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 |
|---|---|---|---|---|
3,659,413 | <p>I'm reading the book "Topological Graph Theory" by Gross and I've gone through a fair bit of it. It seems like the entire book is leading up to being able to imbed a group onto a surface, and I have no idea why you would want to do that.
I am a physics major and not very advanced in math.
Any insight would be appre... | Alexander Gruber | 12,952 | <p>The embeddings one generally encounters first are planar embeddings. A good example is the three cottage problem:</p>
<blockquote>
<p>Suppose there are three cottages on a plane (or sphere) and each needs to be connected to the water, gas, and electricity companies. Without using a third dimension or sending any ... |
3,163,697 | <p>So, I have to approximate a value of square root of e: <span class="math-container">$\sqrt{e}$</span> with a precision of <span class="math-container">$10^{-3}$</span>.
I have calculated the first and second derivative:
So instead of <span class="math-container">$\sqrt{e}$</span> I need to approximate the value for... | E.H.E | 187,799 | <p><span class="math-container">$$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$</span>
so
<span class="math-container">$$e^{0.5}=\sqrt{e}=\sum_{n=0}^{\infty}\frac{0.5^n}{n!}$$</span>
now find the <span class="math-container">$n$</span> according to</p>
<p><span class="math-container">$$\frac{0.5^n}{n!}<10^{-3}$$</span></... |
3,163,697 | <p>So, I have to approximate a value of square root of e: <span class="math-container">$\sqrt{e}$</span> with a precision of <span class="math-container">$10^{-3}$</span>.
I have calculated the first and second derivative:
So instead of <span class="math-container">$\sqrt{e}$</span> I need to approximate the value for... | marty cohen | 13,079 | <p>Since
<span class="math-container">$e^x
=\sum_{k=0}^{\infty} \dfrac{x^k}{k!}
$</span>,
the error after <span class="math-container">$n$</span> terms is
<span class="math-container">$e_n(x)
=\sum_{k=n}^{\infty} \dfrac{x^k}{k!}
$</span>.</p>
<p>If <span class="math-container">$ 0 < x < 1$</span>,
the error can ... |
4,092,994 | <p>The question is</p>
<blockquote>
<p>Find the solutions to the equation <span class="math-container">$$2\tan(2x)=3\cot(x) , \space 0<x<180$$</span></p>
</blockquote>
<p>I started by applying the tan double angle formula and recipricoal identity for cot</p>
<p><span class="math-container">$$2* \frac{2\tan(x)}{1-... | ryang | 21,813 | <p>The error is in your very first line.</p>
<p>As alluded to by @Tavish in the comments, since the double-angle identity for <span class="math-container">$\tan$</span> has domain <span class="math-container">$$\mathbb R\setminus \left\{(2k+1)\frac{\pi}2 \mid k\in\mathbb Z\right\},$$</span> a separate case for odd mult... |
1,453,067 | <p>My friend's professor raised this question in a coaching and he and I tried everything we could think of. But later I thought that since $\sin (2x) $ can have values only between -1 & +1 and anything but +1 makes the equation complex ( keeping in mind that the integral is meant to be non-complex), there is no so... | Lucian | 93,448 | <p><strong>Hint:</strong> For starters, write $\sqrt{\sin2x-1}=\sqrt{-(1-\sin2x)}=i\sqrt{1-\sin2x}~.~$ Then remember the fact that there are well-known <a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Half-angle_formulae" rel="nofollow">trigonometric formulas</a> for $1\pm\cos2t$. So, how do we g... |
2,099,751 | <p>Suppose that $f$ is a continuous function and that $f(0)=1$ and $f(1)=2$. Show that there is $c\in(0,1)$ such that $f(c)=-\ln (2-2c)$. </p>
<p>It is a bit different from the technique I used <a href="https://math.stackexchange.com/questions/237648/a-question-on-intermediate-value-theorem">Here</a>. </p>
<p>Any ide... | Torsten Schoeneberg | 96,384 | <p>No. Let $y$ be an integer and assume $67500y^3-y = y(67500y^2-1)$ is a square. Since for any prime $p$ (EDIT: which divides $y$), the $p$-adic valuation $val_p(67500y^3-y) = val_p(y)$, $y$ has to be a positive square, or the negative of a square, or 0. In the first case, $67500y^2 -1$ must be a square too. But modul... |
2,099,751 | <p>Suppose that $f$ is a continuous function and that $f(0)=1$ and $f(1)=2$. Show that there is $c\in(0,1)$ such that $f(c)=-\ln (2-2c)$. </p>
<p>It is a bit different from the technique I used <a href="https://math.stackexchange.com/questions/237648/a-question-on-intermediate-value-theorem">Here</a>. </p>
<p>Any ide... | Xam | 133,781 | <p>Let's suppose that $67500y^3-y=x^2$ for some integer $x$. If we factorize we get $$y(67500y^2-1)=x^2.$$</p>
<p>Now, if $y<0$, the LHS is negative, but $x^2\ge 0$, so there are no solutions.</p>
<p>If $y>0$, since $\gcd(y, 67500y^2-1)=1$, then it's a consequence of the <em>fundamental theorem of the arithmeti... |
4,300,993 | <p>In this paper <a href="https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf" rel="nofollow noreferrer">Five stages of accepting constructive mathematics</a> on page 484 (shown in the image below) it contrastingly shows the use of the axiom of choice (<span class="math-cont... | Léreau | 351,999 | <p>Let me try to illustrate the difference by giving yet another example.</p>
<p>Let <span class="math-container">$M \neq \emptyset$</span> be some set. Using the definition of equality on sets and the fact that <span class="math-container">$y \in \emptyset$</span> is wrong for any <span class="math-container">$y$</spa... |
3,258,249 | <p><span class="math-container">$\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$</span></p>
<p>It's clear that we can simplify the limit a little bit, after which we get:</p>
<p><span class="math-container">$\lim\limits_{n\to\infty}{(\frac{1}{4})^{n}\sum\limits_{... | J.G. | 56,861 | <p>Let <span class="math-container">$\left[x^j\right]f(x)$</span> denote the <span class="math-container">$x^j$</span> coefficient in <span class="math-container">$f(x)$</span>. You seek<span class="math-container">$$\frac15\lim_{n\to\infty}\left(\left[x^{n-1}\right]\sum_{k=n}^{5n}\left(\frac{4+x}{5}\right)^{k-1}\right... |
258,132 | <p>Consider the following simple example as motivation for my question. If it were the case that, say, the Riemann hypothesis turned out to be independent of ZFC, I have no doubt it would be accepted by many as a new axiom (or some stronger principle which implied it). This is because we intuitively think that if we ... | Gerhard Paseman | 3,402 | <p>My poor understanding of Platonism, as a philosophy of mathematics, is that there is a common universe composed of pure thought, representing a collection of idealized objects, about which people can reason and discover (not invent!) new relationships and previously unconceived objects. For example, any arrangement... |
311,380 | <p>Prove that the relation $x \sim y$ iff $y$ is an element of the connected component of $x$ is an equivalence relation.</p>
<p>This question is confusing me, do I simply go about showing the relation is reflexive, symmetric, and transitive? I don't really see how to do this for this question. Any suggestions or hint... | Brian M. Scott | 12,042 | <p>HINT: Perhaps the easiest way to prove that this is an equivalence relation is to show that the connected components of a space $X$ partition $X$: they are pairwise disjoint, and their union is $X$. It will then follow immediately from the relationship between equivalence relations and partitions that $x\sim y$ is a... |
311,380 | <p>Prove that the relation $x \sim y$ iff $y$ is an element of the connected component of $x$ is an equivalence relation.</p>
<p>This question is confusing me, do I simply go about showing the relation is reflexive, symmetric, and transitive? I don't really see how to do this for this question. Any suggestions or hint... | Henno Brandsma | 4,280 | <p>So define $x \sim y$ iff $y$ is in the connected component of $x$, where the connected component of $x$ is the largest connected subset of $X$ that contains $x$.</p>
<p>To see this is well-defined, we can define for fixed $x$ the collection $\mathcal{C}_x = \left\{C \subset X \mid C \mbox { connected and } x \in X ... |
311,380 | <p>Prove that the relation $x \sim y$ iff $y$ is an element of the connected component of $x$ is an equivalence relation.</p>
<p>This question is confusing me, do I simply go about showing the relation is reflexive, symmetric, and transitive? I don't really see how to do this for this question. Any suggestions or hint... | CopyPasteIt | 432,081 | <p>When I look at this question, I get confused. Nothing like putting the 'horse before the cart'! But if you wipe from your mind the concept of connected components, something more interesting can be examined. So, yes, for educational reasons, I think it is justified to answer a modified question.</p>
<hr>
<p>Defini... |
21,238 | <p>Would someone be able to point me to a good resource explaining step by step the process for solving inhomogenous recurrence relations? (ie something of the form $ a_n = \sum{{b_i}{a_{n-i}}} + f(n)$ )</p>
| Karl Schwede | 3,521 | <p>Although there is a good reason that $(x,y)^2$ has a smooth blow-up. It is a power of an ideal which itself has a smooth blowup. See for example Hartshorne, Algebraic Geometry, Chapter II, Section 7, Exercise 7.11. </p>
<p>I suspect that, on smooth surfaces, one can probably say more, via "Zariski-factorization"... |
21,238 | <p>Would someone be able to point me to a good resource explaining step by step the process for solving inhomogenous recurrence relations? (ie something of the form $ a_n = \sum{{b_i}{a_{n-i}}} + f(n)$ )</p>
| Hailong Dao | 2,083 | <p>Craig Huneke told me about this <a href="https://www.tandfonline.com/doi/abs/10.1080/00927879708825958" rel="nofollow noreferrer">paper</a>: "On the smoothness of blow-ups" (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1446135" rel="nofollow noreferrer">MR1446135</a>, <a href="https://zbmath.o... |
2,945,913 | <p>I have a quick question about simplifying these exponents and then comparing them:</p>
<p><span class="math-container">$8^{\log_2 n}, 2^{3log_2(log_2n)}$</span> and <span class="math-container">$2^{(log_2(n))^2} $</span></p>
<p>I know the third one evaluates to <span class="math-container">$n^{log_2(n)}$</span>, b... | Archis Welankar | 275,884 | <p>Use <span class="math-container">$$\int _a^b f(a+b-x)dx=\int_a^b f(x)dx$$</span> and see the magic happen. The answer is <span class="math-container">$\frac{\pi}{4
}$</span> actually it's true for any non-negative number
.</p>
|
2,945,913 | <p>I have a quick question about simplifying these exponents and then comparing them:</p>
<p><span class="math-container">$8^{\log_2 n}, 2^{3log_2(log_2n)}$</span> and <span class="math-container">$2^{(log_2(n))^2} $</span></p>
<p>I know the third one evaluates to <span class="math-container">$n^{log_2(n)}$</span>, b... | Migos | 591,409 | <p>I would like to remark that your substitution <span class="math-container">$x = \arctan{u}$</span> works too; as follows:</p>
<p>Let <span class="math-container">$u \mapsto {u}^{-1}$</span> then
<span class="math-container">$\displaystyle I(n)=\int_0^\infty\frac{1}{1+{u}^{-n}}\frac{1}{1+u^{-2}} \cdot \frac{1}{u^2}... |
2,604,844 | <p>I have the folowing induction :</p>
<p>"Every graph with n vertices and zero edges is connected"</p>
<ul>
<li><p>Base:</p>
<p>For $n=1$ graph with one vertice is connected, hence a graph with $1$ vertex and zero edge.</p></li>
<li><p>Assumpution:</p>
<p>Every graph with $n-1$ vertices and zero edges is connected... | Tal Singrist | 521,391 | <p>The flaw is when you use this sentence :
<em>For every graph with n vertices and zero edges lets remove one vertice hence we get a graph with n−1 vertices and zero edges, by the assumpution the graph is connected, therefore the original graph is connected</em></p>
<p>for n = 2.
It doesn't work : you get 2 "1 vertic... |
4,194,511 | <p><span class="math-container">$$ \frac {d^2y}{dx^2} + 5\frac{dy}{dx} = 15x^2 $$</span></p>
<p>It's solution is</p>
<p><span class="math-container">$$y = y_{h} + y_{p}$$</span></p>
<p>where <span class="math-container">$y_{h} $</span> is the solution for homogenous equation and <span class="math-container">$y_{p} $</s... | José Carlos Santos | 446,262 | <p>If, while doing your computations, you only del with equivalences, there is nothing to check. An example would be:<span class="math-container">$$(e^z)^2=1\iff e^{2z}=1\iff 2z=2k\pi i\ (k\in\Bbb Z)\iff z=k\pi i\ (k\in\Bbb Z).$$</span>
But if you also have implications which are not equivalences, then, yes, you have t... |
791,719 | <p>I have this inequation:
$$5-3|x-6|\leq 3x -7$$</p>
<p>i solved this this way: </p>
<p>i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - : </p>
<p>1) for $x\geq6$ (positive): </p>
<p>$5-3x+6\leq 3x -7\\
6x\geq30\\
x\geq5$</p>
<p>2) for $x<6$ (negative): </... | Community | -1 | <p>We have for $x>0$:
$$\arctan x+\arctan\left(\frac1x\right)=\frac\pi2$$
and to prove it: let
$$f(x)=\arctan x+\arctan\left(\frac1x\right)$$
and by differentiate it we find $f'(x)=0$ so $f(x)=f(1)=\frac\pi2,\quad\forall x>0.$ </p>
|
362 | <p>What do you guys think of <a href="https://math.stackexchange.com/questions/1187/whats-the-most-effective-ways-of-teaching-kids-times-tables">maths education questions</a>? I wouldn't want the site to be overwhelmed, but they are related to maths. By its nature, many education questions will be subjective and diffic... | Larry Wang | 73 | <p>I see little difference in principle between teaching questions and the multitude of "whats an intuitive definition/explanation/motivation for x" questions. They will draw the same kind of answers.<br>
And I definitely think that questions of the latter form have gotten some very good answers here. I'd like to think... |
1,242,570 | <p>I want to use the standard definition $x_n \rightarrow x$ if for all $\epsilon>0$ there exists $N$ such that if $n>N$ then $|x_N-x|<\epsilon$. </p>
<p>So my claim is $x_n\rightarrow 0$ If I set $N=\epsilon^2,$ then the following expression $|\sqrt{n^2+1}-n-0|<\epsilon$ will hold true. I solved for $N$... | agha | 118,032 | <p>$$\sqrt{n^2+1}-n=\frac{(\sqrt{n^2+1}-n)(\sqrt{n^2+1}+n)}{\sqrt{n^2+1}+n}=\frac{1}{\sqrt{n^2+1}+n}<\frac{1}{n}$$</p>
<p>So you have $|x_n|<\varepsilon$ when $\frac{1}{n}<\varepsilon$.</p>
|
2,717,007 | <p>the curves are $x^2 = 4y$ and $x^2=4y-4$
these are just the same parabolas but the other one is shifted up by one unit.</p>
<p>I have been thinking of 3 possibilities that might be the answer.</p>
<ol>
<li><p>The area is equal to infinite sq. units</p></li>
<li><p>The area is equal zero</p></li>
<li><p>The area is... | DeepSea | 101,504 | <p>You really got a improper integral for the area between those two curves. $\text{Area} = \displaystyle \int_{-\infty}^{\infty}\left(\dfrac{x^2+4}{4}-\dfrac{x^2}{4}\right)dx=\displaystyle \lim_{a\to \infty}\displaystyle \int_{-a}^{a}1dx=+\infty$</p>
|
403,114 | <blockquote>
<p>Let <span class="math-container">$G$</span> be any group, and let <span class="math-container">$Z$</span> be its center.</p>
<p>(a) Show that <span class="math-container">$G/Z\cong \text{Inn}(G)$</span>.</p>
<p>(b) Conclude that <span class="math-container">$\text{Inn}(G)$</span> cannot be a nontrivial ... | Vinicius M. | 20,018 | <p>From $G/Z$ cyclic you can get something stronger: Let $a,b \in G$, then $a = z_1g^n$ and $b = z_2g^m$ for some $z_1,z_2 \in Z$. Now $ab = z_1g^n z_2g^m = z_2g^mz_1g^n = ba$. Thus $G$ is abelian, therefore $G = Z$. What now?</p>
|
172,841 | <blockquote>
<p>Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,
$$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$
when $X$ has : a) a discrete distribution, b) a continuous distribution.</p>
</blockquote>
<p>I assumed that for the case of a continuous distribution, since $F_X (t) = \m... | Did | 6,179 | <p>For <strong>every</strong> nonnegative random variable <span class="math-container">$X$</span>, whether discrete or continuous or a mix of these,
<span class="math-container">$$
X=\int_0^X\mathrm dt=\int_0^{+\infty}\mathbf 1_{X\gt t}\,\mathrm dt=\int_0^{+\infty}\mathbf 1_{X\geqslant t}\,\mathrm dt,
$$</span>
hence, ... |
172,841 | <blockquote>
<p>Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,
$$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$
when $X$ has : a) a discrete distribution, b) a continuous distribution.</p>
</blockquote>
<p>I assumed that for the case of a continuous distribution, since $F_X (t) = \m... | Henry | 6,460 | <p>Copied from <a href="https://stats.stackexchange.com/questions/18438/does-a-univariate-random-variables-mean-always-equal-the-integral-of-its-quanti">Cross Validated / stats.stackexchange</a>:</p>
<p><img src="https://i.stack.imgur.com/mSb6a.png" alt="enter image description here"></p>
<p>where $S(t)$ is the <a hr... |
172,841 | <blockquote>
<p>Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,
$$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$
when $X$ has : a) a discrete distribution, b) a continuous distribution.</p>
</blockquote>
<p>I assumed that for the case of a continuous distribution, since $F_X (t) = \m... | pre-kidney | 34,662 | <p>The function <span class="math-container">$x1[x>0]$</span> has derivative <span class="math-container">$1[x>0]$</span> everywhere except for <span class="math-container">$x=0$</span>, so by a measurable version of the <a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" rel="nofollow norefer... |
3,338,388 | <p>I tried to calculate the expression:
<span class="math-container">$$\lim_{n\to\infty}\prod_{k=1}^\infty \left(1-\frac{n}{\left(\frac{n+\sqrt{n^2+4}}{2}\right)^k+\frac{n+\sqrt{n^2+4}}{2}}\right)$$</span>
in Wolframalpha, but it does not interpret it correctly. </p>
<p>Could someone help me type it in and get the ans... | Somos | 438,089 | <p>The infinite product
<span class="math-container">$$ f(n) := \prod_{k=1}^\infty \left(1-\frac{n}{\left(\frac{n+\sqrt{n^2+4}}{2}\right)^k+\frac{n+\sqrt{n^2+4}}{2}}\right) \tag1$$</span>
surprisingly can be evaluated in closed form as
<span class="math-container">$$ f(n) = \frac1{2-n+\sqrt{n^2+4}}. \tag2$$</span>
The ... |
324,557 | <p>Map the common part of the disks $|z|<1$ and $|z-1|<1$ on the inside of the unit circle. Choose the mapping sot hat the two symmetries are preserved.</p>
<p>I don't really know how to approach this??</p>
<p>Any suggestions on how to start constructing such a linear transformation??</p>
<p>Thanks in advance!... | Martin Brandenburg | 1,650 | <p>$R$ is transitive iff $R \circ R \subseteq R$. But the converse inclusion is not true. For example, take $(\mathbb{N},<)$. Then $1 < 2$, but there is no $n \in \mathbb{N}$ with $1 < n < 2$.</p>
|
324,557 | <p>Map the common part of the disks $|z|<1$ and $|z-1|<1$ on the inside of the unit circle. Choose the mapping sot hat the two symmetries are preserved.</p>
<p>I don't really know how to approach this??</p>
<p>Any suggestions on how to start constructing such a linear transformation??</p>
<p>Thanks in advance!... | dtldarek | 26,306 | <p><strong>Hint:</strong> </p>
<ul>
<li><p>By transitivity you have $R \circ R \subseteq R$, but you do <strong>not</strong> have $R \subseteq R \circ R$.</p></li>
<li><p>Your relation cannot be <a href="http://en.wikipedia.org/wiki/Reflexive_relation" rel="nofollow">reflexive</a> since it would imply $R \subseteq R \... |
4,040,726 | <p><strong>UPDATED:</strong></p>
<p><em>(I originally asked for a proof that the expression was real; I meant to ask for the real algebraic expression that the sum equates to - i.e., an expression in terms of <span class="math-container">$x$</span>, <span class="math-container">$\sigma$</span> and <span class="math-con... | Kavi Rama Murthy | 142,385 | <p>The complex conjugate of the first term is exactly the second term. Hence their sum is real.</p>
<p><span class="math-container">$x^{z}=e^{z \ln x}$</span> and the conjugate of <span class="math-container">$e^{c}$</span> is <span class="math-container">$e^{\overline c}$</span> as seen using power series.</p>
|
4,040,726 | <p><strong>UPDATED:</strong></p>
<p><em>(I originally asked for a proof that the expression was real; I meant to ask for the real algebraic expression that the sum equates to - i.e., an expression in terms of <span class="math-container">$x$</span>, <span class="math-container">$\sigma$</span> and <span class="math-con... | GEdgar | 442 | <p>Your calculation is almost done.
<span class="math-container">$$
2\,{\frac {{{\rm e}^{\sigma\,\ln x }} \big( \sigma\cos
\left( t\ln x \right)+t \sin \left( t\ln
x \right) \big) }{{\sigma}^{2}+{t}^{2}}}
$$</span></p>
|
187,459 | <p>What are all 4-regular graphs such that every edge in the graph lies in a unique-4 cycle?</p>
<p>Among all such graphs, if we impose a further restriction that any two 4-cycles in the graph have at most one vertex in common, then can we characterize them in some way?</p>
<p>When is it possible to draw such a graph... | francois pirot | 118,814 | <p>As a matter of fact, I have encountered this family of 4-regular graphs, where every edges lies in exactly one C4, and no two C4 share more than one vertex.
You obtain any such graph with the following operation:
- begin with any 3-regular graph G of girth at least 5, together with a perfect matching M.
- Construc... |
332,772 | <p>I am looking for a reference or a proof of the following fact:</p>
<p>Let <span class="math-container">$X_{1}\subset X_{2}\subset\dots $</span> be a sequence of (hausdorff) topological spaces indexed by natural numbers such that each <span class="math-container">$X_{i}\subset X_{i+1}$</span> is a closed subspace ... | cellular | 141,114 | <p><a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf" rel="nofollow noreferrer">Here</a> is a reference, proposition 2.4.2 page 49. </p>
|
2,972,938 | <p>When is it possible to make a change of variables in the limit?</p>
<p>For example <span class="math-container">$\lim_{x \to \infty}(\ln x/x)$</span>, can I change <span class="math-container">$x=e^{y}$</span>?</p>
<p>Then <span class="math-container">$\lim_{x \to \infty}(\ln x/x)= \lim_{y \to \infty}(y/e^{y})$</s... | John Hughes | 114,036 | <p>Yes, there's another way. It's certainly not easier, but it works. I suspect it uses a number of ideas that you have not yet encountered as well, but that's how life is sometimes. Anyhow, here goes:</p>
<p>First, you can compute the characteristic polynomial, <span class="math-container">$c(x) = \det(A - xI)$</span... |
3,070,788 | <p>Can anyone explain to me why the variance of the standard normal distribution is 1? I am trying to understand the mechanism behind standardising random variable. While I know minus the variable by the mean is like shifting the graph to make it centre at the origin, I don't know why dividing it by SD makes the variab... | KM101 | 596,598 | <ul>
<li>In the second row, you have <span class="math-container">$K+K+K+K = 28$</span>, which means <span class="math-container">$4K = 28 \iff \color{blue}{K = 7}$</span>.</li>
<li>In the first column, you have <span class="math-container">$Q+K+K+K = 30$</span>, which means <span class="math-container">$Q+3K = 30$</sp... |
34,247 | <p>We will have an election soon to elect 2 new moderators. Continuing the tradition from past elections (<a href="https://math.meta.stackexchange.com/questions/17598/2014-nominations-for-moderator-on-math-se">2014</a>, <a href="https://math.meta.stackexchange.com/questions/27078/2017-moderator-election-nominating-anot... | jasmine | 557,708 | <p>I nominate <a href="https://math.stackexchange.com/users/104041/shaun">Shaun</a> .<span class="math-container">$5,231$</span> posts edited and <span class="math-container">$2,079$</span> helpful flags.He is very active in Maths.SE</p>
|
45,973 | <p>Let $B,C,D \geq 1$ be positive integers and $(b_n)_{n\geq 0}$ be a sequence with $b_0 = 1, b_n = B b_{n-1} + C B^n + D$ for $n \geq 1$.</p>
<p>Prove that </p>
<p>(a) $\sum_{n\geq 0}^\infty b_n t^n$ ist a rational function</p>
<p>(b) identify a formula for $b_n$</p>
<hr>
<p>Hi!</p>
<p>(a)</p>
<p>As I know I ne... | Robert Israel | 8,508 | <p>Hint: If $g(t) = \sum_{n=0}^\infty b_n t^n$, substitute in the recurrence for $b_n$
and express the result in terms of $g(t)$, $t$ and some geometric series.</p>
|
2,847,419 | <p>I know that <br/>
$\sigma , \delta$ be 2 function then <br/>
$1)$ $\sigma \circ \delta$ is onto or one-one if both $\sigma $ and $\delta$ is onto or one one.<br/>
I can prove this fact .
I wanted to find the counterexample for both cases if the converse is not true.
<br/> Any Help will be appreciated </p>
| g.kov | 122,782 | <p>Given
\begin{align}
f'(x)&=k(x+\exp(x))^{k-1} \times (1+\exp(x))
\end{align} </p>
<p>the function $f$ have no critical points for $k=1$
and one critical point for $k>1$ at</p>
<p>\begin{align}
x&=-\operatorname{W}(1)
\approx -0.567143290409784
.
\end{align}</p>
<p><strong>Edit</strong></p>
<p>We nee... |
1,510 | <p>It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? </p>
<p>Are most numbers transcendental?</p>
| engelbrekt | 3,304 | <p>The set of real numbers is uncountable, but the set of algebraic numbers is countable, so "most" real numbers are transcendental in a very strong sense of "most". This is actually a capsule description of Cantor's proof of the existence of transcendental numbers; just note that an uncountable set cannot be empty. Lo... |
1,510 | <p>It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? </p>
<p>Are most numbers transcendental?</p>
| Jason E | 860 | <p>The cool thing about transcendental numbers is that they set the set of real numbers, or more specifically uncountable sets, apart from every other set of numbers. The natural numbers for instance, you can get by counting, the integers by adding closure under subtraction, and the rationals by further adding closure ... |
1,510 | <p>It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? </p>
<p>Are most numbers transcendental?</p>
| Halfdan Faber | 1,320 | <p>Adding to the above, it is worth noting that the transcendental numbers that are commonly known/used in mathematics (e, pi, etc.) belong to the countable subset of transcendental numbers with a finite recursive function generator. The uncountable transcendental numbers, or the random numbers, are those which cannot ... |
1,510 | <p>It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? </p>
<p>Are most numbers transcendental?</p>
| Dan Piponi | 1,233 | <p>I would turn this on its head. It's not that transcendentals are special. It's that the algebraic numbers are special. It's very unusual (in a precisely defined sense) for a real number to be expressible as the solution to an algebraic equation with rational coefficients.</p>
<p>As a related aside, there is an inte... |
1,510 | <p>It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? </p>
<p>Are most numbers transcendental?</p>
| Abhishek Parab | 2,720 | <p>I would like to mention a highly readable <a href="http://www.mast.queensu.ca/~murty/Murty-MathStudent.pdf" rel="nofollow">paper</a> by Ram Murty on transcendental numbers. In a narrative fashion, he explains transcendental numbers from Euler via Lindemann, Hermite and Baker to his own work. He describes their esote... |
1,832,757 | <p>Consider the following premises.</p>
<ol>
<li>If A = B then B = C.</li>
<li>B != C.</li>
<li>If C > D then D < E.</li>
<li>F != G and A = B.</li>
<li>A = B or C > D.</li>
</ol>
<p>Alternatives:</p>
<p>a) F != G</p>
<p>b) F != G and D < E</p>
<p>c) A = B</p>
<p>d) B = C or D < E</p>
<p>e) D < E</p>... | Euler_Salter | 348,937 | <p>(1) and (2) do not imply that A != B. Indeed is an if, so that IF A = B, then we have that A = B = C. However in 2, it is not stated that A=B, we only know that B != C,which implies that A != B.</p>
<p>To prove the equality we see that:
from 4) F!=G and A=B, which implies 5) A=B or C > D.
Hence from 3) we get the... |
770,538 | <p>Let $\alpha$ be cylindrical helix with unit vector $u$, angle $\theta$, and arc length $s$ (measured from $0$). The only curve $\gamma$ such that $$\alpha(t)=\gamma(t)+s(t)\cos(\theta)u$$ is called the cross-section curve of the cylinder where $\alpha$ lies. </p>
<p><strong>$\bf (a)$</strong> How to show $\gamma(t)... | dinosaur | 75,334 | <p>You know that your tanget stars in $(-1|2)$ due to the fact that you consider unit circles and it passes through a point on the circle $P=(a|y)$, where $y=\sqrt{1-a^2}$.</p>
<p>Hence the tangent equation is $t(x)=\frac{y-2}{a+1}x + 2a+y$. You want $t(a)=\sqrt{1-a^2}$, which leads to the equation</p>
<p>$$t(a)=\fra... |
1,450,176 | <p>I would like to evaluate this limit :$$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$$.</p>
<p>I used taylor expansion at $y=0$ , where $x$ go to $\infty$ i accrossed this </p>
<p>problem : ${1}^{-\infty }$ then i can't judge if this limit equal's $1$ , </p>
<p>because it is indeterminate case ,T... | Patrick Stevens | 259,262 | <p>If you don't want to use Taylor expansion, the following will work.</p>
<p>Firstly, we can get rid of the $1$ in the power, because $$\lim_{x \to \infty} x \sin \left(\frac{1}{x} \right) = 1$$</p>
<p>So we actually want $$\lim_{x \to \infty} \left(x \sin \frac{1}{x} \right)^{-x}$$</p>
<p>By continuity, this is $$... |
4,620,310 | <p>Prove <span class="math-container">$\lim_{x\to\infty}\sin x/x=0$</span> using definition of a limit of a function.</p>
<p>I know that the definition of a limit of a fuction <span class="math-container">$f(x)$</span> (when <span class="math-container">$x\to\infty$</span> and <span class="math-container">$f(x)$</span... | Célio Augusto | 600,009 | <p>First, note that what you must prove is that, for every <span class="math-container">$\varepsilon>0$</span> there is a <span class="math-container">$M$</span> such that, for every <span class="math-container">$x>M$</span>, <span class="math-container">$$\left|\frac{\sin x}{x}\right|< \varepsilon,$$</span><... |
3,129,248 | <p>I am solving ordinary differential equation in <span class="math-container">$S'$</span> (dual to Schwartz space) given as:</p>
<p><span class="math-container">$y' + ay = \delta$</span>, where <span class="math-container">$\delta$</span> is a Dirac delta function.</p>
<p>The general solution of homogenous equation ... | Botond | 281,471 | <p>The solution to
<span class="math-container">$$y'+ay=0$$</span>
is in the following form:
<span class="math-container">$$y(x)=C\exp(-ax)$$</span>
So we could solve it for both <span class="math-container">$x<0$</span> and <span class="math-container">$x>0$</span>:
<span class="math-container">$$y_{+}(x)=C_{+}\... |
2,526,695 | <p>I've got following sequence formula:
$ a_{n}=2a_{n-1}-a_{n-2}+2^{n}+4$</p>
<p>where $ a_{0}=a_{1}=0$</p>
<p>I know what to do when I deal with sequence in form like this:</p>
<p>$ a_{n}=2a_{n-1}-a_{n-2}$
- when there's no other terms but previous terms of the sequence.
Can You tell me how to deal with this typ... | DanielV | 97,045 | <p>$$\begin{align}
a_{n+2} - 2a_{n+1} + a_{n} &= 4 \cdot 2^{n} + 4 \\
a_{n+1} - 2a_{n} + a_{n-1} &= 2 \cdot 2^{n} + 4 \\
a_{n} - 2a_{n-1} + a_{n-2} &= 1 \cdot 2^{n} + 4 \\
\end{align}$$</p>
<p>$$
\begin{bmatrix}
1 & -2 & 1 & 0 & 0 \\
0 & 1 & -2 & 1 & 0 \\
0 & 0 & 1 &... |
2,967,615 | <p>Given for example 2 functions,<span class="math-container">$\ n^{100} $</span> and<span class="math-container">$\ 2^n$</span>. I know that<span class="math-container">$\ 2^n$</span> grows faster and that therefore there is some<span class="math-container">$\ n$</span> where it will eventually overtake <span class="m... | ncmathsadist | 4,154 | <p>Let <span class="math-container">$\lambda > 0$</span> and consider the limit
<span class="math-container">$$\lim_{x\to\infty} {x^n\over e^{\lambda x}}.$$</span>
This limit can be reduced using L'hospital's rule to
<span class="math-container">$$ \lim_{x\to\infty} {nx^{n-1}\over \lambda e^{\lambda x}}.$$</span>
R... |
1,836,325 | <p>I am trying to check whether </p>
<p>$f: \mathbb{R} \to \mathbb{R}^\omega$ $f(t) = (t, 2t, 3t, \ldots)$ is continuous or not in the product and box topology. </p>
<p>But I have a feeling I don't have the necessary correct concepts to do these proofs. </p>
<p>I know the product topology:</p>
<ol>
<li><p>Is the co... | Brian M. Scott | 12,042 | <p>Consider the subbasic open set </p>
<p>$$S_{V,n}=\prod_{k\in\Bbb Z^+}U_k\;,$$</p>
<p>where $U_n=V$ is an open set in $\Bbb R$, and $U_k=\Bbb R$ for $k\ne n$. It’s not at all hard to calculate $f^{-1}[S_{V,n}]$: $f(t)\in S_{V,n}$ if and only if $nt\in V$. If we let $g:\Bbb R\to\Bbb R$ be the function $g(t)=nt$, we ... |
2,681,621 | <p>I'm trying to calculate the following limit:</p>
<p>$$\lim_{x\to\pi} \dfrac{1}{x-\pi}\left(\sqrt{\dfrac{4\cos²x}{2+\cos x}}-2\right)$$</p>
<p>I thought of calculating this:</p>
<p>$$\lim_{t\to0} \dfrac{1}{t}\left(\sqrt{\dfrac{4\cos²(t+\pi)}{2+\cos(t+\pi)}}-2\right)$$</p>
<p>Which is the same as:</p>
<p>$$\lim_{... | user284331 | 284,331 | <p>Let $f(x)=\sqrt{\dfrac{4\cos^{2}x}{2+\cos x}}$, the limit $L$ is actually the derivative of $f$ at $x=\pi$, so $$L=\dfrac{1}{2}\left(\dfrac{4\cos^{2}x}{2+\cos x}\right)^{-1/2}\dfrac{(2+\cos x)(-8\sin x)-4(\cos^{2}x)(-\sin x)}{(2+\cos x)^{2}}\bigg|_{x=\pi}=\cdots$$.</p>
|
1,918,071 | <p>Sometimes you will see theorems of the form "Let $H_1, \dots, H_n$. If $A$, then $B$". Sometimes "suppose" or "if" is used instead of "let". Here's an example:</p>
<ol>
<li><p>Let $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.</p></li>
<li><p>Suppose $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.</p></li>
<li><p>If $... | ervx | 325,617 | <p>Let $y=e^{x}$. Then, your equation becomes</p>
<p>$$
y^{2}+y-2=0\implies (y+2)(y-1)=0.
$$</p>
<p>Thus, $y=-2$ or $y=1$. Hence, $e^{x}=-2$ or $e^{x}=1$. Since the first is impossible, it follows that $e^{x}=1\implies x=0$. </p>
|
1,918,071 | <p>Sometimes you will see theorems of the form "Let $H_1, \dots, H_n$. If $A$, then $B$". Sometimes "suppose" or "if" is used instead of "let". Here's an example:</p>
<ol>
<li><p>Let $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.</p></li>
<li><p>Suppose $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.</p></li>
<li><p>If $... | Deepak | 151,732 | <p>From this step:</p>
<p>$e^x(e^x + 1) = 2$,</p>
<p>your solution makes no sense.</p>
<p>Observe that putting $y = e^x$ allows you to transform the original equation to the quadratic $y^2 + y -2 = 0$. This has roots $1$ and $-2$. Only the positive root allows a real solution for $x$, i.e. $e^x = 1$ giving $x=0$ as ... |
2,878,206 | <p>Let $a_n = \frac{9^n}{n + 5^n}$.</p>
<p>At large $n$ value, $a_n$ is expected to behave like $\frac{9^n}{5^n}$, therefore it diverges.</p>
<p>Using the direct comparison test, how can I find $b_n$ (has to be smaller than $a_n$ to prove that $a_n$ diverges)?</p>
| Fred | 380,717 | <p>We have $a_n \ge \frac{1}{n}$ for all $n$.</p>
|
2,885,918 | <p>We consider the following random variable $X$: We have a uniform distribution of the numbers of the unit interval $[0,1]$. After a number $x$ from $[0,1]$ is chosen, numbers from $[0,1]$ are chosen until a number $y$ with $x\leq y$ pops up.</p>
<p>The random variable $X$ counts the number of trials to obtain $y$. H... | Mike Earnest | 177,399 | <p>The event $X=n$ occurs if and only if among the first $n$ draws of the uniform random variable, $U_1,U_2,\dots,U_n$, that $U_n$ is the largest and $U_1$ is the second largest. Assuming $n\ge 2$ (which is true if you want $P(X=n)>0$), then the probability of this is easily seen to be $$\frac{1}n\times \frac{1}{n-1... |
1,448,427 | <p>Consider a lot consisting of 3 blue balls and 1 red ball.</p>
<p>Suppose I pick 2 balls one after another without replacement, now the probability of 2 balls being blue:</p>
<p>$$\frac{3\choose 2}{4 \choose 2}$$</p>
<p>Now taking different approach using conditional probability, the solution is also:</p>
<p>$$\f... | Harish Chandra Rajpoot | 210,295 | <p>HINT: </p>
<p>Given $\vec {BC}=-\frac{2}{5}\vec{AB}$ represents that the vectors $\vec {BC}$ & $\vec {AB}$ should be anti-parallel to each other (i.e. parallel to each other & in opposite directions) but the diagram shows them nonparallel/intersecting so the depiction is not correct. </p>
|
635,195 | <p>I'm trying to calculate the following limit: </p>
<p>$$\mathop {\lim }\limits_{x \to {0^ + }} {\left( {\frac{{\sin x}}{x}} \right)^{\frac{1}{x}}}$$</p>
<p>What I did is writing it as: </p>
<p>$${e^{\frac{1}{x}\ln \left( {\frac{{\sin x}}{x}} \right)}}$$</p>
<p>Therefore, we need to calculate: </p>
<p>$$\matho... | Bill Dubuque | 242 | <p>Here, as often, Taylor series are much more efficient than L'Hopital's rule.</p>
<p>$$\begin{eqnarray}\dfrac{\sin(x)}x \ &=&\ \color{#c00}1\color{#0a0}{\,-\,\dfrac{x^2}6+O(x^4)}\!\!\!\!\!\!\!\\&=&\ \color{#c00}1\, +\,\color{#0a0}z\\
\Rightarrow\ \ x^{-1}\log\left(\dfrac{\sin(x)}x\right) &=&\... |
237,838 | <p>The data are for the model $T(t) = T_{s} - (T_{s}-T_{0})e^{-\alpha t}$,
where $T_0$ is the temperature measured at time 0, and $T_{s}$ is the temperature at time $t=\infty$, or the environment temperature. $T_{s}$ and $\alpha$ are parameters to be determined.</p>
<p>How can I fit my data against this model? I'm try... | JJacquelin | 108,514 | <p>This is a problem of non-linear regression. Usually one solve it thanks to some iterative computation process starting with guessed values of the parameters. The Levenberg–Marquardt algorithm is commonly used.</p>
<p>A non-conventional approach (not iterative, no initial guess) is described in the paper : <a href="... |
1,505,336 | <p>Please can someone explain if this identity is correct:</p>
<p>|a| = $\sqrt{a^2} \ $ <p>
I thought it should be:<p> |a| = $(\sqrt{a})^2\ $</p>
<p>being that the former would produce an answer that is either positive or negative.</p>
<p>Thank you for your help.</p>
<p>PS: The full question was comparing say:</p>
... | Ethan Bolker | 72,858 | <p>You are confusing the standard meaning of the square root symbol with a correct understanding of squares and square roots. There are in fact two solutions to the equation $x^2 = 9$. namely $3$ and $-3$. But the accepted <em>convention</em> is that $\sqrt{9} = 3$, not $\pm 3$.</p>
|
1,326,626 | <p><a href="https://books.google.com/books?id=e-YlBQAAQBAJ&pg=PA144&lpg=PA144&dq=hungerford%20ring%20of%20fractions&source=bl&ots=ztyrAtNHmr&sig=l9Xc098vOOKXs0VP58_lzMkrdBc&hl=en&sa=X&ved=0CCQQ6AEwAWoVChMIi-Xn7fmRxgIVQdCACh3sigQm#v=onepage&q=hungerford%20ring%20of%20fractions&... | David Wheeler | 23,285 | <p>This is the basic idea: suppose we have an integral domain, $R$. We can build a (usually bigger) integral domain $F$, which is "nicer", because it's a field, by taking the field of fractions.</p>
<p>Now we can take the polynomial ring $R[x]$, and create a (usually bigger) ring $F[x]$. We can think of $R[x]$ "sittin... |
635,351 | <p>It is well known that if a series $\sum\limits_{k= 0}^\infty a_k$ converges, then $a_k \to 0$. </p>
<p>However, this is not true for integrals. What makes them different? Is it simply that they are "smoother?" Is there a rigorous way to explain this difference?</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... |
3,669,539 | <p>I’ve been looking at non-trivial solutions of ODEs. I found one and have problems with it.</p>
<p><span class="math-container">$y’(x)=\frac{1}{4}\sqrt{y(x)}$</span></p>
<p><span class="math-container">$y(0)=0$</span></p>
<p>I know the one of the solutions of this ODE is</p>
<p><span class="math-container">$y(x) ... | Community | -1 | <p>The equation is separable (and autonomous),</p>
<p><span class="math-container">$$\frac{y'}{2\sqrt y}=\frac18,$$</span></p>
<p>giving by definite integration</p>
<p><span class="math-container">$$\sqrt y-\sqrt 0=\frac{x-0}8$$</span> or <span class="math-container">$$y=\frac{x^2}{64}$$</span> which is the only sol... |
1,748,547 | <p>Show that if the closed interval $[a,b]$ is covered by finitely many open intervals $(a_1,b_1), ...,(a_n,b_n)$, then $$b-a \le \sum^n_{i=1}(b_i-a_i)$$. </p>
<p>I know that $(a_1,b_1), ...,(a_n,b_n)$ form an open covering of $[a,b]$, and my thought is to show the inequality by mathematical induction, but not sure ho... | copper.hat | 27,978 | <p>Here is a direct proof:</p>
<p>Since the $(a_k,b_k)$ form a cover of $[a,b]$ then for all $x \in [a,b]$
there is some $k$ such that $x \in (a_k,b_k)$.</p>
<p>Hence $1_{[a,b]}(x) \le \sum_k 1_{(a_k,b_k)}(x)$ for all $x$.</p>
<p>It is clear
that both sides are integrable, hence
$b-a = \int 1_{[a,b]}(x)dx \le \int \... |
1,106,325 | <p>In <a href="https://www.encyclopediaofmath.org/index.php/Codimension" rel="nofollow noreferrer">this article about codimension</a> there is the following remark: </p>
<p>The codimension of a subspace $L$ of a vector space $V$ is equal to the dimension of any complement of $L$ in $V$, since all complements have the... | Jesse Madnick | 640 | <p>(1) Let $X$ be a <em>finite-dimensional</em> vector space, $W \subset X$ a vector subspace.</p>
<p>A <em>complement</em> of $W$ in $X$ is any subspace $S \subset X$ such that
$$X = W \oplus S.$$</p>
<p>(2) Let $X$ be a <em>finite-dimensional</em> inner product space, $W \subset X$ a vector subspace.</p>
<p>The <e... |
4,286,983 | <p>Let <span class="math-container">$p,q$</span> be two odd primes. Prove that <span class="math-container">$p$</span> is a primitive root of <span class="math-container">$q$</span> if and only if <span class="math-container">$\frac{x^q-1}{x-1}$</span> is an irreducible polynomial on <span class="math-container">$\math... | user985321 | 985,321 | <p>Just rewrite the solutions above in a simplr way (<br />
This is immediate via considering the corresponding field extensions:<br />
Recall that the multiplicative group of any finite field <span class="math-container">$\mathbb{F}_{p^k}$</span> is cyclic (of order <span class="math-container">$p^k-1$</span>).<br />
... |
3,320,901 | <p>How to find the length of major and minor axis of ellipse given the length of two conjugate diameters and the angle between them?</p>
<p>I am aware about how to construct the ellipse using the above given facts(not by Rytz's Construction). I would like to know, independent of what method of construction one uses, h... | Parcly Taxel | 357,390 | <p>Suppose the conjugate <em>radii</em> are <span class="math-container">$a$</span> and <span class="math-container">$b$</span> and the angle between them is <span class="math-container">$\theta$</span>. Form the following matrix:
<span class="math-container">$$\mathbf A=\begin{bmatrix}a&b\cos\theta\\
0&b\sin\t... |
3,320,901 | <p>How to find the length of major and minor axis of ellipse given the length of two conjugate diameters and the angle between them?</p>
<p>I am aware about how to construct the ellipse using the above given facts(not by Rytz's Construction). I would like to know, independent of what method of construction one uses, h... | Intelligenti pauca | 255,730 | <p>Here's a geometric construction: if <span class="math-container">$MN$</span> and <span class="math-container">$DE$</span> are conjugate diameters, draw line <span class="math-container">$QQ'$</span> through <span class="math-container">$N$</span> perpendicular to <span class="math-container">$DE$</span> (see diagram... |
660,259 | <p>$f(y)=\begin{cases} \frac{b}{y^2}, & y\ge b,\\ 0, & \mbox{elsewhere}\end{cases}$.</p>
<p>is a bona fide probability density function for a random variable, $Y$. Assuming $b$ is a known
constant and $U$ has a uniform distribution on the interval $(0, 1)$, transform $U$ to obtain a random variable with the sa... | Did | 6,179 | <p>The target distribution is characterized by the fact that any random variable $X$ with this distribution is such that, for every $x\geqslant b$,
$$
P(X\geqslant x)=\int_x^\infty f=\int_x^\infty \frac{b}{y^2}\,\mathrm dy=\frac{b}x.
$$
On the other hand, if $Y=\dfrac{b}U$ with $U$ uniform on $(0,1)$, then for every $y... |
292,835 | <p>Does there exist a bijective function $f:{\mathbb R}\rightarrow{\mathbb R}$ that is nowhere-continuous, assuming that both domain and range have the "standard topology"? <sup>1</sup></p>
<p><sub><sup>1</sup> By this I mean the one generated by the open intervals $(a, b) \subset {\mathrm R}$. BTW, if this topo... | Alessandro | 59,474 | <p>This function should work.</p>
<p>$ f(x) = \begin{cases} x & \text{if x $\in \mathbf{Q}$ and $x \neq 0$} \\
-x & \text{if x $\in \mathbf{R-Q}$}\\
\sqrt{2} & \text{if $ x = 0 $} \end{cases}$</p>
<p>You can find more counterexaples of this type in "Counterexamples in analysis" by Gelbaum and Holmsted.</p... |
1,507,290 | <p>Kindly help me understand this statement made by my prof.</p>
<blockquote>
<p>The identity matrix I has the property that any non zero vector <span class="math-container">$V$</span> is an eigenvector of eigenvalue <span class="math-container">$1$</span>.</p>
</blockquote>
<p>My assumption of this statement is that t... | Yes | 155,328 | <p>Let $n \geq 1$; let $\Bbb{F}$ be a field; let $I_{n}$ be the $n \times n$ identity matrix over $\Bbb{F}$. Then there is some $\lambda \in \Bbb{F}$ such that
$$
\det (I_{n} - \lambda I_{n}) = (1-\lambda )^{n} = 0
$$
iff $\lambda = 1$;
hence $1$ is the only eigenvalue of $I_{n}$.
But $v \in \Bbb{F}^{n}$ and $v \neq 0$... |
1,507,290 | <p>Kindly help me understand this statement made by my prof.</p>
<blockquote>
<p>The identity matrix I has the property that any non zero vector <span class="math-container">$V$</span> is an eigenvector of eigenvalue <span class="math-container">$1$</span>.</p>
</blockquote>
<p>My assumption of this statement is that t... | c.sidney | 285,438 | <p>Will Jagy's demonstration is correct. The reason for the "non-zero condition"
is that for any matrix (linear function) :m, zero is a fixed point : m(o) = o, for any matrix:m.</p>
|
1,507,290 | <p>Kindly help me understand this statement made by my prof.</p>
<blockquote>
<p>The identity matrix I has the property that any non zero vector <span class="math-container">$V$</span> is an eigenvector of eigenvalue <span class="math-container">$1$</span>.</p>
</blockquote>
<p>My assumption of this statement is that t... | Ciro Santilli OurBigBook.com | 53,203 | <p><strong>Every vector is an eigenvector of the identity matrix</strong></p>
<p>First remember that even in the simple case where we have distinct eigenvalues and eigenvectors, then eigenvectors are not unique because if <span class="math-container">$v$</span> is an eigenvector, then so is every <span class="math-cont... |
1,610,700 | <blockquote>
<p>$$\int \frac{x-3}{\sqrt{1-x^2}} \mathrm dx$$</p>
</blockquote>
<p>I know that $\int \frac{1}{1-x^2}\mathrm dx=\arcsin(\frac{x}{1})$ but how can I continue from here? </p>
| 5xum | 112,884 | <p>Two hints:</p>
<ul>
<li>$$\frac{x-3}{\sqrt{1-x^2}} = \frac{x}{\sqrt{1-x^2}} - \frac{3}{\sqrt{1-x^2}}$$</li>
<li>You had it a bit wrong: $$\int\frac{1}{\sqrt{1-x^2}} = \arcsin(x)$$</li>
</ul>
|
864,920 | <p>The matrices $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ are well-known to (freely) generate a free group. Some years ago, I read a paper that, if memory serves, sort of summarised what was known about generalisations to pairs of the form $\begin{pmatrix}1&\... | Moishe Kohan | 84,907 | <p>For a slightly more recent reference see </p>
<p>M. Yu. Lyubich, V. V. Suvorov, Free subgroups of $SL_2(C)$ with two parabolic generators
Journal of Soviet Mathematics, 1988, Volume 41, Issue 2, p. 976-979. </p>
|
2,956,791 | <p>There is an equation here:
<span class="math-container">$$\sqrt{x+1}-x^2+1=0$$</span>
Now we want to write the equation <span class="math-container">$f(x)$</span> like <span class="math-container">$h(x)=g(x)$</span> in a way that we know how to draw h and g functions diagram.
Then we draw the h and g function diagra... | Michael Rozenberg | 190,319 | <p>By the Sonnhard's hint we obtain the domain: <span class="math-container">$\{-1\}\cup[1,+\infty)$</span>.</p>
<p><span class="math-container">$-1$</span> is a root and after squaring of the both sides we obtain:
<span class="math-container">$$1=(x+1)(x-1)^2$$</span> or
<span class="math-container">$$x(x^2-x-1)=0,$$... |
2,142,042 | <p>how would you use induction to prove this:</p>
<p>$\sin(x)-sin(3x)+sin(5x)-...+(-1)^{(n+1)}sin[(2n-1)x] = \frac{(-1)^{(n+1)}sin2nx}{2cosx} $</p>
<p>I know how you assume its true for n=k, and then prove for n=k+1, but I get to </p>
<p>Left Hand Side: $\frac{(-1)^{(k+1)}sin2kx}{2cosx}+(-1)^{k+2}sin[(2k+1)x]$ but I... | Taylor Ted | 225,132 | <p>This integral of yours is evaluated using Beta function. <a href="https://en.wikipedia.org/wiki/Beta_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Beta_function</a></p>
|
3,279,544 | <blockquote>
<p><span class="math-container">$50$</span> fighters are standing around in a circle. Every fighter may choose
to fight with the person on its left or its right with equal
probability. One person may be chosen by two different persons and
another may not be chosen at all. Can you expect how many fi... | Arthur | 15,500 | <p>Hint: Let <span class="math-container">$X_i$</span> be the random variable denoting whether person <span class="math-container">$i$</span> is left alone. It is <span class="math-container">$1$</span> if person <span class="math-container">$i$</span> wasn't chosen by anyone, and <span class="math-container">$0$</span... |
3,419,620 | <p>I'm trying to solve the equation
<span class="math-container">$$(z+i)^2=(\sqrt3+i)^3$$</span> but I don't know how to extract
the roots
<span class="math-container">$$(z+i)^2=(\sqrt3+i)^3 \rightarrow (z+i)^2=8i \rightarrow z^2+(2i)z-(8i+1)=0$$</span></p>
<p><span class="math-container">$z_{1,2}=-i \pm \sqrt{8i}$</s... | Quanto | 686,284 | <p>Recast your results,</p>
<p><span class="math-container">$$-i \pm \sqrt{8i}=-i \pm 2\sqrt{2i}=-i \pm 2\sqrt{(1+i)^2}=-i \pm 2(1+i)$$</span></p>
<p>which are <span class="math-container">$2+i$</span> and <span class="math-container">$-2-3i$</span>.</p>
|
351,850 | <blockquote>
<p>A subset $E$ contained in $\mathbb{R}^n$ is such that the function $x \mapsto \left\Vert x\right\Vert^2$ is uniformly continuous on $E$. For $r > 0$, let $E_r$ denote the union of all open balls of radius $r$ contained in $E$. Prove that $E_r$ is bounded for all $r > 0$. Find an example showing ... | Christian Blatter | 1,303 | <p>Given $r>0$ there is a positive $\delta<r$ such that $x$, $y\in E$ with $|y-x|<\delta$ implies $|y|^2-|x|^2<1$. Let $m$ be the center of an $r$-ball $B\subset E_r$. Then $m$ and $y:=m+\delta{m\over2|m|}$ are both in $E$; furthermore $|y-m|<\delta$. It follows that
$$|m|\delta+{\delta^2\over 4}=|y|^2... |
1,645,232 | <blockquote>
<p>Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}$.</p>
</blockquote>
<p>I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking:
\begin{align*}
ax + by &= 0\\
yb &= -ax\\
y &= -ax/b
\end{align*}</p... | YoTengoUnLCD | 193,752 | <p>The answers are equivalent. Your vector is $\vec v=\langle x, -ax/b\rangle$, which works for all $x\in\Bbb R$. </p>
<p>Factor out $x$: $\vec v=x\langle 1,-a/b\rangle$ and write $x=kb$.</p>
<p>Notice that your deductions work only if $b\neq 0$, you should treat that case separately.</p>
|
4,141,378 | <p>In this equation quasi-linear eq <span class="math-container">$\Big\{\exp(f(x,y))\dfrac{\partial f(x,y)}{\partial x} + \dfrac{y}{x} \dfrac{\partial f(x,y)}{\partial y} = 1\Big\}$</span> how <span class="math-container">$f$</span> changes based on <span class="math-container">$x$</span> and <span class="math-containe... | Community | -1 | <p>The set <span class="math-container">$\mathbb{R}^2$</span> is dust. It has no structure. When we draw it as a plane, we are giving it some sort of geometric structure. (This geometric structure is better described as a metric space than a topology, and even better described as a manifold, but we'll stick with topolo... |
1,700,689 | <p>Let $A, B$, and $C$ be sets. If $A\backslash B$ is a subset of $C$, then $A\backslash C$ is a subset of $B$. Is this a direct proof where I let $x$ be an element of $A$ and then work from there? I can't seem to figure out all of the cases. Thanks for help in advance.</p>
| BrianO | 277,043 | <p>This follows from a simple equivalence of propositional formulas:
$$
(p \land \neg q) \to r \equiv (p \land \neg r) \to q. \tag{1}
$$
Take $p := x\in A, q := x\in B, r := x\in C$.
In turn, (1) follows from
$$
(u \land v) \to w \equiv u \to (v \to w) \tag{2}
$$
and
$$
v \to w \equiv \neg v \to \neg w. \tag{3}
$$</p>... |
1,107,317 | <p>I've got this hypergeometric series</p>
<p>$_2F_1 \left[ \begin{array}{ll}
a &-n \\
-a-n+1 &
\end{array} ; 1\right]$</p>
<p>where $a,n>0$ and $a,n\in \mathbb{N}$</p>
<p>The problem is that $-a-n+1$ is negative in this case. So when I try to use Gauss's identity</p>
<p>$_2F_1 \left[ \begin{array}{ll}... | Marko Riedel | 44,883 | <p>We may try to keep it simple.
Suppose we seek to evaluate
$$\sum_{k=0}^n {a-1+k\choose k} {a-1+n-k\choose n-k}.$$
It is immediately apparent that this is a convolution of two ordinary generating functions.
To see what they are re-write the sum as
$$\sum_{k=0}^n {a-1+k\choose a-1} {a-1+n-k\choose a-1}.$$
By the Newto... |
1,273,441 | <p>In Contests in Higher Mathematics: Miklos Schweitzer Competitions,
1962-1991 by Gabor J Szekely, problem F.57 there is the study of
$f~:~[0,\infty)\to (0,\infty)$ such that: $\exists c>0, \forall x>0$,
$f'(x)=cf(x+1)$.</p>
<p>It states, without proof, that for this equation has a solution if and only if
$c\le... | Ward Beullens | 134,047 | <p>This is a partial solution:</p>
<p>You can see that if $c \leq 1/e$ the equation $\ln(a) = c*a$ has a real solution for $a$. This means that the function defined as $f(x) = a^x$ satisfies the functional equation because $f'(x) = \ln(a) * f(x) = c*a*f(x) = c*f(x+1)$. </p>
<p>Now we still have to prove that there ar... |
1,523,287 | <p>We choose a random number from 1 to 10. We ask someone to find what number it is by asking a yes or no question. Calculate the expected value if the person ask if it is $x$ number till you got it right?</p>
<p>I know the answer is around 5 but i can't find how to get there.</p>
<p>I tried $\frac{1}{10}(1)+\frac{1}... | Jantar88 | 285,411 | <p>In brief, "a better method" uses the Chinese Remainder Theorem with $5,7$ as the primes. It is not necessarily more simple...</p>
<p>You should find an integer $a \leftrightarrow (1,0)$ (i.e. $a \equiv 1 \bmod 5$, $a \equiv 0 \bmod 7$) and $b \leftrightarrow (0,1)$. Then your $n = 1\cdot a + 3\cdot b$, maybe also $... |
2,548,942 | <p>What would be the best approach to calculate the following limits </p>
<p>$$ \lim_{x \rightarrow 0} \left (1+\frac {1} {\arctan x} \right)^{\sin x}, \qquad \lim_{x \rightarrow 0} \frac {\tan ^7 x} {\ln (7x+1)} $$
in a basic way, using some special limits, without L'Hospital's rule? </p>
| Rene Schipperus | 149,912 | <p>Use the standard limits
$$\lim\limits_{x\to 0}\frac{\ln(1+x)}{x}=1$$
and
$$\lim\limits_{x\to 0}\frac{\sin x}{x}=1$$ </p>
<p>In the first example take a log first. </p>
<p>Note that the later implies that </p>
<p>$$\lim\limits_{x\to 0}\frac{\tan x}{x}=1$$ </p>
<p>and thus that
$$\lim\limits_{x\to 0}\frac{\arc... |
2,548,942 | <p>What would be the best approach to calculate the following limits </p>
<p>$$ \lim_{x \rightarrow 0} \left (1+\frac {1} {\arctan x} \right)^{\sin x}, \qquad \lim_{x \rightarrow 0} \frac {\tan ^7 x} {\ln (7x+1)} $$
in a basic way, using some special limits, without L'Hospital's rule? </p>
| user | 505,767 | <p>For the second:</p>
<p>$$\frac {\tan ^7 x} {\ln (7x+1)}=\frac {\tan ^7 x} {x^7}\ \frac {x^7} {7x} \ \frac {7 x} {\ln (7x+1)}=1\cdot0\cdot1=0$$</p>
|
2,292,015 | <p>The question is:
the first three terms of an arithmetic series $c_{n}$ are
$$a(1+b), a(1+3b),a(1+5b)$$
I needed to find the common difference in terms of $a$ and $b$ and then find the expression for $c_{n}$.</p>
<p>The final part I struggled with where I have to find $a$ and $b$ and the information given is
$$c_{5... | Chickenmancer | 385,781 | <p>To imagine such a curve you should think bout what is occurring in each coordinate. As $x$ ranges over $[0,5],$ the first or '$x$' coordinate is a version of cosine, which is growing in amplitude, since $x$ is an increasing function. Similarly, with the '$y$' coordinate, however here we have sine with increasing amp... |
2,448,349 | <p>Would the cubic equation that is the best approximation for $e^x$ just be the first 4 terms of its Taylor Series expansion?
$$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}$$
I guess it would depend on the bounds you are inspecting though? I'm not well-versed in dealing with infinities. Is there a cubic equation that is th... | Sangchul Lee | 9,340 | <p>How close a function is to another definitely depends on how you define that notion. The $n$-th Taylor polynomial $T_n$ of $f$ at $x_0$ is the best approximation of $f$ among all degree $n$ polynomials, precisely in the sense that this is a unique polynomial satisfying</p>
<p>$$f(x) = T_n(x) + \mathcal{O}((x-x_0)^{... |
3,754,030 | <p>Let <span class="math-container">$I^n$</span> be the <span class="math-container">$n$</span>-cube <span class="math-container">$[0,1]^n$</span>. Also define two subsets of <span class="math-container">$\partial I^n$</span>:</p>
<ul>
<li><span class="math-container">$A=\{(x_1,\ldots,x_n)\mid x_1=0\}$</span></li>
<li>... | Angina Seng | 436,618 | <p>Let's consider <span class="math-container">$A'=\{x_1=1\}$</span>, the top face, in place of <span class="math-container">$A$</span>
(we can then turn over the cube to swap <span class="math-container">$A$</span> and <span class="math-container">$A'$</span>). So I'll define
an auto-homeomorphism <span class="math-co... |
1,171,492 | <p>I was trying to obtain the square root of $-5-12i$ by the formula for square root (given below) and also by De Moivre's theorem and verify that both give the same result. But the two results are somehow not matching for this complex number. I am writing my solution below in two cases for each method:</p>
<p>Case - ... | user2661923 | 464,411 | <p>Toby's answer has a minor miscue. Also, in the first chapter of Bruce Palka's "An Introduction To Complex Function Theory", an easier algorithm is given.</p>
<p>Let <span class="math-container">$\;z = -5 - 12i,\;$</span> and let <span class="math-container">$\alpha = \text{Arg}(z).$</span> This means that <span c... |
134,205 | <blockquote>
<p>Find the expectation of a Geometric distribution using $\mathbb{E}(X)= \sum_{k=1}^\infty P(X \ge k)$. </p>
</blockquote>
<p>Okay I know how to find the expectation using the definition of the geometric distribution $$P(X=k)= p \cdot(1-p)^{k-1}$$ and I figured that $P(X \ge k)=(1-p)^{k-1}$ but I don't... | Community | -1 | <p>The answer that is here does not address one aspect of the question:</p>
<blockquote>
<p>I figured that $P(X \ge k)=(1-p)^{k-1}$ but I don't know how to show it.[...]</p>
</blockquote>
<p><strong>Here is an hint:</strong> </p>
<p>$$P(X \ge k)=\sum_{i=k}^\infty P(X=i)$$</p>
<p>Now, to evaluate the above sum, yo... |
3,364,404 | <blockquote>
<p>If <span class="math-container">$X\sim \operatorname{Binom}(n,p)$</span> and <span class="math-container">$Y\sim \operatorname{Ber}\left(\frac Xn\right)$</span>, then find <span class="math-container">$E[X\mid Y]$</span>.</p>
</blockquote>
<p>Is there a name for such a random variable <span class="ma... | Graham Kemp | 135,106 | <blockquote>
<blockquote>
<p>If <span class="math-container">$X\sim Binom(n,p)$</span> and <span class="math-container">$Y\sim Ber(X/n)$</span> then find <span class="math-container">$E[X|Y]$</span></p>
</blockquote>
<p>Is there a name for such a random variable <span class="math-container">$Y$</span>, where its distri... |
3,812,087 | <p>Let <span class="math-container">$(M,d)$</span> be a metric space. A set <span class="math-container">$A \subset M$</span> is said to be compact if every open cover of <span class="math-container">$A$</span> has a finite subcover.</p>
<p>Why do we use this definition, rather than the other "definition" whi... | Noah Schweber | 28,111 | <p>That definition <strong>is</strong> the more general definition - it holds in <span class="math-container">$\mathbb{R}^n$</span> and in things very different from <span class="math-container">$\mathbb{R}^n$</span>. (In fact, the more concrete definition isn't even appropriate to arbitrary metric spaces!)</p>
<p>Spec... |
1,201,002 | <p>I´m trying to find a vector $\vec{c} = $ , which is orthogonal to vector $\vec{a}$ and $\vec{b}$:</p>
<p>As far I understood, I have to show that:</p>
<p>$$\langle a,c\rangle=0 $$
$$\langle b,c\rangle=0 $$ </p>
<p>So if I would like to determine an orthogonal vector regarding: \begin{bmatrix}-1\\1\end{bmatrix}
... | Tyr Curtis | 209,743 | <p>Take an arbitrary vector $$c=\begin{bmatrix}x\\y\\z\end{bmatrix}$$ then write down the equations in terms of $xyz$:</p>
<p>$$\langle a,c\rangle = 0;\langle b,c\rangle = 0$$</p>
<p>The solutions to this system will all lie on one line and this gives you the perpendicular vector. </p>
<p><strong>Easy example:</str... |
650,395 | <p>I am given generating functions $f(x)= \frac{x}{1-x}$ or $f(x)=\frac{1}{1+x^{2}}$ or $f(x)=\frac{1}{x^2-5x+6}$ and I am obliged to write sequence which are generated by this functions. What is the fastest algorithm to solve these problems? I have problem with even starting. I will be glad if anyone would be so nice ... | Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> For ordinary generating function, you need to find the coefficient of $x^n$, so you need to find the power series expansion of these functions. See this <a href="https://math.stackexchange.com/questions/279868/causal-inverse-z-transform-of-fibonacci">technique</a>.</p>
|
650,395 | <p>I am given generating functions $f(x)= \frac{x}{1-x}$ or $f(x)=\frac{1}{1+x^{2}}$ or $f(x)=\frac{1}{x^2-5x+6}$ and I am obliged to write sequence which are generated by this functions. What is the fastest algorithm to solve these problems? I have problem with even starting. I will be glad if anyone would be so nice ... | Ethan Splaver | 50,290 | <p>In general given the ratio of two polynomials, apply partial fraction decomposition to the ratio so as to obtain a sum of terms that can be expanded as a binomial series, then collect the coefficients.</p>
|
650,395 | <p>I am given generating functions $f(x)= \frac{x}{1-x}$ or $f(x)=\frac{1}{1+x^{2}}$ or $f(x)=\frac{1}{x^2-5x+6}$ and I am obliged to write sequence which are generated by this functions. What is the fastest algorithm to solve these problems? I have problem with even starting. I will be glad if anyone would be so nice ... | preferred_anon | 27,150 | <p>By the fundamental theorem of algebra, any real-valued polynomial can be factorised into real quadratics. Sometimes these quadratics can be factored further into linear binomials, and if not they are writeableas a sum of two squares. </p>
<p>Given a ratio of polynomials, $\frac{p(x)}{q(x)}$ (where we take the deg... |
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