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This volume contains the proceedings of the virtual workshop on Computational Aspects of Discrete Subgroups of Lie Groups, held from June 14 to June 18, 2021, and hosted by the Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island.The major theme deals with a novel domain of computational algebra: the design, implementation, and application of algorithms based on matrix representation of groups and their geometric properties. It is centered on computing with discrete subgroups of Lie groups, which impacts many different areas of mathematics such as algebra, geometry, topology, and number theory. The workshop aimed to synergize independent strands in the area of computing with discrete subgroups of Lie groups, to facilitate solution of theoretical problems by means of recent advances in computational algebra.

We present a neuro-symbolic visual question answering (VQA) pipeline for CLEVR, which is a well-known dataset that consists of pictures showing scenes with objects and questions related to them. Our pipeline covers (i) training neural networks for object classification and bounding-box prediction of the CLEVR scenes, (ii) statistical analysis on the distribution of prediction values of the neural networks to determine a threshold for high-confidence predictions, and (iii) a translation of CLEVR questions and network predictions that pass confidence thresholds into logic programmes so that we can compute the answers using an answer-set programming solver. By exploiting choice rules, we consider deterministic and non-deterministic scene encodings. Our experiments show that the non-deterministic scene encoding achieves good results even if the neural networks are trained rather poorly in comparison with the deterministic approach. This is important for building robust VQA systems if network predictions are less-than perfect. Furthermore, we show that restricting non-determinism to reasonable choices allows for more efficient implementations in comparison with related neuro-symbolic approaches without losing much accuracy.

This paper starts by settling the computational complexity of the problem of integrating a polynomial function ff over a rational simplex. We prove that the problem is NP\\mathrm {NP}-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We explore our algorithms with some experiments. We conclude the article with extensions to other polytopes and discussion of other available methods.

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. Under the heuristic assumption, justified by experiments, that the correctness of the result is not perturbed by rounding errors, the algorithm runs in time \\[ O ( | D | log 3 ⁡ | D | M ( | D | log 2 ⁡ | D | ) ) ⊆ O ( | D | log 6 + ε ⁡ | D | ) ⊆ O ( h 2 + ε ) O \\left ( \\sqrt {|D|} \\log ^3 |D| \\, M \\left ( \\sqrt {|D|} \\log ^2 |D| \\right ) \\right ) \\subseteq O \\left (|D| \\log ^{6 + \\varepsilon } |D| \\right ) \\subseteq O \\left ( h^{2 + \\varepsilon } \\right ) \\] for any ε > 0 \\varepsilon > 0 , where D D is the CM discriminant, h h is the degree of the class polynomial and M ( n ) M (n) is the time needed to multiply two n n -bit numbers. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials.

Under investigation in this paper is a (3+1)-dimensional nonlinear evolution equation, which was proposed and analyzed to study features and properties of nonlinear dynamics in higher dimensions. Using the Hirota bilinear method, we construct a bilinear Bäcklund transformation, which consists of four equations and involves six free parameters. With test function method and symbolic computation, three sets of lump–kink solutions and new types of interaction solutions are derived, and figures are presented to reveal the interaction behaviors. Setting constraints to the new interaction solution via the test function expressed by “polynomial-cos-cosh,” we simulate the periodic interaction phenomenon. Pfaffian solutions to the (3+1)-dimensional nonlinear evolution equation are obtained based on a set of linear partial differential conditions. According to our results, the diversity of solutions to the (3+1)-dimensional nonlinear evolution equation is revealed.

Abundant exact interaction solutions, including lump-soliton, lump-kink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2 + 1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions.

Recently, during the investigations on planetary oceans, Hirota-Satsuma-Ito-type models have been developed. In this paper, for a (2+1)-dimensional generalized variable-coefficient Hirota-Satsuma-Ito system describing the fluid dynamics of shallow-water waves in an open ocean, non-characteristic movable singular manifold and symbolic computation enable an oceanic auto-Bäcklund transformation with three sets of the oceanic solitonic solutions. The results rely on the oceanic variable coefficients in that system. Future oceanic observations might detect some nonlinear features predicted in this paper, and relevant oceanographic insights might be expected.

With symbolic computation, two classes of lump solutions to the dimensionally reduced equations in (2+1)-dimensions are derived, respectively, by searching for positive quadratic function solutions to the associated bilinear equations. To guarantee analyticity and rational localization of the lumps, two sets of sufficient and necessary conditions are presented on the parameters involved in the solutions. Localized characteristics and energy distribution of the lump solutions are also analyzed and illustrated.

With respect to oceanic fluid dynamics, certain models have appeared, e.g., an extended time-dependent (3+1)-dimensional shallow water wave equation in an ocean or a river, which we investigate in this paper. Using symbolic computation, we find out, on one hand, a set of bilinear auto-Bäcklund transformations, which could connect certain solutions of that equation with other solutions of that equation itself, and on the other hand, a set of similarity reductions, which could go from that equation to a known ordinary differential equation. The results in this paper depend on all the oceanic variable coefficients in that equation.