The discriminants of a series of unimodular singularities of space curves

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.

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On the Complexity of Computing the Topology of Real Algebraic Space Curves

Journal of Systems Science and Complexity ◽

10.1007/s11424-020-9164-2 ◽

2021 ◽

Vol 34 (2) ◽

pp. 809-826

Author(s):

Kai Jin ◽

Jinsan Cheng

Keyword(s):

Algebraic Space ◽

Space Curves

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Interpolation for Brill–Noether space curves

manuscripta mathematica ◽

10.1007/s00229-017-0961-4 ◽

2017 ◽

Vol 156 (1-2) ◽

pp. 137-147 ◽

Cited By ~ 4

Author(s):

Isabel Vogt

Keyword(s):

Space Curves

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Parallel Software to Offset the Cost of Higher Precision

ACM SIGAda Ada Letters ◽

10.1145/3463478.3463483 ◽

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.

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On the Classification of Algebraic Space Curves

Vector Bundles and Differential Equations ◽

10.1007/978-1-4684-9415-0_5 ◽

1980 ◽

pp. 83-112 ◽

Cited By ~ 15

Author(s):

Robin Hartshorne

Keyword(s):

Algebraic Space ◽

Space Curves

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Characterization and construction of helical polynomial space curves

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2003.08.030 ◽

2004 ◽

Vol 162 (2) ◽

pp. 365-392 ◽

Cited By ~ 32

Author(s):

Rida T. Farouki ◽

Chang Yong Han ◽

Carla Manni ◽

Alessandra Sestini

Keyword(s):

Polynomial Space ◽

Space Curves

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Uniqueness of the Geometric Representation in Large Rotation Finite Element Formulations

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4001909 ◽

2010 ◽

Vol 5 (4) ◽

Cited By ~ 10

Author(s):

Ahmed A. Shabana

Keyword(s):

Finite Element ◽

Finite Elements ◽

Computer Aided Design ◽

Curvature Vector ◽

Space Curve ◽

Body Motion ◽

Large Rotation ◽

Space Curves ◽

Curvature Continuity ◽

Finite Element Formulations

Several finite element formulations used in the analysis of large rotation and large deformation problems employ independent interpolations for the displacement and rotation fields. As explained in this paper, three rotations defined as field variables can be sufficient to define a space curve that represents the element centerline. The frame defined by the rotations can differ from the Frenet frame of the space curve defined by the same rotation field and, therefore, such a rotation-based representation can provide measure of twist shear deformations and captures the rotation of the beam about its axis. However, the space curve defined using the rotation interpolation has a geometry that can significantly differ from the geometry defined by an independent displacement interpolation. Furthermore, the two different space curves defined by the two different interpolations can differ by a rigid body motion. Therefore, in these formulations, the uniqueness of the kinematic representation is an issue unless nonlinear algebraic constraint equations are used to establish relationships between the two independent displacement and rotation interpolations. Nonetheless, significant geometric and kinematic differences between two independent space curves cannot always be reduced by using restoring elastic forces. Because of the nonuniqueness of such a finite element representation, imposing continuity on higher derivatives such as the curvature vector is not straight forward as in the case of the absolute nodal coordinate formulation (ANCF) that defines unique displacement and rotation fields. ANCF finite elements allow for imposing curvature continuity without increasing the order of the interpolation or the number of nodal coordinates, as demonstrated in this paper. Furthermore, the relationship between ANCF finite elements and the B-spline representation used in computational geometry can be established, allowing for a straight forward integration of computer aided design and analysis.

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