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.

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.

A novel smart vaccination method is proposed in this paper to distribute a limited number of vaccines among the people of a large community, such as a country, consisting of smaller communities like cities or provinces. The proposed method is comprised of two phases; A vaccine allocation phase and a targeted vaccination phase. In the first phase, the available vaccines are allocated to the communities based on demographics and the effectiveness of each type of vaccine. In the second phase, each community is modelled as a contact graph, and the vaccines available to the community are administered to the individuals whose vaccination has the greatest impact on breaking the chain of transmission. As a result of utilizing the Node2Vec graph Embedding algorithm, the complexity of the proposed method increases linearly with the number of people in the community, as opposed to common centrality based methods, the complexities of which increase with the square or cube of the number of individuals. Furthermore, the proposed method can distribute multiple types of vaccines with different probabilities of effectiveness. The performance of the proposed method is comparable to the common centrality based vaccination methods, while its complexity is lower. The results of the simulation show a 20% decrease in the peak number of infected individuals.