Related papers: Some thoughts about matrix coordinate transformati…
Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous…
Invariant integration of vectors and tensors over manifolds was introduced around fifty years ago by V.N. Folomeshkin, though the concept has not attracted much attention among researchers. Although it is a sophisticated concept, the…
B\"acklund transformations (BTs) are traditionally regarded as a tool for integrating nonlinear partial differential equations (PDEs). Their use has been recently extended, however, to problems such as the construction of recursion…
The unit vectors transformation between the Cartesian and the novel Similar Oblate Spheroidal coordinates, and vice versa, is derived. It can help to transform vector fields between these two types of orthogonal coordinates which can…
We investigate a linear operator associated with a functional equation that arises from studying some class of invariant measures under multidimensional transformations. By examining its iterates, we derive an explicit solution formula for…
This article aims to explore the most recent developments in the study of the Hilbert matrix, acting as an operator on spaces of analytic functions and sequence spaces. We present the latest advances in this area, aiming to provide a…
By the Fourier transformations, any group-invariant functions over finite Abelian groups are transformed into group-invariant functions over the character groups. In this paper, we calculate matrix elements of this transformations under…
We generalize the embedding formalism for conformal field theories to the case of general operators with mixed symmetry. The index-free notation encoding symmetric tensors as polynomials in an auxiliary polarization vector is extended to…
Transformers often struggle with length generalization, meaning they fail to generalize to sequences longer than those encountered during training. While arithmetic tasks are commonly used to study length generalization, certain tasks are…
We define a class of "optimal" coordinate systems by requiring that the deviation from an exact Robertson-Walker metric is "as small as possible" within a given four dimensional volume. The optimization is performed by minimizing several…
In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian…
We consider iterated function systems (finite or countable), together with linear and continuous operators on Hilbert spaces, which enable us to construct Markov-type operators. Under suitable conditions, these Markov-type operators have…
We introduce two new sets of invariant functions of quark mass matrices, which express the constraints on these mass matrices due to knowledge of the quark mixing matrix. These invariants provide a very simple method to test candidate forms…
Based on the matrix realignment and partial transpose, we develop an approach to entangling power and operator entanglement of quantum unitary operators. We demonstrate efficiency of the approach by studying several unitary operators on…
Convolution admits a natural formulation as a functional operation on matrices. Motivated by the functional and entrywise calculi, this leads to a framework in which convolution defines a matrix transform that preserves positivity. Within…
Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most…
Vector is a physical quantity and it does not depend on any co-ordinate system. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The expansion in orthonormal basis is…
The aim of this paper is to present a study on the representations of coordinate, momentum and dispersion operators in the framework of a phase space representation of quantum mechanics that we have introduced and studied in previous works.…
In this paper, we describe the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We…
Boundedness for a class of projection operators, which includes the coordinate projections, on matrix weighted $L^p$-spaces is completely characterised in terms of simple scalar conditions. Using the projection result, sufficient…