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seen Jul 26 at 0:53

a real pythagorean


Jul
26
comment Alternate proof for a theorem on ordered fields
We have $f(x)=\displaystyle\frac1{1+\frac1x}$, and $1/$ reverts the monotonicity (on positive range(!)), while $1+$ doesn't affect it. So it's twisted twice.
Jul
26
answered Matrix of a linear map Questions
Jul
26
comment Matrix of a linear map Questions
Sorry, the question is not clear. What quantifier is placed before the basis $w_1,\dots,w_n$? First you say 'there exists' then 'for every choice of basis'...
Jul
25
comment What is a bilinear form?
A bilinear form is a generalized scalar product. It can be realized by an intermediate (usually symmetric) matrix $\Lambda$ so that the 'generalized scalar product' of vectors $x,y\in\Bbb R^n$ is the matrix product $(x,y)=x^T\Lambda y$.
Jul
19
comment Example of operation that is tree associative, but not generally associative
So, if I understand correctly, you basically want a set equipped with an arbitrary binary operation and define $2^n$-ary operations $a_1a_2\dots a_{2^{k+1}}\ :=\ (a_1a_2\dots a_{2^k})(a_{2^k+1}\dots a_{2^{k+1}})$.
Jul
19
answered What are the elements of Z/10Z and of Z/2Z×Z/5Z, and identify which elements correspond under the map g from Z/10Z to Z/2Z × Z/5Z.
Jul
19
comment finite additivity&countable additivity
Sorry, what is a 'semialgebra' of sets?
Jul
19
comment manifold structure on a finite dimensional real vector space
Come on, $\Bbb R^n$ is the most basic example of an $n$ dimensional manifold. Actually, something is an $n$ dim. manifold iff it locally looks like $\Bbb R^n$.
Jul
17
comment Prove $A \subset \emptyset \iff A = \emptyset$
Hint: $\emptyset\subseteq A$ holds for all sets $A$.
Jul
15
comment A field with characteristic $0$ contains $\mathbb Q$
$T$ is the $F$-internal instance of the rational numbers, it contains exactly the numbers of the form $\pm(1+\dots+1)(1+1+..+1)^{-1}$ where $1$ is the unit of $F$. So, $T$ is isomorphic to $\Bbb Q$, and indeed $T$ is the subfield of $F$, which might be technically (as set) different from $\Bbb Q$, but they are usually identified any way.
Jul
15
comment Kernel of a map into infinite dimensions
Why do you want to repeat the same thing in infinite many coordinates?
Jul
15
answered A field with characteristic $0$ contains $\mathbb Q$
Jul
15
comment How Can I prove that this module is isomorphic to the generators with relations…
$\sum_{g\in G}g$.
Jul
13
comment Alternative axioms of the real numbers
No: A square number cannot be negative in an ordered field, so the characteristic must be zero, hence it contains $\Bbb Q$ as the subfield generated by $1$.
Jul
13
comment If $f\tau$ is continuous for every path $\tau$ in $X$, is $f:X\rightarrow Y$ continuous?
I guess it is the unit interval $[0,1]$.
Jul
13
answered What is the $s$ in the Laplace transform?
Jul
12
comment Write out the operator (A)^2 for A = (d/dx + x)
Have you also tried to write out yourself the double application of $A$ for a function $f$ instead of only reading it?
Jul
11
comment Commuting quaternions
They don't necessarily have, but they equal to their imaginary part, as being purely imaginary.
Jul
11
answered Commuting quaternions
Jul
8
comment If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?
Exactly what do you intend to mean by $(G,A)$ where $G=\Bbb Z^d$ and $A\in\Bbb Z^{d\times d}$?