Related papers: On ultradifferentiable functions
In this note, we construct an algorithm that, on input of a description of a structurally stable planar dynamical flow $f$ defined on the closed unit disk, outputs the exact number of the (hyperbolic) equilibrium points and their locations…
For a finite set $A \subseteq \mathbb{R}^n$, consider a function $u \in \mathrm{BV}_{\mathrm{loc}}^2(\mathbb{R}^n)$ such that $\nabla u \in A$ almost everywhere. If $A$ is convex independent, then it follows that $u$ is piecewise affine…
Recent work showed that a theorem of Joris (that a function $f$ is smooth if two coprime powers of $f$ are smooth) is valid in a wide variety of ultradifferentiable classes $\mathcal C$. The core of the proof was essentially…
The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such…
In this note we extend some of the results of a previous paper \url{arXiv:math/0511593} to algebraically closed fields of finite characteristic. In particular, we show that there is an explicit expression in $n$ and $d$ which is divisible…
We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above…
Let H be the space of quaternions, with its standard hypercomplex structure. Let R(D) be the module of regular functions on D. For every unitary vector p in S^2, R(D) contains the space of holomorphic functions w.r.t. the complex structure…
This article examines differentiability properties of the value function of positioning choice problems, a class of optimisation problems in finite-dimensional Euclidean spaces. We show that positioning choice problems' value function is…
High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of…
In this paper, applying the Fa\`a di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials…
This work provides a systematic study of the variational properties of decomposable functions which are compositions of an outer support function and an inner smooth mapping under certain constraint qualifications. A particular focus is put…
Derivative-matching approximations are constructed as power series built from functions. The method assumes the knowledge of special values of the Bell polynomials of the second kind, for which we refer to the literature. The presented…
We introduce a new setting, the category of $\omega$PAP spaces, for reasoning denotationally about expressive differentiable and probabilistic programming languages. Our semantics is general enough to assign meanings to most practical…
The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several…
Fourier matrices naturally appear in many applications and their stability is closely tied to performance guarantees of algorithms. The starting point of this article is a result that characterizes properties of an exponential system on a…
In this paper, on the basis of a specific question raised in [6], we further continue our investigations on the uniqueness of a meromorphic function with its higher derivatives sharing two sets and answer the question affirmatively.…
This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{P}\Delta(0;1_n)$. We provide a definitive resolution to the Bohr…
It is proved that every function of finite Baire index on a separable metric space $K$ is a $D$-function, i.e., a difference of bounded semi-continuous functions on $K$. In fact it is a strong $D$-function, meaning it can be approximated…
This paper concerns the twice epi-differentiability and parabolic regularity of a class of non-amenable functions, the composition of a piecewise twice differentiable (PWTD) function and a parabolically semidifferentiable mapping. Such…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…