Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find a real-valued vector solution to a system of differential equations Given:
$$\vec x'(t) = \begin{bmatrix}
4&-1\\
13&0
\end{bmatrix} \vec x(t) $$
Evaluating to find eigenvalues:
$$ (4-\lambda)(-\lambda)+13=0 $$$$ (\lambda-2)^2=-9$$ $$\lambda=2\pm3i$$
Finding the eigenvector for the eigenvalue $\lambda = 2+... | Your matrix is actually similar to one of the form $\begin{bmatrix}
2&-3\\
3&2
\end{bmatrix}$ with transition matrix $\begin{bmatrix}
2&3\\
13&0
\end{bmatrix}$ given respectively by the eigenvalues' real and imaginary parts and the transition is given (in columns) by real and imaginary parts of the firs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Are these vectors linearly independent? 3 vectors. I have this vector set S = {(2,-1,2), (1,0,3), (3,-2,1)}. Are they linearly independent?
$$(0, 0, 0) = c_1(2,-1,2) + c_2(1,0,3) + c_3(3,-2,1)$$
Here are the corresponding equations:
$$2c_1 + c_2 + 3c_3= 0$$
$$-c_1 - 2c_3 = 0$$
$$2c_1 + + 3c_2 + c_3 = 0$$
->
$$\begin{bm... | Personally, I wouldn't use matrices for a simple set of equations like these. You have
$2c_1+ c_2+ 3c_3= 0$
$-c_1- 2c_3= 0$
$2c_1+ 3c_2+ c_3= 0$
From the second equation $c_1= -2c_3$. So the first equation is $-4c_3+ c_2+ 3c_3= c_2- c_3= 0$ and the second is
$-4c_3+ 3c_2+ c_3= 3c_2- 3c_3= 0$.
Both of those equati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Integrating sum
*
*$\displaystyle \sum_{k=1}^n k=\dfrac{n(n+1)}{2}$
Which is equivalent of saying, $$1+2+3+4+5+\cdots+n=\dfrac{n(n+1)}{2}$$
Now how do I integrate the left side of the equality and how that would look like?
Integrating the right side gives $\int \dfrac{n(n+1)}{2} \, \mathrm dn =\dfrac{1}{2}(\dfrac{n... | I'm choosing to interpret "taking the integral" as finding an anti-derivative, in the sense of indefinite integrals. So, we're looking for a function $F$ such that
$$F'(n) = 1 + 2 + \ldots + n.$$
In that sense, $\frac{n^3}{6} + \frac{n^2}{4}$ fits the bill; it's a function whose derivative agrees with the sum at intege... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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If $T^9=T^8$, prove that $T^2=T$. Suppose $V$ is a complex inner product space and $T $ is a normal operator on $V$ such that $T^9=T^8$. Prove that $T^2=T$. Give an example of a non-normal $T$ such that $T^9=T^8$ but $T^2 \ne T$.
| The polynomial $x^9 - x^8 = x^8(x-1)$ annihilates $T$ so the minimal polynomial $m_T$ of $T$ divides $x^8(x-1)$.
On the other hand, $T$ is normal and hence diagonalizable so $m_T$ consists only of linear factors. Therefore $$m_T \in \{x, x-1, x(x-1)\}$$
and therefore $0 = T(T-I) = T^2 - T$.
As an example of a non-norm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2914039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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What is this function related with continued fractions? Playing with continued fractions, I came with the idea of iterating the limit of the simplest one:
$$1 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}}}\ = \Phi$$
And then I thoght about iterating the result:
$$\Phi + \cfrac{1}{\Phi+\cfrac{... | I think you can set up a differential equation and solve it.
\begin{eqnarray}
2f'(n) &= -f(n)+\sqrt{f(n)^2+4}\\
\text{d}n&=\frac{2\, \text{d}\! f}{-f+\sqrt{f^2+4}} \\
n &= \frac{1}{4} (f (f + \sqrt{4 + f^2} + 4 \sinh^{-1}\left(\frac{f}{2}\right)
\end{eqnarray}
Solving for $f$ should result in the function you wanted.
W... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2914208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 2
} |
Parabola touching a circle
The parabola $\dfrac {x^2}{d^2}+\dfrac yh=1$ touches the circle $x^2+(y-R)^2=R^2$ at two points, $(\pm p, q)$. It can be easily shown geometrically that $p=\sqrt{2qR-q^2}$. Can it be shown geometrically that $q=d$?
See desmos implementation here. The purple dotted line forms a square.
(See... | In parabola below, $AM=d$, $VM=h$ and $VF=VN=d^2/(4h)$, where $F$ is the focus and $NH$ the directrix of the parabola. We have $FP=PH$ and $\angle FPC=\angle KPC$, because radius $PC$ is normal to the parabola. Hence $CPHF$ is a parallelogram and
$$
CF=PH={h\over d^2} HN^2+VN=
{h\over d^2}(FH^2-FN^2)+{d^2\over4h}={h\ov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2914328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Doubt in high school permutation question(seating arrangement)
$6$ boys and $6$ girls form a line with boys and girls alternating.Find the number of ways of making the line.
Answer $2(6!)^2$
I was trying to solve this question using Permutation and Combination
$$\square B \square B \square B \square B \square B \sq... |
Therefore girls can be chosen in C(7,6)
Note that C(7,6) (out of seven, pick six to include) is the same as C(7,1) (out of seven, pick one to exclude), which is just 7. There are seven boxes, so it doesn't make sense to say that this is how many ways there are of choosing girls. It would make more sense to say that t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2914436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Is this set a basis for $R^3$. How to verify? Say I have $S = {(1,0,-1), (2,1,1), (-3,0,2)}$. Is my method correct for determining if it's a basis? How do I check if I'm right?
We need to determine if $S $ spans $R^3$ and if it's linearly independent.
First, check if $S$ spans $R^3$:
Let $u_1, u_2, u_3$ be a random vec... | Take a look at the zero-components. The second vector can’t be a linear combination of the other vectors. Since these other vectors aren’t parallel, the three vectors are linearly independent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2914509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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The ambigous definition of vacuous truth It is no doubt that the vacuous truth is related to material implication "$P\Rightarrow Q$". We say the material implication statement is true when $P$ is false. However, seems that this is not the definition for vacuously truth. Do we call it vacuously true only when $P$ can n... | *
*I use this inclusive definition:
*
*A vacuous truth is an implication or universally-quantified implication whose antecedent is true or universally true, respectively.
As such, both $$P{\implies}Q$$ and $$\forall x\;[P(x){\implies}Q(x)]$$ are vacuously true. (Wikipedia gives both types of examples.)
*In partic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2914619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
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The limit of $\frac{n^3-3}{2n^2+n-1}$ I have to find the limit of the sequence above.
Firstly, I tried to multiply out $n^3$, as it has the largest exponent.
$$\lim_{n\to\infty}\frac{n^3-3}{2n^2+n-1} =
\lim_{n\to\infty}\frac{n^3(1-\frac{3}{n^3})}{n^3(\frac{2}{n} + \frac{1}{n^2} - \frac{1}{n^3})} =
\lim_{n\to\infty}\... | You have done everything.I just want to represent it graphically.Then you will answer your own question.
Now,you tell me.what are you seeing here?
And ofcourse the answer of your your questions is simply "yes".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2914728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 7,
"answer_id": 6
} |
Showing that $ \lim_{x \to \infty} x(1/2)^x = 0$ Can someone explain to me why $$ \lim\limits_{x \to \infty} x\bigg(\frac{1}{2}\bigg)^x = 0$$
Is it because the $\big(\frac{1}{2} \big)^x$ goes towards zero as $ x $ approaches $\infty$, and anything multiplied by $0 $ included $\infty$ is $0$ ?
Or does this kind of ques... | Let
$a_x = \dfrac{x}{2^x}$.
Then
$\dfrac{a_{x+1}}{a_x}
=\dfrac{\dfrac{x+1}{2^{x+1}}}{\dfrac{x}{2^x}}
=\dfrac{1+1/x}{2}
$.
Therefore,
for $x > 3$,
$\dfrac{a_{x+1}}{a_x}
=\dfrac{1+1/x}{2}
\lt\dfrac{1+1/3}{2}
=\dfrac46
=\dfrac23
$.
Therefore,
for $x > 3$,
$\begin{array}\\
\dfrac{a_x}{a_3}
&=\prod_{y=3}^{x-1}\dfrac{a_{y+1}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Vector norm such that $\| (1,i) \| = 6$
Is there a norm $\| \cdot \|: \mathbb{C}^2 \rightarrow \mathbb{R}^+$ such that $\bigg\| \pmatrix{1\\i} \bigg\| = 6$?
I have come across this question and I don't really know what to look for here. Trying out random norms to see if one of them give $6$ doesn't look like a good i... | Consider
$$\Vert x \Vert = 3\sqrt{x^H x}$$
where $x^H$ is the transpose - conjugation operation, i.e. Hermitian operator. This norm satisfies the three basic norm properties, i.e.
*
*$ \Vert x \Vert = 0 $ only if $x = 0$
*For $\alpha \in \mathbb{C}$,$ \Vert \alpha x \Vert = 3\sqrt{(\alpha x)^H \alpha x} = 3 \sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2915111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Probability problem involving mappings of a finite set into itself. One mapping is selected at random from all the mappings of the set $\{1,2,\ldots,n\}$ into itself. What is the probability that
$(i)$ a specified element $i$ is transformed into another specified element $j$?
$(ii)$ the elements $i_1,i_2,\ldots,i_h$ a... | For (ii) suppose that the $i_k$ are all distinct, and the $j_k$ are all distinct.
There are $n(n-1)(n-2)\cdots(n-h+1)=n!/(n-h)!$ $h$-tuples of distinct elements
from $\{1,2,\ldots,n\}$. It is equally likely that your first $h$-tuple is taken
to any one of them, so the probability you seek is $(n-h)!/n!$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2915174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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The possible values of $x$, if $\tan^{-1} x>\cot^{-1}x$ What are the possible values of $x$, if $\tan^{-1}x >\cot^{-1}x$?
We have $\tan^{-1}x >\cot^{-1}x\implies \tan^{-1}x -\cot^{-1}x>0 \implies \tan^{-1} x-\tan^{-1}1/x>0\implies \tan^{-1}\dfrac{x-1/x}{1-x.1/x}>0$.
What can I do now?
| HINT:
$$\cot^{-1}x=\frac\pi2-\tan^{-1}x\implies \tan^{-1}x-\cot^{-1}x=2\tan^{-1}x-\frac\pi2>0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2915256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Is Taylor Series changing from an uncountable basis to a countable basis? Say we've got an analytic function $f(x)$ from $\mathbb R$ to $\mathbb R$. It has an uncountable number of components in this basis, since there is one value of $f(x)$ for each $x$ and $x$ varies continuously.
When we do a Fourier transform and ... | It is true that the domain ${\mathbb R}$ of a function $f:\>{\mathbb R}\to{\mathbb R}$ has uncountably many points. If you want to produce the function values by a random number generator independently for each point then you would have to call this routine an uncountable number of times. But already a continuous fun... | {
"language": "en",
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"question_score": "2",
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Sum of the series $\sum_{k=0}^n \frac{1}{k+1} \binom{n}{k}$ The task is to transform $\sum_{k=0}^n \frac{1}{k+1} \binom{n}{k}$ into a compact and not recursive formula.
I've done a bunch of similar (as I thought at first) series. I'll describe my method. I name $\sum_{k=0}^n f(k)=a_{n}$, and then describe $a_{n}=a_{n-1... | Hint:
$$\frac1{k+1}\binom{n}{k}=\frac1{n+1}\binom{n+1}{k+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2915549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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An Olympiad question about equality How many $(x, y)$ positive integer pairs have $y^2-x^2=2y+7x+4$ equality?
I can't solve this Olympiad question.
$\textbf{Solution:}$ If above equality is regulated,we obtain $(2x+2y-5)(2x-2y+9)=29$. And, $29$ is prime we can easily find solution.
*
*$\textbf{1-)}$ How can we regu... | Let the factors $u:=y+x$ and $v:=y-x$ appear.
$$uv=\frac{9u-5v}2+4.$$
Then,
$$\left(2u+5\right)\left(2v-9\right)=61.$$
Update to please Misha Lavro:
The factorization of the LHS hints the introduction of these intermediate variables: as the RHS is linear, we know that we can transform to $uv=au+bv+c$, which factors as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2915624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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2 squres with 1 common corner The line HC is a median in triangle $\triangle BCE$
How would I prove that for two arbitrary sized squares any angle $\alpha$, $\angle DIC$ is 90 degrees?
I have tried playing around with this in geogebra but to no avail.
|
Choose point $J$ on line $BC$ such that $BC=CJ$. Let us focus on (green) triangle $DCG$ and (red) triangle $JCE$.
$$\angle DCG = \angle JCE = \angle JCG + 90^\circ$$
$$DC=JC,\space CG=CE$$
By SUS, triangles $DCG$ and $JCE$ are congruent and therefore $JE=DG$ and $\angle CDG = \angle CJE$. But $CD\bot CJ$ so it must ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2915765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the meaning of the difference between the equations of two non-intersecting circles represent? The difference between the equations of two intersecting circles gives a linear equation which represents the common chord or the common tangent.
But what about two non-intersecting circles?
I experimented with a num... | Difference of two non intersecting circles is a line perpendicular to the line joining the centre of the two circles.
The line always lies between the two circles(near to the smaller one) and touches neither of them(for non intersecting circles).
The distance of the line from mid-point of centres is equal to "differenc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to tell if $\sum_{n=1}^\infty\frac{\ln(n)}{n^2}$ converges using Integral Test? So I have this problem:
Determine whether or not the following infinite series converge or diverge. State what test you used.
$$\sum_{n=1}^\infty\frac{\ln(n)}{n^2}$$
So I decided to do the Integral Test for $$\int^\infty_1\frac{ln(n)}... | By integral test we should obtain
$$\int_1^\infty \frac{\ln x}{x^2} dx=\left[-\frac{1+\log x}{x}\right]_1^\infty$$
or by $\ln x=u \implies \frac1x dx=du$
$$\int_0^\infty \frac{u}{e^u} du=\left[-\frac{u+1}{e^u}\right]_0^\infty$$
If you are not forced to use integral test, as an effective alternative, we can use limit co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2915959",
"timestamp": "2023-03-29T00:00:00",
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Does differentiability on $(a, b)$ implies continuity on $[a, b]$? Provided that $f$ is defined on $[a, b]$, is it true that the differentiability of $f$ on $(a, b)$ implies that $f$ is continuous on $[a, b]$ ?
I'm asking this question because I'm reading Salas's calculus and, as he goes trough some theorems, it's not ... | Take the function differentiable on $(0,1); f(x)=x$, $f(0)=f(1)=2$ it is not continuous on $[0,1]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2916085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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What is the proof of finding the LCM of numbers by prime factorization method? Fir example on the internet I found this method to find the LCM :
Let's find the LCM of 30 and 45. One way to find the least common multiple of two numbers is to first list the prime factors of each number.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
Then... | By having all the prime factors of each, one has a common multiple.
By having no additional (prime) factors, one has the least such.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2916209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Elementary divisor rings (Smith normal form) Is the ring $R={\mathbb Z}/n{\mathbb Z}$ an elementary divisor ring? Can you provide an easy proof from well-known results, or a reference to this result?
(Recall that an elementary divisor ring $R$ is one for which every matrix over $R$ is equivalent to a diagonal matrix. T... | This follows from the fact that Smith normal forms over $\Bbb{Z}$ exist.
If $\overline{A}$ is matrix over $\Bbb{Z}_n$ then we can, arbitrarily, lift the entries to integers and form a matrix $A$ with integer entries. By the existence of Smith normal forms over $\Bbb{Z}$ there exists invertible (i.e. determinant $=\pm1$... | {
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Orbit of conjugation on subgroups of $D_8$ Let $X$ be the set of all subgroups of $D_8$ with order $2$. For fixed $g\in D_8$, and for all $x\in X$, conjugation by $g$ is defined by
$$x\mapsto gxg^{-1}$$
What is the orbit of this group action? I have that $X=\{\{e,r^2\},\{e,s\},\{e,sr\},\{e,r^2s\},\{e,rs\}\}$
| Orbit is that of an element of the set $X$ on which the group is acting. For example,
$$\text{Orb}(\{e,r^2\})=g\{e,r^2\}g^{-1}=\{e,gr^2g^{-1}\}.$$
The second element $gr^2g^{-1}$ will run over all conjugates of $r^2$, thus you need the conjugacy class of $r^2$, which is just....... Can you take it from here?
Added expl... | {
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Let $p$ be an odd prime and $P(x)$ a polynomial of degree $p-2$. If $p\mid P(n)+P(n+1)+...+P(n+p-1) \forall n$, must $P$ have integer coefficients? I am stuck on this question from Zeit's book The Art and Craft of Problem Solving.
Question
Let $p$ be an odd prime and $P(x)$ a polynomial of degree at most $p-2$.
If $P(n... | The answer is no. For example, $$P(x):=\frac{x\,(x+1)\,(x+5)}{6}$$
is not in $\mathbb{Z}[x]$. However, for $p=5$, we see that $P(x)$ satisfies the required conditions. To see this, we observe that
$$\sum_{r=0}^4\,P(n+r)=5\,\left(\frac{n(n-1)(n+1)}{6}+2n^2+8n+11\right)$$
for all $n\in\mathbb{Z}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2916501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Proposition 3.6 - " Functional Analysis , Sobolev Spaces and Partial Differential Equations ",Brezis "When $E$ is finite-dimensional, the weak topology $\sigma ( E,E') $ and the usual topology are the same."
We just need to show that every strongly open set ( I think it's the same of open set in the usual topology, tha... | A set $M$ is open if for every $x_0\in M$ it contains a neighborhood $U$ of $x_0$:
$$x_0\in U\subseteq M.$$
Brezis shows that if $E$ is finite dimensional and $U$ is a neighborhood in the strong topology it contains a neighborhood $V$ in the weak topology:
$$x_0\in V\subseteq U\subseteq M$$
thus showing that every stro... | {
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"source": "stackexchange",
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Finding the limit by using the definition of derivative. Here is the problem.
Let $f$ be the function that has the value of $f(1)=1$ and $f'(1)=2$. Find the value of
$$ L = \lim_{x \to 1} {\frac{\arctan{\sqrt{f(x)}-\arctan{f(x)}}}{ \left (\arcsin{\sqrt{f(x)}}-\arcsin{f(x)}\right)^2}} $$
I have tried using
$$
L=\lim... | We shall need several times the following principle: If $f(0)=0$, $\>f'(0)=1$, and $\lim_{x\to0} g(x)=0$, then we may write
$$f\bigl(g(x)\bigr)=g(x)\>h(x)\ ,\tag{1}$$ whereby the function $x\mapsto h(x)$ defined by $(1)$ satisfies $\lim_{x\to0}h(x)=1$.
We only need that $f(x)\nearrow1$ when $x\nearrow 1$ (note that yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2916702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Roll a die probability question
An unbiased six-sided die is to be rolled five times. Suppose all these trials are independent. Let $E_1$ be the number of times the die shows a 1, 2 or 3. Let $E_2$ be the number of times the die shows a 4 or a 5. Find $P(E_1 = 2, E_2 = 1)$.
I have tried to solve this question this wa... | The probability equals:$$\binom5{2,1,2}\left(\frac36\right)^2\left(\frac26\right)^1\left(\frac16\right)^2=\frac5{72}$$(where $\binom5{2,1,2}:=\frac{5!}{2!1!2!}$)
You can look at this as a case of trinomial distribution.
There are $5$ independent experiments and there are not $2$ possible and mutually exclusive outcome... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2916745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $f : [a,b]\to\Bbb R$ is continuous, are there $x_1,x_2\in (a,b)$ such that $\tfrac{f(b)-f(a)}{b-a} = \tfrac{f(x_1)-f(x_2)}{x_1-x_2}$? I just thought about the mean value theorem and wondered whether the following statement is true:
If $f : [a,b]\to\Bbb R$ is continuous, then there are $x_1,x_2\in (a,b)$
such that... | First, without loss of generality assume that $f(a) = 0$(why can we do this?).
Next, set $h(x) = \frac{f(b)(x-a)}{b-a}$ on the interval $[a,b]$. This function is continuous. Note that $h(a) = 0$ and $h(b) = f(b)$.
Therefore, so is $g(x) = h(x) - f(x)$ on $[a,b]$. Note that $g(a) = 0$ and $g(b) = 0$.
Suppose there ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2916829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 1
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Help with Mellin-Barnes Integral (product of two Hypergeometrics) I am trying to prove that
$$\int_0^1 \frac{dz}{z^2} z^{h}\cdot {}_{2}F_{1}(h,h;2h;z) \cdot {}_{2}F_{1}\left(\frac{1+2a}{2},\frac{1-2a}{2};1;\frac{z-1}{z}\right) = -\frac{\Gamma(2h)}{\Gamma{(h)}^2} \cdot \left(\frac{1}{a^2-(h-1/2)^2} \right).$$
To begin I... | The hypergeometric function in contour form is
$${}_{2}F_{1}(a, b; c; x) = \frac{1}{2 \pi i} \, \int_{-i \infty}^{i \infty} \frac{\Gamma(-s) \Gamma(a+s) \Gamma(b+s)}{\Gamma(c + s)} \, ds.$$
The integral in question becomes:
\begin{align}
I &= \int_{0}^{1} {}_{2}F_{1}(p,p;2p;t) \cdot {}_{2}F_{1}\left(\frac{1+2a}{2},\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2916913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Hilbert space and orthonormal basis. Let $H$ be a Hilbert space and let ${e_n} ,\ n=1,2,3,\ldots$ be an orthonormal basis of $H$. Suppose $T$ is a bounded linear oprator on $H$. Then which of the following can not be true?
$$(a)\quad T(e_n)=e_1, n=1,2,3,\ldots$$
$$(b)\quad T(e_n)=e_{n+1}, n=1,2,3,\ldots$$
$$(c)\qua... | You're questions been answered, but if you're curious about examples of bounded linear operators that satisfy the last two resp. Look no further than $l^2(\mathbb{N})$ and consider the left and right shift operators on the standard basis. That is
$$T_L( (a_1, a_2, ... a_n , ... )) = (a_2, a_3, ... a_n,\ ... ) \\
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2917024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Find particular solution of $y''+4y=12$ if the point $(0,5)$ has horizontal tangent line Find the particular solution of $$y''+4y=12$$ if the point $(0,5)$ has horizontal tangent line (parallel to the $x$-axis).
I know the general solution of $y''+4y=12$ is $$y=C_1\cos{(2x)}+C_2\sin{(2x)}+3\quad\text{for some}~C_1,C_2... | The equation of the tangent line at $(0,5)$ is $$y(x) = y'(0)(x-0) + y(0) = y'(0)x + 5$$
This is parallel to the $x$-axis if and only if $y'(0) = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2917181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Onto and one-to-one linear transformations My book seems a bit unclear about onto and one to one linear transformations. Here is the excerpt:
and this is what I understand to be the rank of T:
rank(T) = dimension of column space of the original matrix.
Is the rank of T not the same thing as the dimension of W? Are th... | Yes of course for any matrix the dimension of $\operatorname{Col}(A)$ is equal the dimension of $\operatorname{Row}(A)$ and that number is by definition $\operatorname{rank}(A)$.
Therefore if $\operatorname{rank}(A)$ is equal to the dimension of $W$ it means that the set "columns of $A$" contains a basis for $W$ and th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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It is the set $V(y-\sin(x))\subset k^2$ a variety? It is the set $V(y-\sin(x))\subset k^2$ a variety?
I know that $V(y-\sin(x))=\{...,-4\pi,-3\pi,-2\pi,-\pi,0,\pi,2\pi,3\pi,4\pi,...\}$. Could it be a set of points a variety? My intuition tells me what not. How can I do a formal test, what property should I use? Thank y... | Suppose $X = V(y-\sin(x))\subset \Bbb A^2$ were a variety. Then $X$ also admits a description as $V(f_1,\cdots,f_n)$ for some finite list of nonzero polynomials $f_i(x,y)$, each of which vanishes on $V(y-\sin(x))$.
By examining $f_i(x,0)$, each single-variable polynomial $f_i(x,0)$ should vanish on $V(y-\sin(x))\cap V(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2917394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $2x^3+2x^2-10x+6$ is positive if $x>1$
Show that $2x^3+2x^2-10x+6$ is positive if $x>1$
I have tried to solve it using mean value theorem but it doesn't work. Please anyone help me to solve this.
| Alt. hint: let $\,x-1 = y \gt 0\,$, then:
$$
2x^3+2x^2-10x+6 = 2(y+1)^3+2(y+1)^2-10(y+1)+6=2 y^3 + 8y^2 \gt 0
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2917546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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RSA: Prove/disprove that $M_1^e\equiv M_2^e\ (\textrm{mod}\ n),$ given $M_1\not=M_2$ Given that $M_1\not=M_2, n=pq$ where $p,q$ are large primes and $\gcd(e,\varphi(n))=1$, how to prove/disprove that
$$M_1^e\equiv M_2^e\ (\textrm{mod}\ n)$$
is impossible? That is: I'm thinking about the case that two distinct messages ... | I posted a proof here that RSA is a bijective map from $\mathbb{Z}_n$ to itself, provided that $(\phi(N),e) =1$, and where the message space is the set of classes modulo $N$, the modulus.
So if we assume (as we must) that we are working in that message set, the RSA mapping has an inverse (using the exponent $d$ as desc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2917622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Definite integral of $x\sin^n x$ from $0$ to $\pi/2$ How to find
\begin{equation*}
\int_0^{\pi/2} x\sin^n x dx
\end{equation*}
where $n$ is a positive integer? I tried $y=x-\pi/4$ and that gives
\begin{equation*}
\frac{1}{2^{n/2}}\frac{\pi}{4}\int_{-\pi/4}^{\pi/4} (\sin y+\cos y)^n dy+\frac{1}{2^{n/2}}\int_{-\pi/4}^{\p... | Do the same thing: let the integral be $I_n$,
$$
\begin{align*}
&\phantom{=}\int_0^{\pi/2} x \sin(x)^n \mathrm dx = -\int_0^{\pi/2} x \sin(x)^{n-1}\mathrm d(\cos(x)) \\
&=\left. x \sin(x)^{n-1} \cos(x) \right|_{\pi/2}^0 - \int_{\pi/2}^0 \cos(x) (\sin(x)^{n-1} + (n-1)x \sin(x)^{n-2} \cos(x))\mathrm d x \\
&= 0 + \int_0^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2917725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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How to evaluate $ \int\left[\left(\frac{x}{e}\right)^x+\left(\frac{e}{x}\right)^x\right]\ln(x)\,dx$? $$ \int \left[ \left( \frac {x}{e}\right) ^ x + \left( \frac {e}{x}\right) ^ x \right] \ln(x)\, dx$$
Here I have no clue from where to start with.
I already tried some
*
*substitutions
*taking out ${e} ^ {x}$
But n... | We have
$\frac{d}{dx}\left(x\ (\ln x - 1)\right) = (\ln x - 1) + 1 = \ln x$, and
$\left(\frac xe\right)^x = e^{x(\ln x - 1)}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2917834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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How to show that $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
Say we have a venn diagram with A on the left, intersection C in the middle and B on the right.
I want to show that $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
I know what it can be written as A+B+C, but a lot of the proofs go from:
$P(A\cup B) = P(A) + P(B) + P... | One of the reasons that Venn Diagrams are effective in visualizing probability problems as it poses a rather difficult concept to grasp (probability) in terms of a more intuitive concept (area). So, why don't we use area as a tool to help us prove this concept informally.
Let $P(A)$ be the area enclosed by circle $A$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2917915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Min, max and saddle points of a function I want to find all local minima, maxima and saddle points of the function $f(x,y)=(x-y)(1-xy)$
Therefore I wanted to set the partial derivatives to zero to get all possible points.
First I calculated the derivatives $f_x(x,y) = y^2-2xy+1$ and $f_y(x, y) = -x^2+2xy-1$
From $f_x$ ... | You are ok expressing it as a square root...no more is needed. By the way, using taylor series is a great way of approximating functions. Or using differentials (easier):
Call $f(x)=\sqrt{x}, \space x=9 ,\space dx=1 \space:$
$$f(x+dx)\approx f(x)+f'(x)dx$$
$$\sqrt{10}\approx\sqrt{9}+\frac {1}{2\sqrt{9}}*1$$
$$\sqrt{10}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2918006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Functional equation $f(a)·f(b)=\Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$ I am trying to get all possible solutions of the following functional equation:
$$f(a)·f(b)= \Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$$
Where {} mean fractional part function.
Solutions only need to be valid inside open interval $a,b \in (0,1)$.
I am not ... | We have
$$\Biggl(f\left(\frac{1}{2}\right)\Biggr)^2=\left\{\frac{1}{\left(\frac{1}{2}\right)^2}\right\}=\{4\}=0\,,$$
so
$$f\left(\frac{1}{2}\right)=0\,.$$
Consequently, for $a\in(0,1)$, we get
$$0=f(a)\,f\left(\frac{1}{2}\right)=\left\{\frac{1}{a\cdot\left(\frac12\right)}\right\}=\left\{\frac{2}{a}\right\}\,.$$
However... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2918154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Basics of Quadratic Sieve algorithm I'm trying to understand Quadratic Sieve algorithm for integer factorization, I follow the description in the book "Prime Numbers" by Crandall and Pomerance, specifically the Algorithm 6.1.1. (Even though the question below apply to any description of QS, as far as I can see.)
Please... | $(1)$ The purpose of the roots is to accelerate the search for numbers $x$ with the property that $\ \ x^2\mod N\ \ $ has small factors. The algorithm would also work for random numbers, but the quadratic residues are a significant improvement concerning speed.
$(2)$ A number is $B$-smooth , if it has no prime factor l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2918263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that exists $u_m$ such that $\phi(u_m) = \max\{\phi(u),u\in E\}$
Let $E = \{u:[0,1]\to \mathbb R | u(0)=0, lip(u)\leq 1\}$. Define $\phi(u) = \int_0^1 (u^2(x)-u(x))dx$
Show that exists $u_m\in E$ such that $\phi(u_m) = \max\{\phi(u),u\in E\}$
I intended to show that $\{\phi(u),u\in E\}$ is compact, in order to f... | This is an easy consequence of Arzela -Ascoli Theorem. The set $E$ is equi-continuous and bounded at $0$ which implies it is uniformly bounded. Hence any sequence $\{u_n\} \subset E$ has a subsequence which converges uniformly. Can you now prove that $\{\phi (u):u\in E\}$ is closed?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2918388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Upper bound of sub-gaussian norm of bounded random variable? I am reading the High-Dimensional Probability by Dr.Roman Vershynin , where I stuck on some statement at page 28. where state as below:
Any bounded random variable $X$ is sub-gaussian with: $$\newcommand\norm[1]{\left\lVert#1\right\rVert}
\norm{X}_{\psi_2}\l... | If $t=(\sqrt {\log 2})^{-1} \|X\|_{\infty}$ then $Ee^{\frac {X^{2}} {t^{2}}} \leq e^{\log 2}= 2$ and hence $\|X\|_{\psi_2} \leq t$. I have used the fact that $\|X\|_{\infty}$ is nothing but the essential supremum of $X$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2918470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Summation With Binomial Coefficient I am wondering how to estimate the following summation.
For $p \ge 0$,
$$
\sum_{i=0}^n \binom{2n}{i} (-1)^i (n-i)^p.
$$
When $p$ is a fixed integer this seems easy to do. But what if $p$ is a general real number?
I performed some numerical simulation and it seems this summation equa... | Working with the question asked about $p$ an even integer we evaluate
$$S_{n,p} = \sum_{q=0}^{n} {2n\choose q} (-1)^q (n-q)^p.$$
Note that
$$\sum_{q=n}^{2n} {2n\choose q} (-1)^q (n-q)^p
= \sum_{q=0}^{n} {2n\choose n+q} (-1)^{n+q} (-q)^p
\\ = \sum_{q=0}^{n} {2n\choose 2n-q} (-1)^q (q-n)^p.$$
With $p$ even this is
$$\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2918611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding $f$ satisfies $\limsup_{n\to\infty}\frac{\tau(n)}{f(n)}=1$ Is there a known result on finding the function $f$ satisfies
$$\limsup_{n\to\infty}\frac{\tau(n)}{f(n)}=1?$$
where $\tau(n)$ is the number of factor(s) of $n$.
Related:Prove that $d(n)\leq 2\sqrt{n}$
It shows that $f(n)=O(\sqrt n)$.
Some Ideas
Als... | The best result in this direction is due to S. Wigert (1907):$$\limsup_{n\to\infty}\frac{\log\tau (n) \log\log(n)}{\log(2)\log(n)}=1.$$
It comes as a corollary to certain estimates of $\max_{n\le x} \tau(n)$, the strongest of which is due to S. Ramanujan (1915).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2918726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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There are infinity many numbers when added to Three distinct integers making them pair-wise relatively prime Given $0<a<b<c$ three distinct integers, prove that there exists infinity many numbers $n$ such that $a+n,b+n,c+n$ are relatively prime to each other.
For the case when $(a,b)=(b,c)=(a,c)=1$ which means they are... | Note: this does not work with $4$ numbers. For example, if you start with $(2,3,4,5)$ then whatever you add to each term, two of the set will be even. Whatever argument one constructs must fail if you try to extend it to $4$ terms.
Your remark shows that it would suffice to find a single such $n$.
Without loss of gen... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Non trivial “multiplications” on $\mathbb{R}^2$ What are all the continuous and associative binary maps $M: \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that satisfy $M(x,y+z) = M(x,y) + M(x,z)$?
Besides the 0 constant function, and the identity multiplicative map $M((x_0,x_1),(y_0,y_1)) = (x_0y_0, x_... | Let $z=a+bi$ with $b\ne0$. Then $\mathbb C=\mathbb R[z]$. In particular, $1,z$ is a basis for $\mathbb C$ over $\mathbb R$. In this basis, multiplication is given by
$$
\begin{align}
(x_1,y_1)(x_2,y_2)
&=(x_1+y_1 z)(x_2+y_2 z)
\\&=x_1 x_2 + y_1 y_2 z^2 +(x_1 y_2 +x_2 y_1)z
\\&=(x_1 x_2 - y_1 y_2(a^2+b^2))+(x_1 y_2 +x_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2919015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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prove a function is lower semicontinuous $E = \ell^p$, with $1\le p<\infty$. For $x\in\ell^p$, $x = (x_1,x_2,\dots,x_n,\dots)$, check function
$$\varphi(x) = \begin{cases}\sum_{k=1}^{\infty}k|x_k|^2 &\text{ if } \sum_{k=1}^{\infty}k|x_k|^2<\infty \\
+\infty &\text{ otherwise}\end{cases}$$
is convex, l.s.c.
I have shown... | The function $\phi$ is the supremum of the functions $\phi_n(x) = \sum_{k=1}^n k |x_k|^2$. Each $\phi_n$ is lower semicontinuous, and a supremum of l.s.c. functions is l.s.c.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2919163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
how many numbers in (not-strictly) descending order of string length k Given K numbers $n_1, \, \ldots \,, n_k$ s.t. $0 \leq n_i \leq b$
How many ways are there to write the numbers $n_1$ to $n_k$ such that they are descending. (Descending here is used in the analysis sense)
Given $k=4$ and $b=4$ we could say that:
$0... | Notice that such a string is completely determined by how many times each digit appears in the string. For instance, if we have a descending string of length $6$ consisting of three $3$s, two $2$s, and one $1$, then it must be $333221$ since there is only one way to arrange the digits in descending order.
Let's consid... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it known that $\sum_{i=1}^\infty \frac{if\;(i \pmod n=0)\;then\;(1-n)\;else\;(1)}{i}=log(n)$? I found this general infinite sum:
$\sum_{i=1}^\infty \frac{ \mathtt i \mathtt f \; (i \pmod n = 0) \; \mathtt t \mathtt h \mathtt e \mathtt n \;(1-n) \; \mathtt e \mathtt l \mathtt s \mathtt e \; (1)}{i} = log(n)$
Sample... | You have to start with partial sums. Choose some $M$ to get:
$$
S_M = \sum_{i=1}^{nM} \frac{if\;(i \pmod n=0)\;then\;(1-n)\;else\;(1)}{i}= \sum_{i=1}^{nM} \frac1i - n \sum_{i=1}^M \frac1{n i} = \sum_{i=1}^{nM} \frac1i - \sum_{i=1}^M \frac1{i}
$$
Now observe the well-known limit (see here) $\lim_{k \to \infty}\sum_{i=1... | {
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"timestamp": "2023-03-29T00:00:00",
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Going from Fourier sum to Fourier integral - confusion on intermediate step How does it solve the $A_0$ term?
Can one prove this with an example? I don't know how to prove it to myself with an example
In my lecture notes, my lecturer is trying to justify $C_n$ as weighting the sine and cosine terms in the Fourier integ... | The "trick" is in the sum from $-\infty$ to $\infty$ where $|n|$ is counted once for $|n|=0$ , and twice for $|n| \ge 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2919530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prime and rational numbers I came across the following question while studying that is stumping me. Can anyone please help me solve it?
Let "$a$" be a prime number greater than $10,000$ and let $x=\sqrt{a}$. Which of the following expressions represents a rational number?
F) $x/2$
G) $\sqrt{x}$
H) $2x$
J) $x^2$
K) $x+... | Assume $x$ is rational. Then we can express $x$ as:
$$x=\frac{p}{q}$$
where $p,q$ are co-prime integers. $x=\sqrt a$ can be written as:
$$x^2=a$$
$$\frac{p^2}{q^2}=a$$
$$p^2=aq^2$$
As the prime $a$ divides the RHS, it divides the LHS too. $p=ka$ for some integer $k$. Substituting:
$$k^2a^2=aq^2$$
$$k^2a=q^2$$
Hence, $a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2919671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many ways are there for 50 people to divide them into three groups, A, B and C such that each consists of 20, 18, and 12, respectively? I have tried to solve this problem but I can not figure out where to start. Any help would be appreciated.
Thanks,
EDIT: After another attempt I am leaning towards the answer $\fra... | As indicated in the comments, your answer is correct.
We can select $20$ of the $50$ people for group $A$ in $\binom{50}{20}$ ways, $18$ of the remaining $30$ people for group $B$ in $\binom{30}{18}$, and all $12$ of the remaining $12$ people for group $C$ in $\binom{12}{12}$ ways. Hence, the number of ways of selecti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2919782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Evaluating $\sum_{n=1}^{\infty} \frac{\phi(n)}{7^n + 1}$, where $\phi(n)$ is Euler's totient function
Evaluate $$\sum_{n=1}^{\infty} \frac{\phi(n)}{7^n + 1}$$
where $\phi(n)$ is Euler's totient function.
I found this problem on a Discord server I just joined. The first time I saw this sum, I was daunted. After gath... | Another approach is to use Lambert Series generating function:
$$
\begin{align}
\frac{q}{(1-q)^2} &=\sum_{n=1}^{\infty}\frac{\phi(n)q^n}{1-q^n}=\sum_{n=1}^{\infty}\frac{\phi(n)}{q^{-n}-1} \space\colon\space |q|\lt1\implies \\
\frac{q^2}{(1-q^2)^2} &=\sum_{n=1}^{\infty}\frac{\phi(n)}{q^{-2n}-1} =\frac12\sum_{n=1}^{\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2919929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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$K = \mathbb{Z} / 2 \mathbb{Z}$. How many subspaces does the $K$-vector space $K^2$ have?
Let $K = \mathbb{Z} / 2 \mathbb{Z}$, How many subspace does the $K$-vector space $K^2$ have?
I (hope to) already know the following
*
*When diving a whole number by 2, we can only obtain 0 or 1, so $K = \{0, 1\}$ and $K^2 = \{... | Yes, $\{0\}\times\{0,1\}$ is indeed a $K$-vector subspace. You almost completed your proof, but it has some flaws you should still fix. Remember, to prove that $U\subset V$ is a vector subspace (over some field $F$), you need to prove two things:
*
*For every pair $u_1, u_2\in U$, the element $u_1+u_2$ is also in $U... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2920045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Finding the sum of squares of roots of a quartic polynomial. What is the sum of the squares of the roots of $ x^4 - 8x^3 + 16x^2 - 11x + 5 $ ?
This question is from the 2nd qualifying round of last year's Who Wants to be a Mathematician high school competition which can be seen here:
I know the answer (32) because tha... | Hint:
\begin{align}
\sum_{i=1}^4a_i^2 &= \left(\sum_{i=1}^4a_i \right)^2-2\sum_{i< j}a_ia_j
\end{align}
Also Vieta's formula might help.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2920175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
How to calculate the derivative of $x^x$? I'm trying to follow an example in my textbook.
$$y=x^x$$
$$\ln (y)=\ln (x) \cdot x$$
We want to calculate the derivative with respect to x
The book makes quite a leap here and states that:
$$\frac{y'}{y}=\frac{1}{x}\cdot x+1\cdot \ln(x)$$
Since $y=x^x$ this means that:
$$y'=x^... | You missed out $y'$ in this line:
$$\frac {1}{y}e^{\ln y}=e^{x\ln x}(1+\ln x)$$
as $$\left(e^{\ln y}\right)'=e^{\ln y}\cdot\frac1y\cdot y'=y'$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2920262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Simplicial homology and homeomorphisms In Hatcher's book, in the introduction page of singular homology, he mentions that "it is obvious that homeomorphic spaces have the same singular homology, in contrast to simplicial homology". However I thought that this was also true for simplicial homology (and looking at the co... | Neal's answer explains the problem of different triangulations. Nevertheless it turns out that the simplicial homology $H_*(\mathcal{T})$, where $\mathcal{T}$ is a simplicial complex triangulating $X$, is a topological invariant of $X$. That is, if $X_1, X_2$ are homeomorphic and $\mathcal{T}_i$ are triangulations of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2920378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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In how many ways can ten people be seated in a row so that a certain two of them are not next to each other? What I'm thinking: Find total ways that the ten people can be seated, which is 10!.
Then I take that number and subtract the ways the these two people would be seated next to each other. I do this by treating t... | Call the two special people $A$ and $B$. There are nine seats where the "left" one can sit and the right next to him. There are two such cases $AB$ and $BA$. For each of these conditions there are 8 remaining slots that can be filled by the remaining $8$ folk arbitrarily.
Thus: $9 \cdot 2 \cdot 8! = 725,760$
The t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2920517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Construct an order on a field $\mathbb{Q}^2$ Let $\mathbb{Q}$ be the field of rational numbers. Let $\mathbb{Q}^2=\{(a,b):a,b\in\mathbb{Q}\}$ and define addition and multiplication as follows:
$$
(a,b)+(c,d)=(a+c,b+d)\\
(a,b)\cdot(c,d)=(ac+2bd, ad+bc)
$$
Then $(\mathbb{Q}^2, +, \cdot)$ is a field, and $(0,0)$ is its ze... | The quick and easy answer is to notice that this field is isomorphic to $\mathbb Q[\sqrt{2}]$ by sending $(a,b)$ to $a+b\sqrt 2$, so $(a,b)<(c,d)$ if $a+b\sqrt 2<c+d\sqrt 2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2920634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Why does an indeterminate cause a hole in a rational function and not a vertical asymptote? My intermediate Algebra textbook gave the following example and a graph of this function:
$$f(x)=\frac{2x+1}{2x^2-x-1}$$
The factored form of this is:
$$\frac{2x+1}{(2x+1)(x-1)}$$
I know that a vertical asymptote is caused wh... | Simply,which value of x makes the numerator zero of any function is called zero point and which x value makes denominator zero means for what values of x the function becomes undefined is called poles.Here,we first have to check whether the numerator is zero or not.In this case,if the numerator would not be zero means ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2920739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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String powers. Lemma 1.3.5 in word processing in groups. Epstein.
I can't understand this demonstration.
Why if ${w'}_1$ is different from ´${w'}_2$ then we have that $f(u)$ and $v'$ are powers of some string $z$?
| The way induction is used is not described in much detail. I try to understand and explain it. The base case where $|f(u)|+|f(v)| = 0$ is not even mentioned.
For the induction step, instead of the alphabet $\{u,v\}$ the alphabet $\{u,v'\}$ is used, supposedly with the map $f'(u) = f(u); f'(v') = v'$. Since $f(u)v' = f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2920900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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No analytic function $f$ has modulus $|f(z)|=1/\cosh(\Re z)$ An analytic function $f(z) = f(x+iy)$ in $\mathbb{C}$ cannot have modulus $\frac{A}{\cosh(x)}$ for some constant $A \neq 0$.
Can we do so simply using the Cauchy-Riemann Equations?
I tried working by contradiction:
Say there is such a $f(z)$. Given $f(z) = u(... | So
$$
\begin{align*}
uu_x+vv_x&=-A^2\frac{\sinh x}{\cosh^3 x}\\
-uv_x+vu_x&=0\\
\end{align*}
$$
So
$$
u_x=-u\frac{\sinh x}{\cosh x}
$$
(remember $u^2+v^2=\dfrac{A^2}{\cosh^2 x}$) and similarly $v_x=-v\dfrac{\sinh x}{\cosh x}$. So
$$
u=\frac{1}{\cosh x}\cdot f(y)\text{ and }v=\frac{1}{\cosh x}\cdot g(y)
$$
and there ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2921051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
definite integral $\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx$
$$\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx$$
I tried $2$ or $3$ trigonometric transformations but it did not work. One of them is as follows
$$\frac{\sin^2x\cos^2x}{(\sin x+\cos x)(1-\sin x\cos x)}$$ after ... | Using identities
$$\sin x+\cos x=\sqrt{2}\cos(x-\dfrac{\pi}{4})$$
$$\sin x\cos x=\cos^2(x-\dfrac{\pi}{4})-\dfrac12$$
and then substitution $\dfrac{\pi}{4}-x=u$ gives
\begin{align}
I
&= \int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx \\
&= \int_{0}^{\frac{\pi}{4}}\frac{\sin^2x\cos^2x}{(\sin x+\cos x)(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2921300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Two-player game in $19$ rounds
Aashna and Radhika see the integers $1$ to $211$ written on a blackboard. They alternate turns and in every step each of them wipes out any $11$ numbers until only $2$ numbers are left on the blackboard. If the difference of these $2$ numbers (by subtracting the smaller from the larger) ... | If you chose to play first you would have 10 turns. -> 10 x 11 = 110
If you chose to play second you would have 9 turns. -> 9 x 11 = 99
If you are second and remove every high number starting with 211 while the first removes the low numbers you would stop at 211 - 99 = 112.
The last left numbers would be 112 and one n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2921390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Rank of a Hermitian matrix in terms of Eigen values? I have a complex Hermitian matrix, say W, which is obtained by solving a convex optimization problem. In order for this matrix to be the result of my original problem, W must satisfy the following condition rank ( W ) = 1. When I checked this condition, MATLAB gives ... | If the matrix has three non-zero eigenvalues, then its rank is $3$. The rank of a $3 \times 3$ matrix is $3$ minus the dimension of the eigenspace corresponding to $0$ (see here for more detail).
However, if you look at the three eigenvalues produced, you'll notice that two of them are very small. I'd wager that they a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this Riemann sum formula for definite integral using of prime numbers true? While answering another question in MSE, I had used the following result which I thought was a trivial consequence of the prime number theorem and equidistribution. However, I realized from the comments that many people thought that this w... | Too long for a comment
Divide $[0,1]$ into $$[0,p_1/p_n],[p_1/p_n,p_2/p_n],\cdots,[p_{n-1}/p_n,1]$$
Then, using Riemann sum, we have
$$I:=\int^1_0f(x)dx=\lim_{n\to\infty}\sum^n_{k=1}f\left(\frac{p_k}{p_n}\right)\frac{p_{k+1}-p_k}{p_n}$$
If we assume that $p_j=j\ln j$,
$$I=\lim_{n\to\infty}\sum^n_{k=1}f\left(\frac{p_k}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2921560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 0
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How to prove that $\text{null} \;T_1 \subset \text{null} \;T_2$ implies $\text{dim(range}\; T_1) \geq \text{dim(range}\ T_2)$?
If $W$ is finite dimensional, $T_1, T_2 \in L(V,W)$ and $\text{null}\; T_1 \subset \text{null}\; T_2$ then $$\text{dim}( \text{range}\; T_1) \geq \text{dim}(\text{range}\; T_2)$$
I'm solving ... | Since you can prove it for $V$ finite-dimensional, the easiest way out is to look at the induced linear maps on the quotient $V/\ker T_1\to W$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2921674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inequality for the maximum of the absolute value of two normal distributed random-variables I would like to show following statement:
For $M\geq 2,\ X_1,\dots,X_M\sim^{iid}\mathcal{N}(0,1)$ independent, it holds $P(\max_{i=1,\dots,M}\lvert X_i\rvert\geq y)\leq Me^{-y^2/2}$.
I think it is possible to show it via indu... | Direct proof: Let $G(y)=1-4\Phi(y)(1-\Phi(y))=1+4\Phi(y)^2-4\Phi(y)$, the distribution function for $max(|X_1|,|X_2|)$. The density function $g(y)=8\Phi(y)\phi(y)-4\phi(y)$. Now $\Phi(y)=\frac{1}{2}+\frac{1}{\sqrt{2\pi}}\int_0^y e^{-\frac{u^3}{2}}du$, so $g(y)=\frac{4}{\pi}e^{-\frac{y^2}{2}}\int_0^y e^{-\frac{u^2}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2921798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I find the residue of this removable singularity? We have the following function : $$f(z)=\frac{z^2}{1-\cos z}$$ where $z_0=0$ is a removable singularity since the limit as $z$ goes to $0$ is $2$.
In such cases, in order to find the residue I proceed by trying to find the Laurent series around the singularity ... | Mark's answer is, of course, the best to approach this. Supposing it were not a removable singularity, then what I'll show will still sometimes work. In specific, you'll get the Laurent Series.
Recall Geometric Series.
$$ \frac{x^2}{1-\cos{x}} = x^2 (1+\cos(x)+\cos(x)^2+\dots) = x^2 \sum_{n=0}^\infty \left( \sum_{k=0}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2921927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Expectation of product of more than two independent random variables I am trying to determine whether independence of random variables changes when multiplied with other, potentially dependent variables. The question isn't in a measure-theoretic context.
In particular, I have 3 random variables, which we can call A, B,... | Let's assume you have discrete random variables so that we do not deal with integrals. First use the independence assumption:
$$
\mathbb E(ABC)=\sum_{a,b,c} abc P(A=a,B=b,C=c)=\sum_{a,b,c} abc P(A=a)P(B=b,C=c).
$$
And then use the following identity to get the result:
$$
\sum_{a,b,c} abc P(A=a)P(B=b,C=c)=\sum_{a} a P(A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2922051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Conditional Probability Problem (3 Cards, 1 red-red, 1 red-black, 1 black-black) Our class got into a heated debate over a question that even my professor couldn't answer with certainty. So here's the problem:
"Of three cards, one is painted red on both sides; one is painted black on both sides; and one is painted red ... |
My argument is that when you uncover the red card, the black-black card is no longer considered in the sample space.
It eliminates more than that.
The card is selected (1 from 3), and a side from that card is shown (1 from 2), without bias. So this experiment may be represented using a sample space consisting of s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2922227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Sum of series $\frac{ 4}{10}+\frac{4\cdot 7}{10\cdot 20}+\frac{4\cdot 7 \cdot 10}{10\cdot 20 \cdot 30}+\cdots \cdots$
Sum of the series
$$\frac {4}{10}+\frac {4\cdot7}{10\cdot20}+ \frac {4\cdot7\cdot10}{10\cdot20\cdot30}+\cdots $$
Sum of series $\frac {4}{10}+\frac {4\cdot7}{10\cdot20}+ \frac {4\cdot7\cdot10}{10\cdot... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2922492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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Continuous group actions and transitivity Let $G$ be a topological group and $M$ a topological space. Suppose that $$\circ: M\times G\longrightarrow M$$ is a continuous group action of $G$ on $M$. If $\circ$ is transitive, then we can take an arbitrary $m\in M$ and the set $\mathfrak{H}$ of all cosets of $C_G(m)$ in $G... | This is not true unless you impose further assumptions: both spaces should be locally compact and Hausdorff and $G$ should be second countable. You can find a proof of this fact in the online notes by Paul Garret about 2-Solenoids. See also this related question (where I found the link to the previous notes when I had ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2922631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Any rectangular shape on a calculator numpad when divided by 11 gives an integer. Why? I have come across this fact a very long time ago, but I still can't tell why is this happening.
Given the standard calculators numpad:
7 8 9
4 5 6
1 2 3
if you dial any rectangular shape, going only in right angles and each shape c... | Start at the degenerate rectangle 1111, a multiple of 11.
Each time you move a side of your rectangle by one step in one of the 4 directions (leaving the other side in place), you add or remove one of these numbers (leading zeroes added for clarity):
*
*0011 (horizontally) or 0033 (vertically)
*0110 (horz.) or 0330... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2922749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "86",
"answer_count": 6,
"answer_id": 2
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Prove formula $\operatorname{arctanh} x = \frac12\,\ln \left(\frac{1+x}{1-x}\right)$ Problem
Prove formula $\operatorname{arctanh} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$
Attempt to solve
To start off with definition of functions $\sinh(x)$ and $\cosh(x)$
$$ \cosh(x)=\frac{e^x+e^{-x}}{2} $$
$$ \sinh(x) = \f... | Just to add to the responses already here. You can also show this by solving this integral two different ways, one with trigonometric substitutions and the other with partial fraction decomposition.
$$ \int \frac {dx}{x^2+1} = \int \frac {dx}{x^2+1}$$
$\qquad x \mapsto \tan u$
$$ \int\frac{dx}{(1+ix)(1-ix)}= \int \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2922852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Show that $\phi : H\to HN/N$ defined by $\phi(h)=hN$ is injective. I am trying to show the following:
Show that $\phi : H\to HN/N$ defined by $\phi(h)=hN$ is injective.
Note that $H\leq G$ and $N$ is normal in $G$.
My attempt so far: Let $\phi(h_{1})=\phi(h_{2})$. Then $h_{1}N=h_{2}N$ or there exists an $n\in N$ such... | It's not true. $\phi$ is a homomorphism and you have $h\in Ker(\phi)$ $\iff$ $hN=N$ $\iff$ $h\in H\cap N$. So the kernel is trivial only when $H\cap N=\{e\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2922999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Using Theta Notation $\Theta$ how can I prove that $(13n + 3)(9n + 1)(\log(4n^2 + 100))$ is an element of $\Theta(n^2 \log n)$ I'm having a hard time figuring out what systematic approach I need to follow to solve questions like these :/ Do I expand and use the Max() principle to show that it reduces to the RHS?
Please... | So you would like to show
$$f(n) = \Theta(n^2 \log n)$$
This means that we have to find $c_1,c_2,n_0$ for every $n \geq n_0$, we have
\begin{equation}
c_1 n^2 \log n \leq f(n) \leq c_2 n^2 \log n
\end{equation}
where
\begin{equation}
f(n) = (13n + 3)(9n + 1)(\log(4n^2 + 100))
\end{equation}
Notice that you can writ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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uniform convergence of $f_n = n \chi_{[1/n, 2/n]}$
$f_n = n \chi_{[1/n, 2/n]}$. Show that $f_n$ converge uniformly to $0$.
$f_n = n$ if $1/n \le x \le 2/n$, and $0$ otherwise.
But, I am a little bit confused here.
When $n$ goes to infinity,
$f_n \to \infty$ if $0\le x\le 0 \implies x=0$?
How can we deal with thi... | The given sequence of functions converges pointwise and almost uniformly but NOT uniformly to $f \equiv 0$ on $\mathbb{R}$.
Notice that if $x \in \mathbb{R} \setminus (0,2]$ then $f_n(x)=0$ for every $n \in \mathbb N$.
Suppose $x \in (0,2]$. By the Archimedean Property there is a positive integer $N$ so that $Nx>2$. H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2923333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $f(x) = 2x + 3$ and $g(x) = ax + b$. If $f(g(x)) = x$ for all $x$, determine $a$ and $b$ This is what I know:
$f(x) = 2x + 3$
$g(x) = ax+b$
$f(g(x)) = 2(ax+b) + 3 = x$
I am just not too sure what to do next to find $a$ and $b$.
| As mentioned in the comments, we have $f(g(x)) = 2ax + 2b + 3 = x$. This is only true if $2a = 1$ and $2b + 3 = 0$, otherwise the equality wouldn't hold. Solving those, we get that $a = \dfrac{1}{2}$ and $b = -\dfrac{3}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2923453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Generating Function Sum and Combinotorics Problem:
The sum $\sum_{n=2}^{\infty} \frac{\binom n2}{4^n} ~~=~~ \frac{\binom 22}{16}+\frac{\binom 32}{64}+\frac{\binom 42}{256}+\cdots$ has a finite value. Determine that value.
I am quite stuck on how to do this. Can somebody give me <only> a hint or hints to get going?
Th... | Consider
$$S=\sum_{n=2}^\infty \binom{n}{2}x^n =\frac 12 \sum_{n=2}^\infty n(n+1) x^n=\frac 12 \sum_{n=2}^\infty[n(n-1)+2n] x^n$$
$$S=\frac 12 \sum_{n=2}^\infty n(n-1) x^n+ \sum_{n=2}^\infty n x^n=\frac {x^2}2 \sum_{n=2}^\infty n(n-1) x^{n-2}+x\sum_{n=2}^\infty n x^{n-1}$$
$$S=\frac {x^2}2 \left(\sum_{n=2}^\infty x^n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2923567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to show example where convergence in $L^1$ norm does not hold for continuous functions of the random variables? Suppose a sequence of random variables $X_n$ converges to a random variable $X$ in $L^1$ norm, and that $g: \mathbb{R}\rightarrow\mathbb{R}$ is a continuous function.
It is not necessarily true that $g(X... | Let $x_n = n 1_{[0,{1 \over n^2}]}$ and $g(t) = t^2$.
Then $\|x_n-0\|_1 \to 0$ but $\|g\circ x_n - g \circ 0 \|_1 = 1$ for all $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2923687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find all the positive integers k for which $7 \times 2^k+1$ is a perfect square Find all the positive integers $k$ for which $7 \times 2^k+1$ is a perfect square.
The only value of $k$ I can find is $5$. I am not sure how to find every single one or the proof, I simply used trial and error.
| \begin{align}
a^2 &= 7*2^k + 1 \\
\Rightarrow (a^2 - 1) = (a - 1)(a+1) &= 7*2^k \\
\text{For } k > 2: (\frac{a-1 }{2}) (\frac{a+1}{2}) &= 7*2^{k-2} \\
\text{Let } d = (\frac{a-1 }{2}) &\Rightarrow d + 1 = (\frac{a+1}{2}).\\
\Rightarrow d(d+1) &= 7*2^n (n = k - 2\in \mathbb{N})\\
gcd(d,d+1) = 1 &\Rightarrow (d,d+1) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2923880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Is the substitution of variables allowed in proofs? EDIT: I later realized no one saw that substituting $\sqrt{x} = x$ is not valid because they are not equal, you would have to assume $x$ as a completely different variable (i.e. $a$) and not the same one because they have no relation to each other. However, the solut... | HINT
We have that
$$2y\le \frac{y^2}{x^2}+x^2 \iff2yx^2\le y^2+x^4 \iff x^4-2yx^2+y^2\ge 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2924017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Rank of Matrix Determination $X=(I+ab^T)A(I+ba^T)$;
$A$ is symmetric and positive definite matrix of $n \times n$. $I$ is the Identity matrix of $n \times n$.
$a$ and $b$ are vectors of $n \times 1$.
$a.b \neq -1$ and $a.b \neq 0$
$a$ is not parallel to $Ab$
How do we show that $X-A$ is a rank $2$ matrix ?
Efforts:
$$... | $(X-A)$ is spanned by two vectors only
We can solve this using projector matrices, i.e. matrices of the form
\begin{equation}
P = Z(Z^TZ)^{-1}Z^T
\end{equation}
The number of columns of $X$ will determine the rank of the matrix $X -A $. If we stack in $Z$,
\begin{equation}
Z =
\begin{bmatrix}
a & Ab
\end{bmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2924127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Union of an increasing sequence on connected sets Let $(A_i)$ be a family of strictly increasing connected sets, then I have to prove that $A=\bigcup_{i\in I}A_i$ is connected.
By contradiction, I assume that $A$ is not connected. Then there exist two open, non empty and disjoint sets $U$ and $V$ such that $ A= U\cup V... | @Ashish's comment I think makes it easier to prove. If you have $$ \forall i \in I, [A_i \subset U \text{ or } A_i \subset V]$$ and we know that $\exists i , A_i \subset U$ and $\exists j, A_j \subset V$ (because otherwise $U$ or $V$ would be empty), then your sets can't be increasing $A_i \not\subset A_j$ and $A_j \no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2924403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Trouble understanding how the Transfer Principle is applied for the Extreme Value theorem. I am reading Keisler's Elementary Calculus (which can be downloaded here). I am having trouble understanding his proof sketch of Extreme Value Theorem and how he is applying the Transfer Principle.
For reference, he defines the "... | Let me propose a different interpretation of transfer (in the formulation suggested by m_t_) for the Intermediate Value Theorem.
Consider the following "standard" argument about ordinary real numbers. For all $n \in \mathbb{N}$, it is possible to partition $[a,b]$ into $a, a+\frac{b-a}{n}, \ldots, a+n\frac{b-a}{n}=b$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2924513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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Introductory recursion Proof. Suppose $f(n)$ is the number of strings of length $n$ with symbols from the
alphabet $\{a,b,c,d\}$ with an even number of $a$’s.
(b) Show that $f$ satisfies the recurrence
$f(n+1)=2f(n)+4^n$
My approach was to first to assume the case holds for $n=k$ and then consider the case for $n=k+1$(... | Your approach is quite right, and I believe its level of rigor fits the question.
As you correctly said, considering all possible strings of length $n+1$ with that property yields a partition in two sets: chopping off the last letter, we can either end up with a string of length $n$ which already has an even number of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2924614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Differentiating the sum of increasing nonnegative functions Let $f_n:[0,1] \to \mathbb R_{+}$ be a sequence of nondecreasing functions such that $f_n(0) = 0$ and $\sum_{n=1}^{\infty} f_n(1) \leq \infty$. Show that the sum can be differentiated term by term almost everywhere, i.e. $f' = \sum^{\infty}_{n=1} f'$.
It appea... | I have myself a different solution to the same problem.
Since each $F_n$ is increasing, $F$ is increasing (on $[0, 1]$). Furthermore, this means $F'$ exists on $(0, 1)$. Since we're only asked for a derivative formula to hold a.e. I'm going to discard the endpoints $0$ and $1$ now to avoid technicalities with one-sided... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2924707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Denseness of a preimage. Continous and surjective function Let $F:\mathbb{R^2} \to \mathbb{R}$ surjective and continuous on $\mathbb{R^2} $.
Let $A\subset \mathbb{R}$, where $A$ is a dense set on $\mathbb{R}$.
Is $F^{-1}(A)$ a dense set on $\mathbb{R^2}$ ?
| Let $F(x,y)=xy$ if $x>0$ and $F(x,y)=0$ if $x \leq 0$. Let $A$ be any dense set in $\mathbb R$ which does not contain $0$, say non-zero rationals. Then $F^{-1}(A)$ does not intersect $\{(x,y): x \leq 0\}$ so it is not dense.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2924855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that if in a tetrahedron if two pairs of opposite edges are perpendicular then the third pair is also perpendicular.
Prove that if in a tetrahedron if two pairs of opposite edges are
perpendicular then the third pair is also perpendicular.
Method:
$\vec a+ \vec b + \vec c + \vec d +\vec e + \vec f = 0 \tag0... | Let the tetrahedron $T$ be spanned from $0$ by the three vectors $a$, $b$, $c$. The three equations
$$\eqalign{v+w&=a\cr u\!\qquad+w&=b\cr u+v\qquad &=c\cr}\tag{1}$$
determine three linearly independent vectors $u$, $v$, $w$. From $(1)$ it then follows that the vertices of $T$ are vertices of the parallelepiped $P$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2924988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
connected set of sum of upper semi continuous function Let $C(X)$:space of continuous functions on a compact space.
Consider $f$ and $g :C(X)\rightarrow \mathbb{R}$ are upper semi continuous.
suppose for every $T\in C(X)$ set of $f(T)$ and set $(f+g)(T)$ are closed interval(connected set).Can we say set of $g(T)$ is ... | The answer is no. First ask this question for functions from $\mathbb{R}$ to $\mathbb{R}$. There are easy counterexamples: take $f(t) = t$ and let $g$ be the characteristic function of the interval $(-\infty,0)$. Then the ranges of $f$ and $f + g$ are both $\mathbb{R}$ but the range of $g$ is not connected.
You can tur... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find $n$ elements of $\mathbb{R_{\mathbb{Q}(\sqrt[3]{2})}}$ linearly independent We know that $\mathbb{R}$ is a linear space over $\mathbb{Q}$, denoted by $\mathbb{R_{\mathbb{Q}}}$,where the vector addition is the real numbers and scalar multiplication is the number in $\mathbb{Q}$ times the number in $\mathbb{R}$.
The... | Here’s a way of looking at your problem that makes use of ramification theory. It may be too advanced for your taste, but at least it does provide an answer to your question.
You’re looking for many real quantities, preferably algebraic, that will be linearly independent over the real field $\Bbb Q\left(2^{1/3}\right)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If a fair die is rolled 3 times, what are the odds of getting an even number on each of the first 2 rolls, and an odd number on the third roll? If a fair die is rolled 3 times, what are the odds of getting an even number on each of the first 2 rolls, and an odd number on the third roll?
I think the permutations formula... | Assuming the die is just marked even and odd rather than with numbers, there are eight orderings. They range from all odd to all even: OOO, OOE, OEO, EOO, OEE, EOE, EEO, EEE. In your formula:
$$\frac{n!}{(n-r)!}$$
You are missing that you don't care about the orders of the duplicates. So you need
$$\frac{n!}{(n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Inheriting complex structure from a covering space (Griffiths and Harris) On page 16 of Griffiths and Harris' Principles of algebraic geometry, they write
In general, if $\pi: M\to N$ is a topological covering space and $N$ is a complex manifold, then $\pi$ gives $M$ the structure of a complex manifold as well; if $M$... | Let $X$ be a compact connected Riemann surface such that $\chi(X)$ is negative and divisible by $3$. Then $X$ admits an irregular topological covering $\pi: X\to Y$, where $Y$ is another Riemann surface. On the other hand, by the dimension count, a generic Riemann surface of hyperbolic type does not admit a nontrivial... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
A Probability Problem About Seating Arrangements I'm having trouble with a probability question. It goes, "a class room has three rows of 4 seats. These 12 seats are randomly assigned to 4 male students and 5 female students. a)What is the probability that no female student sits in the front row?"
What I've tried to d... | The correct answer is:$$\frac{\binom85\binom40}{\binom{12}5}=\frac{\binom85}{\binom{12}5}$$
You only have to focus on females and this is the probability that $5$ chairs are selected for them that do not belong to the front row.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Lagrange inversion formula example unclear The following example is from De Bruijn's Asymptotic methods in analysis (page 24). The considered equation is
$x^t = e^{-x}$
The author wants to transform the equation into the form: $w=z/f(z)$, in order to use the Lagrange inversion formula. So he sets $x=1+z$ and $t^{-1}=w... | It's a removable singularity; we commonly identify functions with removable singularities with the function which agrees with them on the original domain and with the singularities removed. It's a removable singularity because the numerator and denominator are analytic in a neighborhood of $z=0$ and both scale as $O(z)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Difference between $\operatorname{Var}(Y)$ and $\operatorname{Var}(Y\mid X)$? What is the difference between $\mathrm{var}(Y)$ and $\mathrm{var}(Y\mid X)$? If $Y = c + \beta X$ and $\operatorname{var}(X)=\sigma^2$, won't both come out to be the same, i.e., $\beta^2\sigma^2$?
| Note that we always have $E[f(X)|X] = f(X)$. Loosely speaking, given $X$ there is no randomness in $f(X)$, so we expect the conditional variance to be zero.
Let $f(x) = c+ \beta x$.
\begin{eqnarray}
\operatorname{var} (f(X)|X) &=& E [ (f(X)-E[f(X)|X])^2 | X] \\
&=& E [ (f(X)-f(X))^2 | X] \\
&=& E[0 | X] \\
&=& 0
\end{e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Finding the minimum or maximum of a bivariate function when $f_{xx}\times f_{yy}-f_{xy}^2=0$. I see that when $f_{xx}\times f_{yy}-f_{xy}^2<0$ then it is a saddle point.
Also when $f_{xx}\times f_{yy}-f_{xy}^2>0$ then it is a minima or maxima.
What is exactly happening when $f_{xx}\times f_{yy}-f_{xy}^2=0$?
| For a regular $f(x,y)$ we have around a point $p_0 = (x_0,y_0)$ with $p = (x,y)$
$$
f(x,y) = f(x_0,y_0) + f_x(p_0)(x-x_0)+f_y(p_0)(y-y_0) + \frac 12(p-p_0)^{\top}J(p_0)(p-p_0)+O(|p-p_0|^3)
$$
If at $p_0$ we have a relative minimum/maximum then $f_x(p_0) = f_y(p_0) = 0$ so the characterization is done with the help of
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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