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Results tagged with mathematics-education
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user 766
For questions in Mathematics Education as a scientific discipline. For more hands-on questions on teaching Mathematics, please use the tag teaching. There is also a Stack Exchange community http://matheducators.stackexchange.com/
64
votes
Real-world applications of mathematics, by arxiv subject area?
math.AC Commutative algebra
Reed-Solomon codes (a type of error correction codes based on polynomials over finite fields - this is why CDs and DVDs still work even after being scratched!)
- 4,713
46
votes
Real-world applications of mathematics, by arxiv subject area?
math.DG Differential geometry
Lie groups are used in robotics (to find the most efficient way to maneuver a robotic arm, for instance).
Spherical trigonometry is essential for navigation (a few centu …
- 4,713
30
votes
Real-world applications of mathematics, by arxiv subject area?
math.AG Algebraic geometry
Elliptic curve cryptography
Motion planning: Configuration spaces of robot arms are semi-algebraic sets, and algebraic geometry (especially the Cylindrical Algebraic Decomp …
- 4,713
31
votes
Real-world applications of mathematics, by arxiv subject area?
math.ST Statistics
almost everywhere
Accurate polling
Fraud detection: whether financial (e.g. via Benford's law) or voter fraud.
Used in the world of finance, economics and gambling on a daily basis …
- 4,713
21
votes
Real-world applications of mathematics, by arxiv subject area?
math.NA Numerical analysis
Linear programming algorithms are used in compressed sensing, which is now being used in MRI and imaging to increase resolution and/or decrease the number of measurements r …
- 4,713
6
votes
Real-world applications of mathematics, by arxiv subject area?
math.GM General Mathematics
The cosmic distance ladder is largely built using elementary geometry (although for some legs of the ladder, more advanced mathematics, e.g. relativity and probability, pl …
- 4,713
380
votes
Examples of common false beliefs in mathematics
The closure of the open ball of radius $r$ in a metric space, is the closed ball of radius $r$ in that metric space.
In a somewhat related spirit: the boundary of a subset of (say) Euclidean space ha …
- 319
103
votes
Examples of common false beliefs in mathematics
In order to show that a polynomial $P \in F[x_1,\ldots,x_n]$ vanishes, it suffices to show that $P(x_1,\ldots,x_n) = 0$ for all $x_1,\ldots,x_n \in F$. True in infinite fields, but very false for sma …
88
votes
Examples of common false beliefs in mathematics
This is perhaps a misunderstood definition rather than a false belief, but:
"A subnet of a net $( x_\alpha )_{\alpha \in A}$ takes the form
$( x_\alpha )_{\alpha \in B}$ for some subset $B$ of $A$. …
- 243
207
votes
Examples of common false beliefs in mathematics
Some false beliefs in linear algebra:
If two operators or matrices $A$, $B$ commute, then they are simultaneously diagonalisable.
(Of course, this overlooks the obvious necessary condition that each …
- 346
17
votes
Real-world applications of mathematics, by arxiv subject area?
math.SP Spectral theory
Spectroscopy (of course)
To avoid bridge collapse, knowing where the resonant frequencies are is extremely important. :-)
Shape and model recognition
Network analysis and sec …
- 43
141
votes
Accepted
Is the boundary $\partial S$ analogous to a derivative?
The surface area $|\partial S|$ of a (bounded, smooth) body $S$ is the derivative of the volume $|S_r|$ of the $r$-neighbourhoods $S_r$ of $S$ at $r=0$:
$$ |\partial S| = \frac{d}{dr} |S_r| |_{r=0}.$ …
- 113
15
votes
Insightful books about elementary mathematics
If first-order logic counts as "elementary mathematics", then I would like to suggest (the relevant chapters of) "Godel, Escher, Bach", by Douglas Hofstadter. (As an aside: Hofstadter's puzzle of enc …
- 114k
94
votes
Accepted
Are there proofs that you feel you did not "understand" for a long time?
As an undergraduate, I learned the Sylow theorems in my algebra classes but could never retain either the statement or proof of these theorems in memory except for short periods of time (and in partic …
- 114k
16
votes
Reference for a nice proof of "undetermined coefficients"
If $L$ has characteristic polynomial $\lambda \mapsto p(\lambda)$, then the conjugated differential operator $e^{-qt} L e^{qt}$ has characteristic polynomial $\lambda \mapsto p(\lambda+q)$. From thi …
- 114k