Continuing previous work, this paper focuses on the summability problem of multivariate rational functions in the mixed case in which both shift and q-shift operators can appear. Our summability criteria rely on three ingredients including orbital decompositions, Sato’s isotropy groups, and difference transformations. This work settles the rational case of the long-term project aimed at developing algorithms for symbolic summation of multivariate functions.

We propose a way to split a given bivariate P-recursive sequence into a summable part and a non-summable part in such a way that the non-summable part is minimal in some sense. This decomposition gives rise to a new reduction-based creative telescoping algorithm based on the concept of integral bases.

Trager’s Hermite reduction solves the integration problem for algebraic functions via integral bases. A generalization of this algorithm to D-finite functions has so far been limited to the Fuchsian case. In the present paper, we remove this restriction and propose a reduction algorithm based on integral bases that is applicable to arbitrary D-finite functions.

Bronstein's lazy Hermite reduction is a symbolic integration technique that reduces algebraic functions to integrands with only simple poles without the prior computation of an integral basis. We sharpen the lazy Hermite reduction by combining it with the polynomial reduction to solve the decomposition problem of algebraic functions. The sharpened reduction is then used to design a reduction-based telescoping algorithm for algebraic functions in two variables.

In the context of New Generation of Information Technology (NGIT) and Emerging Engineering Education (3E) in China, it is a new research hot topic to in evaluating students involvements and skills in engineering practice. Many automated assessment systems were developed specifically for the use case of grading student work. Sometimes, it is subjective evaluation and not always correct. These systems usually emphasize the training results, but neglects the process tracking od the result. In this paper, our automated assessment and evaluation of the contribution of collaborative software engineering training is proposed to evaluate student involving in software engineering training against a rubric feedback. Contributions of our method are automated assessment of laboratory environment and collaborative software development process, and cooperative development contribution model. Finally, the results of automated assessment of workload and its balance is analyzed to illustrate the effect of our software engineering training.

In an earlier paper, the notion of integrality known for algebraic number fields and fields of algebraic functions has been extended to D-finite functions. The aim of the present paper is to extend the notion to the case of P-recursive sequences. In order to do so, we formulate a general algorithm for finding all integral elements for valued vector spaces and then show that this algorithm includes not only the algebraic and the D-finite cases but also covers the case of P-recursive sequences.

We present criteria on the existence of telescopers for trivariate rational functions in four mixed cases, in which discrete and continuous variables appear simultaneously. We reduce the existence problem in the trivariate case to the exactness testing problem, the separation problem and the existence problem in the bivariate case. The existence criteria help us to determine the termination of Zeilberger's algorithm for the input functions studied in this paper.