This paper concerns binary quadratic forms over F[T ]. It develops theory analogous to the theory of binary quadratic forms over Z. Most although not all of the results are almost identical, while some of the proofs require different techniques. In particular, the form class group is determined when the form takes values in a principal ideal domain, and the ideal class group (and class group is...

An internal kink instability is observed to grow and saturate in a line-tied screw pinch plasma. Detailed measurements show that an ideal, line-tied kink mode begins growing when the safety factor q = (4pi2r2B(z))/(mu0I(p)(r)L) drops below 1 inside the plasma; the saturated state corresponds to a rotating helical equilibrium. In addition to the ideal mode, reconnection events are observed to pe...

38. Following the hint, suppose I is a non-zero ideal (since the zero ideal, (0), is clearly principal). Then there exists some a ∈ I with a 6= 0. Then, either a is positive, or −a is positive, and also in I since I is a subring of Z. Thus, I contains positive elements. Now, let S = {x ∈ I |x > 0}. By the Well-Ordering Property of Z, S contains a smallest element, call it c. We claim that, in f...

We introduce the notion of Gröbner S-basis of an ideal of the free associative algebra K〈X〉 over a field K invariant under the action of a semigroup S of endomorphisms of the algebra. We calculate the Gröbner S-bases of the ideal corresponding to the universal enveloping algebra of the free nilpotent of class 2 Lie algebra and of the T-ideal generated by the polynomial identity [x, y, z] = 0, w...

This section deals with the structure theory of complex semisimple Lie algebras. Some references for this material are [He], [Hu], [J], [K1], [K3], and [V]. Let g be a finite-dimensional Lie algebra. For the moment we shall allow the underlying field to be R or C, but shortly we shall restrict to Lie algebras over C. Semisimple Lie algebras are defined as follows. Let rad g be the sum of all th...

Let Z denote a finite collection of reduced points in projective n-space and let I denote the homogeneous ideal of Z . The points in Z are said to be in (i, j)-uniform position if every cardinality i subset of Z imposes the same number of conditions on forms of degree j . The points are in uniform position if they are in (i, j)-uniform position for all values of i and j . We present a symbolic ...

We introduce a new ideal D of the p-adic Galois group-ring associated to a real abelian field and a related ideal J for imaginary abelian fields. Both result from an equivariant, Kummer-type pairing applied to Stark units in a Z p-tower of abelian fields and J is linked by explicit reciprocity to a third ideal S studied more generally in [So3]. This leads to a new and unifying framework for the...

3 Proof of the Composition Theorem Let A be any B-limited adversary that interacts with protocol QP in the real-life model. Our goal is to construct a B-limited adversary H that interacts with protocol Q in the f -hybrid model, such that no environment Z can tell the difference between the two interactions. We will construct H via the following three steps. 1. From A we construct an adversary A...

Let Z denote a finite collection of points in projective n-space and let I denote the homogeneous ideal of Z. The points in Z are said to be in (i, j)-uniform position if every cardinality i subset of Z imposes the same number of conditions on forms of degree j. The points are in uniform position if they are in (i, j)-uniform position for all values of i and j. We present a symbolic algorithm t...

Let S be a standard N-graded polynomial ring over a field k, let I be a multigraded homogeneous ideal of S, and let M be a finitely generated Z-graded Smodule. We prove that the resolution regularity, a multigraded variant of CastelnuovoMumford regularity, of IM is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to the multigraded situation.