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This work presents some outcomes of a theoretical investigation of incompressible high-order networks defined by a generalized graph representation. We study some of their network topological properties and how these may be related to…
A shape optimization problem arising from the optimal reinforcement of a membrane by means of one-dimensional stiffeners or from the fastest cooling of a two-dimensional object by means of ``conducting wires'' is considered. The criterion…
This paper explores the characterization of equivariant linear layers for representations of permutations and related groups. Unlike traditional approaches, which address these problems using parameter-sharing, we consider an alternative…
We establish a structure theorem for degree three codimension one foliations on projective spaces of dimension $n\ge 3$, extending a result by Loray, Pereira, and Touzet for degree three foliations on $\mathbb P^3$. We show that the space…
We study the connections between three seemingly different combinatorial structures - "uniform" brackets in statistics and probability theory, "containers" in online and distributed learning theory, and "combinatorial Macbeath regions", or…
Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying…
In dimensions greater than or equal to three, we establish global uniqueness and obtain reconstruction in the Calderon problem for the Schrodinger equation with certain singular potentials. The potentials considered are conormal of order…
To make sense of the world our brains must analyze high-dimensional datasets streamed by our sensory organs. Because such analysis begins with dimensionality reduction, modelling early sensory processing requires biologically plausible…
The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper…
We consider the solution of linear systems with tensor product structure using a GMRES algorithm. In order to cope with the computational complexity in large dimension both in terms of floating point operations and memory requirement, our…
We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary…
Einstein-Weyl geometry is a triple (D,g,w), where D is a symmetric connection, [g] is a conformal structure and w is a covector such that: (i) connection D preserves the conformal class [g], that is, Dg=wg; (ii) trace-free part of the…
The multidimensional ($n$-D) systems described by Roesser model are presented in this paper. These $n$-D systems consist of discrete systems and continuous fractional order systems with fractional order $\nu$, $0<\nu<1$. The stability and…
We examine the five-dimensional super-de Rham complex with $N = 1$ supersymmetry. The elements of this complex are presented explicitly and related to those of the six-dimensional complex in $N = (1, 0)$ superspace through a specific notion…
A scheme to reduce translational noninvariant quasi-one-dimensional wave guides into singly or multiply connected one-dimensional (1D) lines is proposed. It is meant to simplify the analysis of wave guides, with the low-energy properties of…
Non-invertible symmetries have by now seen numerous constructions in higher dimensional Quantum Field Theories (QFT). In this paper we provide an in depth study of gauging 0-form symmetries in the presence of non-invertible symmetries. The…
The question addressed is whether magnetic materials based on physical small world networks are possible. Physical constraints, such as uniform bond length and embedding in three dimensions, are the new features added to make small world…
The statistical problem of estimating the effective dimension-reduction (EDR) subspace in the multi-index regression model with deterministic design and additive noise is considered. A new procedure for recovering the directions of the EDR…
Let $\mathbb T$ be the differential field of transseries. We establish some basic properties of the dimension of a definable subset of ${\mathbb T}^n$, also in relation to its codimension in the ambient space ${\mathbb T}^n$. The case of…
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ``conformal'' (multicomponent) and Kontsevich models are considered in some detail, together with the…