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It is well known that the laws of Leimkuhler, Mandelbrot, Lotka and the verbal formulation of Bradford's law are equivalent. In this note we prove an analogous framework of laws in which now the law of Zipf occurs instead of Mandelbrot's law. Laws in this group are Brookes' law and the graphical formulation of Bradford's law. We show that both groups of laws are very different in the sense that they are not even equal asymptotically for high rankings.

It is known that a mathematical ecological model and, in general, a particular methodology of modelling, can be considered a literary text written in a formal mathematical language. In this context, stylometric mathematical laws such as Zipf’s (range‐frequency and number‐frequency) can be applied to obtain information parameters in different semantic levels within the same model. Adapts several of these laws and introduces new elements, lexic units, operating and separating units, to carry out several statistical analyses upon two models or texts. The estimated slopes in the regression equations obtained in the present work are compared with the results of previous papers where Mandelbrot’s law was applied and comparisons between them are shown.

In this work, we estimated the different entropies like Shannon entropy, Rényi divergences, Csiszár divergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrot law are used to estimate the Shannon entropy. The Abel–Gontscharoff Green functions and Fink’s Identity are used to construct new inequalities and generalized them for m-convex function.

The Bradford law is explored theoretically by means of a very mixed Poisson model which, it is claimed, elucidates the uncertainties surrounding the law and its applications. It is argued that Bradford succeeded in formulating an empirical regularity which has pure and hybrid forms but that all the variants can be subsumed under a simple logarithmic law which, for reasons explained, escapes exact expression in conventional frequency terms. The theoretical aspects discussed include the hybridity of form, estimations, sampling problems, the stability of ranks, homogeneity of data, and tests of significance. Some numerical examples, some simulated and some drawn from social contexts outside bibliography, are used both to illustrate theoretical issues and also to indicate the wide generality of the Bradford law. Possible applications and developments of the theory are indicated.

A major difficulty in continuous speech recognition research is the lack of effective and objective evaluation of the statistical models of text. Herbert Simon's view for evaluating theories is here applied to the statistical modelling of text. Three significant contributions can be identified. First, a time‐series representation of text is used to identify three well‐known empirical laws of text generation. These laws provide an effective and objective approach for evaluating four leading statistical models of text. Second, it is shown that the Simon‐Yule model of text provides a constructive mechanism for those laws. Third, based on Simon's explanatory processes of imitation and association, an adaptive framework for continuous speech recognition is suggested.

There is general agreement among researchers and practitioners that satisfaction is relative to competitive alternatives. Nonetheless, researchers and managers have not treated satisfaction as a relative construct. The result has been weak relationships between satisfaction and share of wallet in the literature, and challenges by managers as to whether satisfaction is a useful predictor of customer behavior and business outcomes. The purpose of this paper is to explore the best approach for linking satisfaction to share of wallet.

Using data from 79,543 consumers who provided 258,743 observations regarding the brands that they use (over 650 brands) covering 20 industries from 15 countries, various models such as the Wallet Allocation Rule (WAR), Zipf-AE, and Zipf-PM, truncated geometric model, generalization of the WAR and hierarchical regression models are compared to each other.

The results indicate that the relationship between satisfaction and share of wallet is primarily driven by the relative fulfillment customers perceive from the various brands that they use (as gauged by their relative ranked satisfaction level), and not the absolute level of satisfaction.

The findings provide practical insight into several easy-to-use approaches that researchers and managers can apply to improve the strength of the relationship between satisfaction and share of wallet.

This research provides support to the small number of studies that point to the superiority of using relative metrics, and encourages the adoption of relative satisfaction metrics by the academic community.