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Nowadays pedagogical testing technology has become the basic tool for diagnosis and assessment of the level of students' mastery of learning material. Primarily they allow testing the acquired knowledge and skills in their use as a technology of the definite types of problems solution. Thus, the level of logical reasoning development plays a significant role in the successful mastery of many subjects (mathematical courses in particular). So, the problem of objective and measurable criteria for assessing the impact of the level of logical reasoning on the mastery of mathematical subjects is of current concern. We have studied the scientific sources that describe the testing technologies use for assessment of academic achievement as well as the level of logical reasoning development. We have found that the existing methods are based only on the individual work of the teacher with a student and don't suggest any diagnosing technology. The goal of our research is to prove the effectiveness of our method as compared to the traditional one.

A decorated lattice is suggested and the Ising model on it with three kinds of interactions K1, K2, and K3 is studied. Using an equivalent transformation, the square decorated Ising lattice is transformed into a regular square Ising lattice with nearest-neighbor, next-nearest-neighbor, and four-spin interactions, and the critical fixed point is found at K1 = 0.5769, K2 = −0.0671, and K3 = 0.3428, which determines the critical temperature of the system. It is also found that this system and the regular square Ising lattice, and the eight-vertex model belong to the same universality class.

Abstract-For a 3-generated free modular lattice we obtain a set of 11 defining relations and prove that this set is minimal. DOI: 10.3103/S1066369X13100071Keywords and phrases: free modular lattices, defining relations.Recall that the rank of free algebra from some manifold is the cardinal number of the set of its free generators. We concentrate our attention on a free lattice of the rank 3 in the manifold of modular lattices; we denote it by A. Let F be a free lattice of the rank 3 in the manifold of all lattices; let f , g, and h be its free generators, and let ϕ be a homomorphism from F to A. By standard considerations of a universal algebra, elements a = ϕ(f ), b = ϕ(g), and c = ϕ(h) are free generators of the lattice A. Relations defining this lattice in the manifold of all lattices were considered in [1] and [2]. In the paper [1] one has particularly shown that A can be defined by 21 relations. In [2] one has proved that this set of defining relations is not minimal; namely, it was shown there that 15 relations among those mentioned above define the lattice A, moreover, they form the minimal set of defining relations for A. Note that in [1] one has described a set of seven defining relations for a free distributive lattice of the rank 3, and in [3] this set was proved to be minimal.The following assertion is the main result of this paper: There exists a set of 11 defining relations for the lattice A. Note that this set is not a subset of the set of defining relations indicated in [1]. Let us enumerate these relations:*