arXiv research

Study differential properties of matrix square roots in specific cases.

The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures. The…

Study geometric properties of S1 singularities and their deformations.

We study the differential geometric properties of the manifold of non-singular symmetric real matrices endowed with the trace metric; in case of positive definite matrices we describe the full group of isometries

We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set f…

In the present paper we show spectral properties of a littleknown natural Riemannian second-order differential operator acting on differential forms.

In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differen…

In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differen…

GRAND ensures node-level differential privacy for network data.

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational dire…

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

The distance function $\varrho(p,q)$ (or $d(p,q)$) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over $\mathbb R^n$, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…

In this work we systematically analyze general properties of differential equations used as machine learning models. We demonstrate that the gradient of the loss function with respect to to the hidden state can be considered as a generalized momentum conjugate to the hidden state, allowing application of the tools of c…

Study describes how to realize periods of meromorphic differentials with specific properties.

Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.

We count meromorphic differentials with fixed residues and poles of fixed orders.

Private minimum Hellinger distance estimators maintain robustness and efficiency while ensuring privacy.

In this paper we give explicit formulas of differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and differential Euler class. Furthermore, we show that…

We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.

The paper defines and analyzes $abla$-Sobolev spaces and operators on manifolds.

The properties of the Riemann extensions of nonriemannian spaces defined by the first order systems of differential equations are considered.

Deep ResNets exhibit distinct scaling properties with depth, challenging neural ODE models.

This work establishes properties on diffeological structures for set-valued maps and measures.

We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of cuspidal edges. We study the relation of these invariants with several other invar…

We are interested in global properties of systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and finite dimensionality of their cohomology spaces, when acting on various function spaces e.g. smooth, analytic and Gevrey. Extending the …

Study uniformly differentiable graphs in Carnot groups, proving area formulas.

The paper explores the geometry and topology of DNN decision boundaries.

Integrates singular subalgebroids using diffeological groupoids.

We propose a novel deep learning paradigm of differential flows that learn a stochastic differential equation transformations of inputs prior to a standard classification or regression function. The key property of differential Gaussian processes is the warping of inputs through infinitely deep, but infinitesimal, diff…

Study describes how to realize periods of holomorphic differentials with specific properties.

Differential privacy improves AI security, fairness, and learning.

Study differentially private methods for learning Hawkes processes.

Examines WENDy-IRLS algorithm's noise robustness and efficiency in various differential equations.

This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential g…

Characterizes Bonnet surfaces using analytic conditions.

In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equa…

The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.

Neural networks can approximate complex stochastic equations well.

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential in…

In this paper, we study fundamental groups of strata of the moduli space of quadratic differentials. We use certain properties of the Abel-Jacobi map, combined with local surgeries on quadratic differentials, to construct quotient groups of the fundamental groups for a particular family of strata.

We summarize results concerning the Bernstein property of differential equations.

Measures financial resilience using BSDEs and their properties.

Differentially private conformal prediction improves statistical efficiency.

Develops a new method for statistical optimal allocation problems.

Connected boundaries of strata of differentials are always connected in various compactifications.

Constructs a support-preserving homotopy for differential forms with boundary decay estimates.

A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical line bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification of the moduli space of $k$-differentials. In this paper we give a complete description for the compa…

Tensoring $p$-weak differentiable structures preserves their properties.