Related papers: Imaginary Number Bases
In this paper we lay the foundations of a not necessarily rational negative imaginary systems theory and its relations with positive real systems theory and, hence, with passivity. In analogy with the theory of positive real functions, in…
We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division…
In this paper will be introduced large, probably complete family of complex base systems, which are 'proper' - for each point of the space there is a representation which is unique for all but some zero measure set. The condition defining…
Despite the fact that almost all real numbers are absolutely normal---that is, the digits in their expansions to any base occur in all possible configurations with the expected frequency---not one specific example of an absolutely normal…
We study a generalization of additive bases into a planar setting. A planar additive basis is a set of non-negative integer pairs whose vector sumset covers a given rectangle. Such bases find applications in active sensor arrays used in,…
Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in simple situations. This paper concerns binary radix systems, i.e., positional number systems with digits…
The history of complementary observables and mutual unbiased bases is reviewed. A characterization is given in terms of conditional entropy of subalgebras. The concept of complementarity is extended to non-commutative subalgebras.…
Mutually unbiased bases encapsulate the concept of complementarity - the impossibility of simultaneous knowledge of certain observables - in the formalism of quantum theory. Although this concept is at the heart of quantum mechanics, the…
Error operator bases for systems of any dimension are defined and natural generalizations of the bit/sign flip error basis for qubits are given. These bases allow generalizing the construction of quantum codes based on eigenspaces of…
Quantum systems with variables in ${\mathbb Z}(d)$ are considered, and three different structures are studied. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases…
We associate a half-integer number, called {\em the quantum index}, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum…
A process of extending sets which can be used as foundation for an alternative organization for Differential and Integral Calculus is presented.
We study the computational complexity of converting one representation of real numbers into another representation. Typical examples of representations are Cauchy sequences, base-10 expansions, Dedekind cuts and continued fractions.
One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques…
Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with…
Imaginary-time evolution is fundamental for analyzing quantum many-body systems, yet classical simulation requires exponentially growing resources in both system size and evolution time. While quantum approaches reduce the system-size…
Quantified constraints over the reals appear in numerous contexts. Usually existential quantification occurs when some parameter can be chosen by the user of a system, and univeral quantification when the exact value of a parameter is…
We consider the notion of unitary transformations forming bases for subspaces of $M(d,\mathbb{C})$ such that the square of Hilbert-Schmidt inner product of matrices from the differing bases is a constant. Moving from the qubit case,…
The representation of measurements by positive operator valued measures and the description of the most general state transformations by means of completely positive maps are two basic concepts of quantum information theory. These concepts…
Evolution in imaginary time is a prominent technique for finding the ground state of quantum many-body systems, and the heart of a number of numerical methods that have been used with great success in quantum chemistry, condensed matter and…