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Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore…

We proposed a deterministic multidimensional growth model for small-world networks. The model can characterize the distinguishing properties of many real-life networks with geometric space structure. Our results show the model possesses…

The rotation of multi-dimensional arrays, or tensors, is a fundamental operation in computer science with applications ranging from data processing to scientific computing. While various methods exist, achieving this rotation in-place…

Certain aspects of nonrelativistic diffeomorphisms in 2+1 dimensions are investigated. These include a nonrelativistic limit of some relativistic actions in 3 dimensions, the Seiberg-Witten map, a modification of the viscosity tensor in…

Let W be a projective variety of dimension n+1, L a free line bundle on W, X in $H^0(L^d)$ a hypersurface of degree d which is generic among those given by sums of monomials from $L$, and let $f : Y \to X$ be a generically finite map from a…

While going deeper has been witnessed to improve the performance of convolutional neural networks (CNN), going smaller for CNN has received increasing attention recently due to its attractiveness for mobile/embedded applications. It remains…

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost…

Over the last decade, random hyperbolic graphs have proved successful in providing geometric explanations for many key properties of real-world networks, including strong clustering, high navigability, and heterogeneous degree…

We classify quasilinear systems in Riemann invariants whose characteristic webs are linearizable on every solution. Although the linearizability of an individual web is a rather nontrivial differential constraint, the requirement of…

We introduce the pro-tensor network, a categorification of the tensor network, as a fully rigorous yet graphically transparent framework for studying the collection of many many-body theories, which we dub many-many-body theory. We provide…

After identifying the reduced incidence algebra of an arbitrary cobweb poset the very first properties of these algebras are being disclosed.

We introduce two-dimensional tensor network representations of finite groups carrying a 4-cocycle index. We characterize the associated gapped (2+1)D phases that emerge when these anomalous symmetries act on tensor network ground states. We…

We use the Ward identities corresponding to general linear transformations, and derive relations between transport coefficients of $(2+1)$-dimensional systems. Our analysis includes relativistic and Galilean invariant systems, as well as…

We report on initial findings on Gabor systems with multivariate Gaussian window. Unlike the existing characterisation for dimension one in terms of lattice density, our results indicate that the behavior of Gaussians in higher-dimensional…

Consider a codimension $1$ submanifold $N^n\subset M^{n+1}$, where $M^{n+1}\subset\mathbb{R}^{n+2}$ is a hypersurface. The envelope of tangent spaces of $M$ along $N$ generalizes the concept of tangent developable surface of a surface along…

The classical low-dimensional models of thin structures are based on certain a priori assumptions on the three-dimensional deformation and/or stress fields, diverse in nature but all motivated by the smallness of certain dimensions with…

An invariant characterization of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space is given in terms of invariants and covariants of a real binary quartic canonically associated to the characteristic…

We study three aspects of commutation classes of reduced decompositions: the number of commutation classes, the structures of their corresponding graphs, and the enumeration of subnetworks, a concept recently introduced by Warrington [21].…

Dimensionality is one of the most important properties of complex physical systems. However, only recently this concept has been considered in the context of complex networks. In this paper we further develop the previously introduced…

Let L be a link in the 3-sphere that is in thin position but not in bridge position and let P be a thin level sphere. We generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks that P can have,…