Related papers: On the derivative of the Minkowski question mark f…

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \| f(z) \|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…

The Minkowski question mark function is a rich object which can be explored from the perspective of dynamical systems, complex dynamics, metric number theory, multifractal analysis, transfer operators, integral transforms, and as a function…

The family of functions, we investigate in this article, was originally introduced by A.Denjoy and later rediscovered by R Tichy and J. Uitz. We denote the functions of the family by $g_{\lambda}(x),$ where $\lambda\in(0,1)$. The definition…

In this short note, we establish the following result: Let $f:[0,+\infty[\to [0,+\infty[$, $\alpha:[0,1]\to ]0,+\infty[$ be two continuous functions, with $f(0)=0$. Assume that, for some $a>0$, the function $\xi\to…

In this paper I consider the applications of several kinds of approximations of real functions to the problem of verified computation (reliable computing) of the range of implicitly defined real function $x_{n+1} = G(x_{1}, ..., x_{n}),$…

The main objective of this paper is to present Ostrowski's inequality for a broader class of functions and to propose a refinement to the classical version of it. The original Ostrowski's inequality can be stated as follows "If…

Given a weakly compact cardinal $\kappa$, we give an axiomatization of intuitionistic first-order logic over $\mathcal{L}_{\kappa^+, \kappa}$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the…

To each weakly holomorphic modular function $f\not \equiv 0$ for $\mathrm{SL}(2,\mathbb{Z})$, which is non-negative on the geodesic arc $\{e^{it} : \pi/3\leq t\leq 2\pi/3\}$, we attach a $\mathrm{GL}(2,\mathbb{Z})$-invariant map…

We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q<p+2$ with $p \geq 2$. This relation improves existing restrictions when $p \leq N-1$,…

We show that the conclusion of Plachky-Steinebach theorem holds true for intervals of the form $\left]\overline{L}_r'(\lambda),y\right[$, where $\overline{L}_r'(\lambda)$ is the right derivative (but not necessarily a derivative) of the…

Motivated from studies on anomalous diffusion, we show that the memory function $M(t)$ of complex materials, that their creep compliance follows a power law, $J(t)\sim t^q$ with $q\in \mathbb{R}^+$, is the fractional derivative of the Dirac…

In this paper, we study a Landis-type conjecture for the general fractional Schr\"{o}dinger equation $((-P)^{s}+q)u=0$. As a byproduct, we also proved the additivity and boundedness of the linear operator $(-P)^{s}$ for non-smooth…

Suppose $f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$ is convex where $\sigma>0$, and the argmin function $\gamma(x) = \{ \gamma : \inf_y f(x,y) = f(x,\gamma)\}$ exists and is single valued. We will prove $\gamma$ is…

Let $x \in [0,1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x), \cdots,a_n(x),\cdots]$ and $q_n(x)$ be the denominator of its $n$-th convergent. We establish, for any $\alpha,\beta$ in $[0,+\infty]$, the…

Let $f\colon \mathbb{R}_+\to\mathbb{R}$ be a continuous and strictly monotone function. In the main result of this paper, we show that, for a fixed $n\geq 2$, the $n$-variable mean $\mathscr{A}_f \colon \mathbb{R}_+^n \to \mathbb{R}_+$…

We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant derivatives of an order $\alpha \in (0, 1)$. Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by…

The continued fraction expansion of the real number $x=a_0+x_0, a_0\in {\ZZ},$ is given by $0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in {\NN},$ for $n\geq 0.$ The Brjuno function is then $B(x)=\sum_{n=0}^{\infty}x_0x_1...…

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set…

We prove that: I. If $L$ is a $T_1$ space, $|L|>1$ and $d(L) \leq \kappa \geq \omega$, then there is a submaximal dense subspace $X$ of $L^{2^\kappa}$ such that $|X|=\Delta(X)=\kappa$; II. If $\frak{c}\leq\kappa=\kappa^\omega<\lambda$ and…

In this paper, we derive the Dirac equation in the $\kappa$-deformed Minkowski space-time. We start with $\kappa$-deformed Minkowski space-time and investigate the undeformed $\kappa$-Lorentz transformation valid to all order in the…