Quadratic Space Curve Based Cubic Splines

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

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Totally Wild Quadratic Forms of Type E7

10.23943/princeton/9780691166902.003.0015 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Quadratic Space ◽

Trace Map ◽

Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

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Linked Tori, II

10.23943/princeton/9780691166902.003.0006 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Integral Domain ◽

Quadratic Space ◽

Small Observation ◽

Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

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On approximation with the continuous polynomial cubic splines

2020 24th International Conference on Circuits, Systems, Communications and Computers (CSCC) ◽

10.1109/cscc49995.2020.00030 ◽

2020 ◽

Author(s):

I.G. Burova

Keyword(s):

Cubic Splines

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Alignment of three sequences in quadratic space

ACM SIGAPP Applied Computing Review ◽

10.1145/381771.381773 ◽

1993 ◽

Vol 1 (2) ◽

pp. 7-11 ◽

Cited By ~ 6

Author(s):

Xiaoqiu Huang

Keyword(s):

Quadratic Space

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Geometry Design and Contact Ratio Analysis for Circular Arc Parallel-Axis Helical Gears

Volume 11: Systems, Design, and Complexity ◽

10.1115/imece2016-65820 ◽

2016 ◽

Cited By ~ 1

Author(s):

Z. Chen ◽

B. Lei ◽

Q. Zhao

Keyword(s):

Rolling Process ◽

Helical Gear ◽

Space Curve ◽

Impact Factors ◽

Contact Ratio ◽

Circular Arc ◽

Practical Applications ◽

Geometry Design ◽

Gear Mechanism ◽

The Impact

Based on space curve meshing theory, in this paper, we present a novel geometric design of a circular arc helical gear mechanism for parallel transmission with convex-concave circular arc profiles. The parameter equations describing the contact curves for both the driving gear and the driven gear were deduced from the space curve meshing equations, and parameter equations for calculating the convex-concave circular arc profiles were established both for internal meshing and external meshing. Furthermore, a formula for the contact ratio was deduced, and the impact factors influencing the contact ratio are discussed. Using the deduced equations, several numerical examples were considered to validate the contact ratio equation. The circular arc helical gear mechanism investigated in this study showed a high gear transmission performance when considering practical applications, such as a pure rolling process, a high contact ratio, and a large comprehensive strength.

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