Contrary Statements About Mathematics

The preceding paper ‘Strong statements of analysis’by A. R. D. Mathias defends a so-called full-blooded set theorywithout full detail [3]. He again objects to a weak set theorywhich he calls ‘Mac’, in which the usual Zermelo–Fraenkelseparation scheme is required only for formulas with suitably‘restricted’ quantifiers. I had proposed that suchseparation is adequate for all standard uses of set theory inmathematics. But Mathias has not produced any counter examplesof actual mathematics which requires the use of a stronger separation.     

Eric Charles Milner

Eric Charles Milner was born on 17 May 1928 and brought up inLondon. His father was an engineer, but times were hard andwork was often difficult to obtain. So his mother had to helpout by working as a seamstress, and Eric was often looked afterby his grandmother. At the age of 11, he won a scholarship tothe Haberdashers' Aske's Boys' School, but never attended itin its permanent London buildings because the outbreak of theSecond World War caused all London schools and their pupilsto be evacuated to safer parts of the country. As a result,Eric, an only child and knowing none of his new schoolfellows,was billeted at a home near Reading where he was extremely unhappy.In despair, he ran away and returned to London, where, afterunsuccessful attempts to find him another billet, he roamedthe streets and missed school. After some time, he was eventuallyfound another billet where he received kindness and was muchhappier. Despite these disruptions and the other inevitableshortcomings of a war-time education, Eric's intelligence morethan sufficed to surmount such hurdles, and in later life hecould speak and write better than most of us. From 1946 to 1951, Eric attended King's College, London. Hegraduated with First Class Honours in 1949, when he was awardedthe Drew Gold Medal as the most distinguished Mathematics studentin that year, and a Research Studentship. He then studied foran MSc degree, taking ‘Modern algebra’ and ‘Quantummechanics (Wave mechanics)’ as his selected subjects,his supervisors being Richard Rado (then a Reader at King'sCollege) and Professor Charles Coulson. He received the MScdegree, with distinction, in 1950. This was followed by a year'sresearch in quantum mechanics under the supervision of ProfessorCoulson.     

Kunihiko Kodaira

Kunihiko Kodaira, who died on 26 July 1997, was the outstandingJapanese mathematician of the post-war period, his fame establishedby the award of the Fields Medal at the Amsterdam Congress in1954. He was born on 16 March 1915, the son of an agricultural scientistwho at one time was Vice Minister of Agriculture in the JapaneseGovernment and had also played an active role in agriculturaldevelopments in South America. Kodaira studied at Tokyo University,taking degrees in both mathematics and physics. From 1944 to1951 he was an associate professor of physics at the University.His PhD thesis was published in the Annals of Mathematics [18],and it immediately attracted international attention. Essentiallythis filled a significant lacuna in the basic theorem of W.V. D. Hodge on harmonic integrals. Kodaira had worked on thisfor many years but, because of the war, his research was carriedout in isolation from the international community and did notbecome known until much later. Hermann Weyl, who had been a keen supporter of Hodge's work,realised the importance of Kodaira's thesis, and arranged forhim to come to the Institute for Advanced Study in Princetonin 1949. This was the start of Kodaira's 18-year residence inthe United States, a fruitful period which saw the full blossomingof his research, much of it in collaboration with Donald Spencer.Kodaira spent many years at Princeton, divided between the Instituteand the University, but the years 1961–67 were more unsettled,seeing him successively at Harvard, Johns Hopkins and finallyStanford. In 1967 he returned to a professorship at the Universityof Tokyo, where he remained until the normal retiring age. From1975 to 1985 he worked at Gakushuin University, where retirementrestrictions did not apply.     

Weakly Almost Periodic Functions on N with a Negative Base

Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T², T³, ...} of iteratesof a single operator T on the Hilbert space L²(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2^cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2^c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more ‘natural’ than Ruppert's. We also showthat there are enough functions of our type to distinguish 2^cidempotentsin wN.     

Die Habilitation von John von Neumann an der Friedrich-Wilhelms-Universität in Berlin: Urteile über einen ungarisch-jüdischen Mathematiker in Deutschland im Jahr 1927

The mathematician John von Neumann was born in Hungary but principally received his scientific education and socialization in the German science system. He received his Habilitation from the Friedrich-Wilhelms–Universität in Berlin in 1927, where he lectured as a Privatdozent until his emigration to the USA. This article aims at making a contribution to this early part of Neumann’s scientific biography by analyzing in detail the procedure that led to his Habilitation as well as the beginnings of Neumann’s research on functional analysis. An analysis of the relevant sources shows that in Berlin in the year 1927 Neumann was not yet regarded as the outstanding mathematical genius of the 20th century. Furthermore it will be seen that Neumann had great difficulties in developing the fundamental concepts for his path breaking work in spectral theory and only managed to do so with the support of the Berlin mathematician Erhard Schmidt.     

Basil Cameron Rennie

Basil Rennie was born in London on 24 December 1920. He camefrom a long line of engineers, a family tradition that surfacedin much of his later mathematical work. He attended the UniversityCollege School in London, where he obtained a Mathematical Scholarshipat Peterhouse, Cambridge. After graduating in 1941, he foundemployment first with the Rolls Royce Aero Engine division,then with Austin Motor Works. In 1943 he joined the Fleet AirArm of the Royal Navy as a radio mechanic, and he served inthe Pacific Fleet until the end of the war. This was his firstcontact with Australia, and he seems to have liked what he saw. After his service with the Navy, Rennie resumed his studiesat Peterhouse and received a PhD in 1949. Given his strong practicalbent, it is perhaps surprising that he chose lattice theoryas the subject of his thesis; apart from an article [1] in theProceedings of the London Mathematical Society (he became amember in 1947) and a small booklet [2] published at his ownexpense, he never touched lattice theory again. It was at Peterhousethat he took up rowing, an activity which became a life-longinterest. In 1950 Rennie accepted an offer of a senior lectureship atthe University of Adelaide in South Australia. This was a timeof considerable post-war expansion at the University, and itsforward-looking Vice-Chancellor A. P. Rowe recruited a numberof young and promising staff from overseas, some to leadingpositions. For instance, he established a Mathematical Physicsdepartment (unique in Australia) with the 30-year-old H. S.Green as its head, which became one of the most active researchdepartments in Australia.     

Eric John Fyfe Primrose (1920-1998)

Eric Primrose was appointed to a Lectureship in Pure Mathematicsat the University College, Leicester, in 1947 and promoted toa Senior Lectureship in 1954. He came to Leicester direct fromOxford where he had spent the 1946–47 year completinghis degree, which had been interrupted by war service. Ericwas awarded his PhD in 1957 by the University of London. Eric's secondary education was at Chigwell School and in 1939he won an Open Scholarship in Mathematics and went up to StJohn's College, Oxford. He took ‘shortened finals’in 1941 and then entered the RAF as a Technical Officer (Radar),reaching the rank of Flight Lieutenant, having served in variouslocations throughout Europe.     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, N_G(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a ‘perfectisometry’ and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of N_G(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a ‘splendid tilting complex’for p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same ‘p-local structure’, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and ‘lifting’. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and ‘cutting’ by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods.     

Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.     

Groups Quasi-Isometric to H2xR

The most powerful geometric tools are those of differentialgeometry, but to apply such techniques to finitely generatedgroups seems hopeless at first glance since the natural metricon a finitely generated group is discrete. However Gromov recognizedthat a group can metrically resemble a manifold in such a waythat geometric results about that manifold carry over to thegroup [18, 20]. This resemblance is formalized in the conceptof a ‘quasi-isometry’. This paper contributes toan ongoing program to understand which groups are quasi-isometricto which simply connected, homogeneous, Riemannian manifolds[15, 18, 20] by proving that any group quasi-isometric to H²xRis a finite extension of a cocompact lattice in Isom(H²xR) orIsom((2, R)).     

Non-Classical Stems from Classical: N. A. Vasiliev’s Approach to Logic and his Reassessment of the Square of Opposition

In the XIXth century there was a persistent opposition to Aristotelian logic. Nicolai A. Vasiliev (1880–1940) noted this opposition and stressed that the way for the novel – non-Aristotelian – logic was already paved. He made an attempt to construct non-Aristotelian logic (1910) within, so to speak, the form (but not in the spirit) of the Aristotelian paradigm (mode of reasoning). What reasons forced him to reassess the status of particular propositions and to replace the square of opposition by the triangle of opposition? What arguments did Vasiliev use for the introduction of new classes of propositions and statement of existence of various levels in logic? What was the meaning and role of the “method of Lobachevsky” which was implemented in construction of imaginary logic? Why did psychologism in the case of Vasiliev happen to be an important factor in the composition of the new ‘imaginary’ logic, as he called it?      

Brian Hartley, 1939-94

Brian Hartley began his algebraic career as one of Philip Hall'sresearch students in Cambridge. He obtained his Ph.D. in 1964,spent two post-doctoral years in the USA and, on his returnto the United Kingdom, accepted a lectureship in the newly establishedMathematics Department at Warwick University; there he was promotedto a readership in 1973. He was appointed to a chair of puremathematics at the University of Manchester in 1977 and wasHead of the Mathematics Department there from 1982–4.He was elected to the London Mathematical Society in 1968 andserved on Council from 1987–9. He won an EPSRC SeniorResearch Fellowship, but died on 8 October 1994, a few daysafter taking it up. He travelled widely and took a lively interestin other cultures and languages. His intellectual energy, enthusiasmfor algebra, direct manner and dry sense of humour endearedhim to the many mathematical friends he made around the world.He was devoted to mathematics and gave generously of his timeand energy in support of younger colleagues.     

Nathan Jacobson (1910-1999)

Nathan Jacobson, who died on 5 December 1999, was an outstandingalgebraist, whose work on almost all aspects of algebra wasof fundamental importance, and whose writings will exercisea lasting influence. He had been an honorary member of the Societysince 1972. Nathan Jacobson (later known as ‘Jake’ to his friends)was born in Warsaw (in what he describes as the ‘Jewishghetto’) on 5 October 1910 (through an error some documentshave the date 8 September); he was the second son of CharlesJacobson (as he would be known later) and his wife Pauline,née Rosenberg. His family emigrated to the USA duringthe First World War, first to Nashville, Tennessee, where hisfather owned a small grocery store, but they then settled inBirmingham, Alabama, where Nathan received most of his schooling.Later the family moved to Columbus, Mississippi, but the youngNathan entered the University of Alabama in 1926 and graduatedin 1930. His initial aim was to follow an uncle and obtain adegree in law, but at the same time he took all the (not verynumerous) mathematics courses, in which he did so well thathe was offered a teaching assistantship in mathematics in hisjunior (3rd) year. This marked a turning point; he now decidedto major in mathematics and pursue this study beyond College.During his final year at Alabama he applied for admission andfinancial aid to three top graduate schools in the country:Princeton, Harvard and Chicago. He was awarded a research assistantshipat Princeton; after the first year he was appointed a part-timeinstructor for two years, and during his fourth year he wasappointed a Procter Fellow. The stipend was enough to enablehim to make a grand tour of Europe by car in 1935, in the companyof two Princeton fellow-students at the time: H. F. Bohnenblustand Robert J. Walker.     

Righting the early history of computing,or how sausage was made

The short answer to the question just posed seems to be, “Not much.” Since I have given “the long answer” elsewhere,¹² I can summarize it here. Berg could see no point in writing Bromley. What could he write to someone he believed guilty of plagiarism? What could such a letter accomplish? He did, however, write to New York University Press; to all the universities involved, and to the Works’ English publisher (Pickering and Chatto), who said they passed the letter on to Campbell-Kelly (30 June 1990); to a great many professional societies in Australia, England, and the United States; to a great many governmental agencies and some politicians in those countries; to some publications, both academic and popular; to the Pope and several cardinals; and to a miscellany of other individuals. Generally, those in the best position to do something—for example, the three universities involved —did not even answer Berg’s letter. Others often did answer, but their answer was generally that they were in no position to do anything. That was how matters stood when I published my first article on “the Berg Affair”.₁₂ Its publication finally roused those best positioned to answer. Late in 1993, Galler, Bromley, and Campbell-Kelly wrote letters to the editor of Accountability in Research criticizing me for not getting their side of the story before I published Berg’s. Campbell-Kelly threatened the journal’s publisher with a lawsuit if I (or it) did not retract. The three also provided some insight into what their explanation of events might be. Bromley, though listed prominently in ads for the Works, claimed to have had only a small part, merely advising Campbell-Kelly on selection and arrangement of the papers printed in Volumes 2 and 3. Campbell-Kelly confirmed that Bromley took no part in the detailed editing or in the provision of documents. That work was performed by one C.J.D. (“Jim”) Roberts, a “London-based independent scholar” who was “editorial consultant to the Works” (and, apparently, worked directly under Campbell-Kelly). Roberts seems to deserve more public credit than he has so far received. According to Campbell-Kelly, it was Roberts who, making a systematic search for unknown holdings of Babbage, turned up the original of the letter to Quetelet by writing the Royal Library (one “tiny triumph” among many). Campbell-Kelly also claimed that neither he nor Roberts knew of Berg’s prior discovery.     

Multilevel flexible specification of the production function in health economics

^** Email: grassetti{at}stat.unipd.it^*** Email: e.gori{at}dss.uniud.it^**** Email: simona.minotti{at}unicatt.it Previous studies on hospitals' efficiency often refer to quiterestrictive functional forms for the technology (Aigner et al.,1977, J. Econom., 6, 21–37). In this paper, referringto a study about some hospitals in Lombardy, we formulate convenientcorrectives to a statistical model based on the translogarithmicfunction—the most widely used flexible functional form(Christensen et al., 1973, Rev. Econ. Stat., 55, 28–45).More specifically, in order to take into consideration the hierarchicalstructure of the data (as in Gori et al., 2002, Stat. Appl.,14, 247–275), we propose a multilevel model, ignoringfor the moment the one-side error specification, typical ofstochastic frontier analysis (Aigner et al., 1977, J. Econom.,6, 21–37). Given this simplification, however, we areeasily able to take into account some typical econometric problemsas, e.g. heteroscedasticity. The estimated production functioncan be used to identify the technical inefficiency of hospitals(as already seen in previous works), but also to draw some economicconsiderations about scale elasticity, scale efficiency andoptimal resource allocation of the productive units. We willshow, in fact, that for the translogarithmic specification itis possible to obtain the elasticity of the output (regardingan input) at hospital level as a weighted sum of elasticitiesat ward level. Analogous results can be achieved for scale elasticity,which measures how output changes in response to simultaneousinputs variation. In addition, referring to scale efficiencyand to optimal resource allocation, we will consider the resultsof Ray (1998, J. Prod. Anal., 11, 183–194) to our context.The interpretation of the results is surely an interesting administrativeinstrument for decision makers in order to analyse the productiveconditions of each hospital and its single wards and also todecide the preferable interventions.     

Capillary waves past a flat plate in water of finite depth

Free-surface flow past a semi-infinite flat plate in a channelof finite depth is considered. The fluid is assumed to be inviscidand incompressible, and the flow to be two-dimensional and irrotational.Surface tension is included in the dynamic boundary conditionbut the effects of gravity are neglected. It is shown that thereis a three-parameter family of solutions with waves in the farfield and a discontinuity in slope at the separation point.This family includes as particular cases the solutions previouslycomputed by Osborn & Stump (2001, Phys. Fluids, 13, 616–623)and by Andersson & Vanden-Broeck (1996, Proc. R. Soc., 452,1985–1997).     

Ordinal Operations on Surreal Numbers

An open problem posed by John H. Conway in [2] was whether onecould, on his system of numbers and games, ‘... defineoperations of addition and multiplication which will restricton the ordinals to give their usual operations’. Sucha definition for addition was later given in [4], and this paperwill show that a definition also exists for multiplication.An ordinal exponentiation operation is also considered.     

Stability of uniformly Morse-Smale gradient-like numerical methods for flows

In Garay (1996, Numer. Math., 72, 449–479) and Li (1997b,SIAM J. Math. Anal., 28, 381–388), it was shown that thequalitative properties of a Morse–Smale gradient-likeflow are preserved by its numerical approximations. In thispaper, we show that the qualitative properties of a family ofuniformly Morse–Smale gradient-like numerical methodsare preserved by the approximated flow. The techniques usedin the study of the structural stability theorem for diffeomorphismsare the main tools for this work.