Combining the two lines of equalities above yields the claim. \square The lemma readily follows from Claims 3 and 4. \square Next, let us present an example that satisfies the conclusion of Lemma 5.4 (2).Example 5.7Let A:=\left( 1~p_{k+1}~p_{k+1}^2~\cdots ~p_{k+1}^{n-...
When astronomers analyse light from the universe, they see each of these colours as single spectral lines. H-delta is the name of the fourth line – our view of the electron falling from hydrogen’s sixth orbit to its second orbit. The Rydberg formula calculates the wavelength, λ, of the...
In this section we discuss several examples to motivate the study of the problem of deciding when an individual fiber of a local diffeomorphism is a singleton, paving the way for the results to be proved in later sections. In Example 1 we recall the reduction theorems from the algebraic study...
Blue lines show motion and inferred shapes Full size image Discussion Our findings from the pilot data are interesting, to say the least. And while our pilot sample sizes are modest, and there are reasons to question whether the two student samples can be combined, the combined results provide...
So then you have to do things like figure out how to break the expression into lines elegantly, which is something we did a lot of work on in Mathematica. There are a lot of interesting issues, like the fact that if you edit an expression, its optimal line breaking can change all the...
My mathematical campaign will cost me much stubborn thought: I am aware of this after the first few lines of my book. I am entering upon the domain of the abstract, rough ground that can only be cleared by the insistent plow of reflection. The blackboard, excellent for the curves of ana...
Mathematical fictionalists needn't claim that mathematical utterances are analogous to either of these utterances: they needn't claim that when we use mathematical singular terms, we're engaged in a bit of make-believe (along the lines of (1)) or that we're straightforwardly mistaken (along...
The lines that we pick correspond to certain pairs of points in , and hence we can denote them by the edges of a graph on the vertex set . The following proposition is a generalisation of the fact that the -simplex is a strong blocking set. It gives us a general graph theoretic way ...
They spout the usual lines about Fibonacci numbers in nature. $$$ Alex and Eve (2016) IMDB ** Australian romance about a Greek maths teacher and a Lebanese Lawyer. A quick scene about the Greeks having discovered everything, but the arabs having up with algebra. ...
To minimize \(\mathcal {J}\) defined in (3.9), it is sufficient to travel in straight lines from the initial measure \(m_0\), as the following lemma makes clear. For \(m_T \in {\mathscr {P}}_2(\mathbb {R}^d)\), we define ...