Mathematics is really Psychology. Psychology is really Biology. Biology is really Chemistry. Chemistry is really Physics. Physics is really Mathematics.
Hi everyone, small question: How could I read the ring: R[q][[x]] and what's the difference with R[[x]][q]. For the first one, is it the ring of fine sums of x with coefficients in q or the ring of formal power series?
@DavidCardozo $R[q]$ is a ring, and $R[q][[x]]$ is the ring of power series over that ring. Elements are power series in $x$, whose coefficients are polynomials in $q$ with coeffs in $R$.
Similarly, $R[[x]][q]$ is the set of polynomials in $q$, whose coefficients are power series in $x$ over $R$.
@infinitesimal He and I both agreed this is not the time. For therapy to be effective, one's mind must be calm first. And he and I both agreed on the same medication.
One cannot do math when tired. One cannot do therapy when overly anxious.
the statement you quoted before regarding 2-by-2 block matrices with commuting entries was that the following maps are equivalent: 1) taking the determinant of the entire matrix wrt to the underlying field, 2) taking the determinant of the 2-by-2 matrix in the commutative ring, and then taking the determinant of the matrix which results
It has stuff like Malcev's local theorems of group theory, Hilbert's Nullstellensatz of field theory and Steinitz dimension theory for field extensions..doesn't even sound like logic but I guess logic has a lot of algebraic applications
Model theory frequently has algebraic and number theoretic applications. Presumably there's going to be a model theoretic proof of the Nullstellensatz.
@JasperLoy Do you really give a f*ck on what the world think about you? Who is the world? You need to believe in yourself, no need for thr opinion of the world.
@JasperLoy Respect yourself! It's OK to talk to the others about your problems, but don't put much hope on what the others think about you. You have only one option, to trust yourself 100%.
@Ramanewbie It might take even more than that. The only problem is that ask for a better salary than others would do. If they accept ... wow (it's just a matter of money here) :-)
Well, what you do, is choose an $r$. Then you expand the Taylor series for the exponential, and the $q_{r,n}$ polynomial is the $n$-th coefficient (presumably the Taylor series is expanded about $0$).
@robjohn did you try this one? $$\sum_{n=1}^{\infty} \frac{1^{(2011)}+2^{(2011)}+ \cdots + n^{(2011)}}{n^{2014}}$$ I also posted it yesterday. In the numerator we have the falling factorial.
I guess I've been pretty bored today. Otherwise I probably wouldn't have bothered writing this much on a question unlikely to see more activity: math.stackexchange.com/questions/674982/…
Suppose I have a complete k-partite graph G = $K_{x_1,x_2,…,x_k}$. Then the union of all $K_{x_i}$ union G equals the complete graph K_(sum of x_i), correct?
@Vrouvrou your set is connected because it's path connected. For example you can connect $re^{i\theta}$ to $se^{i\psi}$ by first a circular arc and then a radial line.
Because the lenses need to correct what your eye's lens cannot, and focus light on the retina. This means that they cannot be the same, or else only one eye will have correct vision.
Using vector methods show that the distance between two non parallel lines $l_1$ and $l_2$ are given by $$d=\frac{|(\overrightarrow{v}_1 - \overrightarrow{v}_2) \cdot (\overrightarrow{ a}_1 \times \overrightarrow{a}_2)|}{||\overrightarrow{a}_1 \times \overrightarrow{a}_2||}$$ where $\vec{v}_1$ an...
Hi, I really love algebra(that is linear, abstract), however taking two courses in real analysis i loved the rigor of it, especially proving stuff.However i kind of hate differential equations since most courses just make us memorise a bunch of techniques, what area in real analysis do you think i would like
@AntonioVargas, I generally love the theoretical aspects of a subject that is proving stuff finding relations, hence i have a huge love for algebra and combinatorics, one field that i think i might be interested in is Functional Analysis
So it's purely by coincidence that a moderator might stumble upon a series of comments that he/she deems unworthy and delete them, and there isn't much that can be done about those which go unnoticed?
@AntonioVargas what are other interesting areas in Real Analysis since i was causually flipping through a journal of real anlysis in my library almost all of the papers were related to differential equations in one form or another
@G-man oh cool, if you really want to crack IIT with a top rank, then please dont spend much time on Math overflow, seriously IIT is a rat race with the competition sickening, I spent a lot of time on AOPS which led me to getting a not so good rank in IIT. Dammn i should have joned CMI
@Owatch isn't the right thing to say either 'I know a guy' or 'I have a friend', but not 'I know a friend', because him being a friend already establishes that you do,in fact,know him?
for example if you're in $\mathbb{R}^2$, then your matrix is $$A_2(x) + t \left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right)$$ or $$A_2(x) + t \left( \begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix} \right)$$ ?
The following $n\times n$ determinant identity appears as eq. 19 on Mathworld's entry for the Chebyshev polynomials of the second kind:
$$U_n(x)=\det{A_n(x)}\equiv \begin{vmatrix}2 x& 1 & 0 &\cdots &0\\ 1 & 2x &1 &\cdots &0 \\
0 & 1 & 2x &\cdots &0\\0 & 0 & 1 & \ddots & \vdots \\
\vdots & \dd...
