arXiv research

Suppose $f(x,y) + \fracκ{2} \|x\|^2 - \fracσ{2}\|y\|^2$ is convex where $σ>0$, and the argmin function $γ(x) = \{ γ: \inf_y f(x,y) = f(x,γ)\}$ exists and is single valued. We will prove $γ$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.

New method learns binary decision trees efficiently.

New method finds minimum in noisy data, useful for model selection.

Paper proposes new loss functions for training energy networks.

Proposes a new risk measurement method for risk-averse stochastic optimization.

EGO-MDA identifies optimal spectral-bands for process discrimination.

We describe a novel family of models of multi- layer feedforward neural networks in which the activation functions are encoded via penalties in the training problem. Our approach is based on representing a non-decreasing activation function as the argmin of an appropriate convex optimiza- tion problem. The new framewor…

This paper introduces a novel generative encoder (GE) model for generative imaging and image processing with applications in compressed sensing and imaging, image compression, denoising, inpainting, deblurring, and super-resolution. The GE model consists of a pre-training phase and a solving phase. In the pre-training …

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…

Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for the existence of a complete uniform differential structure on a given differenti…

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 define the Simons-Sullivan differential analytic index by translating the Freed-Lott differential analytic index via explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory. We prove the differential Grothendieck-Riemann-Roch theorem in Simons-Sullivan differenti…

We generalize stochastic smoothing for gradient estimation of non-differentiable functions.

The paper proves Gorenstein contractions for multiscale differentials on nodal curves.

Classifies components of strata of k-differentials on Riemann surfaces.

Monotonic differentiable sorting networks improve upon previous methods.

Lecture notes introduce differential geometry using sheaves and differential operators.

Given a unital associatve graded algebra we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-th power (N>1) of the differential of this graded q-differential algebra is equal to zero. We use our approach to construct the graded q-differentia…

Finite intersection numbers between horizontal foliations of quadratic differentials.

We give chain homotopy maps of Khovanov-type link homology of a universal differential. The universal differential, discussed by Mikhail Khovanov, Marco Mackaay, Paul Turner and Pedro Vaz, contains the original Khovanov's differential and Lee's differential. We also consider the conditions of any differential ensuring …

We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds admit a unique differential, coinciding with the usual one. Thus smooth manifolds fo…

Paper solves a class of differential equations with specific solutions.

We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra a…

By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology …

We study two notions of relative differential cohomology, using the model of differential characters. The two notions arise from the two options to construct relative homology, either by cycles of a quotient complex or of a mapping cone complex. We discuss the relation of the two notions of relative differential cohomo…

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differentia…

New privacy framework tailored to specific data distributions.

Studies projective geometry and partial differential equations prolongation.

Develops differential K-theory for noncommutative algebras.

Extends differential geometry concepts to manifolds with super tangent bundles.

In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surfaces without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order k exceeding 2 satisfying certa…

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic ch…

Differentiable pipeline replaces non-differentiable CAE components for shape optimization.

Differentiable programming aids in solving differential equations and their sensitivities.

The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS. We show t…

A brief introduction to exterior differential systems for graduate students familiar with manifolds and differential forms. For complete files, see https://github.com/Ben-McKay/introduction-to-exterior-differential-systems

DiffEqFlux.jl is a library for fusing neural networks and differential equations. In this work we describe differential equations from the viewpoint of data science and discuss the complementary nature between machine learning models and differential equations. We demonstrate the ability to incorporate DifferentialEqua…

Differential K-theory gets a $λ$-ring structure.

We study Cheeger-Simons differential characters and provide geometric descriptions of the ring structure and of the fiber integration map. The uniqueness of differential cohomology (up to unique natural transformation) is proved by deriving an explicit formula for any natural transformation between a differential cohom…

Defines semi-symmetric metric connections on differential forms.

In this paper we first prove that every differential character can be represented by differential form with singularities. Then we lift the Gauss-Bonnet-Chern theorem for vector bundles to differential characters.

In this note we prove some results in flat and differential $K$-theory. The first one is a proof of the compatibility of the differential topological index and the flat topological index by a direct computation. The second one is the explicit isomorphisms between Bunke-Schick differential $K$-theory and Freed-Lott diff…

Researchers address the generation of differential invariants for geometric structures.

This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the wo…

The study characterizes infinite Riemann surfaces and their foliations using quadratic differentials.

Defines fiber-wise linear differential operators on vector bundles.

Equivalence of second order differential operators in vector bundles studied.

Develops a computationally tractable high-dimensional differential privacy estimator.