Related papers: Universal scaling in real dimension
The hyperbolic random graph model (HRG) has proven useful in the analysis of scale-free networks, which are ubiquitous in many fields, from social network analysis to biology. However, working with this model is algorithmically and…
A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$…
We present a new unified theory of critical finite-size scaling for lattice statistical mechanical models with periodic boundary conditions above the upper critical dimension. Our theory is based on recent mathematically rigorous results…
The Universe is a physical object. Physical objects have shapes and sizes. General relativity is insufficient to describe the global shape and size of the Universe: the Hilbert-Einstein equations only treat limiting quantities towards an…
Starting with the large deviation principle (LDP) for the Erd\H{o}s-R\'enyi binomial random graph $\mathcal{G}(n,p)$ (edge indicators are i.i.d.), due to Chatterjee and Varadhan (2011), we derive the LDP for the uniform random graph…
We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the…
When faced with a mathematical model, often the first step is to reduce the complexity of the model by turning variables and parameters into dimensionless quantities. This process is often performed by hand, relying on a skill practiced…
We investigate homogeneity in the special Colombeau algebra. It is shown that strongly scaling invariant functions on the d-dimensional space are simply the constants. On the pierced space, strongly homogeneous functions admit tempered…
It is shown that, in the scaling regime, transport properties of quantum wires with off-diagonal disorder are described by a family of scaling equations that depend on two parameters: the mean free path and an additional continuous…
We address the question of weak versus strong universality scenarios for the random-bond Ising model in two dimensions. A finite-size scaling theory is proposed, which explicitly incorporates $\ln L$ corrections ($L$ is the linear finite…
Consider the long-range percolation model on the integer lattice $\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise $x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$, independently of the…
The metric dimension of a graph is the smallest number of nodes required to identify all other nodes based on shortest path distances uniquely. Applications of metric dimension include discovering the source of a spread in a network,…
For thermal transport in one-dimensional (1D) systems, recent studies have suggested that employing different theoretical models and different numerical simulations under different system's parameter regimes might lead to different…
First steps in incorporating Nottale's scale-relativity principle to string theory and extended objects are taken. Scale Relativity is to scales what motion Relativity is to velocities. The universal, absolute, impassible, invariant scale…
A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on…
Using the geometric entanglement measure, we study the scaling of multipartite entanglement in several 1D models at criticality, specifically the linear harmonic chain and the XY spin chain encompassing both the Ising and XX critical…
We extend our two-scale neural-network method for scalar singularly perturbed problems with one small parameter to dynamical systems with multiple small parameters. To accommodate multiple small parameters, we use a single effective scale…
The analysis of the radial distribution function of a system provides a possible procedure for uncovering interaction rules between individuals out of collective movement patterns. This approach from classical statistical mechanics has…
Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the…