Group-theoretic Language Research Articles

Energetic Couplings in Ferroics

AbstractFerroics include diverse degrees of freedom (such as structural distortions and magnetic moments) among which cross couplings occur, rendering a large variety of interesting phenomena. Determining such couplings, based on symmetry analysis, is not only important to interpret observed phenomena but can also result in novel predictions to be then experimentally checked. Often, such energetic couplings are difficult to construct without a deep knowledge of group theoretical symmetry principles. In the present review, a crash course toward the derivation of energetic couplings, without using much group theoretical language, is provided. Rather, the present approach relies on a graphical technique and suitable symbolic language, which naturally yields some known couplings (resulting in, e.g., spin/dipole canting, magnetically driven polarization, and antipolar/antiferroelectric states). This review also reports and discusses other symmetry‐allowed energetic terms, including some leading to the occurrence of an electric polarization in a variety of materials, and “exotic” ones that generate complex phases and phenomena in, for example, nanostructures and heterostructures.

A Method for the Construction of a Cryptographic Group-theoretic Language

This paper generalizes the theory of universal hashing in the construction of cryptographic protocols using group- theoretic language formalisms. This idea is due to Wegman and Carter who gave a construction in 1981 which is extremely useful when the number of authenticators is small compared to the number of possible source states (plaintext messages) in order to accommodate the situation where we would like to authenticate a sequence of messages with the same key. Unlike previous methods for doing this we do not require that each message in the sequence have a “counter” attached to it. We provide necessary definitions and theory and then give a construction which achieves our goals. Keywords: Universal hashing, projective hash family, cryptographic protocol, group-theoretic language,

Quantification of local geometry and local symmetry in models of disordered materials

Abstractmagnified imageWe suggest two metrics for assessing the quality of atomistic configurations of disordered materials, both of which are based on quantifying the orientational distribution of neighbours around each atom in the configuration. The first metric is that of geometric invariance: i.e., the extent to which the neighbour arrangements are as similar as possible for different atoms, allowing for variations in frame of reference. The second metric concerns the degree of local symmetry. We propose that for a set of configurations with equivalent pair correlations, ranking highly those configurations with low geometric invariance but with high local symmetry selects for structural simplicity in a way that does not rely on formal group theoretical language (and hence long‐range periodic order). We show that these metrics rank a range of SiO2 and a‐Si configurations in an intuitive manner, and are also significantly more sensitive to unphysical features of those configurations in a way that metrics based on pair correlations are not. We also report that implementation of the metrics within a reverse Monte Carlo (RMC) algorithm gives rise to an energy landscape that is too coarse (at least in this initial implementation) for amorphous structure ‘solution’.

The congruence subgroup property for Aut $F_2$: A group-theoretic proof of Asada’s theorem

The goal of this paper is to give a group-theoretic proof of the congruence subgroup property for Aut( F_2 ), the group of automorphisms of a free group on two generators. This result was first proved by Asada using techniques from anabelian geometry, and our proof is, to a large extent, a translation of Asada’s proof into group-theoretic language. This translation enables us to simplify many parts of Asada’s original argument and prove a quantitative version of the congruence subgroup property for Aut( F_2 ).

Unitary Groups Acting on Grassmannians Associated with a Quadratic Extension of Fields

Let (V, H) be an anisotropic Hermitian space of finite dimension over the algebraic closure of a real closed field K. We determine the orbits of the group of isometries of (V, H) in the set of K-subspaces of V . Throughout the paper K denotes a real closed field and K its algebraic closure. Then it is well known (see, for example, [4, Chapter 2], [23]; see also [8]) that K = K(i) with i = √−1. Also we let (V,H) be an anisotropic Hermitian space (with respect to the involution underlying the quadratic field extension K/K) of finite dimension n over K. In this context we consider the natural action of the unitary group U = U(V,H) of isometries of (V,H) on the set Xd of all ddimensional K-subspaces of V . The analogous problem where (V,H) is a symplectic space was treated in [1] (for arbitrary quadratic field extensions). It turns out that, in contrast with the symplectic case, there are infinitely many orbits for the action of the unitary group U on Xd. In group theoretic language the stated problem turns into the determination of the double coset spaces of the form (1) GW \G/U, where G = GL (VK) and GW denotes the parabolic subgroup of G stabilizing a member W ∈ Xd (we write VK to indicate that we are regarding V as a vector space over K). The precise structure of double coset spaces involving classical groups is of great interest in applying the classical Rankin-Selberg method for explicit construction of automorphic L-functions, as Garrett [2] and Piatetski-Shapiro and Rallis [6] worked out. 2000 AMS Mathematics Subject Classification. Primary 51N30, 15A21, Secondary 11E39. Received by the editors on October 13, 2003. Copyright c ©2006 Rocky Mountain Mathematics Consortium

Thurston equivalence of topological polynomials

We answer Hubbard's question on determining the Thurston equivalence class of ``twisted rabbits'', i.e. images of the ``rabbit'' polynomial under n-th powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of n. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.

Universal Hash Proofs and a Paradigm for Adaptive Chosen Ciphertext Secure Public-Key Encryption

We present several new and fairly practical public-key encryption schemes and prove them secure against adaptive chosen ciphertext attack. One scheme is based on Paillier's Decision Composite Residuosity (DCR) assumption, while another is based in the classical Quadratic Residuosity (QR) assumption. The analysis is in the standard cryptographic model, i.e., the security of our schemes does not rely on the Random Oracle model.<br /> <br />We also introduce the notion of a universal hash proof system. Essentially, this is a special kind of non-interactive zero-knowledge proof system for an NP language. We do not show that universal hash proof systems exist for all NP languages, but we do show how to construct very efficient universal hash proof systems for a general class of group-theoretic language membership problems.<br /> <br />Given an efficient universal hash proof system for a language with certain natural cryptographic indistinguishability properties, we show how to construct an efficient public-key encryption schemes secure against adaptive chosen ciphertext attack in the standard model. Our construction only uses the universal hash proof system as a primitive: no other primitives are required, although even more efficient encryption schemes can be obtained by using hash functions with appropriate collision-resistance properties. We show how to construct efficient universal hash proof systems for languages related to the DCR and QR assumptions. From these we get corresponding public-key encryption schemes that are secure under these assumptions. We also show that the Cramer-Shoup encryption scheme (which up until now was the only practical encryption scheme that could be proved secure against adaptive chosen ciphertext attack under a reasonable assumption, namely, the Decision Diffie-Hellman assumption) is also a special case of our general theory.

Irreducible Affine Varieties over a Free Group: I. Irreducibility of Quadratic Equations and Nullstellensatz

The object of this paper (which consists of two parts) is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups. We prove that any variety over a free group F can be defined by a finite number of systems of equations S = 1 in triangular form where quadratic words play the role of leading terms. Algebraically, irreducible varieties are exactly the varieties whose coordinate groups are fully residually F . The crucial point of the classification of fully residually free groups is to prove that the coordinate groups of irreducible varieties are embeddable into Lyndon’s free Z[x]-group F. Since every finitely generated fully residually free group is a free factor of the coordinate group of an irreducible variety, and the group F is fully residually F , we obtain a characterization of finitely generated fully residually free groups as subgroups of F. The group F and its subgroups have been studied extensively during the last several years. In particular, every finitely generated subgroup of F (hence, every finitely generated fully residually free group) can be obtained from free abelian groups of finite rank by finitely many free products with amalgamation and HNN-extensions of the type, where amalgamated and associated subgroups are free abelian of finite rank. In particular, this implies that every finitely generated fully residually free group is finitely presented. There are three parts to this paper: algebraic geometry over free groups, the theory of free exponential groups, and Makanin-Razborov’s machinery to deal with equations over free groups [11],[14],[13]. The algebraic geometry approach has been shown to be very usefull in dealing with equations over groups. It provides necessary topological means and a method to transcribe geometric notions into pure group-theoretic language. Following Baumslag, Myasnikov, Remeslennikov [1] we use the standard algebraic geometry notions such as variety, Zariski topology, irreducibility of varieties, radicals and coordinate groups. Some of the ideas of the algebraic geometry approach go back to R. Lyndon [9], E. Rips, J. Stallings [16].

Tensor Products of Representations

If G is a reductive group in characteristic zero with two irreducible representations V1 and V2, we may form the tensor product representation V1 0 V2 where G acts diagonally. The problem is to decompose V1 0 V2 into its irreducible components with their multiplicities. Of course this may be done using Weyl's character formula but this is a very difficult calculation. This paper presents a general method for doing this calculation in a more efficient way. For the general linear group the method was explained in [2] with a minimum of group-theoretic language. For the other classical groups this paper will give explicit formulas. The method of this paper may be used for exceptional groups but for combinatorial reasons it does not seem worthwhile at this point to write an explicit formula. Perhaps it would be more useful to have a computer program for the calculation. As an after thought I have included an appendix on how to decompose an irreducible representation of G when restricted to a large reductive subgroup. The case of the general linear group is well-known.

Glueball calculations with the t expansion.

We discuss the way glueball states can be calculated using the t-expansion technique for the SU(3) lattice gauge theory in 3+1 space-time dimensions. Starting with strong-coupling wave functions we specify the simplest glueball structures that can be analyzed. They are described in a general group-theoretical language. We also present an alternative construction in terms of electric and magnetic fields. We then explain the use of the connected matrix formalism for this problem and discuss the results for the ${0}^{++}$ and ${1}^{+\mathrm{\ensuremath{-}}}$ glueballs that have been analyzed before. We present new results for ${0}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{-}}}$, ${1}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{-}}}$, and ${2}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{-}}}$ glueballs, for which we calculated the t expansion to order ${t}^{6}$. They exhibit a scaling behavior leading to masses which are of the order of twice that of the scalar glueball.

Chapter 2. Outline of the Mode-Mode Coupling Theory

The main purpose of this chapter is to recapitulate an essential part of the mode-mode coupling theoryu~a> by putting great emphasis on clarifying the basic idea, because it has been published in separate papers in a splitted form. The theory of the dressed n-quasiparticle (nQP) mode was developed for the purpose of taking account of the anharmonicity effects automatically, since it had become dear by means of the boson-expansion method4>. 5> that the anharmonicity effects neglected in the random-phase approximation (RPA) play a decisive role in such a finite quantal system as nuclei. Furthermore it has been developed into a microscopic theory capable to treat the dynamical mutual interplay between the pairing and the quadrupole correlation which are generally considered to be the basic correlations in transitional nuclei. On the other hand, there is a phenomenological approach called the inter­ acting boson model (IBM) by Arima and Iachello.6> In this model two kinds of bosons are used; they may be regarded to be introduced in order to embrace the main dyrramical correlations in nuclei. It can successfully classify many experimental data by using a group theoretical language. There have also been some efforts to give a microscopic foundation of the phenomenologi­ cal IBM by using the single-j-shell model.n.s> As extensively discussed in Chapter 1, the dynamical interplay between the pairing and the quadrupole degrees of freedom might play a decisive role for the mechanism of the phase transition from spherical to deformed nuclei and the microscopic structure of the collective excited states could be changed from nucleus to nucleus. Such a dynamical interplay can never be described by the simple group theory (e.g. the IBM), where the microscopic structure of the collective boson is supposed to be unchangeable. In order to make clear the structural change of the collective excited states, therefore, we have to construct a theory which can treat the dynamical interplay between the

On the use of the imaginary-mass representations of the Poincaré group in scattering amplitudes

The imaginary mass representations of the Poincaré group are used in theS-matrix theory in order to describe two-particle elastic-scattering amplitudesF(s, t) for fixed momentum transfert<0. In this first Part, I, we give the definitions of the various group-theoretical quantities in terms of scattering kinematics. At this stage we can formulate in a precise covariant language a «phase-shift analysis in thet-channel» using the imaginary-mass representations of the Poincaré group. This is the natural starting formula for further theoretical investigations. We express very simply the fixed-t dispersion relations in the group-theoretical language. Analyticity of the scattering amplitude is stated in a simple and transparent manner, as a relation between real and imaginary parts. The asymptotic behaviour at high energies is very simple and Regge-pole like.