Definitions Suppose $A = U \cap \{ x \mathop : \boldsymbol \Theta^m ( \| V \|, x ) \geq 1 \}$, $X$ is the set of $x \in U$ such that $\mathrm{Tan}^m ( \| V \|, x )$ is an $m$ dimensional plane, $Y$ is the vectorspace of symmetric endomorphisms of $\mathbf R^n$, the tangent plane map $\tau : X \to Y$ is defined by requiring that $\tau (x) \in Y$ is the canonical orthogonal projection of $\mathbf R^n$ onto $\mathrm{Tan}^m ( \| V \|, x )$ for $x \in X$, $\Theta = \boldsymbol \Theta^m ( \| V \|, \cdot )$, and the trace $T : \mathrm{Hom} ( \mathbf R^n, Y) \to \mathbf R^n$ satisfies $T ( g ) = \sum_{i=1}^n g(u_i)(u_i)$ whenever $g \in \mathrm{Hom} ( \mathbf R^n, Y )$ and $u_1, \ldots, u_n$ form an orthonormal basis of $\mathbf R^n$. ...

Definition Whenever $a \in \mathbf R^n$, $0 < r < \infty$, $\mathbf B(a,r) \subset U$, and $T \in \mathbf G(n,m)$, we define \begin{align*} \phi_q (a,r,T) & = r^{-m/q} \big ( {\textstyle\int_{\mathbf B(a,r) \times \mathbf G(n,m)}} | S_\natural - T_\natural |^q \, \mathrm d V \, (x,S) \big)^{1/q}\ \text{for $1 \leq q < \infty$}, \\ \phi_\infty (a,r,T) & = \inf \big \{ t \mathop : V ( ( \mathbf B(a,r) \times \mathbf G (n,m) ) \cap \{ (x,S) \mathop : | S_\natural - T_\natural | > t \} ) = 0 \big \}. \end{align*} This quantity is usually referred to as the tilt-excess with exponent $q$ of the varifold $V$ in $\mathbf B(a,r)$ with respect to $T$. ...

Problem. Are there infinitely many natural numbers $n$, for which \[ 2^n+1=p.q, \] where $p$ and $q$ are different prime numbers ?

Suggestion. Numerical experiments show that there are infinitely many numbers $n$ with the above property. Are there an asymptotic for these $n$ ? ...

Problem: Is the scaling limit of ASM convex? ...

Pairs problem: Given $t \geq 2$ pairs of vertices of a regular $n$-gon ($n \geq 3$ integer) with $2t \leq n$, is it possible to rotate the individual pairs to mutually disjoint copies? What is the largest number $t = t(n)$ of pairs that always rotate apart, regardless of the data? For instance, if $n \geq 8$ is a multiple of $4$ and $t = \frac{n}{4} + 1$ then $t$ pairs can be rotated apart. For $5 \leq n \leq 38$ computations show that the following proposal works: $t =\frac{n}{3}$ if $n$ is divisible by $3$ and $t =\lceil \frac{n+2}{3} \rceil$ otherwise. For $n \leq 11$, or if $n$ is a multiple of $3$, or if $n = 14, 20$, this boundary is the desired $t(n)$. The value is not the best possible for other $n \leq 32$. A peculiar result is that $n$ is of type $8x$ or $8x + 2$ if and only if the $n$-gon can be partitioned into $\frac{n}{2}$ pairs with all different diameters $1,2,\ldots,\frac{n}{2}$. Difficulties in determining the exact value of $t(n)$ are seemingly due to the appearance of pairs having the same diameter. ...

We consider finite, undirected graphs that may contain parallel edges but no loops. If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a coloring of edges of $G$ with edges of $H$, such that any three edges incident to a vertex of $G$ are colored by a similar triple of edges of $H$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. If $C$ is a set of connected cubic graphs, then we say that $C$ is hereditary, if for any $G$, $H$, we have $H\prec G$, $H\in C$ implies $G\in C$. ...

Denote by $I_\kappa^n(v)$ the perimeter (i.e. $(n-1)$-dimensional volume of the boundary) of a round ball of volume $v$ in the simply connected Riemannian manifold $X_\kappa^n$ of dimension $n$ and constant curvature $\kappa$. It is known that every domain $D\subset X_\kappa^n$ has perimeter at least $I_\kappa^n(\lvert D\rvert)$, where $\lvert D\rvert$ denotes the volume of $D$. ...