Combinatorics of Necklaces and “Hermite Reciprocity”

Combinatorial proof of an explicit formula for dimensions of spaces of semi-invariants of regular representations of finite cyclic groups is obtained. Using bicolored necklaces, a certain reciprocity law following from this formula is also derived combinatorially.     

Cohomology with coefficients in stacks of abelian 2-groups

Cohomology of a topological space with coefficients in stacks of abelian 2-groups is invented. This theory extends the classical sheaf cohomology. An application is given to twisted sheaves.     

Funayama’s theorem revisited

Funayama’s theorem states that there is an embedding e of a lattice L into a complete Boolean algebra B such that e preserves all existing joins and meets in L iff L satisfies the join infinite distributive law (JID) and the meet infinite distributive law (MID). More generally, there is a lattice embedding e: L → B preserving all existing joins in L iff L satisfies (JID), and there is a lattice embedding e: L → B preserving all existing meets in L iff L satisfies (MID). Funayama’s original proof is quite involved. There are two more accessible proofs in case L is complete. One was given by Grätzer by means of free Boolean extensions and MacNeille completions, and the other by Johnstone by means of nuclei and Booleanization. We show that Grätzer’s proof has an obvious generalization to the non-complete case, and that in the complete case the complete Boolean algebras produced by Grätzer and Johnstone are isomorphic. We prove that in the non-complete case, the class of lattices satisfying (JID) properly contains the class of Heyting algebras, and we characterize lattices satisfying (JID) and (MID) by means of their Priestley duals. Utilizing duality theory, we give alternative proofs of Funayama’s theorem and of the isomorphism between the complete Boolean algebras produced by Grätzer and Johnstone. We also show that unlike Grätzer’s proof, there is no obvious way to generalize Johnstone’s proof to the non-complete case.     

Metric properties of quality criteria space for multicriteria optimization problems

The problem of the construction of an object functioning in the regime of optimum performance at the design stage is reduced to the solution of the problem of multicriterion optimization, where the quality criteria are chosen to be its most essential characteristics (parameters). At the same time in all methods of multicriterion optimization the vector quality criterion is considered basically in the linear Euclidean space. Actually, in most cases, the criterion space is non-Euclidean - it is curved. Therefore, such setting cannot give results adequately reflecting the processes running in real systems.In order for the design system to really satisfy the optimality requirements the authors of the given paper offer an absolutely new approach to the solution of the problems of multicriterion optimization based on the definition of the quality criteria space and on finding an invariant corresponding to the distance between any two points of that space.The idea of the study of the metric properties of the quality criteria space and their use in solving problems of multicriterion optimization was offered in the work [1]. But that idea, due to its complexity, has not been completely realized until now. When solving such problems the quality criteria space was automatically identified with the Euclidean space with corresponding metrics. In the general case this could not give results adequately reflecting the processes occurring in real systems.In the present paper metric properties of space criteria are studied for the first time, using as the main instrument the mathematical apparatus of tensor analysis, Riemannian geometry, differential equations in partial derivatives etc. Boundary problems relative to the components of the metric tensor of the n-dimensional space of the phenomenon states enabling to determine its metric properties are posed. The knowledge of the metric tensor furthers the objective appraisal of the phenomenon state and the definition of the optimal state.     

Moduli algebras of some non-semiquasihomogeneous singularities

Under some additional restrictions we find dimensions and bases of moduli algebras of isolated singularities of polynomials in n variables that are sums of n monomials of equal weighted degrees and one monomial of lower degree.     

Classification of Abelian Track Categories

We show that each category enriched in Abelian groupoids is a linear track extension and hence is determined up to weak equivalence by a characteristic chomology class. We also discuss compatibility with coproducts.     

Characteristic classes and transfer relations in cobordism

Decompositions of products of the Ray elements by free generators of small dimensions in the symplectic cobordism ring are obtained. In particular it is stated that most of the -dimensional generators, for small, after multiplication by the Ray elements , , land in the ideal generated by Ray elements of low dimension.

     

The Steenrod Algebra and Theories Associated to Hopf Algebras

We show that the homotopy category of products of Z/p-Eilenberg–Mac Lane spaces is an -algebra which algebraically is determined by the Steenrod algebra considered as a Hopf algebra with unstable structure.     

Secondary derived functors and the Adams spectral sequence

Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of “additive groupoid enriched categories”, in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E₃-term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E₃-term explicitly by an algorithm.