Transformation of Quotient Values for their use as Continuous Cladistic Characters

Recent advances in cladistic technology have produced novel methods for introducing morphological data into cladistic analyses, such as the landmark and continuous character functions in the software TNT and RevBayes. While these new methods begin to address the problem of representing morphology, there has been little consideration of how to transform and code the operational taxonomic units’ (OTUs) dimensions into the datamatrix. Indeed, angles, serial counts, percentages and quotient values can be used as continuous characters, but little has been said about how coding these data affect the trees discovered. Logically, counts of elements and angles measured off specimens may be coded directly into continuous character matrices but percentages and quotient values are more problematic, being transformed data. Quotient values and percentages are the simplest way of representing proportional differences between two dimensions and reducing the effect of inter-taxonomic magnitude differences. However, both are demonstrated to be problematic transformations that produce continuous characters with weighted states that are non-representative of morphological variation. Thus, two OTUs may be represented as less/more similar morphologically than other OTUs that display the same degree of morphological variation. Furthermore, the researcher’s choice of which dimension is the divisor and dividend will have a similar affect. To address this problem, a trigonometric solution and a logarithmic solution have been proposed. Another solution called linear transposition scaling (LTS) was recently presented, with the intention of best representing and coding observable morphological variation. All three methods are reviewed to establish the best way to represent and code morphology in a cladistic analysis using continuous characters.

BORN–INFELD GRAVITY REVISITED

In this paper, we investigate the behavior of linearized gravitational excitation in the Born–Infeld gravity in AdS3 space. We obtain the linearized equation of motion and show that this higher-order gravity propagate two gravitons, massless and massive, on the AdS3 background. In contrast to the R2 models, such as TMG or NMG, Born–Infeld gravity does not have a critical point for any regular choice of parameters. So the logarithmic solution is not a solution of this model, due to this one cannot find a logarithmic conformal field theory as a dual model for Born–Infeld gravity.

Eigenfunctions and Fundamental Solutions of the Fractional Two-Parameter Laplacian

We deal with the following fractional generalization of the Laplace equation for rectangular domains(x,y)∈(x0,X0)×(y0,Y0)⊂ℝ+×ℝ+, which is associated with the Riemann-Liouville fractional derivativesΔα,βu(x,y)=λu(x,y),Δα,β:=Dx0+1+α+Dy0+1+β, whereλ∈ℂ,(α,β)∈[0,1]×[0,1]. Reducing the left-hand side of this equation to the sum of fractional integrals byxandy, we then use the operational technique for the conventional right-sided Laplace transformation and its extension to generalized functions to describe a complete family of eigenfunctions and fundamental solutions of the operatorΔα,βin classes of functions represented by the left-sided fractional integral of a summable function or just admitting a summable fractional derivative. A symbolic operational form of the solutions in terms of the Mittag-Leffler functions is exhibited. The case of the separation of variables is also considered. An analog of the fractional logarithmic solution is presented. Classical particular cases of solutions are demonstrated.

Mathematical Miscellanea

While browsing through the earliest French treatise on logarithms—Traicté des logarithmes (Paris, 1626) by Denis Henrion (c.1590-1640), Professor of Mathematics at Paris, my attention was caught by the method he adopted for the logarithmic solution of an oblique triangle when the three sides are given.