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We explore model-based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the… (More)

The systematics of indices of physico-chemical properties of codons and amino acids across the genetic code are examined. Using a simple numerical labelling scheme for nucleic acid bases, A=(-1,0), C=(0,-1), G=(0,1), U=(1,0), data can be fitted as low order polynomials of the six coordinates in the 64-dimensional codon weight space. The work confirms and… (More)

It is possible to consider stochastic models of sequence evolution in phylogenetics in the context of a dynamical tensor description inspired from physics. Approaching the problem in this framework allows for the well developed methods of mathematical physics to be exploited in the biological arena. We present the tensor description of the homogeneous… (More)

- P D Jarvis, R C King
- 2006

We derive group branching laws for formal characters of subgroups Hπ of GL(n) leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings GL(n) ↓ GL(n − 1) , GL(n) ↓ O(n) and GL(2n) ↓ Sp(2n) fixing a vector vi , a symmetric tensor gij = gji and an antisymmetric tensor fij = −fji , respectively, are… (More)

The purpose of this article is to show how the isotropy subgroup of leaf permutations on binary trees can be used to systematically identify tree-informative invariants relevant to models of phylogenetic evolution. In the quartet case, we give an explicit construction of the full set of representations and describe their properties. We apply these results… (More)

Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the " Dirichlet Hopf algebra of arithmetics " by dualizing the addition and multiplication maps. Then we study… (More)

- P D Jarvis, S O Morgan
- 2008

The Schrödinger-Robertson inequality for relativistic position and momentum operators X µ , Pν , µ, ν = 0, 1, 2, 3 , is interpreted in terms of Born reciprocity and 'non-commutative' relativistic position-momentum space geometry. For states which saturate the Schrödinger-Robertson inequality, a typol-ogy of semiclassical limits is pointed out, characterised… (More)

Recent work has discussed the importance of multiplicative closure for the Markov models used in phylogenetics. For continuous-time Markov chains, a sufficient condition for multiplicative closure of a model class is ensured by demanding that the set of rate-matrices belonging to the model class form a Lie algebra. It is the case that some well-known Markov… (More)

A model is presented for the structure and evolution of the eukaryotic and vertebrate mitochondrial genetic codes, based on the representation theory of the Lie superalgebra A(5,0) approximately sl(6/1). A key role is played by pyrimidine and purine exchange symmetries in codon quartets.

The supersymmetric model of 1] for the evolution of the genetic code is elaborated. Energy considerations in nucleic acid strand modelling, using sl(2) polarity spin and sl(2=1) family box quartet symmetry, lead for the case of codons and anticodons to assignments of codons to 64-dimensional sl(6=1) ' A(5; 0) multiplets. In a previous paper 1] we… (More)