Related papers: Slices of the Takagi function

K. Igusa and G. Todorov introduced the $\varphi$ function which generalizes the notion of projective dimension. We study the behavior of the $\varphi$ function for cyclic Nakayama algebras of infinite global dimension. We prove that the…

On a convex bounded open set, we prove that Poincar\'e-Sobolev constants for functions vanishing at the boundary can be bounded from below in terms of the norm of the distance function in a suitable Lebesgue space. This generalizes a result…

The curve graphs are not locally finite. In this paper, we show that the curve graphs satisfy a property which is equivalent to graphs being uniformly locally finite via Masur--Minsky's subsurface projections. As a direct application of…

We define the notion of a twisted topological graph algebra associated to a topological graph and a $1$-cocycle on its edge set. We prove a stronger version of a Vasselli's result. We expand Katsura's results to study twisted topological…

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{\"u}ller space. For any isotopy class of closed curves $\gamma$, we compute the first three derivatives of the length function…

In 1901, Severi proved that if $Z$ is an irreducible hypersurface in $\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then $Z$ is either a quadratic hypersurface or a scroll of planes. We prove a discretized…

A salami is a connected, locally finite, weighted graph with non-negative Ollivier Ricci curvature and at least two ends of infinite volume. We show that every salami has exactly two ends and no vertices with positive curvature. We moreover…

The Bar\'at-Thomassen conjecture asserts that there is a function $f$ such that for every fixed tree $T$ with $t$ edges, every graph which is $f(t)$-edge-connected with its number of edges divisible by $t$ has a partition of its edges into…

We quantify the pointwise doubling properties of self-similar measures using the notion of pointwise Assouad dimension. We show that all self-similar measures satisfying the open set condition are pointwise doubling in a set of full…

We derive the partition function of 5d ${\cal N}=1$ gauge theories on the manifold $S^3_b \times \Sigma_{\frak g}$ with a partial topological twist along the Riemann surface, $\Sigma_{\frak g}$. This setup is a higher dimensional uplift of…

The Reduction Map Theorem in H. Tsuji's work on numerical trivial fibrations is corrected and proven. To this purpose various definitions of Tsuji's new intersection numbers for pseudo-effective line bundles equipped with a positive…

Let $X:=\mathbb{A}^{n}_{R}$ be the $n$-dimensional affine space over a discrete valuation ring $R$ with fraction field $K$. We prove that any pointed torsor $Y$ over $\mathbb{A}^{n}_{K}$ under the action of an affine finite type group…

We give a new proof of Cartan's fixed point theorem using topological fixed point theory. For an odd dimensional, simply connected and complete manifold having non-positive curvature, we further prove that every isometry with finite order…

We prove that for $1\le k<d$, if $E$ is a Borel subset of $\mathbb{R}^d$ of Hausdorff dimension strictly larger than $k$, the set of $(k+1)$-volumes determined by $k+2$ points in $E$ has positive one-dimensional Lebesgue measure. In the…

Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$. Let $X$ be a smooth completion of $Y$ such that the boundary $X-Y$ is the support of an effective…

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$,…

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote…

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.

Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has…

Let $F$ be a Bedford-McMullen carpet defined by independent exponents. We prove that $\overline{\dim}_B (\ell \cap F) \leq \max \lbrace \dim^* F -1,0 \rbrace$ for all lines $\ell$ not parallel to the principal axes, where $\dim^*$ is…