Application of first-order logic to identify organizers and perpetrators of illegal actions in teams of a limited circle of people

There are three main families of inference algorithms in first-order logic: direct inference and its application to deductive databases and production systems; backward inference procedures and logic programming systems; theorem proving systems based on the resolution method. When solving specific problems, the most effective algorithms are those that allow you to cover all the facts and axioms and must be taken into account in the process of inference. An example is considered in which it is necessary to prove the guilt of a person in murder. On the basis of statements, a knowledge base is formed from expressions, with the help of which an expression of first-order logic is compiled and proved using direct logical inference. The proof of the reasoning obtained in direct inference using the proof tree is given. However, direct inference provides for the implementation of all admissible stages of logical inference based on all known facts. The article also considers a method based on the resolution when implementing the reverse inference, taking into account the expression obtained in the direct inference. This expression is converted into a conjunctive normal formula using the laws of Boolean algebra and is proved by the elimination of events using the conjunction operation.

On the projectivity of proper normal curves over valuation domains

A theorem of Lichtenbaum states, that every proper, regular curve [Formula: see text] over a discrete valuation domain [Formula: see text] is projective. This theorem is generalized to the case of an arbitrary valuation domain [Formula: see text] using the following notion of regularity for non-noetherian rings introduced by Bertin: the local ring [Formula: see text] of a point [Formula: see text] is called regular, if every finitely generated ideal [Formula: see text] has finite projective dimension. The generalization is a particular case of a projectivity criterion for proper, normal [Formula: see text]-curves: such a curve [Formula: see text] is projective if for every irreducible component [Formula: see text] of its closed fiber [Formula: see text] there exists a closed point [Formula: see text] of the generic fiber of [Formula: see text] such that the Zariski closure [Formula: see text] meets [Formula: see text] and meets [Formula: see text] in regular points only.

New results on K-frames

In this paper, we introduce some new [Formula: see text]-frames related to the adjoint of a [Formula: see text]-frame and sum of two [Formula: see text]-frames in Hilbert spaces. We prove that adjoint of a normal [Formula: see text]-frame is also a [Formula: see text]-frame.

Effect of Alternative Fillers on the Properties of Rubber Compounds

The paper is focused on the influence of alternative fillers on rubber compounds properties. Three different types of powder fillers, drinking water treatment sludge (DWTS), perlite and calcium carbonate, were mixed into rubber compound mixtures. The mixtures were composed of STR20, EPDM, zinc oxide, steric acid, paraffin wax, 2-mercaptobenzothiazole (MBT), sulphur, Wingstay L, and filler. The mixtures were mixed in a Kneader type mixer at temperature of 70°C and then continuously mixed using a two-roll mill at temperature of 70°C. The relationships between type and the amount of filler versus properties of rubber compounds were demonstrated. The results showed that tensile and elongation at break of rubber compounds gradually decreased with increasing the amount of filler. Rubber compounds filled with small particle size filler possessed higher tensile strength and elongation at break than those filled with large particle size filler. Values of DIN abrasion loss of rubber compounds prepared under proper mixing condition were not more than 300 mm3. Under appropriate condition, the rubber compounds with DWTS, perlite and calcium carbonate provided sufficiently high shore A hardness (not less than 50 Shore A hardness). Finally, alternative fillers such as DWTS and perlite were expected to replace calcium carbonate in normal formula.

Finite groups with σ-Frobenius condition for non-normal σ-primary subgroups

Let [Formula: see text] be a partition of the set [Formula: see text] of all primes and [Formula: see text] a finite group. A group is said to be [Formula: see text]-primary if it is a finite [Formula: see text]-group for some [Formula: see text]. We say that a [Formula: see text]-subgroup [Formula: see text] of [Formula: see text] satisfies the [Formula: see text]-Frobenius condition in [Formula: see text] if [Formula: see text] is a [Formula: see text]-group. In this paper, we determine the structure of finite groups in which every non-normal [Formula: see text]-primary subgroup satisfies the [Formula: see text]-Frobenius condition.

Some results on normal GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds

The purpose of this paper is to study normal [Formula: see text]-lightlike submanifolds of indefinite nearly Kaehler manifolds. We find some necessary and sufficient conditions for an isometrically immersed [Formula: see text]-lightlike submanifold of an indefinite nearly Kaehler manifold to be a normal [Formula: see text]-lightlike submanifold. Further, we derive a characterization theorem for holomorphic bisectional curvature of a normal [Formula: see text]-lightlike submanifold of an indefinite nearly Kaehler manifold.

Groups with an automorphism which inverts at least one-third of the elements

Let [Formula: see text] be a finite group and [Formula: see text] be the set of the elements [Formula: see text] of [Formula: see text] such that [Formula: see text] where [Formula: see text]. In this paper, we give strong restrictions on the structure of the quotients [Formula: see text] where [Formula: see text] is a finite group with an automorphism [Formula: see text] such that [Formula: see text] and [Formula: see text] is the largest normal [Formula: see text]-subgroup of [Formula: see text].

A new Composition-Diamond lemma for associative conformal algebras

Let [Formula: see text] be the free associative conformal algebra generated by a set [Formula: see text] with a bounded locality [Formula: see text]. Let [Formula: see text] be a subset of [Formula: see text]. A Composition-Diamond lemma for associative conformal algebras is first established by Bokut, Fong and Ke in 2004 [L. A. Bokut, Y. Fong and W.-F. Ke, Composition-Diamond Lemma for associative conformal algebras, J. Algebra 272 (2004) 739–774] which claims that if (i) [Formula: see text] is a Gröbner–Shirshov basis in [Formula: see text], then (ii) the set of [Formula: see text]-irreducible words is a linear basis of the quotient conformal algebra [Formula: see text], but not conversely. In this paper, by introducing some new definitions of normal [Formula: see text]-words, compositions and compositions to be trivial, we give a new Composition-Diamond lemma for associative conformal algebras, which makes the conditions (i) and (ii) equivalent. We show that for each ideal [Formula: see text] of [Formula: see text], [Formula: see text] has a unique reduced Gröbner–Shirshov basis. As applications, we show that Loop Virasoro Lie conformal algebra and Loop Heisenberg–Virasoro Lie conformal algebra are embeddable into their universal enveloping associative conformal algebras.

On the size of paged recursive trees

A paged recursive tree is constructed as a recursive tree except that it depends on an integer parameter [Formula: see text] representing a page capacity, small subtrees with size [Formula: see text]. We investigate the number of nodes [Formula: see text] (size of the tree) in paged recursive trees built from labels [Formula: see text]. The expectation and variance of [Formula: see text] are derived, and it is also shown that [Formula: see text] is asymptotically normal. [Formula: see text] as [Formula: see text] by applying the contraction method.

On normal subgroups with consecutive G-class sizes

Let [Formula: see text] be a group and [Formula: see text] be a normal subgroup of [Formula: see text]. If the set [Formula: see text] is composed by consecutive integers, then [Formula: see text] is either nilpotent or a quasi-Frobenius group with abelian kernel and complements. This is a generalization of Theorem 2 of [A. Beltrán, M. J. Felipe and C. G. Shao, [Formula: see text]-divisibility of conjugacy class sizes and normal [Formula: see text]-complements, J. Group Theory 18 (2015) 133–141].