Tootfinder

Tensor Forms of Derivatives of Matrices and their applications in the Solutions to Differential Equations
Yiran Xu, Guangbin Wang, Changqing Xu
https://arxiv.org/abs/2509.08429 …

Tensor Forms of Derivatives of Matrices and their applications in the Solutions to Differential Equations
We introduce and extend the outer product and contractive product of tensors and matrices, and present some identities in terms of these products. We offer tensor expressions of derivatives of tensors, focus on the tensor forms of derivatives of a matrix w.r.t. another matrix. This tensor form makes possible for us to unify ordinary differential equations (ODEs) with partial differential equations (PDEs), and facilitates solution to them in some cases. For our purpose, we also extend the outer …

Replaced article(s) found for math.AG. https://arxiv.org/list/math.AG/new
[1/1]:
- Differential Forms and Hodge Structures on Singular Varieties
Donu Arapura, Scott Hiatt

Super homology groups of differential forms and vector fields on Euclidean line
Kentaro Mikami, Tadayoshi Mizutani
https://arxiv.org/abs/2509.04698 https://

Super homology groups of differential forms and vector fields on Euclidean line
The arXiv:2105.09738 claims several stuffs. In particular, we recall the following two. (1) Vector fields and differential forms become a Lie superalgebra structure for each manifold. (2) For an n-dimensional Euclidean space, vector fields and differential forms with polynomial coefficients become a double weighted Lie uperalgebra. By using Euler vector field, the Betti numbers are 0 except the last one if the primary weight and the secondary weight are different. Now, a simple question arises:…

Differential forms, open-closed maps, and Gromov-Witten axioms
Pavel Giterman, Jake P. Solomon, Sara B. Tukachinsky
https://arxiv.org/abs/2509.06034 https://

Differential forms, open-closed maps, and Gromov-Witten axioms
We construct open-closed maps on various versions of Hochschild and cyclic homology of the Fukaya $A_\infty$ algebra of a Lagrangian submanifold modeled on differential forms. The $A_\infty$ algebra may be curved. Properties analogous to Gromov-Witten axioms are verified. The paper is written with applications in mind to gravitational descendants and obstruction theory.

On the de Rham theorem in the globally subanalytic setting
Annette Huber, Tobias Kaiser, Abhishek Oswal
https://arxiv.org/abs/2508.03499 https://arxiv.org/…

On the de Rham theorem in the globally subanalytic setting
For globally subanalytic manifolds we define de Rham complexes of globally subanalytic differential forms and of constructible differential forms. Whereas the de Rham theorem does not hold for the former in the non-compact case, it does hold for the latter in full generality. We deduce that the constructible de Rham cohomology groups are canonically isomorphic to the classical ones. We stress that our results apply already in the $C^1$-setting.

Lepage equivalents for second order Lagrangians and applications: $2D$ modified higher order Boussinesq-type equations
Marcella Palese, Fabrizio Zanello
https://arxiv.org/abs/2510.07090

Lepage equivalents for second order Lagrangians and applications: $2D$ modified higher order Boussinesq-type equations
In the frame of the Lagrangian formalism on $r$-order prolongations of fibered manifolds and related structures such as (prolongation of) projectable vector fields, (sheaves of) differential forms and contact structures, we propose a Lagrangian two-field derivation of $2D$ modified Boussinesq equations, obtained as coupled systems of Euler--Lagrange (E-L) equations for the two fields. By means of a recursive formula involving geometric integration by parts formulae, we construct extended `full'…

A unified framework for the analysis, numerical approximation and model reduction of linear operator equations, Part I: Well-posedness in space and time
Moritz Feuerle, Richard L\"oscher, Olaf Steinbach, Karsten Urban
https://arxiv.org/abs/2508.05407

A unified framework for the analysis, numerical approximation and model reduction of linear operator equations, Part I: Well-posedness in space and time
We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic and hyperbolic partial differential equations. This general approach incorporates known weak variational formulations as well as novel space-time variational forms of the hyperbolic wave equation. The main concept is completion and extension of operators starting from the strong form of the problem. This paper lays the theoretical foundation for a un…

Canonical differential equations and intersection matrices
Claude Duhr, Sara Maggio, Franziska Porkert, Cathrin Semper, Yoann Sohnle, Sven F. Stawinski
https://arxiv.org/abs/2509.17787

Canonical differential equations and intersection matrices
Differential equations are one of the main approaches to evaluate multi-loop Feynman integrals. The construction of a canonical or $\varepsilon$-factorised basis for multi-loop integrals remains a key bottleneck for this approach, especially when the differential equation involves non dlog-forms. Recently, several methods have been proposed to find $\varepsilon$-factorised differential equations. Many of them introduce new functions that are themselves defined as iterated integrals. If and when…

Dynamics of Linear Scalar Perturbation in f(Q) f(T) class of f(Q,T) gravity
Houda Filali, Rachid Ahl Laamara, Mohamed Bennai, El Hassan Saidi
https://arxiv.org/abs/2508.03495 ht…

Dynamics of Linear Scalar Perturbation in f(Q)+f(T) class of f(Q,T) gravity
In this work, we study the f(Q,T) model of symmetric teleparallel modified gravity in the framework of cosmological perturbation theory. Using a general approach, we extract the differential matter density equation then we simplify it as a second-order equation by considering the sub-Hubble approximation. Our analysis is then based on two different forms of f(Q,T) that we study in a classic approach and again using Holographic dark energy. Our initial results yield a significant divergence from…

Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties
Dmitrii Adler, Valery Gritsenko
https://arxiv.org/abs/2509.25954 https://

Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties
We study modular differential equations (MDEs) of high orders for weak Jacobi forms and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for the elliptic genus of six-dimensional manifolds with a trivial first Chern class. We prove that the minimal possible order of the MDE for the elliptic genus of a strict six-dimensional Calabi--Yau variety is four, and find MDEs of order 7 for hyperkähler varietie…

Asymptotic analysis of the Allen-Cahn equation with dynamic boundary conditions of Cahn-Hilliard Type
Pierluigi Colli, Takeshi Fukao
https://arxiv.org/abs/2508.02021 https://

Asymptotic analysis of the Allen-Cahn equation with dynamic boundary conditions of Cahn-Hilliard Type
Problems for partial differential equations coupled with dynamic boundary conditions can be viewed as a type of transmission problem between the bulk and its boundary. For the heat equation and the Allen-Cahn equation, various forms of such problems with dynamic boundary conditions are studied in this paper. In the case of the Cahn-Hilliard equation in the bulk, several models have been proposed in which the boundary equations and conditions differ. Recently, the vanishing surface diffusion lim…

Replaced article(s) found for math.CV. https://arxiv.org/list/math.CV/new
[1/1]:
- On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversel...
Sheng Rao, Runze Zhang

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…

The singularity category as a stable module category
Xiao-Wu Chen, Zhengfang Wang
https://arxiv.org/abs/2509.01056 https://arxiv.org/pdf/2509.01056

The singularity category as a stable module category
We investigate the stabilization $\mathcal{S}$ of the module category over an artinian ring $Λ$ by formally inverting the tensor endofunctor given by the bimodule of relative noncommutative differential $1$-forms. It turns out that $\mathcal{S}$ is a Frobenius abelian category, which is equivalent to the category of finitely presented modules over the zeroth component $L_0$ of the Leavitt ring $L$. It follows that $L_0$ is an FC ring in the sense of Damiano, which is usually not quasi-Frobeniu…

Non-existence of higher-order conformal fundamental forms in odd dimensions
Samuel Blitz
https://arxiv.org/abs/2509.20554 https://arxiv.org/pdf/2509.20554

Non-existence of higher-order conformal fundamental forms in odd dimensions
Conformal fundamental forms populate a minimal generating set for low differential order invariants of conformal hypersurface embeddings. In this work we complete the characterization of conformal fundamental forms by proving the general non-existence of higher-order conformal fundamental forms when the embedded hypersurface is even dimensional.

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.

On a class of third order differential equations describing pseudospherical or spherical surfaces
Mingyue Guo, Jing Kang, Zhenhua Shi, Zhiwei Wu
https://arxiv.org/abs/2508.20515

On a class of third order differential equations describing pseudospherical or spherical surfaces
In this paper, we study third order nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness of connection 1-forms, we present a classification of equations with the type $u_t - u_{xxt} = λu^2 u_{xxx} + G(u, u_x, u_{xx}) (λ\in\mathbb{R})$, which describe pseudospherical or spherical surfaces. We show that series of typical soliton equations belong to certain subclass, such as the generalized Camassa-Holm equation, which gives a geometric explan…

Using Dynamical Systems Theory to Quantify Complexity in Asymptotic Lenia
Ivan Yevenko, Hiroki Kojima, Chrystopher L. Nehaniv
https://arxiv.org/abs/2508.02935 https://

Using Dynamical Systems Theory to Quantify Complexity in Asymptotic Lenia
Continuous cellular automata (CCAs) have evolved from discrete lookup tables to continuous partial differential equation (PDE) formulations in the search for novel forms of complexity. Despite innovations in qualitative behavior, analytical methods have lagged behind, reinforcing the notion that emergent complexity defies simple explanation. In this paper, we demonstrate that the PDE formulation of Asymptotic Lenia enables rigorous analysis using dynamical systems theory. We apply the concepts …

Conforming lifting and adjoint consistency for the Discrete de Rham complex of differential forms
Daniele A. Di Pietro, J\'er\^ome Droniou, Silvano Pitassi
https://arxiv.org/abs/2509.21449

Conforming lifting and adjoint consistency for the Discrete de Rham complex of differential forms
Discrete de Rham (DDR) methods provide non-conforming but compatible approximations of the continuous de Rham complex on general polytopal meshes. Owing to the non-conformity, several challenges arise in the analysis of these methods. In this work, we design conforming liftings on the DDR spaces, that are right-inverse of the interpolators and can be used to solve some of these challenges. We illustrate this by tackling the question of the global integration-by-part formula. By non-conformity o…

A note on the Maxwell's eigenvalues on thin sets
Francesco Ferraresso, Luigi Provenzano
https://arxiv.org/abs/2510.01846 https://arxiv.org/pdf/2510.018…

A note on the Maxwell's eigenvalues on thin sets
We analyse the Maxwell's spectrum on thin tubular neighborhoods of embedded surfaces of $\mathbb R^3$. We show that the Maxwell eigenvalues converge to the Laplacian eigenvalues of the surface as the thin parameter tends to zero. To achieve this, we reformulate the problem in terms of the spectrum of the Hodge Laplacian with relative conditions acting on co-closed differential $1$-forms. The result leads to new examples of domains where the Faber-Krahn inequality for Maxwell's eigenvalues fails…

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 …

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…

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…

The exterior derivative and the mean value equality in $\mathbb{R}^n$
Daniel Fadel, Henrique N. S\'a Earp, Tom\'as S. R. Silva
https://arxiv.org/abs/2510.00999 https://

The exterior derivative and the mean value equality in $\mathbb{R}^n$
This survey revisits classical results in vector calculus and analysis by exploring a generalised perspective on the exterior derivative, interpreting it as a measure of "infinitesimal flux". This viewpoint leads to a higher-dimensional analogue of the Mean Value Theorem, valid for differential $k$-forms, and provides a natural formulation of Stokes' theorem that mirrors the exact hypotheses of the Fundamental Theorem of Calculus - without requiring full $C^1$ smoothness of the differential for…

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…

A new approach to the bialgebra theory for relative Poisson algebras
Guilai Liu, Chengming Bai
https://arxiv.org/abs/2509.11282 https://arxiv.org/pdf/2509.…

A new approach to the bialgebra theory for relative Poisson algebras
It is natural to consider extending the typical construction of relative Poisson algebras from commutative differential algebras to the context of bialgebras. The known bialgebra structures for relative Poisson algebras, namely relative Poisson bialgebras, are equivalent to Manin triples of relative Poisson algebras with respect to the symmetric bilinear forms which are invariant on both the commutative associative and Lie algebras. However, they are not consistent with commutative and cocommut…

Lorentz Covariance of the $4d$ Nonlinear Higher-Spin Equations via BRST
O. A. Gelfond, M. A. Vasiliev
https://arxiv.org/abs/2509.22852 https://arxiv.org/pd…

Lorentz Covariance of the $4d$ Nonlinear Higher-Spin Equations via BRST
We propose a BRST extension of the higher-spin gauge theory in $AdS_4$ with the BRST operator associated with the local Lorentz symmetry. Our construction supports manifest local Lorentz covariance and is applicable both to any homotopy scheme of the perturbative analysis including the recently proposed differential homotopy and to the variety of further extended higher-spin models in $AdS_3$ and $AdS_4$ with higher differential forms and Coxeter higher-spin algebras.

The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory
Sergiu I. Vacaru, El\c{s}en V. Veliev
https://arxiv.org/abs/2508.10939 …

The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory
The gravitational field equations in general relativity (GR) consist of a sophisticated system of nonlinear partial differential equations. Solving such equations in some generic off-diagonal forms is usually a hard analytic or numeric task. Physically important solutions in GR were constructed using a diagonal ansatz for metrics with a maximum of 4 independent coefficients. The Einstein equations can be solved in exact or parametric forms determined by some integration constants for correspond…

The Geometry of Loop Spaces V: Fundamental Groups of Geometric Transformation Groups
Yoshiaki Maeda, Steven Rosenberg
https://arxiv.org/abs/2510.01566 https://

The Geometry of Loop Spaces V: Fundamental Groups of Geometric Transformation Groups
We use differential forms on loop spaces to prove that the fundamental group of certain geometric transformation groups is infinite. Examples include both finite and infinite dimensional Lie groups. The finite dimensional examples are the conformal group of $S^{4k+1}$ for a family of nonstandard metrics, and the group of pseudo-Hermitian transformations of a compact CR manifold. Infinite dimensional examples include the group of strict contact diffeomorphisms of a regular contact manifold, and …

Automated Runge-Kutta-Nystr\"om time stepping for finite element methods in Irksome
Robert C. Kirby, Scott P. MacLachlan, Pablo D. Brubeck
https://arxiv.org/abs/2508.20255 …

Automated Runge-Kutta-Nyström time stepping for finite element methods in Irksome
Irksome is a library based on the Unified Form Language (UFL) that automates the application of Runge-Kutta time-stepping methods for finite element spatial discretizations of partial differential equations (PDEs). This paper describes recent updates to Irksome that allow users to express semidiscrete forms of PDEs that contain second-order temporal derivatives, whence it generates stage-coupled variational problems to be solved at each time step for Runge-Kutta-Nyström methods. Firedrake then…

Replaced article(s) found for math.DG. https://arxiv.org/list/math.DG/new
[1/1]:
- Characteristic forms of complex Cartan geometries III: G-structures
Benjamin McKay

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.

Kinematic flow from the flow of cuts
Ross Glew, Andrzej Pokraka
https://arxiv.org/abs/2508.11568 https://arxiv.org/pdf/2508.11568

Kinematic flow from the flow of cuts
The wavefunction coefficients of conformally coupled scalars in power-law FRW cosmologies satisfy differential equations governed by a set of simple combinatorial rules known as the kinematic flow. In this paper we derive the kinematic flow, expressed using a set of differential forms referred to as the cut basis, from a geometric perspective, relying solely on the cosmological hyperplane arrangement and without invoking bulk physics. Each element of the cut basis corresponds to the positive ge…

Replaced article(s) found for math.DG. https://arxiv.org/list/math.DG/new
[1/1]:
- Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy
Ao Sun, Jinxin Xue

Development and Analysis of Chien-Physics-Informed Neural Networks for Singular Perturbation Problems
Gautam Singh, Sofia Haider
https://arxiv.org/abs/2509.10945 https://…

Development and Analysis of Chien-Physics-Informed Neural Networks for Singular Perturbation Problems
In this article, we employ Chien-Physics Informed Neural Networks (C-PINNs) to obtain solutions for singularly perturbed convection-diffusion equations, reaction-diffusion equations, and their coupled forms in both one and two-dimensional settings. While PINNs have emerged as a powerful tool for solving various types of differential equations, their application to singular perturbation problems (SPPs) presents significant challenges. These challenges arise because a small perturbation parameter…

Hodge-de Rham Theory on Higher-Dimensional Level-L Sierpinski Gaskets
Sze-Man Ngai, Shui-Hong Zhou
https://arxiv.org/abs/2508.12319 https://arxiv.org/pdf/2…

Hodge-de Rham Theory on Higher-Dimensional Level-L Sierpinski Gaskets
This paper extends the Hodge-de Rham theory of Aaron \textit{et al.} [Commun. Pure Appl. Anal. {\bf 13} (2014)] to higher-dimensional level-$l$ Sierpinski gaskets $SG_{\ell}^{n},$ providing a framework for analyzing differential forms and Laplacians on these fractal structures. We construct a sequence of graphs approximating $SG_{\ell}^{n}$ and define $k$-forms, de Rham derivatives, and their duals on these graphs. We prove that the extension of a $1$-form on a generation-$m$ graph to a $1$-for…

Submanifolds with boundary in sub-Riemannian Heisenberg Groups
Marco Di Marco, Davide Vittone
https://arxiv.org/abs/2508.15634 https://arxiv.org/pdf/2508.1…

Submanifolds with boundary in sub-Riemannian Heisenberg Groups
We discuss the notion of submanifolds with boundary with intrinsic $C^1$ regularity in sub-Riemannian Heisenberg groups and we provide some examples. Eventually, we present a Stokes' Theorem for such submanifolds involving the integration of Rumin's differential forms in Heisenberg groups.