Related papers: Smooth maps on convex sets

For any symplectic manifold, Hamiltonian diffeomorphism group contains a subset which consists of times one flows of autonomous(time-independent) Hamiltonian vector fields. Polterovich and Shelukhin proved that the complement of autonomous…

Let $C$ be a compact convex subset of $\mathbb{R}^n$, $f:C\to\mathbb{R}$ be a convex function, and $m\in\{1, 2, ..., \infty\}$. Assume that, along with $f$, we are given a family of polynomials satisfying Whitney's extension condition for…

In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called $w^*$-locally relatively compact sets was introduced recently by Fonf, Pallares, Troyanski and the…

Theories of rough sets and soft sets are powerful mathematical tools for modelling various types of vagueness. Hybrid model combining a rough set with a soft set which is called soft rough set proposed by Feng et al. [3] in 2010. In this…

We make an attempt to develop "noncommutative algebraic geometry" in which noncommutative affine schemes are in one-to-one correspondence with associative algebras. In the first part we discuss various aspects of smoothness in affine…

We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave functions on such spaces satisfy several strong continuity properties.

In this paper we study a subclass of subcartesian space-the orbit space of a proper action of Lie group on smooth manifold. We show that continuous functions on orbit space can be approximated by smooth functions.

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where…

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an…

In this paper, we investigate properties of set-valued mappings that establish connection between the values of this map at two arbitrary points of the domain and the value at their midpoint. Such properties are, for instance, Jensen…

We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map $f:D\to D'$ close to a boundary regular contact point $p\in \de…

A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse…

It is known that the classical Banach--Stone theorem does not extend to the class of $AC(\sigma)$ spaces of absolutely continuous functions defined on compact subsets of the complex plane. On the other hand, if $\sigma$ is restricted to the…

We classify the most common local forms of smooth maps from a smooth manifold L to the plane. The word "local" can refer to locations in the source L, but also to locations in the target. The first point of view leads us to a classification…

For any complex scheme X or any dg category, there is an associated K-theory presheaf on the category of complex affine schemes. We study real smooth functions on this presheaf, defined by Kan extension, and show that they are closely…

Recently, versions of neural networks with infinite-dimensional affine operators inside the computational units (``neural operator'' networks) have been applied to learn solutions to differential equations. To enable practical computations,…

We introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosicky. On elastic spaces there is a natural…

This article introduces innovative classes of open sets in \(\mathbb{R}^{N}\), where \(N=2, 3\), characterized by a geometric property associated with the inward normal. The focus lies on proving compactness results for the Hausdorff…

In this note, we construct examples of bounded smooth convex domains with no non-trivial analytic discs on the boundary which possess a holomorphic self-map without fixed points so that the iterates do not converge to a point (that is, the…

The class of special generic maps is a natural class of smooth maps containing Morse functions on spheres with exactly two singular points and canonical projections of unit spheres. We find new restrictions on such maps on $6$-dimensional…