Related papers: Quantitative visibility estimates for unrectifiabl…

We extend a result proved in \cite{Col} for mirror symmetries of planar systems to measure-preserving non-linear reversibilities of $n$-dimensional systems, dropping the analyticity and nondegeneracy conditions.

We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\colon X \to…

We verify the existence of a purely unrectifiable set in which the typical Lipschitz function has a large set of full differentiability points. The example arises from a construction, due to Cs\"ornyei, Preiss and Ti\v{s}er, of a universal…

The classical Besicovitch projection theorem states that if a planar set $E$ with finite length is purely unrectifiable, then almost all orthogonal projections of $E$ have zero length. We prove a quantitative version of this result: if…

Let $K$ be a centrally symmetric spherical and simplicial polytope, whose vertices form a $\frac{1}{4n}-$net in the unit sphere in $\mathbb{R}^n$. We prove a uniform lower bound on the norms of all hyperplane projections $P: X \to X$, where…

Symmetry is conventionally described in a contrariety manner that the system is either completely symmetric or completely asymmetric. Using group theoretical approach to overcome this dichotomous problem, we introduce the degree of symmetry…

Let $\beta>1$. Define a class of similitudes \[S=\left\{f_{i}(x)=\dfrac{x}{\beta^{n_i}}+a_i:n_i\in \mathbb{N}^{+}, a_i\in \mathbb{R}\right\}.\] Let $\mathcal{A}$ be the collection of all the self-similar sets generated by the similitudes…

We sketch a tentative proof of P-completeness for the $\beta$-convertibility problem on untyped planar (a.k.a. ordered or non-commutative) $\lambda$-terms.

We look at the average sum of the Euler's phi function $\phi{(n)}$ and it's relation with the visibility of a point from the origin.We show that $\forall{\hspace{0.05in}{k} \ge{1}},k\in\mathbb{N},\exists$ a $k$$\times$$k$ grid in the 2D…

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…

We improve the best known lower bound for the dimension of radial projections of sets in the plane. We show that if $X,Y$ are Borel sets in $\R^2$, $X$ is not contained in any line and $\dim_H(X)>0$, then $$\sup\limits_{x\in X} \dim_H(\pi_x…

Let $K$ be a number field with ring of integers $\mathcal{O}$. Two lattice points ${\bf x, y}\in \mathcal{O}^m$ with $m\geq 2$ are said to be visible from one another if $\gcd((x_i-y_i),\ldots, (x_m-y_m))=\mathcal{O}$, where $(x_i-y_i)$ is…

Estimates of the fractal dimension $D$ of the set of galaxies in the universe, based on ever improving data sets, tend to settle on $D\approx 2$. This result raised a raging debate due to its glaring contradiction with astrophysical models…

Symmetry is ubiquitous throughout nature and can often give great insights into the formation, structure and stability of objects studied by mathematicians, physicists, chemists and biologists. However, perfect symmetry occurs rarely so…

A well known notion of $k$-rectifiable set can be formulated in any metric space using Lipschitz images of subsets of $\mathbb{R}^k$. We prove some characterizations of $k$-rectifiability, when the metric space is an arbitrary homogeneous…

Supersymmetry does not dictate the way we should quantize the fields in the supermultiplets, and so we have the freedom to quantize the Standard Model (SM) particles and their superpartners differently. We propose a generalized quantization…

Let $\Pi_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound…

We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2…

For every pattern $P$, consisting of a finite set of points in the plane, $S_{P}(n,m)$ is defined as the largest number of similar copies of $P$ among sets of $n$ points in the plane without $m$ points on a line. A general construction,…

We prove that in any Banach space the set of windows in which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants that are independent of the curve, the dimension of the space, and the…