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.

In this paper, we first present a new type of the concept of open sets by expressing some properties of arbitrary mappings on a power set. With the generalization of the closure spaces in categorical topology, we introduce the GENERALIZED topological spaces and the concept of GENERALIZED continuity and become familiar with weak and strong structures for GENERALIZED topological spaces. Then, introducing the concept of the GENERALIZED EMBEDDING and the GENERALIZED injection, we study Császár product of GENERALIZED spaces in the category of GENERALIZED topological spaces. Using by the tools of category theory, we describe the results of classifying on the GENERALIZED injective spaces in which these spaces are characterized as GENERALIZED EMBEDDING of Császár product with the product topology of two points Sierpinski space. Finally, the GENERALIZED dual-injection spaces as the objects of a special subcategory of the GENERALIZED topological spaces are studied for which all singlepoint subsets are closed.

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.

In the image stegenography, the distortion of stego image will be increased by increasing the number of embedded bits and this makes the enemy be much more suspicious about the existence of secret information in the images. One can use the matrix EMBEDDING technique to alleviate this problem which can be done by dividing the cover image into some blocks and limiting the changes in each block. However, the EMBEDDING capacity will be limited by this technique. In this article, a solution to this problem (limited EMBEDDING capacity), a modified version of Fan’s method, is presented. In this technique, the maximum number of changes in each block is chosen to be 2 instead of 1 in Fan’s technique which leads to reducing the blocks length and increasing the EMBEDDING capacity. The results revealed that for a given capacity, the stego image distortion is decreased and the EMBEDDING capacity is increased in compared with the other existing methods in this field. Also, the stego images have high PSNR along with high vision quality which leads to more security.

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.

In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of GENERALIZED differential squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have high accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.

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.