Tootfinder

Continued fractions with large prime partial quotients
Gerardo Gonz\'alez Robert, Mumtaz Hussain, Benjamin Ward, Lauren White
https://arxiv.org/abs/2507.17167

Continued fractions with large prime partial quotients
We determine the Lebesgue measure and Hausdorff dimension of various sets of real numbers with infinitely many partial quotients that are both large and prime, thus extending the well-known theorems by Łuczak (1997) and Huang-Wu-Xu (2020). To this end, we obtain new asymptotics on the tail end of the almost prime zeta function. Our results include some recent work by Schindler-Zweim{ü}ller (2023).

The higher order partial derivatives of Okamoto's function with respect to the parameter
Pieter Allaart, Nathan Dalaklis, Kiko Kawamura, Matthew Ortiz, Jiajie Zheng
https://arxiv.org/abs/2506.17737

The higher order partial derivatives of Okamoto's function with respect to the parameter
Let $\{F_a: a\in(0,1)\}$ be Okamoto's family of continuous self-affine functions, introduced in [{\em Proc. Japan Acad. Ser. A Math. Sci.} {\bf 81} (2005), no. 3, 47--50]. This family includes well-known ``pathological" examples such as Cantor's devil's staircase and Perkins' continuous but nowhere differentiable function. It is well known that $F_a(x)$ is real analytic in $a$ for every $x\in[0,1]$. We introduce the functions \[ M_{k,a}(x):=\frac{\partial^k}{\partial a^k}F_a(x), \qquad k\in\mat…

PICore: Physics-Informed Unsupervised Coreset Selection for Data Efficient Neural Operator Training
Anirudh Satheesh, Anant Khandelwal, Mucong Ding, Radu Balan
https://arxiv.org/abs/2507.17151

PICore: Physics-Informed Unsupervised Coreset Selection for Data Efficient Neural Operator Training
Neural operators offer a powerful paradigm for solving partial differential equations (PDEs) that cannot be solved analytically by learning mappings between function spaces. However, there are two main bottlenecks in training neural operators: they require a significant amount of training data to learn these mappings, and this data needs to be labeled, which can only be accessed via expensive simulations with numerical solvers. To alleviate both of these issues simultaneously, we propose PICore…

Partitioning the electronic wave function using deep variational Monte Carlo
Mat\v{e}j Mezera, Paolo A. Erdman, Zeno Sch\"atzle, P. Bern\'at Szab\'o, Frank No\'e
https://arxiv.org/abs/2506.18574

Partitioning the electronic wave function using deep variational Monte Carlo
We propose a novel wave function partitioning method that integrates deep-learning variational Monte Carlo with ansätze based on generalized product functions. This approach effectively separates electronic wave functions (WFs) into multiple partial WFs representing, for example, the core and valence domains or different electronic shells. Although our ansätze do not explicitly include correlations between individual electron groups, we show that they accurately reproduce the underlying physi…

DDFEM: A Python Package for Diffuse Domain Methods
Luke Benfield, Andreas Dedner
https://arxiv.org/abs/2507.16964 https://arxiv.org/pdf/2507.16964

DDFEM: A Python Package for Diffuse Domain Methods
Solving partial differential equations (PDEs) on complex domains can present significant computational challenges. The Diffuse Domain Method (DDM) is an alternative that reformulates the partial differential equations on a larger, simpler domain. The original geometry is embedded into the problem by representing it with a phase-field function. This paper introduces ddfem, an extensible Python package to provide a framework for transforming PDEs into a Diffuse Domain formulation. We aim to make …

The $\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\frac{n}{2}$ and beyond
Jonah A. J. Duncan, Luc Nguyen
https://arxiv.org/abs/2507.16394

The $σ_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\frac{n}{2}$ and beyond
Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with smooth non-empty boundary $\partial M$. Let $Γ\subset\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $Γ$ satisfying standard assumptions. Denoting by $A_{g_u}$ the Schouten tensor of a conformal metric $g_u = u^{-2}g_0$, we show that the associated fully nonlinear Loewner-Nirenberg problem \begin{align*} \begin{cases} f(λ(-g_u^{-1}A_{g_u})) = \frac{1}{2}, \quad λ(-g_u…

Learning Iterated Function Systems from Time Series of Partial Observations
Emilia Gibson, Jeroen S. W. Lamb
https://arxiv.org/abs/2508.13794 https://arxiv…

Learning Iterated Function Systems from Time Series of Partial Observations
We develop a methodology to learn finitely generated random iterated function systems from time-series of partial observations using delay embeddings. We obtain a minimal model representation for the observed dynamics, using a hidden variable representation, that is diffeomorphic to the original system.

Homomorphism Calculus for User-Defined Aggregations
Ziteng Wang, Ruijie Fang, Linus Zheng, Dixin Tang, Isil Dillig
https://arxiv.org/abs/2508.15109 https://

Homomorphism Calculus for User-Defined Aggregations
Data processing frameworks like Apache Spark and Flink provide built-in support for user-defined aggregation functions (UDAFs), enabling the integration of domain-specific logic. However, for these frameworks to support \emph{efficient} UDAF execution, the function needs to satisfy a \emph{homomorphism property}, which ensures that partial results from independent computations can be merged correctly. Motivated by this problem, this paper introduces a novel \emph{homomorphism calculus} that can…

Pricing Options on Forwards in Function-Valued Affine Stochastic Volatility Models
Jian He, Sven Karbach, Asma Khedher
https://arxiv.org/abs/2508.14813 https://

Pricing Options on Forwards in Function-Valued Affine Stochastic Volatility Models
We study the pricing of European-style options written on forward contracts within function-valued infinite-dimensional affine stochastic volatility models. The dynamics of the underlying forward price curves are modeled within the Heath-Jarrow-Morton-Musiela framework as solution to a stochastic partial differential equation modulated by a stochastic volatility process. We analyze two classes of affine stochastic volatility models: (i) a Gaussian model governed by a finite-rank Wishart process…

BGD domains in p.c.f. self-similar sets II: spectral asymptotics for Laplacians
Qingsong Gu, Hua Qiu
https://arxiv.org/abs/2507.14858 https://

BGD domains in p.c.f. self-similar sets II: spectral asymptotics for Laplacians
Let $K$ be a p.c.f. self-similar set equipped with a strongly recurrent Dirichlet form. Under a homogeneity assumption, for an open set $Ω\subset K$ whose boundary $\partial Ω$ is a graph-directed self-similar set, we prove that the eigenvalue counting function $ρ^Ω(x)$ of the Laplacian with Dirichlet or Neumann boundary conditions (Neumann only for connected $Ω$) has an explicit second term as $x\to +\infty$, beyond the dominant Weyl term. If $\partialΩ$ has a strong iterated structure, …

Concepts for Composing Finite Element Function Space Bases
Christian Engwer, Carsten Gr\"aser, Steffen M\"uthing, Simon Praetorius, Oliver Sander
https://arxiv.org/abs/2508.10125

Concepts for Composing Finite Element Function Space Bases
Finite Element discretizations of coupled multi-physics partial differential equation models require the handling of composed function spaces. In this paper we discuss software concepts and abstractions to handle the composition of function spaces, based on a representation of product spaces as trees of simpler bases. From this description, many different numberings of degrees of freedom by multi-indices can be derived in a natural way, allowing to adapt the function spaces to very different da…

On global regular axially-symmetric solution to the Navier-Stokes equations in a cylinder
Wies{\l}aw J. Grygierzec, Wojciech M. Zaj\k{a}czkowski
https://arxiv.org/abs/2507.14964

On global regular axially-symmetric solution to the Navier-Stokes equations in a cylinder
We consider the axisymmetric Navier-Stokes equations in a finite cylinder $Ω\subset\mathbb{R}^3$. We assume that $v_r$, $v_φ$, $ω_φ$ vanish on the lateral part of boundary $\partialΩ$ of the cylinder, and that $v_z$, $ω_φ$, $\partial_zv_φ$ vanish on the top and bottom parts of the boundary $\partialΩ$, where we used standard cylindrical coordinates, and we denoted by $ω={\rm curl}\, v$ the vorticity field. We use $H^3$ Sobolev estimates for the modified stream function (stream functio…

Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops
Wen Chang, Sibylle Schroll
https://arxiv.org/abs/2508.14860 https://

Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops
We prove that the entropy of the Serre functor $\mathbb{S}$ in the partially wrapped Fukaya category of a graded surface $Σ$ with stops is given by the function sending $t \in \mathbb{R}$ to $ h_t(\mathbb{S}) = (1-\min Ω)t$, for $t\geq 0$, and to $h_t(\mathbb{S})=(1-\max Ω)t$, for $t\leq 0$, where $Ω= \{\frac{ω_1}{m_1} \ldots, \frac{ω_b}{m_b},0\}$, and $ω_i$ is the winding number of the $i$th boundary component $\partial_iΣ$ of the surface with $b$ boundary components and $m_i$ stops on…

Warped products over one-dimensional base spaces and the RCD condition
Christian Ketterer
https://arxiv.org/abs/2506.10809 https://ar…

Warped products over one-dimensional base spaces and the RCD condition
We prove the Riemannian curvature-dimension condition $\mathsf{RCD}(KN,N+1)$ for an $N$-warped product $B\times_f^N F$ over a one-dimensional base space $B$ with a Lipschitz function $f: B\rightarrow \mathbb R_{\geq 0}$, provided (1) $f$ is a $Kf$-concave function, (2) $f$ satisfies a sub-Neumann boundary condition $\frac{\partial f}{\partial n}\geq 0$ on $\partial B\backslash f^{-1}(0)$ and $F$ is a compact metric measure space satisfying (3) the condition $\mathsf{RCD}(K_F (N-1), N)$ with $K_…

Learning from Preferences and Mixed Demonstrations in General Settings
Jason R Brown, Carl Henrik Ek, Robert D Mullins
https://arxiv.org/abs/2508.14027 https://

Learning from Preferences and Mixed Demonstrations in General Settings
Reinforcement learning is a general method for learning in sequential settings, but it can often be difficult to specify a good reward function when the task is complex. In these cases, preference feedback or expert demonstrations can be used instead. However, existing approaches utilising both together are often ad-hoc, rely on domain-specific properties, or won't scale. We develop a new framing for learning from human data, \emph{reward-rational partial orderings over observations}, designed …

Physics-inspired neural networks as quasi inverse of quantum channels
Sameen Aziz, Muhammad Faryad
https://arxiv.org/abs/2506.11716 https://

Physics-inspired neural networks as quasi inverse of quantum channels
Quantum channels are not invertible in general. A quasi-inverse allows for a partial recovery of the input state, but its analytical results are found only in a restricted space of its parameters. This work explores the potential of neural networks to find the quasi-inverse of qubit channels for any values of the channel parameters while keeping the quasi-inverse as a physically realizable quantum operation. We introduce a physics-inspired loss function based on the mean of the square of the mo…

Characterizing circle graphs with binomial partial Petrial polynomials
Ruiqing Feng, Qi Yan, Xuan Zheng
https://arxiv.org/abs/2507.02421 https://

Characterizing circle graphs with binomial partial Petrial polynomials
The partial Petrial polynomial was first introduced by Gross, Mansour, and Tucker as a generating function that enumerates the Euler genera of all possible partial Petrials on a ribbon graph. Yan and Li later extended this polynomial invariant to circle graphs by utilizing the correspondence between circle graphs and bouquets. Their explicit computation demonstrated that paths produce binomial polynomials, specifically those containing exactly two non-zero terms. This discovery led them to pose…

MPC-based Coarse-to-Fine Motion Planning for Robotic Object Transportation in Cluttered Environments
Chen Cai, Ernesto Dickel Saraiva, Ya-jun Pan, Steven Liu
https://arxiv.org/abs/2507.11211

MPC-based Coarse-to-Fine Motion Planning for Robotic Object Transportation in Cluttered Environments
This letter presents a novel coarse-to-fine motion planning framework for robotic manipulation in cluttered, unmodeled environments. The system integrates a dual-camera perception setup with a B-spline-based model predictive control (MPC) scheme. Initially, the planner generates feasible global trajectories from partial and uncertain observations. As new visual data are incrementally fused, both the environment model and motion planning are progressively refined. A vision-based cost function pr…

Measurement of the Born cross section for $e^ e^- \to p K^- K^- \bar{\Xi}^ $ at $\sqrt{s} =$ 3.5-4.9 GeV
BESIII Collaboration, et al.
https://arxiv.org/abs/2508.11276 https://…

Measurement of the Born cross section for $e^+e^- \to p K^- K^- \barΞ^+$ at $\sqrt{s} =$ 3.5-4.9 GeV
Using $e^+ e^-$ collision data corresponding to a total integrated luminosity of 20 ${\rm fb}^{-1}$ collected with the BESIII detector at the BEPCII collider, we present a measurement of the Born cross section for the process $e^+e^- \to p K^-K^-\barΞ^{+}$ at 39 center-of-mass energies between 3.5 and 4.9 GeV with a partial reconstruction technique. By performing a fit to the dressed cross section of $e^{+}e^{-}\to p K^- K^-\barΞ^{+}$ with a power law function for continuum production and one…

Stochastic intrinsic gradient flows on the Wasserstein space
Panpan Ren, Michael R\"ockner, Feng-Yu Wang, Simon Wittmann
https://arxiv.org/abs/2506.12755

Stochastic intrinsic gradient flows on the Wasserstein space
We construct stochastic gradient flows on the $2$-Wasserstein space $\mathcal P_2$ over $\mathbb R^d$ for energy functionals of the type $W_F(ρd x)=\int_{\mathbb R^d}F(x,ρ(x))d x$. The functions $F$ and $\partial_2 F$ are assumed to be locally Lipschitz on $\mathbb R^d\times (0,\infty)$. This includes the relevant examples of $W_F$ as the entropy functional or more generally the Lyapunov function of generalized porous media equations. First, we define a class of Gaussian-based measures $Λ$ o…

Generalization of Gaussian third-order Jacobsthal numbers and their new families
Gamaliel Morales
https://arxiv.org/abs/2508.11650 https://arxiv.org/pdf/25…

Generalization of Gaussian third-order Jacobsthal numbers and their new families
In this study, we introduce the generalized Gaussian third-order Jacobsthal numbers with arbitrary initial values and discuss two particular cases, namely, Gaussian third-order Jacobsthal and Gaussian modified third-order Jacobsthal numbers. In this paper we discuss several of its algebraic properties such as Binet's formula, partial sum, generating function, negative subscript elements, d'Ocagne's and Cassini's identities. Furthermore, we study and introduce a new generalization of this sequen…

Multiple sums with the M\"obius function
William D. Banks, Igor E. Shparlinski
https://arxiv.org/abs/2506.08787 https://arxiv.or…

Multiple sums with the Möbius function
We establish nontrivial bounds for bilinear sums involving the Möbius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as Möbius-function analogues of the ternary Goldbach problem. By contrast, the binary versions of our results remain out of reach, much like the binary Goldbach problem. Nevertheless, we make partial progress in this direction by restricting the range of the third variable as far as possible.

On data-driven robust distortion risk measures for non-negative risks with partial information
Xiangyu Han, Yijun Hu, Ran Wang, Linxiao Wei
https://arxiv.org/abs/2508.10682 http…

On data-driven robust distortion risk measures for non-negative risks with partial information
In this paper, by proposing two new kinds of distributional uncertainty sets, we explore robustness of distortion risk measures against distributional uncertainty. To be precise, we first consider a distributional uncertainty set which is characterized solely by a ball determined by general Wasserstein distance centered at certain empirical distribution function, and then further consider additional constraints of known first moment and any other higher moment of the underlying loss distributio…

Solutions And Gradient Of The Conformal Ricci Bourguignon Soliton On Vaidya Spacetime
Ayaan Abdur Rehman, Narayan S Iyer, Naeem Ahmed Pundeer
https://arxiv.org/abs/2508.09985 ht…

Solutions And Gradient Of The Conformal Ricci Bourguignon Soliton On Vaidya Spacetime
In this work, we derive the complete and explicit solution for the conformal Ricci-Bourguignon soliton on Vaidya spacetime. We provide the closed-form expression for the vector field and establish the necessary conditions for the existence of the scalar potential, for which we also derive an explicit form. Our solution to the underlying system of linear partial differential equations proves that such solitons exist if and only if the mass function vanishes, forcing the metric to reduce to flat …

Stochastic parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function
Piotr Gwiazda, Jakub Wo\'znicki, Aneta Wr\'oblewska-Kami\'nska, Aleksandra Zimmermann
https://arxiv.org/abs/2506.13305

Stochastic parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function
We consider a stochastic parabolic partial differential equation with Dirichlet boundary conditions, multiplicative stochastic noise, and a monotone parabolic operator A. The growth and coercivity of A is controlled by a general N-function M, which depends on time, and spatial variable, but we do not assume any regularity with respect to the former. We show the existence of weak solutions to such system. As auxiliary result, we also provide the proof for the Itô's formula in Orlicz spaces. Thi…

LevelSetPy: A GPU-Accelerated Package for Hyperbolic Hamilton-Jacobi Partial Differential Equations
Lekan Molu
https://arxiv.org/abs/2507.11542 https://

LevelSetPy: A GPU-Accelerated Package for Hyperbolic Hamilton-Jacobi Partial Differential Equations
This article introduces a software package release for geometrically reasoning about the \textit{safety} desiderata of (complex) dynamical systems via level set methods. In emphasis, safety is analyzed with Hamilton-Jacobi equations. In scope, we provide implementations of numerical algorithms for the resolution of Hamilton-Jacobi-Isaacs equations: the spatial derivatives of the associated value function via upwinding, the Hamiltonian via Lax-Friedrichs schemes, and the integration of the Hamil…

On Higher Order Busy Beaver Function
Zining Cao
https://arxiv.org/abs/2507.20321 https://arxiv.org/pdf/2507.20321…

On Higher Order Busy Beaver Function
In this paper, we extend Busy Beaver function to a class of higher order Busy Beaver functions based on Turing oracle machine. We prove some results about the relation between decidability of number theoretical formula and higher order Busy Beaver functions, and the relation between computability of max-min partial recursive functions and higher order Busy Beaver functions. We also present some conjectures on higher order Busy Beaver functions.

Neural Actor-Critic Methods for Hamilton-Jacobi-Bellman PDEs: Asymptotic Analysis and Numerical Studies
Samuel N. Cohen, Jackson Hebner, Deqing Jiang, Justin Sirignano
https://arxiv.org/abs/2507.06428

Neural Actor-Critic Methods for Hamilton-Jacobi-Bellman PDEs: Asymptotic Analysis and Numerical Studies
We mathematically analyze and numerically study an actor-critic machine learning algorithm for solving high-dimensional Hamilton-Jacobi-Bellman (HJB) partial differential equations from stochastic control theory. The architecture of the critic (the estimator for the value function) is structured so that the boundary condition is always perfectly satisfied (rather than being included in the training loss) and utilizes a biased gradient which reduces computational cost. The actor (the estimator f…

Prokhorov Metric Convergence of the Partial Sum Process for Reconstructed Functional Data
Tim Kutta, Piotr Kokoszka
https://arxiv.org/abs/2506.21172 https:…

Prokhorov Metric Convergence of the Partial Sum Process for Reconstructed Functional Data
Motivated by applications in functional data analysis, we study the partial sum process of sparsely observed, random functions. A key novelty of our analysis are bounds for the distributional distance between the limit Brownian motion and the entire partial sum process in the function space. To measure the distance between distributions, we employ the Prokhorov and Wasserstein metrics. We show that these bounds have important probabilistic implications, including strong invariance principles an…

Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric L\'evy noise
Juan J. Jim\'enez
https://arxiv.org/abs/2507.12656 https…

Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise
In this article, we investigate the existence and uniqueness of random-field solutions to the elliptic SPDE $-\mathcal{L}u=\dotξ$ on a bounded domain $D$ with Dirichlet boundary conditions $u=0$ on $\partial D$, driven by symmetric Lévy noise $\dotξ$. Under general sufficient conditions on the coefficients of the second-order operator $\mathcal{L}$, we prove the existence of a mild solution via the corresponding Green's function and show that the same framework applies to the spectral fracti…

Computability of a Whitney Extension
Andrea Brun, Guido Gherardi, Alberto Marcone
https://arxiv.org/abs/2507.02113 https://arxiv.org/…

Computability of a Whitney Extension
We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if $F \subseteq \mathbb{R}^n$ is a closed set represented so that the distance function $x \mapsto d(x,F)$ can be computed, and $(f^{(\bar{k})})_{|\bar{k}| \le m}$ is a Whitney jet of order $m$ on $F$, then we can compute $g \in C^{m}(\mathbb{R}^n)$ such that $g$ and its partial derivatives coincide on $F$ with the corresponding functions of $(f^{(\bar{k})})_{|…

This https://arxiv.org/abs/2311.07309 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Non-singular extensions of circle-valued Morse functions
In this paper, we consider the non-singular extension problem for circle-valued Morse functions on closed orientable surfaces. The problem asks, given a circle-valued Morse function $f\colon M\to S^{1}$ on a closed orientable surface $M$, under what condition there exist a compact orientable 3-dimensional manifold $N$ with $\partial N = M$ and a submersion $G\colon N \to S^{1}$ such that $G|_{\partial N}=f$. We provide necessary and sufficient conditions for the existence of a non-singular…

Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations
Difeng Cai, Paulina Sep\'ulveda
https://arxiv.org/abs/2506.23344

Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations
The appearance of singularities in the function of interest constitutes a fundamental challenge in scientific computing. It can significantly undermine the effectiveness of numerical schemes for function approximation, numerical integration, and the solution of partial differential equations (PDEs), etc. The problem becomes more sophisticated if the location of the singularity is unknown, which is often encountered in solving PDEs. Detecting the singularity is therefore critical for developing …

Functional analysis and partial differential equations in spectral Barron spaces
Mourad Choulli, Shuai Lu, Hiroshi Takase
https://arxiv.org/abs/2507.06778 …

Functional analysis and partial differential equations in spectral Barron spaces
Spectral Barron spaces, constituting a specialized class of function spaces that serve as an interdisciplinary bridge between mathematical analysis, partial differential equations (PDEs), and machine learning, are distinguished by the decay profiles of their Fourier transform. In this work, we shift from conventional numerical approximation frameworks to explore advanced functional analysis and PDE theoretic perspectives within these spaces. Specifically, we present a rigorous characterization …

A degree-counting formula for a Keller-Segel equation on a surface with boundary
Mohameden Ahmedou, Zhengni Hu, Heming Wang
https://arxiv.org/abs/2506.12783

A degree-counting formula for a Keller-Segel equation on a surface with boundary
In this paper, we consider the following Keller-Segel equation on a compact Riemann surface $(Σ, g)$ with smooth boundary $\partialΣ$: \[ -Δ_g u = ρ\Big(\frac{V e^u}{\int_Σ V e^u \mathrm{d} v_g} - \frac{1}{|Σ|_g}\Big) \text{ in } Σ, \quad \text{ with } \partial_{ν_g} u = 0 \text{ on } \partial Σ, \] where $V$ is a smooth positive function on $Σ$ and $ρ> 0$ is a parameter. We perform a refined blow-up analysis of bubbling solutions and establish sharper a priori esti…

Real non-attractive fixed point conjecture for complex harmonic functions
Mohd Vaseem
https://arxiv.org/abs/2507.18414 https://arxiv.org/pdf/2507.18414

Real non-attractive fixed point conjecture for complex harmonic functions
We prove the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions. A harmonic function $f=h+\overline{g}$ is polynomial (rational) if both $h$ and $g$ are polynomials (rational functions) of degree at least 2. We show that every such function with a super-attracting fixed point has a $\mathfrak{h}$-fixed point $ζ=μ+\overlineω$ such that the real parts of its multipliers satisfy $\text{Re}(\partial_z h(μ)) \geq 1$ and $\text{Re}(\partial_z g(ω)) …

This https://arxiv.org/abs/2501.07514 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_eco…

Estimating Sequential Search Models Based on a Partial Ranking Representation
The rapid growth of online shopping has made consumer search data increasingly available, opening up new possibilities for empirical research. Sequential search models offer a structured approach for analyzing such data, but their estimation remains difficult. This is because consumers make optimal decisions based on private information revealed in search, which is not observed in typical data. As a result, the model's likelihood function involves high-dimensional integrals that require intensi…

This https://arxiv.org/abs/2503.02041 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csCE_…

Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems
Deep learning has been extensively employed as a powerful function approximator for modeling physics-based problems described by partial differential equations (PDEs). Despite their popularity, standard deep learning models often demand prohibitively large computational resources and yield limited accuracy when scaling to large-scale, high-dimensional physical problems. Their black-box nature further hinders the application in industrial problems where interpretability and high precision are cr…

S-shaped Utility Maximization with VaR Constraint and Partial Information
Dongmei Zhu, Ashley Davey, Harry Zheng
https://arxiv.org/abs/2506.10103 https://

S-shaped Utility Maximization with VaR Constraint and Partial Information
We study S-shaped utility maximisation with VaR constraint and unobservable drift coefficient. Using the Bayesian filter, the concavification principle, and the change of measure, we give a semi-closed integral representation for the dual value function and find a critical wealth level that determines if the constrained problem admits a unique optimal solution and Lagrange multiplier or is infeasible. We also propose three algorithms (Lagrange, simulation, deep neural network) to solve the prob…

Gradient properties of $\varphi^3$ in $d=6-\varepsilon$
Lorenzo Benfatto, Omar Zanusso
https://arxiv.org/abs/2507.20761 https://arxiv.org/pdf/2507.20761

Gradient properties of $φ^3$ in $d=6-\varepsilon$
The renormalization group flow of the multiscalar interacting $φ^3$ theory in $d=6$ dimensions is known to have a gradient structure, in which suitable generalizations of the beta functions $B^{I}$ emerge as the gradient of a scalar function $A$, $\partial_I A = T_{IJ} B^J $, with a nontrivial tensor $T_{IJ}$ in the space of couplings. This has been shown directly to three loops in schemes such as $\overline{\rm MS}$ and can be argued in general by identifying $A$ with the coefficient of the t…

The hydrodynamic thermal continuum, with applications to stratified atmospheres and 1D coronal loop models
Rony Keppens, Jordi De Jonghe, Adrian Kelly, Nicolas Brughmans, Hans Goedbloed
https://arxiv.org/abs/2506.23591

The hydrodynamic thermal continuum, with applications to stratified atmospheres and 1D coronal loop models
Using both analytical and numerical means, we demonstrate that linear stability analysis of a hydrodynamic stratified atmosphere or a 1D coronal loop model in non-adiabatic settings features a thermal continuum corresponding to highly localized eigenfunctions. This thermal continuum can be precomputed, involving the net heat-loss function and its partial derivatives, and is the generalization of the thermal instability introduced by~\citet{Parker1953}. We account for a thermal imbalance, direct…

Partial Information for Inverse Spectral Uniqueness in Vibration System with Multiple Frozen Arguments
Lung-Hui Chen
https://arxiv.org/abs/2507.20184 https://

Partial Information for Inverse Spectral Uniqueness in Vibration System with Multiple Frozen Arguments
In this paper, we investigate the inverse spectral problem of the Sturm-Liouville operator with many frozen arguments fixed at the points $\{a_{1}, a_{2},\ldots,a_{N}\}$ in $(0,π)$. We start with counting the zeros or the eigenvalues of characteristic function, and then discuss how certain information provided a priori on the point set $\{a_{1}, a_{2},\ldots,a_{N}\}$ would affect the uniqueness or non-uniqueness of this vibration system with many frozen points. The knowledge at the frozen or r…

Enhanced Thermodynamic Framework for Third-Order Galaxy Correlation Functions: From Theory to Observations
Sameer Choudhary, Naseer Iqbal
https://arxiv.org/abs/2506.23829

Enhanced Thermodynamic Framework for Third-Order Galaxy Correlation Functions: From Theory to Observations
We present a comprehensive thermodynamic derivation and observational validation of the third-order galaxy correlation function, extending the seminal work of Iqbal et al [ 1, 2]. Using the statistical mechanics framework for gravitating systems, we derive the governing partial differential equation and provide explicit analytical solutions. Our enhanced framework includes: (1) comparison with N-body simulations and SDSS/BOSS galaxy survey data showing excellent agreement (\c{hi}2/dof = 1.27), …

Expansion of a bivariate symmetric mean in the neighborhood of the first bisector
Bakir Farhi
https://arxiv.org/abs/2506.07601 https://

Expansion of a bivariate symmetric mean in the neighborhood of the first bisector
In this paper, we investigate the behavior of a bivariate mean $M$ near the first bisector by establishing, in several significant cases, an important expansion of $M$ derived from the Taylor expansion of a single-variable function. These expansions are made explicit for a number of classical means. This motivates the introduction of the concept of the characteristic function $Q_M$ of a mean $M$, defined as the second partial derivative of $M$ with respect to its first variable, evaluated along…

Modeling turbulent and self-gravitating fluids with Fourier neural operators
Keith Poletti, Stella S. R. Offner, Rachel A. Ward
https://arxiv.org/abs/2507.23662 https://

Modeling turbulent and self-gravitating fluids with Fourier neural operators
Neural Operators (NOs) are a leading method for surrogate modeling of partial differential equations. Unlike traditional neural networks, which approximate individual functions, NOs learn the mappings between function spaces. While NOs have been predominantly tested on simplified 1D and 2D problems, such as those explored in prior works, these studies fail to address the complexities of more realistic, high-dimensional, and high-dynamic range systems. Moreover, many real-world applications invo…

Extensions of Schr\"odinger operators that generate $C_0$ contraction semigroups
Lawrence Frolov
https://arxiv.org/abs/2506.00231 https://

Extensions of Schrödinger operators that generate $C_0$ contraction semigroups
Consider a non-relativistic quantum particle with wave function $ψ$ in a bounded $C^2$ region $Ω\subset \mathbb{R}^n$, and suppose detectors are placed along the boundary $\partial Ω$. Assume the detection process is irreversible, its mechanism is time independent and also hard, i.e., detections occur only along the boundary $\partial Ω$. Under these conditions Tumulka argued that the dynamics of $ψ$ must be governed by a $C_0$ contraction semigroup that weakly solves the Schrödinger equa…

The Glider Equation for Asymptotic Lenia
Hiroki Kojima, Ivan Yevenko, Takashi Ikegami
https://arxiv.org/abs/2508.04167 https://arxiv.org/pdf/2508.04167

The Glider Equation for Asymptotic Lenia
Lenia is a continuous extension of Conway's Game of Life that exhibits rich pattern formations including self-propelling structures called gliders. In this paper, we focus on Asymptotic Lenia, a variant formulated as partial differential equations. By utilizing this mathematical formulation, we analytically derive the conditions for glider patterns, which we term the ``Glider Equation.'' We demonstrate that by using this equation as a loss function, gradient descent methods can successfully dis…

Energy function approximations for differential algebraic polynomial systems of Stokes-type
Hamza Adjerid, Jeff Borggaard
https://arxiv.org/abs/2507.18909 https://

Energy function approximations for differential algebraic polynomial systems of Stokes-type
Energy functions are generalizations of controllability and observability Gramians to nonlinear systems and as such find applications in both nonlinear balanced truncation and feedback control. These energy functions are solutions to the Hamilton-Jacobi-Bellman (HJB) equations partial differential equations defined over spatial dimensions determined by the number of state variables in the nonlinear system. Thus, they cannot be resolved with local basis functions for problems of even modest dime…

Gradient regularity for widely degenerate elliptic partial differential equations
Michael Strunk
https://arxiv.org/abs/2506.11708 https://

Gradient regularity for widely degenerate elliptic partial differential equations
In this paper, we investigate the regularity of weak solutions $u\colonΩ\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $Ω$}, \end{equation*} whose ellipticity degenerates in a fixed bounded and convex set $E\subset\mathbb{R}^n$ with $0\in \mathrm{Int}\, E$. Here, $Ω\subset\mathbb{R}^n$ denotes a bounded domain, and $\mathcal{F} \colon Ω\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ is a function with the properties:…

This https://arxiv.org/abs/2409.15683 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qu…

Quantum DeepONet: Neural operators accelerated by quantum computing
In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as deep operator network (DeepONet), which learn mappings between infinite-dimensional function spaces, promise efficient computation of PDE solutions for a new condition in a single forward pass. However, classical DeepONet entails quadratic complexity concerni…

Sound Field Reconstruction Using Physics-Informed Boundary Integral Networks
Stefano Damiano, Toon van Waterschoot
https://arxiv.org/abs/2506.03917 https:/…

Sound Field Reconstruction Using Physics-Informed Boundary Integral Networks
Sound field reconstruction refers to the problem of estimating the acoustic pressure field over an arbitrary region of space, using only a limited set of measurements. Physics-informed neural networks have been adopted to solve the problem by incorporating in the training loss function the governing partial differential equation, either the Helmholtz or the wave equation. In this work, we introduce a boundary integral network for sound field reconstruction. Relying on the Kirchhoff-Helmholtz bo…

Modified L\"uscher zeta-function and the modified effective range expansion in the presence of a long-range force
Rishabh Bubna, Hans-Werner Hammer, Bai-Long Hoid, Jin-Yi Pang, Akaki Rusetsky, Jia-Jun Wu
https://arxiv.org/abs/2507.18399

Modified Lüscher zeta-function and the modified effective range expansion in the presence of a long-range force
An efficient numerical algoritm is proposed for the calculation of the modified Lüscher zeta-function in the presence of a long-range force. Using the formalism developed in Ref.~\cite{Bubna:2024izx} for the analysis of synthetic data on the finite-volume energy levels in a toy model, it is demonstrated that, in contrast to the standard Lüscher approach, the truncation of the higher partial waves has very little effect on the final result. Furthermore, the regularization and renormalization o…

Counting edges of different types in a local graph of a Grassmann graph
Ian Seong
https://arxiv.org/abs/2506.18700 https://arxiv.org/…

Counting edges of different types in a local graph of a Grassmann graph
Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick vertices $x,y$ of $J_q(n,k)$ such that $1<\partial(x,y…

Nonparametric Bayesian Inference for Stochastic Reaction-Diffusion Equations
Randolf Altmeyer, Sascha Gaudlitz
https://arxiv.org/abs/2507.06857 https://

Nonparametric Bayesian Inference for Stochastic Reaction-Diffusion Equations
We consider the Bayesian nonparametric estimation of a nonlinear reaction function in a reaction-diffusion stochastic partial differential equation (SPDE). The likelihood is well-defined and tractable by the infinite-dimensional Girsanov theorem, and the posterior distribution is analysed in the growing domain asymptotic. Based on a Gaussian wavelet prior, the contraction of the posterior distribution around the truth at the minimax optimal rate is proved. The analysis of the posterior distribu…

On Ricci Solitons with Isoparametric Potential Functions
Hung Tran, Kazuo Yamazaki
https://arxiv.org/abs/2506.03545 https://arxiv.org…

On Ricci Solitons with Isoparametric Potential Functions
This paper studies a complete gradient Ricci soliton with an isoparametric potential function. Our first theorem asserts that, for the steady case, there is a critical level set of codimension greater than one. This is consistent with construction of cohomogeneity one models with singular orbits. There is a partial result for the shrinking case. We also study a particular ansatz of popular interest and obtain asymptotic behaviors.

PDE-aware Optimizer for Physics-informed Neural Networks
Hardik Shukla, Manurag Khullar, Vismay Churiwala
https://arxiv.org/abs/2507.08118 https://arxiv.org/pdf/2507.08118 https://arxiv.org/html/2507.08118
arXiv:2507.08118v1 Announce Type: new
Abstract: Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical constraints into the loss function. However, standard optimizers such as Adam often struggle to balance competing loss terms, particularly in stiff or ill-conditioned systems. In this work, we propose a PDE-aware optimizer that adapts parameter updates based on the variance of per-sample PDE residual gradients. This method addresses gradient misalignment without incurring the heavy computational costs of second-order optimizers such as SOAP. We benchmark the PDE-aware optimizer against Adam and SOAP on 1D Burgers', Allen-Cahn and Korteweg-de Vries(KdV) equations. Across both PDEs, the PDE-aware optimizer achieves smoother convergence and lower absolute errors, particularly in regions with sharp gradients. Our results demonstrate the effectiveness of PDE residual-aware adaptivity in enhancing stability in PINNs training. While promising, further scaling on larger architectures and hardware accelerators remains an important direction for future research.
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Resonant Structures in $p{}^7\mathrm{Be}$ Scattering and Their Connection to the Astrophysical $S$-Factor
Anil Khachi
https://arxiv.org/abs/2506.20349 http…

Resonant Structures in $p{}^7\mathrm{Be}$ Scattering and Their Connection to the Astrophysical $S$-Factor
In this paper, we employ the Variable Phase Approach (VPA) to obtain the scattering phase shifts \( δ(E, r) \), amplitude function \( A(r) \), and radial wavefunction \( u(r) \) for various channels involved in the astrophysical reaction \( {}^7\mathrm{Be}(p,γ)^8\mathrm{B} \). Using the extracted phase shifts, we compute the total and partial cross sections. It is observed that the peaks in the partial cross section correspond to resonant states in the compound nucleus, which also manifest as…

Perimeter on a manifold, with applications to partial differential equations
Satyanad Kichenassamy (DMA)
https://arxiv.org/abs/2507.04740 https://

Perimeter on a manifold, with applications to partial differential equations
The perimeter of a measurable subset of $\mathbb R^N$ is the total variation of its characteristic function. We generalize this notion to a subset $E$ of a closed Riemannian manifold. We show that the perimeter of $E$ is the limit of the hear kernel regularization of its characteristic function. A generalization of the isoperimetric inequality and of the Fleming-Rishel formula follow. These results are applied to a quasilinear elliptic problem in $\mathbb R^N$ for which the usual symmetrization…

Enumeration of Hopf Monoids and Supersolvable Convex Geometries
Yichen Ma
https://arxiv.org/abs/2506.00380 https://arxiv.org/pdf/2506…

Enumeration of Hopf Monoids and Supersolvable Convex Geometries
We consider the Hopf monoid of convex geometries, which contains partial orders as a Hopf submonoid, and investigate combinatorial invariants constructed from canonical characters on them. Each invariant consists of a pair: a polynomial and a more general quasisymmetric function. We provide combinatorial descriptions of the polynomial invariants and prove combinatorial reciprocity theorems for polynomials that generalize order and enriched order polynomials within a unified framework. For the q…

This https://arxiv.org/abs/2405.20907 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The Muckenhoupt condition
The goal of this paper is to unify the theory of weights beyond the setting of weighted Lebesgue spaces in the general setting of quasi-Banach function spaces. We prove new characterizations for the boundedness of singular integrals, pose several conjectures, and prove partial results related to the duality of the Hardy-Littlewood maximal operator. Furthermore, we give an overview of the theory applied to weighted variable Lebesgue, Morrey, and Musielak-Orlicz spaces.

Is Quantum Mechanics a proper subset of Classical Mechanics?
Khaled Mnaymneh
https://arxiv.org/abs/2508.00044 https://arxiv.org/pdf/2508.00044

Is Quantum Mechanics a proper subset of Classical Mechanics?
Quantum mechanics is widely regarded as a complete theory, yet we argue it is a tractable projection of a deeper, computationally-inaccessible classical variational structure. By analyzing the coupled partial differential equations of the Hamilton type 1 principal function, we show that classical action-based dynamics are generally undecidable, paralleling spectral gap undecidability in quantum systems. In near Kolmogorov-Arnold-Moser systems, stability hinges on Diophantine conditions that are…

Extended Interface Physics-Informed Neural Networks Method for Moving Interface Problems
Ran Bi, Weibing Deng, Yameng Zhu
https://arxiv.org/abs/2508.01463 https://

Extended Interface Physics-Informed Neural Networks Method for Moving Interface Problems
Physics-Informed Neural Networks (PINNs) have emerged as a powerful class of mesh-free numerical methods for solving partial differential equations (PDEs), particularly those involving complex geometries. In this work, we present an innovative Extended Interface Physics-Informed Neural Network (XI-PINN) framework specifically designed to solve parabolic moving interface problems. The proposed approach incorporates a level set function to characterize the interface, which can be obtained either …

This https://arxiv.org/abs/2501.16815 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Best Subset Selection: Optimal Pursuit for Feature Selection and Elimination
This paper introduces two novel criteria: one for feature selection and another for feature elimination in the context of best subset selection, which is a benchmark problem in statistics and machine learning. From the perspective of optimization, we revisit the classical selection and elimination criteria in traditional best subset selection algorithms, revealing that these classical criteria capture only partial variations of the objective function after the entry or exit of features. By form…

Quantitative analysis of cell size control mechanisms
Shuqi Fan, Jinzhi Lei
https://arxiv.org/abs/2505.19416 https://arxiv.org/pdf/25…

Quantitative analysis of cell size control mechanisms
Cell size control is crucial for maintaining cellular function and homeostasis. In this study, we develop a first-order partial differential equation model to examine the effects of three key size control mechanisms: the sizer, timer, and adder. Each mechanism is incorporated into the model through distinct boundary conditions. Exact solutions for these mechanisms are derived using the method of characteristics, allowing us to explore how the steady-state size distribution depends on control pa…

This https://arxiv.org/abs/2504.18632 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Backward stochastic differential equations with nonlinear Young driver
We study backward stochastic differential equations (BSDEs) involving a nonlinear Young integral of the form $\int_{t}^{T}g(Y_{r})η(dr,X_{r})$, where the driver $η(t,x)$ is a space-time Hölder continuous function and $X$ is a diffusion process. Solutions to such equations provide a probabilistic interpretation of the solutions to stochastic partial differential equations (SPDEs) driven by space-time noise. We first obtain the existence and uniqueness of the solutions to these BSDEs under the…

Fast prediction of the hydrodynamic QGP evolution in ultra-relativistic heavy-ion collisions using Fourier Neural Operators
David Stewart, Joern Putschke
https://arxiv.org/abs/2507.23598

Fast prediction of the hydrodynamic QGP evolution in ultra-relativistic heavy-ion collisions using Fourier Neural Operators
Recent research in machine learning has employed neural networks to learn mappings between function spaces on bounded domains termed ``neural operators''. As such, these operators can provide alternatives to standard numerical methods for partial differential equation (PDE) solutions. In particular, the Fourier Neural Operator (FNO) has been shown to map solutions for classical fluid flow problems with accuracy competitive with traditional PDE solvers and with much greater computing speed. This…

Fun/cursed classical computability fact of the day: https://cs.stackexchange.com/a/172967
TL;DR: there is a decidable set S of programs such that (classically) every partial computable function is computed by a program in S, and the halting problem on the empty input is (uniformly) decidable for programs in S. And S can be described quite intuitively!

Uniqueness of diffeomorphic minimizers of $L^p$-mean distortion
Yizhe Zhu
https://arxiv.org/abs/2507.20597 https://arxiv.org/pdf/2507.20597

Uniqueness of diffeomorphic minimizers of $L^p$-mean distortion
We study the $L^p$-mean distortion functionals, \[{\cal E}_p[f] = \int_\mathbb Y K^p_f(z) \; dz, \] for Sobolev homeomorphisms $f: \overline{\mathbb Y}\xrightarrow{\rm onto} \overline{\mathbb X}$ where $\mathbb X$ and $\mathbb Y$ are bounded simply connected Lipschitz domains, and $f$ coincides with a given boundary map $f_0 \colon \partial \mathbb Y \to \partial \mathbb X$. Here, $K_f(z)$ denotes the pointwise distortion function of $f$. It is conjectured that for every $1 < p < \infty$, the f…

Existence and nonexistence of solutions for singular quadratic quasilinear equations
David Arcoya, Jos\'e Carmona, Tommaso Leonori, Pedro J. Mart\'inez-Aparicio, Luigi Orsina, Francesco Petitta
https://arxiv.org/abs/2508.06375

Existence and nonexistence of solutions for singular quadratic quasilinear equations
We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -Δu + \frac{|\nabla u|^2}{u^γ} = f & \mbox{in } Ω,\newline \hfill u=0 \hfill & \mbox{on } \partial Ω, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N $, $γ> 0$ and $f$ is a function which is strictly positive on every compactly contained subs…

This https://arxiv.org/abs/2406.13539 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qfi…

Robust Lambda-quantiles and extremal distributions
In this paper, we investigate the robust models for $Λ$-quantiles with partial information regarding the loss distribution, where $Λ$-quantiles extend the classical quantiles by replacing the fixed probability level with a probability/loss function $Λ$. We find that, under some assumptions, the robust $Λ$-quantiles equal the $Λ$-quantiles of the extremal distributions. This finding allows us to obtain the robust $Λ$-quantiles by applying the results of robust quantiles in the literature. …

Muckenhoupt-weighted $L_q(L_p)$ boundedness for time-space fractional nonlocal operators
Yong Zhen Yang, Yong Zhou
https://arxiv.org/abs/2507.01890 https:/…

Muckenhoupt-weighted $L_q(L_p)$ boundedness for time-space fractional nonlocal operators
Based on the $ϕ(Δ)$-type operator studied by Kim \cite[\emph{Adv.~Math.}]{Kim2}, where $ϕ$ is a Bernstein function, we establish weighted $L_{q}(L_{p})$ estimates for solutions to the following fractional evolution equation: $$ \partial_{t}^αw(t,x) = ϕ(Δ)w(t,x) + h(t,x), \quad t > 0, \; x \in \mathbb{R}^{d}, $$ where $\partial_{t}^α$ denotes the Caputo derivative of $0 < α< 1$. To be specific, for all $1 < p, q < \infty$, we demonstrate that $$ \int_{0}^{\infty} \left( \int_{\mathbb{R}^…

Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces
Utsav Dewan
https://arxiv.org/abs/2506.00881

Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces
We study the Carleson's problem on Damek-Ricci spaces $S$ for dispersive equations: \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} +Ψ(\sqrt{-\mathcal{L}} )u=0\:,\: (x,t) \in S \times \mathbb{R} \:, \\ u(0,\cdot)=f\:,\: \text{ on } S \:, \end{cases} \end{equation*} where $\mathcal{L}= Δ$, the Laplace-Beltrami operator or $\tildeΔ$, the shifted Laplace-Beltrami operator, so that the corresponding phase function $ψ$ satisfies for some $a \in (0,1)$, the large frequency as…

Fractional Infinity Laplacian with Obstacle
Samer Dweik, Ahmad Sabra
https://arxiv.org/abs/2507.04328 https://arxiv.org/pdf/2507.0432…

Fractional Infinity Laplacian with Obstacle
This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, Ω\\ u=g &\qquad on\, \partial Ω\end{cases},$$ with $$L[u](x)=\sup_{y\in Ω,\,y\neq x}\dfrac{u(y)-u(x)}{|y-x|^α}+\inf_{y\in Ω,\,y\neq x} \dfrac{u(y)-u(x)}{|y-x|^α},\qquad 0<α<1.$$ Under the assumptions that $f$ is a continuous and monotone functi…

Wavelet based solutions to the Poisson and the Helmholtz equations on the $n$-dimensional unit sphere
Ilona Iglewska-Nowak, Piotr Stefaniak
https://arxiv.org/abs/2507.03451

Wavelet based solutions to the Poisson and the Helmholtz equations on the $n$-dimensional unit sphere
We present a method of solving partial differential equations on the $n$-dimensional unit sphere using methods based on the continuous wavelet transform derived from approximate identities. We give an explicit analytical solution to the Poisson equation and to the Helmholtz equations. For the first one and for some special values of the parameter in the latter one, we derive a closed formula for the generalized Green function.

Solutions with large number of peaks for a slightly supercritical nonlinear equation in dimension three
Yixing Pu
https://arxiv.org/abs/2506.01652 https://…

Solutions with large number of peaks for a slightly supercritical nonlinear equation in dimension three
We investigate the existence of solutions to the semilinear equation with a slightly supercritical exponent in dimension three, \begin{align*} -Δu=K(x) u^{5+μ},\quad u>0 ~\text{in}~ \mathbf{B}, \quad u=0 ~\text{on}~ \partial \mathbf{B}, \end{align*} where $μ>0$, $\mathbf{B}$ is the unit ball in $\mathbb{R}^3$, $K(x)$ is a nonnegative radial function under suitable condition on $K$. We prove the existence of positive multi-peak solutions for $μ>0$ small enough. All peaks of our solut…

PDE methods for extracting normal vector fields and distance functions of shapes
Takahiro Hasebe, Jun Masamune, Hiroshi Teramoto, Takayuki Yamada
https://arxiv.org/abs/2506.19323 …

PDE methods for extracting normal vector fields and distance functions of shapes
Partial differential equations can be used for extracting geometric features of shapes. This article summarizes recent methods to extract the normal vector field from an elliptic equation proposed by Yamada and from the heat equation, and also a method to extract the (signed) distance function from an elliptic equation that generalizes Varadhan's in 1967.