Approximation of quadratic irrationals and their pierce expansions

In this article two aims are pursued: on the one hand, to present a rapidly converging algorithm for the approximation of square roots; on the other hand and based on the previous algorithm, to find the Pierce expansions of a certain class of quadratic irrationals as an alternative way to the method presented in 1984 by J.O. Shallit; we extend the method to find also the Pierce expansions of quadratic irrationals of the form $2 (p-1) (p - \sqrt{p^2 - 1})$ which are not covered in Shallit's work.Quadratic irrationals, Pierce series

A total order in [0,1] defined through a 'next' operator

A `next' operator, s, is built on the set R1=(0,1]-{ 1-1/e} defining a partial order that, with the help of the axiom of choice, can be extended to a total order in R1. Besides, the orbits {sn(a)}n are all dense in R1 and are constituted by elements of the same arithmetical character: if a is an algebraic irrational of degree k all the elements in a's orbit are algebraic of degree k; if a is transcendental, all are transcendental. Moreover, the asymptotic distribution function of the sequence formed by the elements in any of the half-orbits is a continuous, strictly increasing, singular function very similar to the well-known Minkowski's ?(×) function.Total orders, pierce series, singular functions

Recull de la Comissió de Cetologia de la Institució Catalana d'Història Natural. II: anys 1974 i 1975

This second report of the Comissió de Cetologia of the Institució Catalana d'Història Natural includes 34 records of Cetaceans in the Mediterranean area between the French border in the North and Valencia in the South, and also from the Balearic Islands. From other coasts of the Spanish state there are few citations as there are no periodical surveys. We are trying to create local groups all along the Spanish state coasts to avoid this lack of information. The existence of six strandings of striped dolphin (Stenella coeruleoalba) obliges to discuss its incidence in the past and in the present. It seems that now the striped dolphin is becoming a very common species in the Catalan Mediterranean coast. The situation of the common dolphin (Delphinus delphis) would be opposed. For the remaining species the situation is the same that CASINOS & VERICAD (1976) evidenced. In each case the locality, date, morphological and biological data, preserved material, references and observer are noted. A short description of the specimens collected is following: Balaenoptera physalus 1. Stranded dead specimen on Ca16 des Mort (Formentera Island). Physeter macrocephalus 2. Stranded on platja del Grau (Plana Alta). Preserved: right flipper. 3. Stranded dead specimen on Favaritx (Menorca Island). 4. Stranded dead specimen on Palmeral beach (l'Alacanti), after a storm. 5. Stranded on Os Castros (Lugo, Bay of Biscay). 6. Stranded on Badalona (Barcelones). Ziphius cavirostris 7. Stranded on La Magdalena (near Santander, Bay of Biscay). Delphinus deiphis. 8. Caught outside Es Castell (Menorca Island). Stenella coeruleoalba. 9. Stranded dead specimen on Castelldefels (Baix Llobregat). Preserved: complete skeleton and internal and external parasites. 10. Caught by local fishermen off Cubelles (Garraf). Preserved: skull and flippers. 11. Stranded dead specimen on bay of Pollenga (Mallorca Island). It showed injuries. Preserved: complete skeleton, internal and external parasites and stomach contents (cephalopod beaks). 12. Stranded on Sitges (Garraf). It showed injuries. Preserved: stomach contents (cephalopod beaks of the families Gonatidae and Sepiolidae and the species Loligo vulgaris Lam.). External parasites. 13. Stranded on Orpesa (Plana Alta). Preserved: skull and flippers. 14. Stranded on Sant Pere Pescador (Alt Emporda), after a storm. Preserved: complete skeleton. Tursiops truncatus 15. Skeleton found in Punta del Fangar (Delta de l’Ebre). Preserved: skull. 16. Stranded on Punta Negra (Mallorca Island). 17. Skull found in the marshes of the Guadalquivir River, Atlantic Ocean. 18. Stranded on Castelldefels (Baix Llobregat). Preserved: skull, half lower jaw and a vertebra. 19. Stranded on Badalona (Barcelones). 20. Pieces of a skeleton found in Tarragona (Tarragones). Preserved: right lower jaw, a scapula, a rib and an intervertebral disc. 21. Skull found in Punta de la Banya (Delta de l’Ebre). Preserved: skull. 22. Dead stranded on Liencres (Asturies, Bay of Biscay). 23. Stranded on Motril (Granada, Mediterranean). 24. Caught on Torrevieja (Baix Segura). Grampus griseus 25. Stranded on RincOn de la Victoria, near Malaga, Mediterranean. 26. A group of 10-14 specimens observed swimming (40º 25' N, 2º10' E). Pictures were taken. 27. Stranded dead specimen on Garraf (Garraf). Indetermined specimens 28. Small delphinid stranded on La Mola (Formentera Island). 29. Small delphinid (Stenella coeruleoalba or Delphinus deiphis?) caught by local fishermen off Barcelona. Bougth at the Central Fish Market of Barcelona. 30. Delphinid stranded on Port d'Es Canonge (Mallorca Island). 31. Delphinid (Tursiops truncatus?) stranded on the Coll d'En Rabassa (Mallorca Island). 32. Small delphinid stranded on Cala Gaidana (Menorca Island). 33. Cetacean stranded on Torrevieja (Baix Segura). 34. Small cetacean stranded on Alboraia (l'Horta)

A singular function and its relation with the number systems involved in its definition

Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it permits us to prove that its derivative, as it also happens for many other non-decreasing singular functions from [0,1] to [0,1], when it exists can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k* =5.31972, and ?'(x) exists then ?'(x)=0. In the same way, if the same average is less than k**=2 log2(F), where F is the golden ratio, then ?'(x)=infinity. Finally some results are presented concerning metric properties of continued fraction and alternated dyadic expansions.Singular function, number systems, metric number theory

Fermat's treatise on quadrature: A new reading

The Treatise on Quadrature of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, R x+m/n dx, or under a higher hyperbola, R x-m/n dx— with the appropriate limits of integration in each case—, has a second part which was not understood by Fermat’s contemporaries. This second part of the Treatise is obscure and difficult to read and even the great Huygens described it as 'published with many mistakes and it is so obscure (with proofs redolent of error) that I have been unable to make any sense of it'. Far from the confusion that Huygens attributes to it, in this paper we try to prove that Fermat, in writing the Treatise, had a very clear goal in mind and he managed to attain it by means of a simple and original method. Fermat reduced the quadrature of a great number of algebraic curves to the quadrature of known curves: the higher parabolas and hyperbolas of the first part of the paper. Others, he reduced to the quadrature of the circle. We shall see how the clever use of two procedures, quite novel at the time: the change of variables and a particular case of the formula of integration by parts, provide Fermat with the necessary tools to square very easily curves as well-known as the folium of Descartes, the cissoid of Diocles or the witch of Agnesi.History of mathematics, quadratures, integration methods