Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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How to show that: if $n\ln\left(1+a/n\right)\geqslant k\ln\left(1+a/k\right)$ then $n\geqslant k$? Let $a>0$ and $n,k$ positive integers.
If
$$n\ln\left(1+a/n\right)\geqslant k\ln\left(1+a/k\right),$$
then
$$n\geqslant k.$$
I tried by contrapositive by I do not get much. If $n<k$ then I would have
$$\ln\left(1+a/n\ri... | Let
$f(x)
=x \ln(1+a/x)
$.
$\begin{array}\\
f'(x)
&=\ln(1+a/x)+x(\ln(1+a/x))'\\
&=\ln(1+a/x)+x\frac{(1+a/x)'}{1+a/x}\\
&=\ln((x+a)/x)+x\frac{-a/x^2}{1+a/x}\\
&=-\ln(x/(x+a))-\frac{a}{x+a}\\
&=-\ln(1-a/(x+a))-\frac{a}{x+a}\\
&\gt a/(x+a)-\frac{a}{x+a}
\qquad\text{since }-\ln(1-z) > z \text{ for }z > 0\\
&= 0\\
\end{arra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1687467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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least squares problem SVD consider the least squares problem $$\min_{x\in \mathbb{R}^n} \|Ax - b \|_2^2 + \|Lx \|_2^2, L \in \mathbb{R}^{n\times n}.$$ I am asked to show that the solution of this least squares problem is the same as the solution to
$$(A^TA + L^TL)x = A^Tb$$
My attempt: for the least squares problem $$\... | Let $\tilde{A}= \begin{bmatrix} A \\ L\end{bmatrix}$,
$\tilde{b}= \begin{bmatrix} b \\ 0 \end{bmatrix}$, then the problem reduces
to $\min {1 \over 2} \| \tilde{A} x - \tilde{b} \|$ for which you know the
necessary & sufficient condition for a minimum to be
$\tilde{A}^T (\tilde{A} x- \tilde{b}) = 0$.
Since $\tilde{A}^T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1687549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Prove cosh(x) and sinh(x) are continuous. I failed this task at my univiersity and i do not understand why. No feedback was given. I have to prove that cosh(x) and sinh(x) are continious. I proved it for cosh(x) and said the same principles could be applied to sinh(x). Here is my argument:
$cosh(x) = \frac{e^x + e^{-x}... | Your reasoning looks good, except when it comes to $e^{-x}$. True, that dividing $1$ by $e^{x}$ is still continuous, but why? The reason is that $e^{x}\neq 0$ for all $x\in\mathbb{R}$, and hence $e^{-x}=\frac{1}{e^{x}}$ is continuous as well since $e^{x}$ is.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1687815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove: Let $f$ be continuous on $[a,b]$, then suppose $f(x) \geq 0$ for all $x \in [a,b]$ Prove: Let $f$ be continuous on $[a,b]$, then suppose $f(x) \geq 0$ for all $x \in [a,b]$. if there exists a point $c \in [a,b]$, such that $f(c) > 0. then $$\int_{a}^{b} f(x) dx>0$.
This is what i have so far,
since $f(x)$ is con... | First prove that $f(x) \geqslant 0$ implies $\displaystyle \int_a^bf(x) \, dx \geqslant 0$.
This follows because for any partition $Q$ and lower Darboux sum $L(Q,f)$
$$0 \leqslant L(Q,f) \leqslant \sup_{P} L(P,f) = \int_a^bf(x) \, dx.$$
As you observed, if there is at least one point $c \in [a,b]$ where $f$ is contin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1687911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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On the solution of Volterra integral equation I got stuck with some strange point, solving Volterra integral equation:
$$
\int_0^t (t-s)f(s) ds =\sqrt{t}.
$$
The solution can be obtained by ssuccessive differnetiation
$$
\int_0^t f(s)ds=\frac{1}{2\sqrt{t}}, \quad \mbox{and then}
$$
$$
f(t)=-\frac{1}{4t\sqrt{t}}
$$
But... | Notice that integration by part gives $$\int_{0}^{t}(t-s)\,f(s) \, ds = \left[(t-s)\int_{0}^{s}f(u)du \right]_{s=0}^{s=t} +\int_{0}^{t}\int_{0}^{s}f(u)\,du\,ds = V^{2}(f)(t)$$
where $V^{2}(f)$ is understood as the composition of Volterra operatorer $$V(f)(t):= \int_{0}^{t}f(u) \, du$$ with itself.
Now for the Volterra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1688018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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If $p$ is a prime number in $Z$, how do you show $\langle p^n \rangle$ is a primary ideal in $Z$ Suppose $ab \in \langle p^n \rangle = I$. How do you show either $a \in I$ or $b^m\in I$. It has been some time since I've studied this and would appreciate if someone can help me recall how the usual argument goes.
Edit: ... | The statement $ab\in \langle p^n\rangle$ means that $p^n$ divides $ab$. So $p|ab$. So if $p \nmid a$, then $p$ must divide $b$, i.e. $b^n \in \langle p^n\rangle$. So $\langle p^n\rangle$ is primary!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1688106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find no of nuts and raisins. Grandmother made 20 gingerbread biscuits for her grandchildren. She decorated them
with raisins and nuts. First she decorated 15 cakes with raisins and then 15 cakes with nuts.
At least how many cakes were decorated both with raisins and nuts?
| You have $20$ cakes in total, if she decorated $15$ cakes with raisins the least possible number of cakes decorated with nuts and raisins is to decorate first the cakes that have no raisins and then decorate those that already have raisins. So the minimum number of cakes with both raisins and nuts is:
$$15 - 5 = 10$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1688268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Basic probability : the frog riddle - what are the chances? A few days ago I was watching this video The frog riddle and I have been thinking a lot about this riddle.
In this riddle you are poisoned and need to lick a female frog to survive.
There are 2 frogs behind you and basically, you have to find what are your cha... | Since you can lick both frogs the order in which we place the frogs are irrelevant. there are only two possibilities FM and MM. FF being eliminated. The chances are 50%. Knowing which one is the male, saves you one lick but the probability for the other one being a female is still 50%. (Same situation for the boy-g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1688332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
How to think about negative infinity in this limit $\lim_{x \to -\infty} \sqrt{x^2 + 3x} - \sqrt{x^2 + 1}$
Question:
calculate:
$$\lim_{x \to -\infty} \sqrt{x^2 + 3x} - \sqrt{x^2 + 1}$$
Attempt at a solution:
This can be written as:
$$\lim_{x \to -\infty} \frac{3 + \frac{1}{x}}{\sqrt{1 + \frac{3}{x}} + \sqrt{1 + \fra... | The reason your sign has changed from what it should be, is you illegally pulled something out of the square roots on the denominator.
$$\sqrt{a^2b}=|a|\sqrt{b}$$
the absolute value sign being essential.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1688435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Find $\lim_{x \to 0} x \cdot \sin{\frac{1}{x}}\cos{\frac{1}{x}}$ $$\lim_{x \to 0} x \cdot \sin{\frac{1}{x}}\cos{\frac{1}{x}}$$
I don't solve this kind of limits, I can't try anything because it seems difficult to me.
| Let $\epsilon>0$ be given. Now consider $ |x\sin \frac{1}{x}\cos\frac{1}{x}|\leq|x|<\delta=\epsilon$, since sin and cos functions are bounded. So, the required limit is equal to 0.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1688508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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what is the shortest distance between a parabola and the circle? what is the shortest distance between the parabola and the circle?
the equation of parabola is $$y^2=4ax$$
and the equation of circle is $$x^2+y^2-24y+81=0$$
if you can show graphically it will be more helpful!! thanks
| HINT...find the general equation of the normal to the parabola at the point $P(at^2, 2at)$ and find the value of $t$ for which this normal passes through the centre of the circle. Then you can find the closest point on the parabola (with this value of $t$), and the rest is just considering distances and the radius of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1688639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integral $\int \sqrt{\frac{x}{2-x}}dx$ $$\int \sqrt{\frac{x}{2-x}}dx$$
can be written as:
$$\int x^{\frac{1}{2}}(2-x)^{\frac{-1}{2}}dx.$$
there is a formula that says that if we have the integral of the following type:
$$\int x^m(a+bx^n)^p dx,$$
then:
*
*If $p \in \mathbb{Z}$ we simply use binomial expansion, other... | Let me try do derive that antiderivative. You computed:
$$f(x)=\underbrace{-2\arcsin\sqrt{\frac{2-x}{2}}}_{f_1(x)}\underbrace{-\sqrt{2x-x^2}}_{f_2(x)}.$$
The easiest term is clearly $f_2$:
$$f_2'(x)=-\frac{1}{2\sqrt{2x-x^2}}\frac{d}{dx}(2x-x^2)=\frac{x-1}{\sqrt{2x-x^2}}.$$
Now the messier term. Recall that $\frac{d}{dx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1688762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Extending Functions in Sobolev Spaces If $U\subset W$ then every function in $L^p (U)$ can be extended to a function in $L^p (W)$, for example by setting it to be 0 outside of $U$.
However, not every continuous or differentiable function on $U$ can be extended to a continuous or differentiable function on $W$. For e... | This depends on what $U$ is; the term Sobolev extension domain was introduced for such $U$. To see why this matters, take $U$ to be the unit disk in $\mathbb{R}^2$ with a radial slit. Then a function that has different boundary limits on two sides of the slit cannot be extended to a $W^{1,1}$ function on $\mathbb{R}^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges? My friend gave me this puzzle:
What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges?
... | I am just passing by here. Perhaps the idea is to have something elegant, but this is just straightforward: an integral in polar coordinates with the help of Mathematica. I reuse the picture of the solution of Zubin. Let J be the midpoint of CD. By symmetry, we can restrict the analysis to CFJ. Without loss of generali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1688936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "129",
"answer_count": 4,
"answer_id": 3
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What's the formula for this series for $\pi$? These continued fractions for $\pi$ were given here,
$$\small
\pi = \cfrac{4} {1+\cfrac{1^2} {2+\cfrac{3^2} {2+\cfrac{5^2} {2+\ddots}}}}
= \sum_{n=0}^\infty \frac{4(-1)^n}{2n+1}
= \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots\tag1
$$
$$\small
\pi = 3 + \cf... | The third one should be obtained from $4.1.40$ in A&S p.68 using $z:=ix$ (from Euler I think not sure) :
$$-2\,i\,\log\frac{1+ix}{1-ix} = \cfrac{4x} {1+\cfrac{(1x)^2} {3+\cfrac{(2x)^2} {5+\cfrac{(3x)^2} {7+\ddots}}}}
$$
Except that the expansion of the function at $x=1$ is simply your expansion for $(1)$.
Some neat var... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1689040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Proof of an identity that relates hyperbolic trigonometric function to an expression with euclidean trigonometric functions.
Given a line $r$ and a (superior) semicircle perpendicular to $r$, and an arc $[AB]$ in the semicircle, I need to prove that
$$
\sinh(m(AB)) = \frac{\cos(\alpha)+\cos(\beta)}{\sin(\alpha)\sin(\b... | Using the definition of the hyperbolic sine and hyperbolic cosine functions, we have
$$
\sinh m(AB)
= \frac{e^{m(AB)} - e^{-m(AB)}}{2}
= \frac{(AA'\cdot BB')^2 - (BA'\cdot AB')^2}{2(AA'\cdot BB')(BA'\cdot AB')} \\
\cosh m(AB)
= \frac{e^{m(AB)} + e^{-m(AB)}}{2}
= \frac{(AA'\cdot BB')^2 + (BA'\cdot AB')^2}{2(AA'\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1689160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Change of variable in $\int_{r_{0}}^{r_{1}}\frac{dr}{r(1-r^{2})}=\int_{0}^{2\pi}dt $ In Strogatz's Book Nonlinear Dynamics and Chaos the example 8.7.1 we have the vector field $\dot{r}=r(1-r^2)$ , $\dot{\theta}=1$ given in polar coordinates. Let $r_0$ and $r_1$ points in the positive real axis. We know that after a t... | Hint: If you have $r(t)$ such that $r(0) = r_0$, $r(2\pi) = r_1$, you can for sure integrate this thing as:
$$ \int_{0}^{2\pi} \frac{\dot{r}(t)}{r(t)(1-r(t)^2)} \, dt = \int_{r_0}^{r_1} \frac{dr}{r(1-r^2)}. $$
So, in terms of substitution formula it means that $\varphi(t) = r(t)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1689251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Lindeberg Condition for a sequence of discrete random variables.
Let $X_1,X_2,...$ be independent and for any n $\ge 1$ and $\alpha>0$
$$X_n = \left\{
\begin{array}{rl}
n^\alpha & \text{with } Pr(X_n= n^\alpha) = \frac{1}{2n^{2\alpha}},\\
-n^\alpha & \text{with }Pr(X_n= -n^\alpha) = \frac{1}{2n^{2\alpha}},\\
0 & \t... | The condition we have to check for Lindeberg's condition is thtat for all positive $\varepsilon$,
$$
\lim_{n \rightarrow \infty} \frac{\sum_{k=1}^{n} \mathbb E\left[X_k^2 I_{\{|X_k|>\epsilon B_n\}}\right]}{B_n^2} = 0,
$$
and since $B_n=\sqrt n$, this is equivalent to
$$
\lim_{n \rightarrow \infty} \frac{\sum_{k=1}^{n} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1689455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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False: if $C$ is closed then closure of interior of $C$ is equal to $C$? If $C$ is a closed set in a metric space $(X,d)$, then $\overline{C^\circ} = C$
I know that this is false, but I'm having trouble coming up with a good
counterexample to show that it doesn't work. Ideas?
Edit:
Wow, the answers are so simple! Major... | Take $X=\mathbb{R}$ with the standard metric. A singleton $\{x\}$ is closed. But what is the interior of $\{x\}$? And the closure of that?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving monotonicity and convergence of a sequence $a_n=(1+{1 \over 2}+{1 \over 3}+....+{1 \over n})-\ln(n)$
Show that $a_n$ is bounded and monotone and hence convergent.
I know that the $-\ln(n)$ portion will be monotonically decreasing. I think I need to somehow show that $1+{1 \over 2}+{1 \over 3}+....+{1 \over n}>... | This answer may be off-topic; so, please, forgive me if this is the case.
$$\sum_{i=1}^n\frac 1i=H_n$$ the rhs being the harmonic number. For large values of $n$, the asymptotic expansion is $$H_n=\gamma +\log (n)+\frac{1}{2 n}-\frac{1}{12
n^2}+O\left(\frac{1}{n^3}\right)$$ This makes $$a_n=\sum_{i=1}^n\frac 1i-\log... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1689642",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Show that $\overline{f(\overline{z})}$ is holomorphic on the domain $D^*:=\{\overline z: z\in D\}$ using Cauchy Riemann equation. Please do not vote to close it as I want to find errors in my proof, which cannot be rectified on previously answered question.
I want a different proof using Cauchy Riemann equation.
Let $D... | We have to prove that the function
$$g(w):=\overline{f(\bar w)}$$
is holomorphic on $D^*$. To this end fix a point $w\in D^*$ and consider a variable complex increment vector $W$ attached at $w$. Then
$$g(w+W)-g(w)=\overline{f(\bar w+\bar W)-f(\bar w)}=\overline{f'(\bar w)\bar W+o(|\bar W|)}\qquad(W\to0)\ .$$
It follow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1689734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Pullback of an invertible sheaf through an isomorphism Consider an isomorphism of schemes $(f,f^{\#})(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$. Moreover let $\mathcal F$ be an invetible sheaf on $Y$ and let $f^{*}\mathcal{F}$ be its pullback.
Is it true that $\chi(\mathcal{F})=\chi(f^{*}\mathcal{F})$? Clearly $\chi(\cdot... | Yes, because $f^\ast$ is an exact functor if $f$ is an isomorphism.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1689849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Rearrangement of Students (flaw in my solution)
There are 11 students in a class including A, B and C. The 11 students have to form a straight line. Provided that A cannot be the first person in the line, what is the probability that in any random rearrangement of line, A comes before B and C.
For eg, this is a valid... | Short way:
As OP has remarked, by symmetry, the probability that $A$ comes before $B$ and $C$ is $\dfrac13$.
The only object that need concern us is the one immediately preceding $A$.
(Others won't affect the probability computation)
There are $8$ ways with the constraints, as against $10$ unconstrained ways for this o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1689953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 4
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Does $a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + ... + \frac{1}{\sqrt{n^2+2n-1}}$ converge? $a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + ... + \frac{1}{\sqrt{n^2+2n-1}}$
and I need to check whether this sequence converges to a limit without finding the limit itself. I think about using... | On the one hand,
$$a_n \ge \mbox{smallest summand} \times \mbox{number of summands}= \frac{1}{\sqrt{n^2+2n-1}}\times n .$$
To deal with the denominator, observe that
$$n^2+2n-1 \le n^2+2n+1=(n+1)^2.$$
On the other hand,
$$a_n \le \mbox{largest summand}\times \mbox{number of summands} = \frac{1}{\sqrt{n^2+n}}\time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1690092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Clarification on variance and expected value problem Suppose that X is a random variable where:
$P(X = 1)$ = 1/2
$P(X = 2) = $1/4
$P(X = 4)$ = 1/4
Suppose Y is another random variable that takes values from the set $$Y = {{1, 2, 4}}$$
but
the probabilities that it takes each value are unknown and some of them could ... | The expectation is maximized when the probability that the random variable $Y$ attains its largest value is maximized. Since there are no constraints, this is achieved when $Y=4$ with probability $1$ and $Y=1,2$ with probability $0$. Formally, in this case $$E[Y]=1\cdot P(Y=1)+2\cdot P(Y=2)+4\cdot P(Y=4)=0+4\cdot1=4$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1690196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Does there exist a computable number that is normal in all bases? Following up on this exchange with Marty Cohen...
Almost all numbers are normal in all bases (absolutely normal), but there are only a countable number of computable numbers, so it is plausible that none of them are absolutely normal. Now I don't expect... | Below are a couple of papers for what you want. For more, google computable absolutely normal.
Verónica Becher and Santiago Figueira, An example of a computable absolutely normal number, Theoretical Computer Science 270 #1-2 (6 January 2002), 947-958. [Another copy here.]
Verónica Becher, Pablo Ariel Heiber, and Theodo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1690285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Prove that $\sin(x) + \cos(x) \geq 1$ $\forall x\in[0,\pi/2]: \sin{x}+\cos{x} \ge 1.$
I am really bad at trigonometric functions, how could I prove it?
| you can consider the function $f(x) = \sin{x} +\cos{x} -1$ then see that in $[0,\pi /4)$ ,
$f'(x) >0$
and in $(\pi/4, \pi/2], f'(x) <0$
now see what happens at $0$ and $\pi/2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1690381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 6
} |
Hint for $\lim_{n\rightarrow\infty} \sqrt[n]{\prod_{i=1}^n\frac{1}{\cos\frac{1}{i}}}$. How to calculate the following limit:
$$\lim_{n\rightarrow\infty}\sqrt[n]{\prod_{i=1}^n\frac{1}{\cos\frac{1}{i}}}$$
thanks.
| Hint:
$$\lim_{n \to \infty} a_n ^{1/n} = \lim_{n \to \infty} \frac{a_{n+1}}{a_n},$$
and
$$\lim_{n \to \infty} \frac{1}{\cos[1/(n+1)]} = 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1690470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
What is the simplified form of $\frac{2 \cos 40^\circ -1}{\sin 50^\circ}$? I just encountered the following multiple choice question on a exam. It looks simple but surprisingly I couldn't decipher it! So I decided to mention it here. :)
What is the simplified form of $\dfrac{2 \cos 40^\circ -1}{\sin 50^\circ}$?
... | Consider
$$
2\sin50^\circ\sin10^\circ=\cos(50^\circ-10^\circ)-\cos(50^\circ+10^\circ)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1690643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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How to show that $(W^\bot)^\bot=W$ (in a finite dimensional vector space) I need to prove that if $V$ is a finite dimensional vector space over a field K with a non-degenerate inner-product and $W\subset V$ is a subspace of V, then:
$$
(W^\bot)^\bot=W
$$
Here is my approach:
If $\langle\cdot,\cdot\rangle$ is the non-de... | Hint It follows from the definition that $(W^\perp)^\perp \subset W$.
Hint 2: For every subspace $U$ of $V$ you have
$$\dim(U)+ \dim(U^\perp)=\dim(V)$$
What does this tells you about $\dim(W)$ and $\dim (W^\perp)^\perp$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1690797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
$C^{1}[0,1]$ is not Banach under $\|\cdot\|_{\infty}$ This is a curiosity from a reading a text that offered no proof. Why is $(C^{1}[0,1], \|\cdot\|_{\infty})$ not Banach?
| By Stone—Weierstrass, any continuous function can be uniformly approximated on $[0,1]$ by a sequence of polynomials.
Take any continuous function in $C^0[0,1] \setminus C^1[0,1]$, i.e. $f$ that is continuous but not continuously differentiable on $[0,1]$. Such functions exist.
Now, take a sequence $(P_n)_n$ of polynomi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the probability of drawing 1 red pen and 1 green pen? There are 3 blue pens, 2 red pens, 3 green pens and you're drawing two pens at random. What's the probability that 1 will be red and another will be green?
What I tried doing:
$$\frac{\binom{2}{1}\binom{3}{1}\binom{3}{0}}{\binom{8}{2}} = \frac{3}{14}$$
answe... | Consider two slots where the balls will be placed. The total number of events would be $8$ choices for the first slot and $7$ for the second, i.e. $7\times 8 = 56$.
Now, consider your situation. We can get red balls for this slot in $2$ scenarios, and then for the second slot we need to have a green ball whose possibil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1691065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Show that the polynomial $(x-1)(x-2) \cdots (x-n)-1$ is irreducible on $\mathbb{Z}[x]$ for all $n \geq 1$
Show that the polynomial $h(x)=(x-1)(x-2) \cdots (x-n)-1$ is irreducible in $\mathbb{Z}[x]$ for all $n \geq 1$.
This problem seems to be hard to solve. I thought I could use Eisenstein in developping this polynom... | David's observation that if $f=gh$, then $g(k)=-h(k)=1$ or both $=-1$ for each of $k=1,2,\ldots , n$ is spot on. So both $g$ and $h$ take at least one of these values at least $n/2$ times. If we now take the polynomial of smaller degree (let's say it's $g$), so $\deg g=m\le n/2$, then the only way to avoid a constant $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1691140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find an explicit pair of vectors $(u,v)$ in $V$ that span a hyperbolic plane $W$ inside $V$. Consider the symmetric form $\langle\ ,\ \rangle$ on $V=\mathbb{F}_7^3$ defined by the symmetric matrix $$A= \begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 2 \end{pmatrix} \in M_{3 \times 3}(\mathbb{F}_7).$$
Find an explicit... | If $\;\begin{pmatrix}x\\y\\z\end{pmatrix}\;$ is isotropic, then $$(x\;y\;z)A\begin{pmatrix}x\\y\\z\end{pmatrix}=(x\;y\;z)\begin{pmatrix}2x\\y+z\\y+2z\end{pmatrix}=2x^2+y^2+2yz+2z^2=0\iff$$
$$\iff2x^2+(y+z)^2+z^2=0$$
The above has only the trivial solution over $\;\Bbb Q\;$ or $\;\Bbb R\;$, say, but over $\;\Bbb C\;$ fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1691267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to evaluate the value of $\begin{bmatrix}\vec{l},\vec{m},\,\vec{n}\end{bmatrix} \begin{bmatrix}\vec{a},\vec{b},\,\vec{c}\end{bmatrix}$ Lets the value of $\,\begin{bmatrix}\vec{l},\vec{m},\,\vec{n}\end{bmatrix}$ is $\,\vec{l}.\left(\vec{m}\times\vec{n}\right)$.
We have to show that
$$
\begin{bmatrix}\vec{l},\vec{m},... | Recall that the product of determinants of two $n\times n$ matrices is equal to the determinant of the product of these matrices:
$$ \det\left(A\right)\det\left(B\right) = \det\left(AB\right), $$
and that the determinant of a matrix is equal to the determinant of its transpose:
\begin{align}
\det\left(A\right) &= \det\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1691382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Which groups $G$ has the property that for all subgroups $H$ , there is a surjective map from $G$ to $H$? I tried many examples , but i can't find any counterexample .
But I guess there are many counter examples , and specific sorts of groups or subgroups have this property (e.g abelian groups or normal subgroups).
Thu... | Take $G = F(\{x_1,\dots, x_n\})$, the free group on $n$ generators. The commutator subgroup $G' = [G,G]$ is a free group of infinite rank and thus $G$ cannot surject onto $G'$, as it simply doesn't have enough generators (it has finite rank).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1691561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Morphism epimorphism if and only if surjective In the category of sets, I want to prove that a morphism is an epimorphism if and only if it is surjective.
In both directions, I'm having a hard time approaching this problem.
This is how far I got.
$$\text{Morphism is epimorphism} \implies \text{Morphism is surjective}:$... | Some hints:
*
*For epic $\Rightarrow$ surjective, let $Y=B \cup \{ \star \}$ (where $\star \not \in B$), let $\beta$ be the identity on $B$, and let $\beta'$ be the map which sends everything in the image of $\phi$ to itself, and everything not in the image of $\phi$ to $\star$. See what happens.
*For surjective $\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Is $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID? As the title suggests, I'm interested whether $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID or not. Assume $p$ is prime.
My feeling is that it is a PID, since $\mathbb{Z}/(p)$ is cyclic an morally if an ideal is generated by elements of $\mathbb{Z}/(p)$, it's enough to consider the element w... | $\mathbb Z[C_p]$ is not even a domain.
Take for instance $p=2$. Then there is an element $u\in C_2$ such that $u^2=1$ but $u\ne \pm1$.
More generally, if $G$ has an element $u$ of finite order, then $\mathbb Z[G]$ is not a domain because $(u-1)(u^{n-1}+\cdots+u+1)=u^n-1=0$. In particular, $\mathbb Z[G]$ is never a doma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1691808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the limit: $\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$
Find the limit: $$\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$$
I have no idea what to do, but I thought that this is the limit of the derivative of $f(x)=\left(1+x\right)^{1/x}$, as $x$ tends to 0. Any help?
| If Taylor series (the "right" approach) are not yet available, let's use L'Hospital's Rule.
When we differentiate the top we get
$$(1+x)^{1/x}\left(\frac{x/(1+x)-\ln(1+x)}{x^2}\right).$$
The front part safely has limit $e$, so we only need to find
$$\lim_{x\to 0}\frac{x/(1+x)-\ln(1+x)}{x^2}.$$
One round of L'Hospital... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1691913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Bayesian probability on Bernoulli distribution Let $D$ be a Bernoulli distribution with $P[X=1] = \theta$ (and so $P[X=0]=1-\theta$). Let $\chi = \{0,1\}$ be an iid sample drawn from $D$. Assume a prior distribution on $\theta$, with $\theta$ uniformly distributed between 0 and .25.
What is the value of $p(\theta)$ for... | I'd say it's natural that you're confused.
What is the value of $p(\theta)$ for $\theta=\frac{1}{8}$?
is slightly confusing. First, as you rightly noted, $\theta$ is a continuous random variable, so $p(\theta)$ is actually a density function.
Then, let's guess that "value of $p(\theta)$ for $\theta=\frac{1}{8}$" sim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Fundamental theorem on flows lee's book 2nd edition I am reading Lee's book Introduction to smooth manifolds 2nd edition chapter 9 the fundamental theorem on flows.
In the proof of the fundamental theorem on flows the author defines $t_0=\inf\{t\in\mathbb{R}:(t,p_0)\notin W\}$, and argues that since $(0,p_0)\in W$, we ... | My understanding is the same as yours: it should be $t_0=\sup\{t\in\mathbb R:(t,p_0)\in W\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
how to prove that invertible matrix and vectors span the same space? Given $M$ is an invertible matrix, and {$\vec{v_1}...\vec{v_k}$} spans $R^n$,
then {A$\vec{v_1}...A\vec{v_k}$} also spans $R^n$
What does matrix invertibility have to do with span?
| From scratch:
If $\vec v\in \mathbb R^n$, then so is $A^{-1}\vec v$, and so
$A^{-1}\vec v=\sum_{i=1}^{k}c_i\vec v_i$ for some $c_i\in \mathbb R$. Upon multiplying by $A$ you get
$AA^{-1}\vec v=\vec v=A\sum_{i=1}^{k}c_i\vec v_i=\sum_{i=1}^{k}c_iA\vec v_i$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Linear Algebra - Real Matrix and Invertibility Let $M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ be a real matrix $2n\times 2n$ with $A,B,C,D$ real matrices $n\times n$ that are commutative to each other. Show that $M$ is invertible if and only if $AD-BC$ is invertible.
| Hint: try considering
$$
\begin{pmatrix}
D(AD-BC)^{-1} & -B(AD-BC)^{-1} \\
-C(AD-BC)^{-1} & A(AD-BC)^{-1}
\end{pmatrix}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Differential Equation solve differential equation $$ \frac{dy}{ dx} =\frac{(3x-y-6)}{(x+y+2)}$$
I tried to do this but it´s first order and posible is separable variables
| We can solve the differential equation given by
$$\frac{dy}{dx}=\frac{3x-y-6}{x+y+2} \tag 1$$
in a straightforward way. Rearranging $(1)$ reveals
$$x\,dy+y\,dx+(y+2)\,dy+(6-3x)\,dx=0\tag 2$$
Next, we integrate $(2)$ and write
$$\int (x\,dy+y\,dx)\,+\int (y+2)\,dy\,+\int (6-3x)\,dx=C \tag 3$$
Noting that $(x\,dy+y\,dx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Quotients in Ceilings and Floors How would I simplify the expression
$\lceil\frac{2x + 1}{2}\rceil - \lceil\frac{2x + 1}{4}\rceil + \lfloor\frac{2x + 1}{4}\rfloor$
I've tried writing the expression without floors or ceilings, but with no success. I also tried some casework on the parity of x.
| We would need to first look at the last two terms. The floor and ceil of a number are equal if and only if it is an integer. So if $\frac{2x + 1}{4} \in \mathbb{Z}$, i.e. if $x = \frac{4n - 1}{2}$ where $n$ is an integer, then the last two terms cancel out. Otherwise, the floor of the number is $1$ more than the ceil o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof by induction: $F_{3n-1}$ is even. Note: For this problem, I am using the definition of Fibonacci numbers such that $F_0 = 1, F_1 = 1, F_2 = 2$, and so on.
Here's my current work.
Proof.
The proof is by induction on n.
Basis step: For $n = 1, F_2 = 2$. $2|2$, so the statement is true for $n = 1$.
Induction hypothe... | $F_{3k+2}=F_{3k+1}+F_{3k}$ and $F_{3k+1}=F_{3k}+F_{3k-1}$
So $F_{3k+2}=2F_{3k}+F_{3k-1}$
Since $F_{3k}$ is an integer, if $F_{3k-1}$ is even then $F_{3k+2}$ must be even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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How prove that: $[12\sqrt[n]{n!}]{\leq}7n+5$? How prove that: $[12\sqrt[n]{n!}]{\leq}7n+5$,$n\in N$ I know $\lim_{n\to \infty } (1+ \frac{7}{7n+5} )^{ n+1}=e$ and $\lim_{n\to \infty } \sqrt[n+1]{n+1} =1$.
| By AM-GM
$$\frac{1+2 + 3 + \cdots + n}{n} \ge \sqrt[n]{1 \times 2 \times 3 \times \cdots \times n}$$
$$\implies \frac{n+1}2 \ge \sqrt[n]{n!} \implies 6n+6 \ge 12\sqrt[n]{n!}$$
But $7n+5 \ge 6n+6$ for $n \ge 1$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Why is it impossible to move from the corner to the center of a $3 \times 3 \times 3$ cube under these conditions? There is a $3 \times 3 \times 3$ cube block, starting at the corner, you are allowed to take 1 block each time, and the next one must share a face with the last one. Can it be finished in the center?
This ... | An approach coloring the cubies.
Let's color the individual cubes red and blue. We paint them in a "checker" (the 3D version, that is) pattern, so that the corners are red and the middles of each sides are blue, the centers of each face red, and the center of the cube blue, like this (created with POV-Ray):
Now let's ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Under what conditions is $J\cdot M$ an $R$-submodule of $M$? I have that $M$ is an $R$-module where $R$ is commutative and unitary ring. Supposing that $J$ is an ideal of $R$, when is the set $J \cdot M$ an $R$-submodule of $M$?
I have to check the two axioms of an $R$-submodule. First that $(J\cdot M,+)$ is a subgroup... | Based on your comment describing the context of the situation, (thank you for that, by the way) it's clear that the text intends to use the standard definition of the product:
$JM:=\{\sum j_i m_i\mid j\in J, m\in M, i\in I \text{ for a finite index set $I$ }\}$
This is what is intended when looking at any sort of ideal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A Teacher wrote either of words $PARALLELOGRAM$ or $PARALLELOPIPED$ A Teacher wrote either of words $PARALLELOGRAM$ or $PARALLELOPIPED$ on board but due to malfunction of marker words are not properly written and only two consecutive letters $RA$ are visible, then the chance that the written word is $PARALLELOGRAM$ is ... | Counting pairs of consecutive positions as $1-2,\;\; 2-3$ etc,
there are $12$ such positions in parallelogram, and $13$ in parallelopiped,
thus P(saw RA) $= 2/12$ in parallelogram and $1/13$ in parallelopiped
and P(word "parallelogram" | saw $RA) = \dfrac{2/12}{2/12 + 1/13}=$
Continue....
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1692988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is $\mathbb{Z}_p[\mathbb{Z}_p]$ a PID?
Is $\mathbb{Z}_{p}[G]$ a PID, where $G=(\mathbb{Z}_{p},+)$ is the additive group of the $p$-adics $\mathbb{Z}_{p}$?
I am studying a paper where the authors implicitly use that claim, but it is unclear to me. (I am a little bit embarassed by the fact that I cannot solve this myse... | This isn't true; in fact, $\mathbb{Z}_p[G]$ is not even Noetherian. For instance, take the augmentation ideal $I$, i.e. the ideal generated by $\{g-1:g\in G\}$. If $I$ were finitely generated, there would be a finite subset $F\subset G$ such that $I$ is generated by the elements $g-1$ for $g\in F$. But if $H\subsete... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1693102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Trying to understand the Nabla Operator I'm trying to wrap my head around the following line done in my physics textbook:
$\vec\nabla f(r) = \begin{pmatrix}
f'(r) \frac{\partial r}{\partial x}\\
f'(r) \frac{\partial r}{\partial y}\\
f'(r) \frac{\partial r}{\partial z}
\end{pmatrix}$
Where $r$ represents the distance ... | It is a consequence of the derivative of function.
You have $f(r)$, with $r=\sqrt{x^2+y^2+z^2}$
Then $\dfrac{\partial f(r)}{\partial x}=\dfrac{\partial f(r)}{\partial r}\times\dfrac{\partial r}{\partial x}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1693193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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$3$-adic expansion of $- \frac{9}{16}$ I get the $3$-adic expansion to be $1+1 \cdot 3+2 \cdot 3^2 +2 \cdot 3^3 + 0 \cdot 3^4+\cdots$. I'm trying to work out a pattern of the coefficients and think it is $1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, 0,...$. To show this I need to show that $1+1 \cdot 3+2 \cdot 3^2 +2 \cdot 3^3 + 0... | I guess I’m on a long-term rant to urge people to write their $p$-adic numbers as ordinary $p$-ary expansions extending (potentially) infinitely to the left. In your case, that would be ternary expansion, so, just as you learned in elementary school, sixteen comes out as $121;\,$. I like to use a semicolon for the radi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1693273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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What is the limit of $\frac{\prod\mathrm{Odd}}{\prod\mathrm{Even}}?$ Does $\pi$ show up here?
What is this limit
$$
\frac{1\times3\times5\times\cdots}{2\times4\times6\times8\times\cdots} = \lim_{n \rightarrow \infty}\prod_{i=1}^{n}\frac{(2i-1)}{2i}
$$
I remember that it was something involving $\pi$.
How can I co... | Here's a formula which I found embedded in an old C program. I don't know where this comes from, but it converges to Pi very quickly, about 16 correct digits in just 22 iterations:
$\pi = \sum_{i=0}^{\infty}{
\frac{6(\prod{2j-1})}
{(\prod{2j})(2i+1)(2^{2i+1})}}$
(Each product is for j going from 1 to i. When i is 0, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1693377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 7,
"answer_id": 6
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Prove $(\mathbf{A}+\mathbf{u}\mathbf{v}^{\text{T}})^{-1}=\cdots$ Problem:
Assuming $\mathbf{A}$ and $\mathbf{A}+\mathbf{u}\mathbf{v}^{\text{T}}$ are nonsingular, prove
\begin{equation}
(\mathbf{A}+\mathbf{u}\mathbf{v}^{\text{T}})^{-1}=\mathbf{A}^{-1}-\frac{\mathbf{A}^{-1}\mathbf{uv}^{\text{T}}\mathbf{A}^{-1}}{(1+\mathb... | As a side note, I prefer rewriting the equality as
$$
(I+xv^T)^{-1} = I - \frac{xv^T}{1+v^Tx}.
$$
where $x=A^{-1}u$. The merit of doing so is that, we immediately see why the inverse of the rank-1 update of a matrix is a rank-1 update of the inverse: by Cayley-Hamilton theorem, $(I+xv^T)^{-1}$ is a polynomial in $I+xv^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1693523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Uniform approximation by even polynomial Proposition
Let $\mathcal{P_e}$ be the set of functions $p_e(x) = a_o + a_2x^2 +
\cdots + a_{2n}x^{2n}$, $p_e : \mathbb{R} \to \mathbb{R}$
Show that all $f:[0,1]\to\mathbb{R}$ can be uniformly approximated by
elements in $\mathcal{P_e}$
Attempt:
Since we are talking about... | Two standard approaches would be either to use the Stone-Weierstraß theorem, and note that the algebra of even polynomials satisfies the premises of that theorem, or to look at the isometry $S\colon C^0([0,1],\mathbb{R}) \to C^0([0,1],\mathbb{R})$ given by
$$S(f) \colon t \mapsto f(\sqrt{t}).$$
Approximate $S(f)$ with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1693629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Using linear algebra to find resonance frequency and normal oscillations and motion I am stuck part way through the following and not sure how or if finding eigenvalues will help with finding modes of oscillations:
Consider the system of three masses and two ideal elastic bands:
$(m)$---$k$---$(2m)$---$2k$---$(m)$ [$m... | Without using the condition $m\ddot{x}+m\ddot{y}+m\ddot{z}=0$, just write your 3D system as
$$ k\left[ \begin {array}{ccc} -1&1&0\\1&-3&2
\\ 0&2&-2\end {array} \right] \left[ \begin {array}{c} x\\y\\z\end {array} \right]=m\left[ \begin {array}{c} \ddot{x}\\\ddot{y}\\\ddot{z}\end {array} \right]$$
Now compute the frequ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1693733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
determinant of a very large matrix in MATLAB I have a very large random matrix which its elements are either $0$ or $1$ randomly. The size of the matrix is $5000$, however when I want to calculate the determinant of the matrix, it is either $Inf$ or $-Inf$. Why it is the case (as I know thw determinant is a real number... | If the determinant is needed, then a numerically reliable strategy is to compute the $QR$ decomposition of $A$ with column pivoting, i.e. $AP = QR$, where $P$ is a permutation matrix, $Q$ is an orthogonal matrix and $R$ is an upper triangular matrix. In MATLAB the relevant subroutine is 'qr'. Then the determinant of $A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1693784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Analytic Number Theory: Problem in Bertrand’s postulate I am trying to learn Bertrand’s postulate. I can not understand two steps
*
*Why $\displaystyle\sum_{n \leq x}\log n=\sum_{e \leq x} \psi\left(\frac{x}{e}\right)$,
where $\psi(x)=\displaystyle\sum_{p^\alpha \leq x, \alpha \geq 1}\log p$?
*$\displaystyle\sum_{... | This question has an answer to the first part of your question. Chebyshev's original proof is online and you don't have to rely on my translation of his argument. In case the link breaks, it is Vol. I of Chebyshev's Oeuvres, p. 49, Memoire Sur Les Nombres Premiers at p. 53.
For the second part, begin with
$\hspace{55m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1693959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Problem understanding half wave symmetry I am trying to understand half wave symmetry.
I understand the first (a) graphical image is half wave symmetry but (b) seems like even symmetry and (c) seems to be odd symmetry. I am unable to find the difference. Please guide.
Edit: I read my question and it seems to be confus... | You seem to be assuming that it is an either/or situation. It isn't. A wave can be all three: odd (OR even), have half wave symmetry, and also have quarter wave symmetry. All your examples have half wave symmetry, (b) is even in addition to having half wave symmetry, while (c) is odd in addition to having half wave s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1694156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Does there exist a continuous injection from $[0,1)$ to $(-1,1)$? Does there exist a continuous injective or surjective function from $[0,1)$ to $(-1,1)$ ? I know there is no continuous bijection from $[0,1)$ to $(-1,1)$ , but am stuck with only injective continuous or surjective continuous . Please help . Thanks in ad... | $$
f(x) = x.
$$
This is a continuous injection from $[0,1)$ onto $[0,1)$ and so a continuous injection from $[0,1)$ into $(-1,1)$.
For a continuous surjection, let's do it piecewise. Say a function $g$ is piecewise linear and $g(0)=0$, $g(1/2) = 0.9$, $g(3/4)=-0.9$, $g(7/8) = 0.99$, $g(15/16) = -0.99$, $g(31/32) = 0.9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1694251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$\pm$ sign in $y=\arcsin\frac{x}{\sqrt{1+x^2}}$ If:
$$y=\arcsin\frac{x}{\sqrt{1+x^2}}$$
Then:
$$\sin(y)=\frac{x}{\sqrt{1+x^2}}$$
$$\cos^2(y)=1-\sin^2(y)=\frac{1}{1+x^2}$$
$$ \tan^2(y)=\sec^2(y)-1=1+x^2-1=x^2$$
Therefore I would say:
$$\tan(y)=\pm x$$
However, my calculus book says (without the $\pm$):
$$\tan(y)=x$$
Q... | The sign of $\tan(\theta)$ is not uniquely determined from an equation $\sin(\theta) = a$, which has two solutions with opposite signs for the tangent. Under any convention for choosing one of the two $\theta$'s as the value of $\arcsin(a)$, the tangent is uniquely determined. The convention consistent with what you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1694328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Inversion of mean value theorem Given two functions $f,g: A \subset \mathbb{R} \rightarrow \mathbb{R}$ , $f,g \in C^1(A)$, I want to study the number of points of intersection of these two functions.
So I can take $h(x) := f(x)-g(x)$, $h \in C^1(A)$, and thus I want to study the roots of that function $h$. If I have tw... | A sufficient condition such that $h'(\alpha)=0$ implies $h(x_1)=h(x_2)$ for some $x_1 \neq x_2$ is:
$h''$ exists on $A$ (not necessarily continuous) and $h''(\alpha)\neq 0$.
Proof: It suffices to show that $h$ has a local extremum in $\alpha$. Suppose $h''(\alpha) < 0$.
Claim: There is an $x_1 < \alpha$ such that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1694418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to evaluate $\sum_{k=1} ^{n-1} \frac{\sin (k\theta)}{\sin \theta}$ How to evaluate $$\sum_{k=1} ^{n-1} \frac{\sin (k\theta)}{\sin \theta}$$
Any help ? I tried to use difference method. But I'm not getting there.
| For a different approach:
$$\sum_{k=1}^{n-1}\frac{\sin(k\theta)}{\sin(\theta)}=\frac1{\sin(\theta)}\sum_{k=0}^{n-1}\sin(k\theta)$$
$\Im$ means the imaginary part.
$$=\frac1{\sin(\theta)}\Im\sum_{k=0}^{n-1}(\cos(k\theta)+i\sin(k\theta))$$
$$=\frac1{\sin(\theta)}\Im\sum_{k=0}^{n-1}e^{k\theta i}$$
$$=\frac1{\sin(\theta)}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1694543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Mod of a random variable I had this problem where I wanted to generate random variables (discrete) in a way that certain numbers were more probable than others (basically geometric) but since I wanted to use this number as an array index, I wanted it to be bounded between $[0,n)$, where $n$ could be anything between $5... | For any $k$ such that $1\leq k\leq n-1$:
\begin{align}
P(Y=k) &= \sum_{j=0}^{\infty} P(X=k+jn) \\
&= \sum_{j=0}^{\infty} q^{k+jn-1}p \\
&= q^{k-1}p \sum_{j=0}^{\infty} \left(q^{n}\right)^j \\
&= \dfrac{q^{k-1}p}{1-q^{n}} = \dfrac{P(X=k)}{1-q^{n}}. \\
\end{align}
Also, the special case of $Y=0$ since $X=0$ can't occur:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1694658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Let $A$ be a matrix, and let $B$ be the result of doing a row operation to $A$. Show that the null space of $A$ is the same as the null space of $B$. Let $A$ be a matrix, and let $B$ be the result of doing a row operation to $A$. Show that the null space of $A$ is the same as the null space of $B$.
Any ideas of what ... | Doing a row operation on $A$ can be obtained by multiplying $A$ by an invertible matrix $E$. So $B=EA$ for a suitable invertible matrix $E$. If $Av=0$, then $Bv=EAv=0$. Since $A=E^{-1}B$, the converse inclusion also holds.
The matrix $E$ is obtained from the identity $m\times m$ matrix (if $A$ is $m\times n$) by applyi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1694771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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Show that $q'$ is the quotient of euclidean division of the number $n$ by $ab$ Let $n\in N$ and $a,b\in N$:
1- $q$ the quotient of euclidean division of $n$ by $a$
2- $q'$ the quotient of euclidean division of $q$ by $b$
Show that $q'$ is the quotient of euclidean division of $n$ by $ab$
I thought about unicity of $(... | The hypotheses mean that
*
*$n=aq+r$, with $0\le r<a$
*$q=bq'+r'$, with $0\le r'<b$
Therefore
$$
n=aq+r=a(bq'+r')+r=(ab)q'+(ar'+r)
$$
and you want to show that $0\le ar'+r<ab$.
Until now your argument is sound.
Clearly $ar'+r\ge0$. Suppose $ar'+r\ge ab$; then $r\ge a(b-r')$, in particular
$$
a(b-r')<a
$$
that mea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1694855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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The definition of continuity and functions defined at a single point In this question, the answers say that $\lim_{x \to p} f(x) = f(p) \Longleftrightarrow f \ \text{is continuous at} \ p$ fails if $f$ is only defined at a single point. Let us consider $f:\{1\} \rightarrow \mathbb{R}$. This is continuous at $1$, yet s... | Topological Assessment
If $f: \mathbb{R} \to \mathbb{R}$ is defined at a single point then its image must also be a single point, so that the function is defined as $f(x_0) = y_0$ for some $x_0, y_0 \in \mathbb{R}$.
I'm not sure about the answer and post this more for feedback. It seems that as a function $f: \mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Solving a Probability Question A student goes to the library. The probability that she checks out (a) a work of fiction is 0.40, (b) a work of non-fiction is 0.40,and (c) both fiction and non-fiction is 0.20. What is the probability that the student checks out a work of fiction, non-fiction, or both?
I am trying to und... | The question is ambiguous. The probabilities are stated in a way that makes it seem that the probability that the student checks out a single nonfiction book is $0.4,$ and similarly, the probability that the student checks out a single fiction book is $0.4.$ If we consider it in this way, the probability is in fact $0.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Understanding how this derivative was taken
I am pouring water into a conical cup 8cm tall and 6cm across the top.
If the volume of the cup at time t is $V(t)$, how fast is the water
level ($h$) rising in terms of $V'(t)$?
The solution in the book is:
Take the water volume, given by
$$\frac{1}{3}\pi(\frac{3}{8})... | To answer in words, making GoodDeeds point more clear.
Notice how if you differentiate the whole term with respect to t
$$
V = \frac{1}{3}{\pi}(\frac{3}{8})^2h^3
$$
use chain rule on the h such that the expression becomes
$$
h'*constants *3h^2
$$
where h' is the derivative of h with.
All thats left is to deal with the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to find $\int \frac {\sec x}{1+ \csc x} dx $?
How to find $\int \frac {\sec x}{1+ \csc x} dx $ ?
Well it reduces to $ \int \frac {\sin x}{\cos x (1+\sin x)} dx $ .Any hints next ?
I'm looking for a short and simple method without partial fractions if possibe.
| Hint:
$$ \int \frac {\sin x}{\cos x (1+\sin x)} dx =\int\frac{\sin x\cos x}{\cos^2x(1+\sin x)}dx=\int\frac{\sin x\cos x}{(1-\sin^2 x)(1+\sin x)}dx\int\frac{\sin x\cos x}{(1+\sin x)^2(1-\sin x)}dx$$
Take $u=\sin x$ and use partial fractions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is $|f(a) - f(b)| \leqslant |g(a) - g(b)| + |h(a) - h(b)|$? when $f = \max\{{g, h}\}$ Let $f = \max\{{g, h}\}$ where all 3 of these functions map $\mathbb{R}$ into itself. Is it true that $|f(a) - f(b)| \leqslant |g(a) - g(b)| + |h(a) - h(b)|$? I'm thinking it can be proven by cleverly adding and subtracting inside of ... | If you dont want to deal with multiples cases :
\begin{array}{lcl}
|f(a)-f(b)| & = & \left| \max\{g(a),h(a)\} - \max\{g(b),h(b)\} \right| \\
& = &\left| \frac{g(a)+h(a)+|g(a)-h(a)|}{2} - \frac{g(b)+h(b)+|g(b)-h(b)|}{2} \right| \\
& = & \left| \frac{g(a)-g(b)}{2} + \frac{h(a)-h(b)}{2} + \frac{|g(a)-h(a)|-|g(b)-h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to show $\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}$ How to show the below equation ?
$$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}
~~~~~(t\in \mathbb Z^+)$$
| One may write
$$
\begin{align}
\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}&=\sum\limits_{i=t+1}^\infty \frac{1}{2^{i-t-1}i}\qquad (i=k+t+1)\\\\
&=\sum\limits_{i=1}^\infty \frac{1}{2^{i-t-1}i}-\sum\limits_{i=1}^t \frac{1}{2^{i-t-1}i}\\\\
&=2^{t+1}\sum\limits_{i=1}^\infty \frac{1}{2^{i}i}-2^{t+1}\sum\limits_{i=1}^t \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Sum of minors of matrix If $ k $ is a diagonal minor of a matrix $ A \in M_{m \times n}(\mathbb{C}) $ then $ k $ has the following form:
$$ M_{i_{1}, i_{2}, ..., i_{k}}^{i_{1}, i_{2}, ..., i_{k}}(A) = \begin{vmatrix}
a_{i_{1}, i_{1}} & a_{i_{1}, i_{2}} & ... & a_{i_{1}, i_{k}}\\
a_{i_{2}, i_{1}} & a_{i_{2}, i_{2}} & .... | Suppose that $m \leqslant n$. If needed ($m<n$), append rows of zeroes to $A$ and columns of zeroes to $B$ to change them into square $n\times n$ matrices $A'$ and $B'$. The only non-zero (diagonal) $k \times k$ minors of $A'B'$ are the same as the non-zero (diagonal) $k \times k$ minors of $AB$, hence the sum for $AB$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Solving for $k$ when $\arg\left(\frac{z_1^kz_2}{2i}\right)=\pi$ Consider $$|z|=|z-3i|$$
We know that if $z=a+bi\Rightarrow b=\frac{3}{2}$
$z_1$ and $z_2$ will represent two possible values of $z$ such that $|z|=3$. We are given $\arg(z_1)=\frac{\pi}{6}$
The value of $k$ must be found assuming $\arg\left(\frac{z_1^kz_2... | You already know that $\arg(z_1)=\frac{\pi}{6}$, and moreover $\arg(z_2)=\frac{5\pi}{6}$ and $\arg \left(\frac{1}{2i}\right)=\frac{-\pi}{2}$. Multiplying complex numbers results in adding their arguments (modulo $2\pi$) so you get the equation
$$\arg\left(\frac{z_1^kz_2}{2i}\right)=k\frac{\pi}{6}+\frac{5\pi}{6}-\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Definition 10.3 from PMA Rudin
It's an excerpt from Rudin's book. I can't understand the following moments:
1) Why he considers continuous function with compact support? Why compactness is so important?
2) Why equation (3) has the meaning? Why $f$ is zero on the complement of $I^k$?
3) Why integral in (3) is indepen... | Basically, the answer to 1) is that the Riemann integral is defined only on rectangles, so you need to be able to enclose the set of points $x$ where $f(x)\ne 0$ in a giant rectangle and then integrate over that rectangle. That's what he's doing with the integral over the $k$-cell $I^k$. I'm not sure what you mean by 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Intuition: Why will $3^x$ always eventually overtake $2^{x+a}$ no matter how large $a$ is? I have a few ways to justifiy this to myself.
I just think that since $3^x$ "grows faster" than $2^{x+a}$, it will always overtake it eventually. Another way to say this is that the slope of the tangent of $3^x$ will always event... | $2^{x+a} = 2^x 2^a$, so the problem reduces down to $3^x$ surpassing $c 2^x$ for any positive $c$. Dividing both sides by $2^x$ results in $(3/2)^x$ surpassing $c$, which can be achieved using logarithm and monotonicity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1695949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Poisson Distribution - Poor Understanding So I have this question about poisson distribution:
"The number of computers bought during one day from shop A is given a Poisson distribution mean of 3.5, while the same for another shop B is 5.0, calculate the probability that a total of fewer than 10 computers are sold from ... | If you sell fewer than 10 computers in 5 out of 5 consecutive days then you must have also sold 4 out of 5.
It would be nice if questions like this made it explicit if they mean "in exactly 4 out of 5 days" or "in at least 4 out of 5 days" but there we are!
Here they mean "in at least 4 out of 5 days" so it's the proba... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1696074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Let $ a \neq 0,$ $b$ integers, show that $N_{a,b} := \{a + nb: n \in \mathbb{Z}\}$ is basis for some topology on $\mathbb{Z}.$ It is Hausdorff? Each $N_{a,b}$ is closed, why?
I am having trouble in how to show that each $N_{a,b}$ is closed... I saw in wikipedia that $N_{a,b}^c = \mathbb{Z} - \cup_{j=1}^{a-1}N_{a,b+j}.$... | First of all, there is an error in your question: denoting
$$N_{a,b}= \{ a+nb : n \in \Bbb{Z} \} = a+b \Bbb{Z}$$
you need $b \neq 0$ and not $a \neq 0$ (in fact, for $b=0$ this is just the singleton $\{ a \}$: this would generate the discrete topology on $\Bbb{Z}$).
You can see that $N_{a,b}$ is simply a coset of the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1696184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Structure sheaf consists of noetherian rings Let $X\subseteq \mathbb{A}^n$ be an affine variety. The ring $k[x_1,\ldots,x_n]$ is noetherian because of Hilbert's basis theorem.
The coordinate ring $k[X]=k[x_1,\ldots,x_n]/I(X)$ is noetherian because ideals of $k[X]$ are of the form $J/I(X)$, where $J\supseteq I(X)$ is a... | There is a counterexample in section 19.11.13 of Ravi Vakil's Foundations of Algebraic Geometry https://math216.wordpress.com/
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1696350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Why does dividing a number with $n$ digits by $n$ $9$'s lead to repeated decimals? For example, $\frac{1563}{9999} = 0.\overline{1563}$.
Why does that make sense from the way the number system works?
I can vaguely see that since the number $b$ with $n$ $9$'s is always greater than the dividend $a$ of $n$ digits (excep... | This is because the example can be rewritten as
\begin{align*}\frac{1563}{9999}&=0.1563\cdot\frac 1{1-10^{-4}}=0.1563(1+0.0001+0.00000001+\dots)\\
&=0.1563+0.00001563+0.000000001563+\dots
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1696455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Beautiful geometry: Laser bouncing of walls of a semicircle Consider a semicircle with diameter $AB$. A beam of light exits from $A$ at a $58^{\circ}$ to the horizontal $AB$, reflects off the arc $AB$ and continues reflecting off the "walls" of the semicircle until it returns to point $A$.
How many times does the beam ... | By reflection across the diameter we can map what happens to the beam inside inside the semicircle to a full circle; them beam contacting the diameter and bounces off in the semicircle is equivalent to the beam passing through the diameter of the full circle.
We know that when the beam hits a wall, it reflects so that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1696557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$\lambda^2$ is an eigenvalue of $T^2$ Let $T:V\rightarrow V$ be a linear map. If $\lambda^2$ is an eigenvalue of $T^2$, then $\lambda$ or $-\lambda$ is an eigenvalue of T.
I have then $(T\circ T )(v) = T^2 (v) = \lambda^2 v$, but $\lambda^2 = (-\lambda)(-\lambda)$ or $(\lambda)(\lambda)$.
Any hints
| Suppose $T^2 v = \lambda^2 v$, then
$(T-\lambda I)(T+\lambda I) v = 0$.
If $(T+\lambda I) v = 0$ then $-\lambda$ is an eigenvalue corresponding
to eigenvector $v$, otherwise
$\lambda$ is an eigenvalue corresponding to eigenvector $(T+\lambda I) v$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1696624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Integrating $\int^2_{-2}\frac{x^2}{1+5^x}$ $$\int^2_{-2}\frac{x^2}{1+5^x}$$
How do I start to integrate this?
I know the basics and tried substituting $5^x$ by $u$ where by changing the base of logarithm I get $\frac{\ln(u)}{\ln 5}=x$, but I got stuck.
Any hints would suffice preferably in the original question and not... | Hint:
$$\frac1{1+5^{-x}} + \frac1{1+5^x} = 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1696811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Finding CDF from PDF I need to find the CDF from a given PDF.
The PDF is given by
$f_R(r)=2\lambda\pi r\exp(-\lambda\pi r^2)$.
What is its corresponding CDF $F_Y(y)$?
| $$
f_R(r)=2\lambda\pi r\exp(-\lambda\pi r^2)
$$
This function is non-negative only if $r\ge0$. We have
$$
\int_0^\infty \exp(-\lambda\pi r^2) (2\lambda\pi r \,dr) = \int_0^\infty \exp(-u) \, du = 1,
$$
so the support must be all of $[0,\infty)$. The c.d.f. is
$$
F_R(r) = \Pr(R\le r) = \int_0^r \exp(-\lambda\pi s^2) (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1696904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Number of ways of making a die using the digits $1,2,3,4,5,6$
Find the number of ways of making a die using the digits $1,2,3,4,5,6$.
I know that $6!$ is not the correct answer because some arrangements can be obtained just by rotation of the dice. So there will be many repetitions. I tried by fixing any two opposite... | Hint if you fix an arrangement then it has $6$ similar arrangements you see why by rotating in $2$ directions X,Z so answer should be $\frac{180}{6}=30$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1697164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Number of generators of a given ideal.
Let $I=\langle 3x+y, 4x+y \rangle \subset \Bbb{R}[x,y]$. Can $I$ be generated by a single polynomial?
My approach: If $I$ can be generated by a single polynomial, then the two "apparent" generators, $(3x+y),(4x+y)$ are dependent in this fashion:
There exist two polynomials $p(x... | It is not enough to consider only polynomials $p(x)$ and $q(x)$ in one variable. Assume that $I$ is principal, and let $f$ be a generator in $I$. We need to prove or disprove that $f(x,y)=(3x+y)p(x,y)+(4x+y)q(x,y)$ for some polynomials $p,q \in \mathbb{R}[x,y]$. Indeed, we obtain a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1697255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Notation for sets that do not overlap Is there notation to describe, say a set that consists entirely of two mutually exclusive subsets?
Say $ D = D_1 \cup D_2 $, how to indicate that $D_1$ and $D_2$ do not overlap?
| $D_1 \sqcup D_2$ and $D_1 \stackrel{\circ}\cup D_2$ [possibly with the circle or dot lower within the 'cup'] are both used, to denote the union of $D_i$ when you want to emphasize that the $D_i$ are disjoint. Sometimes "disjoint union" is meant to be an operation in its own right, guaranteeing disjointness of the opera... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1697461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to prove that a depth-first algorithm labels every vertex of G? I understand exactly how a depth-first search/algorithm works. You start at the root, and then go to the left most node, and go down as far as you can until you hit a leaf, and then start going back and hitting the nodes you did not hit before.
I get ... | If the search didn't visit all vertices, there is a non-empty set of vertices that it visited and a non-empty set of vertices that it didn't visit. Since the graph is connected, there is at least one edge between a visited vertex and a non-visited vertex. But when the algorithm reached the visited vertex, it successive... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1697560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
closed form for $I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$ $$I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$$
for $n=1$ I tried to use $\arctan x=u$
and by notice that $$\frac{1+\tan u}{1-\tan u}=\cot\left ( \frac{\pi}{4}-u \right )$$... | For $n=2$ we have, integrating by parts, $$I\left(2\right)=\int_{0}^{\pi/4}x^{2}\cot\left(x\right)dx=\frac{\pi^{2}}{16}\log\left(\frac{1}{\sqrt{2}}\right)-2\int_{0}^{\pi/4}x\log\left(\sin\left(x\right)\right)dx
$$ and now we can use the Fourier series of $\log\left(\sin\left(x\right)\right)$ $$\log\left(\sin\left(x\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1697651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Trigonometric inequality in sec(x) and csc(x) How can I prove the following inequality
\begin{equation*}
\left( 1+\frac{1}{\sin x}\right) \left( 1+\frac{1}{\cos x}\right) \geq 3+%
\sqrt{2},~~~\forall x\in \left( 0,\frac{\pi }{2}\right) .
\end{equation*}%
I tried the following
\begin{eqnarray*}
\left( 1+\frac{1}{\sin x}... | Expand the expression to get
$$\left(1+\frac{1}{\sin x}\right)\left(1+\frac{1}{\cos x}\right)=1+\frac{1}{\sin x}+\frac{1}{\cos x}+\frac{1}{\sin x\cos x}$$
Then using the identity $\sin x\cos x = \frac{1}{2}\sin 2x$, rewrite as
\begin{eqnarray*}
\left( 1+\frac{1}{\sin x}\right) \left( 1+\frac{1}{\cos x}\right) &=
& 1+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1697753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Simultaneous Diagonalization of two bilinear forms I need to diagonalize this two bilinear forms in the same basis (such that $f=I$ and $g$=diagonal matrix):
$f(x,y,z)=x^2+y^2+z^2+xy-yz $
$g(x,y,z)=y^2-4xy+8xz+4yz$
I know that it is possible because f is positive-definite, but I don't know how can I do it
| I wonder if the method the OP was attempting is simply Lagrange's method from multivariable calculus. Because a quadratic form takes its extreme values on the unit circle (or any circle) at eigenvectors, and because we can diagonalize a symmetric matrix using a basis of orthonormal eigenvectors, we look for the station... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1697846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
If X ∼ N(0, σ2), find the pdf of Y = |X|. If X ∼ N(0, $σ^2$
), find the pdf of Y = |X|.
So far I have
$F_Y(y) = P(\lvert x \rvert < y) = P(-y < x < y) = F_X(y) - F_X(-y)$
but I don't know where to go from there
| In fact, this distribution has a name: Half-normal distribution.
$$
F_Y(y) = F_X(y) - F_X(-y)\\
=F_X(y) - (1-F_X(y)) = 2F_X(y) - 1
$$
then differentiate for the pdf
$$f_Y(y) = \frac{\sqrt{2}}{\sigma\sqrt{\pi}}\exp \left( -\frac{y^2}{2\sigma^2} \right), \quad y>0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1697981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$a,b,c,d,e$ are positive real numbers such that $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$, find the range of $e$.
$a,b,c,d,e$ are positive real numbers such that $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$, find the range of $e$.
My book tells me to use tchebycheff's inequality
$$\left(\frac{a+b+c+d}{4}\right)^2\le \... | As @ChenJiang stated, its a case of cauchy's inequality
$$\left(\frac{a+b+c+d}{4}\right)^2\le \frac{a^2+b^2+c^2+d^2}{4}$$
$$(a+b+c+d)^2\le 4(a^2+b^2+c^2+d^2)$$
$$(8-e)^2\le 4(16-e^2)$$
$$5e^2-16e\le 0$$
$$e(5e-16)\le 0$$
$$\implies 0\le e\le \frac{16}{5}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1698058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Can anybody help me with math expressions? So , I am in $7^{th}$ grade and my teacher gave me some really hard homework. What I have to do is use math expressions that equals each number between $1$ and $100$ , only using the numbers $1,2,3,4$. I really need help on this. Can anybody help?
| I'm not entirely sure what you are asking but I can take a gander:
Suppose you want to express, say, $63$ using only the numbers $1,2,3$, and $4$. We know that $63=60+3$, so all we have to do is express $60$. We also know that $60=3\times 20$, and $20$ is just $4+4+4+4+4$. We can write
$$63=3\times(4+4+4+4+4)+3$$
There... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1698159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to find a function whose curl is $(7e^y,8x^7 e^{x^8},0)$? How to find a function whose curl is $(7e^y,8x^7 e^{x^8},0)$?
I've tried several integration but can't find a trivial form.
| Since $\vec\nabla \times \vec F(x,y,z) $ has an x component that depends only on y and a y component that depends only on x and no z component
A good guess is that $\vec F(x,y,z) $ takes the form
$\vec F(x,y,z) = (0,0,g(x)+h(y)) $
$g(x) $ and $h(y) $ can be found easily by integration.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1698257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Subtle Proof Error I'm having trouble seeing the error in the following "proof":
$$ (-4)=(-4)^1=(-4)^\frac{2}{2}=[(-4)^2]^\frac{1}{2}=[16]^\frac{1}{2}=4$$ therefore $(-4)=4$.
Obviously this is incorrect, but I'm not seeing where the error is occuring. I appreciate any help. Thanks.
| $x^{\frac{1}{2}}$ gives two values, both additively inverse to each other, on all nonzero real numbers $x$. You can think of an equation like $2^\frac{1}{2} = \pm \sqrt{2}$ as saying that the square root of $2$ is "$\sqrt{2}$ or $-\sqrt{2}$."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1698349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show that every nearly compact space is almost compact space but the converse is not true I am learning about the almost compact space and nearly compact space. I know that every nearly compact space is almost compact space but the converse is not true in general. So i need an example of almost compact space which is n... | I hope that the followings works. Let $X := \{a, a_n, b_n, c_n, c: n ∈ ω\}$ such that all the points are distinct. Let us consider the topology generated by sets $U_n := \{a_n, b_n, c_n\}$ for $n ∈ ω$, $A := \{a, a_n: n ∈ ω\}$, $C := \{c_n, c: n ∈ ω\}$.
$X$ is almost compact since any open cover contains the sets $A$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1698491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Absolute value of complex number This question might be very simple, but I can't visualize how to get the absolute value of this complex number ($j$ is the imaginary unit):
$$\frac{1-\omega^2LC}{1-\omega^2LC+j\omega LG}$$
Thanks
| Assuming all other symbols are real numbers, it might help to first multiply top and bottom by the complex conjugate of the denominator, then expand the denominator. This will give you a complex number of the form $x+jy$, which you should then be able to find the modulus.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1698630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Where did i go wrong in trying to find the intervals where y is increasing and decreasing? Question: Find the intervals in which the following function is strictly increasing or decreasing: $(x+1)^3(x-3)^3$
The following was my differentiation:
$y = (x+1)^3(x-3)^3$
$\frac1y \frac{dy}{dx} = \frac3{x+1} + \frac3{x-3}$ (T... | $$\dfrac{dy}{dx}=3\{(x+1)(x-3)\}^2(x+1+x-3)$$
Now for real $x,\{(x+1)(x-3)\}^2\ge0$
So, the sign of $x+1+x-3$ will dictate the sign of $\dfrac{dy}{dx}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1698734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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