Related papers: Fractional Local Dimension
The fractal dimension of a liquid column is a crucial parameter in several models describing the main features of the primary break-up occurring at the interface of a liquid phase surrounded by the gas-flow. In this work, the deformation of…
The thesis deals with applications of fractional calculus to fractals. It introduces the notion of local fractional derivative (LFD). Fractal and multifractal functions have been studied in the thesis using LFD. New kind of equations are…
In a previous paper (arXiv:1608.02262), we used computer-assisted methods to find explicit expressions for the moments of the size of a uniform random (n,n+1)-core partition with distinct parts. In particular, we conjectured that the…
We establish pointwise and distributional fractal tube formulas for a large class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A relative fractal drum (or RFD, in short) is an ordered pair $(A,\Omega)$ of subsets…
A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions…
Let $B^H = \{ B^H(t), t\in\mathbb{R}^N \}$ be an $(N,d)$-fractional Brownian sheet with Hurst index $H=(H_1,\dotsc,H_N)\in (0,1)^N$. The main objective of the present paper is to study the Hausdorff dimension of the image sets $B^H(F+t)$,…
Direct comparison of three-dimensional (3D) objects is computationally expensive due to the need for translation, rotation, and scaling of the objects to evaluate their similarity. In applications of 3D object comparison, often identifying…
Motivated by the Polynomial Freiman-Ruzsa (PFR) Conjecture, we develop a theory of locality in sumsets, with applications to John-type approximation and sets with small doubling. First we show that if $A \subset \mathbb{Z}$ with $|A+A| \le…
The fractional moment method, which was initially developed in the discrete context for the analysis of the localization properties of lattice random operators, is extended to apply to random Schr\"odinger operators in the continuum. One of…
This article introduces the novel notion of dimension preserving approximation for continuous functions defined on $[0,1]$ and initiates the study of it. Restrictions and extensions of continuous functions in regards to fractal dimensions…
This paper continues work begun in \cite{M1}, in which we introduced a theory of Gowers uniformity norms for singular measures on $\mathbb{R}^d$. There, given a $d$-dimensional measure $\mu$, we introduced a $(k+1)d$-dimensional measure…
High speed data transmission is enabled by time and wavelength division multiplexing. Here is introduced fractional dimension order of Nyquist pulses sequences for orthogonal time division multiplexing. Firstly, with a representation of the…
The Point Pair Feature (Drost et al. 2010) has been one of the most successful 6D pose estimation method among model-based approaches as an efficient, integrated and compromise alternative to the traditional local and global pipelines.…
Dimension theory lies at the heart of fractal geometry and concerns the rigorous quantification of how large a subset of a metric space is. There are many notions of dimension to consider, and part of the richness of the subject is in…
In this paper, we consider modular local polynomials. These functions satisfy modularity while they are locally defined as polynomials outside of an exceptional set. We prove an inequality for the dimension of the space of such forms when…
Let $\mathcal{P}$ be an $n$-point subset of Euclidean space and $d\geq 3$ be an integer. In this paper we study the following question: What is the smallest (normalized) relative change of the volume of subsets of $\mathcal{P}$ when it is…
The metric dimension has been introduced independently by Harary, Melter and Slater in 1975 to identify vertices of a graph G using its distances to a subset of vertices of G. A resolving set X of a graph G is a subset of vertices such…
We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of…
Given $[n]=\{1,2,\ldots,n\}$, a partial order $\preceq$ on $[n]$, a label map $\pi : [n] \rightarrow \mathbb{N}$ defined by $\pi(i) = k_i$ with $\sum_{i=1}^{n}\pi (i) = N$, the direct sum $ \mathbb{F}_{q}^{k_1} \oplus…
The local descriptors have gained wide range of attention due to their enhanced discriminative abilities. It has been proved that the consideration of multi-scale local neighborhood improves the performance of the descriptor, though at the…