χ)$ its associated $L$-function, and let $ψ(x,q,a)$ be Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We that under the assumption of an exceptional character $χ$ with $L(1,χ)=o\left((\log D)^{-5}\right)$, for any $q

New formulas for Macdonald polynomials via the multispecies exclusion and zero range processes
Olya Mandelshtam
https://arxiv.org/abs/2508.04666 https://ar…

New formulas for Macdonald polynomials via the multispecies exclusion and zero range processes
We describe some recently discovered connections between one-dimensional interacting particle models and Macdonald polynomials. The first such model is the multispecies asymmetric simple exclusion process (ASEP) on a ring, linked to the symmetric Macdonald polynomial $P_λ(X;q,t)$ through its partition function. Through this connection, a new formula was found for $P_λ$ by generalizing multiline queues, which were introduced by Martin in 2018 to compute stationary probabilities of the ASEP. Th…

On some new sharp results on Toeplits operators in the unit disk
R. F. Shamoyan, V. A. Bednazh
https://arxiv.org/abs/2508.01859 https://arxiv.org/pdf/2508.…

On some new sharp results on Toeplits operators in the unit disk
We provide new sharp results on the action of Toeplitz operators from Triebel and Besov spaces to new BMOA-type function spaces on the unit disk. In this paper we consider $s\geq 1$ case in previous papers $s<1$ was covered fof $BMOA_{s,q}$ and $BMOA_{s}^p$ spaces. We modify a little our previously known proofs.

Scarcity of partition congruences on semiprime progressions
Scott Ahlgren, Olivia Beckwith
https://arxiv.org/abs/2508.19512 https://arxiv.org/pdf/2508.1951…

Scarcity of partition congruences on semiprime progressions
In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+β)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results which show that such congruences are scarce in a precise sense. Here we improve one of our results when $r=1$; in particular we prove (outside of trivial cases) that the set of primes $Q$ such that there exists $β\in \mathbb{Z}$ with $p(\ell Q n+β)\equiv 0\pmod \…

Gradient regularity for double-phase orthotropic functionals
Stefano Almi, Chiara Leone, Gianluigi Manzo
https://arxiv.org/abs/2507.18474 https://arxiv.org…

Gradient regularity for double-phase orthotropic functionals
We prove higher integrability for local minimizers of the double-phase orthotropic functional \[ \sum_{i=1}^{n}\int_Ω\left(\left|u_{x_i}\right|^p+a(x)\left| u_{x_i}\right|^q\right)dx \] when the weight function $a \geq0$ is assumed to be $α$-Hölder continuous, while the exponents $p, q$ are such that $2 \leq p \leq q$ and $\frac{q}{p} < 1 + \fracα{n}$. Under natural Sobolev regularity of~$a$, we further obtain explicit Lipschitz regularity estimates for local minimizer…

The stable homology of Hurwitz modules and applications
Aaron Landesman, Ishan Levy
https://arxiv.org/abs/2510.02068 https://arxiv.org/pdf/2510.02068

The stable homology of Hurwitz modules and applications
We show that the homology of modules for Hurwitz spaces stabilizes and compute its stable value. As one consequence, we compute the moments of Selmer groups in quadratic twist families of abelian varieties over suitably large function fields. As a second consequence, we deduce a version of Bhargava's conjecture, counting the number of $S_d$ degree $d$ extensions of $\mathbb F_q(t)$, for suitably large $q$. As a third consequence, we deduce that the homology of Hurwitz spaces associated to racks…

Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials
Tewodros Amdeberhan, Leonid G. Fel, Ken Ono
https://arxiv.org/abs/2507.12352

Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials
Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series $F(X)$. Using the weight 1 form $θ(q)^2$ and $F(X)=e^{X/2}$, we obtain a sequence $\{Y_n(q)\}$ of weight $n$ quasimodular forms on $Γ_0(4)$ whose symmetric functi…

Robin Problems of Elliptic Equations on Rough Domains: H\"older Regularity, Green's Functions, and Harmonic Measures
Jiayi Wang, Dachun Yang, Sibei Yang
https://arxiv.org/abs/2509.23073

Robin Problems of Elliptic Equations on Rough Domains: Hölder Regularity, Green's Functions, and Harmonic Measures
Let $n\ge 2$ and $s\in (n-2,n)$. Assume that $Ω\subset \mathbb{R}^n$ is a one-sided bounded non-tangentially accessible domain with $s$-Ahlfors regular boundary and $σ$ is the surface measure on the boundary of $Ω$, denoted by $\partial Ω$. Let $β$ be a non-negative measurable function on $\partial Ω$ satisfying $β\in L^{q_0}(\partial Ω,σ)~\text{with}~ q_0 \in(\frac{s}{s+2-n},\infty]~\text{and}\ β\ge a_0~\text{on}~E_0\subset \partial Ω, $ where $a_0$ is a given positive constant and …

Hausdorff dimension of the graph of the error-sum function of continued fractions
Min Woong Ahn
https://arxiv.org/abs/2507.16536 https://

Hausdorff dimension of the graph of the error-sum function of continued fractions
We study the unweighted error-sum function $\mathcal{E}(x) \coloneqq \sum_{n \geq 0} ( x- p_n(x)/q_n(x) )$, where $p_n(x)/q_n(x)$ is the $n$th convergent of the continued fraction expansion of $x \in \mathbb{R}$. We prove that the Hausdorff dimension of the graph of $\mathcal{E}$ is exactly $1$. Our proof is number-theoretic in nature and involves Möbius inversion, summation over coprime convergent denominators, and precise upper bounds derived via continued fraction recurrence relations. As a…

Divisibility properties of weighted $k$ regular partitions
Debika Banerjee, Ben Kane
https://arxiv.org/abs/2508.20573 https://arxiv.org/pdf/2508.20573

Divisibility properties of weighted $k$ regular partitions
We study a generalized class of weighted $k$-regular partitions defined by \[ \sum_{n=0}^{\infty} c_{k, r_1, r_2}(n) q^n = \prod_{n=1}^{\infty} \frac{(1 - q^{nk})^{r_1}}{(1 - q^n)^{r_2}}, \] which extends the classical $k$-regular partition function $b_k(n)$. We establish new infinite families of Ramanujan-type congruences, divisibility results, and positive-density prime sets for which $c_{k, r_1, r_2}(n)$ vanishes modulo a given prime. These results generalize recent work on $5$-regular parti…

Falling stars: a fall-decorated rational shuffle theorem
Alessandro Iraci, Roberto Pagaria, Giovanni Paolini
https://arxiv.org/abs/2508.20935 https://arxiv…

Falling stars: a fall-decorated rational shuffle theorem
In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a $(m+k) \times (n+k)$ rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and non-decorated rectangular objects. We prove a symmetric function formula for the $q,t$-generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to $e_{m,n+km}$ and, condi…

Coh zeta functions for inert quadratic orders
Yifeng Huang
https://arxiv.org/abs/2507.21966 https://arxiv.org/pdf/2507.21966…

Coh zeta functions for inert quadratic orders
We study the Coh zeta function for a family of inert quadratic orders, which we conjecture to be given by $t$-deformed Bressoud $q$-series. This completes a trilogy connecting the zeta functions of ramified and split quadratic orders to the classical Andrews--Gordon and Bressoud identities, respectively. We provide strong evidence for this conjecture by deriving the first explicit formulas for the finitized Coh zeta function of the simplest order in the family, and for the $t=1$ specialization …

Effective Brauer-Siegel theorems for Artin $L$-functions
Peter J. Cho, Robert J. Lemke Oliver, Asif Zaman
https://arxiv.org/abs/2510.02309 https://arxiv.or…

Effective Brauer-Siegel theorems for Artin $L$-functions
Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $ζ_K(s)$ and used this to give effective upper and lower bounds on the residue of $ζ_K(s)$ at $s=1$. We extend Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin $L$-functions at $s=1$ that are, up to the value of implied constants, as strong as could reasonably be…

Algorithms for Modular Parametrizations of Elliptic Curves over $\mathbb{Q}$
SanMin Wang
https://arxiv.org/abs/2509.14747 https://arxiv.org/pdf/2509.14747

Algorithms for Modular Parametrizations of Elliptic Curves over $\mathbb{Q}$
Let \( E \) be a complex elliptic curve with conductor \( N \) and modular invariant \( j(E) \in \mathbb{Q} \). We construct a class of modular polynomials $F_N(x,j)$ that relate the modular function $x$ on $X_0(N)$ to the $j$-invariant $j$, where $x$ is obtained by composing the first coordinate function of $E$ with the modular parametrization $φ: X_0(N) \rightarrow E$. Using $F_N(x,j)$, we can precisely determine the poles of $φ$, compute exact values of $φ$ at cusps, and develop an algori…

Permutation of values of irrationality measure functions
Victoria Rudykh
https://arxiv.org/abs/2507.20416 https://arxiv.org/pdf/2507.20416

Permutation of values of irrationality measure functions
For an irrational number $α\in\mathbb{R}$ we consider its irrationality measure function $$ ψ_α(t) = \min_{1\le q\le t,\, q\in\mathbb{Z}} \| qα\|. $$ Let $\boldsymbolα = (α_1, \dots, α_n)$ be $n$-tuple of pairwise independent irrational numbers. For each $t \in \mathbb{R}_{>1}$ irrationality measure functions $ψ_{α_1}, \dots, ψ_{α_n}$ can be written in an increasing order $$ψ_{α_{v_1}}(t) > ψ_{α_{v_2}}(t) > \dots > ψ_{α_{v_{n-1}}}(t) > ψ_{α_{v_n}}(t).$$ We consider …

The imaginary case of the nonabelian Cohen--Lenstra heuristics
Yuan Liu, Ken Willyard
https://arxiv.org/abs/2507.21558 https://arxiv.org/pdf/2507.21558

The imaginary case of the nonabelian Cohen--Lenstra heuristics
For a finite group $Γ$, we study the distribution of the Galois group $G_{\emptyset}^{\#}(K)$ of the maximal unramified extension of $K$ that is split completely at $\infty$ and has degree prime to $|Γ|$ and $\textit{Char}(K)$, as $K$ varies over imaginary $Γ$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. In the function field case, we compute the moments of the distribution of $G_{\emptyset}^{\#}(K)$ by counting points on Hurwitz stacks. In order to understand the probability o…

Integer Partitions With Restricted Distinct Parts
Rinchin Drema, Nipen Saikia
https://arxiv.org/abs/2508.12739 https://arxiv.org/pdf/2508.12739

Integer Partitions With Restricted Distinct Parts
For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for $Q_{t}^{s}(n)$ for some particular values of $s$ and $t$ by employing $q$-series and theta function identities.

Digital functions along Squares of Prime Numbers
Michael Drmota, Jo\"el Rivat
https://arxiv.org/abs/2509.17474 https://arxiv.org/pdf/2509.17474…

Digital functions along Squares of Prime Numbers
In the last 20 years the Gelfond conjectures concerning the well distribution of the sum-of-digits function along prime numbers and along squares have been solved and these results, which are strongly connected with the Sarnak conjecture, were generalized to $q$-multiplicative functions and automatic sequences. In this paper we study a combination of both challenges and prove a Prime Number Theorem for $q$-multiplicative functions along squares which can be rewritten into a well distribution re…

Partitions and Hecke images
Florian Breuer, Fabien Pazuki
https://arxiv.org/abs/2509.21603 https://arxiv.org/pdf/2509.21603

Partitions and Hecke images
We obtain a new family of relations satisfied by the partition function. In contrast with most partition relations, these involve non-trivial roots of unity. We present two proofs, one using the fact that the discriminant modular form is a multiplicative Hecke eigenform, and one direct proof using q-series.

Analytic ranks of twists of Carlitz modules -- a survey of results
A. Grishkov, D. Logachev
https://arxiv.org/abs/2509.16160 https://arxiv.org/pdf/2509.161…

Analytic ranks of twists of Carlitz modules -- a survey of results
We give in this paper a survey of results obtained in our earlier papers, and state explicitly some problems of further research, for example: are the analytic ranks bounded, or not? Twists of Carlitz modules are parametrized by polynomials over finite fields $\Bbb F_q$. The analytic rank of a twist is the order of zero of its L-function at a point. The set of polynomials of degree $\le m$ such that the analytic ranks of the corresponding twists are $\ge i$ is $X(m,i)(\Bbb F_q)$ where $X(m,i)$ …