arXiv subjects

Jon A. Sjogren

Publications and source records attributed to Jon A. Sjogren.

6 recordsLinked to original sources

arXiv2020

Parity of n-Frames with Application to Non-Procrustean Orthogonalization

The space of complete orthonormal frames in Euclidean space is not path connected. In fact it has exactly two path components, containing respectively the coordinate frame of n standard coordinates and the frame with two coordinates reversed. The matrices corresponding to these frames have determinant one and minus one respectively. We wish to avoid the implement of the determinant function in many variables, and rather work with these compact spaces by means of fibrations and covering maps. The important fibrations are deletion of the final vector from the frame, and selection of the first vector. The long homotopy sequence for this fibration implies that the equivalence classes of loops form the group of two elements. One generator of this group is a circle of Givens rotations. Dimension three is critical for the proof. It is handled by the Rodrigues formula, together with Invariance of Domain. Another explicit proof that the mapping from unit quaternions is surjective to the main path-component is due to Itzhack. Thus the Orthogonal Group is not path connected, using the Homotopy Lifting Theorem for coverings. But the Givens loop performs an identity lifting, so it cannot be the group generator. As an application we review how quoted algorithms from the astronautical literature, related to the Wahba guidance problem, give rise to an orthogonalization of any matrix that is metrically close to some rotation. This method has a bias in the Frobenius norm, but under realistic conditions serves as well as a Procrustean method.

math.FA↗

arXiv2018

Complex Axis and de Medeiros' Campo Vetorial

The Complex Axis theorem states that any endomorphism of a finite-dimensional complex vector space affords an eigen-vector (or "invariant axis"). A geometric proof of this geometric result was given by A. de Medeiros, transforming the endomorphism into a topological self-map with Lefschetz number not equal to zero. We give a dual version of this proof, which may be more uniform, and does not rely on the need to do any calculation of an Euler characteristic or Lefschetz number. A vector field on Projective space is read off directly from the coordinates ("entries") of the given endomorphism (complex square matrix). A bordism is defined between such vector fields by means of Stokes' Theorem applied to a real manifold-with-boundary. This is the principle behind Hopf's lemma relating the Gauss map and the index of a vector field. All vector fields of the de Medeiros type are co-bordant to the Milnor-Hopf vector field. This latter comes from a non-derogatory, real diagonal endomorphism, so clearly possesses an eigen-vector. Therefore so has the given arbitrary endomorphism. The main theorem on complex polynomials naturally follows, using the companion matrix, secular polynomial reciprocity. The geometric Complex Axis derivation is meant to avoid determinants or "general position" arguments.

math.FA↗

arXiv2017

Homogeneous Systems and Euclidean Topology

The Theorem on Invariance of Domain due to L.E.J. Brouwer states that one connected, compact (Hausdorff) m-dimensional manifold embedded into another actually realizes a homeomorphism. This fundamental result is relevant to Functional Analysis, as the classical Gelfand-Mazur Theorem, as well as the real form of the Fundamental Theorem of Algebra, can both be derived easily from it. Our main tool is the m-dimensional borsuk-Ulam Theorem: a certain real vector must be found, which is established by means of solving a real homogeneous system of equations as in the Theorem of Be'zout. We emphasize the ideal-theoretic approach to the latter, based on work of Kapferer and vander Waerden from the 1920s. A modern explanation combines resultant- and non-resultant oriented methods in projective geometry. The technical point involves algebraic independence (over the rational numbers) of the System coefficients. Invariance of Domain follows from the fact that on a sphere, "any odd mapping is essential" (B-U), as well as "an injection of a ball is homotopic to an odd mapping". This homotopy avoids a neighborhood of the Origin, giving the required open set at the Origin of the image ball.

math.FA↗

arXiv2015

Polynomial Roots and Open Mappings

The "openness" of a complex polynomial mapping is discussed and applied to the Fundamental Theorem of Algebra. In this category fall proofs of S. Wolfenstein, R.L. Thompson, J. Milnor, and S. Reich-S. Smale. These proofs take into account the critical points of the polynomial. New elementary proofs of openness due to D. Reem and F.S. Cater make possible a very short proof of FTA without reference to zeros of the derivative. We regard Gauss's Helmstedt Thesis (1799) as an exercise in applied differential topology, and fill out a synopsis published by S. Gersten and J. Stallings. The polynomial function may be perturbed or re-aligned (work of Martin, Savitt and Singer) to eliminate critical values so we work with a configuration of real, plane algebraic curves. The treatment we give to the Implicit Function Theorem uses contractive operators on the Banach algebra of convergent power series according to W. Walter. In addition to Functional Analysis, the key geometric insight is a topological transversality result that leads to a rigorous demonstration that the plane curves intersect as Gauss and A. Ostrowski said they would. Working in the category of analytic mappings makes uniqueness of "implicit" solutions more transparent, and the polynomial (and harmonic) nature of the curves imposes finiteness on the number of extrema, coming from a well-known application of the Be'zout Theorem. We think this provides a clarification of Gauss's statement that an algebraic curve entering a domain, must leave again, that is more concrete than declaring such a curve to be a "one-dimensional manifold".

math.CV↗

arXiv2015

Complex Odd-Dimensional Endomorphism and Topological Degree

A linear mapping upon real n-dimensional space, where the dimension n is odd, has a real eigenvalue-eigenvector pair. The corresponding statement for complex vector spaces holds true for any dimension n, but should be easy to demonstrate when n is again odd. Derksen uses this result in an induction chain to prove the Fundamental Theorem of Algebra using matrix representations. In fact the question is closely tied to topological issues. We discuss proofs coming from vector field theory and characteristic classes. But a new proof uses rather elementary constructions. A complex linear n-mapping without an eigenvector gives rise to three real 2n-mappings, such that all transformations (mappings) in their (non-trivial) linear span have full rank (are linear isomorphisms). The theory of matrices of fixed rank shows that we obtain a 2n-plane bundle equation over the real projective plane. The equation states that the sum of trivial line bundles is essentially the same as the (same rank) sum of Hopf line bundles. To show that this yields a contradiction makes use of the homotopy theory of vector bundles over a base of low dimension, the two-dimensional Borsuk-Ulam theorem, and the invariance of topological degree for self-mappings of an (embedded) two-sphere in three-space.

math.FA↗

arXiv2015

Real Polynomial Rings and Domain Invariance

Recent proofs of classical theorems in polynomial algebra and functional analysis are discussed, which use tools from the topology of real manifolds. Simpler proofs were discovered in the new century, of the Hilbert Nullstellensatz, and the Gelfand-Mazur Theorem. We give a related proof that an irreducible real polynomial has degree 2 or less, Gauss's form of the Fundamental Theorem of Algebra. It has been debated whether an elementary proof for FTA can be found, using the Brouwer Fixed-Point Theorem as its "analytical" component. In the present case the analytic or topological tool employed is Brouwer's Theorem on Invariance of Domain, which derives from his Fixed-Point Theorem. A corollary of Domain Invariance is that an injective mapping of one compact manifold to another (connected) one of the same dimension, is in fact surjective and a homeomorphism. The desired result (FTA) comes from the fact that a real sphere and its (quotient) projective space of the same dimension are homeomorphic only when this dimension equals 1. This proof joins a class of proofs that depend on Euclidean fixed-point theory, and also the class of proofs that involve no field extensions or methods of complex analysis.

math.GT↗