There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0×10(-21). It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203,000 years, equivalent to a significance greater than 5.1σ. The source lies at a luminosity distance of 410(-180)(+160)  Mpc corresponding to a redshift z=0.09(-0.04)(+0.03). In the source frame, the initial black hole masses are 36(-4)(+5)M⊙ and 29(-4)(+4)M⊙, and the final black hole mass is 62(-4)(+4)M⊙, with 3.0(-0.5)(+0.5)M⊙c(2) radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.

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The Observation of Gravitational Waves from a Binary Black Hole Merger

EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

Dyson-Schwinger equations and their application to hadronic physics

Several phenomenological models of hadrons are derived from quantum chromodynamics (QCD). Functional-integral methods are used to obtain an approximate bilocal-field representation of the QCD generating functional. The degenerate vacuum configurations of the action for the bilocal fields have a rich topological structure arising from the dynamically broken chiral symmetry. Restricting the bilocal-field fluctuations to those of local fields results in a local-field bosonization of QCD, from which we obtain values for several meson parameters. A nontopological-soliton ansatz for hadronic states leads to a generalized bag-model action which as special cases contains the chiral-bag-model action and the MIT-bag-model action. All parameters in these actions are calculable and in particular the MIT bag constant is calculated. This work extends our previous calculation of the bag constant.

Soliton bag models of hadrons from QCD.

From QCD we derive a three-body Faddeev-type formulation of baryons, as qqq colour-singlet states bound by gluon exchange, which is covariant, has dynamically hidden chiral symmetry and incorporates the colour dynamics. The formulation exploits the dynamical role of colour "3 diquark substructure in baryons to simplify computations. For non-zero current quark masses the jP = ~ + and ~ - baryon octet mass formulae are shown to satisfy the Gell-Mann-Okubo and the Coleman-Glashow multiplet mass relationships.

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Baryon Structure and QCD

Baryons in QCD are three-quark color-singlet bound states and any two of the quarks are necessarily in a 3-bar color state. These 3-bar diquark states form bound states which, if they are of sufficiently low mass, will play a significant dynamical role in the structure of baryons. Here we estimate these diquark masses by using an approximate homogeneous Bethe-Salpeter (BS) equation. A new technique for determining the mass of a bound state from the BS equation, which avoids the need to numerically solve the integral equation, is introduced and its validity demonstrated by application to various meson states. Our diquark mass values are compared with those determined from analysis of experimental data on inclusive production of protons.

Calculation of diquark masses in QCD

Introduction Non-Process and Process Physics Syntactical and Semantic Information Self-Referential Neural Networks Quantum Homotopic Field Theory Emergent Universe A New Theory of Gravity Dark Matter Effect Gravitational Attractors Detection of Absolute Motion Gravitational In-Flow Modern Interferometers Looking Back Conclusion References Index.

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Reginald T. Cahill

Papers

Soliton bag models of hadrons from QCD.

Baryon Structure and QCD

Calculation of diquark masses in QCD

Process Physics: From Information Theory To Quantum Space And Matter

The effective action for the Goldstone modes in a global colour symmetry model of QCD