Linear independence of cables in the knot concordance group

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

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Convergence properties of the Broyden-like method for mixed linear–nonlinear systems of equations

Numerical Algorithms ◽

10.1007/s11075-020-01060-y ◽

2021 ◽

Author(s):

Florian Mannel

Keyword(s):

Linear Independence ◽

Affine Subspace ◽

Systems Of Equations ◽

Nonlinear Systems Of Equations ◽

Convergence Properties ◽

Initial Matrix ◽

First Time ◽

Special Case ◽

Dimensional Mapping ◽

Finite Number Of Iterations

AbstractWe consider the Broyden-like method for a nonlinear mapping $F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ F : ℝ n → ℝ n that has some affine component functions, using an initial matrix B0 that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping $G:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ G : ℝ d → ℝ d , where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B0, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by $\frac {\sqrt {5}+1}{2}$ 5 + 1 2 if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

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Student attitudes towards linear algebra: an attempt to roll back the fog

Teaching Mathematics and its Applications An International Journal of the IMA ◽

10.1093/teamat/hraa012 ◽

2020 ◽

Author(s):

Barry J Griffiths ◽

Samantha Shionis

Keyword(s):

Student Perceptions ◽

Linear Algebra ◽

Student Attitudes ◽

Linear Independence ◽

Negative Emotions ◽

Practical Applications ◽

Key Concepts ◽

The Subject ◽

Roll Back

Abstract In this study, we look at student perceptions of a first course in linear algebra, focusing on two specific aspects. The first is the statement by Carlson that a fog rolls in once abstract notions such as subspaces, span and linear independence are introduced, while the second investigates statements made by several authors regarding the negative emotions that students can experience during the course. An attempt is made to mitigate this through mediation to include a significant number of applications, while continually dwelling on the key concepts of the subject throughout the semester. The results show that students agree with Carlson’s statement, with the concept of a subspace causing particular difficulty. However, the research does not reveal the negative emotions alluded to by other researchers. The students note the importance of grasping the key concepts and are strongly in favour of using practical applications to demonstrate the utility of the theory.

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A converse to linear independence criteria, valid almost everywhere

The Ramanujan Journal ◽

10.1007/s11139-014-9662-8 ◽

2015 ◽

Vol 38 (3) ◽

pp. 513-528 ◽

Cited By ~ 4

Author(s):

S. Fischler ◽

M. Hussain ◽

S. Kristensen ◽

J. Levesley

Keyword(s):

Linear Independence ◽

Almost Everywhere

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On independence of iterated Whitehead doubles in the knot concordance group

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500037 ◽

2018 ◽

Vol 27 (01) ◽

pp. 1850003

Author(s):

Kyungbae Park

Keyword(s):

Correction Term ◽

Khovanov Homology ◽

Floer Theory ◽

Knot Concordance ◽

Torus Knot ◽

Linearly Independent ◽

Heegaard Floer

Let [Formula: see text] be the positively clasped untwisted Whitehead double of a knot [Formula: see text], and [Formula: see text] be the [Formula: see text] torus knot. We show that [Formula: see text] and [Formula: see text] are linearly independent in the smooth knot concordance group [Formula: see text] for each [Formula: see text]. Further, [Formula: see text] and [Formula: see text] generate a [Formula: see text] summand in the subgroup of [Formula: see text] generated by topologically slice knots. We use the concordance invariant [Formula: see text] of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the [Formula: see text]-invariant of knot Floer theory and the [Formula: see text]-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of [Formula: see text] for any [Formula: see text] generalizing a result for [Formula: see text] of Hedden, Kim and Livingston.

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