For a given measure space (X, B, m) we construct all measure spaces (Y, C, l) in which (X, B, m) is embeddable. The construction is modeled on the ultrafilter construction of the Stone-Cech compactification of a completely regular topological space.Under certain conditions the construction simplifies. Examples are given when this simplification occurs.

One of the important features of an interconnection network is its ability to efficiently simulate programs or parallel algorithms written for other architectures. Such a simulation problem can be mathematically formulated as a graph embedding problem. In this paper we compute the lower bound for dilation and congestion of embedding onto wheel-like networks. Further, we compute the exact dilation of embedding wheellike networks into hypertrees, proving that the lower bound obtained is sharp. Again, we compute the exact congestion of embedding windmill graphs into circulant graphs, proving that the lower bound obtained is sharp. Further, we compute the exact wirelength of embedding wheels and fans into 1, 2-fault hamiltonian graphs. Using this we estimate the exact wirelength of embedding wheels and fans into circulant graphs, generalized Petersen graphs, augmented cubes, crossed cubes, Möbius cubes, twisted cubes, twisted n-cubes, locally twisted cubes, generalized twisted cubes, odd-dimensional cube connected cycle, hierarchical cubic networks, alternating group graphs, arrangement graphs, 3-regular planer hamiltonian graphs, star graphs, generalised matching networks, fully connected cubic networks, tori and 1-fault traceable graphs.

Purpose: The aims of the paper are to obtain necessary and sufficient conditions of existence and smoothness for non-polynomial spline spaces of fifth order, to establish the uniqueness of the Bφ-spline spaces in the class C4 among mentioned spaces (under condition of fixed grid), and to prove the embedding of the Bφ-spline spaces corresponding to embedded grids.Methods: In the paper, the approximation relations with initial grid and with complete chain of vectors are applied to obtain the minimal spline spaces. Usage of locally orthogonal chain of vectors gives opportunity to construct special approximation relations from which the initial space of Bφ splines is constructed.Results: Deletion of a knot from initial grid gives a new grid, and as result, a new space of Bφ splines is embedded in the initial space mentioned above.Conclusions: Consequent deletion of the knots (one by one) generates the sequence of the embedded spaces of Bφ splines. Obtained results are successfully proved. They may be applied to spline-wavelet decompositions.

Background: The link prediction issue is one of the most widely used problems in complex network analysis. Link prediction requires knowing the background of previous link connections and combining them with available information. The link prediction local approaches with node structure objectives are fast in case of speed but are not accurate enough. On the other hand, the global link prediction methods identify all path structures in a network and can determine the similarity degree between graph-extracted entities with high accuracy but are time-consuming instead. Most existing algorithms are only using one type of feature (global or local) to represent data, which not well described due to the large scale and heterogeneity of complex networks. Methods: In this paper, a new method presented for Link Prediction using node embedding due to the high dimensions of real-world networks. The proposed method extracts a smaller model of the input network by getting help from the deep neural network and combining global and local nodes in a way to preserve the network's information and features to the desired extent. First, the feature vector is being extracted by an encoder-decoder for each node, which is a suitable tool for modeling complex nonlinear phenomena. Secondly, both global and local information concurrently used to improve the loss function. More obvious, the clustering similarity threshold considered as the local criterion and the transitive node similarity measure used to exploit the global features. To the end, the accuracy of the link prediction algorithm increased by designing the optimization operation accurately. Results: The proposed method applied to 4 datasets named Cora, Wikipedia, Blog catalog, Drug-drug-interaction, and the results are compared with laplacian, Node2vec, and GAE methods. Experimental results show an average accuracy achievement of 0. 620, 0. 723, 0. 875, and 0. 845 on the mentioned datasets, and confirm that the link prediction can effectively improve the prediction performance using network embedding based on global similarity.

Letf ¹1; 3 be a positive integer. We prove that there exists a numerical semigroup S with embedding dimension three such that f is the Frobenius number of S. We also show that the same fact holds for affine semigroups in higher dimensional monoids.