Visualization of Escher-like Spiral Patterns in Hyperbolic Space

Spirals, tilings, and hyperbolic geometry are important mathematical topics with outstanding aesthetic elements. Nonetheless, research on their aesthetic visualization is extremely limited. In this paper, we give a simple method for creating Escher-like hyperbolic spiral patterns. To this end, we first present a fast algorithm to construct Euclidean spiral tilings with cyclic symmetry. Then, based on a one-to-one mapping between Euclidean and hyperbolic spaces, we establish two simple approaches for constructing spiral tilings in hyperbolic models. Finally, we use wallpaper templates to render such tilings, which results in the desired Escher-like hyperbolic spiral patterns. The method proposed is able to generate a great variety of visually appealing patterns.

Computer simulation of the difference between Lasso and Lars

In order to correct the error that Efron et al pointed out that the modified Lars is a fast algorithm for lasso problem, computer simulation is carried out with data. The simulation results show that when a variable is removed first and then entered in the lasso problem, the variables selected by lasso and Lars cannot be the same. At this time, no matter how to modify the size of the variables selected by Lars, the lasso problem cannot be solved. The computer simulation method is used to correct the errors of Efron et al in this paper, and contribute to development of world data computing.

Fast Algorithm for Calculating Transitive Closures of Binary Relations in the Structure of a Network Object

The article proposes a fast algorithm for constructing the transitive closures between all pairs of nodes in the structure of a network object, which can have both directional and non-directional links. The algorithm is based on the disjunctive addition of the elements of certain rows of the adjacency matrix, which models (describe) the structure of the original network object. The article formulates and proves a theorem that using such a procedure, the matrix of transitive closures of a network object can be obtained from the adjacency matrix in two iterations (traversal) on such an array. An estimate of the asymptotic computational complexity of the proposed algorithm is substantiated. The article presents the results of an experimental study of the execution time of such an algorithm on network structures of different dimensions and with different connection densities. For this indicator, the developed algorithm is compared with the well-known approaches of Bellman, Warshall-Floyd, Shimbel, which can also be used to determine the transitive closures of binary relations of network objects. The corresponding graphs of the obtained dependences are given. The proposed algorithm (the logic embedded in it) can become the basis for solving problems of monitoring the connectivity of various subscribers in data transmission networks in real time when managing the load in such networks, where the time spent on routing information flows directly depends on the execution time of control algorithms, as well as when solving other problems on the network structures.