Related papers: What are kets?
In machine learning, the use of an artificial neural network is the mainstream approach. Such a network consists of layers of neurons. These neurons are of the same type characterized by the two features: (1) an inner product of an input…
In this paper we explore the notion of inner function in a broader context of operator theory.
We introduce real vector spaces composed of set-valued maps on an open set. They are also complete metric spaces, lattices, commutative rings. The set of differentiable functions is a dense subset of these spaces and the classical gradient…
The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin-Barnes integral. The fundamental idea is to transform these integral…
It is shown that the main geometrical objects involved in all the symmetries or supersymmetries of the Dirac operators in curved manifolds of arbitrary dimensions are the Killing vectors and the Killing-Yano tensors of any ranks. The…
In accelerator physics, the concept of impedance is popularly used to describe the interactions of charged particles inside a bunch or between bunches in a train. Standard formulations of impedance assume that the driving charge has a…
It is shown that scalar product of two vectors can be introduced in any geometry (metric space) independently of possibility of the linear space introduction. In general, linear properties of scalar product are restricted. Domain of…
Loop braid groups characterize the exchange of extended objects, namely loops, in three dimensional space generalizing the notion of braid groups that describe the exchange of point particles in two dimensional space. Their interest in…
On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts'. Also, one might think of algebras as being used to add more data to basic…
This paper suggests a statistical framework for describing the relations between the physical and conceptual entities of a brain-like model. Features and concept instances are put into context, where the paper suggests that features may be…
By median we mean a scheme that inputs three element of a lattice, and outputs an element that is an average of the three inputs in a certain sense. The medians of a given finite lattice form a new lattice that is usually larger than the…
In [8] a notion of generalized Hadamard product was introduced. We show that when certain kinds of tensors interact with the eigenvalues of symmetric matrices the resulting formulae can be nicely expressed using the generalized Hadamard…
We propose a routing algorithm that takes a sequence of vectors and computes a new sequence with specified length and vector size. Each output vector maximizes "bang per bit," the difference between a net benefit to use and net cost to…
Probing neural models for the ability to perform downstream tasks using their activation patterns is often used to localize what parts of the network specialize in performing what tasks. However, little work addressed potential mediating…
Multisets are sets that allow repetition of elements. As such, multisets pave the way to a number of interesting possibilities of theoretical and applied nature. In the present work, after revising the main aspects of traditional sets, we…
A framed surface is a smooth surface in the Euclidean space with a moving frame. By using the moving frame, we can define Bertrand framed surfaces as the same idea as Bertrand framed curves. Then we find the caustics and involutes as…
Let $A_1=K < X, Y | [Y,X]=1>$ be the (first) Weyl algebra over a field $K$ of characteristic zero. It is known that the set of eigenvalues of the inner derivation $\ad (YX)$ of $A_1$ is $\Z$. Let $ A_1\ra A_1$, $X\mapsto x$, $Y\mapsto y$,…
Several studies have been carried out on revealing linguistic features captured by BERT. This is usually achieved by training a diagnostic classifier on the representations obtained from different layers of BERT. The subsequent…
Founding our analysis on the Geneva-Brussels approach to the foundations of physics, we provide a clarification and classification of the key concept of observation. An entity can be observed with or without a scope. In the second case, the…
We give a detailed exposition of the "vectorized" notation for dealing with quantum operations. This notation is used to highlight the relationships between representations of completely-positive dynamics. Vectorization considerably…