Related papers: Difference between three quantities
In this paper, we prove that the inequalities $\alpha [1/3 Q(a,b)+2/3 A(a,b)]+(1-\alpha)Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b) <\beta [1/3 Q(a,b)+2/3 A(a,b)]+(1-\beta)Q^{1/3}(a,b)A^{2/3}(a,b)$ and $\lambda [1/6 C(a,b)+5/6…
An elementary but useful fact is that the numerator of the difference of two consecutive Farey fractions is equal to one. For triples of consecutive fractions the numerators of the differences are well understood and have applications to…
The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates,…
A new mathematics, the constructive one, characterizes a singular limit as undecidable. Hence, a singular limit between two theories actually represents a difference between two different kinds of mathematics. This particular situation…
We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of…
Divergence functions play a key role as to measure the discrepancy between two points in the field of machine learning, statistics and signal processing. Well-known divergences are the Bregman divergences, the Jensen divergences and the…
This paper presents and philosophically assesses three types of results on the observational equivalence of continuous-time measure-theoretic deterministic and indeterministic descriptions. The first results establish observational…
The product of two complex-valued harmonic function is not in general complex-valued harmonic function. In this paper we show that if a complex-valued harmonic function is the product of two complex-valued harmonic functions, then it is the…
In the work the fair division problem for two participants in presence of both divisible and indivisible items is considered. The set of all divisions is formally described; it is demonstrated that fair (in terms of Brams and Taylor)…
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not…
Products and tensor products are linked by a universal property. Imposing the invariance of the laws of Nature under tensor composition along with Leibniz identity determines quantum and classical mechanics algebraic structure through the…
The concepts of similarity and distance are crucial in data mining. We consider the problem of defining the distance between two data sets by comparing summary statistics computed from the data sets. The initial definition of our distance…
Observables have a dual nature in both classical and quantum kinematics: they are at the same time \emph{quantities}, allowing to separate states by means of their numerical values, and \emph{generators of transformations}, establishing…
In contrast to the well-known solution of the two-body problem through the use of the concept of reduced mass, a solution is proposed involving separation of potentials. It is shown that each of the two point bodies moves in its own…
The uncertainty principle and entanglement are two fundamental, but yet not well understood, features of quantum theory. The uncertainty relation reflects the capability limit in acquiring the knowledge of different physical properties of a…
The resolvability of equations in integers containing truncated Newton's binomial, is determined by the divisibility of the binomial by the characteristic parameters of the equation, which most often is the binomial exponent. Two types of…
The concept of measurement is discussed. It is argued that counting process in mathematics is also measurement which requires a basic unit. The idea of scale is put forward. The basic unit itself, which are composed of the infinitesimal of…
Incompatibility between conjugate variables and complementary pictures comes in two kinds, exclusive of one another. The first kind is unconditional, and the second conditional on quantum's indivisibility. We employ this distinction to…
Complementarity is one of the main features of quantum physics that radically departs from classical notions. Here we consider the limitations that this principle imposes due to the unpredictability of measurement outcomes of incompatible…
Most of the fundamental characteristics of quantum mechanics, such as non-locality and contextuality, are manifest in discrete, finite-dimensional systems. However, many quantum information tasks that exploit these properties cannot be…