Probabilistic models are becoming increasingly important in analysing the huge amount of data being produced by large-scale DNA-sequencing efforts such as the Human Genome Project. For example, hidden Markov models are used for analysing biological sequences, linguistic-grammar-based probabilistic models for identifying RNA secondary structure, and probabilistic evolutionary models for inferring phylogenies of sequences from different organisms. This book gives a unified, up-to-date and self-contained account, with a Bayesian slant, of such methods, and more generally to probabilistic methods of sequence analysis. Written by an interdisciplinary team of authors, it aims to be accessible to molecular biologists, computer scientists, and mathematicians with no formal knowledge of the other fields, and at the same time present the state-of-the-art in this new and highly important field.

We present sparse-graph codes appropriate for use in quantum error-correction. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparse-graph codes keep the number of quantum interactions associated with the quantum error-correction process small: a constant number per quantum bit, independent of the blocklength. Third, sparse-graph codes often offer great flexibility with respect to blocklength and rate.We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.Our aim in this paper is to create useful quantum error-correcting codes. To be useful, we think a quantum code must have a large blocklength (since quantum computation becomes interesting only when the number of entangled qubits is substantial), and it must be able to correct a large number of errors. From a theoretical point of view, we would especially like to find, for any rate R, a family of error-correcting codes with increasing blocklength N, such that, no matter how large N is, the number of errors that can be corrected is proportional to N. (Such codes are called 'good' codes.) From a practical point of view, however, we will settle for a lesser goal: to be able to make codes with blocklengths in the range 500-20, 000 qubits and rates in the range 0.1-0.9 that can correct the largest number of errors possible.While the existence of 'good' quantum error-correcting codes was proved by Calderbank and Shor (1996), their method of proof was non-constructive. Recently, a family of asymptotically good quantum codes based on algebraic geometry has been found by Ashikhmin et al. (2000) (see also Ling et al. (2001) and Matsumoto (2002)); however to the best of our knowledge no practical decoding algorithm (i.e., an algorithm for which the decoding time is polynomial in the blocklength) exists for these codes. Thus, the task of constructing good quantum error-correcting codes for which there exists a practical decoder remains an open challenge.This stands in contrast to the situation for classical error-correction, where practicallydecodable codes exist which, when optimally decoded, achieve information rates close to the Shannon limit. Low-density parity-check codes (Gallager 1962;Gallager 1963) are an example of such codes. A regular low-density parity-check code has a parity-check matrix H in which each column has a small weight j (e.g., j = 3) and the weight per row, k is also uniform (e.g., k = 6). Recently low-density parity-check codes have been shown to have outstanding performance (MacKay and Neal 1996;MacKay 1999) and modifications to their construction have turned them into state-of-the-art codes, both at low rates and large blocklengths (Luby et al. 2001;Richardson et al. 2001;Davey and MacKay 1998) and at high rates and short blocklengths (MacKay and Davey 2000). The sparseness of the paritycheck matrices makes the codes easy to encode and decode, even when co...

We propose that the function of dream sleep (more properly rapid-eye movement or REM sleep) is to remove certain undesirable modes of interaction in networks of cells in the cerebral cortex. We postulate that this is done in REM sleep by a reverse learning mechanism (see also p. 158), so that the trace in the brain of the unconscious dream is weakened, rather than strengthened, by the dream.

We argue that cortical maps, such as those for ocular dominance, orientation and retinotopic position in primary visual cortex, can be understood in terms of dimension-reducing mappings from many-dimensional parameter spaces to the surface of the cortex. The goal of these mappings is to preserve as far as possible neighbourhood relations in parameter space so that local computations in parameter space can be performed locally in the cortex. We have found that, in a simple case, certain self-organizing models generate maps that are near-optimally local, in the sense that they come close to minimizing the neuronal wiring required for local operations. When these self-organizing models are applied to the task of simultaneously mapping retinotopic position and orientation, they produce maps with orientation vortices resembling those produced in primary visual cortex. This approach also yields a new prediction, which is that the mapping of position in visual cortex will be distorted in the orientation fracture zones.

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by 2 kd 2 n , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation Uµ+ν ⊂ Uµ ⊗ Uν of the unitary group U(d), for highest weights µ, ν and µ + ν. Let ξµ be the state obtained by tracing out Uν. Then ξµ is close to a convex combination of the coherent states Uµ(g)|vµ , where g ∈ U(d) and |vµ is the highest weight vector in Uµ.For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our "half" theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.

Determining the relationship between composite systems and their subsystems is a fundamental problem in quantum physics. In this paper we consider the spectra of a bipartite quantum state and its two marginal states. To each spectrum we can associate a representation of the symmetric group defined by a Young diagram whose normalised row lengths approximate the spectrum. We show that, for allowed spectra, the representation of the composite system is contained in the tensor product of the representations of the two subsystems. This gives a new physical meaning to representations of the symmetric group. It also introduces a new way of using the machinery of group theory in quantum informational problems, which we illustrate by two simple examples.

The notion of weak measurement provides a formalism for extracting information from a quantum system in the limit of vanishing disturbance to its state. Here we extend this formalism to the measurement of sequences of observables. When these observables do not commute, we may obtain information about joint properties of a quantum system that would be forbidden in the usual strong measurement scenario. As an application, we provide a physically compelling characterisation of the notion of counterfactual quantum computation.

The hormone auxin is transported through many plant tissues with a definite velocity. It is thought that certain channels, or pumps, located at the basal ends of cells, are responsible for the hormone’s transport. It is also known that auxin will induce veins when applied to suitable tissues. T. Sachs has suggested that it is the flow of the hormone that induces vessels. He suggests that discrete strands form because the transport capacity of a pathway increases with the flux that that pathway carries, leading to a canalization of flow. I cast this in the form of a more specific hypothesis: I suppose the permeability for the transport of auxin through the basal plasmalemma of a cell (by means of whatever kind of pump or channel) to increase with flux. I then show that discrete veins will form provided that the transport permeability increases rapidly enough with flux, and provided that the movement of auxin is not too polar, in the sense that there is a substantial amount of diffusive movement of auxin in addition to polar transport. The same hypothesis offers an explanation for the loops of veins found under certain conditions.