Affine cubic functions. IV. Functions on C 3 , nonsingular at infinity

1981 ◽  
Vol 302 (1470) ◽  
pp. 415-455 ◽  
Keyword(s):  
Detailed Analysis ◽  
Critical Points ◽  
Final Section ◽  
Critical Values ◽  
Complete List ◽  
Level Surface ◽  
Cubic Surface ◽  
Cubic Curve ◽  
Analysis Of Cases ◽  
Plane At Infinity

The object of this paper is to classify cubic functions f on C 3 according to their singularities. A level surface of such a function extends to a cubic surface in projective 3-space. The intersections S^,Tœ of S and its Hessian quartic T with the plane at infinity are the same for all levels. We assume throughout that is a nonsingular cubic curve. In §3 we show how the equisingularity class of Tœ determines the number and multiplicities of critical points of f .In § 2 we investigate n T^, and show that the equisingularity class of the pair (S^Tœ) determines that of f. Next we study the case when some point of has polar quadric a plane-pair; complete enumerations are given in §5 for the case when contains a line, and in §6 for when it contains an Eckardt point of S. In the final section we give a detailed analysis of cases when f has just two critical values, and show how to obtain a complete list of types of functions f.

The shape of musical improvisation

2017 ◽  
Author(s):  
Milton Mermikides ◽  
Eugene Feygelson
Keyword(s):  
Nineteenth Century ◽  
Theoretical Model ◽  
Detailed Analysis ◽  
Final Section ◽  
Theoretical Framework ◽  
Classical Period ◽  
Violin Concerto ◽  
Fourth Section ◽  
Musical Space ◽  
The Third

This chapter presents practitioner–researcher perspectives on shape in improvisation. A theoretical framework based in jazz improvisational pedagogy and practice is established, and employed in the analysis of examples from both jazz and classical-period repertoire. The chapter is laid out in five sections. The first section provides a brief overview of improvisational research, while the second discusses the concept of improvisation as ‘chains-of-thought’ (a logical narrative established through the repetition and transformation of musical objects). The third reflects upon improvisation as the limitation and variation of a changing set of musical parameters. Using this concept, the fourth section builds a theoretical model of improvisation as navigation through multidimensional musical space (M-Space). The final section uses this model in a detailed analysis of the nineteenth-century violinist Hubert Léonard’s cadenza for Beethoven’s Violin Concerto Op. 61.

The Role of the Inter-American Court of Human Rights in Monitoring Compliance with Judgments

2020 ◽  
Vol 12 (1) ◽  
pp. 178-184
Author(s):  
Pablo Saavedra Alessandri
Keyword(s):  
Human Rights ◽  
Detailed Analysis ◽  
Analysis Of Cases ◽  
Scarcity Of Resources

Abstract This practice note considers the role of the Inter-American Court in promoting compliance with its judgments. It shows how creative the Court has been, despite scarcity of resources, in helping states to comply with judgments. It discusses some opportunities and challenges through a detailed analysis of cases and of tools currently in use.

Baker Domains of Period Two for the Family $f_{\lambda,\mu}(z)=\lambda e^{z} + \frac{\mu}{z}$

2014 ◽  
Vol 24 (05) ◽  
pp. 1450059 ◽  
Author(s):  
Marco A. Montes de Oca Balderas ◽  
Guillermo J. F. Sienra Loera ◽  
Jefferson E. King Davalos
Keyword(s):  
Critical Points ◽  
Critical Values ◽  
Fatou Set ◽  
The Family

Our main theorem establishes that the Fatou set of the functions fλ,μ(z) = λez + μ/z contains a two-cycle of Baker domains {F∞, F0} if Re(λ) < 0 and |Im(λ)| < 1/2|Re(λ)|. We show that under [Formula: see text] every point in F∞ tends to infinity and in F0 tends to zero. Moreover, if |Im(λ)| < 1/2|Re(λ)| - 4, the set F∞ contains infinitely many critical points of fλ,μ and F0 contains infinitely many critical values; also, infinitely many critical values of [Formula: see text] are contained in both F∞ and F0. Finally, the images of the Baker domains are displayed for some parameters.

The Situation over ℤ: Results

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
Critical Points ◽  
Monic Polynomial ◽  
Critical Values

This chapter treats an integer monic polynomial f(x) ɛ ℤ[x] of degree n ≤ 2 which, over ℂ, is “weakly supermorse,” meaning that it has n distinct roots in ℂ, its derivative f'(x) has n − 1 distinct roots (the critical points) α‎ᵢ ɛ ℂ, and the n − 1 values f(α‎ᵢ) (the critical values) are all distinct in ℂ.

Objektive Zurechnung und Tatherrschaft

2019 ◽  
Author(s):  
Rebecca von Atens
Keyword(s):  
Detailed Analysis ◽  
Third Parties ◽  
Analysis Of Cases ◽  
Legal Forms ◽  
The Relationship

For decades, cases of autonomous self-endangerment and consensual endangerment by third parties have been treated as constellations of objective attribution by jurisprudence as well as by the prevailing theory, whereby their contentual differentiation is made by means of the criterion of authority of action from the dogmatics of participation. According to the author, this is astonishing, since, according to the prevailing theory, the legal forms of objective attribution and participation are to be distinguished from one another with regard to dogmatics and criminal systematics. Therefore, she conducts a detailed analysis of cases of self-endangerment and of endangerment by third parties, as well as of the relationship between the two categories. The author concludes that the prevailing dualistic understanding of attribution and participation is not convincing and that instead, a uniform offence, which includes the category of objective attribution as well as that of participation, should be assumed.

Perpendicular bisectors, duality and local symmetry of plane curves

1995 ◽  
Vol 125 (1) ◽  
pp. 181-194 ◽  
Author(s):  
Peter Giblin ◽  
Farid Tari
Keyword(s):  
Simple Closed Curve ◽  
Local Symmetry ◽  
Closed Curve ◽  
Critical Values ◽  
Parameter Family ◽  
Complete List ◽  
Plane Curves ◽  
Perpendicular Bisector ◽  
Symmetry Set

For a smooth, simple closed curve α in the plane, the perpendicular bisector map P associates to each pair of distinct points (p, q) on α the perpendicular bisector of the chord joining p and q. To a pair (p, p), the map P associates the normal to α at p. The set of critical values of this map is the union of the dual of the symmetry set of α and the dual of the evolute. (The symmetry set is the locus of the centres of circles bitangent to α.) We study the mapP and use it to give a complete list of the transitions which take place on the dual of the symmetry set and the dual of the evolute, as α varies in a generic one-parameter family of plane curves.

scholarly journals The presentation of Jesus in the missionary speeches of Acts and the mission of the church

2014 ◽  
Vol 35 (1) ◽  
Author(s):  
Christoph W. Stenschke
Keyword(s):  
Detailed Analysis ◽  
Final Section ◽  
Comprehensive Approach ◽  
Methodological Issues ◽  
Pauline Theology ◽  
Detailed Interpretation ◽  
Book Of Acts ◽  
The Impact ◽  
The Church

This article first discusses the methodological issues involved in examining the portrayal of Jesus in the missionary speeches of the book of Acts and the nature of these speeches. This is followed by a detailed analysis of the presentation of Jesus, following a chronological line: Jesus� origin, his ministry, suffering, death and burial, his resurrection, exaltation, present ministry and parousia. The analysis is supplemented by the various portrayals of Jesus in the narrative of Acts. Afterwards, a detailed interpretation of this portrayal is offered, that is, its emphases (namely the saving significance of all of Jesus� life, the pervasive motif of the fulfilment of Scripture, Jesus as the agent of God, and the Jewishness of his life and ministry, focussing on Israel), the consequences that are drawn from this portrayal, the impact of the audiences on the presentation of Jesus, and the use of Christological titles. A final section reflects on the implications of the portrayal of Jesus in the missionary speeches of Acts for the witness and proclamation of the church. This comprehensive approach accounts for the length of the article.Intradisciplinary and/or interdisciplinary implications: Some research has criticised Luke-Acts for its alleged lack of Pauline theology and the �depth� of Paul�s interpretation of the life, death and resurrection of Jesus. This article shows that Acts in its own way offers a significant summary and Christological and soteriological interpretation of the life of Jesus and its enduring significance. The article also reflects on the significance of this interpretation for the present day presentation of Jesus and provides some necessary corrections.

Bifurcation Analysis and Pattern Selection of Solutions for the Modified Swift–Hohenberg Equation

2020 ◽  
Vol 30 (11) ◽  
pp. 2030031
Author(s):  
Yuncherl Choi ◽  
Taeyoung Ha ◽  
Jongmin Han
Keyword(s):  
Bifurcation Analysis ◽  
Critical Points ◽  
Trivial Solution ◽  
Numerical Results ◽  
Critical Values ◽  
Pattern Selection ◽  
Periodic Condition ◽  
Pattern Formations ◽  
Selection Of

In this paper, motivated by [Peletier & Rottschäfer, 2004; Peletier & Williams, 2007], we study the dynamical bifurcation of the modified Swift–Hohenberg equation endowed with an evenly periodic condition on the interval [Formula: see text]. As [Formula: see text] crosses over the critical points, the trivial solution bifurcates to an attractor and some new patterns of solutions emerge. We provide detailed descriptions of all possible final patterns of solutions on the overlapped intervals of [Formula: see text], which emerge after a gap collapses to a point. We also compute all critical values of [Formula: see text], [Formula: see text] and [Formula: see text] precisely, which are responsible for bifurcation and pattern formations. We finally provide numerical results that explain the main theorems.

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