Language Of Group Theory Research Articles

Some Results on Polish Groups

We prove that no quantifier-free formula in the language of group theory can define the ℵ1-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group. Finally, we prove some results on the structure of the group of automorphisms of a locally finite group: firstly, we prove that it is not the case that every group of automorphisms of a graph of power λ is the group of automorphism of a locally finite group of power λ; secondly, we conjecture that the group of automorphisms of a locally finite group of power λ has a locally finite subgroup of power λ, and reduce the problem to a problem on p-groups, thus settling the conjecture in the case λ = ℵ0.

Orderable groups, elementary theory, and the Kaplansky conjecture

Abstract We show that each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We then consider the relationship with the Kaplansky group rings conjecture and show that {{\mathcal{K}}} , the class of groups which satisfy the conjecture, is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in {{\mathcal{K}}} or more generally two torsion-free groups are universally equivalent.

Loop Representation of Wigner’s Little Groups

Wigner’s little groups are the subgroups of the Lorentz group whose transformations leave the momentum of a given particle invariant. They thus define the internal space-time symmetries of relativistic particles. These symmetries take different mathematical forms for massive and for massless particles. However, it is shown possible to construct one unified representation using a graphical description. This graphical approach allows us to describe vividly parity, time reversal, and charge conjugation of the internal symmetry groups. As for the language of group theory, the two-by-two representation is used throughout the paper. While this two-by-two representation is for spin-1/2 particles, it is shown possible to construct the representations for spin-0 particles, spin-1 particles, as well as for higher-spin particles, for both massive and massless cases. It is shown also that the four-by-four Dirac matrices constitute a two-by-two representation of Wigner’s little group.

The determining number of Kneser graphs

Graph Theory A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G is the minimum cardinality of a determining set of G. This paper studies the determining number of Kneser graphs. First, we compute the determining number of a wide range of Kneser graphs, concretely Kn:k with n≥k(k+1) / 2+1. In the language of group theory, these computations provide exact values for the base size of the symmetric group Sn acting on the k-subsets of 1,..., n. Then, we establish for which Kneser graphs Kn:k the determining number is equal to n-k, answering a question posed by Boutin. Finally, we find all Kneser graphs with fixed determining number 5, extending the study developed by Boutin for determining number 2, 3 or 4.

Crystallography and phase equilibria a review: Part III—Second-order transitions and approximations

In this third part of the review of the relevance of crystallography to phase equilibria, the distinction between firstorder and second-order transitions is the first item to be considered. In a first-order transition G/ is discontinuous; however, in a second-order transition G/ is continuous, but G/d 2 is discontinuous. Here G is the Gibbs energy and is a state variable such as temperature, pressure, or composition (T, P, or X). Plots of G versus T are shown in Fig. III-1 to illustrate the difference between firstand second-order transitions. In Fig. III-1(a), the curve for the -symmetry structure crosses the curve for the -symmetry structure at a point where the Gibbs energies are the same, but the slopes of the two curves are different, dG /dT dG /dT. The juncture is the transition temperature, Tt. It is obvious that has the lower Gibbs energy at T Tt has the lower Gibbs energy and is the stable structure. At Tt the two phases can exist in a two-phase equilibrium. Since dG/dT −S, it is also obvious that iS is not zero. Thus, the combination of iG 0 and iS 0 characterizes a first-order transition with two distinct phases in equilibrium at Tt. For comparison Fig. III-1(b) illustrates the G versus T behavior of a second-order reaction. In this figure, the and -symmetry structures have Gibbs energies that differ at low temperatures but merge at a point where both G G and dG /dT dG /dT and at temperatures above the merger point, the symmetry difference disappears and the Gibbs energies are indistinguishable. The transition temperature is the point of merger. Figure III-1(b) characterizes a second-order transition, and there is no phase change but rather a transition between two symmetries of the same single phase. The second-order transition contrasts with the first-order transition in that both tG and tS equal zero in the second-order transition. The symmetries of the two structures in a second-order transition are related in a manner that allows the possibility of the structures to continuously change from one to the other. The specific conditions that must apply before a secondorder transition can occur were first delineated by Landau and Lifshitz on the mathematical basis of group theory. Their result in the language of group theory necessitates that all of four conditions apply: • Space groups of the structures in a second-order transition must be in a group-subgroup relationship. • The difference between the particle density functions of the two structures must be a basis function, or combination of basis functions, of the irreducible representation of the higher-symmetry space group. • It must not be possible to form a totally symmetric third-order combination of such basis functions. • The low-symmetry structure must be locked in by symmetry to the high-symmetry space group (even if incommensurate).

Discretizing continuous wavelet transforms using integrated wavelets

We present integrated wavelets as a method for discretizing the continuous wavelet transform. Using the language of group theory, the results are presented for wavelet transforms over semidirect product groups. We obtain tight wavelet frames for these wavelet transforms. Further integrated wavelets yield tight families of convolution operators independent of the choice of discretization of scale and orientation parameters. Thus these families can be adapted to specific problems. The method is more flexible than the well-known dyadic wavelet transform. We state an exact algorithm for implementing this transform. As an application the enhancement of digital mammograms is presented.

Wavelet Analysis over Abelian Groups

In this paper we want to show how the language of group theory can be used to unify the different multiresolution analysis for signals on R n , Z n , Z n / N Z n , or any other translation invariant grid such as the hexagonal grid or the quincunx grid in two dimensions. In particular, we want to give a fairly general construction of an infinite dimensional manifold of finite impulse response QMF that works for data on any regular grid in arbitrary dimensions; that is, we give a (possibly incomplete) parametrization of paraunitary filterbanks in arbitrary dimensions.

Radial and latitudinal variations of the magnetic field strength in the outer heliosphere

This paper analyzes the radial, latitudinal, and temporal variations of the magnetic field in the outer heliosphere, using the observations from Voyagers 1 and 2 and Pioneers 10 and 11. The radial variation of the magnetic field strength measured during 1973‐1989 from 1 to 19 AU is compared with that predicted by Parker's spiral field model. The speed deficit (V ‐ V1)/V1 between 1 and 19 AU is (+0.0027 ± 0.0026)/AU for the Voyager data and (0.0099 ± 0.0024)/AU for the Pioneer 11 data. Thus the Voyager data are consistent with no radial variation of the speed, as assumed in deriving Parker's equations for the spiral field, but the Pioneer data are not. The magnetic flux deficit in the outer heliosphere relative to 1 AU is measured by the quantity D ≡ (A ‐ A1)/A1 versus R. For the Voyager data the 1 AU intercept of the best fit line to these data is (0.029 ± 0.031), which is consistent with zero, as required by the definition of D. For the Pioneer 11 data the corresponding 1 AU intercept is (−0.068 ± 0.044), which implies a flux deficit of (7% ± 4%) at 1 AU, suggesting either a systematic error in the data or statistical uncertainties greater than we estimated. The magnetic flux deficit is given by the slope the best fit line to D versus R. The Voyager data give a slope −(0.0027 ± 0.0036)/AU, which is consistent with zero, indicating no flux deficit in the Voyager data. The Pioneer data give a slope (0.0040 ± 0.0042)/AU, which is also consistent with zero. Since the best fit line to the Voyager observations of D versus R gives a value of D at 1 AU consistent with zero, we may set D(1) = 0 and obtain D(R) = [(0.0001 ± 0.0021)/AU] × (R ‐ 1)(AU), consistent with no flux deficit between 1 and 19 AU from 1977 through 1985. For the years 1986 and 1987 the Voyager data give a nonzero flux deficit equal to (−0.264 ± 0.075)/AU and (−0.171 ± 0.070)/AU, respectively. The Pioneer 11 data show no significant flux deficit during 1986 (+0.19 ± 0.22)/AU. The flux deficit observed by Voyager 2 during these 2 years near solar minimum might be related to the presence of a heliospheric vortex street at this time. The magnetic field strength measured by Voyager 1 at high latitudes is smaller than the magnetic field strength measured by Voyager 2 near the heliographic equator beyond 21 AU. During 1986, when the heliospheric vortex street wrapped field lines in spirals around the axes of the vortices, the source magnetic field strength at 27°N was greater than the source field strength at ∼1°N. During 1988 and 1989, when the vortex street was no longer present, the source field strength at ∼30°N was less than that at ∼3°N to 4°N. The topology of the spiral interplanetary magnetic field is determined by a defect in the heliosphere, the rotating Sun. The language of group theory can describe this and other more complex defects and vector fields in the heliosphere.

QE nil-2 groups of exponent 4

The notion of a QE (quantifier-eliminable) group comes from logic: a group G is QE if every formula of the first-order language of group theory is equivalent in G to a formula without quantifiers. In the context of uniformly locally finite (ULF) groups, this logical property has an interesting algebraic equivalent: a ULF group G is QE if and only if every isomorphism between finite subgroups of G is induced by an automorphism of G. In this paper we show that there exist 2No countable ULF QE groups. It is well-known, and easy to verify, that every ULF QE group is Nocategorical. The existence of 2Ko countable &categorical groups is a recent result of Berline and Cherlin [ 11. Actually, Berline and Cherlin construct, for any prime p, 2No countable QE nilrings of characteristic p (necessarily Nocategorical). Using the Mal’cev correspondence [5] they then obtain, for every odd p, 2”” countable &-categorical nonabelian nil-2 groups of exponent p. However, it is easy to see that these groups are not QE, and in fact it can be shown [2] that every QE nonabelian nilpotent p-group must have exponent 4. The result we prove here is:

Minimal and prime models of complete theories of torsion free abelian groups

We shall characterize the complete first order theories of torsion free abelian groups having algebraically prime models, elementarily prime models, minimal models, etc., using the well known invariants from [9], [3]. This study was suggested by the questions raised in [2]. Group here will always mean abelian group. For reference on groups see Fuchs [5], [6], for model theory Chang-Keisler [1]. All theories considered are theories in the first order language of group theory. A group G is minimal (strictly minimal) if H < G implies H = G (H-= G), and elementarily prime (algebraically prime) if G is elementarily (algebraically) embeddable in every H G. p, q always denote prime numbers, P the set of all primes. The p-rank of a group G, p-rk(G), is the dimension of G/pG as a vector space over Z/pZ. We define tf(p, G): = min {p-rk(G), ~o}. By the results of [9], [3], two nontrivial torsion free abelian groups G and H are elementarily equivalent iff for all primes p tf(p, G) = tf(p, H). This gives a natural 1-1 correspondence between the complete theories of nontrivial torsion free abelian groups and the functions /3 : P --~ ~o + 1. /3 will always stand for such a function, and the corresponding complete theory will be denoted by T 0. S o is the set of primes p for which/3(p) ~ 0. For more definitions see section 3. Let 13 be a function from P to o) + 1. We shall study the following questions: Does T o have an elementarily (algebraically) prime model? Does T o have a minimal (strictly minimal) model, and if this is the case, how many such models does T o have? Do the minimal and strictly minimal models of T o coincide? These questions have been studied for the special case of the theory of the