scholarly journals Computing the Chow Variety of Quadratic Space Curves

2016 ◽  
pp. 130-136
Author(s):  
Peter Bürgisser ◽  
Kathlén Kohn ◽  
Pierre Lairez ◽  
Bernd Sturmfels
Keyword(s):  
Quadratic Space ◽  
Space Curves ◽  
Chow Variety

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

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.

Totally Wild Quadratic Forms of Type E7

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.

Linked Tori, II

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.

Particle and Astro-physics Challenge Kant’s Phenomenolism

1998 ◽  
pp. 323-333
Author(s):  
Lawrence H. Starkey
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Physical Basis ◽  
Primary Sources ◽  
One Dimension ◽  
Space Curves ◽  
Slowing Down ◽  
Parity Conservation ◽  
Charge Parity ◽  
Einstein’S General Relativity

For two centuries Kant's first Critique has nourished various turns against transcendent metaphysics and realism. Kant was scandalized by reason's impotence in confronting infinity (or finitude) as seen in the divisibility of particles and in spatial extension and time. Therefore, he had to regard the latter as subjective and reality as imponderable. In what follows, I review various efforts to rationalize Kant's antinomies-efforts that could only flounder before the rise of Einstein's general relativity and Hawking's blackhole cosmology. Both have undercut the entire Kantian tradition by spawning highly probable theories for suppressing infinities and actually resolving these perplexities on a purely physical basis by positing curvatures of space and even of time that make them reëntrant to themselves. Heavily documented from primary sources in physics, this paper displays time’s curvature as its slowing down near very massive bodies and even freezing in a black hole from which it can reëmerge on the far side, where a new universe can open up. I argue that space curves into a double Möbius strip until it loses one dimension in exchange for another in the twin universe. It shows how 10-dimensional GUTs and the triple Universe, time/charge/parity conservation, and strange and bottom particle families and antiparticle universes, all fit together.

scholarly journals Interpolation for Brill–Noether space curves

2017 ◽  
Vol 156 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Isabel Vogt
Keyword(s):  
Space Curves

scholarly journals Parallel Software to Offset the Cost of Higher Precision

2021 ◽  
Vol 40 (2) ◽  
pp. 59-64
Author(s):  
Jan Verschelde
Keyword(s):  
Power Series ◽  
Parallel Algorithms ◽  
Series Expansions ◽  
Use Case ◽  
Double Precision ◽  
Algebraic Space ◽  
Space Curves ◽  
Computational Overhead ◽  
Power Series Expansions ◽  
The Cost

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

scholarly journals Chirality in Geometric Algebra

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1521
Author(s):  
Michel Petitjean
Keyword(s):  
Geometric Algebra ◽  
Orthogonal Group ◽  
Matrix Representation ◽  
Quadratic Space ◽  
Complex Numbers ◽  
Chiral Algebra

We define chirality in the context of chiral algebra. We show that it coincides with the more general chirality definition that appears in the literature, which does not require the existence of a quadratic space. Neither matrix representation of the orthogonal group nor complex numbers are used.

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