Additional Abelian Research Articles

Splittings of cyclic groups and perfect shift codes

A splitting (M,S) of an additive Abelian group G consists of a set of integers M and a subset S/spl sub/G such that every nonzero element g/spl isin/G can be uniquely written as m/spl middot/h for some m/spl isin/M and h/spl isin/S. Splittings M={/spl plusmn/1,/spl middot//spl middot//spl middot/,/spl plusmn/k} correspond to perfect k-shift codes used in the analysis of run-length-limited codes correcting single peak shifts. We shall determine the set S for splittings of cyclic groups Z/sub p/, p prime, by M={1,a,/spl middot//spl middot//spl middot/,a/sup r/,b,/spl middot//spl middot//spl middot/,b/sup s/} and M={/spl plusmn/1,/spl plusmn/a,/spl middot//spl middot//spl middot/,/spl plusmn/a/sup r/,/spl plusmn/b,/spl middot//spl middot//spl middot/,/spl plusmn/b/sup s/}. This yields new conditions on the existence of perfect 3- and 4-shift codes. Further, it can be shown that splittings of Z/sub p/ by {/spl plusmn/1,/spl plusmn/2,/spl plusmn/3} exist exactly if Z/sub p/ is also split by {1,2,3}.

MULTIDIMENSIONAL ABELIAN AND TAUBERIAN COMPARISON THEOREMS

Theorems in which a specified asymptotic behavior of the quotient of two (generalized) functions leads to a conclusion about the asymptotic behavior of the quotient of integral transforms of them are called Abelian comparison theorems. The theorems converse to them are called Tauberian comparison theorems. This article concerns some Abelian and Tauberian comparison theorems for generalized functions with supports in pointed cones. The Laplace transform is used as an integral transform. It is shown that additional Abelian conditions are needed for the validity of Abelian theorems in the multidimensional case. Bibliography: 5 titles.

Dual potentials in non-Abelian gauge theories.

Motivated by the possibility that confinement and superconductivity are similar phenomena, dual potentials are introduced into Yang-Mills theory in two different ways. Both are extensions of Zwanziger's two-potential formalism for Abelian charges and monopoles to the non-Abelian case. In the first approach the dual potentials carry a color index and there is a rather simple, although nonlocal, dual-variable formulation. In the second approach dual variables are introduced into the so-called Abelian projection of the SU(2) Yang-Mills theory. An interesting feature is that the quartic contact interactions are absent and there is a special gauge choice for which the theory takes on a ``purely electromagnetic'' form. More important, however, is the appearance of an additional Abelian magnetic gauge symmetry the dynamical breaking of which may be associated with confinement.

A class of resolvable incomplete block designs

Abstract A class of resolvable incomplete block designs with three and four replications is obtained here starting from an additive Abelian group.

A gauge-invariant, spontaneously broken spinor theory without genuine gauge and Higgs fields.—II

In terms of a 4-component spinor-isospinor Weyl-type constituent field, a unique Lagrangian ~det[χ(n)χ*(n)] can be constructed on a space-time lattice with the large symmetry groupU1⊗SL4,C,loc—or essentially a subgroupU1⊗SU4,loc by taking the quantization condition into account—as invariance group. The invariance under local transformations in this framework is established without introducing additional vector gauge fields. In the continuum limit the Lagrangian obtains the form of a non-Abelian gauge-invariant spinor Lagrangian where effective vector gauge fields and certain effective Higgstype fields occur on the same level as, respectively, vectorial and scalar (or tensorial) bilinear local composites of the constituent spinor field. Physical spinor fields appear as trilinear local composites. The effective Lagrangian—with appropriate interpretation given elsewhere—reflects the salient features of the Glashow-Weinberg-Salam model, but contains additional Abelian and non-Abelian gauge-type interactions.

Goldstone Bosons as Bound States in the Quark-Gluon Model

We examine the implications of a Nambu-Goldstone realization of chiral symmetry in the quark-gluon model. The context of this examination is that of a renormalizable, finite theory, so eigenvalue conditions are assumed to be satisfied. We discuss the solutions to the Schwinger-Dyson gap equation for the fermion self-energy $\ensuremath{\Sigma}({p}^{2})$ that exhibit spontaneous breaking of the vacuum symmetry. In the leading-order Bethe-Salpeter approximation the boundary conditions to the homogeneous, linear integral equations stipulate the vacuum symmetry. It is shown how the Goldstone bosons emerge as bound states, as suggested by Nambu and Jona-Lasinio. We also examine the Goldstone alternative in the Bethe-Salpeter equation for fermion-fermion scattering. Explicit symmetry breaking is introduced by additional Abelian vector gluons coupling to hypercharge and isospin besides baryon number. The eigenvalue condition for the fine-structure constant is consequently model-dependent but takes a simple form. We also consider the influence of explicit symmetry breaking on the ground-state mesons and indicate how the solutions to the eigenvalue problem regulate the structure of symmetry breaking.

Complex Number Algebra as a Simple Case of Heaviside Operational Calculus

In [1] the author offered the following postulational basis for an algebraic structure called a heaviside operational calculus. Let G be an additive Abelian group of operands. Let E be a set of endomorphisms on G. If a nonnull element, U, of G has the property that every element of G is the map of U under some endomorphism in E, we call U a unit element of G relative to E. Suppose G and E satisfy the following postulates: 1. The endomnorphisms in E are permutable. 2. G contains at least one unit element, U, relative to E. 3. Any endomorphisin in E that sends a unit element U into a nonnull operand sends every nonnull operand in G into a nonnull operand. It follows, as shown in [1], that E is an integral domain under the usual definitions of equality, sums, and products for endomorphisms. For some choices of G and E, E will also be a field. G and E (or G and a class of operators associated with E) determine a heaviside operational calculus. For example, let G be the class of entering, sectionally continuous functions of t. The unit step function, {u(t) }, will serve as a unit operand for postulate 2. Given a member, f, of G we find the following endomorphism on G that sends u intof:

A Ring of Sequences Generated by Rational Functions

Proof of the Theorem. The additive identity of the ring is { 0 } and the multiplicative identity is I I }, generated by 1/(1 -x). Since the generator of {an+bn } is found by adding the generators of { an } and { bn, }, it is clear that the sequences of D form an additive Abelian group. In fact, the only ring property which is not immediately obvious is that the term by term product of any two sequences in D is also in D. To prove this we recall an observation due to Hadamard [5]: If A (x) = fanXn, B(x) = ZbnXn, and C(x) = EZanbnXn, then

A Homological characterization of certain Abelian groups

Let G be a monoid; that is to say, G is a set such that with each pair σ, τ of elements of G there is associated a further element of G called the ‘product’ of σ and τ and written as στ. In addition it is required that multiplication be associative and that G shall have a unit element. The so-called ‘Homology Theory’† associates with each left G-module A and each integer n (n ≥ 0) an additive Abelian group Hn (G, A), called the nth homology group of G with coefficients in A. It is natural to ask what can be said about G if all the homology groups of G after the pth vanish identically in A. In this paper we give a complete answer to this question in the case when G is an Abelian group. Before describing the main result, however, it will be convenient to define what we shall call the homology type of G. We write Hn(G, A) ≡ 0 if Hn(G, A) = 0 for all left G-modules A.

On the Realizability of Homotopy Groups

It should be stated at once that we allow dim PT = ??. The same question, with the restriction dim PT < CC Iis, presumably, much more difficult. By a polyhedron we shall mean one of the kind defined on p. 316 of [1]. Such a polyhedron, P, is covered by a simplicial complex, K. The latter need not be star-finite but the topology of P is such that a sub-set, X C P, is closed if, and only if, X n a is closed for each (closed) simplex, o, of the K. We shall describe P as locally finite if, and only if, K is star-finite. The combinatorial processes used here are described in [1], [2] and, more clearly, I hope, in [3]. By a we shall mean a connected membrane complex of the kind introduced on p. 1211 of [2]. These complexes, which are treated in some detail in [3], may be infinite provided they cover polyhedra, in our sense of the word. Let iri be an arbitrary (multiplicative) group and let r2 , 1r3, ... be a sequence of arbitrary (additive) Abelian groups, each of which admits ir1 as a group of operators. If the number of elements in each group i-xn is countable we shall describe the set irn } as countable. We prove: THEOREM. There is a polyhedron, P, which is locally finite if Jr,, I} is countable, whose homotopy groups are related to {7rn} by isomorphisms (onto), f'n :7,(P) a-*r (n 1, 2, ),such that

The Ring of Endomorphisms for which every Subgroup of an Abelian Group is Invariant

Given an additive Abelian group A, the invariance of a subgroup B of A for an endomorphism a of A is a dyadic relation ap B between endomorphisms and subgroups of A. From every dyadic relation, two reciprocal lattices of closed sets are derived*.