This paper studies the integration problem in differential fields that may involve quantities reminiscent of the classical Weierstrass ℘ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of ℘.

Elaborating on an approach recently proposed by Mark van Hoeij, we continue to investigate why creative telescoping occasionally fails to find the minimal-order annihilating operator of a given definite sum or integral. We offer an explanation based on the consideration of residues.

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive sequence and its shifts. There is a distinction between so-called normal and so-called special polynomials. Under the assumption that the corresponding difference field has no unnatural constants, we are able to predict the normal polynomials appearing in the denominator of a potential closed form. We can also handle the numerator. Our method is incomplete so far as we cannot predict the special polynomials appearing in the denominator. However, we do have some structural results about special polynomials for the setting under consideration.

Next-POI recommendation aims to explore from user check-in sequence to predict the next possible location to be visited. Existing methods are often difficult to model the implicit association of multi-modal data with user choices. Moreover, traditional methods struggle to fully explore the variation of user preferences at variable time intervals. To tackle these limitations, we propose a Multi-Modal Temporal Knowledge Graph-aware Sub-graph Embedding approach (Mandari). We first construct a novel Multi-Modal Temporal Knowledge Graph. Based on the proposed knowledge graph, we integrate multi-modal information and leverage the graph attention network to calculate sub-graph prediction probability. Next, we implement a temporal knowledge mining method to model the segmentation and periodicity of user check-in and obtain temporal prediction probability. Finally, we fuse temporal prediction probability with the previous sub-graph prediction probability to obtain the final result. Extensive experiments demonstrate that our approach outperforms existing state-of-the-art methods.