Task: A polynomial can be factorized to the product of linear polynomials, but only for orders up to 4. Numerical solution is possible, but it is imprecise and division leaves a remainder. Also this is a subject of numerical anaysis. Task: Linear algebra: “A system of linear equations Ax = b has a unique solution x = A−1b if and only if det A 6= 0.” Programming: “Real numbers cannot be tested f...

Several representations of the symmetric group, arising from different combinatorial, algebraic and geometric constructions, have lead to the same character, up to multiplication by the sign character: the homology of partition lattice (cf. [5, 7, 13]), the top component of a special quotient of the Stanley-Reisner ring of this same lattice [4], the top component of the cohomology algebra of th...

The de nition of Standard ML provides a form of generic equality which is inferred for certain types, called equality types, on which it is possible to de ne an equality relation in ML. However, the standard de nition is incomplete in the sense that there are interesting and useful types which are not inferred to be equality types but for which an equality relation can be de ned in ML in a unif...

Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as first-order logic, the lambda-calculus, or process calculi. Nominal algebra has a semantics in nominal sets (sets with a finitely-supported permutation actio...

indq 6 indqξ for any ξ ∈ q . It is not always true that indq = indqξ, see Example 1.1 below. However, it was conjectured by ELASHVILI that if q = g is semisimple, then this equality always holds. It easily seen that it suffices to prove the equality indg = indgξ only for the nilpotent elements ξ ∈ g≃ g∗. The conjecture was recently proved by CHARBONNEL [3]. A proof for the classical Lie algebra...

In this work we study the differential aspects of noncommutative geometry for magnetic $C^*$-algebra which is a 2-cocycle deformation group $\mathbb{R}^2$. This algebra intimately related to Quantum Hall Effect in continuous, and our results aim provide new geometric interpretation Kubo's formula. Taking inspiration from ideas developed by Bellissard during 80's, build an appropriate Fredholm m...

Let n E Z+ and let K be a field. Let ~ be a partial order on {1, 2, ... , n}. Let An(:::;) be the matrix incidence algebra consisting of those n x n matrices A = (ai,j) with entries in K, satisfying ai,j 0 whenever i 1:. j. For a subset £ ~ An (::5), a necessary and sufficient condition that the algebra generated by £ u {I} is An(::5) is that (i) for every 1 :::; i, j :::; n with i =1= j, there...

In this paper, we present an extended relational calculus and algebra for a functional logic deductive database language. The extended relational calculus is based on the relational first-order logic, by adding constraints in the form of equalities and disequalities over complex (partially defined and possibly infinite) values and interpreted functions. In addition, we propose the notion of saf...

We study the action of the symmetric group Σn on a tensor product of n − 1 copies of a commutative Hopf algebra A, defined by the second author [8]. We show that for ‘nice’ Hopf algebras, the cohomology algebra H∗(Σn;A⊗n−1) is independent of the coproduct if n.(n − 2)! is invertible in the ground ring. Let A be a graded, connected, unital, counital, associative, coassociative Hopf algebra. In s...

Let R be a Noetherian local ring and Ω an arbitrary R-module of finite depth and finite projective dimension. The flat dimension of Ω is at least depth(R)−depth(Ω) with equality in the following cases: (i) Ω is finitely generated over some Noetherian local R-algebra S; (ii) dim(R) = 1; (iii) dim(R) = 2 and Ω is separated; (iv) R is CohenMacaulay, dim(R) = 3 and Ω is complete.