Extension of the partition sieve

We demonstrate the correspondence which lies behind certain partition identities used by Andrews in his partition sieve. This leads to an extension of his methods and a generalization of his results.     

Boscovich's geometrical principle of continuity,and the “mysteries of the infinity”

In this paper we give a detailed account of Boscovich's geometrical principle of continuity. We also compare his ideas with those of his forerunners and successors, in order to cast some light on his possible sources of inspiration and to underline the elements of novelty in his approach to the subject.     

The rejected parts of Brouwer's dissertation on the foundations of mathematics

Brouwer launched his intuitionist attack on the formalistic trends in mathematics in his now famous dissertation “On the Foundations of Mathematics” in 1904. In the autumn of 1976, the author found what turned out to be the first version of Brouwer's dissertation. He also discovered that his supervisor disapproved and rejected what Brouwer considered to be the most important part of his dissertation. The extreme solipsistic views held by Brouwer in his later years are well known. The first version of the dissertation shows that these views were held before Brouwer began his intuitionist campaign and that they determined his philosophy of mathematics. The deleted parts not found in the final version of the dissertation are presented here in an English translation. In a brief introduction the author gives some of the historical background, based partly on the correspondence between Brouwer and his supervisor, Korteweg.     

On the life and work of Cyrus Derman

This article provides a brief biographical synopsis of the life of Cyrus Derman and a comprehensive summary of his research. Professor Cyrus Derman was known among his friends as Cy.     

A different perspective of the teaching philosophy of RL Moore

Dr RL Moore was undoubtedly one of the finest mathematics teachers ever. He developed a unique teaching method designed to teach his students to think like mathematicians. His method was not designed to convey any particular mathematical knowledge. Instead, it was designed to teach his students to think. Today, his method has been modified to focus on using student participation toward the goal of the conveyance of mathematical knowledge rather than on Dr Moore's goal of teaching students to think. This article proposes that undergraduates would be better served if they took at least one course using Dr Moore's original method and his original goal.     

On the decomposition map of Grothendieck groups

This article is a part of the doctoral dissertation of the author at the University of Minnesota under the guidance of Professor Morton E. Harris. The author wishes to express his gratitude to Professor Harris for his advice and suggestions     

Asymptotics of the discrete spectrum of a perturbed Hill operator

One considers a perturbed one-dimensional Hill operator. One gives a formula for the asymptotics of the discrete spectrum in the gap of the continuous spectrum and conditions under which this formula is valid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 188–189, 1985.In conclusion the author expresses his deep gratitude to his scientific adviser M. Sh. Birman for the formulation of the problem and for his constant interest in this note.     

Operational Research at the Crossroads: Cecil Gordon and the Development of Post-War OR

Cecil Gordon is not now a name to conjure with in operational research. If he is remembered, it is for his classic ‘Planned Flying, Planned Maintenance’ study during World War II. Yet in 1948 his prominence was such that the Athenaeum dinner party which led to the foundation of the OR Club (and subsequently Society) consisted of Blackett, Goodeve, Tizard-and Gordon. This paper reconstructs his life and work, focussing in particular on his attempts, in the Board of Trade and elsewhere, to establish operational research within the civilian government machine. There are lessons to be learnt from what he was trying to do (an alternative direction for OR?) and the reasons for its failure at that time.     

Investigation of the asymptotic behavior of Green's function for the waveguide problem of propagation near the surface of a conducting sphere

The field of a point source in the waveguide at a point slightly above the waveguide is investigated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 138–154, 1975.In conclusion, the author expresses his gratitude to his scientific advisor V. S. Buldyrev.     

From anomaly to fundament: Louis Poinsot's theories of the couple in mechanics

In 1803 Louis Poinsot published a textbook on statics, in which he made clear that the subject dealt not only with forces but also with ‘couples’ (his word), pairs of coplanar non-collinear forces equal in magnitude and direction but opposite in sense. His innovation was not understood or even welcomed by some contemporary mathematicians. Later he adapted his theory to put forward a new relationship between rectilinear and rotational motion in dynamics; its reception was more positive, although not always appreciative of the generality. After summarising the creation of these two theories and noting their respective receptions, this paper considers his advocacy of spatial and geometrical thinking in mechanics and the fact that, despite its importance, historians of statics who cover his period usually ignore his theory of couples.     

Looking back to the roots of partially correct constructs: The case of the area model in probability

We use the notion Partially Correct Constructs (PaCCs) for students’ constructs that partially match the mathematical principles underlying the learning context. A frequent expression of partial construction of mathematical principles is that a student’s words or actions provide an inaccurate or misleading picture of the student’s knowledge. In this study, we analyze the learning process of a grade 8 student, who learns a topic in elementary probability. The student successfully accomplishes a sequence of several tasks without apparent difficulty. When working on a further task, which seems to require nothing beyond his proven competencies, he encounters difficulties. Using the epistemic actions of the RBC model for abstraction in context as tracers, we analyze his knowledge constructing processes while working on the previous tasks, and identify some of his constructs as PaCCs that are concealed in these processes and explain his later difficulties. In addition, our research points to the complexity of the knowledge structures students are expected to deal with in their attempts to learn an elementary mathematical topic with understanding.     

John Reynolds of the Mint: A mathematician in the service of king and commonwealth

In his youth, John Reynolds showed a talent for arithmetic and was destined for a career as a mathematician at the Tower Mint in London. He became skilled in the algorithms needed to determine the correct relationship between the weight and purity of coins and their values. This was a matter of national importance, and his work came to the attention of King James I, who reigned from 1603 to 1625, and his chief ministers, including Robert Cecil and Francis Bacon. It seemed that John might attain high office himself, but the murky administration of the early Stuart period cast its shadow over his career. Nevertheless, for the next forty years he continued to play a major part in the nation's affairs. He produced books of tables for the valuation of coins in the commercial world, and for the highly technical work of the assayers. Also, he was actively involved in the production of standard measures and instruments used by the excise officers. His life and works illustrate how mathematical ideas were employed by the English government in the period of the early Stuart kings and the Commonwealth.     

E.W. Chittenden and the early history of general topology

E.W. Chittenden's work and its influence on the early history of general topology are examined. Particular attention is given to his work in metrization theory and its role in the background of the Aleksandrov-Uryson Metrization Theorem. A recounting of Professor Chittenden's career, a list of his students and his publications and a chronology in the early history of General Topology are also included.     

Sieve of war: the legacy of Jitsuro Nagura

This paper contains a biographical sketch of the late Japanese number theorist Jitsuro Nagura (1901–68) based on material provided by his eldest son Riichi Nagura, and a synopsis of his two surviving mathematical papers, the first a well-known article on primes in short intervals, and the second an unpublished draft of a proof relating to complex polynomials.     

Negative spectrum of the Schrödinger operational equation

The negative part of the spectrum of Schrödinger's equation with a completely continuous operational potential is studied. Several conditions are obtained for the discreteness and boundedness of the negative spectrum, already known for the scalar and matrix cases.Translated from Matematicheskie Zametki, Vol. 7, No. 6, pp. 753–763, June, 1970.In conclusion the author wishes to express his gratitude to M. Sh.Birman for his valuable advice and his help in this work.     

Computing solutions of the multiclass network equilibrium problem with affine cost functions

Annals of Operations Research - We consider a non-atomic congestion game on a graph, with several classes of players. Each player wants to go from his origin vertex to his destination vertex at the...     

On the inversion of functional operators in a space of functions bounded on the axes

A theorem is proved on the inversion of operators of the form (d/dt) + A, where A is a linear operator in a space of vector functions bounded on the axes.Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 269–274, March, 1972.The author wishes to express his appreciation to M. A. Krasnosel'skii for his assistance.     

Some Problems with the Russellian Open Future

In a recently published paper, Todd (Mind, 125(499), pp. 775–798, 2016a) advocates a novel treatment of future contingents. On his view, all statements concerning the contingent future are false. He motivates his semantic postulates by considerations in philosophy of time and modality, in particular by the claim that there is no actual future. I present a number of highly controversial consequences of Todd’s theory. Inadequacy of his semantics might indirectly serve as an argument against the philosophical view underpinning his proposal.     

Number of invariant measures with maximal entropy for translation in the space of sequences

An example is constructed of a topologically transitive dynamic system with positive entropy, having any assigned finite number of ergodic measures with respect to each of which the metric entropy is equal to the topological entropy.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 291–302, March, 1971.The author expresses his gratitude to B. M. Gurevich for his valuable discussions of the work and D. V. Anosov for his comments on the work.