Related papers: Interpretations and Representations of Classical T…
The power of multivariate functions is their ability to model a wide variety of phenomena, but have the disadvantages that they lack an intuitive or interpretable representation, and often require a (very) large number of parameters. We…
An assessment is given as to the extent to which pure unitary evolution, as distinct from environmental decohering interaction, can provide the transition necessary for an observer to interpret perceived quantum dynamics as classical. This…
For classical knots, there is a concept of (semi)meander diagrams; in this short note we generalize this concept to virtual knots and prove that the classes of meander and semimeander diagrams are universal (this was known for classical…
The paper presents a REDUCE program for the simplification of tensor expressions that are considered as formal indexed objects. The proposed algorithm is based on the consideration of tensor expressions as vectors in some linear space. This…
Tensor hierarchies are algebraic objects that emerge in gauging procedures in supergravity models, and that present a very deep and intricate relationship with Leibniz (or Loday) algebras. In this paper, we show that one can canonically…
We present a classical analog of the quantum metric tensor, which is defined for classical integrable systems that undergo an adiabatic evolution governed by slowly varying parameters. This classical metric measures the distance, on the…
Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity…
Techniques in which words are represented as vectors have proved useful in many applications in computational linguistics, however there is currently no general semantic formalism for representing meaning in terms of vectors. We present a…
We study the structure of tensor representations of the classical infinite-dimensional locally finite Lie algebras $gl_\infty$, $sl_\infty$, $sp_\infty$ and $so_\infty$. In contrast with the finite-dimensional case, these tensor…
Tensor networks (TNs) have been gaining interest as multiway data analysis tools owing to their ability to tackle the curse of dimensionality and to represent tensors as smaller-scale interconnections of their intrinsic features. However,…
The BSSN and standard gauge equations are written in covariant form with respect to spatial coordinate transformations. The BSSN variables are defined as tensors with no density weights. This allows us to evolve a given set of initial data…
Tensor harmonics are a useful mathematical tool for finding solutions to differential equations which transform under a particular representation of the rotation group $\mathrm{SO}(3)$. The aim of this work is to make use of this tool also…
Following the guidelines of classical differential geometry the `building material' for the tensor calculus in non-commutative geometry is suggested. The algebraic account of moduli of vectors and covectors is carried out.
We present general formulas for transverse and transverse-traceless (TT) symmetric tensors in flat spaces. TT tensors in conformally flat spaces can be obtained by means of a conformal transformation.
Classical dimensional analysis is one of the cornerstones of qualitative physics and is also used in the analysis of engineering systems, for example in engineering design. The basic power product relationship in dimensional analysis is…
Discrete interaction models for the classical harmonic oscillator are used for introducing new mathematical generalizations in the usual continuous formalism. The inverted harmonic potential and generalized discrete hyperbolic and…
The magnetic field is traditionally presented as a (pseudo)vector quantity, tied closely to the cross product. Though familiar to experts, many students find these ideas challenging and full of subtleties. Building on earlier work in…
A new object, called the velocity tensor, is introduced. It allows to formulate a generally covariant mechanics. Some properties of the velocity tensor are derived.
We study the wave function of a tensor model in the canonical formalism by Hamiltonian Monte Carlo method for Lie group symmetric or nearby values for the argument of the wave function, and show that there emerge Lie-group symmetric…
We prove a theorem on scalar-valued functions of tensors, where ``scalar'' refers to absolute scalars as well as relative scalars of weight $w$. The present work thereby generalizes an identity referred to earlier by Rosenfeld in his…