Related papers: Superdifferential of the Takagi function

A fundamental open question asking whether all real-valued strongly quasiconvex functions defined on $\mathbb R^n$ are necessarily continuous, akin to their convex counterparts, is answered in detail in this paper. Among other things, we…

We study a discrete analogue of the classical multivariate Gaussian distribution. It is supported on the integer lattice and is parametrized by the Riemann theta function. Over the reals, the discrete Gaussian is characterized by the…

We give necessary and sufficient conditions for a real-valued quasiconvex function f on a Baire topological vector space X (in particular, Banach or Frechet space) to be continuous at the points of a residual subset of X. These conditions…

There are many functions which are continuous everywhere but not differentiable at some points, like in physical systems of ECG, EEG plots, and cracks pattern and for several other phenomena. Using classical calculus those functions cannot…

In this paper, we study the uniqueness of the differential-difference polynomials of entire functions on $\mathbb{C}^{n}$. We prove the following result: Let $f(z)$ be a transcendental entire function on $\mathbb{C}^{n}$ of hyper-order less…

We construct a separate continuous function $f\colon\mathbb{Q}\times\mathbb{Q}\to[0,1] $ and a dense subset $D\subseteq \mathbb{Q}\times\mathbb{Q}$ such that $f[D]$ is not dense in $f[\mathbb{Q}\times\mathbb{Q}]$, in other words, $f$ is…

In the framework of superanalysis we get a functions theory close to complex analysis, under a suitable condition (A) on the real superalgebras in consideration. Under the condition (A), we get an integral representation formula for the…

We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation".…

We study two-dimensional subshifts whose horizontal trace (a.k.a. projective subdynamics) contains only points of finite support. Our main result is a classification result for such subshifts satisfying a minimality property. As…

Let $T$ be an absolutely continuous polynomially bounded operator, and let $\theta$ be a singular inner function. It is shown that if $\theta(T)$ is invertible and some additional conditions are fulfilled, then $T$ has nontrivial…

We construct uncountably generated algebras inside the following sets of special functions: Sierpi\'nski-Zygmund functions, perfectly everywhere surjective functions and nowhere continuous Darboux functions. All conclusions obtained in this…

We study a fractional differentiation operator for functions on the conjugate space to an infinite extension of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. In particular, a…

Subdifferentials (in the sense of convex analysis) of matrix-valued functions defined on $\mathbb{R}^d$ that are convex with respect to the L\"{o}wner partial order can have a complicated structure and might be very difficult to compute…

Let $\mathfrak{p}_{\mathbb{P}_r}(n)$ denote the number of partitions of $n$ into $r$-full primes. We use the Hardy-Littlewood circle method to find the asymptotic of $\mathfrak{p}_{\mathbb{P}_r}(n)$ as $n \to \infty$. This extends previous…

This paper presents a necessary and sufficient condition for a real-valued function defined on an open and convex subset of a Banach space to be quasi-concave, and a sufficient condition for such a function to be strictly quasi-concave.…

This work provides calculus for the Fr\'echet and limiting subdifferential of the pointwise supremum given by an arbitrary family of lower semicontinuous functions. We start our study showing fuzzy results about the Fr\'echet…

We present a simple proof of the maximal monotonicity of the subdifferential operator in general Banach spaces. Using the Fitzpatrick function the Rockafellar surjectivity theorem follows as a corollary.

We prove that functions that are homogeneous of degree $\alpha \in (0,\,1)$ on $\mathbb{R}^n$ and have nowhere vanishing Hessian determinant cannot change sign.

For a function space $X(\OO)$ satisfying weak assumptions we prove that the generic function in $X(\OO)$ is totally unbounded, hence non-extendable. We provide several examples of such spaces; they are mainly localized versions of classical…

Near every point of a real-analytic set in $\mathbb R^n$, we make use of Hironaka's resolution of singularity theorem to construct a family of continuous functions in $W^{1, 1}_{loc}$ such that their weak derivatives have (removable)…