Using a singular value decomposition of a beamline matrix, composed of many beam position measurements for a large number of pulses, together with the measurement of pulse-by-pulse beam properties or machine attributes, the contributions of each v ariable to the beam centroid
Section I IntroductionThe purpose of this paper is to investigate the relation between the structures of primary groups and their large subgroups. All groups are abelian. This research was motivated by a study ot the monumental paper "Homomorphisms of Primary Abelian Groups" [8] by R. S. PIERCE in which the theory of large subgroups was introduced. We shall state some of the fundamental results of this theory in section I.In section II we collect an assortment of facts, some of which appear ill the literature, some of which must be classified as folklore and some of which are new. The principal result of this section is a new representation theorem for large subgroups of p-groups.Section III is devoted to the problem: Give necessary and sufficient conditions for a p-group G to be quasi-isomorphic to every large subgroup of G. The results of this section show that unless the group G satisfies certaiD restrictive conditions on its Ulm invariants it contains a large subgroup L which is not quasi-isomorphic to G.Although a large subgroup ot a group G is not m general quasi-isomorphic to G the theorems of section IV show that the structure ot a p-group G is in many cases preserved by its large subgroups. Some of our results can be stated as follows. If L is a large subgroup of a group G then: 1. L is a direct sum of cyclic groups if and only if G is a direct sum ot cyclic groups; 2. L is totally projective if and only if G is totally projective; 3. L is torsion-complete Jf and only if G is torsion-complete; 4. L is quasi-closed if and only if G is quasi-closed.Our notation is that of FUCHS [1] with the following exceptions. We use A + B and Y, zAz to denote group unions. Direct sums we denote by AOB and | Direct products we denote by H~Aa. We use the symbol Zx to denote the cyclic group generated by the element x.The following definitions and theorems were given by PIERCE in [8]. All groups are p-groups. A large subgroup L of a group G is a fully invariant subgroup of G such that B+L = G for every basic subgroup B of G. It is easily shown that if L is a large subgroup of a group G then pnL is a large subgroup of G for every nonnegative integer n.If x is an element of a group G define the U'sequence of x to be U(x) = = (ha(x), hG(px ), hG(p2x), ...). Our heights are either non-negative integers or the symbol ~. Define a U-sequence for the group G to be any strictly increasing sequence of integers (no, hi, rt2, .-.) such that ni + 1
First experimental results from the final focus test beam (FFTB) are reported. The vertical dimension of a 47-GeV electron beam from the SLAC linac has been reduced at the focal point of the FFTB by a demagnification of 320 to a beam height of approximately 70 nm.
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