While the unit group of an imaginary quadratic field is very simple, the unit group of a real quadratic field has nontrivial structure. Its study involves some geometry and analysis, but also it relates to Pell's equation and continued fractions, topics from elementary number theory.

The primary aim of this paper is to look into the game related practices and significances of games. This perspective is applied to examining the pleasures derived from different games and to analyse the different strategies developed by children and their families to situate and control game playing. Research was conducted among 10– 12-year-old children in Finland during spring and summer 2003...

As mentioned several times in class, the arithmetic of quaternionic Shimura varieties is strongly controlled by the behavior of the class of CM points (just as in the case of modular curves). Just as for elliptic curves, if o is an order in an imaginary quadratic field K, one has a notion of a QM-surface with o-CM. Namely, let ι : O ↪→ A be a QM structure on an abelian surface (we are still wor...

In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on [15], where the case of real-valu...

Let p be a prime number. We say that a number field F satisfies the condition (H ′ pn) when any abelian extension N/F of exponent dividing p has a normal integral basis with respect to the ring of p-integers. We also say that F satisfies (H ′ p∞) when it satisfies (H ′ pn) for all n ≥ 1. It is known that the rationals Q satisfy (H ′ p∞) for all prime numbers p. In this paper, we give a simple c...

For the case of 4 points in Euclidean space, we present a computer aided proof of Conjectures II and III made by Atiyah and Sutcliffe regarding Atiyah’s determinant along with an elegant factorization of the square of the imaginary part of Atiyah’s determinant.

We calculate spin, charge, and azimuthal asymmetries in deeply virtual Compton scattering at leading twist-two level. The measurement of these asymmetries gives access to the imaginary and real part of all deeply virtual Compton scattering amplitudes. We note that a consistent description of this process requires taking into account twist-three contributions and we give then a model dependent e...

It is almost always the case that the elementary matrices generate the special linear group SLn over a ring of integers in a number field. The only exceptions to this rule occur for SL2 over rings of integers in imaginary quadratic fields. The surprise is compounded by the fact that, in these cases when elementary generation fails, it actually fails rather badly: the group E2 generated by the e...

The robustness of the factorization theorem for total cross sections, σ nn /σ γp = σ γp /σ γγ , originally proved by Block and Kaidalov[1] for nn (the even portion of pp and ¯ pp scattering), γp and γγ scattering, is demonstrated. Factorization theorems for the nuclear slope parameter B and ρ, the ratio of the real to the imaginary portion of the forward scattering amplitude, are derived under ...

We show that, up to isomorphism, there are only finitely many totally real function fields which have any totally imaginary extension of a given ideal class number.