Graham Higman's PORC theorem

Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer $n$ the number of groups of order $p^{n}$ is bounded by a polynomial in $p$, and he formulated his famous PORC conjecture about the form of the function $f(p^{n})$ giving the number of groups of order $p^{n}$. In the second of these two papers he proved that the function giving the number of $p$-class two groups of order $p^{n}$ is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general theorem contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof. In this note I give a proof of Higman's general theorem written in the light of these recent developments

Phase Lags and Coherence of X-Ray Variability in Black Hole Candidates

The ``low'' (hard or ``non-thermal'') state of black hole candidates is sometimes modelled via an optically thick, hot Compton cloud that obscures a softer input source such as an accretion disk. In these models the observed output spectra consist entirely of photons reprocessed by the cloud, making it difficult to extract information about the input spectra. Recently Miller (1995) has argued that the Fourier phase (or time) lag between hard and soft X-ray photons in actuality represents the phase lags intrinsic to the input source, modulo a multiplicative factor. The phase lags thus would be a probe of the input photon source. In this paper we examine this claim and find that, although true for the limited parameter space considered by Miller, the intrinsic phase lag disappears whenever the output photon energy is much greater than the input photon energy. The remaining time lags represent a ``shelf'' due to differences between mean diffusion times across the cloud. As pointed out by Miller, the amplitude of this shelf -- which is present even when the intrinsic time lags remain -- is indicative of the size and temperature of the Compton cloud and is a function of the two energies being compared. However, we find that with previous instruments such as Ginga the shelf, if present, was likely obscured by counting noise. A more sensitive measure of Compton cloud parameters may be obtainable from the coherence function, which is derived from the amplitude of the Fourier cross power spectral density. This function has been seen to exponentially decrease at high Fourier frequencies in Cygnus X-1. Coherence loss is characteristic of Compton clouds that undergo large variations of size and/or temperature on time scales longer than about 10 seconds. We argue that observing phase lags and coherenceComment: 14 pages, uuencoded postscript, accepted for publication in Monthly Notice

Non-PORC behaviour of a class of descendant pp-groups

We prove that the number of immediate descendants of order $p^10$ of $G_p$ is not PORC (Polynomial On Residue Classes) where $G_p$ is the $p$-group of order $p^9$ defined by du Sautoy's nilpotent group encoding the elliptic curve $y^2=x^3-x$. This has important implications for Higman's PORC conjecture

The writer as storyteller?

African Studies Seminar series. Paper presented 7 March, 1988In recent times, new schools of literary theory have appeared in South Africa. First came the materialists to challenge the liberal hegemony, and then the structuralists, who, in some of the later versions of theory, see the materialists themselves as too much wedded to liberalism and humanism. And then there is also Njabulo Ndebele, who seems to be something of a phenomenon in himself. Perhaps this is because he is African, not white. Literary theory in South Africa is, of course, largely monopolised by white academics, and this has no doubt, some consequences for the character of th resultant theory. It is noticeable, for example, with perhaps few exceptions, that materialist and structuralist theorists in South Africa derive their conceptual apparatus from the West, intact and ready-formed. All that remains is to apply it, as well as may be, to the local material. In other words, such local theorists art in an essentially pupilage relationship to the theorists of the West, who are vastly more sophisticated, inventive and original. Indeed, it could be said that the real theoretical work is being done in the West, and only imitated here, in a muffled sort of way. This is not said with the intention of deriding the efforts of local academics who attempt to grapple with and apply materialist and structuralist theories. I am, after all, one of those involved! There seems to be no other way, and this way does offer a certain scope. Imitation is never simply repetition, and perhaps this imitation is never, in any case, simply imitation. If local critical theory is derivative, this is a reflection on the nature of the relationship between South Africa and the West, on the nature of the South African education system, and on the separation of the upper reaches of the education system from the major realities of South African life. The separation of academic life from social life makes it very difficult, if not impossible, to be conceptually productive. All this is by way prelude to some consideration of the distinctiveness of Ndebele's contribution to literary theory in South Africa....

Japan and the issue of nuclear energy

As a resource poor country, Japan needs to import 84% of its energy requirements. In 2010, Japan generated 1,080 billion kWh gross - with 27% from coal, 27% from gas, 27% from nuclear, 9% from oil and 7% from hydro. Following the March 2011 Fukushima meltdown disaster, 70% of Japanese citizens in 2012 want nuclear power to be reduced and 80% disapprove of the Government's handling of the Fukushima crisis. Business interests emphasize the inexpensive cost of nuclear generated electricity compared to other sources - in terms of the 859.7 billion kWh of electrictiy consumed in Japan in 2010, nuclear generated electricity was 1.06 times cheaper than coal; 12 times cheaper than LNG; and 40 times cheaper than oil. Such arguments, however, are wearing very thin in the public mind. Some 20,000 protestors gathered outside the Diet Building on 29 July 2012 to insist on ending all nuclear energy. Feeling was so high that the protestors broke through police barriers in political demonstrations the nature of which have not been seen since the 1960s. Japan presently has the capability to generate 222 GW of electricity from wind turbines; 70 GW from geothermal plants; 26.5 GW from additional hydro capacity; and 4.8 GW from solar energy - a potential total of 323.3 GW or 115% of Japan's 2010 energy usage. In 2010, Japan generated 282 GW of total installed electricity; the third highest level of consumption in the world behind only the United States and the People's Republic of China

Dividends, stock repurchases and signaling: evidence from U.S. panel data

This paper exploits yearly accounting data from 1977 to 1994 to test the relative signaling power of dividends and net stock repurchases. The specification controls for potential agency cost and asset dissipation effects. Specifically, we regress changes in future income before extraordinary items on changes in dividends, changes in net stock repurchases, and a host of control variables. We also split the sample at 1981 to measure the impact of changes in the relative taxation of distribution methods. For the full twenty-year sample, only dividend changes are correlated with changes in future income. Moreover, the dividend coefficient and the repurchases coefficient differ statistically different in every future income equation. Splitting the sample reveals that the pre-1981 subsample drives the full-sample results. Put another way, the empirical link between changes in dividends and changes in future income vanishes just as a revision of the tax law reduced the tax disadvantage of dividend distributions. This evidence supports the notion that, at least for a period in time, firms deliberately exposed shareholders to punitive taxation to signal favorable prospects.Corporate governance

China, the United Nations and regime stability

Since assuming its seat in the UN Security Council in 1971, China has gradually moved to use its veto to protect pariah states (with which it does business) and to check United States policy on matters deemed not to be in China's interests (the protection or recognition of Taiwan.) China's UN diplomacy is coloured far more by pragmatism than it is by ideology - its voting being shaped by world perceptions as well as strategic considerations. Domestically, China experiences some 150,000 to 180,000 protest demonstrations and riots a year. Yet, despite widespread corruption and massive inequalities of income and wealth, the Chinese Communist Party (CCP) retains its iron grip on power - economic, health, environmental and political destabilizing factors notwithstanding. Essentially, the CCP controls, imprisons or executes all who seek to reform it or who dare to openly oppose it. Significantly, Mao Ze Dong, often eulogized as a great benefactor, killed 78 million of his own citizens (through starvation or violence), making him a greater mass murderer than the Nazi leader, Adolph Hitler, who killed 17 million German civilian opponents - some 458.8% fewer than Mao's tally of victims

Reality between the lines

The Poem is a hybrid historical narrative and an imaginative reconstruction of the Edo Period - one of the most important socio-cultural eras in Japan's history. The Poem examines the power and wealth of samurai, daimyos and shoguns; the rigid class system of feudal Japan; the despised but ultimately wealthy position of merchants (often trading in ivory and in silk garments); the expulsion of all foreigners - excepting the Dutch and the Portuguese (whose movements were tightly controlled and closely watched); the expanding popularity of Kabuki Theatre and the Ukiyo-e (or "floating world") of widely-frequented pleasure centres in Edo and Kyoto; the adoption of rice as the basic unit of currency (prior to the acceptance of coins and paper notes); and finally, the fate of the last shogun who lost all political and military power in the four day Battle of Toba-Fushimi - ushering in the Meiji Restoration of 1868 and the reassertion of the power of the divine god-head emperor