The records of the Phenylketonuria Register were examined to determine the factors associated with early intellectual progress in children who had received a diet low in phenylalanine from soon after birth. A total of 1031 children were born between 1964 and 1980 and started treatment before they were 4 months of age, and 808 of them were followed up prospectively. In 263 children born between ...

In this paper, we first define the notion of finitely-cosmall quotient (singly-cosmall quotient) morphisms. Then give a characterization new concept. We show that an epimorphism p:Y→U is if and only for any right R-module Z morphism g:Z→Y such pg finitely-copartial isomorphism (singly-copartial isomorphism) from to Y with codomain U finitely (singly) split epimorphism. also investigate relation...

Inspired by the classical concept of height of a prime ideal in a ring, we proposed in a precedent paper the notion of height of a prime hyperideal in a Krasner hyperring. In this note we first generalize some results concerning the height of a prime hyperideal in a Noetherian Krasner hyperring, with the intent to extend this definition to the case of a general hyperideal in a such hyperring. T...

For every local ring R in what follows, mR denotes the maximal ideal. Let Λ̃ be a complete, regular, local Noetherian ring, let E ⊂ m Λ̃ be an ideal, and denote the quotient by Λ. Thus, Λ is also a complete, local Noetherian ring. Denote the residue field Λ/mΛ by k. Denote by C = CΛ the category whose objects are Λ-algebras A such that (i) A is a local, Artin ring with mΛA ⊂ mA, and (ii) the indu...

BACKGROUND Associations between genotype and intellectual outcome in patients with phenylketonuria are complicated because intelligence is influenced by many variables, including environmental factors and other genetic determinants. Intellectual changes with age, both on and after relaxation of diet, vary within the patient population. This study aims to determine whether a significant associat...

Let G be a locally compact abelian group with compact open subgroup H . The best known example of such a group is G = Qp, the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H = Zp, the ring of padic integers. Classical wavelet theories, which require a non-trivial discrete subgroup for translations, do not apply to G, which may not have such a subg...

6 Introduction to Affine Schemes 2 6.1 Rings and Modules of Fractions . . . . . . . . . . . . . . . . . 2 6.2 The Spectrum of a Unital Commutative Ring . . . . . . . . . 5 6.3 The Spectrum of a Quotient Ring . . . . . . . . . . . . . . . . 7 6.4 The Spectrum of a Ring of Fractions . . . . . . . . . . . . . . 9 6.5 Intersections of Prime Ideals . . . . . . . . . . . . . . . . . . . 11 6.6 Topolo...

Entangling properties of a mixed 2-qubit system can be described by the local homogeneous unitary invariant polynomials in elements of the density matrix. The structure of the corresponding invariant polynomial ring for the special subclass of states, the so-called mixed X−states, is established. It is shown that for the X−states there is an injective ring homomorphism of the quotient ring of S...

We show that the WZW fusion rings at finite levels form a projective system with respect to the partial ordering provided by divisibility of the height, i.e. the level shifted by a constant. From this projective system we obtain WZW fusion rings in the limit of infinite level. This projective limit constitutes a mathematically welldefined prescription for the ‘classical limit’ of WZW theories w...