qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
48,864 | <p>I can't resist asking this companion question to the <a href="https://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking"> one of Gowers</a>. There, Tim Dokchitser suggested the idea of Grothendieck topologies as a fundamentally new insight. But Gowers' original motivation is to ... | Ivan Meir | 7,113 | <p>I think that many conjectures from number theory (which I think count as insights) might be more obvious to a computed than a human being since they would have access to a huge amount of empirical data from which to discover patterns and obtain estimates as to the probability that something is true or plausible. Thi... |
340,264 | <p>Given that</p>
<p>$L\{J_0(t)\}=1/(s^2+1)$</p>
<p>where $J_0(t)=\sum\limits^{∞}_{n=0}(−1)n(n!)2(t2)2n$,</p>
<p>find the Laplace transform of $tJ_0(t)$. </p>
<p>$L\{tJ_0(t)\}=$_<strong><em>_</em>__<em>_</em>__<em>_</em>___<em></strong>---</em>___?</p>
| Lubin | 17,760 | <p>There are so many ways of approaching this that one can get dizzy contemplating them all. You can use the Binomial series for $(1+z)^{1/2}$, plugging in $2/y^2-1$, for example, to get $\sqrt2/y$, then multiply that result by $y$.</p>
<p>Or you can use Newton-Raphson directly. I'll illustrate how it goes for $p=7$, ... |
3,076,504 | <p>The problem states: </p>
<p>Right Triangle- perimeter of <span class="math-container">$84$</span>, and the hypotenuse is <span class="math-container">$2$</span> greater than the other leg. Find the area of this triangle. </p>
<p>I have tried different methods of solving this problem using Pythagorean Theorem and... | poetasis | 546,655 | <p><span class="math-container">$\\ \textbf{Finding triples, given perimeter using Euclid's formula}$</span> where <span class="math-container">$P=perimeter$</span></p>
<p><span class="math-container">$$P=(m^2-n^2 )+2mn+(m^2+n^2 )=2m^2+2mn\implies n=\frac{P-2m^2}{2m}\quad where \quad \biggl\lceil\frac{\sqrt{P}}{2}\big... |
3,365,112 | <p>Pretty simple, for <span class="math-container">$a,b \in \mathbb R$</span>, show that <span class="math-container">$|a-b|<\frac{|b|}{2}$</span> implies <span class="math-container">$\frac{|b|}{2}<|a|$</span>. I can see this graphically on the number line, but I can't seem to show it algebraically.</p>
<p>I'm ... | cmk | 671,645 | <p>Since <span class="math-container">$|a-b|<|b|/2,$</span> we know that <span class="math-container">$$-|b|/2<a-b<|b|/2,$$</span> or <span class="math-container">$$b-|b|/2<a<b+|b|/2,$$</span> from the definition of the absolute value. Try to consider what happens if you choose <span class="math-containe... |
729,444 | <p>Let be two lists $l_1 = [1,\cdots,n]$ and $l_2 = [randint(1,n)_1,\cdots,randint(1,n)_m]$ where $randint(1,n)_i\neq randint(1,n)_j \,\,\, \forall i\neq j$ and $n>m$. How I will be able to found the number of elements $x\in l_1$, to select, such that the probability of $x \in l_2$ is $1/2$?. I'm trying using the bi... | Brian Fitzpatrick | 56,960 | <p>If you don't know what an eigenvalue is, and if you're not worried about elegance, then here is a more direct approach (assuming you're working over a field not of characteristic two).</p>
<p>There exist scalars $\lambda_{ij}$ for $1\leq i,j\leq n$ such that for every basis $\{v_1,\dotsc,v_n\}$ of $V$ we have
$$
\b... |
2,155,652 | <p>I have a question regarding this proof my professor gave us. For the third property, I understand the proof up to the sentence "If $x \in E'$, i.e., x is a limit point of E."
Well, I also understand that if x is not in F, then x can't be a limit point since F is closed.
After that, I don't fully understand it. Could... | manofbear | 230,268 | <p>If $x$ is a limit point of $E$, that means there exists a sequence $(x_n)\subset E$ for which $x_n\neq x$ for all $n$, and $\lim x_n=x$. Notice that the same sequence $(x_n)\subset F$ satisfies the same conditions, so $x$ is a limit point of $F$. Since $F$ is closed, it contains its limit points. So $x\in F$.</p>
|
809,516 | <p>I need to calculate </p>
<p>$$\lim_{x \to \infty} \frac{((2x)!)^4}{(4x)! ((x+5)!)^2 ((x-5)!)^2}.$$</p>
<p>Even I used Striling Approximation and Wolfram Alpha, they do not help.</p>
<p>How can I calculate this?</p>
<p>My expectation of the output is about $0.07$.</p>
<p>Thank you in advance.</p>
| Claude Leibovici | 82,404 | <p>Using directly Stirling approximation of the factorial $$\begin{align}
\Gamma(n+1) \approx \sqrt{2 \pi} \ n^{n+1/2} \ e^{-n}
\end{align}$$ the expression becomes $$\frac{((2x)!)^4}{(4x)! ((x+5)!)^2 ((x-5)!)^2}\approx\sqrt{\frac{2}{\pi }} (x-5)^{9-2 x} x^{4 x+\frac{3}{2}} (x+5)^{-2 x-11}$$ which, for large values of ... |
1,555,548 | <p>There are $8$ people and they want to sit in a bus which has $2$ single front seats and $4$ sets of $3$ seats with $1$ person that is always the designated driver. How many ways are there for the people to sit in the bus?</p>
<p>I solved it by using:</p>
<p>$6!*(\binom{9}{3}) - 4((6*5*4*3)*2(\binom{4}{2})+(6*5*4*3... | mathochist | 215,292 | <p>If the specific seat matters, it seems like there should be an $8$ way choice to choose the driver, then in the remaining $14$ seats, we have to choose $7$ for the remaining passengers, which gives $8{14 \choose 7}$.</p>
|
622,278 | <p>What is the relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$ where $a_{n} \in \mathbb{C} \ \forall n$ ?</p>
<p>Where can I find some references about this topic ?</p>
| benh | 115,596 | <p>Let $a_n\neq 0$ for all $n\in \Bbb N$, then $$\prod_{n=1}^\infty a_n \text{ converges} \Leftrightarrow \sum_{n=1}^\infty \log{a_n} \text{ converges}$$
Moreover, for $a_n \neq -1$, we have an equivalence $$\sum_{n=1}^\infty \log(1+a_n) \text{ converges absolutely} \Leftrightarrow \sum_{n=1}^\infty a_n \text{ converge... |
1,811,812 | <p>I've to prove that $Q_8/Z(Q_8)$ is isomorphic to Klein's $4$-group. I know that
$Q_8/Z(Q_8)$ has order $4$. But, I'm not able to get the elements of $Q_8/Z(Q_8)$. Please help me in representing $Q_8/Z(Q_8)$ in Roster form.</p>
| carmichael561 | 314,708 | <p>I'm not sure what roster form is, but $1,i,j,k$ is a complete system of coset representatives for $Z(Q_8)=\{\pm 1\}$.</p>
|
1,811,812 | <p>I've to prove that $Q_8/Z(Q_8)$ is isomorphic to Klein's $4$-group. I know that
$Q_8/Z(Q_8)$ has order $4$. But, I'm not able to get the elements of $Q_8/Z(Q_8)$. Please help me in representing $Q_8/Z(Q_8)$ in Roster form.</p>
| rschwieb | 29,335 | <p>Here's another way to get at it. There is a popular group theory exercise that says <a href="https://math.stackexchange.com/q/546971/29335">if $G/Z(G)$ is cyclic, then $G$ is Abelian</a>.</p>
<p>You know that $Q_8$ is nonabelian, and that there are only two groups of order $4$ possible...</p>
|
65,658 | <p>Suppose $X_i$'s are i.i.d, with the density distribution $f(x) = e^{-x}$, $x \geq 0$. I was able to show that
$$P(\limsup X_n/\log{n} =1)=1$$ using Borel-Cantelli.</p>
<p>Define $M_n=\max \{X_1,\ldots,X_n\}$, can I claim $M_n/\log{n} \rightarrow 1$ a.s. in this case? Is it still true in general without knowing the ... | Robert Israel | 8,508 | <p>If $F(x) = 1 - e^{-x}$ for $x > 0$ is the CDF of each $X_i$, the CDF of $M_n$ is $F_{M_n}(x) = F(x)^n = (1 - e^{-x})^n$ for $x > 0$. Note that $\ln(F_{M_n}(x)) = n \ln(1 - e^{-x})$ and since $- t - t^2 < \ln(1-t) < -t$ for $0 < t < .683$, for any $c>0$ we have $-n^{1-c} - n^{1-2c} \le \ln(F_(M_... |
4,513,368 | <p>The following question seems to be quite simple, but I am having a hard time to prove it rigorously.</p>
<p>Consider <span class="math-container">$n\in\mathbb{N}$</span> vertices, for example <span class="math-container">$\{v_1,\ldots, v_n\}$</span>. I have some further information on these vertices, namely, that an... | Prem | 464,087 | <p>Either I am missing something or the Question is missing something.</p>
<p>Let <span class="math-container">$V=\{a,b,c,u,v,w,x,y,z\}$</span> & Edges are <span class="math-container">$\{ab,bc,uu,uv,vw,ww,wx,wy,xy,zz\}$</span> where each Vertex is connected to something, maybe itself.</p>
<p>In set <span class="ma... |
3,910,013 | <p>I'm preparing for a high school math exam and I came across this question in an old exam.</p>
<p>Let <span class="math-container">$f(x) = \dfrac{1}{2(1+x^3)}$</span>.</p>
<p><span class="math-container">$\alpha \in (0, \frac{1}{2})$</span> is the only real number such that <span class="math-container">$f(\alpha) = \... | AlanD | 356,933 | <p>Hint: Since <span class="math-container">$f(\alpha)=\alpha$</span>, you are trying to prove
<span class="math-container">$$
|u_{n+1}-\alpha|\leq \frac 12|u_n-\alpha|,
$$</span>
which is equivalent to showing
<span class="math-container">$$
|f(u_n)-f(\alpha)|\leq \frac 12|u_n-\alpha|
$$</span>
or
<span class="math-co... |
133,604 | <p>"A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia</p>
<p><img src="https://i.stack.imgur.com/Rl1WS.gif" alt="cycloid animation"></p>
<p>In many calculus books I have, the cycloid, in parametric form, is used in examples to find arc lengt... | Ross Millikan | 1,827 | <p>The center of the circle moves along a horizontal line at constant velocity. If we want the cusps to be at $y=0$, that means the center should be $(x_c,y_c)=(rt,r)$. Then we add on the location of the point on the rim relative to the center. This will be something like $(r\cos t, r\sin t)$ but we still need to g... |
133,604 | <p>"A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia</p>
<p><img src="https://i.stack.imgur.com/Rl1WS.gif" alt="cycloid animation"></p>
<p>In many calculus books I have, the cycloid, in parametric form, is used in examples to find arc lengt... | J. M. ain't a mathematician | 498 | <p>Here is a cartoon depiction of what Robert and Ross were showing, courtesy of <a href="http://books.google.com/books?hl=en&id=EbVrWLNiub4C&pg=PA78" rel="noreferrer">Stan Wagon</a>:</p>
<p><img src="https://i.stack.imgur.com/Yd6sv.gif" alt="rolling penny"></p>
|
133,604 | <p>"A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia</p>
<p><img src="https://i.stack.imgur.com/Rl1WS.gif" alt="cycloid animation"></p>
<p>In many calculus books I have, the cycloid, in parametric form, is used in examples to find arc lengt... | louie mcconnell | 131,577 | <p>This book is a great resource. See pdf page 599, actual page 567.</p>
<p><a href="http://www.marystarhigh.com/apps/download/7vb7ETI4n4RtLFWDnZw0xNfQRUSB1swoBHQpP7i1l9pXZS1Y.pdf/Precalculus%20Book.pdf" rel="nofollow">http://www.marystarhigh.com/apps/download/7vb7ETI4n4RtLFWDnZw0xNfQRUSB1swoBHQpP7i1l9pXZS1Y.pdf/Preca... |
138,091 | <p>I am trying to compute an explicit formula using Mathematica for the following multinomial expression:</p>
<blockquote>
<p>\begin{equation} \sum_{n_{1}+n_{2}+...+n_{M}=N}^{M} {N \choose
n_{1},n_{2},...,n_{M }} \cdot n_{i} = ? \end{equation}</p>
</blockquote>
<p>where $i={1,2,...,M}$ and using </p>
<pre><code>... | Mr.Wizard | 121 | <p>Following Carl Woll's comment, correcting $m$ and $n$, and providing a more efficient form:</p>
<pre><code>n = 17;
m = 9;
p = IntegerPartitions[n, {m}, Range[0, n]];
Sum[Total[Permutations[x][[All, 1]] * Multinomial @@ x], {x, p}]
n m^(n - 1)
</code></pre>
<blockquote>
<pre><code>31501343210481297
315013432104... |
1,201,900 | <p>This is a rather soft question to I will tag it as such.</p>
<p>Basically what I am asking, is if anyone has a good explanation of what a homomorphism is and what an isomorphism is, and if possible specifically pertaining to beginner linear algebra.</p>
<p>This is because, in my courses we have talked about vector... | Community | -1 | <p>A <em>homomorphism</em> is a structure-preserving mapping.<br>
An <em>isomorphism</em> is a bijective homomorphism.</p>
<p>"Structure" can mean many different things, but in the context of linear algebra, almost exclusively means the vectorial structure -- i.e. all those rules about addition and scalar multiplicati... |
2,410,517 | <p>I feel like I'm missing something very simple here, but I'm confused at how Rudin proved Theorem 2.27 c:</p>
<p>If <span class="math-container">$X$</span> is a metric space and <span class="math-container">$E\subset X$</span>, then <span class="math-container">$\overline{E}\subset F$</span> for every closed set <spa... | Trevor Gunn | 437,127 | <p>If $x$ is a limit point of $E$ then $x = \lim x_n$ for some sequence $x_n \in E \setminus \{x\}$. If $E \subseteq F$ then $x_n \in F \setminus \{x\}$ so we can also say that $x$ is a limit point of $F$. Therefore</p>
<p>$$ E' \subseteq F' \subseteq F. $$</p>
|
894,159 | <p>I was assigned the following problem: find the value of $$\sum_{k=1}^{n} k \binom {n} {k}$$ by using the derivative of $(1+x)^n$, but I'm basically clueless. Can anyone give me a hint?</p>
| Community | -1 | <p>Note that $$(1+x)^n= \sum_{k=0}^{n} \binom {n} {k} x^k$$ </p>
<blockquote class="spoiler">
<p> $$f'(x)=n \cdot (1+x)^{n-1}= \sum_{k=1}^{n} k \binom {n} {k} x^{k-1}$$
Whence, we have:
$$\sum_{k=1}^{n} k \binom {n} {k} = f'(1) = n \cdot 2^{n-1}$$</p>
</blockquote>
|
2,809,686 | <p>Let S={1,2,3,...,20}. Find the probability of choosing a subset of three numbers from the set S so that no two consecutive numbers are selected in the set.
"I am getting problem in forming the required number of sets."</p>
| Graham Kemp | 135,106 | <p>Count the total ways to select any three numbers from the set of twenty.</p>
<p>Count the ways to select a number, its immediate successor, and any other number <em>that is not the first number's immediate predecessor</em> (to avoid overcounting). There will be two cases to consider: first select $1$ or firs... |
112,226 | <p>Prove that there are exactly</p>
<p>$$\displaystyle{\frac{(a-1)(b-1)}{2}}$$ </p>
<p>positive integers that <em>cannot</em> be expressed in the form </p>
<p>$$ax\hspace{2pt}+\hspace{2pt}by$$</p>
<p>where $x$ and $y$ are non-negative integers, and $a, b$ are positive integers such that $\gcd(a,b) =1$.</p>
| Brian M. Scott | 12,042 | <p>Call an integer <em>bad</em> if it cannot be represented as an integer combination of $a$ and $b$ with non-negative coefficients. There are $(a-1)(b-1)$ non-negative integers less than $(a-1)(b-1)$, and you know that all of the bad integers are among them. Take a look at a simple example. Suppose that $a=4$ and $b=7... |
2,909,480 | <p>Please notice the following before reading: the following text is translated from Swedish and it may contain wrong wording. Also note that I am a first year student at an university - in the sense that my knowledge in mathematics is limited.</p>
<p>Translated text:</p>
<p><strong>Example 4.4</strong> Show that it ... | Hagen von Eitzen | 39,174 | <p>First line: Just expand the third power:
$$ (n+1)^3=(n+1)(n+1)(n+1)=n^3+3n^2+3n+1$$
(and then subtract $n+1$, of course).</p>
<p>Second line: We know by induction hypothesis that $n^3-n=3b$ (for some $b$)</p>
<p>Third line: extract the common factor $3$.</p>
|
3,154,407 | <p>I would like to define a function whose domain is any multiset of real numbers and image is a real number.</p>
<p>To my understanding, the domain of a function that can be applied on any set of real numbers is the power set <span class="math-container">$\mathcal{P}(\mathbb{R})$</span>. Is it correct? If yes, is the... | Gary Moon | 477,460 | <p>I think you'd probably be best served by defining your own notation. You can write multisets as sets using various conventions (see Wikipedia page), and so you could perhaps view the function as having domain <span class="math-container">$\mathcal{P}(\mathbb{R}) \times \mathcal{P}(\mathbb{R}^2)$</span> (I believe). ... |
3,006,511 | <p>Let <span class="math-container">$c = \{ x = \{x_k\}_{k=1}^{\infty} \in l^\infty \vert \exists \lim_{k \to \infty} x_k \in \mathbb{C} \}$</span>. </p>
<p>Let <span class="math-container">$x_n \in c$</span>, with <span class="math-container">$x_n \to x = \{x_k\}$</span> with the sup norm. </p>
<p>I want to prove th... | alepopoulo110 | 351,240 | <p>As said in the comments, you should improve your notation. It will help.</p>
<p>Let <span class="math-container">$(x_n)\subset c$</span> with <span class="math-container">$x_n=(x_n^1,x_n^2,\dots)$</span> for any <span class="math-container">$n$</span>.</p>
<p>Let <span class="math-container">$x=(b_1,b_2,\dots)$</s... |
760,767 | <p>I don't understand the last part of this proof:</p>
<p><a href="http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup" rel="nofollow">http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup</a></p>
<p>where they say: $p \nmid \left[{N : P \cap N}\right]$, t... | ayuanx | 246,849 | <p>Interesting topic.</p>
<p>First to @mathguy, $\int r d\theta = r θ$ is wrong. Because usually $r$ is a function of $\theta$. so $\int r d\theta =\int r(\theta)d\theta$.</p>
<p>The reason why the concept of "<em>curve length</em> $r d\theta$" can apply to area intergration in polar coordinates $Area=\int \frac{1}{2... |
2,498,628 | <p>This was a question in our exam and I did not know which change of variables or trick to apply</p>
<p><strong>How to show by inspection ( change of variables or whatever trick ) that</strong></p>
<p><span class="math-container">$$ \int_0^\infty \cos(x^2) dx = \int_0^\infty \sin(x^2) dx \tag{I} $$</span></p>
<p>Co... | Quanto | 686,284 | <p>Note that
<span class="math-container">$\int_0^\infty(\mathcal{L} g)(x)f(x)\,dx = \int_0^\infty g(x)(\mathcal{L}f)(x)\,dx$</span>
with <span class="math-container">$f(x) =\sin x -\cos x$</span>, <span class="math-container">$g(x)=\frac1{\sqrt x}$</span> and their transformations <span class="math-container">$\mathca... |
44,771 | <p>A capital delta ($\Delta$) is commonly used to indicate a difference (especially an incremental difference). For example, $\Delta x = x_1 - x_0$</p>
<p><strong>My question is: is there an analogue of this notation for ratios?</strong></p>
<p>In other words, what's the best symbol to use for $[?]$ in $[?]x = \dfrac... | Amad | 625,606 | <p>The greek letter <span class="math-container">$\Delta$</span> stands for <em>Difference</em>
So the symbol for the multiplicative increase of variable should be related to letter <span class="math-container">$Q$</span> in greek which is the ancient Qoppa letter and stands for <em>Quotient</em>
<a href="https://i.sta... |
1,053,506 | <p>I had thought that the ultra-metric property was just a rule that someone made up, that if applied shows some bizarre behavior. I however came across these notes: <a href="http://www.math.harvard.edu/~thorne/all.pdf" rel="nofollow">Lecture notes</a> and it seems that the ultra-metric property is actually derived fro... | Slade | 33,433 | <p>This has nothing to do with the usual Archimedean/Euclidean metric. We can define a norm (hence a metric) on $\mathbb{Q}$ as follows: let $\| a \|_p$ be $c^{-\nu_p (a)}$. Here $\nu_p (a)$ is the largest number of powers of $p$ dividing $a$ (possibly negative, or $\infty$ for $a=0$). So things are small exactly wh... |
3,712,128 | <p>I've been reading about combinatorial games, specifically about positions in such games can be classified as either winning or losing positions. However, what I'm not sure about now is how I can represent draws using this: situations where neither player wins or loses. Do I use a winning position or losing position ... | Z Ahmed | 671,540 | <p><span class="math-container">$(a)$</span>: two parallel non-intersecting lines, (b) intesecting lines.</p>
<p>In the case of three planes (equations) when non two are parallel and no two are idendtical: Take <span class="math-container">$z=k$</span> and solve for %x, y$ punt these in the third equation.
One of the... |
295,618 | <p>Problem A: Please fill each blank with a number such that all the statements are true:</p>
<p>0 appears in all these statements $____$ time(s)<br>
1 appears in all these statements $____$ time(s)<br>
2 appears in all these statements $____$ time(s)<br>
3 appears in all these statements $____$ time(s)<br>
4 appears ... | Ross Millikan | 1,827 | <p>Douglas Hofstadter wrote about these problems years ago. He suggested a useful approach is to fill in the blanks with something, then just count and refill, iterating to convergence. On the first one, I got trapped in a loop between $1741111121$ and $1821211211$ and for the second a loop between $254311311150$ and... |
101,098 | <p>I apologize in advance because I don't know how to enter code to format equations, and I apologize for how elementary this question is. I am trying to teach myself some differential geometry, and it is helpful to apply it to a simple case, but that is where I am running into a wall.</p>
<p>Consider $M=\mathbb{R}^2$... | Jeremy Mann | 23,200 | <p>Yes, the fact that the tangent space at a point isn't actually R^2 cannot be emphasized enough. Another way to think of the tangent space at a point is through equivalence classes of differentiable curves through the given point, with the relation being that two curves are equivalent if the curve composed with your... |
3,852,952 | <p>Given a projective space <span class="math-container">$\mathbb{P}^n(\mathbb{C})$</span>, I can consider the Grasmannian of lines <span class="math-container">$G(2,n+1)$</span>, which has a structure of projective variety inside <span class="math-container">$\mathbb{P}^N$</span>, where <span class="math-container">$... | mathma | 270,091 | <p>I don't know if I'm understanding your question right, but here are some ideas. The Grassmanian <span class="math-container">$G(1,n)$</span> can be seen as the projective <span class="math-container">$n-1$</span> dimensional space, because lines are seen as points. Then a line in projective space comes from a plane ... |
359,277 | <p>Can you find function which satisfies $f(ab)=\frac{f(a)}{f(b)}$? For example $log(x)$ satisfies condition $f(ab)=f(a)+f(b)$ and $x^2$ satisfies $f(ab)=f(a)f(b)$?</p>
| Luke Mathieson | 35,289 | <p>You could take a relatively trivial function like $f(x) = 1$. Or a slightly more general version that takes everything to the identity.</p>
|
359,277 | <p>Can you find function which satisfies $f(ab)=\frac{f(a)}{f(b)}$? For example $log(x)$ satisfies condition $f(ab)=f(a)+f(b)$ and $x^2$ satisfies $f(ab)=f(a)f(b)$?</p>
| baharampuri | 50,080 | <p>Let us reformulate the question as classify all maps $f : G \rightarrow H$ which need not be group morphism that satisfies the condition $f(ab)=f(a)f(b)^{-1}$</p>
<p>A simple calculation shows that $f(e)=f(x)f(x^{-1})^{-1}= f(x^{-1})f(x)^{-1}$ or we have $f(x)=f(e)^{-1}f(x^{-1})=f(e)f(x^{-1})$ or $f(e)^{-1}=f(e)$ N... |
3,534,566 | <p>I want to know if my answer is equivalent to the one in the back of the book. if so what was the algebra? if not then what happened?</p>
<p><span class="math-container">$$x^2y'+ 2xy = 5y^3$$</span></p>
<p><span class="math-container">$$y' = -\frac{2y}{x} + \frac{5y^3}{x^2}$$</span></p>
<p><span class="math-contai... | user577215664 | 475,762 | <p>You made a sign mistake here:
<span class="math-container">$$\frac{-1}{2}v'+\frac{2}{x}v = \frac{5}{x^2}$$</span>
And also
<span class="math-container">$$\left (\frac 1 {y^2} \right )'=-2\frac {y'}{y^3}$$</span>
<span class="math-container">$$\implies v'=-2\frac {y'}{y^3}$$</span>
Another way:
<span class="math-cont... |
2,871,949 | <p>Let $X_1, X_2, X_3, X_4$ be independent Bernoulli random variables. Then
\begin{align}
Pr[X_i=1]=Pr[X_i=0]=1/2.
\end{align}
I want to compute the following probability
\begin{align}
Pr( X_1+X_2+X_3=2, X_2+X_4=1 ).
\end{align}
My solution: Suppose that $X_1+X_2+X_3=2$ and $X_2+X_4=1$. Then $(X_2, X_4)=(0,1)$ or $(... | BGM | 297,308 | <p>Just add as a supplementary technique:</p>
<p>You may also try to use the law of total probability with conditioning on $X_2$, since only $X_2$ are in common of the two events, then make use of the independence:</p>
<p>$$ \begin{align} &\Pr\{X_1 + X_2 + X_3 = 2, X_2 + X_4 = 1\} \\
=& \Pr\{X_1 + X_2 + X_3 =... |
63,525 | <p>I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with uniform distribution on the unit circle. Consider the random polynomial $P(z)$ given by
$$
P(z)=\prod_{i=1}^{n}(z−z_i).
$... | Joe Silverman | 11,926 | <p>I have a couple of comments regarding Chandru's answer, but they're too long to fit in the comment box, so I'm making this a separate answer. First, the quantity
$$
F(P) = \int_0^{2\pi} \log|P(e^{i\phi})| d\phi
$$
is not non-negative without some further assumption. The easiest is to assume that the polynomial $P$... |
3,919,284 | <p>Let <span class="math-container">$S^1 \subseteq \mathbb{R}^2$</span> be the unit circle circle <span class="math-container">$\mathbb{V}(X^2+Y^2-1)$</span>. Let <span class="math-container">$S$</span> be an infinite subset of <span class="math-container">$S^1$</span>. I want to show that
<span class="math-container">... | Sabino Di Trani | 82,009 | <p>The ring <span class="math-container">$\mathbb{R}[x,y]$</span> has dimension 2 and consequently a prime ideal containing <span class="math-container">$x^2+y^2-1$</span> must be a maximal one.</p>
<p>Moreover, the ring <span class="math-container">$\mathbb{R}[x,y]$</span> is Noetherian and every ideal has a finite n... |
3,919,284 | <p>Let <span class="math-container">$S^1 \subseteq \mathbb{R}^2$</span> be the unit circle circle <span class="math-container">$\mathbb{V}(X^2+Y^2-1)$</span>. Let <span class="math-container">$S$</span> be an infinite subset of <span class="math-container">$S^1$</span>. I want to show that
<span class="math-container">... | KReiser | 21,412 | <p>Sabino Di Trani's answer is good. Here is a solution which is perhaps more hands-on.</p>
<p>Suppose <span class="math-container">$f\in \Bbb R[x,y]/(x^2+y^2-1)$</span> vanishes on an infinite subset of <span class="math-container">$S^1$</span>. We'll show it's actually zero, which shows that the Zariski closure of an... |
1,530,848 | <p>Let $F(\mathbb{R})$ be the set of all functions $f : \mathbb{R} → \mathbb{R}$. Define pointwise addition and multiplication as follows. For any $f$ and $g$ in $F(\mathbb{R})$ let:</p>
<p>(i) $(f + g)(s) = f(x) + g(x)$ for all $x \in \mathbb{R}$</p>
<p>(ii) $(f · g)(s) = f(x) · g(x)$ for all $x \in \mathbb{R}$</p>
... | Slade | 33,433 | <p>This is not really an answer, but it was getting too long to be a comment.</p>
<p>Mathematics draws much of its power from deep, sometimes mysterious dualities between geometry and algebra, so I do not think there is any way to understand the relationship between geometric intuition and symbol manipulation in gener... |
3,450,713 | <p>Let <span class="math-container">$a+3b=7$</span> and <span class="math-container">$c=3$</span>. Then value of <span class="math-container">$a+3(b+c)$</span> is</p>
<p>A) <span class="math-container">$10$</span></p>
<p>B) <span class="math-container">$16$</span> </p>
<p>C) <span class="math-container">$21$</span><... | fleablood | 280,126 | <p>Method 1: Distribute first then substitute what you recognize:</p>
<p><span class="math-container">$a+3(b+c) =$</span></p>
<p><span class="math-container">$a + 3b + 3c = $</span></p>
<p><span class="math-container">$(a + 3b) + 3(c)= $</span> (and we know <span class="math-container">$c=3$</span> and <span class=... |
3,450,713 | <p>Let <span class="math-container">$a+3b=7$</span> and <span class="math-container">$c=3$</span>. Then value of <span class="math-container">$a+3(b+c)$</span> is</p>
<p>A) <span class="math-container">$10$</span></p>
<p>B) <span class="math-container">$16$</span> </p>
<p>C) <span class="math-container">$21$</span><... | suhbell | 592,879 | <p>For questions like these where you cannot solve for each value to then find the solution, you are supposed to substitute some form of the given equation in for the variable. You are given a number for c so substitute that into the equation <span class="math-container">$ a+3(b+c) $</span> first.
<span class="math-co... |
1,478,314 | <p>In this particular case, I am trying to <strong>find all points $(x,y)$ on the graph of $f(x)=x^2$ with tangent lines passing through the point $(3,8)$</strong>. </p>
<p>Now then, I know the <a href="http://www.meta-calculator.com/online/?panel-102-graph&data-bounds-xMin=-10&data-bounds-xMax=10&data-bo... | Ian Miller | 278,461 | <p>A point on the graph is $(x,x^2)$. The slope from that point to $(3,8)$ is given by: $\frac{x^2-8}{x-3}$. This has to be equal to the derivative at the point for it to be a tangent. So:
$$\frac{x^2-8}{x-3}=2x$$
$$x^2-8=2x^2-6x$$
$$0=x^2-6x+8$$
$$0=(x-2)(x-4)$$
So $x=2$ or $x=4$.
So the points are $(2,4)$ and $(4,16)... |
27,965 | <p>I'm looking at <a href="https://math.stackexchange.com/questions/2669893/calculating-the-sums-of-series">this question</a>. I gave the answer that was accepted. Please bear in mind that, when I answered this question, it was a different edit. In particular, there were more parts to the question.</p>
<p>The reason I... | hardmath | 3,111 | <p>I think better of <em>thorough</em> answers to poorly stated Questions than of fragmentary "hint" posts.</p>
<p>I suspect there is a temptation for many to give hasty replies to substandard Questions, accompanied by frequent rationalizations that "I didn't want to give the OP a full solution that can be copy-pasted... |
730,929 | <p>Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements.</p>
<p>I am not sure how to do this one, but furthermore, what does $a$ being algebraic over $F$ of degree $n$ mean? Does it mean the polynomial ... | user134824 | 134,824 | <p>The statement "$a$ is algebraic over $F$ of degree $n$" means two things together:</p>
<ol>
<li>$a$ is the root of some polynomial in $F[x]$ (that is, the coefficients of the polynomial lie in $F$) that has degree $n$.</li>
<li>Every other nonzero polynomial in $F[x]$ for which $a$ is a root has degree at least $n$... |
24,318 | <p>I have an expression as below:</p>
<pre><code>Equations = 2.0799361919940695` x[1] + 3.3534325557330327` x[1]^2 -
4.335179297091139` x[1] x[2] + 1.1989715511881491` x[2]^2 -
3.766597877399148` x[1] x[3] - 0.33254815073371535` x[2] x[3] +
1.9050048836042945` x[3]^2 + 1.1386715715291826` x[1] x[4] +
2... | bill s | 1,783 | <p>How about:</p>
<pre><code>apply[func_] := Module[{}, xvalues = Range[0, 500, 2.5];
points1 = Map[func1, xvalues];
Do[If[points1[[i]] < 0, points1[[i]] = 0], {i, 1, Length[points1], 1}];
table1 = Transpose[{xvalues, points1}]];
</code></pre>
<p>Now you call the function apply with your desired funcX ... |
4,316,780 | <p>Let <span class="math-container">$X$</span> be a real Banach space. Let <span class="math-container">$J \colon X \to 2^{X^*}$</span> be its (normalized) duality map,
<span class="math-container">$$ J(x) = \{ x^* \in X^* \colon \langle x^* , x \rangle =||x|| \ ||x^*||, \ || x^* ||=||x|| \} , \ x \in X.$$</span>
As... | Evangelopoulos Phoevos | 739,818 | <p>Here is a different approach. For <span class="math-container">$r >0$</span> consider the unit functional <span class="math-container">$y^*_r = j(x+r^{-1}y)/\|j(x+r^{-1}y)\|$</span>. Then
<span class="math-container">\begin{align*}
\|x\| &\le \|x+r^{-1}y\| = \frac{1} {\| j(x+r^{-1}y) \|}\langle j(x+r^{-1}y),... |
122,503 | <p>It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is <a href="http://arxiv.org/pdf/1009.0843v1.pdf">this one</a> by László Erdös, but the closest the quantum Brownian motion comes to appearing is in this conjecture (p. 30):</p>
<blockquote>
<p>[<b>Quantum Browni... | Jess Riedel | 5,789 | <p>(In words explained below:) <strong>Quantum Brownian motion (QBM) is a class of possible dynamics for an open, quantum, continuous degree of freedom in which the reduced dynamics are specified by a quadratic Hamiltonian and linear Lindblad operators in the phase-space variables $x$ and $p$.</strong></p>
<p>Consider... |
4,130,809 | <p>I have questions regarding the proof that I made about the following statement: "Let <span class="math-container">$(X,\tau_{X})$</span> be a topological space and <span class="math-container">$\lbrace \infty\rbrace$</span> an object that doesn't belong to X. Define <span class="math-container">$Y=X\cup\lbrace\i... | SolubleFish | 918,393 | <blockquote>
<p>Since <span class="math-container">$C_\alpha \in \tau_\infty$</span> for every <span class="math-container">$α$</span>, then <span class="math-container">$Y−C_\alpha$</span> is compact and closed in <span class="math-container">$X$</span> for every <span class="math-container">$\alpha$</span>.</p>
</blo... |
774,209 | <p>I got stuck to find a fair formula to calculate the average ranking of the items that I found after consecutive searches, look:</p>
<p><img src="https://i.stack.imgur.com/5LazY.jpg" alt="enter image description here" /></p>
<p>If I calculate the simple average of the item2 for example I get 1,33 as a result, not eve... | Kathystats | 700,749 | <p>The arithmetic mean is a parametric descriptive statistic that presumes normality in the data. Ordinal data (rank-ordered data) does not meet the assumptions of normality, and you cannot use an arithmetic mean to describe the central tendency—only the median.</p>
|
2,348,131 | <p>In our class, we encountered a problem that is something like this: "A ball is thrown vertically upward with ...". Since the motion of the object is rectilinear and is a free fall, we all convene with the idea that the acceleration $a(t)$ is 32 feet per second square. However, we are confused about the sign of $a(t)... | crankk | 202,579 | <p>You can model this on a linear one dimensional space, the "heigth" $x$ of the particle. Different forces are acting on the ball, on the one hand the gravitation, which is directed to the "ground", on the other hand in the beginning a force is applied towards the "top" as the ball is thrown in that direction. You can... |
2,348,131 | <p>In our class, we encountered a problem that is something like this: "A ball is thrown vertically upward with ...". Since the motion of the object is rectilinear and is a free fall, we all convene with the idea that the acceleration $a(t)$ is 32 feet per second square. However, we are confused about the sign of $a(t)... | Crazy | 449,016 | <p>The acceleration due to gravity is always downward. The convention is so. </p>
<p>Case I:<strong>Downward is the positive direction</strong>.</p>
<p>Let's us examine your case where the ball is falling vertically from the sky. Normally, we take the direction of initial motion of the ball to be positive. This happe... |
3,612,016 | <p>Consider <span class="math-container">$(\Bbb R\setminus\{-1\},*)$</span>, where
<span class="math-container">$$a*b:=ab+a+b \qquad a,b \in \Bbb R\setminus\{-1\}.$$</span>
We have to prove that it's an Abelian group. While it's easy to show how the properties of associativity, identity element, inverse element and com... | Ivo Terek | 118,056 | <p>You have to show that <span class="math-container">$ab+a+b$</span> is not equal to <span class="math-container">$-1$</span>, if <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are both not equal to <span class="math-container">$-1$</span> to begin with. This can be shown by assumi... |
3,612,016 | <p>Consider <span class="math-container">$(\Bbb R\setminus\{-1\},*)$</span>, where
<span class="math-container">$$a*b:=ab+a+b \qquad a,b \in \Bbb R\setminus\{-1\}.$$</span>
We have to prove that it's an Abelian group. While it's easy to show how the properties of associativity, identity element, inverse element and com... | Akash Yadav | 474,986 | <p>Hint : </p>
<p>If <span class="math-container">$ab+a+b=-1$</span>, then either <span class="math-container">$a=-1$</span> or <span class="math-container">$b=-1$</span>.</p>
|
1,076,292 | <p>I wish to use two points say $(x_1$,$y_1)$ and $(x_2$,$y_2)$ and obtain the coefficients of the line in the following form: $$ Ax + By + C = 0$$</p>
<p>Is there any direct formula to compute.</p>
| DeepSea | 101,504 | <p>$\text{slope} = m = \dfrac{y_2-y_1}{x_2-x_1} \Rightarrow y - y_1 = m(x-x_1) = \dfrac{y_2-y_1}{x_2-x_1}\left(x-x_1\right) \Rightarrow (x_2-x_1)y - y_1(x_2-x_1) = (y_2-y_1)x - x_1(y_2-y_1) \Rightarrow -(y_2-y_1)x + (x_2-x_1)y -y_1(x_2-x_1) +x_1(y_2-y_1)=0$. This gives the formula:</p>
<p>$A = y_1-y_2$</p>
<p>$B = x_... |
1,076,292 | <p>I wish to use two points say $(x_1$,$y_1)$ and $(x_2$,$y_2)$ and obtain the coefficients of the line in the following form: $$ Ax + By + C = 0$$</p>
<p>Is there any direct formula to compute.</p>
| Community | -1 | <p>Here are a few ways of finding values of $A$, $B$, and $C$.</p>
<p><strong>Method 1:</strong> A line is the set of points $\vec{x}$ such that
$$\vec{a}\cdot \vec{x_0} = \vec{a} \cdot \vec{x}$$
where $x_0$ is an arbitrary point on the line and $\vec{a}$ is a nonzero vector perpendicular to the line. We know that t... |
3,738,508 | <p>If <span class="math-container">$G$</span> is order <span class="math-container">$p^2q$</span>, where <span class="math-container">$p$</span>, <span class="math-container">$q$</span> are primes, prove that either a Sylow <span class="math-container">$p$</span>-subgroup or a Sylow <span class="math-container">$q$</sp... | quasi | 400,434 | <p>Assume <span class="math-container">$x^4+2$</span> factors in <span class="math-container">$\mathbb{Z}_5[x]$</span> as
<span class="math-container">$$x^4+2=(x^2+ax+b)(x^2+cx+d)$$</span>
If the <span class="math-container">$\text{RHS}$</span> is expanded,</p>
<ul>
<li>The coefficient of the <span class="math-containe... |
1,301,476 | <p>(Cross-posted in <a href="https://matheducators.stackexchange.com/q/8173/"><strong>MESE 8173</strong></a>.) </p>
<p>I want to start to do mathematical Olympiad type questions but have absolutely no knowledge on how to solve these apart from my school curriculum. I'm $16$ but know maths up to the $18$ year old level... | Keith | 244,951 | <p>I understand your reluctance to focus solely on contest books. Often some of the "tricks" used are really applications of some more general theory in disguise. Greater clarity is sometimes brought by studying math at a higher level or at least in a more orderly way (from books where the exposition is directed at a g... |
2,280,133 | <p><em>I need help to understand, some steps of the proof of this theorem.</em> </p>
<p><strong>(Kolmogorov-M. Riesz-Fréchet)</strong> Let $\mathcal{F}$ be a bounded set in $L^p(\mathbb{R}^N)$ with $1\leq p < \infty$. Assume that </p>
<p>\begin{equation}
\lim\limits_{|h|\longrightarrow 0 }\|\tau_hf-f\|_p=0 \ ... | Felix B. | 445,105 | <p>If $\rho_n$ is a probability density function and $\mu_n(B)=\int_B\rho_n(x)dx$, then
\begin{align}\int |f(x-y)-f(x)|\rho_n(y)dy&=\int|f(x-y)-f(x)|d\mu_n\\
&\le \left(\int|f(x-y)-f(x)|^{p}d\mu_n\right)^{1/p}\left(\int |1|^{q}d\mu_n\right)^{1/q}\\
&=\left(\int|f(x-y)-f(x)|^{p}\rho_n(y)dy\right)^{1/p}
\end{... |
2,433,174 | <p>I'm struggling with the following (<em>is it true?</em>):</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a set and denote <span class="math-container">$\aleph(X)$</span> the <em><strong>cardinality</strong></em> of <span class="math-container">$X$</span>. Suppose that <span class="math-container"... | Peter Szilas | 408,605 | <p>$T:\mathbb{R^2} \rightarrow \mathbb{R^2}$.</p>
<p>$T(x,y) \mapsto (x',y')$,</p>
<p>$x'=4x-y$; $y'=3x-2y.$</p>
<p>Solving for $x,y$ in terms of $x',y'$.</p>
<p>$2x'-y'=5x$; and $ 3x' -4y' =5y$.</p>
<p>The line $x+2y = 6$ is transformed:</p>
<p>$5x +10y =30 \rightarrow$</p>
<p>$(2x'-y' ) + 2 (3x'-4y') =30.$</... |
2,604,825 | <p>So I have a problem (two problems, actually) that a friend helped me out with, I'm able to work out the components of this problem but get lost when I have to bring it all together... so what I have is this</p>
<p>$f(t)=t^{2}e^{-2t}+e^{-t}\cos(3t)+5$</p>
<p>Simple enough. I got:</p>
<p>$$\mathcal{L}[t^2]=\frac{2}... | Olivier Oloa | 118,798 | <p>One may use the <a href="https://math.stackexchange.com/questions/980551/how-to-show-sin-x-geq-frac2x-pi-x-in-0-frac-pi2">standard identity</a>
$$
\sin x\ge \frac{2x}{\pi},\qquad x \in \left[0,\frac \pi2 \right],
$$ giving
$$\int_0^{\frac{\pi}{2}}\sqrt{\sin x}\:dx >\int_0^{\frac{\pi}{2}}\sqrt{ \frac{2x}{\pi}}\:... |
4,214,474 | <p>Consider <span class="math-container">$\mathbb{R}^\omega$</span> (countably infinite product of <span class="math-container">$\mathbb{R}$</span>) with the uniform metric.</p>
<p>Let <span class="math-container">$A$</span> be the set of infinite bounded sequences of <span class="math-container">$\mathbb{R}$</span>, i... | Huy Nguyen | 922,437 | <p>You are almost there. Suppose <span class="math-container">$y=(y_k) \in B_d(x, \frac{M}{2})$</span> is bounded which means that there exsits <span class="math-container">$N > 0$</span> such that <span class="math-container">$|y_k| \leq N$</span> for all <span class="math-container">$k$</span>. But since <span cl... |
1,238,783 | <p>I am currently in high school where we are learning about present value. </p>
<p>I struggle with task like these: Say you get 6% interest each year, how much interest would that be each month?</p>
| Alberto Debernardi | 140,199 | <p>In order to compute the monthly interest, you know that in a year (which has 12 months) you get $6\%$ interest. Call $a$ the (decimal) monthly interest. Thus,
$$
(1+a)^{12}=1.06.
$$
Then you just solve the equation for $a$.</p>
|
58,947 | <p>Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?</p>
<p><strong>Edit:</strong> To rule out the case where $X$ has infinite genus, perhaps one could add the hypothesis that the topological space... | Sylvain Bonnot | 15,673 | <p>You should probably check the following article:
Migliorini, Luca, "On the compactification of Riemann surfaces".
Here is the Mathscinet review about it:
"In this paper the author studies some questions concerning the compactifications of Riemann
surfaces. It is proved that if $X$ is an open connected Riemann surface... |
624,715 | <p>Find this follow ODe solution
$$y''-y'+y=x^2e^x\cos{x}$$</p>
<p>I konw solve this follow three case
$$y''-y'+y=x^2\cos{x}$$
$$y''-y'+y=e^x\cos{x}$$
$$y''-y'+y=x^2e^x$$</p>
<p>But for $f(x)=x^2e^x\cos{x}$, I can't.</p>
<p>Thank you very much!</p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displayst... |
142,127 | <p>A simplex is regular if its all edges have the same length.</p>
<p>How to test in Mathematica whether a <code>Simplex</code> is regular or not, without checking all the edges manually? I'm not really familiar with loops in Mathematica. I also can't find in the documentation how to access the vertices of a <code>Sim... | kglr | 125 | <pre><code>ClearAll[regSimplexQ]
regSimplexQ = Equal @@ PropertyValue[{MeshRegion[#, Simplex[{1, 2, 3, 4}]] & @@ #, 1},
MeshCellMeasure] &;
regSimplexQ@Simplex[{{0, 1, 0}, {1, 0, 0}, {0, 0, 1}, {1, 1, 1}}]
</code></pre>
<blockquote>
<p>True</p>
</blockquote>
<pre><code>regSimplexQ@Simplex[{{0, 0, 1}, {1,... |
1,573,262 | <p>What's an example of a convergent, yet unbounded sequence?</p>
<p>I'm having trouble of thinking how to do this. I want to use a piecewise function, but I feel like there might be something easier than this.</p>
| Adhvaitha | 228,265 | <p>Consider the sequence of functions on the interval $(0,1)$ given by $f_n(x) = nx^n$. Note that $f_n(x)$ converges to $0$ point wise on $(0,1)$. However, $f_n(x)$ is unbounded on $(0,1)$.</p>
|
1,573,262 | <p>What's an example of a convergent, yet unbounded sequence?</p>
<p>I'm having trouble of thinking how to do this. I want to use a piecewise function, but I feel like there might be something easier than this.</p>
| MC989 | 292,147 | <p>Rise is correct. I can't believe I didn't see this sooner. all convergent sequences are bounded. I was confusing this with convergent functions, which are different. (i.e. $f(x)= \frac {1}{x}$)</p>
|
3,704,633 | <p>Evaluate: <span class="math-container">$$\lim_{n \to \infty} \sqrt[n]{\frac{n!}{\sum_{m=1}^n m^m}}$$</span>
In case it's hard to read, that is the n-th root. I don't know how to evaluate this limit or know what the first step is... I believe that: <span class="math-container">$$\sum_{m=1}^n m^m$$</span> doesn't ha... | Fnacool | 318,321 | <p>Let <span class="math-container">$a_n= (n! / \sum_{m=1}^n m^m)^{1/n}$</span>. </p>
<p>Observe that <span class="math-container">$n^n \le \sum_{m=1}^n m^m \le n n^n$</span>. </p>
<p>Then <span class="math-container">$$(\frac{n!}{n^n})^{1/n} \frac{1}{n^{1/n}} \le a_n \le (\frac{n!}{n^n})^{1/n}.$$</span> </p>
<p>S... |
1,620,686 | <p>Prove that this works for all $x$ and and only some $y$
$$\sqrt{(x-1)^2-(y+2)^2}=0.$$</p>
<p>This is as far as I got so far</p>
<p>Difference of squares:</p>
<p>$\sqrt{(x-1-y-2)(x-1+y+2)}=0$<br>
$\sqrt{x-y-3}\sqrt{x+y+1}=0$</p>
<p>Therefore $x-y-3=0 \implies y=x-3$ </p>
<p>$x+y+1=0$ and $y=-1-x$</p>
<p>I ju... | Ross Millikan | 1,827 | <p>What you must mean (and it <em>really</em> helps to write it clearly) is that for all $x$ you can find a $y$ that makes the statement true and you can find another $y$ that makes the statement false. So given an $x$ you can exhibit a $y$ that makes the statement true-just solve for $y$<br>
$$(x-1)^2=(y+2)^2\\y+2=\p... |
1,285,273 | <p>Looking for hints to find the orthnormal basis for the null space/range of the following matrix</p>
<p>$A = \frac{1}{3}\left( \begin{array}{ccc}
2 & -1 & -1 \\
-1 & 2 & -1 \\
-1 & -1 & 2 \end{array} \right)$</p>
| BruceET | 221,800 | <p>This is an extended Comment, not an answer. Actually, I'm not
sure there <em>is</em> an answer along the lines you seek. First, although
the proof of the CLT is in terms of MGFs, I'm not sure MGFs
are going to be helpful judging how well sums or averages of
iid observations match the normal distribution. Second, the... |
126,901 | <p>How to evaluate this determinant $$\det\begin{bmatrix}
a& b&b &\cdots&b\\ c &d &0&\cdots&0\\c&0&d&\ddots&\vdots\\\vdots &\vdots&\ddots&\ddots& 0\\c&0&\cdots&0&d
\end{bmatrix}?$$</p>
<p>I am looking for the different approaches.</p>
| Davide Giraudo | 9,849 | <p>If the dimension of the matrix is $2$, it's only $ad-bc$. If it's $\geq 3$, and $d=0$ the determinant is $0$. If $d\neq 0$, do $C_1\leftarrow C_1-\frac cdC_j$, $2\leq j\leq n$. The first column becomes $\pmatrix{a-(n-1)\frac{bc}d\\\ 0\\\ \vdots\\\ 0}$, and the determinant is $d^{n-1}\left(a-(n-1)\frac{bc}d\right)=d^... |
2,517,469 | <p>Let $P$ be a projective module and $P=P_1+N$, where $P_1$ is a direct summand of $P$ and $N$ is a submodule. Show that there is $P_2\subseteq N$ such that $P=P_1\oplus P_2$. </p>
<p>I know that there is a submodule $P'$ of $P$ such that $P=P_1\oplus P'$. I wanted to consider the projection from this to $P_1$ and us... | Guest | 90,271 | <p>The condition $P = P_1 + N$ implies that the natural map $N \to P/P_1$ is surjective. As $P$ is projective this means the quotient map $P \to P/P_1$ factors through $N$, so there is a homomorphism $P \to N$ such that the composition $P \to N \to P/P_1$ is the quotient map. Since the quotient map gives an isomorphi... |
2,284,451 | <blockquote>
<p><span class="math-container">$A$</span> and <span class="math-container">$B$</span> alternately throw a pair of coin. The player who throws head two times first will win.</p>
<p>A has the first throw. The find chance of winning <span class="math-container">$A$</span> is</p>
</blockquote>
<p>Attempt: Let... | true blue anil | 22,388 | <p>Assuming that each throw <strong>two coins</strong> alternately (the more difficult formulation), one way to solve is to consider the <strong>odds</strong> for the first two rounds.<br>
[ Subsequent rounds of $2$ will only add some common multiplier, <strong>odds</strong> won't change]</p>
<p>$\Bbb P$(A wins on fir... |
140,358 | <p>Let $X$ and $Y$ be two topological spaces with $C(X) \cong C(Y)$ (where $C(X)$ is the ring of all continuous real valued functions on $X$). I know that we can not conclude that $X$ and $Y$ are homeomorphic. But I wonder how independent $X$ and $Y$ could be ? For example is there any forced relation between their car... | Joseph Van Name | 22,277 | <p>Since we are talking about rings of continuous functions, I will only talk about completely regular spaces in this problem. It is well known that for completely regular spaces $X$, $C(X)\simeq C(Y)$ if and only if $\upsilon X\simeq\upsilon Y$ where $\upsilon X$ denotes the Hewitt-realcompactification of a space $X$.... |
2,309,721 | <p>The problem is: Prove that $7|x^2+y^2$ only if $7|x$ and $7|y$ for $x,y∈Z$.</p>
<p>I found a theorem in my book that allows to do the following transformation:
if $a|b$ and $a|c$ -> $a|(b+c)$</p>
<p>So, can I prove it like this: $7|x^2+y^2 =>7|x^2, 7|y^2 => 7|x*x, 7|y*y => 7|x, 7|y$ ?</p>
<p>I am no... | Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>Work in the field $\;\mathbf Z/7\mathbf Z=\{0,\pm1,\pm2,\pm3\}$, compute all squares, then all sums of squares (a <em>Pythagoras' table</em> will be useful) and check $$x^2+y^2\equiv 0\mod7\iff (x\equiv 0)\wedge (y\equiv 0)\mod 7.$$</p>
|
155,429 | <p>Consider a square skew-symmetric $n\times n$ matrix $A$. We know that $\det(A)=\det(A^T)=(-1)^n\det(A)$, so if $n$ is odd, the determinant vanishes.</p>
<p>If $n$ is even, my book claims that the determinant is the square of a polynomial function of the entries, and Wikipedia confirms this. The polynomial in questi... | mike stone | 252,564 | <p>By continuity, we can assume that $A$ can always be reduced to block diagonal form with blocks
$$
\left(\begin{matrix} 0 &\lambda_i\cr -\lambda_i &0 \end{matrix} \right)
$$
on the diagonal. In this case computing the determinant gives $\prod \lambda^2_i$ and computing the Pfaffian gives $\prod \lambda_i$, s... |
3,785,967 | <p>Let <span class="math-container">$E(R)_X$</span> denote the expected return of asset <span class="math-container">$X$</span>.</p>
<p>Given a market with only 3 assets; <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and <span class="math-container">$C$</span>, the following three thi... | tommik | 791,458 | <blockquote>
<p>or do I need to somehow incorporate the probabilities in to this formula?</p>
</blockquote>
<p>Yes, You do need.</p>
|
162,655 | <p>Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat? </p>
<p>I am especially interset in the case of Lorentzian Manifold whose sign signature is (- ,+ ,+ , + ). Of course, the example is not constricted in L... | Dietrich Burde | 32,332 | <p>For an example, let $N$ be a compact complex hypersurface of degree $m+1$ of the complex projective space $\mathbb{CP}^m$ with complex dimension $m\ge 3$ (for $m=3$ this is a complex $K3$ surface). The first Chern class of $N$ vanishes, and hence $N$ admits a Ricci-flat but <em>nonflat</em> Riemannian metric, by a t... |
73,383 | <p>The problem is:
$$\displaystyle \lim_{(x,y,z) \rightarrow (0,0,0)} \frac{xy+2yz+3xz}{x^2+4y^2+9z^2}.$$</p>
<p>The tutor guessed it didn't exist, and he was correct. However, I'd like to understand why it doesn't exist.</p>
<p>I think I have to turn it into spherical coordinates and then see if the end result depen... | Chris Taylor | 4,873 | <p>Let's take the limit $z\to0$ first, getting</p>
<p>$$\lim_{x,y\to 0} \frac{xy}{x^2+4y^2}$$</p>
<p>Now consider what happens if you take the limit along $y=x$:</p>
<p>$$\lim_{x,y\to 0} \frac{xy}{x^2+4y^2} = \lim_{x\to0} \frac{x^2}{5x^2} = \frac{1}{5}$$</p>
<p>and along $y=2x$:</p>
<p>$$\lim_{x,y\to 0} \frac{xy}{... |
733,280 | <p>I cannot understand why $\log_{49}(\sqrt{ 7})= \frac{1}{4}$. If I take the $4$th root of $49$, I don't get $7$.</p>
<p>What I am not comprehending? </p>
| Mr.Fry | 68,477 | <p>Solve: $49^x = \sqrt{7} \Rightarrow 49^{2x} = 7 \Rightarrow \frac{\log7}{\log 49} = 2x \Rightarrow \frac{1}{2}=2x \Rightarrow x = \frac{1}{4}$</p>
|
2,813,595 | <p>which of the following can be expressed by exact length but not by exact number?</p>
<p>(i) $ \sqrt{10} $</p>
<p>(ii) $ \sqrt{7} $</p>
<p>(iii) $ \sqrt{13} \ $</p>
<p>(iv) $ \ \sqrt{11} \ $</p>
<p><strong>Answer:</strong></p>
<p>I basically could not understand th question.</p>
<p>What is meant by expressi... | Peter Szilas | 408,605 | <p>Correct me if wrong .</p>
<p>$√n$ is constructible, $n \in \mathbb{Z^+}$.</p>
<p>$n=1$, ok.</p>
<p>Assume $√n$ is constructible.</p>
<p>Step: </p>
<p>Show that $\sqrt{n+1}$ is constructible.</p>
<p>Pythagorean Theorem:</p>
<p>$(\sqrt{n})^2+1= n+1= (\sqrt{n+1})^2$, i.e.</p>
<p>construct a right triangle with ... |
57,988 | <p>I am a programmer/analyst with limited (and pretty rusty) knowledge of math.</p>
<p>"Just for the heck of it" I have decided to try my hand at <a href="http://spectrum.ieee.org/automaton/robotics/artificial-intelligence/you-you-can-take-stanfords-intro-to-ai-course-next-quarter-for-free" rel="nofollow">Stanford's i... | Juan S | 2,219 | <p>For Linear Algebra I think that the (free) book by <a href="http://joshua.smcvt.edu/linearalgebra/" rel="nofollow">Heffron</a> is pretty good. </p>
<p>For probability I don't know too much, but maybe the first half of <a href="http://rads.stackoverflow.com/amzn/click/0521540364" rel="nofollow">Tijms</a> book</p>
|
3,109,001 | <p>I want to compute the Picard group of <span class="math-container">$\mathbb{Z}[\sqrt{-19}]$</span>, which is not a Dedekind domain. The problem is that I don't even know where to begin.</p>
<p>Any ideas would be helpful.</p>
<p>Thanks</p>
<p>Edit- Since there's some confusion by what I mean by the "Picard group".... | DanLewis3264 | 480,329 | <p>You want to use the Kummer-Dedekind Theorem to factor the prime ideals of norm at most the Minkowski bound. </p>
<p>I recommend you follow 2.6.1 of <a href="https://jrsijsling.eu/notes/ant-notes.pdf" rel="nofollow noreferrer">https://jrsijsling.eu/notes/ant-notes.pdf</a></p>
|
3,109,001 | <p>I want to compute the Picard group of <span class="math-container">$\mathbb{Z}[\sqrt{-19}]$</span>, which is not a Dedekind domain. The problem is that I don't even know where to begin.</p>
<p>Any ideas would be helpful.</p>
<p>Thanks</p>
<p>Edit- Since there's some confusion by what I mean by the "Picard group".... | pisco | 257,943 | <p>In case of imaginary quadratic field, there is a nice correspondence between the Picard group of orders with discriminant <span class="math-container">$D$</span> and reduced primitive quadratic form with the same discriminant. More precisely, the form <span class="math-container">$ax^2+bxy+cy^2$</span> corresponds t... |
307,701 | <p>Show that if $G$ is a finite group with identity $e$ and with an even number of elements, then there is an $a \neq e$ in $G$, such that $a \cdot a = e$.</p>
<p>I read the solutions here <a href="http://noether.uoregon.edu/~tingey/fall02/444/hw2.pdf" rel="nofollow">http://noether.uoregon.edu/~tingey/fall02/444/hw2.p... | i.a.m | 60,195 | <p>let $e,a_1,a_2,...,a_n$ be the elements of the group since the number of these elements is even we get the number of these elemnts $a_1,a_2,...,a_n$ is odd, now start putting every element with its inverse in a set lets say $\{a_i,a_j\}$ since the number is odd you will be left with one element call it $a$ and you ... |
4,132,402 | <p>Can this be solved without trigonometry?</p>
<blockquote>
<p><span class="math-container">$AB$</span> is the base of an isosceles <span class="math-container">$\triangle ABC$</span>. Vertex angle <span class="math-container">$C$</span> is <span class="math-container">$50^\circ$</span>. Find the angle between the alt... | Blue | 409 | <p>As <a href="https://math.stackexchange.com/a/4132518/409">@quasi's answer</a> suggests, the target angle almost-certainly isn't rational, so avoiding trig is unlikely.</p>
<p>That said, there's a pretty quick trigonometric approach to the target:</p>
<p><a href="https://i.stack.imgur.com/d7cfi.jpg" rel="nofollow nor... |
947,618 | <p>For T: V2->V2</p>
<p>T maps each point with polar coordinate (r.theta) to each point with polar coordinate (r,2theta) and T maps 0 onto itself.</p>
<p>Hi,</p>
<p>I was trying to do this by letting r= square root of x^2 + y^2 and theta=arctan(y/x) </p>
<p>but I failed.</p>
<p>can anybody please explain it? </p>
| Slade | 33,433 | <p>$T$ is surjective but not injective. For a linear transformation from a finite-dimensional vector space to itself, this is impossible.</p>
|
947,618 | <p>For T: V2->V2</p>
<p>T maps each point with polar coordinate (r.theta) to each point with polar coordinate (r,2theta) and T maps 0 onto itself.</p>
<p>Hi,</p>
<p>I was trying to do this by letting r= square root of x^2 + y^2 and theta=arctan(y/x) </p>
<p>but I failed.</p>
<p>can anybody please explain it? </p>
| Ben Grossmann | 81,360 | <p>We can deduce that $T$ is not linear because
$$
T[(1,0)] + T[(1,\pi)] = [(1,0) + (1,0)] = (2,0)
$$
But
$$
T[(1,0) + (1,\pi)] = T[0] = 0
$$</p>
|
1,663,838 | <p>Show that a positive integer $n \in \mathbb{N}$ is prime if and only if $\gcd(n,m)=1$ for all $0<m<n$.</p>
<p>I know that I can write $n=km+r$ for some $k,r \in \mathbb{Z}$ since $n>m$</p>
<p>and also that $1=an+bm$. for some $a,b \in \mathbb{Z}$</p>
<p>Further, I know that $n>1$ if I'm to show $n$ is... | ThisIsNotAnId | 24,567 | <p>What you've written on the right of the iff. is the definition of a prime number. You have just stated it using more concise notation.</p>
<p>According to the <a href="http://mathworld.wolfram.com/PrimeNumber.html" rel="nofollow">wolfram page</a> for <em>Prime Number</em>,</p>
<blockquote>
<p>A prime number (or ... |
3,354,990 | <p>I have points and limits of a function and even the shape of the function and I'm looking for the function, something that very interesting for me how could I control the curve of the function?</p>
<p>(1) <span class="math-container">$\lim\limits_{x \to inf} f(x) = 1 $</span></p>
<p>(2) <span class="math-container... | Kavi Rama Murthy | 142,385 | <p>One such function is <span class="math-container">$1-(1-c^{2}x^{2})^{+}$</span> where <span class="math-container">$y^{+}$</span> denotes <span class="math-container">$\max \{y,0 \}$</span>. </p>
<p>Edit: this function is convex on <span class="math-container">$(0,\frac 1 c)$</span>. To make it concave consider <sp... |
2,930,413 | <p>The problem is as shown. I tried using gradient and Hessian but can not make any conclusions from them. Any ideas?</p>
<p><span class="math-container">$$\max x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}$$</span></p>
<p>subject to</p>
<p><span class="math-container">$$\sum_{i=1}^nx_i=1,\quad x_i\geq 0,\quad i=1,2,\ldots,n,$... | Ahmed S. Attaalla | 229,023 | <p>The potential energy of the spring is $\frac{1}{2}x^2$ so,</p>
<p>$$\frac{1}{2}x^2+\frac{1}{2} \left(\frac{dx}{dt} \right)^2=E$$</p>
<p>It follows,</p>
<p>$$(\frac{dx}{dt})^2=2E-x^2$$</p>
<p>Supposing that $\frac{dx}{dt} \geq 0$ (ie spring is moving in positive direction), we get:</p>
<p>$$\frac{dx}{dt}=\sqrt{2... |
323,783 | <p>How do I evaluate this definite integral?
$$\int_{0}^{\frac{\pi}{12}}{\sin^4x \, \cos^4x\, \operatorname{d}\!x}$$
I know this is a trig. function. </p>
| Community | -1 | <p>From <a href="https://math.stackexchange.com/questions/20397/striking-applications-of-integration-by-parts/20481#20481">here</a>, we have
$$\int_0^{\pi/2} \sin^{2n}(x) dx = \dfrac{2n-1}{2n} \cdot \dfrac{2n-3}{2n-2} \cdots \dfrac34 \cdot \dfrac12 \cdot\dfrac{\pi}2$$
Hence,
$$\int_0^{\pi/2} \sin^4(x) \cos^4(x) dx = \d... |
743,473 | <p>A long Weierstrass equation is an equation of the form
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$
Why are the coefficients named $a_1, a_2, a_3, a_4$ and $a_6$ in this manner, corresponding to $xy, x^2, y, x$ and $1$ respectively? Why is $a_5$ absent?</p>
| Alex | 38,873 | <p>OK, here's my suggestion: for 1) use the law of total probability
$$
P_M=P_{M|L}P_L + P_{M|L'}P_{L'}
$$
Once you've done that, use $P_{M'}=1-P_M$ for 2):
$$
P_{L|M'}=\frac{P_{M'|L}P_L}{P_{M'}}
$$</p>
|
2,798,598 | <p>We have the series $\sum\limits_{n=1}^{\infty} \frac{(-1)^n n^3}{(n^2 + 1)^{4/3}}$. I know that it diverges, but I'm having some difficulty showing this. The most intuitive argument is perhaps that the absolute value of the series behaves much like $\frac{n^3}{\left(n^2\right)^{4/3}} = \frac{1}{n^{-1/3}}$, which div... | user | 505,767 | <p>Note that</p>
<p>$$ \left|\frac{(-1)^n n^3}{(n^2 + 1)^{4/3}}\right|\sim {n^\frac13}\to \infty$$</p>
<p>thus the given series diverges since each terms in the limit diverges.</p>
<p>Recall indeed for convergence we always need that, as necessary condition, that $|a_n|\to 0$.</p>
|
283,824 | <p>Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Let $V$ be a linear subspace of dimension $k$ and let $P_V(x)$ be the orthogonal projection of $x$ onto $V$.
I have seen quoted in the literature that
\begin{align}
\mathbb{P}[|\left\| P_V(x)\right\|_2 - \sqrt{k/n} | \le \epsilo... | Dirk | 9,652 | <p>Actually, the answer is almost in the paper you linked. There the author refers to </p>
<blockquote>
<p>Duality theorems for marginal problems, Hans G. Kellerer, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, November 1984, Volume 67, Issue 4, pp 399–432, <a href="https://link.springer.com/arti... |
73,629 | <p>I want to use <code>Listplot</code> with <code>Tooltip</code>that displays <code>Position</code>of the element I'm hovering over.</p>
<pre><code>data={{0,1},{1,3},{2,2}};
ListPlot[Tooltip[data]]
</code></pre>
<p>This is displaying the value of the element. Can I use the <code>Position</code>function in the tooltip... | kglr | 125 | <pre><code>data = Sort@RandomInteger[10, {10, 2}];
tts = {Directive[Red, 16, Bold], CellFrame -> 3, CellFrameMargins -> 5};
ListPlot[MapIndexed[Tooltip[#, First@#2, TooltipStyle -> tts] &, data],
PlotStyle -> PointSize[Large], Frame -> True, AxesOrigin -> {0, 0}]
</code></pre>
<p><img s... |
4,358 | <p>I've been reading a bit about how the set of bounds changes for a set depending on what superset one works with. I considered the sets $S\subseteq T\subseteq\mathbb{Q}$ and worked out a few contrived examples:</p>
<p>If $S=T=$ {$x\in\mathbb{Q}\ | \ x^2\lt 2$}, so here $S$ is not bounded above in $T$, but it is boun... | Dan Ramras | 1,392 | <p>For the first question, let $S$ be the set of all rationals less than $\pi$. Then $S$ has no least upper bound in $\mathbb{Q}$. On the other hand, if $T = S\cup {4}$, then 4 is the least upper bound of $S$ inside $T$.</p>
<p>For the second question, let $S$ be the set of all rationals strictly less than 1, and le... |
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