Some Special Complex Multiplications in Two Variables Using Hilbert Singular Moduli

Number Theory ◽  
1991 ◽  
pp. 75-83
Author(s):  
Harvey Cohn
Keyword(s):  
Singular Moduli ◽  
Complex Multiplications ◽  
Special Complex

Vampix, a multimicroprocessor control system adaptable to special complex machines

1987 ◽  
Vol 21 (1-5) ◽  
pp. 325-330 ◽  
Author(s):  
Dipl Ing. Drazen FLEGO ◽  
Herbert Schweinzer
Keyword(s):  
Control System ◽  
Special Complex

scholarly journals Monitoring the dynamic behaviors of the Bosporus Bridge by GPS during Eurasia Marathon

2007 ◽  
Vol 14 (4) ◽  
pp. 513-523 ◽  
Author(s):  
H. Erdoğan ◽  
B. Akpınar ◽  
E. Gülal ◽  
E. Ata
Keyword(s):  
Frequency Domain ◽  
Element Model ◽  
Traffic Load ◽  
Suspension Bridges ◽  
Engineering Structures ◽  
Low Frequencies ◽  
Temperature Changes ◽  
Bridge Responses ◽  
Special Complex ◽  
Short Time

Abstract. Engineering structures, like bridges, dams and towers are designed by considering temperature changes, earthquakes, wind, traffic and pedestrian loads. However, generally, it can not be estimated that these structures may be affected by special, complex and different loads. So it could not be known whether these loads are dangerous for the structure and what the response of the structures would be to these loads. Such a situation occurred on the Bosporus Bridge, which is one of the suspension bridges connecting the Asia and Europe continents, during the Eurasia Marathon on 2 October 2005, in which 75 000 pedestrians participated. Responses of the bridge to loads such as rhythmic running, pedestrian walking, vehicle passing during the marathon were observed by a real-time kinematic (RTK) Global Positioning System (GPS), with a 2.2-centimeter vertical accuracy. Observed responses were discussed in both time domain and frequency domain by using a time series analysis. High (0.1–1 Hz) and low frequencies (0.00036–0.01172 Hz) of observed bridge responses under 12 different loads which occur in different quantities, different types and different time intervals were calculated in the frequency domain. It was seen that the calculated high frequencies are similar, except for the frequencies of rhythmic running, which causes a continuously increasing vibration. Any negative response was not determined, because this rhythmic effect continued only for a short time. Also when the traffic load was effective, explicit changes in the bridge movements were determined. Finally, it was seen that bridge frequencies which were calculated from the observations and the finite element model were harmonious. But the 9th natural frequency value of the bridge under all loads, except rhythmic running could not be determined with observations.

On complex multiplications

2002 ◽  
pp. 52-59
Author(s):  
Goro Shimura
Keyword(s):  
Complex Multiplications

scholarly journals R. Schumann’s romantic “Fantastic Pieces” op. 88 – on the way to the piano trio

2019 ◽  
Vol 16 (16) ◽  
pp. 126-142
Author(s):  
Lidiya Sokolova
Keyword(s):  
Subsequent Development ◽  
Theoretical Background ◽  
Middle Part ◽  
Musical Instruments ◽  
Solo Piano ◽  
Piano Trio ◽  
Ensemble Techniques ◽  
The Fantastic ◽  
Special Complex ◽  
Ensemble Analysis

Introduction. The article analyzes R. Schumann’s “Fantastic Pieces” op. 88 as a creative debut in the subsequent development of the piano trio genre. The “Fantastic Pieces” op. 88 are the first composer’s creative experience in combining such musical instruments. Theoretical Background. The analysis of musicological literature did not reveal any special research dedicated to this score, but only its references in E. Karelina’s (1996) thesis research and D. Zhitomirsky’s (1964) monograph. Thus, this article is the first special research of the compositional and ensemble analysis of R. Schumann’s “Fantastic Pieces”, op. 88. The objectives of the research: to analyze compositional-dramatic and ensemble features of R. Schumann’s “Fantastic Pieces”, op. 88, to identify their specific features. The object of the research: R. Schumann’s chamber-instrumental creative activity. The subject of the research: to identify the value of R. Schumann’s “Fantastic Pieces”, op. 88 in the further development of the piano trio genre. Methods: musical-theoretical, aimed at analyzing the musical text of the chosen work; genre-stylistic, allowing to identify the compositional-dramatic and ensemble features of R. Schumann’s “Fantastic Pieces”, op. 88. Research material: R. Schumann’s “Fantastic Pieces” op. 88 for piano, violin and cello. Results and Discussion. The first experience in mastering the piano trio genre of R. Schumann occurred in the “Fantastic Pieces” op. 88, composed in 1842. This was the first composer’s experience in combining such musical instruments. This work is a cycle of four pieces: “Romance”, “Humoresque”, “Duetto” and “Final”. It is noteworthy that R. Schumann used these names in other works. It is useful enough to recall his 3 piano romances op. 28, Humoresque op. 20a, 4 marches op. 76. Despite the four-part character of the work, this composition does not coincide with the sonata-symphonic structure, but is organized according to the suite principle. R. Schumann’s different vision of the trio-ensemble genre is represented by a clear differentiation of works with an individual composition. Therefore, the cycles op. 88 and op. 132 receive program names: “Fantastic Pieces” and “Fairytale Narratives”, respectively, and the trio with the classical (sonata) organization of the cycle acquire sequence numbers and are referred to as “piano trios”. The very names of the parts in the cycle reveal the opposition of two metaphoric spheres, characteristic of the romantic era: lyrical and genre-scherzo ones. The paired relationship of these metaphoric spheres stands out particularly. Such a metaphoric doubling gives the matching modes the rondality features within the whole cycle. This metaphoric paired relationship between the parts allows you to single out two macro parts in a cycle. The first macro part is represented by the lyric “Romance” and the scherzo “Humoresque”, the second one – by the tender song “Duetto” and the marching “Final”. At the same time, the macro parts demonstrate individual features of one or another semantic type. The metaphoric opposition of romantic pieces is also enhanced by tempo and ear-catching contrast. Such an alternation of various metaphoric types gives the entire cycle the features of a kaleidoscopic suite. The proportion of genre parts stands out particularly, which is manifested in both their scale and complexity of the compositional organization. Thus, the lyrical parts are represented by tripartite forms. Quick genre pieces are composed in various forms (“Humoresque” is created in a complex tripartite form with a developed polythematic middle part, and the “Final” is in a rondo form, with the links acting as refrains). Despite the romantic nature of the cycle organization, the “Fantastic Pieces” tone plan is a classical one: a-moll –F-dur– d-moll – a-moll (A-dur). Conclusions. It was revealed that the suite-based principle of composition organization and genre-stylistic features of the cycle (opposition of lyrical and genre-scherzo metaphoric spheres) connect the romantic “Fantastic Pieces” op. 88 with its piano miniatures of the 1830s. Ensemble analysis of R. Schumann “Fantastic Pieces”, op. 88 showed that in his first work for the piano trio, the composer “transplanted” solo piano works into a poly-timbre ensemble, taking it in the context of piano music. At the same time, the composer did not reduce the role of strings to the “service” function, but actively used all their melodic and proper ensemble possibilities in the chosen trio. For example, if “Romance” ensemble demonstrated the piano domination, “Humoresque” – the parity of the instruments, then in “Duetto” primacy was given to stringed instruments. In “Final”, each section of the musical form is highlighted by an appeal to one of the main ensemble techniques. A series of altering various semantic spheres, defining the suite properties of the “Fantastic Pieces”, subordinates the ensemble properties used by the composer. For each number and even its individual sections, their special complex was chosen, which in different semantic contexts had a metaphoric-semantic meaning. It was revealed that the organizing means of creating the ensemble in the R. Schumann’s trio was the polyphonic technique presented in his work in a wide variety, which would later be widely developed in his piano trios.

scholarly journals Singular moduli spaces of stable vector bundles on P3

1996 ◽  
Vol 172 (2) ◽  
pp. 477-482 ◽  
Author(s):  
Rosa Maria Miró-Roig
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
Stable Vector Bundles ◽  
Singular Moduli

scholarly journals LINEAR INDEPENDENCE OF POWERS OF SINGULAR MODULI OF DEGREE THREE

2018 ◽  
Vol 99 (1) ◽  
pp. 42-50
Author(s):  
FLORIAN LUCA ◽  
ANTONIN RIFFAUT
Keyword(s):  
Number Theory ◽  
Number Field ◽  
Linear Independence ◽  
Singular Moduli ◽  
Positive Integers

We show that two distinct singular moduli $j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$, such that for some positive integers $m$ and $n$ the numbers $1,j(\unicode[STIX]{x1D70F})^{m}$ and $j(\unicode[STIX]{x1D70F}^{\prime })^{n}$ are linearly dependent over $\mathbb{Q}$, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.

Notes on Ramanujan’s singular moduli

1999 ◽  
pp. 7-16 ◽  
Author(s):  
Bruce Berndt ◽  
Heng Huat Chan
Keyword(s):  
Singular Moduli

scholarly journals Equations with powers of singular moduli

2019 ◽  
Vol 15 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Antonin Riffaut
Keyword(s):  
Rational Number ◽  
The Other ◽  
Other Hand ◽  
Singular Moduli ◽  
Cm Points ◽  
The One ◽  
Straight Lines ◽  
Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

scholarly journals EVALUATION OF COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND AT SINGULAR MODULI

2006 ◽  
Vol 10 (6) ◽  
pp. 1633-1660 ◽  
Author(s):  
Habib Muzaffar ◽  
Kenneth S. Williams
Keyword(s):  
Elliptic Integrals ◽  
Singular Moduli ◽  
Complete Elliptic Integrals
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