Related papers: Locally piecewise affine functions and their order…

A subset $B \subset Y$ is constructible if it is an element of the smallest family that contains all open sets and is stable under finite intersections and complements. A function $f : X \to Y$ is said to be piece-wise closed if $X$ can be…

We prove that a (globally) subanalytic p-adic function which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken…

Let $\phi_1,\ldots,\phi_n:[0,1]\to (0,1)$ be Lipschitz contractions. Let $I=[0,1)$, $x_0=0$ and $x_n=1$. We prove that for Lebesgue almost every $(x_1,...,x_{n-1})$ satisfying $0<x_1<\cdots <x_{n-1}<1$, the piecewise contraction $f:I\to I$…

Let $V$ be a vector space over a finite field $k$. We give a condition on a subset $A \subset V$ that allows for a local criterion for checking when a function $f:A \to k$ is a restriction of a polynomial function of degree $<m$ on $V$. In…

We describe a method for approximating a single-variable function $f$ using persistence diagrams of sublevel sets of $f$ from height functions in different directions. We provide algorithms for the piecewise linear case and for the smooth…

Motivated by ill-posed PDEs such as $\mathrm{div} (v) = F$ we study locally convex topologies $\mathcal{T}_{\mathcal{C}}$ on real vector spaces $X$ that are a ``localized'' version of a locally convex topology $\mathcal{T}$ to members of a…

It is known that piecewise affine surface homeomorphisms always have measures of maximal entropy. This is easily seen to fail in the discontinuous case. Here we describe a piecewise affine, globally continuous surface map with no measure of…

Deep networks with continuous piecewise affine activations induce polyhedral partitions of the input space, making the number of realized affine regions a natural measure of expressive capacity and a key determinant of how well the model…

We show that any given function can be approximated with arbitrary precision by solutions of linear, time-fractional equations of any prescribed order. This extends a recent result by Claudia Bucur, which was obtained for time-fractional…

We show that on a Riemann surface lamination locally embedded in $\mathbb{C}^2$, $C^1$ functions (in the sense of the $C^1$ structure of the lamination) are uniform limits of ambient $C^1$ functions, with $L^p$ control on the derivatives…

We construct a differentiable locally Lipschitz function $f$ in $\mathbb{R}^{N}$ with the property that for every convex body $K\subset \mathbb{R}^N$ there exists $\bar x \in \mathbb{R}^N$ such that $K$ coincides with the set $\partial_L…

In this article we study fine regularity properties for mappings of finite distortion. Our main theorems yield strongly localized regularity results in the borderline case in the class of maps of exponentially integrable distortion.…

In the present work we study a problem of finding a global minimum of a piecewise affine function. We employ optimality conditions for the problem in terms of coexhausters and use them to state and prove necessary and sufficient conditions…

Let $\mathcal{P}$ be a property of function $\mathbb{F}_p^n \to \{0,1\}$ for a fixed prime $p$. An algorithm is called a tester for $\mathcal{P}$ if, given a query access to the input function $f$, with high probability, it accepts when $f$…

Functions in a Sobolev space are approximated directly by piecewise affine interpolation in the norm of the space. The proof is based on estimates for interpolations and does not rely on the density of smooth functions.

We present a new piecewise linear regression methodology that utilizes fitting a difference of convex functions (DC functions) to the data. These are functions $f$ that may be represented as the difference $\phi_1 - \phi_2$ for a choice of…

In this paper, we prove that a continuous $\mathcal F$-plurisubharmonic functions defined in an $\mathcal F$-open set in $\mathbb C^n$ is $\mathcal F$-maximal if and only if it is $\mathcal F$-locally $\mathcal F$-maximal.

We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating…

In this article we study the local structure of the Fibonacci Partition Function by relating it to a cocycle over an irrational rotation.

We show that $C(X)$ admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided $X$ is Fedorchuk compact of spectral height 3. In other words $X$ admits a fully closed map $f$ onto a metric compact $Y$ such…