Classical Divergence Research Articles

Generalization of the Kullback–Leibler divergence in the Tsallis statistics

In this paper, we present a generalization of the Kullerback–Leibler (KL) divergence in form of the Tsallis statistics. In parallel with the classical KL divergence, several important properties of this new generalization, including the pseudo-additivity, positivity and monotonicity, are shown. Moreover, some strengthened estimates on the positivity of the new divergence and the information loss during transformations are obtained.

Towards understanding the ultraviolet behavior of quantum loops in infinite-derivative theories of gravity

In this paper we will consider quantum aspects of a non-local, infinite-derivative scalar field theory—a toy model depiction of a covariant infinite-derivative, non-local extension of Einstein’s general relativity which has previously been shown to be free from ghosts around the Minkowski background. The graviton propagator in this theory gets an exponential suppression making it asymptotically free, thus providing strong prospects of resolving various classical and quantum divergences. In particular, we will find that at one loop, the two-point function is still divergent, but once this amplitude is renormalized by adding appropriate counter terms, the ultraviolet behavior of all other one-loop diagrams as well as the two-loop, two-point function remains well under control. We will go on to discuss how one may be able to generalize our computations and arguments to arbitrary loops.

Upscaling Interpretation of Nonlocal Fields, Gradients, and Divergences

Recently, nonlocal generalizations of the classical gradient, divergence, and curl have been introduced to examine nonlocal field problems, and develop a nonlocal vector calculus. Here we introduce, by definition, the concept of a nonlocal field variable and relate it to its classical local counterpart. Subsequently, we relate the concept of measurement via a convolution (an upscaling) to the nonlocal field variables and the nonlocal operators. It is shown via Fourier transform that the nonlocal gradient and divergence can be thought of as a very special convolution averaging of their classical counterparts. A nonlocal self-diffusion equation is upscaled and written in terms of nonlocal operators.

From a generalised Helmholtz decomposition theorem to fractional Maxwell equations

The main objective of this paper is to propose a new generalisation of the Helmholtz decomposition theorem for both fractional time and space, which leads to four equations generalising the Maxwell equations that emerge as particular case. To get these results the well-known classical vectorial operators, gradient, divergence, curl, and laplacian are generalised to fractional orders using Grünwald–Letnikov approach.

Aspects of Semiclassical Theory in the Presence of Classical Chaos

Topics concerning the rigorous validity of semiclassical approximations in the presence of classical chaos are discussed from three viewpoints: time-evolution (how long semiclassical behavior can be ascertained in the propagator, with the use of reduction methods and quantum maps), eigenvalues (emphasizing the importance of the two separate divergences in trace formulas, the classical divergence from chaos and the quantum divergence from ℏ-corrections), and eigenfunctions (which display their semiclassical behavior most explicitly in a “stellar” representation over the phase space). All areas are still characterized by a scarcity of proven results and of mathematical investigation tools against an increasing set of observations.

Generalized Symmetric Divergence Measures and the Probability of Error

There are three classical divergence measures exist in the literature on information theory and statistics. These are namely, Jeffryes-Kullback-Leiber (Jeffreys, 1946; Kullback and Leibler, 1951) J-divergence. Sibson-Burbea-Rao (Sibson, 1969), Jensen-Shannon divegernce, (Burbea and Rao, 1982), and Taneja (1995). Arithmetic-Geometric divergence. These three measures bear an interesting relationship among each other. The divergence measures like Hellinger (1909) discrimination, symmetric χ2-divergence, and triangular discrimination are also known in the literature. In this article, we have considered generalized symmetric divergence measures having the measures given above as particular cases. Bounds on the probability of error are obtained in terms of generalized symmetric divergence measures. Study of bounds on probability of error is extended for the difference of divergence measures.

Seven Means, Generalized Triangular Discrimination, and Generating Divergence Measures

Jensen-Shannon, J-divergence and Arithmetic-Geometric mean divergences are three classical divergence measures known in the information theory and statistics literature. These three divergence measures bear interesting inequality among the three non-logarithmic measures known as triangular discrimination, Hellingar’s divergence and symmetric chi-square divergence. However, in 2003, Eve studied seven means from a geometrical point of view, which are Harmonic, Geometric, Arithmetic, Heronian, Contra-harmonic, Root-mean square and Centroidal. In this paper, we have obtained new inequalities among non-negative differences arising from these seven means. Correlations with generalized triangular discrimination and some new generating measures with their exponential representations are also presented.

Constrained Minimization Algorithms

In this paper, we consider the inverse problem of restoring an unknown signal or image, knowing the transformation suffered by the unknowns. More specifically we deal with transformations described by a linear model linking the unknown signal to an unnoisy version of the data. The measured data are generally corrupted by noise. This aspect of the problem is presented in the introduction for general models. In Section 2, we introduce the linear models, and some examples of linear inverse problems are presented. The specificities of the inverse problems are briefly mentionned and shown on a simple example. In Section 3, we give some information on classical distances or divergences. Indeed, an inverse problem is generally solved by minimizing a discrepancy function (divergence or distance) between the measured data and the model (here linear) of such data. Section 4 deals with the likelihood maximization and with their links with divergences minimization. The physical constraints on the solution are indicated and the Split Gradient Method (SGM) is detailed in Section 5. A constraint on the inferior bound of the solution is introduced at first; the positivity constraint is a particular case of such a constraint. We show how to obtain strictly, the multiplicative form of the algorithms. In a second step, the so-called flux constraint is introduced, and a complete algorithmic form is given. In Section 6 we give some brief information on acceleration method of such algorithms. A conclusion is given in Section 7.

Film flow for power-law fluids: Modeling and linear stability

The paper deals with modeling of a power-law fluid film flowing down an inclined plane for small to moderate Reynolds numbers. A model, accurate up to second order [first order] for dilatant [pseudoplastic] fluids is proposed to describe the nonlinear behavior of the flow. The modeling procedure consists of a combination of the lubrication theory and the weighted residual approach using an appropriate projection basis. A suitable choice of weighting functions allows a significant reduction of the dimension of the problem. The resulting model is naturally unique, i.e., independent of the particular form of the trial functions. Reduced models are proposed for the evolution of the local film thickness and flow rate; their linear spectra are compared to that obtained from the full Orr–Sommerfeld numerical solution. To obtain the latter, a new formulation of the eigenvalue problem is proposed to overcome the classical divergence of the apparent viscosity at the free surface. The full model and its reduced forms all have the advantage of the Benney like model close to criticality. Far from the instability threshold the full model continues to follow the Orr–Sommerfeld solution up to sufficiently large Reynolds numbers and gives better predictions than the depth averaging model. An incomplete regularization procedure is performed to cure the rapid divergence of the reduced two-equation model. Due to its relative simplicity the latter might be preferred in practice to the full model, at least at the initial stage of the nonlinear regime. It is also shown that the convective nature of the instability is not affected by the variation of the power law index.

Progress in the Self-Similar Turbulent Flame premixed combustion model

This paper is devoted to premixed combustion modeling in turbulent flow. First, we briefly remind the main features of the Self-Similar Turbulent Flame model that was more extensively developed in a former paper. Then, we carefully describe some improvements of the model. The determination of the turbulent flame velocity is based on the observed self-similarity of the turbulent flame and uses the local flame brush width as a fundamental parameter, which must be retrieved. With respect to the former version, we now derive more rigorously how the density variation has to be taken into account in the width retrieving function. We reformulate the diffusion term as a classical flux divergence term. We enforce the compatibility of the model for the limit of weak turbulence. We include a contracting effect of the source term, thus allowing to give a stationary mono-dimensional asymptotic solution with a finite width. We also include in a preliminary form, a stretch factor, which proves to be useful for controlling the flame behavior close to the flame holder and near the walls. The model implementation in the Star-CD CFD code is then tested on three different flame configurations. Finally, we shortly discuss the model improvements and the simulation results.

The derivation of a numerical diagnostic model for the forcing of the geopotential

AbstractSimply applying the divergence operator to the horizontal velocity equation in spherical‐isobaric coordinates yields a more solvable linear geopotential equation (than the geopotential tendency and the omega equations) for diagnosing the geopotential and its anomalies responsible for global floods and droughts. This geopotential model is freed from the zero‐denominator problem caused by the observed neutral stability which affects the geopotential tendency and omega equations. The geopotential equation's forcing terms 2–5 (the direct operation results of the horizontal advection term in the velocity equation) can identify the synoptic signals mixed up by the classical divergence equation's quadratic and Jacobian terms (the manipulated results of the above terms 2–5). Solving this geopotential equation with the successive‐over‐relaxation method and the neutral stability as calculated from the reanalysis data produces reasonably good reconstructions of the global and local geopotential fields. The additional advantages of the present geopotential model over the geopotential tendency, omega and divergence models are that the geopotential model reveals and explains the following mechanisms which received less attention in previous studies. (1) The saddle patterns favour the high systems. (2) The equatorial easterlies favour the low systems. (3) The contributions of ageostrophic processes associated with jets and saddle flows to the transitions of weather patterns might be overlooked by the geopotential tendency model in which neutral stability and static instability in cold domes as well as jet‐induced inertial instability are smoothed out artificially. Copyright © 2008 Royal Meteorological Society

Noncommutative radial waves

We study radial waves in a (2+1)-dimensional noncommutative scalar field theory, using operatorial methods. The waves propagate along a discrete radial coordinate and are described by finite series deformations of Bessel-type functions. At radius much larger than the noncommutativity scale , one recovers the usual commutative behaviour. At small distances, classical divergences are smoothed out by noncommutativity.

The Divergence Theorem for Divergence Measure Vectorfields on Sets with Fractal Boundaries

A divergence measure vectorfield is an ℝ n valued measure on an open subset U of ℝ n whose weak divergence in U is a (signed) measure. The paper uses the product rule for the product of the divergence measure by a function from W 1,∞ (U) 3 established in Šilhavý [Šilhavý, M., submitted, 2007] to prove the divergence theorem for the divergence measure vectorfields on bounded open sets U. It is shown that the surface integral of the normal component of the vectorfield occurring in the classical divergence theorem has to be replaced by a continuous linear functional on the space of Lipschitz functions on the boundary1 the volume integral contains the duality pairing occurring in the product rule. The boundary of U is arbitrary, it can be even fractal in the sense that the normal to ∂U cannot be defined.

Classical Divergence of Nonlinear Response Functions

The time divergence of classical nonlinear response functions reveals the fundamental difficulty of dynamic perturbation based on classical mechanics. The nature of the divergence is established for systems in regular motions using asymptotic decomposition of Fourier integrals. The asymptotic analysis shows that the divergence cannot be removed by phase-space averaging such as the Boltzmann distribution function. The implications of this study are discussed in the context of the conceptual development of quantum-classical correspondence in dynamic response.

Geometric Hodge star operator with applications to the theorems of Gauss and Green

The classical divergence theorem for an -chainlet. When combined with the general Stokes' theorem ([H1, H2]) \[\int_{\partial A} \omega = \int_A d \omega\] this result yields optimal and concise forms of Gauss' divergence theorem \[\int_{\star \partial A}\omega = (-1)^{(k)(n-k)} \int_A d\star \omega\] and Green's curl theorem \[\int_{\partial A} \omega = \int_{\star A} \star d\omega.\]

On the Amount of Information Resulting from Empirical and Theoretical Knowledge

We present a mathematical model allowing formally define the concepts of empirical and theoretical knowledge. The model consists of a finite set P of predicatesand a probability space (, S, P) over a finite set called ontology which consists of objects ! for which the predicates _ 2 P are either valid (_(!) = 1) or not valid (_(!) = 0). Since this is a first step in this area, our approach is as simple as possible, but still nontrivial, as it is demonstrated by examples. More realistic approach would be more complicated, based on a fuzzy logic where the predicates _ 2 P are valid on the objects ! 2 to some degree (0 _ _(!) _ 1). We use the classical information divergence to introduce the amount of information in empirical and theoretical knowledge. By an example is demonstrated that information in theoretical knowledge is an extension of the “sematic information” introduced formerly by Bar Hillel and Carnap as an alternative to the information of Shannon.

Renormalization group flows for brane couplings

Field theories in the presence of branes encounter localized divergences that renormalize brane couplings. The sources of these brane-localized divergences are understood as arising either from broken translation invariance, or from short distance singularities as the brane thickness vanishes. While the former are generated only by quantum corrections, the latter can appear even at the classical level. Using as an example six-dimensional scalar field theory in the background of a 3-brane, we show how to interpret such classical divergences by the usual regularization and renormalization procedure of quantum field theory. In our example, the zero thickness divergences are logarithmic, and lead classically to non-trivial renormalization group flows for the brane couplings. We construct the tree level renormalization group equations for these couplings as well as the one-loop corrections to these flows from bulk-to-brane renormalization effects.

Rectifiability and perimeter in the Heisenberg group

In this paper, we fully extend to the Heisenberg group endowed with its intrinsic Carnot-Caratheodory metric and perimeter the classical De Giorgi's rectifiability divergence theorems.

Linear and nonlinear response functions of the Morse oscillator: Classical divergence and the uncertainty principle

The algebraic structure of the quantum Morse oscillator is explored to formulate the coherent state, the phase-space representations of the annihilation and creation operators, and their classical limits. The formulation allows us to calculate the linear and nonlinear quantum response functions for microcanonical Morse systems and to demonstrate the linear divergence in the corresponding classical response function. On the basis of the uncertainty principle, the classical divergence is removed by phase-space averaging around the microcanonical energy surface. For the Morse oscillator, the classical response function averaged over quantized phase space agrees exactly with the quantum response function for a given eigenstate. Thus, phase-space averaging and quantization provide a useful way to establish the classical-quantum correspondence of anharmonic systems.

Local Government Systems: From Historic Divergence towards Convergence? Great Britain, France, and Germany as Comparative Cases in Point

The author aims at a comparative analysis of the development of local government systems in Britain, France, and Germany. First, he makes the point that, during the historical evolution of the local government systems of the three countries over the last century, their institutional profiles have exhibited an almost classical divergence. Against this historical background the author pursues answers to the questions of whether, to what degree, and why the local government systems of the three countries have, in their more recent development, shown institutional convergence. Among the factors possibly fostering such convergence, the following are highlighted in the paper: the internationalisation (‘globalisation’) of socioeconomic and political challenges and institutional responses to them.