Density profiles in the raise and peel model with and without a wall; physics and combinatorics

We consider the raise and peel model of a one-dimensional fluctuating interface in the presence of an attractive wall. The model can also describe a pair annihilation process in disordered unquenched media with a source at one end of the system. For the stationary states, several density profiles are studied using Monte Carlo simulations. We point out a deep connection between some profiles seen in the presence of the wall and in its absence. Our results are discussed in the context of conformal invariance ( c = 0 theory). We discover some unexpected values for the critical exponents, which are obtained using combinatorial methods. We have solved known ( Pascal`s hexagon) and new (split-hexagon) bilinear recurrence relations. The solutions of these equations are interesting in their own right since they give information on certain classes of alternating sign matrices.

Quasi-perfect state transfer in a bosonic dissipative network

In this paper we propose a scheme for quasi-perfect state transfer in a network of dissipative harmonic oscillators. We consider ideal sender and receiver oscillators connected by a chain of nonideal transmitter oscillators coupled by nearest-neighbour resonances. From the algebraic properties of the dynamical quantities describing the evolution of the network state, we derive a criterion, fixing the coupling strengths between all the oscillators, apart from their natural frequencies, enabling perfect state transfer in the particular case of ideal transmitter oscillators. Our criterion provides an easily manipulated formula enabling perfect state transfer in the special case where the network nonidealities are disregarded. We also extend such a criterion to dissipative networks where the fidelity of the transferred state decreases due to the loss mechanisms. To circumvent almost completely the adverse effect of decoherence, we propose a protocol to achieve quasi-perfect state transfer in nonideal networks. By adjusting the common frequency of the sender and the receiver oscillators to be out of resonance with that of the transmitters, we demonstrate that the sender`s state tunnels to the receiver oscillator by virtually exciting the nonideal transmitter chain. This virtual process makes negligible the decay rate associated with the transmitter line at the expense of delaying the time interval for the state transfer process. Apart from our analytical results, numerical computations are presented to illustrate our protocol.

Matrix Representation of the Stationary Measure for the Multispecies TASEP

In this work we construct the stationary measure of the N species totally asymmetric simple exclusion process in a matrix product formulation. We make the connection between the matrix product formulation and the queueing theory picture of Ferrari and Martin. In particular, in the standard representation, the matrices act on the space of queue lengths. For N > 2 the matrices in fact become tensor products of elements of quadratic algebras. This enables us to give a purely algebraic proof of the stationary measure which we present for N=3.

Decay of geometry for Fibonacci critical covering maps of the circle

We study the growth of Df `` (f(c)) when f is a Fibonacci critical covering map of the circle with negative Schwarzian derivative, degree d >= 2 and critical point c of order l > 1. As an application we prove that f exhibits exponential decay of geometry if and only if l <= 2, and in this case it has an absolutely continuous invariant probability measure, although not satisfying the so-called Collet-Eckmann condition. (C) 2009 Elsevier Masson SAS. All rights reserved.

Symmetry breaking in the genetic code: Finite groups

We investigate the possibility of interpreting the degeneracy of the genetic code, i.e., the feature that different codons (base triplets) of DNA are transcribed into the same amino acid, as the result of a symmetry breaking process, in the context of finite groups. In the first part of this paper, we give the complete list of all codon representations (64-dimensional irreducible representations) of simple finite groups and their satellites (central extensions and extensions by outer automorphisms). In the second part, we analyze the branching rules for the codon representations found in the first part by computational methods, using a software package for computational group theory. The final result is a complete classification of the possible schemes, based on finite simple groups, that reproduce the multiplet structure of the genetic code. (C) 2010 Elsevier Ltd. All rights reserved.

Symmetry in biology: from genetic code to stochastic gene regulation

Mathematical models, as instruments for understanding the workings of nature, are a traditional tool of physics, but they also play an ever increasing role in biology - in the description of fundamental processes as well as that of complex systems. In this review, the authors discuss two examples of the application of group theoretical methods, which constitute the mathematical discipline for a quantitative description of the idea of symmetry, to genetics. The first one appears, in the form of a pseudo-orthogonal (Lorentz like) symmetry, in the stochastic modelling of what may be regarded as the simplest possible example of a genetic network and, hopefully, a building block for more complicated ones: a single self-interacting or externally regulated gene with only two possible states: ` on` and ` off`. The second is the algebraic approach to the evolution of the genetic code, according to which the current code results from a dynamical symmetry breaking process, starting out from an initial state of complete symmetry and ending in the presently observed final state of low symmetry. In both cases, symmetry plays a decisive role: in the first, it is a characteristic feature of the dynamics of the gene switch and its decay to equilibrium, whereas in the second, it provides the guidelines for the evolution of the coding rules.

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

Motivated in part by the study of Fadell-Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n >= 1, the class of m-string braid groups B(m)(S(2)\{x(1), ... , x(n)}) of the n-punctured sphere includes the usual Artin braid groups B(m) (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type (C) over tilde (for n = 3). We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of B(m) is completely determined for all m is an element of N (respectively, for all m not equal 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients. When n >= 2, we prove that the lower central series (respectively, derived series) of B(m)(S(2)\{x(1), ... , x(n)}) is constant from the commutator subgroup onwards for all m >= 3 (respectively, m >= 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B(2)(S(2)\{x(1), x(2)}) admits various interpretations, as the Baumslag-Solitar group BS(2, 2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B(2)(S(2)\{x(1), x(2)}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product Z(2) * Z. Further, its lower central series quotients Gamma(i)/Gamma(i+1) are direct sums of copies of Z(2), the number of summands being determined explicitly. In the case m >= 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Gamma(i)/Gamma(i+1) are 2-elementary finitely-generated groups.

Skew-symmetric identities of octonions

Quadratic alternative superalgebras are introduced and their super-identities and central functions on one odd generator are described. As a corollary, all multilinear skew-symmetric identities and central polynomials of octonions are classified. (C) 2008 Elsevier B.V. All rights reserved.

Polynomial identities for the ternary cyclic sum

We simplify the results of Bremner and Hentzel [J. Algebra 231 (2000) 387-405] on polynomial identities of degree 9 in two variables satisfied by the ternary cyclic sum [a, b, c] abc + bca + cab in every totally associative ternary algebra. We also obtain new identities of degree 9 in three variables which do not follow from the identities in two variables. Our results depend on (i) the LLL algorithm for lattice basis reduction, and (ii) linearization operators in the group algebra of the symmetric group which permit efficient computation of the representation matrices for a non-linear identity. Our computational methods can be applied to polynomial identities for other algebraic structures.

On moduli spaces for abelian categories

We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.

The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Let n >= 3. We classify the finite groups which are realised as subgroups of the sphere braid group B(n)(S(2)). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B(n)(S(2)): Z(2(n-1)); the dicyclic groups of order 4n and 4(n - 2); the binary tetrahedral group T*; the binary octahedral group O*; and the binary icosahedral group I(*). We give geometric as well as some explicit algebraic constructions of these groups in B(n)(S(2)) and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi`s classification of the torsion elements of B(n)(S(2)) and explain how the finite subgroups of B(n)(S(2)) are related to this classification, as well as to the lower central and derived series of B(n)(S(2)).

Free groups in central simple algebras without Tits` theorem

Let R be a noncommutative central simple algebra, the center k of which is not absolutely algebraic, and consider units a,b of R such that {a,a(b)} freely generate a free group. It is shown that such b can be chosen from suitable Zariski dense open subsets of R, while the a can be chosen from a set of cardinality \k\ (which need not be open).

A stronger Dunford-Pettis property

We discuss a strong version of the Dunford-Pettis property, earlier named (DP*) property, which is shared by both l(1) and l(infinity) It is equivalent to the Dunford-Pettis property plus the fact that every quotient map onto c(0) is completely continuous. Other weak sequential continuity results on polynomials and analytic mappings related to the (DP*) property are shown.

Finite-dimensional non-associative algebras and codimension growth

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties. (C) 2010 Elsevier Inc. All rights reserved.

REALIZING PROFINITE REDUCED SPECIAL GROUPS

Special groups are an axiomatization of the algebraic theory of quadratic forms over fields. It is known that any finite reduced special group is the special group of some field. We show that any special group that is the projective limit of a projective system of finite reduced special groups is also the special group of some field.