Multidimensional Arrays, Indices and Kronecker Products

Much of the algebra that is associated with the Kronecker product of matrices has been rendered in the conventional notation of matrix algebra, which conceals the essential structures of the objects of the analysis. This makes it difficult to establish even the most salient of the results. The problems can be greatly alleviated by adopting an orderly index notation that reveals these structures. This claim is demonstrated by considering a problem that several authors have already addressed without producing a widely accepted solution.

Joaquin Rodrigo

Joaquín Rodrigo y Vidre (b. 1901–d. 1999) was born in Sagunto, in the province of Valencia, Spain, on 22 November 1901, the feast day of St. Cecilia, patron saint of music. Rodrigo was a productive composer for over six decades, and his roughly two hundred works include masterpieces for orchestra, chamber ensemble, chorus, solo voice, piano, and especially guitar, an instrument on which he was not proficient but the one with which his legacy is inextricably connected. Indeed, by far and away his most famous work is the Concierto de Aranjuez for guitar and orchestra, composed in 1938–1939 and premiered in 1940. The middle movement’s main theme has provided inspiration for a whole assortment of arrangements by jazz artists such as Miles Davis and Chick Corea, and it has often been quoted in music for film and television. The unfortunate consequence of this melody’s viral popularity, however, is that it has tended to put much of his other music in the shade. Rodrigo’s achievement is remarkable because he lost his eyesight at age three as a result of diphtheria. Fortunately, the family moved to the city of Valencia in 1906, where advanced institutions for educating the deaf and blind were located. He received excellent training in piano, violin, and composition, as well as in regular academic subjects. Rodrigo became proficient in reading Braille notation for both words and music. He would eventually use a machine to type up his musical ideas in Braille, which he would then dictate to an assistant to write out in conventional notation. The music would thereafter be played at the piano so that he could hear it and make any necessary changes. In 1927 he moved to Paris to continue his studies in composition with Paul Dukas, at the École Normale de Musique. It was during his years in Paris that he met and married Victoria Kamhi, a Sephardic Jewess from Istanbul who was studying piano there. Composer and wife returned to Spain for good in 1939, settling in Madrid after the end of the Spanish Civil War and on the eve of World War II. The successful premiere of the Concierto de Aranjuez cemented his reputation as a leading figure in Spanish music. He also became a music critic for Radio Nacional, the newspaper Pueblo, and he assumed administrative responsibilities for ONCE, the national organization for the blind. Rodrigo’s international reputation began to grow steadily during the 1950s, and he would be the recipient of numerous awards and honorary degrees, at home and abroad. He passed away at the age of ninety-seven, two years after Victoria. His legacy is preserved and promoted by the Fundación Victoria y Joaquín Rodrigo in Madrid (see online), which maintains his apartment as both a museum and a research archive. It is headed by the composer’s daughter, Cecilia (b. 1941).

NICOLA LEFANU (b. 1947)But Stars Remaining (1970)

This chapter discusses a very early work by Nicola Lefanu, But Stars Remaining. This song highlights the composer’s flair and professionalism, revealing a keen awareness of practical issues and of the concerns of the performer. The piece has clear notation, with careful attention to detailed interpretative niceties. Although written originally for soprano, the work could also suit a flexible tenor. The composer envisages it being sung spontaneously, as from a high rock, with echoes reverberating, so there are plenty of gaps in the music to allow for spatial resonance. There are a good many innovative vocal devices and gestures for the singer to get used to, and the composer has annotated these meticulously. Conventional notation is mostly employed, apart from a few freer passages, where note-stems are absent.

RUTH CRAWFORD SEEGER (1901–1953)Five Songs (1929)

This chapter considers the works of Ruth Crawford Seeger, whose music employs the fine settings of poet Carl Sandburg. The music is subtly distinctive, though straightforward to the eye. Words are set with unaffected simplicity and piano writing is idiomatic and evocative. Mostly fragmentary, understated vocal lines are underpinned by contrastingly elaborate, flowing keyboard writing. On its own, the voice part should pose few technical problems, except that of maintaining a clear tone. Crawford Seeger shows real mastery of the art song in the balanced relationship between the two performers. She habitually played her own accompaniments, which clearly require a confident player. Conventional notation is employed throughout, with time (but not key) signatures.

ALISON BAULD (b. 1944)Banquo’s Buried (1982)

This chapter discusses the first of a trilogy of solo scenas from Shakespeare. Although the music may appear simple on the page, it is highly disciplined, and conceived with a clear idea of its desired interpretation. Bauld likes, where possible, to adapt and fine-tune details when working with performers, to make sure that any ambiguities of notation (often to do with variations of pitched speech) can be clarified. This is very much performers’ music, as it feels immediate and vital. Her love of Shakespeare is furthermore central to her art, and her sense of the timing of declamatory verse is acute, responding to the myriad possibilities of verbal inflections. She employs conventional notation in the main, but occasionally dispenses with time signatures for passages of rapid speech or whispering.

Index-Based Notation for Random Variable and Probability Space

In a conventional notation used in many studies, a probability space and state space of random variables is identified by its symbol. However, such a notation makes a formula ambiguous in a large equation. This letter proposes to use an index set to identify the probability space and state space of random variables. It is shown that the proposed notation can increase the generality of formulas without ambiguity.

El currículo del número en educación infantil. Un análisis desde una perspectiva internacional

En este artículo se realiza un análisis de las orientaciones internacionales sobre la enseñanza del número en general y la adquisición del sentido numérico en particular, durante la etapa de educación infantil. Estas orientaciones sirven de base para analizar el currículo nacional y ofrecer algunos andamios para avanzar hacia el desarrollo del pensamiento numérico en las primeras edades. Se concluye que es necesario centrar las prácticas docentes en la comprensión del número, en las distintas representaciones de este (evitando la insistencia en la enseñanza de la notación convencional) y en el significado de las operaciones elementales. The number curriculum in early childhood education. An analysis from an international perspective This article carries out an analysis of international guidelines on teaching numbers in general and the acquisition of number sense particularly, during Early Childhood Education. These guidelines provide a basis for analysing the national curriculum and for offering some scaffolding to contribute towards the development of numerical thinking in early ages. The study concludes that teaching practice needs to be centred on understanding numbers, on different representations of them (avoiding the insistence on teaching conventional notation) and on the meaning of basic operations.Handle: http://hdl.handle.net/10481/40448WOS-ESCINº de citas en WOS (2017): 1 (Citas de 2º orden, 0)