Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Show limit of function $\frac{xy(x+y)}{x^2-xy+y^2}$ for $(x,y)\to(0,0)$ Show $$\lim_{(x,y)\to (0,0)} xy \frac{(x+y)}{x^2-xy+y^2}=0$$
If I approach from $y=\pm x$, I get $0$. Is that sufficient?
| No, that is not sufficient. That can only show that if the limit exists, then it should equal $0$. But it doesn't show that the limit exists.
Change to polar coordinate system, so that
$$
\lim_{(x,y)\to (0,0)} \frac{xy(x+y)}{x^2-xy+y^2} = \lim_{r\to 0} \frac{ (r^2 \sin \theta \cos \theta) (r\sin \theta + r\cos \theta)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
$f(x) = (\cos x - \sin x) (17 \cos x -7 \sin x) $ $f(x) = (\cos x - \sin x) (17 \cos x -7 \sin x)$
Determine the greatest and least values of $\frac{39}{f(x)+14}$ and state a value of x at which greatest values occurs.
Do I just use a graphing calculator for this? Is there a way I could do this without a graphing calc... | HINT:
$$f(x)=17\cos^2x+7\sin^2x-24\sin x\cos x=\dfrac{17(1+\cos2x)+7(1-\cos2x)-24\sin2x}2$$
$$=12+5\cos2x-12\sin2x=12+\sqrt{12^2+5^2}\cos\left(2x+\arctan\dfrac{12}5\right)$$
Now for real $x,-1\le\cos\left(2x+\arctan\dfrac{12}5\right)\le1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What type of number is this $\frac{\sqrt2}{2}$? $$\frac{\sqrt{2}}{2}$$
In this monomial, an irrational number is divided by a rational number. However this is not a general case but can any one tell me that when we divide an irrational number or multiply an irrational number or its multiplicative inverse by a rational ... | If $p$ is a rational (non-zero) number, and $x$ is irrational, then
$$
px, \quad \frac px, \quad \frac xp, \quad p+x, \quad p-x
$$
are all irrational. This can be proven by contradiction. Assume, for instance, that $px$ is a rational number, call it $q$. That is, set $q = px$. Now divide through by $p$, and we get $$\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Integrals with the special functions $Ci(x)$ and $erf(x)$ I'm looking for the solutions of the following two integrals:
$$I_1=\int\limits_0^\infty dx\, e^{-x^2}\text{Ci}(ax)$$
and
$$I_2=\int\limits_0^\infty dx\, e^{-ax}\text{erf}(x)$$
with
$$\text{Ci}(x)=\int\limits_\infty^x\frac{\cos(t)}{t}dt$$
and
$$\text{erf}(x)=\fr... | For $I_2$, please note that this integral is just the Laplace transform of $\text{Erf}(x)$ w.r.t. to the variable $a$
Furthermore the Laplace transform of an integral $\mathcal{L}(\int f(x) dx)$ is just $\mathcal{L}(f(x))/a$
Using this we find that
$$
I_2=\frac{2}{a\sqrt{\pi}}\int_0^{\infty}e^{- a x}e^{-x^2}dx
$$
Co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Given angles and area, how to find sides of a spherical triangle? So, given angles and area, how to find the sides of a spherical triangle?
I only know that the angles uniquely determine the sides, but what is the relation?
| For angles A, B, and C and corresponding sides a, b, and c of a spherical triangle:
cos A = -cos B cos C + sin B sin C cos a
cos B = -cos C cos A + sin C sin A cos b
cos C = -cos A cos B + sin A sin B cos c
Just plug in your numbers and solve for the sides.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Probability Question (Colored Socks)
In a drawer Sandy has 5 pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 10 socks in the drawer. On Tuesday Sandy selects 2 of the remaining 8 socks at random and on Wednesday two of the remaining 6 socks at random. The pr... | Partial solution
Probability of getting matching socks on day 1 is $\frac19$.
If they were not matching, we have 3 pairs and 2 individual socks left. We must select a 'pairable' sock in our first pick, so $\frac68$ for that, and the other pick must be the other sock of the same pair, so $\frac17$. Therefore, probabilit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
Fibonacci sequence and the Principle of Mathematical Induction Consider the Fibonacci sequence, $F_n$. Prove that $2 ~\vert~ F_n$ if and only if $3 ~\vert~ n$, using the principle of mathematical induction.
I know that I have to prove two implications here. Looking at the first implication (if $2 ~\vert~ F_n$, then $3... | $$f_{n+3}=f_{n+2}+f_{n+1}=f_{n+1}+f_n+f_{n+1}=2f_{n+1}+f_n$$
Since $2f_{n+1}$ is even, you get $f_{n+3}$ is even if and only if $f_n$ is even.
The statement follows now by induction : Check $P(1), P(2), P(3)$ and prove $P(n) \Rightarrow P(n+3)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Proving divisibility of $a^3 - a$ by $6$ As part of a larger proof, I need to show why $a^3-a$ is always divisible by $6$. I'm having trouble getting started.
| Consider the following options:
*
*$a\equiv0\pmod6 \implies a^3-a\equiv 0-0\equiv6\cdot 0\equiv0\pmod6$
*$a\equiv1\pmod6 \implies a^3-a\equiv 1-1\equiv6\cdot 0\equiv0\pmod6$
*$a\equiv2\pmod6 \implies a^3-a\equiv 8-2\equiv6\cdot 1\equiv0\pmod6$
*$a\equiv3\pmod6 \implies a^3-a\equiv 27-3\equiv6\cdot 4\equiv0\pmo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Solutions of the Laplace equation How do I find solutions $u=f(r)$ of the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ that depend only on the radial coordinate $r= \sqrt{x^2+y^2}$
| You should also use the polar coordinate version of the Laplace operator
$$
\Delta u
= {\partial^2 u \over \partial r^2}
+{1 \over r} {\partial u \over \partial r}
+ {1 \over r^2} {\partial^2 u \over \partial \theta^2}
$$
and trying for $u(r, \phi) = f(r)$. This reduces to solving the ODE
$$
0 = g'(r) +{1 \over r} {g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Conditional expectation is $L_p$ norm reducing I am trying to follow the proof of
\begin{align*}
E \left(|E(X\mid \mathcal{B} )|^p \right) \le E(|X|^p)
\end{align*}
which goes as follows.
Let $f(t)=|t|^p$ which is convex for $p\ge 1$ then
\begin{align*}
E \left(f(E(X\mid \mathcal{B} )) \right) &\le E \left(E(f(X) |\m... | Using the definition of conditional expectation, we know that for any random variable $Y$ $$E(E(Y|\mathcal{B})) = \int_{\Omega} E(Y|\mathcal{B}) dP = \int_{\Omega} Y dP = E(Y)$$ This is called the tower property of conditional expectation and intuitively it "gets rid of the $\mathcal{B}." $ Now just apply this to $Y=f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If f is surjective then f is not a right divisor of zero Let $R$ be a ring and $M$ a R-module. For $r\in R$ define $f:M\to M$ by $f(s)=sr$.
Show that $f$ is injective if and only if $r$ is not a right zero divisor.
I have done a similar problem to this in the past, involving surjectivity instead of injectivity and came... | $f$ is injective if and only if
$f(s)=0 \Rightarrow s=0$ if and only if
$sr=0 \Rightarrow s=0$ if and only if
$r$ is not a right zero divisor.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Linear approximation with two variables The problem I have is this:
Use suitable linear approximation to find the approximate values for given functions
at the points indicated:
$f(x, y) = xe^{y+x^2}$ at $(2.05, -3.92)$
I know how to do linear approximation with just one variable (take the derivative and such), but wit... | Yes. Denoting the partial derivatives by $f'_x$ and $f'_y$, the formula is:
$$f(x_0+h,y_0+k)=f(x_0,y_0)+f'_x(x_0,y_0)h+f'_y(x_0,y_0)k+o\bigl(\lVert(h,k)\rVert\bigr).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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For which $n$ does $n\mid 1^n+2^n+\cdots+n^n$? Find all the natural numbers $n$ such that
$$n\mid 1^n+2^n+\cdots+n^n.$$
We know through Faulhaber's formula, that
$$\sum_{k=1}^{n}k^n=\frac1{n+1}\sum_{k=0}^n\binom{n+1}{k}B_k n^{n+1-k},$$
where $B_k$ is a Bernoulli number. I checked few dozen values of $n$ and it seems t... | To see tnat odd numbers always work, it is enough to use the formula $a^n+b^n = (a+b)(a^{n-1}-a^{n-2}b+...+b^n)$. Thus, if $n$ is odd, $n | k^n+(n-k)^n$ and therefore it divides the sum $1^n+...+n^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Expressing as $z(t) = x(t) + iy(t)$ My exercise requires to express the equation of a line connecting two point, $z_1 = -3 +2i$ and $z_2 = 3 - 5i$, as $z(t) = x(t) + iy(t)$.
We know that the equation for a line is $$ Re\left [ (m+i)z + b \right ] = 0$$ where $z = x+iy$
The slope is calculated as $$m = \frac{Im(z_1) - ... | The form your exercise is requesting is often called the "parametric" version of the equation, just to throw the terminology out there.
To find the slope, think in terms of the growth of the line. This is expressed by $z_2-z_1$, the vector between the two points, or $6-7i$, with the calculation done.
All we need to d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the distance between a point and a parabola with different methods I'm trying to find the shortest distance from point $(3,0)$ to the parabola $y= x^2$ using the method of Lagrange Multipliers (my practice), and by "reducing to unstrained problem in one variable" (assignment). (I think the sheet might have mean... | Draw a picture.
For either method, we want to minimize $(x-3)^2+y^2$, subject to $y=x^2$.
Substituting $x^2$ for $y$ in $(x-3)^2+y^2$, we find that we want to minimize $f(x)=(x-3)^2+x^4$. Note that $f'(x)=2(x-3)+4x^3$. The critical points of $f(x)$ are where $2(x-3)+4x^3=0$, or equivalently
$$2x^3+x-3=0.$$
In general... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Two complicated limits: $\lim_{x\to 0}\frac{e^{ax}-e^{bx}}{\sin(cx)}$ and $\lim_{x\to 0} x(a^{\frac1x}-1)$ I need to solve these 2 limits ( without using L'Hospital's Rule) , but I can't figure out how to go about them:
Let $a \neq b$, $c \neq 0$.
$$\lim_{x\to 0}\frac{e^{ax}-e^{bx}}{\sin(cx)}$$
Also, let $a>0$, $a \neq... | In short: factor the smaller exponential out. Use l'Hopital's rule on what remains.
One iteration will do: exponentials stay, and $sin(cx)$ will become $c\cos (cx)$, which goes to $1$ instead.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Finiteness in a Hausdorff space where every open subspace is compact Let be $X$ a Hausdorff space where every open subspace of $X$ is compact. I need to prove that $X$ is finite.
As $X$ is Hausdorff, I have that for every distinct elements of $X$, exist disjoint neighborhoods in $X$ and I try to use the set of all thes... | This seems to rely on the following two facts about Hausdorff spaces.
*
*All singleton sets are closed. (This holds for the wider class of T1-spaces.)
*All compact subsets are closed.
So given $x \in X$, $X \setminus \{ x \}$ is open, hence compact by assumption, hence closed by the above. Therefore the singleton... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Hankel transform with Bessel functions of the second kind The Hankel transform is defined for Bessel functions of the first kind (see e.g. http://en.wikipedia.org/wiki/Hankel_transform)
I would like to know if it is possible to define a Hankel transform with Hankel functions, or alternatively with Bessel functions of t... | Some of the Bessel function class transforms. A general Google, or Google Scholar, search will yield some results linked to publications.
Hankel transform
\begin{align}
f(y) = \int_{0}^{\infty} f(x) \, J_{\nu}(xy) \, \sqrt{xy} \, dx
\end{align}
Y-transform
\begin{align}
f(y) = \int_{0}^{\infty} f(x) \, Y_{\nu}(xy) \, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Sums of Consecutive Cubes (Trouble Interpreting Question) Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums.
Any suggestions on what techniques should be used to start the problem?
Also, when the question is phrased like tha... | First, you've noticed that each side of the equation has to sum up to $3042$.
Since $11^3+12^3=3059>3042$, each one of them has to be on a different group.
Since $3042$ is even, each group must contain an even number of odd numbers.
So each group must contain $0$ or $2$ or $4$ or $6$ odd numbers.
With $0$ odd numbers o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find a polynomial $p(x,y)$ with image all positive real numbers Find a polynomial $p(x,y)$ such that
*
*for each $(x,y) \in \mathbb{R}^2$ we have $p(x,y) > 0$, and
*for each $a>0$ $\exists (x,y)$ : $p(x,y) = a$.
I see that $p(x,y)$ must have constant, but how can choose $p$ that achieves each positive value?
May... | Hint One way to produce a (real) polynomial $p$ that only takes nonnegative values is to write a sum of squares $$p(x, y) := q(x, y)^2 + r(x, y)^2,$$ and we can ensure that the sum is always positive by insisting that the squared quantities $q(x, y), r(x, y)$ are never simultaneously zero. On the other hand, we can ens... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Simplify $\sin \arctan x$, assuming $x>0$ Now, naturally I understand that $$\arctan x = \arcsin \frac{x}{\sqrt{x^2+1}}$$ and therefore I should have just $$\sin \arctan x = \frac{x}{\sqrt{x^2 + 1}}.$$ However, I am interested as to the effect of restraining the domain to $\{x > 0\}$, as I believe the function may simp... | It cannot be simplified further. There is nothing simpler than $ x/ \sqrt{ 1 -x^2} $ next in simplicity only to $ \tan^{-1}x $. Draw a triangle to look at the sides $ (\sqrt{1+x^2}, 1,x )$ for yourself. Domain is $ (-\infty, \infty) $.Choose only the positive part if you need only positive argument. Also there is atan... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Suppose that $A$ is a connected subset of a space $X$ and that $A\subseteq B \subseteq \bar A$. Prove $B$ is connected. I think I can prove the closure of $A$, that is $\bar A$, is connected, as there are many other threads on this site. I am then just not sure how to make the jump to show formally that B is connected... | If $B$ is not connected then non-empty sets $U,V$ exist with $U\cap\overline{V}=\varnothing=\overline{U}\cap V$ and $B=U\cup V$.
Then the sets $U\cap A$ and $V\cap A$ are disjoint, open in $A$ and are covering $A$.
So one of these sets must be empty since $A$ is connected.
If $U\cap A=\varnothing$ then $A\subseteq V... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do I find the value of this weird expression? How can I find the value of the expression $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^...}} $? I wrote a computer program to calculate the value, and the result comes out to be 2 (more precisely 1.999997). Can anyone explain what's happening? Is there any general method to calcula... | You can rewrite your expression as $$\sqrt2^{\sqrt2^{...}}=2^{(\frac{1}{2})^{2^{...}}}$$
Clearly multiplying the powers out you end up with $$2^{1^{1^{...}}}=2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1204388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Show that if $\gcd(r,s_1) =\gcd(r,s_2) = 1$, then $\gcd(r,s_1s_2) = 1$ Never mind the question. I want to try to solve that on my own. What I want to understand is how this: "Hint. $1 = ar + bs_1,\ 1 = ar + bs_2$" relates to solving it.
I'm a little confused by this statement, especially since it applies to integers. ... | Integers can be negative, too. You can have $2 \times 5 + ( -3) \times 3= 1$ for example.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1204643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate $\int x^3(x^2+7)\ dx$ I'm trying to find the indefinite integral of $$\int x^3(x^2+7)\ dx$$ and I've seem to have forgotten how to do it in this case. So if anyone can refresh my memory, I'd appreciate it.
| First expand the integrand.
It simplifies to $x^5+7x^3$. Then we have that
$$\int x^5 + 7x^3=\frac{x^6}{6}+\frac{7x^4}{4}+C$$
This is done by the Power Rule, i.e, $$\int x^n\ dx=\frac{x^{n+1}}{n+1}+C$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Orthogonal complement of subspace $W = span(5,1+t)$ I have this subspace of $P_2(\mathbb R)$ and I need to find its orthogonal complemente, using the inner product defined as $$<p(t),q(t)> = \int_o^1 p(t)q(t) dt$$
So I'm assuming the vector $$v = a+bt+ct^2$$ as being the vector such that $$<v,5> = 0\\<v, 1+t> = 0$$
So ... | Your computations are correct. Now you want to find one nonzero solution of
$$
\begin{cases}
6a+3b+2c=0\\
18a+10b+7c=0
\end{cases}
$$
Multiply the first equation by $3$ and subtract it from the second, getting
$$
b+c=0
$$
so $b=-c$; then $6a-3c+2c=0$ or $6a=c$. Thus you get a nonzero solution by taking $c=1$ (or any no... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving Any connected subset of R is an Interval Common Proof:
Suppose $S$ is not an interval of $R$.
Then by Interval Defined by Betweenness, $∃x,y∈S$ and $z\in R∖S$ such that $x<z<y$.
Consider the sets $A_1=S∩(−∞,z)$ and $A_2=S∩(z,+∞)$.
Then $A_1,A_2$ are open by definition of the subspace topology on S.
Neither is ... | Do you know what the subspace topology is? The larger set is $\Bbb R$ with the usual topology. Since $S \subseteq \Bbb R$, we can equip $S$ with the subspace topology, i.e., the topology where each $V$ that is open in S is of the form $V = S \cap U$ with $U \subseteq \Bbb R$ open in $\Bbb R$.
Since $A_{1} = S \cap (-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1205039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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operation to get a diagonal matrix from a vector In many programs you can create diagonal matrix from a vector, like diag function in Matlab and DiagonalMatrix function in Mathematica. I'm wondering whether we can use matrix product (or hadamard product, kronecker product, etc) of a vector and identity matrices to crea... | You can use
$$(xe^T) \odot I_n = \mathrm{diag}(x)$$
Where $\odot$ is the hadamard product and $e^T = (1,1,\ldots)\in\mathbb R^n$. The hadamard product basically masks away all off-diagonal elements and $xe^T$ has $x$ as its diagonal.
| {
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"timestamp": "2023-03-29T00:00:00",
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What are the irreducible curves on the blow up of $\mathbb{P}^{2}$? On the blow-up of $2$-dimensional complex projective space, $\mathbb{P}^2$, I know that
$$Pic(\mathbb{\tilde{P}^2})=\mathbb{Z}[\tilde{H}]+\mathbb{Z}[E]$$
where $\tilde{H}$ is the blow-up of the hyperplane bundle of $\mathbb{P}^2$, and $E$ is the except... | They are either strict transforms of irreducible curves from $\mathbb{P}^2$, or they are the exceptional divisor $E$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Euclidean norm of complex vectors I am working on a proof: One has two vectors, $u,v \in \mathbb C^n$, such that $u \cdot v=0$ . I am trying to prove that
$$|u + v|^2 = |u|^2 + |v|^2.$$
I am a little stuck on how to do $u + v$ dotted with the conjugate of $u + v$. Is there anything special I can do with this?
| Here is how.
$$ ||u+v||^2 = \langle u+v,u+v \rangle = \langle u,u \rangle + \langle v,v \rangle + \langle u,v \rangle + \langle v,u \rangle = ||u||^2+||v||^2 +0+0. $$
Note:
$$ \langle u,v \rangle = u \cdot v $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $f$ and $g$ are both uniformly continuous, show that $\max(f, g)$ is uniformly continuous My friend asked me this question and I gave him a sketch of proof. My idea is that to construct a function
$$h = \begin{cases}
f-g & \textrm{if $f \ge g$}\\
0 & \textrm{if $f < g$}
\end{cases}$$
and show that $h$ is uniformly c... | It is much simpler if you use the formula (proved here)
$$\max(f, g) = \frac{1}{2}\left(f + g + |f - g| \right)$$ and then we can reason in the following manner. Sum and difference of two uniformly continuous functions is uniformly continuous. Hence both $(f + g)$ and $(f - g)$ are also uniformly continuous. If we note... | {
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To find right cosets of H in G where G= and H=$$ ,where o(G)=10 To find right cosets of H in G where G= and H=$<a^{2}>$ ,where o(G)=10
Since order of $G =10$ , so $a^{10}=e$ .We have
$G= { a,a^{2},a^{3},a^{4},a^{5},a^{6},a^{7},a^{8},a^{9},e}$
and
$H = {a^{2} ,a^{4},a^{6},a^{8},e}$
So $Ha={a^{3},a^{5},a^{7},a^{9},e}$
$... | Almost. The only difference is that the coset need not be a group. The only change to be made is $Ha=\{a,a^3,a^5,a^7,a^9\}$. Then as you've noticed every element is in either $Ha$ or $H$. It is a nice exercise to prove that being in a coset is an equivalence relation and the that cosets partition the group. In this way... | {
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Limit of $\sqrt{25x^{2}+5x}-5x$ as $x\to\infty$ $\hspace{1cm} \displaystyle\lim_{x\to\infty} \left(\sqrt{25x^{2}+5x}-5x\right) $
The correct answer seems to be $\frac12$, whereas I get $0$.
Here's how I do this problem:
$$ \sqrt{25x^{2}+5x}-5x \cdot \frac{\sqrt{25x^{2}+5x}+5x}{\sqrt{25x^{2}+5x}+5x} = \frac{25x^2+5x - 2... | If we start where you left off:
$$\dfrac{5x}{\sqrt{25x^{2}+5x}+5x}$$
we can factorize the square root by $5x$:
$$\dfrac{5x}{\sqrt{5x(5x+1)}+5x}$$
Take this outside of the root, and re-factorize the denominator:
$$\dfrac{5x}{\sqrt{5x}(\sqrt{5x+1}+\sqrt{5x})}$$
Cancel by $\sqrt{5x}$:
$$\dfrac{\sqrt{5x}}{\sqrt{5x+1}+\sqrt... | {
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Numerical methods to solve nonlinear system of inequalities? I know some methods to solve nonlinear system of equaltites: Relaxation Method, Newton method, nonlinear Jacobi method, nonlinear Seidel method.
Is it exist some analogous method to solve nonlinear systems of inequaltites?
| I'm answering from my question after couple years....
Yes, there are various ways to solve numerically objective plus inequalities.
But what is deeply wrong in question is that word non-linearity should not bring panic, in fact the problem is not in linear/non-linear but in convex/non-convex. In USSR it was realized in... | {
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Does the totient function reach all its values when restricted to odd numbers? This question might be a duplicate, if so, I apologise in advance. It is simple, but answering it is probably harder :)
Is it true, that the $\phi(n)$ function(Euler's totient function) takes on all of it's values, when $n$ is an odd integer... | The answer is no. Let $m = 2^kn$, with $k=3$ or $4<k<15$ and $n$ odd. Then $\phi(m) = \phi(2^k)\phi(n)=2^{k-1}\phi(n)$. I claim there is no odd number $x$ with $\phi(x) = \phi(m) = 2^{k-1}\phi(n)$.
Let $x$ be an odd number. It can be written as $\Pi_{i=1}^n p_i^{e_i}$, where because $x$ is odd, none of the $p_i$ is $2$... | {
"language": "en",
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How to calculate the area covered by any spherical rectangle? Is there any analytic or generalized formula to calculate area covered by any rectangle having length $l$ & width $b$ each as a great circle arc on a spherical surface with a radius $R$? i.e. How to find the area $A$ of rectangle in terms of length $l$, widt... | Assume we are working on a sphere of radius $1$, or consider the lengths in radians and the areas in steradians.
Extend the sides of length $l$ until they meet. This results in a triangle with sides
$$
w,\quad\frac\pi2-\frac l2,\quad\frac\pi2-\frac l2
$$
The Spherical Law of Cosines says that
$$
\begin{align}
\cos(A)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find $f(x)$ for $f'(x) = f(x) \ln(f(x))$ and $f(0) = 1$ $$f'(x) = f(x) \ln\big(f(x)\big)$$
$$f(0) = 1, \qquad f(x) > 0$$
I am studying for finals on my own and this exercise is really bothering me because I can't seem to solve it. If I divide by $f(x)$ I get $\ln'f(x) = \ln f(x)$ but then I don't know what to do. Any ... | writing your equations as $$\frac{dy}{dx} = y \ln y $$ you can see that it is separable. so $$\frac{dy}{y\ln y} = dx \to x = \int_1^y \frac{dy}{y \ln y} = \int_1^y \frac{d \,( \ln y)}{\ln y} = \ln (\ln y)\big|_1^y = diverges$$
there is trouble with the initial value $f = 1$ at $x = 0.$
| {
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surface area of 'cylinder' with the top cut at an angle I don't know what the name for this shape is, so in essence it is a cylinder, radius at base $r$, which has had a wedge of the top cut off at an angle so that rather than a circle the upper face is an ellipse. its height at the top of the slanted ellipse is $h_{ma... | sorry, hadn't really thought it through. although it is a wave, it goes above the average height just as much as it goes below it, so it is similar to the volume calculation: $2\pi r \frac{h_{min}+h_{max}}2$. I haven't calculated the surface area of the ellipse yet.
EDIT:
thanks for ellipse formula. semiminor axis $=r$... | {
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Are all convergent sequences bounded and monotone? I know of the monotone convergence theorem, but does this mean that sequences converge only if they are bounded and monotone?
| In general, you can choose some Cauchy sequence $\{a_n\}$ in $\mathbb R$ which is alternating, or "jumps around." Then $\{a_n\}$ is bounded and converges but is not monotone.
For example:
$\{a_n\} = \frac 1n$ for odd $n$, $0$ for even $n$
$\{b_n\} = \frac 1 {n^2} \sin (n) $
| {
"language": "en",
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In triangle ABC, ∠ACB = 60. AD and BE are angle bisectors. Prove AE + BD = AB. Here is a diagram if it will be helpful:
| The main idea is to express everything in terms of the triangle sides and then obtain something trivial (or at least easy-to-prove).
For this, just use the property that the angle bisector divides the opposite side in the ratio of the neighbour sides i.e. $\frac{AE}{EC}=\frac{AB}{BC}$.
Then (with the usual notations fo... | {
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proving that the set of all english words is countble. This is the question :
Prove that the set of all the words in the English language is countble (the set's cardinality is אo)
A word is defined as a finite sequence of letters in the English language.
I'm not really sure how to start this. I know that a finite union... | There are $26$ letters in the English language.
Consider each letter as one of the digits on base $27$:
*
*$A=1$
*$B=2$
*$C=3$
*$\dots$
*$Z=26$
Then map each word to the corresponding integer on base $27$, for example:
$\text{BAGDAD}=217414_{27}=2\cdot27^5+1\cdot27^4+7\cdot27^3+4\cdot27^2+1\cdot27^1+4\cdot27^0... | {
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How to find a list of summands and factors adding up to a total? I am neither a mathematician nor do I have an idea on how to write down my problem in accurate mathematic formulas. Please feel free to edit my question into shape and remove this paragraph. Also I am unsure about that tags that apply to this question - p... | Just to make things simpler. I'll assign each number a name:
125: a number
70: b number
55: c number
375:final number
1.Split all the number into their prime factors except for your final number:
5x5x5+2x5x7+11x5=375
*Take the total of their numbers away from your final number (375) and find the prim... | {
"language": "en",
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Tychonoff's theorem for $[0,1]^\mathbb{R}$ According to Tychonoff's theorem any uncountable product of compact spaces is compact with respect to product topology.
Then $[0,1]^\mathbb{R}$, the space of all functions defined on $\mathbb{R}$ taking values in $[0,1]$ is compact w.r.t. the product topology.
Consider the fun... | The product topology is the topology of pointwise convergence; your functions converge pointwise to the zero function.
You are correct that $[0, 1]^{\mathbb{R}}$ is not sequentially compact; I believe $f_n(x) = |\sin nx|$ is an explicit counterexample, but I haven't checked it carefully. But neither sequential compact... | {
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Divergence of $\prod_{n=2}^\infty(1+(-1)^n/\sqrt n)$. Looking looking for a verification of my proof that the above product diverges.
$$\begin{align}
\prod_{n=2}^\infty\left(1+\frac{(-1)^n}{\sqrt n}\right) & =\prod_{n=1}^\infty\left(1+\frac1{\sqrt {2n}}\right)\left(1-\frac1{\sqrt{2n+1}}\right)\\
& =\prod_{n=1}^\infty\l... | I would like for some approval to my answer, please.
Show that $$\prod_2^\infty(1+\frac{(-1)^k}{\sqrt k})$$ diverges even though$$\sum_2^\infty \frac{(-1)^k}{\sqrt k}$$ converges.
$\prod_2^\infty(1+\frac{(-1)^k}{\sqrt k})$ diverges if and only if $\sum _2^\infty\log(1+\frac{(-1)^k}{\sqrt k})$ diverges.
Indeed, by Lei... | {
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Conditional Expectations Given Sum of I.I.D. Given that $X_1,...Xn$ are all identical independent random variables.
$\mathbb{E}(X_1|\sum_{k=1}^{n}X_k)$ = ?
I am unsure how to proceed on this one. I know the default relation: $\mathbb{E}(X|Y)$ = $\mathbb{E}(X*I_{[Y=y]})\over\mathbb{P}(Y=y))$, where I is an indicator fun... | For the late comers for this question. I think clear solution of the problem might be as follows:
Let $Y = \sum_{i=1}^n X_i$, by using the property of conditional expectation $\mathbb{E}[g(Y)|Y] = g(Y)$ (i.e.$\mathbb{E}[Y|Y] = Y$),
we can write
$$ Y = \mathbb{E}[X_1 + X_2 + \cdots + X_n |Y] = \mathbb{E}[X_1|Y]+ \mat... | {
"language": "en",
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Functions and Sets
Robert, Susan, and Thomas are the sole contestants in a lottery in which two prizes will be awarded. Three tickets with their names on them are placed in a hat. The person whose name is on the first ticket drawn wins prize #1. That ticket is placed back in. Then the second ticket is drawn. You can w... | For a,it is OK for a function to map different inputs to the same output. For functions of the reals, $f(x)=0$ is a fine function-it takes any real as an input and returns $0$. A function just has to return a single value for any input. You haven't done the last sentence of the problem, showing how $B^A$ can represen... | {
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"timestamp": "2023-03-29T00:00:00",
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Induction proof verificiation P(n) = in a line of n people show that somewhere in the line a woman is directly in front of a man. The first person will always be a woman and the last person in the line will always be a man
I wanted to prove this with induction and I was a little unsure of how to express this problem in... | Your two cases are far from exhausting the possibilities. If $k=10$, for instance, the number of women can be anywhere from $1$ through $10$, and so can the number of men. You need to come up with an argument that covers all possibilities.
Try this: remove the man at the end of the line. Either the new line has a man a... | {
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Are the following real numbers constructible? 1) $\sqrt[4]{5+\sqrt2}$
2)$\sqrt[6]{2}$
3) $3/(4+\sqrt13)$
4) $3+\sqrt[5]{8}$
From what I know, a number is constructible if it can be converted in a finite number of steps using only the operations addition, subtraction, multiplication, division, and square roots.
This in... | I think that
all your answers are correct
for the reasons stated.
The only quibble I might have
is for the non-constructible ones,
where the particular numbers are
non-constructible,
while,
for example,
$\sqrt[3]{8}$
is obviously constructible.
| {
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If $A,B>0$ and $A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$. If $A,B>0$ and $\displaystyle A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$.
$\bf{My\; Try::}$ Given $$\displaystyle A+ B = \frac{\pi}{3}$$ and $A,B>0$.
So we can say $$\displaystyle 0< A,B<\frac{\pi}{3}$$. No... | $$
\tan A\tan B=\tan A\tan\big(\frac{\pi}{3}-A\big)=\tan A.\frac{\sqrt{3}-\tan A}{1+\sqrt{3}\tan A}=1-1+\frac{\sqrt{3}\tan A-\tan^2A}{1+\sqrt{3}\tan A}\\
=1+\frac{-1-\sqrt{3}\tan A+\sqrt{3}\tan A-\tan^2A}{1+\sqrt{3}\tan A}=1-\frac{1+\tan^2A}{1+\sqrt{3}\tan A}=1-\frac{\sec^2A}{1+\sqrt{3}\tan A}\\
=1-\frac{2}{2\cos^2A+\s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Interior of a set $A$ is dense in $X$ iff $A$ is dense in $X$ I would like to check if the following statement is true:
Interior of a set $A$ is dense in $X$ iff $A$ is dense in $X$
It seems to me that they are equivalent, but I couldn't prove or find a counterexample for it.
Thanks!
| $\mathbb Q$ is dense in $\mathbb R$ but interior of $\mathbb Q$ is empty.
| {
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General versions for quadrics of Pascal and Brianchon theorems I am looking for a generalization to quadrics (with proofs) of Pascal's and Brianchon's theorems. It´s for Three dimensional analytical geometry.
I would be very thankful if you could point me towards one.
| Tried to add this as a comment, but comments not working right now:
The question is an exact duplicate of this one.
But, anyway, start with the classical geometry texts. Salmon, Sommerville, and books like that.
Sommerville, Analytical Geometry of Three Dimensions.
Snyder & Sisam, Analytical Geometry of Space
Salmon, ... | {
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Subtraction of a number I have a number $x$. If I remove the last digit, I get $y$. Given $x-y$, how can I find $x$?
For example x=34 then y=3 given 34-3=31, I have to find 34.
if x=4298 then y=429 , given 4298-429 = 3869 . how can I find 4298 from given 3869?
| Suppose :
$$x=\sum_{k=0}^na_k10^k $$
then :
$$y=\sum_{k=0}^{n-1}a_{k+1}10^k $$
so :
$$x-y=a_0+\sum_{k=1}^na_k(10^k-10^{k-1}) $$
$$x-y=a_0+\sum_{k=1}^n9a_k10^{k-1}=a_0+\frac{9}{10}\sum_{k=1}^n9a_k10^{k} $$
$$x-y=a_0+\frac{9}{10}(x-a_0)=\frac{9x+a_0}{10}$$
Edit : The first part doesn't give the answer to the question (bu... | {
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Prove that for $n \gt 6$, there is a number $1 \lt k \lt n/2$ that does not divide $n$ My nine year old asked this question at lunch today: Is there a number that is divisible by everything that is half or less than the number?
I immediately answered, "No. I mean, 6. But not for any number bigger than 6."
So I tried to... | Since $n>6$, we have that $2<3 <n/2$, so $n$ is divisible by $2$ and $3$.
Since $n=2\cdot\frac{n}{2}$ and $\frac{n}{2}-1<\frac{n}{2}$, $\frac{n}{2}-1$ divides $n$. $\frac{n}{2}−1$ must have the next smallest codivisor, which is $3$, so must be equal to $\frac{n}{3}$.
Solving $\frac{n}{2}-1=\frac{n}{3}$ gives $n=6$, ... | {
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prime numbers and divisibility question - number theory I have a fairly short and straightforward question I would like to ask. Suppose $p_{1},p_{2},...,p_{n}$ are prime numbers. Suppose $X = p_{1}p_{2}...p_{n} + 1$. If $X$ comes out to be a composite number, then why is it true then that none of the $p_{1},p_{2},...,p... | Formally assume that $p_i$ divides $X=bp_i+1$ so there exists an integer $a$ such that $X=a.p_i$and then $p_i(a-b)=1$ so $p_i$ divides $1$ which is impossible.
| {
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Prove that if $L$ is regular then $f(L)$ is regular Prove that if $L$ is regular then $f(L)$ is regular.
$$f(L)=\{w: \text{every prefix of $w$ of odd length $\in$ L } \}$$
So my attemption is:
Let $M= (Q, \Sigma, \delta, q_0,F) $ will be DFA recognizing $L$
Let construct $M'=(Q', \Sigma,\delta', q_0', F')$ recognizing ... | You can prove the result without using automata. Let $A$ be the alphabet, and let $$
K = \{ u \in A^* \mid \text{ every prefix of $u$ of odd length is in $L$} \}
$$
Let $K^c$ be the complement of $K$ in $A^*$. Then
\begin{align}
K^c &= \{ u \in A^* \mid \text{ there exists a prefix of $u$ of odd length in $L^c$} \}\\
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1207974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving Gaussian curvature $\leq 0$ on line Given some regular surface $S$ that contains a line $L$, I need to prove that the Gaussian curvature $K\leq 0$ at all points of $L$. I am thinking that if I could show that no points on $L$ can be elliptical $K>0$ then I would be set. But how can I do that?
| Hint: The Gaussian curvature is the product of the principal curvatures. What do the principal curvatures minimize/maximize?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How many pairs of positive integers $(n, m)$ are there such that $2n+3m=2015$? I know that $m$ must be odd and $m\le671$. Also, $n\le1006$. I can't go any further, any help?
| $$\begin{array} {|cc|} \hline
1 & 2014 \\
{\color{red}{ 2 }} & {\color{blue}{ 2013 }} \\
3 & 2012 \\
{\color{red}{ 4 }} & 2011 \\
5 & {\color{blue}{ 2010 }} \\
{\color{red}{ 6 }} & 2009 \\
7 & 2008 \\
{\color{red}{ 8 }} & {\color{blue}{ 2007 }} \\
9 & 2006 \\
{\color{red}{ 10 }} & 2005 \\
11 &... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Big List of examples of recreational finite unbounded games What are some examples of mathematical games that can take an unbounded amount of time (a.k.a. there are starting positions such that for any number $n$, there is a line of play taking $>n$ times) but is finite (every line of play eventually ends.) Also, it wo... | One example is the ring game, which is defined here - in essence, the game (or a slight variant thereon) can be described as:
Start with a Noetherian ring $R$. At each turn, replace $R$ with a quotient thereof. If a player can make no legal moves (i.e. $R$ is a field, thus has no proper quotients).
One can notice tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1208320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Can I make an assumption about arbitrary numbers in a proof? Assume $ x_1, x_2... x_{10} $ are different numbers and $ y_1, y_2... y_{10} $ are some arbitrary numbers. Prove that there exists some unique polynomial of degree not exceeding 9 such that:
$P(x_1)=y_1,P(x_2)=y_2...P(x_{10})=y_{10}$
Proof to the problem res... | Your assumption is incorrect because proving the existence of such a polynomial is an important part of the problem you are working on.
Here is a hint: Suppose you were given the problem with only two pairs $(x_1,y_1)$ and $(x_2,y_2)$. Then how would you find such a polynomial? Think in terms of a straight line. You ar... | {
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How to resolve $x \in A \wedge x \notin A $? Let A and B be two sets. Then
$A \setminus B = \{x: x\in A \wedge x\notin B\}$
$A \setminus B = \{x: x\in A \wedge x\notin A \cap B\}$
How can one prove that two logical statements are equal?
Suppose $x \in A \setminus B$.
Then,
$x \in A \wedge x \notin A \cap B$
$x \in A \... | Your implication $\{x \in A \wedge x \notin A \cap B\}\Rightarrow \{x \in A \wedge x \notin A \wedge x\notin B\}$ is not true; By De Morgan's laws it should be$$\{x \in A \wedge x \notin A \cap B\}\Rightarrow \{x \in A \wedge (x \notin A \lor x\notin B)\}$$
Now proof.
You want to prove:
$$\{x: x\in A \wedge x\noti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1208461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find a point on the line $-2x + 6y - 2 = 0$ that is closest to the point $(0,4).$ I understand that you would use the distance formula here, but I'm confused as to how you calculate x and y after that. Thanks.
| If $(h,k)$ be any point on on $-2x+6y-2=0\iff x=3y-1$ we have $h=3k-1$
If $d$ is the distance between $(0,4);(3k-1,k)$
$d^2=(3k-1)^2+(k-4)^2=10k^2-14k+17=10\left(k^2-\dfrac75k\right)+17$
$=10\left(k-\dfrac7{10}\right)^2+17-10\cdot\left(\dfrac7{10}\right)^2$
$\ge17-10\cdot\left(\dfrac7{10}\right)^2$
The equality occurs ... | {
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What exactly am I being asked in this question? I don't need the answer, just the interpretation.
Write a program that inputs a whole number N and outputs the percentage of relatively prime pairs of numbers a, b in the range 1 to N.
For some reason, I'm having difficulty understanding the question. Do I need to calc... | I think you should interpret the question as finding, for two independent random variables $X,Y$ each uniformly distributed over the set $[n]=\{1,2,\ldots,n\}$, the probability that $\gcd(X,Y)=1$. While other interpretations of the question are possible, this seems the most natural one to me, and in addition it is comp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Vector bundle of dimension $\leqslant n$ on $n$-connected space is trivial I wonder whether any vector bundle of dimension $\leqslant n$ on an $n$-connected CW-complex is trivial? It seems that,
*
*the complex can be given cellular structure with exactly one 0-dimensional cell and no other cells of dimension $\leqsl... | Your proposed argument fails at the third step. It is just not true that if $X$ is $n$-connected then every $S^{n-1}$-bundle over it is trivial.
The desired statement is not even true rationally: the rational homotopy groups of $BO(n)$ are not hard to calculate and in particular for $n \ge 3$,
$$\pi_{4n-4}(BO(2n)) \ot... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to compute Dottie number accurately? Dottie number is root of this equation : $cos \alpha = \alpha$, $\alpha \approx 0.73908513321516064165531208767\dots$.
I wonder how can I compute it ? I have tried to do it with an approximating formula:
$\alpha = \frac{5\pi^2}{\alpha^2 + \pi^2} - 4$
I have solved this equation ... | Taylor series of order 2 gives a simple quadratic in $\alpha$:
$$\alpha=1-\alpha^2/2\implies \alpha=0.\color{red}{73}2..$$
Of order 4 gives a bi-quadratic (there's a formula to solve roots of a polynomial of degree less than 5) in $\alpha^2$:
$$\alpha=1-\alpha^2/2+\alpha^4/4\implies 0.\color{red}{739}2..$$
Fairly accur... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1208845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is $\sum_{n\ge 1} \sin(n^2)/n$ convergent?
Is the series
$$\sum_{n\ge 1} \frac{\sin(n^2)}{n}$$
convergent?
My thoughts so far:
1) This is an alternating series so the integration test does not work here.
2) The Weyl inequality roughly says $$\sum_{n\le N} \sin(n^2)$$ is $O(N^{1/2+\epsilon})$, so the Dirichlet tes... | You are on the right track. The key is to consider partial sums:
$$ S_N = \sum_{n=1}^{N}\frac{\sin(n^2)}{n} $$
then find a good rational approximation of $\pi$ depending on $N$, apply Weyl bound (or Weyl differencing technique) to estimate $\sum_{n=1}^{k}e^{in^2}$ and finish through partial summation.
Details on page ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integral inequality 5 How can I prove that:
$$8\le \int _3^4\frac{x^2}{x-2}dx\le 9$$
My teacher advised me to find the asymptotes, why? what helps me if I find the asymptotes?
| Direct approach is not so painful:
$$\int_{3}^{4}\frac{x^2}{x-2}\,dx = \int_{1}^{2}\frac{(x+2)^2}{x}\,dx = \int_{1}^{2}\left(x+4+\frac{4}{x}\right)\,dx =\frac{11}{2}+4\log 2 $$
and by the Hermite-Hadamard inequality we have:
$$ \log 2 =\int_{1}^{2}\frac{dx}{x} \in \left(\frac{2}{3},\frac{3}{4}\right) $$
so:
$$\int_{3}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1209138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What are the transitive groups of degree $4$? How can I find all of the transitive groups of degree $4$ (i.e. the subgroups $H$ of $S_4$, such that for every $1 \leq i, j \leq 4$ there is $\sigma \in H$, such that $\sigma(i) = j$)? I know that one way of doing this is by brute force, but is there a more clever approach... | We know subgroups of $S_4$ come in only a few different orders: $1,2,3,4,6,8,12,$ and $24$.
If we use the orbit stabilizer theorem, we have that $|H| = |\operatorname{Orb}_H(x)| \cdot |\operatorname{Stab}_H(x)|$; this limits us to only four possible subgroup orders, as $|\operatorname{Orb}_H(x)|$ can only be one thing.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Inequality and Induction: $\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$ I needed to prove that
$\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$, $\forall n \geq 1$ .
I've atempted by induction.
I proved the case for $n=1$ and assumed it holds for some $n$.
The left-side of the n+1 case... | $$\begin{align}\frac{1}{2}.\frac{3}{4}. ... .\frac{2n-1}{2n}.\frac{2n+2-1}{2n+2} &<\frac{2n+1}{\sqrt{2n+1}(2n+2)}\\~\\&=\frac{\sqrt{2n+1}}{(2n+2)}\\~\\&=\frac{\sqrt{(2n+1)(2n+3)}}{(2n+2)\sqrt{2n+3}}\\~\\&=\frac{\sqrt{4n^2+8n+3}}{(2n+2)\sqrt{2n+3}}\\~\\&\lt \frac{\sqrt{4n^2+8n+3+\color{blue}{1}}}{(2n+2)\sqrt{2n+3}}\\~\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do I use Rouche's theorem here? Suppose I had the polynomial $f(z) = z^5+3z+1$ and I want to find the number of complex roots in the first quadrant.
How would I use Rouche's theorem? or is there a simpler way. I was thinking of comparing it to $z$ because I checked on wolfram that it actually only has 1 root, bu... | In order to use Rouché to find the number of roots of $f(z) = z^5 + 3 z + 1$ (counted by multiplicity) in the region inside a simple closed contour $C$, you need to find $g(z)$ such that you know the answer for $g$ and $|f(z) - g(z)|$ is relatively small (specifically $ < |f(z)| + |g(z)|$, in the best version of the th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Taylor Expansion of Inverse of Difference of Vectors I am trying to derive the multipole moment of a gravitational potential, but I'm getting stuck on some math I believe. So basically the problem is finding the Taylor Expansion for $$\frac{1}{|\mathbf{x}-\mathbf{x'}|}=\frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}.$$ T... | The multivariable Taylor expansion is probably most easily written as
$$ f(\mathbf{x+h}) = \sum_{k=0}^{\infty} \frac{1}{k!} \sum_{i_1,i_2,\dotsc,i_k=1}^n (\partial_{x_{i_1}} \partial_{x_{i_2}} \dotsm \partial_{x_{i_k}} f(\mathbf{x}) ) h_{i_1} h_{i_2} \dotsm h_{i_k} $$
(which keeps all terms of the same order together),... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$. If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
Factorising we get $a^{pq} -a^q -a^p +a =a^p(a^q -1) - a(a^{q-1}-1)$ and we know $p \mid a^{p-1}-1$ and $q... | For prime $q,$
$$a^{pq}-a^q-a^p+a=[(\underbrace{a^p})^q-(\underbrace{a^p})]-[a^q-a]$$
Now by Fermat's Little Theorem, $b^q\equiv b\pmod q$ where $b$ is any integer
Set $b=a^p, a$
Similarly for prime $p$
Now if $p,q$ both divides $a^{pq}-a^q-a^p+a,$ the later must be divisible by lcm$(p,q)$
| {
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"source": "stackexchange",
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Is my solution correct? Differentiate $y = \left(x + \left(x + \sin^2 x\right)^4\right)^6$
Differentiate $y = \left(x + \left(x + \sin^2 x\right)^4\right)^6$.
I am unsure if I skipped a step in the end. Please take a look at my work. Criticize my strategy, point out obvious math deficiencies, or why I am missing the ... | derivative of $\sin^2(x)$ is $2\sin(x)\cos(x)$, also don't omit parentheses
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I find the period of the sine function $y = 20\sin\left[\frac{5 \pi}{2}\left(\frac{x -2}{5}\right)\right] + 100$ Using Desmos I can see the period is $0.8$ but how do I get there?
I understand that the period is $2\pi/$co-efficient of $x$ but the $-2/5$ is throwing me off.
| Shifts are to be neglected, hence, if $T$ is a period, we have
$$
\frac{5\pi}{2}T=2\pi,
$$
which gives $T=0.8$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Measurable set and continuous function exercise Let $E \subset \mathbb R^n$ be a measurable set. Show that $f:\mathbb R_{\geq 0} \to \mathbb R$, given by $f(r)=|E \cap B(0,r)|$ is continuous.
So, given a fixed $r_0 \in \mathbb R_{\geq 0}$ and given $\epsilon$, I want to prove there is $\delta>0 : |r_0-x|<\delta \implie... | Let $s \in \Bbb R_{\ge 0}$. For all $r > s$,
$$0 \le f(r) - f(s) = |E\cap [B(0,r)\setminus B(0,s)]| \le |B(0,r) \setminus B(0,s)| = C_n(r^n - s^n),$$
where $C_n$ is a constant depending only on $n$. Since $r^n - s^n \to 0$ as $r \to s$, it follows that $f(r) \to f(s)$ as $r\to s^{+}$. A similar argument shows that $f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1209835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Vector Analysis - Finding perpendicular vectors Given the vector:
$$
\vec{u}=-7\hat{i}-7\hat{j}+7\hat{k}
$$
I need to find another vector $\vec{v} $ that is parallel to the $xy$ plane and perpendicular to the vector $\vec{u} $ .
How can I do it?
I know that any vector that is perpendicular to $\vec{u}$ must satisfy $\v... | If you have a vector on the Euclidean plane $\vec{v}=(a,b)$ then the vector $\vec{u}=(-b,a)$ is perpendicular to $\vec{v}.$ We can apply the same idea in $3$-dimensions:
If we have a vector $\vec{v}=(a,b,c)$ then the following three vectors $$\vec{u}_1=(-b,a,0),\quad \vec{u}_2=(-c,0,a),\quad \vec{u}_3=(0,-c,b),$$ are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1209927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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elementary deduction on limit of sequence Let $(a_n)$ be a convergent sequence and $M$ a real number such that $a_n ≤ M$ for each $n$. Using the previous question, or otherwise, prove that $\lim_{n\to \infty}a_n≤M$.
I tried the "version" where $a_n > M$ and was able to arrived at a solution but this one seems like a to... | For every $\epsilon>0$ there is $N$ such that $|\lim_na_n-a_n|<\epsilon$.
Then $\lim_na_n-a_n<\epsilon$, for $n>N$.
We deduce that $\lim_na_n<a_n+\epsilon\leq M+\epsilon$.
Since we have obtained that for all $\epsilon>0$, $\lim_na_n<M+\epsilon$, therefore $\lim_na_n\leq M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Baby Rudin Theorem 9.17 In proof of Theorem 9.17 Rudin writes:
$$
|f(x+h)-f(x)-Ah|\leq \frac{1}{2}|Ah| \tag{23}
$$
It follows that
$$
|f(x+h)-f(x)|\geq \frac{1}{2}|Ah| \tag{24}
$$
I note that $f$ is a $C'$ mapping of a subset $E$ of $R^n$ into $R^m$ and $A = f'(A)$. I don't see how (24) follows from (23).
:
| show |u+2v|<|v|-> |u|>|v|
Played with this in 2 dimensions and its true.
Assume |u|<|v|
Draw the vector 2v and from its tip draw a circle of radius |u|. Then u+2v is a vector from the base of 2v to a point on the circle whose length is always greater than |v|. Contradiction. Therefore |u|>|v|.
In n dimensions you would... | {
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Limit of sequences and integers If $a$ is a non zero real number , $x \ge 1$ is a rational number and $(r_n)$ is a sequence of positive integers such that $\lim _{n \to \infty}ax^n-r_n=0$ , then is it true that $x$ is an integer ?
| According to the Wikipedia article on Pisot–Vijayaraghavan numbers, this is true. In fact, if $x$ is algebraic (not necessarily rational) then it has to be a Pisot number, and in particular an algebraic integer. A rational integer is of course a bona fide integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Counting the number of pairs of $(i,j)$ such that $\operatorname{lcm}(i,j)=n$ Pairs $(i,j)$ such that $\operatorname{lcm}(i,j)=n$
how many pair $(i,j)$ can be formed such that $\operatorname{lcm}(i,j)=n$. here $i\leq n$ and $j\leq n$.
here $n$ is a integer number less than $10^{14}$.
| This can get you started.
Factor $n$ into its prime decomposition:
$$n=p_1^{a_1}p_2^{a_2}\dots p_k^{a_k}$$
All the possible $i$'s where $\operatorname{lcm}(i,j)=n$ are given by
$$i=p_1^{b_1}p_2^{b_2}\dots p_k^{b_k}$$
where $0\le b_r\le a_r$. This gives $a_r+1$ choices for the exponent for $p_r$, and the choices are ind... | {
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"timestamp": "2023-03-29T00:00:00",
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How to explain to a 14-year-old that $\sqrt{(-3)^2}$ isn't $-3$? I had this problem yesterday. I tried to explain to the kid this: $$\sqrt{(-3)^2} = 3,$$ and he immediately said: "My teacher told us that we can cancel the square with the square root, so it's $$\sqrt{(-3)^2} = -3."$$
He has a lot of problems with maths,... | I think the kid has misunderstood something his teacher said about cancelling. A math student of any age is bound to misunderstand his teacher at some point or other. When I was in high school, long, long ago, Mr. Jones was fond of saying that such and such equation has no real solutions, equations like, say, $x^2 + 9 ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving the implication $\lim_{n\to\infty} a_n a_{n+1} =0 \Rightarrow \lim_{n\to\infty} a_n=0$ Let $\{a_n\}$ be a sequence such that for all $n$, $a_n>0$. I have to prove that if $\lim_{n\to\infty} a_n a_{n+1} =0$ then $\lim_{n\to\infty} a_n=0$. Here's my reasoning: $\forall\varepsilon>0,\exists k$ such that $\forall n... | As said that's not correct. I think you should better argue by contradiction that $\lim_{n\to\infty}a_n\neq 0$. WLOG the limit exists and $$\lim_{n\to\infty} a_n=\lim_{n\to\infty}a_{n+1}=l>0\Rightarrow \lim a_na_{n+1}=l^2>0$$ in contradiction which means that necessarily $\lim_{n\to\infty}a_n=0$ .
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding $\lim\limits_{n \to \infty}(\sin(1)\cdot \sin(2) \cdot \ldots \cdot \sin(n))$ Problem: Find $\lim_{n \to \infty}(\sin(1)\cdot \sin(2) \cdot \ldots \cdot \sin(n))$.
My idea: I suppose it to be $0$, but how might I go about proving this?
| $$\sin(k)\sin(k+1)=\frac{1}{2} [\cos(1)-\cos(2k+1)]$$
Now use the fact that $.5 < \cos(1) < .6$ to conclude that
$$\left| \sin(k) \sin(k+1)\right| < 0.8$$
and
$$\left| \sin(1)\cdot \sin(2) \cdot \ldots \cdot \sin(n) \right| \leq 0.8^{\frac{n}{2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
} |
implicit differentiation solving for $y\prime$ I'm supposed to implicitly differentiate the following and give the answer in terms of $y\prime$. $$\tan(x-y)={y \over 1+x^2}$$
$${ (1+x^2)y\prime - 2xy \over (1+x^2)^2 }$$
How do I solve for $y \prime$?
Edit:
After a ludicrous amount of algebra, i finally ended up at
$$ {... | Differentiating both sides from
$$\tan (x - y(x)) = \frac{{y(x)}}{{{x^2} + 1}}$$
gives:
$$\frac{1}{{{{\cos }^2}(x - y(x))}} - \frac{{y'(x)}}{{{{\cos }^2}(x - y(x))}} = \frac{{y'(x)}}{{{x^2} + 1}} - \frac{{2xy(x)}}{{{{\left( {{x^2} + 1} \right)}^2}}}$$
which can be solved and simplified for ${y'(x)}$ like so:
$$y'(x) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$P( A\triangle B ) = 0 \Rightarrow P(A)=P(B)=P(A\cup B) + P(A \cap B) $
I want to prove the following statement;
$$P( A\triangle B ) = 0 \Rightarrow P(A)=P(B)=P(A\cup B) \color{blue}{=} P(A \cap B) $$
What I did is that
$$P(A\triangle B) =P((A\setminus B ) \cup (B\setminus A))=0$$
$$=P((A\cap B^c)\cup (B\cap A^c))... | The probabilities have to look like this, from which everything follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Completing System of Vectors to an Orthogonal Basis I want to complete ((1, -2, 2, -3), (2, -3, 2, 4)) to an orthgonal basis. I honestly don't really know how to do this. This a practice problem to a test, were we haven't done problems related to this. I know that an orthogonal basis has that for basis $(v_1, ... , ... | here is what you can do. you already have two vectors $u = (1, -2, 2, -3)^T, v = (2,- 2, 3, 4)^T$ that are orthogonal. we will first find four orthogonal vectors; making them of unit length is easier.
what we will do is pick a vector $a$ and find the projection on to the space spanned by $u, v$ and subtract it. we can ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Having trouble with the substitution method. I have to evaluate the integral:
I see that with u being equal to $4x$ I will get that $du/dx$ will equal $4$ and lead to $du=4dx$. However I am stuck when I plug in the u values to get $\sec u\cos u\,du$. The problem states to use the subsitution method but it looks like I... | i think you can use the change of variable $u = \cos 4x, du = -4 \sin 4x \, dx$ instead.
then $$\int \sec 4x \tan 4x \, dx = \int \frac{\sin 4x}{\cos^2 4x} \, dx
\to -\frac 14\int \frac{du}{u^2} = \frac{1}{4u} = \frac 14\sec4x + C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Convergence of Types Theorem
(Convergence of Types Theorem) Suppose that $F_n(u_nx+v_n) \Rightarrow F(x)$ and $F_n(a_nx+b_n) \Rightarrow G(x)$, where $u_n>0, a_n>0$ and $F$ an $G$ are non-degenerate. Then there exist $a>0$ an $b \in\mathbb R$ such that $a_n/u_n \to a$ and $(b_n-v_n)/u_n \to b$, and $F(ax+b) = G(x).$
... | It seems that Lemma 3 is applied with $\widetilde{F_n}(t):=F_n(a_nt)$ instead of $F_n$ (which is allowed by Lemma 2) and the sequence $(b_n/a_n)_{n\geqslant 1}$ instead of $(b_n)_{n\geqslant 1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Spectrum of a bilateral shift Let $u$ be a bilateral shift on Hilbert space $\ell^2(\Bbb Z)$. As for unilateral shifts, the spectrum of $u$ does not contain any eigenvalue. Also $u$ is unitary, so $\sigma(u) \subset \Bbb S$ ($\Bbb S$ means unit circle). How can I show that $\sigma(u)=\Bbb S$?
| Hint: Let $e_n$ $(n\in \mathbb Z)$ be the standard basis in $\ell^2(\mathbb Z)$. For $\lambda \in {\mathbb T}$, let
$$ \xi_n=\frac{1}{2n+1}\bigl( \lambda^{-n} e_{-n}+\lambda^{-n+1} e_{-n+1}+\cdots+e_0+\cdots+\lambda^{n} e_n\bigr)\qquad (n\in \mathbb Z). $$
Note that $(\xi_n)$ is a sequnce of vectors with norm $1$. Cons... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1211076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
If every functional of $f$ is smooth, is $f$ smooth? Let E be a (real or complex) Banach space and suppose $f: \mathbb{R}^n \rightarrow E$ has the property that $\lambda \circ f$ is $C^\infty$ for every bounded linear functional $\lambda \in E^\ast$. Does it follow that $f$ is also $C^\infty$?
I didn't really encounte... | See Theorem 5.0.3 in these notes by Paul Garrett, where this is shown for weakly smooth $f$ on a closed real interval.
It seems he isn't using anything specific to the fact that $f$ is defined on a dimension $1$ space, so I think the proof generalizes to $\mathbb R^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1211231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
In the sequence $1,3,7,15,31\ldots$ each term is $2\cdot\text{immediately preceding term}+1$. What is the $n$-th term? I readily see that it is $2^n-1$, but how can I deduce the $n$-th term from the given pattern i.e. $n$-th term $= 2\cdot(n-1)\text{th term} + 1$ without computation.
| Here is an explicit proof by induction. Given that $a_{n+1}=2a_n+1$ for $n=0,1,...$, with $a_0=1$ prove the proposition $\mathcal P(n):\;\,a_n=2^n-1$ for $n=0,1,...$ .
We are given that $\mathcal P(n)$ holds for $n=0$. Suppose now that we know $\mathcal P(n)$ for $n=k$; that is, $\mathcal P(k):\;\,a_k=2^k-1$. Now $a_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1211315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Find the Critical Points: $f(x) =(x^2-1)^3$ This question probably has more to do with my Algebra skills than Calculus. Nonetheless, can someone explain why the factored "term" is not set to zero (0) [second picture]. Thanks in advance.
| $f(x)=(x^2-1)^3$ is a sixth-degree polynomial with $x=\pm 1$ being triple zeroes.
Moreover, $f(x)$ is an even function, hence $f'(x)$ (that is a fifth-degree polynomial) has double zeroes in $x=\pm 1$ and a simple zero in $x=0$. The critical points of $f(x)$ may occur only at the zeroes of the derivative or at the endp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1211447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Started this problem but can't finish it: Showing pointwise convergence for this summation I know how to start this problem but am having trouble finishing the end of it. Any help would be great! Thanks
We let $g_n: E \rightarrow \mathbb{R}$ be continuous functions for $1 \leq n \leq N$ and we let $a_k^{(n)}$ be $N$ c... | Hint: |$\sum_{n=1}^{N} a_k^{\{n\}}g_n(x)$ - $\sum_{n=1}^{N} a_ng_n(x)$| = $|\sum_{n=1}^N (a_k^{\{n\}} - a_n) g_n(x)|$.
Additionally, $|\sum_{n=0}^N x_n| \leq \sum_{n=0}^N |x_n|$ for any $x_n$ (this is the triangle inequality).
Can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1211629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Topology, Showing that two metric spaces are topologically equivalent Can someone verify if this is true?
$X=\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$
. $d$
is the standard metric on $\mathbb{R}$
and $d'(x,y)=d\left(\tan(x),\tan(y)\right)$
. We want to show that $\left(X,d\right)$
and $\left(X,d'\right)$
are... | I will just comment on this paragraph here:
Since $\tan(x)$ is a monotonic function and continuous, there exists
$0<r′\in\mathbb{R}$ such that the the greatest $\delta>0$ that
satisfies $d(x_0,y)<\delta\Rightarrow d(\tan(x_0),\tan(y))<r'$ , is
smaller than $r$ . Since the function is monotonic $r'$ exists.
I do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1211733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Another optimization problem I am having trouble figuring out a next step in an optimization problem
the question is to find the max and min values of $f(x,y)=\frac{x+y}{2+x^2+y^2}$
I calculated $f_x$ and $f_y$ and set both them equal to zero, and the only possibility you get is x=y. I dont know how else to find it aft... | Setting $f_x=0$ and $f_y=0$ gives $-x^2-2xy+y^2+2=0$ and $-y^2-2xy+x^2+2=0$, so
subtracting these equations gives $2y^2-2x^2=0, \;\;y^2=x^2,\; $ and so $y=\pm x$.
1) If $y=x$, substituting into the first equation gives $x^2=1$ so $x=\pm 1$.
2) If $y=-x$, substituting into the first equation gives $x^2=-1$, so there is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1211844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How many ways to get at least one pair in a seven card hand? This was the question and answer I saw:
How many different seven-card hands are there that contain two or more cards of
the same rank?
Solution:
There are C(52,7) total hands. To subtract the ones that don’t have pairs, we observe
that such hands have cards ... | Consider a hand which has two pairs, for example
$$\clubsuit A,\,\diamondsuit A,\,\heartsuit 10,\diamondsuit10,\,\spadesuit7,\,\spadesuit K,\,\diamondsuit 2.$$
Your method will count this hand twice:
*
*choose ace, then clubs and diamonds, then $\heartsuit 10,\diamondsuit10,\,\spadesuit7,\,\spadesuit K,\,\diamondsui... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1212173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Application of Differentiation (Doesn't understand) It's given the cubic equation $x^3-12x-5=0$. Show graphically that the iteration $x_{n+1}=\sqrt[3]{12x_n+5}$ should be used to find the most negative root and the positive root, and the iteration $x_{n+1}=\dfrac{x^3_n-5}{12}$ should be used to find the other root.
Th... | The idea to find approximations of roots of $f(x) = 0$ starts with finding an integer $a$ such that a root lies between $a$ and $(a + 1)$. To ensure that this is so we need to guess some integers $a$ such that $f(a)f(a + 1) < 0$ i.e. $f(a)$ and $f(a + 1)$ are of opposite signs. Then by Intermediate Value Theorem there ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1212275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Power Set of a Power Empty Set Find ℙ(ℙ(ℙ(∅))).
I know that ℙ(∅) = {∅}.
Then, ℙ(ℙ(∅)) = {∅, {∅}, {∅,{∅}}?
so, ℙ(ℙ(ℙ(∅))) = {∅,{∅, {∅}, {∅,{∅}}}?
Is it? Will it be ok if someone explain to me this concept?
| I think it's easier if you consider $P(\varnothing)$ to be a one-element set, say $\{1\}$. Then it's pretty clear that $P(P(\varnothing))$ has to be precisely $\{ \varnothing, \{1\}\}$. So $P(P(P(\varnothing))) = \{\varnothing, \{\varnothing\}, \{\{1\}\}, \{\varnothing, \{1\}\}\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1212362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Number of strictly increasing and decreasing 4 digit numbers? I have two questions:
(a) Count the 4 digit numbers whose digits decrease strictly from left to right and;
(b) Count the 4 digit numbers whose digits increase strictly from left to right
I have the answers which are $10 \choose 4$ and $9 \choose 4$ respectiv... | (a) Since the digits must strictly decrease from left to right, there are 10C4 such numbers. For any selection of four digits from the ten digits, there is exactly one way of arranging those four digits in a strictly decreasing order from left to right. We need not worry about leading zeroes because the order is decrea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1212455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
If $A$ and $B$ are similar, why does $\text{rank}(A) = \text{rank}(B)$? Suppose $A$ and $B$ are similar matrices over $\mathbb{C}^n$. Why do we have $\text{rank}(A) = \text{rank}(B)$?
| If $A$ and $B$ are similar then $A = PBP^{-1}$ for some invertible matrix $P$. Because $P$ is invertible its null space is trivial, i.e. $nullity(P) = \{0\}$.
Now suppose that $Ax = PBP^{-1}x = 0$. Using the result above $PBP^{-1}x = 0$ if and only if $PBx = 0$; Again $PBx = 0$ if $Bx = 0$. The latter result means that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1212578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
$(\delta,\varepsilon)$ Proof of Limit I wish to prove that $\lim_{x\to 2} {x+1 \over x+2} = {3 \over 4} $.
The $(\delta,\varepsilon)$ limit definition in this case is:
$\forall \epsilon >0, \exists \delta >0$ such that $0<|x-2|<\delta \Rightarrow |{x+1 \over x+2} - {3 \over 4}| < \epsilon.$
Thus, I need to provide a $\... | Let $\varepsilon>0$ be given. Consider $|\dfrac{x+1}{x+2}-\dfrac{3}{4}|=|1-\dfrac{1}{x+2}-\dfrac{3}{4}|=|\dfrac{1}{4}-\dfrac{1}{x+2}|$
We want to find a $\delta>0$ such that whenever $|x-2|<\delta$ , the above expression $<\varepsilon$.
So start with $|\dfrac{1}{4}-\dfrac{1}{x+2}|<\varepsilon\iff|\dfrac{x-2}{4(x+2)}|<\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1212672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
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