A web based tool to merge geometric morphometric data from multiple characters

Geometric morphometrics （GM） is an important method of shape analysis and increasingly used in a wide range of scientific disciplines. Presently, a single character comparison system of geometric morphometric data is used in almost all empirical studies, and this approach is sufficient for many scientific problems. However, the estimation of overall similarity among taxa or objects based on multiple characters is crucial in a variety of contexts （e.g. （semi-）automated identification, phenetic relationships, tracing of character evolution, phylogenetic reconstruction）. Here we propose a new web-based tool for merging several geometric morphometrics data files from multiple characters into a single data file. Using this approach information from multiple characters can be compared in combination and an overall similarity estimate can be obtained in a convenient and geometrically rigorous manner. To illustrate our method, we provide an example analysis of 25 dung beetle species with seven Procrustes superimposed landmark data files representing the morphological variation of body features： the epipharynx, right mandible, pronotum, elytra, hindwing, and the metendosternite in dorsal and lateral view. All seven files were merged into a single one containing information on 649 landmark locations. The possible applications of such merged data files in different fields of science are discussed.

Geometric morphometrics:Current and future in China

Although there were many ancient Chinese mathematicians contributed a lot on geometry, Geometric morphometrics （GM） in modern concept was not firstly proposed by Chinese. The super capability of geometric morphometries in scientific computing and problem solving has gained a lot of attentions in the world. Until early of 21 centuries, geometric morphometries was introduced into China. Since then, GM was rapidly applied in many research fields. However, it is a pity that GM is still not well-known in China as many works are published out of China. Thus, the special issue ＂Geometric morphometrics： Current shape and future directions＂ is organized. The present issue presents a series of contributions in this scientific field. In future, there will be many considerable new developing fields on GM needed to pay more attentions, for instances, 3D geometric morphometrics, 4D analysis, visualization of amber, new machine developing, new software developing, automatic identification system, etc. Once these technical bottle-necks on 3D data collecting and merging geometric morphometric data from multiple characters could be solved, the automatic identification system and other fields based on Big Data would come true.

On the Number of Solutions of Certain Equations over Finite Fields

Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N（a1x12 ＋ ···＋anxn2 = bx1 ···xs） denote the number of solutions（x1,...,xn） of the equation a1x12 ＋···＋ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F＊q,b ∈ F＊q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.

On Incomplete Gaussian Sums

In this paper it is proved that sum from N=N+1 to N+H x（n）ψ（n）（?）ε H1-（1/r）q(（1/4（r-）1）+ε, where r=4,q is a prime power,χ and ψ are multiplicative and additive characters modulo q respectively,with χ nontrivial.