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Imagine we have a linear or affine space of real symmetric $n\times n$ matrices:
$$\mathcal{A}=\{A_0+c_1 A_1+\ldots+c_m A_m: c\in \mathbb{R}^m\}$$
The question is how to efficiently test if it contains matrix of given eigenspectrum? Equivalently: of given characteristic polynomial?
Trying to exp...
It is known that to compute the Lebesgue outer measure $\lambda^*$ of irrationals $\Bbb{I}$ we make use of the following property:
$$\lambda^*(\Bbb{I}) +\lambda^* (\Bbb{Q}) = \lambda^* (\Bbb{R})$$
and noting that since $\Bbb{Q}$ has zero measure being a countable union of singletons and using c...
In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line ℝ that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. The Smith-Volterra-Cantor set is topologically equivalent to the middle-thirds Cantor set.
== Construction ==
Similar to the construction of the Cantor set, the Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval [0, 1].
The...
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence depends on the axiom of choice. Assuming the existence of an inaccessible cardinal, Solovay constructed a model of Zermelo–Fraenkel set theory (ZF) without the axiom of choice, where all sets of real numbers are Lebesgue measurable.
== Measurable sets ==
Certain sets have a definite 'length' or 'mass'. For instance, the interval [0...
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