The solution of the problem of enumeration of the n-paths in a digraph has so far been attempted through an indirect approach of enumerating the redundant chains. The approach has yielded an algorithm for determination of the general formula for the matrix of redundant n-chains and also a partial recurrence formula for the same. This paper presents a direct approach to the problem. It gives a recurrence relation expressing the matrix of n-paths of a digraph in terms of the matrices of (n − 1)-paths of its first-order subgraphs. The result is exploited to give an algorithm for computing the matrix of n-paths. The algorithm is illustrated with a 6 × 6 matrix.

We describe system verification tests and early science results from the pulsar processor (PTUSE) developed for the newly commissioned 64-dish SARAO MeerKAT radio telescope in South Africa. MeerKAT is a high-gain (${\sim}2.8\,\mbox{K Jy}^{-1}$) low-system temperature (${\sim}18\,\mbox{K at }20\,\mbox{cm}$) radio array that currently operates at 580–1 670 MHz and can produce tied-array beams suitable for pulsar observations. This paper presents results from the MeerTime Large Survey Project and commissioning tests with PTUSE. Highlights include observations of the double pulsar $\mbox{J}0737{-}3039\mbox{A}$, pulse profiles from 34 millisecond pulsars (MSPs) from a single 2.5-h observation of the Globular cluster Terzan 5, the rotation measure of Ter5O, a 420-sigma giant pulse from the Large Magellanic Cloud pulsar PSR $\mbox{J}0540{-}6919$, and nulling identified in the slow pulsar PSR J0633–2015. One of the key design specifications for MeerKAT was absolute timing errors of less than 5 ns using their novel precise time system. Our timing of two bright MSPs confirm that MeerKAT delivers exceptional timing. PSR $\mbox{J}2241{-}5236$ exhibits a jitter limit of $<4\,\mbox{ns h}^{-1}$ whilst timing of PSR $\mbox{J}1909{-}3744$ over almost 11 months yields an rms residual of 66 ns with only 4 min integrations. Our results confirm that the MeerKAT is an exceptional pulsar telescope. The array can be split into four separate sub-arrays to time over 1 000 pulsars per day and the future deployment of S-band (1 750–3 500 MHz) receivers will further enhance its capabilities.

In presenting experimental results on turbulent wall jets, most workers have adopted as basic scales the slot height b and jet exit velocity Uj. Although this is a natural choice, experience with other flows like jets and boundary layers, where the details of the initial conditions are eventually (if not always rapidly) forgotten by the flow, prompts the question: cannot a gross parameter like the jet momentum flux Mj rather than Uj and b separately, suffice to determine the flow, at least in some region where the wall jet may be said to have attained a “fully-developed” state? If the answer to the question is yes, it would give a more compact and practically useful description of the flow than is currently available.

The aim of the present paper is to construct error correcting quantum codes from classical error correcting group codes by using the Schur orthogonality relations for characters of a finite abelian group and the Knill–Laflamme criterion for a quantum code to correct a pre-assigned family of errors. This is an abstraction and generalization of Shor's example of a nine qubit single error correcting quantum code.

We construct several examples of positive definite functions, and use the positive definite matrices arising from them to derive several inequalities for norms of operators.

Let [Ascr] be a unital von Neumann algebra of operators on a complex separable Hilbert space ([Hscr]0, and let {Tt, t [ges ] 0} be a uniformly continuous quantum dynamical semigroup of completely positive unital maps on [Ascr]. The infinitesimal generator [Lscr] of {Tt} is a bounded linear operator on the Banach space [Ascr]. For any Hilbert space [Kscr], denote by [Bscr]([Kscr]) the von Neumann algebra of all bounded operators on [Kscr]. Christensen and Evans [3] have shown that [Lscr] has the form

formula here

where π is a representation of [Ascr] in [Bscr]([Kscr]) for some Hilbert space [Kscr], R: [Hscr]0 → [Kscr] is a bounded operator satisfying the ‘minimality’ condition that the set {(RX−π(X)R)u, u∈[Hscr]0, X∈[Ascr]} is total in [Kscr], and K0 is a fixed element of [Ascr]. The unitality of {Tt} implies that [Lscr](1) = 0, and consequently K0 = iH−½R*R, where H is a hermitian element of [Ascr]. Thus (1.1) can be expressed as

formula here

We say that the quadruple ([Kscr], π, R, H) constitutes the set of Christensen–Evans (CE) parameters which determine the CE generator [Lscr] of the semigroup {Tt}. It is quite possible that another set ([Kscr]′, π′, R′, H′) of CE parameters may determine the same generator [Lscr]. In such a case, we say that these two sets of CE parameters are equivalent. In Section 2 we study this equivalence relation in some detail.

It is shown that the family of representations {jt, t ∈ ℝ+} of the universal enveloping algebra U of the N-dimensional unitary group which is generated by the N-dimensional number process of quantum stochastic calculus can be expressed in the form

where ψ is a bijective linear map from U onto the space S of symmetric tensors over the Lie algebra, and It is the iterated (chaotic) integral on S. The chaotic product * is defined by the formula

and satisfies

This work generalizes and completes earlier results on the centre of U.

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

Some sufficient conditions for the reconstructability of separable graphs are given proceeding along the lines suggested by Bondy, Greenwell and Hemminger. It is shown that the structure and automorphism group of a central block plays an important role in the reconstruction.

An elementary and self-contained account of analytic Jordan decomposition of matrix-valued analytic functions is given. An integral representation for their eigenvalues is obtained. This leads to estimates of the differences in eigenvalues and the number of points of degeneracy.

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

In this paper we use a generalized form of Polya's theorem (1) to obtain generating functions for the number of ordinary graphs with given partition and for the number of bicoloured graphs with given bipartition. Both the points and lines of the graphs are taken as unlabelled. These graph enumeration problems were proposed by Harary in his review article (4). Read (7, 8) solved the problem for unlabelled general graphs and labelled ordinary graphs.