Related papers: Generalized cut operation associated with higher o…
In this paper, we present a new method to derive formulas for the generating functions of interval orders, counted with respect to their size, magnitude, and number of minimal and maximal elements. Our method allows us not only to…
We generalize the definition of convolution of vectors and tensors on the 2-sphere, and prove that it commutes with differential operators. Moreover, vectors and tensors that are normal/tangent to the spherical surface remain so after the…
In this paper, we show that contraction operations preserve the homology of $n$D generalized maps, under some conditions. Removal and contraction operations are used to propose an efficient algorithm that compute homology generators of $n$D…
In this paper, we develop a new concept of Global Curvature Bound for an arbitrary nonlinear operator between abstract metric spaces. We use this notion to characterize the global complexity of high-order algorithms solving composite…
A notion of super operator system is defined which generalizes the usual notion of operator systems to include certain unital involutive operator spaces which cannot be represented completely isometric as a concrete operator system on some…
A definition for functions of multidimensional arrays is presented. The definition is valid for third-order tensors in the tensor t-product formalism, which regards third-order tensors as block circulant matrices. The tensor function…
Capturing complex high-order interactions among data is an important task in many scenarios. A common way to model high-order interactions is to use hypergraphs whose topology can be mathematically represented by tensors. Existing methods…
We construct the ($\beta$-deformed) higher order total derivative operators and analyze their remarkable properties. In terms of these operators, we derive the higher order constraints for the ($\beta$-deformed) Hermitian matrix models. We…
We generalize the classical notion of adjoint of a linear operator and the Aron-Schottenloher notion of adjoint of a homogeneous polynomial. The general notion is shown to enjoy several properties enjoyed by the classical ones, nevertheless…
In the current era of vast data and transparent machine learning, it is essential for techniques to operate at a large scale while providing a clear mathematical comprehension of the internal workings of the method. Although there already…
We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation…
Well-known operations defined on a non-degenerate inner product vector space are extended to the case of a degenerate inner product. The main obstructions to the extension of these operations to the degenerate case are (1) the index…
In this paper we provide an explicit general construction of higher homotopy operations in model categories, which include classical examples such as (long) Toda brackets and (iterated) Massey products, but also cover unpointed operations…
In the paper we investigate an algorithmic associative binary operation $*$ on the set $\mathcal{LR}_1$ of Littlewood-Richardson tableaux with entries equal to one. We extend $*$ to an algorithmic nonassociative binary operation on the set…
These notes provide an introduction to the algebra and geometry of differential operators and jet bundles. Their point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalisations of Lie…
We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many…
In this short note we construct a simple cut-and-join operator representation for Kontsevich-Witten tau-function that is the partition function of the two-dimensional topological gravity. Our derivation is based on the Virasoro constraints.…
A common task in robotics is unloading identical goods from a tray with rectangular grid structure. This naturally leads to the idea of programming the process at one grid position only and translating the motion to the other grid points,…
Upper and lower quantum functionals, introduced by Christandl, Vrana and Zuiddam (STOC 2018, J. Amer. Math. Soc. 2023), are families of monotone functions of tensors indexed by a weighting on the set of subsets of the tensor legs. Inspired…
In this paper, we define generalized Casimir operators for a loop contragredient Lie superalgebra and prove that they commute with the underlying Lie superalgebra. These operators have applications in the decomposition of tensor product…