ON THE SINGULARITIES OF QUOTIENTS BY 1-FOLIATIONS

Degrees of Quantum Function Algebras at Roots of 1

We classify all possible zero-divisor graphs of a particular family of quotients of $$\mathbf{Z}_4[x,y,w,z]$$Z4[x,y,w,z]. As the 90 quotients vary, we obtain a total of 7 graphs, corresponding to seven isomorphism classes, and one of these graphs provides a new example which contradicts Beck's conjecture on the chromatic number of a zero-divisor graph. The algebraic analysis is strongly supported by the combinatorial setting, as already shown in a previous paper, where the graph-theoretical tools were presented and successfully applied to $$\mathbf{Z}_4[x,y,z]$$Z4[x,y,z]--therefore, the just smaller case--in order to get a deeper knowledge of the classical counterexample to Beck's conjecture.

We construct a space W X W_X of fine moduli for the substructures of an arbitrary compact complex space X X . A substructure ( X , A ) (X,\mathcal {A}) of X X is given by a subalgebra A \mathcal {A} of the structure sheaf O X \mathcal {O}_X with the additional feature that ( X , A ) (X,\mathcal {A}) is also a complex space; ( X , A ) (X,\mathcal {A}) and ( X , A ′ ) (X,\mathcal {A’}) are called equivalent if and only if A \mathcal {A} and A ′ \mathcal {A’} are isomorphic as subalgebras of O X \mathcal {O}_X . Since substructures are quotients, it is only natural to start with the fine moduli space Q X Q_X of all complex-analytic quotients of X X . In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of Q X Q_X , we realize that its open subset W X W_X consisting of all substructures turns out to be a complex space which has the required universal property.

It is well-known that quantum algebras at roots of unity are not quasi-triangular. They indeed do not possess an invertible universalR-matrix. They have, however, families of quotients, on which no obstructiona priori forbids the existence an universalR-matrix. In particular, the universalR-matrix of the so-called finite dimensional quotient is already known. We try here to answer the following questions: are most of these quotients equivalent (or Hopf equivalent)? Can the universalR-matrix of one be transformed to the universalR-matrix of another using isomorphisms?

In this paper, we characterize the family of quotients of internally quasicontinuous functions. Moreover, we study cardinal invariants related to quotients in the case of internally quasicontinuous functions and the complement of this family.

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a proper ideal of [Formula: see text]. In this paper, we study some algebraic and homological properties of a family of rings [Formula: see text], with [Formula: see text], that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. Specially, we study when [Formula: see text] is a von Neumann regular ring, a semisimple ring and a Gaussian ring. Also, we study the classical global and weak global dimensions of [Formula: see text]. Finally, we investigate some homological properties of [Formula: see text]-modules and we show that [Formula: see text] and [Formula: see text] are Gorenstein projective [Formula: see text]-modules, provided some special conditions.

In this paper reforms of both present Italian personal income taxation and family allowances are evaluated with the aim to make the tax benefit system able to better support family burden. Two kind of reforms are considered: family quotient, based on the change of the tax unit from persons to families, and a new generalized family allowance, based on equivalent income and absorbing current family tax credits and allowances. After an evaluation of some limits of family quotient (higher effective marginal tax rate for spouse with lower income and a reduced support for low income families) and the current mix of tax credits and allowances (families with higher equivalent income can receive higher support for family burden), the comparison among two specific reforms is performed using a microsimulation model built on Bank of Italy survey about the Italian household and personal incomes. Results shows that it is possible to shape quotient family based reforms with redistributive effects, but also that the new generalized family allowance here presented is able to better support bottom quintiles of population, with or without dependent children.

Family is the first and main place for the growth and development of children. If the condition in the family is good and happy so that the children will grow well. To develop SQ in the family, parents can develop by: duty, care, knowledge, personal change, brotherhood and dedicated leadership. The first place to grow spiritual quotient or spiritual intelligence is family. The children that are grown in high spiritual quotient family environment will be high spiritual quotient people also.