On the characterization of Shannon's entropy by Shannon's inequality

1. In [2,5,6,7] a.o. several interpretations of the inequality for all such that were given and the following was prove

A classification of bisymmetric polynomial functions over integral domains of characteristic zero

We describe the class of n-variable polynomial functions that satisfy Acz\'el's bisymmetry property over an arbitrary integral domain of characteristic zero with identity

Origin of Complex Quantum Amplitudes and Feynman's Rules

Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. In this paper, we show that the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.Comment: v2: Clarifications, and minor corrections and modifications. Results unchanged. v3: Minor changes to introduction and conclusio

Generalized Convexity and Inequalities

Let R+ = (0,infinity) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 in M, we say that a function f : R+ to R+ is (m1,m2)-convex if f(m1(x,y)) < or = m2(f(x),f(y)) for all x, y in R+ . The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1,m2)-convexity on m1 and m2 and give sufficient conditions for (m1,m2)-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.Comment: 17 page

Preassociative aggregation functions

The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions, where preassociativity is a generalization of associativity recently introduced by the authors. These axiomatizations are based on existing characterizations of some noteworthy classes of associative operations, such as the class of Acz\'elian semigroups and the class of t-norms.Comment: arXiv admin note: text overlap with arXiv:1309.730

Zeroth Law compatibility of non-additive thermodynamics

Non-extensive thermodynamics was criticized among others by stating that the Zeroth Law cannot be satisfied with non-additive composition rules. In this paper we determine the general functional form of those non-additive composition rules which are compatible with the Zeroth Law of thermodynamics. We find that this general form is additive for the formal logarithms of the original quantities and the familiar relations of thermodynamics apply to these. Our result offers a possible solution to the longstanding problem about equilibrium between extensive and non-extensive systems or systems with different non-extensivity parameters.Comment: 18 pages, 1 figur

Consistency of the Shannon entropy in quantum experiments

The consistency of the Shannon entropy, when applied to outcomes of quantum experiments, is analysed. It is shown that the Shannon entropy is fully consistent and its properties are never violated in quantum settings, but attention must be paid to logical and experimental contexts. This last remark is shown to apply regardless of the quantum or classical nature of the experiments.Comment: 12 pages, LaTeX2e/REVTeX4. V5: slightly different than the published versio

Properties of Classical and Quantum Jensen-Shannon Divergence

Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the most important divergence measure of information theory, Kullback divergence. As opposed to Kullback divergence it determines in a very direct way a metric; indeed, it is the square of a metric. We consider a family of divergence measures (JD_alpha for alpha>0), the Jensen divergences of order alpha, which generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha is the square of a metric for alpha lies in the interval (0,2], and that the resulting metric space of probability distributions can be isometrically embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a symmetrized and smoothed version of quantum relative entropy and can be extended to a family of quantum Jensen divergences of order alpha (QJD_alpha). We strengthen results by Lamberti et al. by proving that for qubits and pure states, QJD_alpha^1/2 is a metric space which can be isometrically embedded in a real Hilbert space when alpha lies in the interval (0,2]. In analogy with Burbea and Rao's generalization of JD, we also define general QJD by associating a Jensen-type quantity to any weighted family of states. Appropriate interpretations of quantities introduced are discussed and bounds are derived in terms of the total variation and trace distance.Comment: 13 pages, LaTeX, expanded contents, added references and corrected typo

The Schr\"oder functional equation and its relation to the invariant measures of chaotic maps

The aim of this paper is to show that the invariant measure for a class of one dimensional chaotic maps, $T(x)$, is an extended solution of the Schr\"oder functional equation, $q(T(x))=\lambda q(x)$, induced by them. Hence, we give an unified treatment of a collection of exactly solved examples worked out in the current literature. In particular, we show that these examples belongs to a class of functions introduced by Mira, (see text). Moreover, as a new example, we compute the invariant densities for a class of rational maps having the Weierstrass $\wp$ functions as an invariant one. Also, we study the relation between that equation and the well known Frobenius-Perron and Koopman's operators.Comment: 9 page

Entropic uncertainty relations for extremal unravelings of super-operators

A way to pose the entropic uncertainty principle for trace-preserving super-operators is presented. It is based on the notion of extremal unraveling of a super-operator. For given input state, different effects of each unraveling result in some probability distribution at the output. As it is shown, all Tsallis' entropies of positive order as well as some of Renyi's entropies of this distribution are minimized by the same unraveling of a super-operator. Entropic relations between a state ensemble and the generated density matrix are revisited in terms of both the adopted measures. Using Riesz's theorem, we obtain two uncertainty relations for any pair of generalized resolutions of the identity in terms of the Renyi and Tsallis entropies. The inequality with Renyi's entropies is an improvement of the previous one, whereas the inequality with Tsallis' entropies is a new relation of a general form. The latter formulation is explicitly shown for a pair of complementary observables in a $d$-level system and for the angle and the angular momentum. The derived general relations are immediately applied to extremal unravelings of two super-operators.Comment: 8 pages, one figure. More explanations are given for Eq. (2.19) and Example III.5. One reference is adde