Proofs and algebra in al-Fārisī's commentary

The Persian mathematician al-Fārisī (late thirteenth century) wrote a commentary on a practical arithmetic book as a means of giving techniques associated with mental reckoning a foundation in proofs modeled on those in Euclid's number theory books. One problem with this intercultural project is the incompatibility of Euclidean and Arabic numbers, while another is the occasional inadequacy of Euclid's mode of representing numbers via lines labeled with letters. Like others, al-Fārisī found a partial solution to the former by identifying fractions with ratios of integers, and for the latter he turned to the algebra of polynomials to work through one proof. To properly interpret this proof, Arabic algebra is situated in its contemporary mathematical context.     

Twenty-One Algebraic Normal Forms of Citrabhānu

In this article we examine Citrabhānu's (fl. 1530) theory of algebraic normal forms up to the third degree as handed down to us by his pupil Śankara and make a survey of their history in India.Copyright 1998 Academic Press.[formula]Copyright 1998 Academic Press.[formula]Copyright 1998 Academic Press.AMS subject classification: 01A32     

Varāhamihira's pandiagonal magic square of the order four

I have demonstrated in this paper that Varāhamihira (ca. 550), an Indian authority of astronomy, astrology, and divination, utilized a pandiagonal magic square of sixteen cells in prescribing how to prepare perfumes from sixteen original substances. I have also tried to reconstruct the original magic square underlying Varāhamihira's square, which consists of two sets of the series [1, 2, 3, 4, 5, 6, 7, 8], and have suggested a probable historical link between Indian and Islamic magic squares.     

Abū al-Wafā’ Latinus? A study of method

This article studies the legacy in the West of Abū al-Wafā’s Book on those geometric constructions which are necessary for craftsmen. Although two-thirds of the geometric constructions in the text also appear in Renaissance works, a joint analysis of original solutions, diagram lettering, and probability leads to a robust finding of independent discovery. The analysis shows that there is little chance that the similarities between the contents of Abū al-Wafā’s Book and the works of Tartaglia, Marolois, and Schwenter owe anything to historical transmission. The commentary written by Kamāl al-Dīn Ibn Yūnus seems to have had no Latin legacy, either.     

SYMMETRIES， BāCKLUND TRANSFORMATIONS AND BOUNDARY－INITIAL VALUE PROBLEMS FOR NONLINEAR CHROMATOGRAPH

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On simplices in diameter graphs in ℝ4

Agraph G is a diameter graph in ?^d if its vertex set is a finite subset in ?^d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ?⁴ contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ?⁴, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

     

Regular n-simplices in R~n with vertices in Z~n

In this paper the Ⅰ and Ⅱ regular n-simplices are introduced. We prove that the sufficient and necessary conditions for existence of an Ⅰ regular n-simplex in Rn are that if n is even then n = 4m(m + 1), and if n is odd then n = 4m + 1 with that n + 1 can be expressed as a sum of two integral squares or n = 4m - 1, and that the sufficient and necessary condition for existence of a Ⅱ regular n-simplex in Rn is n = 2m2 - 1 or n = 4m(m+1)(m 6 N). The connection between regulars-simplex in Rn and combinational design is given.     

Voter model in a random environment in ? d

We consider the voter model with flip rates determined by {?? _e , e ?? E _d }, where E _d is the set of all non-oriented nearest-neighbour edges in the Euclidean lattice ? ^d . Suppose that {?? _e , e ?? E _d } are independent and identically distributed (i.i.d.) random variables satisfying ?? _e ? 1. We prove that when d = 2, almost surely for all random environments, the voter model has only two extremal invariant measures: ?? ₀ and ?? ₁.     

On the\bar \partial equation in weightedL 2 norms in ℂ1equation in weightedL 2 norms in ℂ1

For a large class of subharmonicφ, the equation is studied in . Pointwise upper bounds are derived for the distribution kernels of the canonical solution operator and of the orthogonal projection onto the space of entire functions inH. Existence theorems inL ^p norms are derived as a corollary. A class of counterexamples, related to the failure of to be analytic-hypoelliptic on certain CR manifolds, is discussed. Communicated by Steven Krantz     

Self-stabilizing Processes in Multi-wells Landscape in ℝ d -Invariant Probabilities

The aim of this work is to analyze the stationary measures for a particular class of non-Markovian diffusions, the self-stabilizing processes. All the trajectories of such a process attract each other. This permits to exhibit a non-uniqueness of the stationary measures in the one-dimensional case, see Herrmann and Tugaut (Stoch. Process. Their Appl. 120(7):1215–1246, 2010). In this paper, the extension to general multi-wells lansdcape in general dimension is provided. Moreover, the approach for investigating this problem is different and needs fewer assumptions. The small-noise limit behavior of the invariant probabilities is also given.     

Time allocated to mathematics in post-primary schools in Ireland: are we in double trouble?

Mathematics educators and legislators worldwide have begun placing a greater emphasis on teaching mathematics for understanding and through the use of real-life applications. Revised curricula have led to the time allocated to mathematics in effected countries being scrutinised. This has resulted in policy-makers and educationalists worldwide calling for the inclusion of double class periods on the mathematics timetable. Research from the United States suggests that the introduction of double or block periods allow for the objectives of revised curricula to be realized. The aim of this study, which is set in the school context, is first to ascertain if schools in Ireland are scheduling double periods for mathematics at both lower post-primary level (Junior Cycle) and upper post-primary level (Senior Cycle). It also seeks to determine if there is a link between teachers’ levels of satisfaction with the time allocated to mathematics and the provision of double periods and to get insights from teachers in relation to their opinions on what can be achieved through the introduction of such classes. Questionnaires were sent to 400 post-primary schools (approximately 1600 teachers) which were selected using stratified sampling techniques. It was found that 8.7% of mathematics teachers reported the provision of double periods at Junior Cycle while 55% reported that double periods were included on their timetable at Senior Cycle. The study also identified a link between teachers’ levels of satisfaction with the time allocated to mathematics and the provision of double periods. Finally, teachers felt that double periods allowed for new teaching methodologies, which were promoted by the revised curricula, to be implemented and teaching for understanding was also more feasible. In essence, it was found that double periods have an influence on the mathematical experience of post-primary students as well as the teaching approaches employed.     

Free interpolation in Hölder classes in quasiconformal domains

The free interpolation problem for Hölder classes on a simply connected bounded domain whose boundary is a quasiconformal curve is studied. The necessary and sufficient conditions on a closed set $E \subset \overline G $ under which the whole Hölder space on E is generated by the restrictions of analytic functions of the Hölder class to the domain G are discussed. Bibliography: 7 titles.     

Micros in mathematics education—Uses of spreadsheets in CAL

In this paper it is shown that teachers and students can study the mathematical problem through commercially available powerful spreadsheet software programs without being bogged down by the programming problem. It is illustrated by two examples. One is taken from a differentiable function the derivative of which is discontinuous at the origin. The other example uses Simpson's rule to find the area under the standard normal distribution.     

Two dimensional scrolls contained in quadric cones in ℙ5

Smooth, complex, ruled surfaces embedded in ℙ⁵ as linearly normal scrolls, such that they are contained in a quadric cone, are considered. Rational scrolls and some elliptic scrolls are shown to be the only ones contained in cones of rank 5. Results on scrolls contained in cones of lower ranks are also obtained.     

On dilworth’s theorem in the in finite case

The following theorem is proved: Letc be an infinite cardinal. There exists a partially ordered set of cardinalc, which contains no infinite independent subset, and which is not decomposable into less thanc chains.