@MatheinBoulomenos I mean in the appropriate pair of bases. Namely, choose a basis for $\{ v_i \}$ for $V$, then that induces the basis $P( \{v_i \})$ for $W$. These guys should be stretched/rotated the same way I think
@PawełKusz Letting $h = \alpha k$ parameterize the line we're approaching the origin along, then the limit $ \sim -k^2\frac{1}{\alpha \sqrt{1+\alpha^2}}$
maybe the intuition is something like this: why should the limit exist if you approach the value in something like a parabola or a weird spiral if it only exists for straight lines?
@KevinDriscoll But yeah the continuous versions (like the rational function Dami showed you earlier) essentially work like that, just a bit smoothed out
(and Dami's was $1$ on a parabola instead of a circle)
(and by "continuous" I mean "continuous away from the bad point")
Apparently people in Roy Moore's campaign don't understand how math works. Doug Jones has 49.9% of the vote. Roy Moore has 48.4%. There were 1.7% write-in votes. Moore's campaign was just arguing that the write-in votes has to be verified to make sure that they're legitimate votes for a person who could actually be a US Senator from Alabama.
And they're banking on that changing the margin of victory enough so that it shrinks from 1.5% to below 0.5% to trigger an automatic recount.
Except..... that's impossible by disqualifying write-ins. The margin is $\frac{\text{Jones vote} - \text{Moore vote}}{\text{Total votes}}$ if the denominator goes down by disqualifying write-ins, the margin goes UP, not down.....
I know you all know this I'm just kind of flabbergasted
Unless I'm being incredibly tired because its late and am overlooking something simple here
hopefully I'm just missing something obvious, but why does $p(p-1) = (a-1)(a^2+a+1)$ and $p$ is prime, mean $(a^2+a+1)/p \equiv 3 mod (a-1)$ ?? Additional context if necessary: https://math.stackexchange.com/questions/2561102/find-values-of-p-such-that-p2-p1-is-cubic https://artofproblemsolving.com/community/c6h35935p3629211
@PPenguin as mentioned in the comments, either (i) $p\mid(a-1)$, in which case $(a^2+a+1)\mid(p-1)$, or (ii) $p\mid(a^2+a+1)$, in which case $(a-1)\mid(p-1)$. The former is impossible as $p\mid(a-1)<(a^2+a+1)\mid(p-1)$ would entail $p<p-1$. Thus, $p\mid(a^2+a+1)$, and $p\equiv 1$ mod $(a-1)$. As such, $$ \frac{a^2+a+1}{p}\equiv\frac{1+1+1}{1} \mod{(a-1)} $$ since $a\equiv1$ mod $(a-1)$.
Hi! I want to make sure about the meaning of "distribution of input". since we have possible choice of input and usually probability is distributed through these possible choice of input. So the distribution of input is the set of possible choice.
Do you have any argument!
I'm asking this question because I read this following statement: But the truth is that typically (e.g. in most practical applications), very little if anything is known about the input distribution.
Let $A,B \in M_n(\mathbb R)$ be such that $A+B=AB$. Then $AB=BA$.
$A+B=AB$ and $B+A=BA \implies AB=BA(\because $ Matrix addition is commutative). Am I correct? I am not confident with the argument. because It had appeared in the PhD Screening test.
TIFR-GS 2018-PhD, Int-PhD Screening Test.
Check whether the statement true or false
The set of nilpotent matrices in $M_3(\mathbb R)$ spans $M_3(\mathbb
R)$ considered as an $\mathbb R$- vector space.
I think the statement is false. Nilpotent matrices have zero diagonal entries. so it can...
@AlessandroCodenotti I think the statement is false. Right? but I don't have counterexamples also. I am weak in proving. How to improve the problem-solving skills in Pure Mathematics? Some areas I am good. some areas I am bad.
Please suggest some elementary books that improve my proving skills. I have gone through a bad curriculum structure. Please help me.
yes. I have M.Sc. I graduated this year. my convocation has over :) Being a graduate, that doesn't mean he/she has good knowledge and base. If it would be correct, Everyone will be a good researcher.
2
I had side business during my graduation. After completing the course, I was attracted to differential geometry and General Theory of relativity. I wish to master in the proof technique. I really don't know, how? I was lazy during my graduation. Please give advise, how to master in the proof techniques.
Anyway @Leaky the right approach is to think about transcendence basis, the "kill a fly with a nuclear bomb" approach is Steinitz's theorem that $\sf{ACF}_0$, the theory of algebraically closed field of characteristic $0$ is $\kappa$-cathegorical for all uncountable $\kappa$
On the one hand, there are plenty of students that focus very hard on simply passing a course rather than truly understanding it; and who take as many "easy" courses as possible
On the other hand, at plenty of institutions, especially with high tuition fees, there can be a mentality that since the since the student paid so much to study, he should graduate; and staff is discouraged to fail students
Hello will anyone tell me is tan x considered continuous if we remove points such as 90 deg from domain I think it will still be discontinuous due to limit as x goes towards 90 deg from both sides!
@Leaky isnt there already a definition.. oh maybe there might be many definitions, i was told that limits from both sides must exist finite, and that function must be defined at that point. Yeah limits of tan x as x goes to 90 deg becomes +- infinity
@Tasty I reckon that the answer in test was wrong. So if 90 degree is removed from domain, we will not be considering lhl and rhl to 90 degree too. That makes tan x continuous on its domain mayb
I guess the proof idea is something like that. A generator of $H_n(M, M - p)$ is represented by a singular simplex $\Delta^n \to M$ with image of interior containing $p$
@TastyRomeo but I have studied in prestigious institutions in India. I thought that Maths means Calculus and Calculus means maths. I was interested in Physics. I thought that Entire mathematics required for doing physics. amateur thinking after schooling. :) I paid my 5 years for that. I luckly completed M.sc. :) I wish to do PhD in GTR only.
@samjoe are you in CBSE?.then it is hell. They don't teach basics of higher mathematics. I am not complaining the board. but, my calculus is very good :). It was my mistake. I should have read what is Higher Mathematics during my high school itself. I shouldn't judge from the curriculum framework. :)
@MikeMiller I guess then the question boils down to patching up a consistent choice of element of each stalk to a section of the sheaf. I wonder if you can just do this "abstractly" or whatever
Well, yeah, you can do that abstractly. I guess that's not the issue. I suppose the technicality lies in seeing $H_n(M)$ as a group of sections of the sheaf? There's a morphism $H_n(M) \to \Gamma$ given by sending each homology class $\alpha$ to the section $s : M \to \tilde{M}$ defined by $s(x) = \alpha_x$
JEE is the big curse for Indian mathematics :)(My opinion). I don't wish to give a negative impression to my institution. It might be my own mistake. I don't want to reveal my institution. sorry
if you have potential. institutions don't matter :)
I want to prove that $h$ is a parametrisation of hyperboloid $S$. To do this, I need to show that $h$ is surjective onto $S$. I see that $S$ is just a set of pairs of circles of varying radii. I see that for a given circle, I can always find some $(u,v)$ in the domain of $h$ such that it "draws" that exact circle. So then, is that enough to say that $h$ is surjective. First time I am trying this method. Is it sound?
I'm looking through the "matrix cookbook" and I'm somewhat confused by the notation in equation 37 there, since it's not explained earlier. The book I'm referring to can be found here (https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) What is meant by (∂X) Y there?
read the set theory from the first chapter of Analysis and algebra book. I hope that will help you. ask your doubts to experts. group discussion is the best thing for Higher mathematics topics. please be sure that you are not deviating from the curriculum frameworks of CBSE.
:)
all the best
these are the things that i haven't done during my schooling. so, I am suffering now. :)
I almost don't dare to ask, but what is just $\partial X$? I mean, I know partial derivatives of functions, but now that I'm dealing with matrices I'm completely lost
I'm working on an optimization problem for the first time and now my lacking knowledge of calculus comes to bite me. If I understand it correctly I need to get the derivative of my objective function with respect to the target variable and then get the gradient of that. I'll post my (edited) version here later for anyone interested
Consider a non-ufd ring $M = Z(\sqrt n)$ for a squarefree integer $n$.
Denote the elements as $a + b \sqrt n$
( $a,b$ are integers )
Define the quadrant as the elements with $a,b $ non-negative.
Define the $m$ th element of the quadrant by $C(a,b) = m$ where $C$ is the cantor pairing function....
@Pseudohuman Recent chess engine developed by Google that was only given the rules of chess (no strategy), a reinforcement learning algorithm, lots of computing power, and four hours to train
Let $X$ be an uncountable set, let $\frak{M}$ be the collection of all sets $E \subseteq X$ such that either $E$ is countable or $E^c$ is countable, and define $\mu (E) = 0$ in the first case, $\mu (E) = 1$ in the second case. Porve that $\frak{M}$ is a $\sigma$-algebra in $X$ and that $\mu$ i...
I am reading a book, and there before proving that there are infinite primes, they ask this: "What is wrong with the following argument?-There are infinitely many positive integers. Each of them factors into primes by the fundamental theorem of arithmetic. Hence there must be infinitely many primes."
I can't see exactly where there is flaw in argument.
Claim: $f_n : X \to [0,\infty]$ is measurable for every $n$, $f_1 \ge f_2 \ge ...$, $f_n(x) \to f(x)$ for every $x \in X$, and $f_1 \in L^1(\mu)$ implies $$\lim_{n \to \infty} \int_X f_n d \mu = \int_X f d \mu$$. Proof: Since the functions are nonnegative, $|f_n(x)| \le f_1(x)$ for all $x \in X$ and every $n \in \Bbb{N}$. Lebesgue's Dominated Convergence Theorem says that $\displaystyle \lim_{n \to \infty} \int_X f_n d \mu = \int_X f d \mu$. QED
Hi. I get the idea of looking at the central binomial coefficient as in the top answer of mathoverflow.net/questions/258711/…, but what if, say, I have $\sum_{k=0}^K\binom{K}{k}^2 K^{-\alpha}$ for some $\alpha\in(0,2)$. Then because of $K^{-\alpha}$ the main contribution is from $k=0$ to some $x$. My attempts show that taking $x=C K^{\beta}$ with some $\beta<1/2$ might work, but I don't see how to show that.. any ideas?
So we have $F$ with characteristic $p>0$ and $E/F$ an algebraic extension with the property that whenever $e_1,\cdots,e_n\in E$ are linearly independent over $F$ also $e_1^p,\cdots,e_n^p$ are linearly independent over $F$
I'm asked to prove that $E$ is a separable extension
Given three solutions $y_1, y_2, y_3$ to an non-homogeneous 2nd order linear equation, how can you construct a fundamental set of solutions to the corresponding homogeneous equation?
Given the matrix
$$
A =
\begin{bmatrix}
1&1&1&1\\
a&b&c&d\\
a^{2}&b^{2}&c^{2}&d^{2}\\
a^{3}&b^{3}&c^{3}&d^{3}
\end{bmatrix}
$$
How can I prove that $\det(A)=(d-c)(d-b)(d-a)(c-b)(c-a)(b-a)$ ?
And as a second question, what is the general formula for this type of matrix and how can I prove i?
...
I'd very much like to solve it myself, so if anyone could give me a push in the right direction, without having to write down the full answer, I'd really appreciate that.
Can I ask somebody just a quick question please? I am writing a small project on the tower of Hanoi and I have to write an abstract. Should I explain the the abstract what the tower of Hanoi is?
"The Tower of Hanoi is an old puzzle in which one must move a tower made from different sized discs from one peg to another, all the time keeping smaller discs on top of the larger ones" or something like that
@BalarkaSen ok thank you. It turns out today my professor told us that if someone S is able to give a new proof for the Tychonoff theorem, then S will become famous, very famous. Is that really true??
Let $X$ be an uncountable set, let $\frak{M}$ be the collection of all sets $E \subseteq X$ such that either $E$ is countable or $E^c$ is countable, and define $\mu (E) = 0$ in the first case, $\mu (E) = 1$ in the second case. Porve that $\frak{M}$ is a $\sigma$-algebra in $X$ and that $\mu$ i...
@BalarkaSen I've only seen one, I remember the general idea and I think I could work out the details if needed, but it's one of those results where the proof isn't very enlightening and you just remember the theorem
TIFR-GS 2018-PhD, Int-PhD Screening Test.
Check whether the statement true or false
The set of nilpotent matrices in $M_3(\mathbb R)$ spans $M_3(\mathbb
R)$ considered as an $\mathbb R$- vector space.
I think the statement is false. Nilpotent matrices have zero diagonal entries. so it can...
Consider a non-ufd ring $M = Z(\sqrt n)$ for a squarefree integer $n$.
Denote the elements as $a + b \sqrt n$
( $a,b$ are integers )
Define the quadrant as the elements with $a,b $ non-negative.
Define the $m$ th element of the quadrant by $C(a,b) = m$ where $C$ is the cantor pairing function....
Let $X$ be an uncountable set, let $\frak{M}$ be the collection of all sets $E \subseteq X$ such that either $E$ is countable or $E^c$ is countable, and define $\mu (E) = 0$ in the first case, $\mu (E) = 1$ in the second case. Porve that $\frak{M}$ is a $\sigma$-algebra in $X$ and that $\mu$ i...
Given the matrix
$$
A =
\begin{bmatrix}
1&1&1&1\\
a&b&c&d\\
a^{2}&b^{2}&c^{2}&d^{2}\\
a^{3}&b^{3}&c^{3}&d^{3}
\end{bmatrix}
$$
How can I prove that $\det(A)=(d-c)(d-b)(d-a)(c-b)(c-a)(b-a)$ ?
And as a second question, what is the general formula for this type of matrix and how can I prove i?
...
