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Rust Implementation of Finite Element Exterior Calculus on Coordinate-Free Simplicial Complexes
Luis Wirth
https://arxiv.org/abs/2506.02429 https://…

Rust Implementation of Finite Element Exterior Calculus on Coordinate-Free Simplicial Complexes
This thesis presents the development of a novel finite element library in Rust based on the principles of Finite Element Exterior Calculus (FEEC). The library solves partial differential equations formulated using differential forms on abstract, coordinate-free simplicial complexes in arbitrary dimensions, employing an intrinsic Riemannian metric derived from edge lengths via Regge Calculus. We focus on solving elliptic Hodge-Laplace eigenvalue and source problems on the nD de Rham complex. We …

Quantum jet Hopf algebroids by cotwist
Xiao Han, Shahn Majid
https://arxiv.org/abs/2507.02848 https://arxiv.org/pdf/2507.02848…

Quantum jet Hopf algebroids by cotwist
We introduce a cotwist construction for Hopf algebroids that also entails cotwisting or `quantisation' of the base and which is dual to a previous twisting construction of Xu. Whereas the latter applied the construction to the algebra of differential operators on a classical base $B$, we show that the dual of this is the algebra of sections $J^\infty(B)$ of the jet bundle and hence the latter forms a Hopf algebroid, which we identify as a quotient of the pair Hopf algebroid $B\otimes B$. This c…

A proof of the Fields Conjectures
Satoshi Murai, Brendon Rhoades, Andy Wilson
https://arxiv.org/abs/2505.24027 https://arxiv.org/pdf/…

A proof of the Fields Conjectures
The {\em superspace ring} of rank $n$ is the algebra $Ω_n$ of differential forms on affine $n$-space. The algebra $Ω_n$ is bigraded with respect to polynomial and exterior degree and carries a natural action of the symmetric group $\mathfrak{S}_n$. Modding out by $\mathfrak{S}_n$-invariants with vanishing constant term yields the {\em superspace coinvariant ring} $SR_n$. We prove that, as an ungraded $\mathfrak{S}_n$-module, the space $SR_n$ is isomorphic to the sign-twisted permutation actio…

Symmetric star transforms and the algebraic geometry of their dual differential operators
Gaik Ambartsoumian, Asher Auel, Mohammad Javad Latifi Jebelli
https://arxiv.org/abs/2507.22138

Symmetric star transforms and the algebraic geometry of their dual differential operators
The star transform is a generalized Radon transform mapping a function on $\mathbb{R}^n$ to the function whose value at a point is the integral along a union of rays emanating from the point in a fixed set of directions, called branch vectors. We show that the injectivity and inversion properties of the star transform are connected to its dual differential operator, an object introduced in this paper. We prove that if the set of branch vectors forms a symmetric shape with respect to the action …

Berezinian expansion and super exterior powers
Maheshan Ekanayaka, Ekaterina Shemyakova
https://arxiv.org/abs/2506.16549 https://arxi…

Berezinian expansion and super exterior powers
In the supergeometric setting, the classical identification between differential forms of top degree and volume elements for integration breaks down. To address this, generalized notions of differential forms were introduced: pseudo-differential forms and integral forms (Bernstein-Leites), and $r|s$-forms (Voronov-Zorich). The Baranov-Schwarz transformation transforms pseudo-differential forms into $r|s$-forms. Also, integral $r$-forms are isomorphic to $r|m$-forms for a superman…

Covariant Lie Derivatives and (Super-)Gravity
Grigorios Giotopoulos
https://arxiv.org/abs/2507.00140 https://arxiv.org/pdf/2507.00140…

Covariant Lie Derivatives and (Super-)Gravity
The slightly subtle notion of covariant Lie derivatives of \textit{bundle-valued} differential forms is crucial in many applications in physics, notably in the computation of conserved currents in gauge theories, and yet the literature on the topic has remained fragmentary. This note provides a complete and concise mathematical account of covariant Lie derivatives on a spacetime (super-)manifold $M,$ defined via choices of lifts of spacetime vector fields to principal $G$-bundles over it, or eq…

The unequal-mass three-loop banana integral
Sebastian P\"ogel, Toni Teschke, Xing Wang, Stefan Weinzierl
https://arxiv.org/abs/2507.23594 https://arxi…

The unequal-mass three-loop banana integral
We compute the three-loop banana integral with four unequal masses in dimensional regularisation. This integral is associated to a family of K3 surfaces, thus representing an example for Feynman integrals with geometries beyond elliptic curves. We evaluate the integral by deriving an $\varepsilon$-factorised differential equation, for which we rely on the algorithm presented in a recent publication. Equipping the space of differential forms in Baikov representation by a set of filtrations inspi…

Differential operators on Hermitian modular forms on $\mathrm{u}(n, n)$
Nobuki Takeda
https://arxiv.org/abs/2506.18236 https://arxiv.…

Differential operators on Hermitian modular forms on $\mathrm{u}(n, n)$
We construct explicit differential operators on hermitian modular forms, extending methods developed for Siegel modular forms. These differential operators are closely related to the two-variable spherical pluriharmonic polynomials. We construct explicit bases for the space of such polynomials and use them to build concrete operators. As an application, we derive an exact pullback formula for hermitian Eisenstein series.

Replaced article(s) found for math.AT. https://arxiv.org/list/math.AT/new
[1/1]:
- Anderson duality of topological modular forms and its differential-geometric manifestations
Yuji Tachikawa, Mayuko Yamashita

The superspace coinvariant ring of type B
Sutanay Bhattacharya
https://arxiv.org/abs/2505.24122 https://arxiv.org/pdf/2505.24122

The superspace coinvariant ring of type B
Given the rank $n$ superspace $Ω_n$, the ring of polynomial-valued differential forms on $\mathbb C^n$, one can define an action of hyperoctahedral group $\mathfrak B_n$ on it. This leads to a superspace coinvariant ideal $SR_n^B$, defined as the quotient of $Ω_n$ by two-sided ideal generated by all $\mathfrak B_n$ invariants with vanishing constant terms. We derive the Hilbert series of $SR^B_n$ conjectured by Sagan and Swanson, and prove an operator theorem that yields a concrete descriptio…

This https://arxiv.org/abs/2312.03657 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Multisymplecticity in finite element exterior calculus
We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which generalizes the more familiar symplectic conservation law for Hamiltonian systems of ODEs, and which is connected with physically-important reciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We characterize hybrid FEEC methods whose numer…

Normal forms for ordinary differential operators, II
Junhu Guo, A. B. Zheglov
https://arxiv.org/abs/2506.19799 https://arxiv.org/pdf/…

Normal forms for ordinary differential operators, II
In this paper, which is a follow-up of our first paper "Normal forms for ordinary differential operators, I", we extend the explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups obtained earlier to analogous parametrisation of torsion free sheaves of arbitrary rank with vanishing cohomology groups on projective irreducible curves. As an illustration of our theorem we calculate one explicit example of such parametrisation, n…

Physics-Informed Neural Networks: Bridging the Divide Between Conservative and Non-Conservative Equations
Arun Govind Neelan, Ferdin Sagai Don Bosco, Naveen Sagar Jarugumalli, Suresh Balaji Vedarethinam
https://arxiv.org/abs/2506.22413

Physics-Informed Neural Networks: Bridging the Divide Between Conservative and Non-Conservative Equations
In the realm of computational fluid dynamics, traditional numerical methods, which heavily rely on discretization, typically necessitate the formulation of partial differential equations (PDEs) in conservative form to accurately capture shocks and other discontinuities in compressible flows. Conversely, utilizing non-conservative forms often introduces significant errors near these discontinuities or results in smeared shocks. This dependency poses a considerable limitation, particularly as man…

The $L_\infty$ structure of Free Differential Algebras and $d=11$ Supergravity
L. Castellani, R. D'Auria
https://arxiv.org/abs/2507.20344 https://arxiv…

The $L_\infty$ structure of Free Differential Algebras and $d=11$ Supergravity
We discuss free differential algebras (FDA's), a generalization of the Cartan-Maurer equations for the group manifold vielbein, appropriate for theories containing $p$-forms ($p >1$). Their dual formulation is an extension of Lie algebras, called $L_\infty$ algebras, and we illustrate this duality in a simple example. Finally we review the FDA structure and the dual $L_\infty$ structure of $d=11$ supergravity.

Exactness, Cohomology, and Uniqueness in First-Order Differential Equations
Hemanta Mandal
https://arxiv.org/abs/2507.16457 https://a…

Exactness, Cohomology, and Uniqueness in First-Order Differential Equations
This paper investigates the relationship between the solvability of first-order differential equations and the topology of the underlying domain through the lens of de\,Rham cohomology. We analyze the conditions under which a closed 1-form associated with a first-order ODE admits a global potential, thereby reducing the problem to the exactness of differential forms. While it is well known that exactness is guaranteed on simply connected domains, we show that triviality of the first de\,Rham co…

Nonlinear Hardy-Stein type identities for harmonic functions relative to symmetric integro-differential operators
Tomasz Klimsiak, Andrzej Rozkosz
https://arxiv.org/abs/2507.18308

Nonlinear Hardy-Stein type identities for harmonic functions relative to symmetric integro-differential operators
We show identities of Hardy-Stein type for harmonic functions relative to integro-differential operators corresponding to general symmetric regular Dirichlet forms satisfying the absolute continuity condition. The novelty is that we consider operators of mixed type containing both local and nonlocal component. Moreover, the identities are proved for compositions of harmonic functions and general convex functions. We also provide some conditional identities, i.e. identities for ratios of harmoni…

Analytical transit light curves for power-law limb darkening: a comprehensive framework via fractional calculus and differential equations
Farrukh A. Chishtie, Mohammad I. Saeed, Shaukat N. Goderya
https://arxiv.org/abs/2506.18860

Analytical transit light curves for power-law limb darkening: a comprehensive framework via fractional calculus and differential equations
We present the first complete analytical framework for computing exoplanetary transit light curves with arbitrary power-law limb darkening profiles $I(μ) \propto μ^α$, where $α$ can be any real number greater than $-1/2$, including the physically important non-integer cases. While the groundbreaking work of Agol et al. (2020) provided exact analytical solutions for polynomial limb darkening through recursion relations, stellar atmosphere models often favor power-law forms with fractional ex…

On the domains of first order differential operators on the Sierpi\'{n}ski gasket
Waldemar Schefer
https://arxiv.org/abs/2506.08679 https://

On the domains of first order differential operators on the Sierpiński gasket
We study the first order structure of $L^2$-differential one-forms on the Sierpiński gasket. We consider piecewise energy finite functions related to one-forms, normal parts, and show self-similarity properties of one-forms as in the case of energy finite functions on the Sierpiński gasket. We introduce first order differential operators, taking functions into functions, that can be understood as total derivatives with respect to some reference function or form. The main result is a pointwise…

Levy Laplacian on manifold and heat flows of differential forms
Boris Volkov
https://arxiv.org/abs/2507.13013 https://arxiv.org/pdf/2…

Levy Laplacian on manifold and heat flows of differential forms
The Levy Laplacian is an infinite-dimensional differential operator, which is interesting for its connection with the Yang-Mills gauge fields. The article proves the equivalence of various definitions of the Levy Laplacian on the manifold of $H^1$-paths on a Riemannian manifold. The heat equation with the Levy Laplacian is considered. The tendency of some solutions of this heat equation to the locally constant functionals as time tends to infinity is studied. These solutions are constructed usi…

Embeddings of Sobolev spaces of differential forms
Vladimir Gol'dshtein, Yaroslav Kopylov, Roman Panenko
https://arxiv.org/abs/2507.05851 https://

Embeddings of Sobolev spaces of differential forms
We continue the~study of embedding between different classes of Sobolev spaces of differential forms started in 2006 in a~paper by Gol'dshtein and Troyanov. As in this paper, our study is based on relations between $L_{q,p}$-cohomology and Sobolev type inequalities. The~main results are estimates for the norms of the embedding operators for $q=p$ and $p>\frac{n-1}{k-1}$ in the~Euclidean $r$-ball $B(r)$ and its bi-Lipschitz images. The~compactness of such operators also studied.

Stochastic Processes and Diffusion Equations
Helder Rojas
https://arxiv.org/abs/2506.13913 https://arxiv.org/pdf/2506.13913

Stochastic Processes and Diffusion Equations
In these lecture notes, we explore the mathematical preliminaries and foundational concepts that connect stochastic processes with partial differential equations. We begin by investigating Brownian motion, which serves as a model for random fluctuations and is deeply connected to the heat equation. This connection forms the basis for understanding diffusion phenomena, where the probability distribution of Brownian motion evolves according to the heat equation over time. To extend this…

The magnetohydrodynamical system in bounded $\mathscr{C}^1$ domains of dimension $n\ge3$
Sylvie Monniaux (I2M,FAMSI - IRL 2032)
https://arxiv.org/abs/2507.18195 https://

The magnetohydrodynamical system in bounded $\mathscr{C}^1$ domains of dimension $n\ge3$
Existence of mild solutions for the magnetohydrodynamical system in $\mathscr{C}^1$ domains is established in critical spaces in dimension $n\ge 3$. The proof relies on recent regularity results on the Stokes operator in $\mathscr{C}^1$ domains and a Leibniz-like formula for differential forms proved here in Section 2.

Biharmonic Steklov problems with Neumann boundary conditions and spectral inequalities on differential forms
Rodolphe Abou Assali
https://arxiv.org/abs/2507.05049

Biharmonic Steklov problems with Neumann boundary conditions and spectral inequalities on differential forms
We introduce three biharmonic Steklov problems on differential forms with Neumann boundary conditions and show that they are elliptic. We prove the existence of a discrete spectrum for each of those problems and give associated variational characterizations for their eigenvalues. We establish eigenvalue estimates known as Kuttler-Sigillito inequalities, relating the eigenvalues of these problems with the eigenvalues of the Steklov, Dirichlet and Neumann standard problems as well as the biharmon…

Hodge Duals in Spherical Compactifications
Arash Azizi
https://arxiv.org/abs/2507.06324 https://arxiv.org/pdf/2507.06324

Hodge Duals in Spherical Compactifications
We present a general formalism for computing the Hodge dual of differential forms in arbitrary dimensions subject to a spherical constraint. This problem arises naturally in Kaluza-Klein compactifications, where sphere reductions demand careful treatment of differential forms constrained to lie on embedded submanifolds. We derive an explicit expression for the Hodge dual of a $p$-form in the presence of such constraints and validate our general ansatz through illustrative examples in three dime…

Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below
Colette Ann\'e (LMJL), Junya Takahashi (GSIS Mathematics)
https://arxiv.org/abs/2506.11579

Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below
For each degree p and each natural number k $\ge$ 1, we construct a oneparameter family of Riemannian metrics on any oriented closed manifold with volume one and the sectional curvature bounded below such that the k-th positive eigenvalue of the Hodge-Laplacian acting on differential p-forms converges to zero. This result imposes a constraint on the sectional curvature for our previous result in [AT24].

Electromagnetism: an intrinsic approach to Hadamard's method of descent
Giuliano Angelone, Elisa Ercolessi, Paolo Facchi, Rocco Maggi, Giuseppe Marmo, Saverio Pascazio, Francesco V. Pepe
https://arxiv.org/abs/2506.05255

Electromagnetism: an intrinsic approach to Hadamard's method of descent
We present a systematic geometric framework for the dimensional reduction of classical electromagnetism based on the concept of descent along vector fields of invariance. By exploring the interplay between the Lie derivative and the Hodge star operator, we implement descent conditions on differential forms that reduce Maxwell's equations in four-dimensional spacetime to electromagnetic theories in lower dimensions. We also consider multiple descent along pairwise commuting vector fields of inva…