Related papers: An Infinite, Converging, Sequence of Brocard Poris…
Bootstrap percolation on a graph with infection threshold $r\in \mathbb{N}$ is an infection process, which starts from a set of initially infected vertices and in each step every vertex with at least $r$ infected neighbours becomes…
Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^d$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,\dots,…
We study percolation problems of overlapping objects where the underlying geometry is such that in D-dimensions, a subset of the directions has a lattice structure, while the remaining directions have a continuum structure. The resulting…
Given a sequence (a_k) = a_0, a_1, a_2,... of real numbers, define a new sequence L(a_k) = (b_k) where b_k = a_k^2 - a_{k-1} a_{k+1}. So (a_k) is log-concave if and only if (b_k) is a nonnegative sequence. Call (a_k) "infinitely…
This work is devoted to the study of families of infinite parabolic iterated function systems (PIFS) on a closed interval parametrized by vectors in $\mathbb{R}^d$ with overlaps. We show that the Hausdorff dimension and absolute continuity…
A symplectic manifold $(M,\omega)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and…
We study oriented percolation on random causal triangulations, those are random planar graphs obtained roughly speaking by adding horizontal connections between vertices of an infinite tree. When the underlying tree is a geometric…
The Burling sequence is a sequence of triangle-free graphs of unbounded chromatic number. The class of Burling graphs consists of all the induced subgraphs of the graphs of this sequence. In the first and second parts of this work, we…
In this paper, we characterise the family of finite arc-transitive bicirculants. We show that every finite arc-transitive bicirculant is a normal $r$-cover of an arc-transitive graph that lies in one of eight infinite families or is one of…
We consider random filtered complexes built over marked point processes on Euclidean spaces. Examples of our filtered complexes include a filtration of $\check{\textrm{C}}$ech complexes of a family of sets with various sizes, growths, and…
We introduce a growth process which samples sections of uniform infinite causal triangulations by elementary moves in which a single triangle is added. A relation to a random walk on the integer half line is shown. This relation is used to…
In this paper, we construct the Bessel line ensemble, a countable collection of continuous random curves. This line ensemble is stationary under horizontal shifts with the Bessel point process as its one-time marginal. Its finite…
We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards…
Given a sequence of complex square matrices, $a_n$, consider the sequence of their partial products, defined by $p_n=p_{n-1}a_{n}$. What can be said about the asymptotics as $n\to\infty$ of the sequence $f(p_n)$, where $f$ is a continuous…
A $3$-Prismatoid is the convex hull of two convex polygons $A$ and $B$ which lie in parallel planes $H_A, H_B\subset\mathbb{R}^3$. Let $A'$ be the orthogonal projection of $A$ onto $H_B$. A prismatoid is called nested if $A'$ is properly…
Classical bond percolation theory studies the conditions for a given point in a random graph to be connected to infinity, or "escape" to infinity, via a sequence of random edges. In this work, we present a higher-dimensional generalization…
We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of…
This is an investigation into the exact forms of images of point charges in 2D conducting ellipses and 3D prolate spheroids. For an ellipse with exterior point source, we compare two previous expressions for the analytic continuation inside…
Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family naturally arise in the analysis of the numerical range and Blaschke products. We examine the behaviour of such polygons when the inscribed conic…
We solve the long-standing existence problem of vertex-primitive 2-arc-transitive digraphs by constructing an infinite family of such digraphs.