Related papers: Exact formula on upper box dimension of generic H\…

We derive, from conformal invariance and quantum gravity, the multifractal spectrum f(alpha,c) of the harmonic measure (or electrostatic potential, or diffusion field) near any conformally invariant fractal in two dimensions, corresponding…

This paper gives the pointwise H\"older (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These…

We apply the structure theory of finite dimensional algebras in order to deduce dimension formulas for spaces of period numbers, i.e., complex numbers defined by integrals of algebraic nature. We get a complete and conceptually clear answer…

The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the…

A new upper bound is given for the dimension of the space of holomorphic cusp forms of weight one and prime level $q$: $$ \hbox{dim}\, S_1(q) << q^{11/12} \log^4{q} $$ with an absolute implied constant.

We prove for an arbitrary complex $^*$-algebra $A$ that every topologically irreducible $^*$-representation of $A$ on a Hilbert space is finite dimensional precisely when the Lebesgue decomposition of representable positive functionals over…

We study generic fractal properties of bounded self-adjoint operators through lower and upper generalized fractal dimensions of their spectral measures. Two groups of results are presented. Firstly, it is shown that the set of vectors whose…

We establish sharp higher-order H\"older regularity estimates up to the boundary for solutions to equations of the form $\partial_t u-Lu=f(t,x)$ in $I\times\Omega$ where $I\subset\mathbb{R}$, $\Omega\subset\mathbb{R}^n$ and $f$ is H\"older…

There are many research papers dealing with fractal dimension of real-valued fractal functions in the recent literature. The main focus of the present paper is to study fractal dimension of complex-valued functions. This paper also…

This work aims to extend the existing results on the Hausdorff dimension of the classical thick point sets of a Gaussian free field (GFF) to a more general class of exceptional sets. We adopt the circle or sphere averaging regularization to…

Let $\mathfrak{g}$ be a complex Kac-Moody algebra, with Cartan subalgebra $\mathfrak{h}$. Also fix a weight $\lambda\in\mathfrak{h}^*$. For $M(\lambda)\twoheadrightarrow V$ an arbitrary highest weight $\mathfrak{g}$-module, we provide a…

In this paper, we generalize the weighted Fourier transform with respect to a function, originally proposed for the one-dimensional case in \cite{Dorrego}, to the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. We develop a comprehensive…

We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper…

Model complexity is an important factor to consider when selecting among graphical models. When all variables are observed, the complexity of a model can be measured by its standard dimension, i.e. the number of independent parameters. When…

Let \((a_n)_{n \in \mathbb{N}}\) be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension $d \geq 1$, we establish asymptotic upper bounds for the maximal gap in the set of dilates…

A generalized family of transcendental (non-polynomial entire) functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to one. Further, there exist multiply connected wandering domains,…

The main purpose of this paper is to compute all irreducible spherical functions on $G={SL}(2,{\mathbb C})$ of arbitrary type $\delta\in \hat K$, where $K={SU}(2)$. This is accomplished by associating to a spherical function $\Phi$ on $G$ a…

In this paper we have introduced two new classes $\mathcal{H}\mathcal{M}(\beta, \lambda, k, \nu)$ and $\overline{\mathcal{H}\mathcal{M}} (\beta, \lambda, k, \nu)$ of complex valued harmonic multivalent functions of the form $f = h +…

We prove higher H\"older regularity for solutions of equations involving the fractional $p-$Laplacian of order $s$, when $p\ge 2$ and $0<s<1$. In particular, we provide an explicit H\"older exponent for solutions of the non-homogeneous…

This work is about global H\"older regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the…