In this article we study inequalities of ideal norms. We prove that in a subring R of a number field every ideal can be generated by at most 3 elements if and only if the ideal norm satisfies N(IJ) ≥N(I)N(J) for every pair of non-zero ideals I and J of every ring extension of R contained in the normalization ˜R .

In this paper we give a module-theoretic description of the isomorphism classes of abelian varieties A isogenous to Br, where the characteristic polynomial g of Frobenius of B is an ordinary square-free q-Weil polynomial, for a power q of a prime p, or a square-free p-Weil polynomial with no real roots. Under some extra assumptions on the polynomial g we give an explicit description of all the isomorphism classes which can be computed in terms of fractional ideals of an order in a finite product of number fields. In the ordinary case, we also give a module-theoretic description of the polarizations of A.

Objective

We aimed to examine whether impaired olfaction is associated with cognitive decline and indicators of neurodegeneration in the brain of dementia-free older adults.

Methods

Within the Rush Memory and Aging Project, 380 dementia-free participants (mean age = 78 years) were followed for up to 15 years, and underwent MRI scans. Olfactory function was assessed using the Brief Smell Identification Test (B-SIT) at baseline, and categorized as anosmia (B-SIT <6), hyposmia (B-SIT 6-10 in men and 6-10.25 in women), and normal (B-SIT 10.25-12 in men and 10.5-12 in women). Cognitive function was annually assessed with a battery of 21 tests, from which composite scores were derived. Structural total and regional brain volumes were estimated. Data were analyzed using linear regression and mixed-effects models.

Results

At study entry, 138 (36.3%) had normal olfactory function, 213 (56.1%) had hyposmia, and 29 (7.6%) had anosmia. In multiadjusted mixed-effects models, hyposmia (beta = -0.03, 95% confidence interval [CI] -0.05 to -0.02) and anosmia (beta = -0.13, 95% CI -0.16 to -0.09) were associated with faster rate of cognitive decline compared to normal olfaction. On MRI, impaired olfaction (hyposmia or anosmia) was related to smaller volumes of the hippocampus (beta = -0.19, 95% CI -0.33 to -0.05), and in the entorhinal (beta = -0.16, 95% CI -0.24 to -0.08), fusiform (beta = -0.45, 95% CI -0.78 to -0.14), and middle temporal (beta = -0.38, 95% CI -0.72 to -0.01) cortices.

Conclusion

Impaired olfaction predicts faster cognitive decline and might indicate neurodegeneration in the brain among dementia-free older adults.

In this thesis we address the problem of developing effective algorithms to compute isomorphism classes of polarized abelian varieties over a finite field and of fractional ideals of an order in a finite product of number fields.

There are well-known methods to efficiently compute the classes of invertible ideals of an order in a number field, but not much has previously been known about non-invertible ideals. In Paper I we produce algorithms to compute representatives of all ideal classes of an order in a finite product of number fields. We also extend a theorem of Latimer and MacDuffee about  conjugacy classes of integral matrices.

There are equivalences established by Deligne and Centeleghe-Stix between the category of abelian varieties over a finite field and the category of finitely generated free abelian groups with an endomorphism satisfying some easy-to-state axioms, which in certain cases can be described in terms of fractional ideals of orders in finite products of number fields. In Paper II we use this method to produce an algorithm that computes the isomorphism classes of abelian varieties in an isogeny class determined by an ordinary square-free q-Weil polynomial or by a square-free p-Weil polynomial with no real roots (where p denotes a prime and q is a power of a prime). In the ordinary case we also produce an algorithm that computes the polarizations up to isomorphism and the automorphism groups of the polarized abelian varieties. If the polarization is principal, we can compute a period matrix of the canonical lift of the abelian variety.

In Paper III we extend the description of the second paper to the case when the Weil polynomial is a power of a square-free polynomial which fulfills the same requirements as in Paper II.

In Paper IV we use the results of the second and third papers to study questions related to base-field extension of the abelian varieties over finite fields.

Deligne and Howe described polarized abelian varieties over finite fields in terms of finitely generated free Z-modules satisfying a list of easy to state axioms. In this thesis we address the problem of developing an effective algorithm to compute isomorphism classes of (principally) polarized abelian varieties over a finite field, together with their automorphism groups. Performing such computations requires the knowledge of the ideal classes (both invertible and non-invertible) of certain orders in number fields. Hence we describe a method to compute the ideal class monoid of an order and we produce concrete computations in dimension 2, 3 and 4.

There are well known algorithms to compute the class group of the maximal order of a number field K and the group of invertible ideal classes of a non-maximal order R. In this paper we explain how to compute also the isomorphism classes of non-invertible ideals of an order R in a finite product of number fields K. In particular we also extend the above-mentioned algorithms to this more general setting. Moreover, we generalize a theorem of Latimer and MacDuffee providing a bijection between the conjugacy classes of integral matrices with given minimal and characteristic polynomials and the isomorphism classes of lattices in certain Q-algebras, which under certain assumptions can be explicitly described in terms of ideal classes.