arXiv research

We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss t…

The main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset $A$ of the Euclidean space and for eve…

The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher order rectifiability of the set of differentiabili…

New principle for optimal control with higher order differential constraints.

Noncommutative geometry connects higher order connections to quantization.

Recalls and refines the concept of algebraically rectifiable curves.

Study proves higher-order conformal forms don't exist in odd dimensions.

The purpose of this article is to present the theory of higher order connections on vector bundles from a viewpoint inspired by projective differential geometry.

This paper deals with the enumeration of the higher order non-trivial compositions of the differential operations and the directional derivative in the space $R^n$ ($n \geq 3$). We present the recurrences for a counting the higher order non-trivial compositions.

Motivated by obtaining a consistent mathematical description for the radiation reaction of point charged particles in linear classical electrodynamics, a theory of generalized higher order tensors and differential forms is introduced. The generalization of some fundamental notions of the differential geometry and the t…

Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The '' local linear algebra '' defined in Part I is generalized by '' higher order local linear algebra ''. The underlying combinatorial object of such higher algebra is the natural n-dimensional hyper-cu…

GENIE accelerates DDM synthesis with higher-order solvers.

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second varia…

Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations …

Using supervector fields and graded forms along a morphism, we study the geometry of ordinary differential superequations, extend the formalism of higher order Lagrangian mechanics to the graded context and prove a generalization of Noether's theorem.

We study higher-order conservation laws of the non-linearizable elliptic Poisson equation $\frac{{\partial}^2 u}{\partial z \partial \bar{z}} = -f(u)$ as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for diffe…

In this paper we consider successive iterations of the first-order differential operations in space ${\bf R}^3.$

New symmetry dimensions for higher order ODEs are identified.

The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher--order in…

After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5…

Geometric AD framework simplifies derivative computation in JAX.

New minimal surfaces in 4D space derived from parametric equations.

Higher-order ODE solvers improve deep learning performance.

Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a $k$-differential with at least one pole of order at least $k…

In this paper we determine a number of meaningful compositions of higher order of a set of functions, which is considered in Malesevic (1998), in implicit and explicit form. Results which are obtained are applied to the vector analysis in order to determine the number of meaningful differential operations of higher ord…

The study sets limits on the complexity of Klein geometries.

We study differential forms and their higher-order generalizations by interpreting them as functions on map spaces. We get a series of approximations of "generalized manifolds" (i.e. of sheaves and stacks) somewhat akin to Taylor series.

In this note we review some results regarding higher order elliptic differential operators on manifolds without boundary.

This paper is being replaced by another of the author's that contains a brief summary of the problem of positivity of Green's functions, heat kernels, and principal eigenvalues of higher-order elliptic differential operators.

In this paper we present a recurrent relation for counting meaningful compositions of the higher-order differential operations on the space $R^{n}$ (n=3,4,...) and extract the non-trivial compositions of order higher than two.

Paper proves higher-order flow matching preserves optimality in generative modeling.

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker (GRW) spacetimes. In particular, we consider the following question: Under what conditions must a compact spacelike hypersurface with constant higher order mean curvatur…

Expands differential geometry to higher-order infinitesimals.

The need to efficiently calculate first- and higher-order derivatives of increasingly complex models expressed in Python has stressed or exceeded the capabilities of available tools. In this work, we explore techniques from the field of automatic differentiation (AD) that can give researchers expressive power, performa…

The aim of this paper is to propose an unambiguous intrinsic formalism for higher-order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories, which implies the existence of different Cartan forms and Legendre transformations. We propose a differential-geo…

We present a system for the automatic differentiation of a higher-order functional array-processing language. The core functional language underlying this system simultaneously supports both source-to-source automatic differentiation and global optimizations such as loop transformations. Thanks to this feature, we demo…

Paper tackles stochastic control with mean and higher-order moments, finding Nash equilibria.

We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher order theory naturally involves higher algebra (n-fold groupoids).(conceptual, topol…

This paper studies a particular class of higher order conformally invariant dif- ferential operators and related integral operators acting on functions taking values in particular finite dimensional irreducible representations of the Spin group. The differential operators can be seen as a generalization to higher spin …

Study privacy vs. utility in estimating network parameters with aggregated data.

A method to estimate high order derivatives of data distributions from samples.

The paper proposes a method to compute higher infinitesimals in numerical and symbolic analysis.

Higher-order geometry modifies Newtonian dynamics and predicts anomalies in spacecraft motion.

Closed-form relations and approximations for SE(3) derivatives for robust numerical simulations.

ξ-torch simplifies physics-informed learning by providing differentiable functionals.

The moments of spatial probabilistic systems are often given by an infinite hierarchy of coupled differential equations. Moment closure methods are used to approximate a subset of low order moments by terminating the hierarchy at some order and replacing higher order terms with functions of lower order ones. For a give…

A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an $(n+1)$-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations o…

In this paper we analyze the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete …