Modern fact databases contain abundant data classified through several classifications. Typically users must consult these classifications in separate manuals or files thus making their effective use difficult. Contemporary database systems do little to support deductive use of classifications. In this paper we show how deductive data management techniques can be applied to the utilization of d...

For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomatic-deductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence...

We study GIT semistability of Hilbert points of Milnor algebras of homogeneous forms. Our first result is that a homogeneous form F in n variables is GIT semistable with respect to the natural SL(n)-action if and only if the gradient point of F, which is the first non-trivial Hilbert point of the Milnor algebra of F, is semistable. We also prove that the induced morphism on the GIT quotients is...

We study Kähler functions on the Hilbert ball and and the algebra of Kähler functions. We introduce quantum mechanics and operator algebras on the Hilbert ball by using the theory of geometrical quantum mechanics.

Characterization of the *-subalgebras in the algebra of bounded operators acting on Hilbert space is presented. Sufficient conditions for the existence of a faithful representation in pre-Hilbert space of a *-algebra in terms of its Groebner basis are given. These conditions are generalization of the unshrinkability of monomial *-algebras introduced by C. Lance and P. Tapper. The applications t...

We compute the C*-algebra generated by a group of composition operators acting on certain reproducing kernel Hilbert spaces over the disk, where the symbols belong to a non-elementary Fuchsian group. We show that such a C*-algebra contains the compact operperators, and its quotient is isomorphic to the crossed product C*-algebra determined by the action of the group on the boundary circle. In a...

An element u of a norm-unital Banach algebra A is said to be unitary if u is invertible in A and satisfies ‖u‖ = ‖u−1‖ = 1. The norm-unital Banach algebra A is called unitary if the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A . If X is a complex Hilbert space, then the algebra BL(X) of all bounded linear operators on X is unitary by the Russo–Dye th...

D. Buşneag 1 defined pseudo valuation on a Hilbert algebra and proved that every pseudo valuation induces a pseudometric on a Hilbert algebra. Also, D. Buşneag 2 provided several theorems on extensions of pseudo valuations. C. Buşneag 3 introduced the notions of pseudo valuations valuations on residuated lattices, and proved some theorems of extension for these using the model of Hilbert algebr...

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C-algebra is represented as a function algebra on the set of pure states with a noncommutative ∗-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space P(H) of a Hilbert space H. The space P(H) is an infinit...

We study the tangent space at a monomial point M in the Hilbert scheme that parameterizes all ideals with the same Hilbert function as M over an exterior algebra.