Related papers: A Hilbert space approach to effective resistance m…

We develop a generalized resistance geometry based on Kron reduction and effective resistance for directed graphs, paralleling classical undirected graph theory. For strongly connected directed graphs, we prove a Fiedler--Bapat identity…

We derive a criterion that is convenient for applications and exactly characterizes the test function space on which the operator realization of a given series of Wick powers of a free field is possible. The suggested derivation does not…

Boundary analysis is developed for a rich class of generally infinite weighted graphs with compact metric completions. These graph completions have totally disconnected boundaries. The classical notion of $\epsilon$-components and the…

The resistance between arbitrary two nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulas for two-point resistances are deduced for…

We establish a geometric inequality relating the Dirichlet energy $E_1(f)$ and the bienergy $E_2(f)$ of smooth maps \[ f : (M,g) \to (\overline{M},\overline{g}) \] between Riemannian manifolds. Assume that $(M,g)$ is a compact, connected…

For a finite point set $E\subset \mathbb{R}^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k + 1$ points in E such that the distance between a pair of points is specified if the…

Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations)…

Let $G$ be a connected graph. The resistance distance between two vertices $u$ and $v$ of $G$, denoted by $R_{G}[u,v]$, is defined as the net effective resistance between them in the electric network constructed from $G$ by replacing each…

Let $(N,\rho)$ be a Riemannian manifold, $S$ a surface of genus at least two and let $f\colon S \to N$ be a continuous map. We consider the energy spectrum of $(N,\rho)$ (and $f$) which assigns to each point $[J]\in \mathcal{T}(S)$ in the…

A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Persistence diagrams do not carry intrinsic spectral or kernel structures, so applications typically use…

For a Reproducing Kernel Hilbert Space on a complex domain we give a formula that describes the Hermitean metrics on the domain which are pull-backs of some metric on the (dual of) the RKHS via the evaluation map. Then we consider the…

Let $A$ be a set and $V$ a real Hilbert space. Let $H$ be a real Hilbert space of functions $f:A\to V$ and assume $H$ is continuously embedded in the Banach space of bounded functions. For $i=1,\cdots,n$, let $(x_i,y_i)\in A\times V$…

It is shown that the natural framework for the solutions of any Schrodinger equation whose spectrum has a continuous part is the Rigged Hilbert Space rather than just the Hilbert space. The difficulties of using only the Hilbert space to…

For a compact set $E \subset \mathbb R^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k+1$ points in $E$ such that the distance between a pair of points is specified if the corresponding…

We provide new results on the structure of optimal transportation networks obtained as minimizers of an energy cost functional consisting of a kinetic (pumping) and material (metabolic) cost terms, constrained by a local mass conservation…

We present a novel framework for variable selection in Fr\'echet regression with responses in general metric spaces, a setting increasingly relevant for analyzing non-Euclidean data such as probability distributions and covariance matrices.…

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in $Q$-polynomial distance-regular…

We investigate the AC conductivity of binary random impedance networks, with emphasis on its dependence on the ratio of the complex conductances of both phases. We propose an algorithm to determine the conductance of a finite network, in…

There are different measures to classify a network's data set that, depending on the problem, have different success. For example, the resistance distance and eigenvector centrality measures have been successful in revealing ecological…

The possible functional forms of the effective conductivity sigma_{eff} of the randomly inhomogeneous two-phase system at arbitrary values of concentrations are discussed. A new functional equation, generalizing the duality relation, is…