The problem discussed in this paper was stated by Alexander O. Ivanov and Alexey A. Tuzhilin in 2009. It stands at the intersection of the theories of Gromov minimal fillings and Steiner minimal trees. Thus, it can be considered as one-dimensional stratified version of the Gromov minimal fillings problem. Here we state the problem; discuss various properties of one-dimensional minimal fillings,...

We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed an arbitrary smooth volume. The we obtain, which is equivalent to half-tube formula, of local nature. It can thus be applied any surface region not containing points. provide geometrical interpretation the coefficients appearing expansion, and compute them on some relevan...

the wiener index $w(g)$ of a connected graph $g$‎ ‎is defined as $w(g)=sum_{u,vin v(g)}d_g(u,v)$‎ ‎where $d_g(u,v)$ is the distance between the vertices $u$ and $v$ of‎ ‎$g$‎. ‎for $ssubseteq v(g)$‎, ‎the {it steiner distance/} $d(s)$ of‎ ‎the vertices of $s$ is the minimum size of a connected subgraph of‎ ‎$g$ whose vertex set is $s$‎. ‎the {it $k$-th steiner wiener index/}‎ ‎$sw_k(g)$ of $g$ ...

We investigate the behaviour of the remainder term R(E) in the Weyl formula #fnjE n Eg = Vol(M) (4) d=2 ?(d=2 + 1) E d=2 + R(E) for the eigenvalues E n of a Schrr odinger operator on a d-dimensional compact Rieman-nian manifold all of whose geodesics are closed. We show that R(E) is of the form E (d?1)=2 (p E), where (x) is an almost periodic function of Besicovitch class B 2 which has a limit ...

There exists an extensive literature on Wiener sausages; see e.g. [25] and the references therein. Nevertheless, relatively little is known so far about the geometry of Wiener sausages due to the complex nature of their realizations. One possible description of the geometric structure of (sufficiently regular) subsets of R is given by their d + 1 intrinsic volumes or Minkowski functionals inclu...

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from 120◦. This notion arises in numerical approximations of minimum Steiner trees. We investigate theworst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It ha...

The emph{Harary index} $H(G)$ of a connected graph $G$ is defined as $H(G)=sum_{u,vin V(G)}frac{1}{d_G(u,v)}$ where $d_G(u,v)$ is the distance between vertices $u$ and $v$ of $G$. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ ...

Recently D. Buchholz and R. Verch have proposed a method for implementing in algebraic quantum field theory ideas from renormalization group analysis of short-distance (high energy) behavior by passing to certain scaling limit theories. Buchholz and Verch distinguish between different types of theories where the limit is unique, degenerate, or classical, and the method allows in principle to ex...

For a given convex (semi-convex) function u, defined on a nonempty open convex set Ω ⊂ Rn, we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more general...

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