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 |
|---|---|---|---|---|
1,577,090 | <p>How can I determine if the function is one-to-one .. I know that any odd function is 1-1 and any even function is NOT 1-1 but what about functions that are neither of those, like $x^3+5$ or $x^3+x^2+3$. How can I determine whether it is a one-to-one? </p>
| SchrodingersCat | 278,967 | <p>There are $3$ distinct ways that I can remember as of now:</p>
<ol>
<li>If the function is $y=f(x)$, then check the value of $f'(x)$. If $f'(x)\ge 0$ $\forall$ $x \in \{\mathbb{R}\}$, or $f'(x) \le 0$ $\forall$ $x \in \{\mathbb{R}\}$ , then $f(x)$ is one-one.</li>
<li>You can draw the graph of the function and perf... |
2,649,907 | <p>As the titles states I have to determine weather or not polynomials of the form $a_{0}+a_{1}x$ is a subspace of $P_{3}$, the polynomials of the form $a_{0}+a_{1}x$ have $a_{0}$ and $a_{1}$ as real numbers.</p>
<p>So since the polynomial is of the third degree, the entire polynomial would look like:</p>
<p>$a_{0}+a... | Matt Samuel | 187,867 | <p>To test closure under addition, you want to already start with polynomials in the alleged subspace. For example,
$$(a_0+a_1x)+(b_0+b_1x)=(a_0+b_0)+(a_1+b_1)x$$
This is still in the same form (meaning it has no higher degree terms), so the subset is indeed closed under addition. </p>
|
1,310,460 | <p>Given a finite connected graph $G$, I can make a finite number of cuts on the edges to obtain a tree. What is the most efficient algorithm to perform this procedure? </p>
<p>Thanks,
Vladimir</p>
| TravisJ | 212,738 | <p>Hint: </p>
<p>2) is called sequential compactness (and there is a well-known theorem relating sequential compactness to compactness).</p>
<p>3) $\frac{1}{2}\sin(x)+\frac{1}{2}$ maps $\mathbb{R}$ to $[0, 1]$ and is continuous.</p>
<p><strong>Edit</strong></p>
<p>This is a hint assuming the question said: there is... |
2,019,049 | <p><a href="https://i.stack.imgur.com/SULGd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SULGd.png" alt="enter image description here"></a></p>
<blockquote>
<p>Let $\theta \in \mathbb{R}$. Prove: if $\prod\limits_{k=1}^{20} (\cos(k\theta) + i \sin(k\theta)) = i$, then there exists an integer $... | Chinny84 | 92,628 | <p>$$
\cos kx + i \sin k x = \mathrm{e}^{ikx}
$$
so we have
$$
\prod_{k=1}^{20}\mathrm{e}^{ikx} = \mathrm{e}^{i\sum_{n=1}^{20}nx} = i
$$
so we have
$$
\cos (x\sum_{n=1}^{20}n) + i\sin(x\sum_{n=1}^{20}n) = i
$$
or we have
$$\sin(x\sum_{n=1}^{20}n) = 1 \\ \cos (x\sum_{n=1}^{20}n) = 0$$
solve for $x$ using $\sum_{i=1}^n... |
2,518,523 | <p>So I've stumbled upon this problem:</p>
<blockquote>
<p>Compare the numbers:</p>
<p><span class="math-container">$$9^{8^{8^9}} \text{ and }\,8^{9^{9^8}}$$</span></p>
</blockquote>
<p>I got this in a test and I had no idea what the answer to the problem is... Can someone give me the answer and how you can find it? It... | prog_SAHIL | 307,383 | <p>On my conclusion, $$ 9^{8^{8^9}} > 8^{9^{9^8}} $$</p>
<p>To explain this,</p>
<p>Manually we know, $$ 9\ln(8) > 8\ln(9) $$</p>
<p>therefore, $$ 8^9 > 9^8 $$
$$ 8^9 \approx 3(9^8) $$</p>
<p>now $\ln(9)=2.19$ </p>
<p>$\ln(8)=2.07$ </p>
<p>i.e. $ \frac{\ln(9)}{\ln(8)}=1.05 $,</p>
<p>therefore $$ 8^9\ln(... |
4,346,765 | <p>While I was solving an integral using Feynman Integration, I came across the following differential equation:</p>
<p><span class="math-container">$$y’’’’-y’’+y=0$$</span></p>
<p>I tried substituting <span class="math-container">$y$</span> with an exponential function which failed. Can someone else show me how to sol... | Jyrki Lahtonen | 11,619 | <p>Supplementing reuns's excellent answer with the following.</p>
<p><strong>Result:</strong> If <span class="math-container">$n\equiv2\pmod4$</span> and <span class="math-container">$-1$</span> is not a square in the field <span class="math-container">$K=\Bbb{F}_q$</span>, then the space <span class="math-container">$... |
1,923,761 | <p>I have a problem in which I need to discover $f$ knowing that $$\left\{\begin{matrix}f(1,y)-f(0,y)=y\\f(x,1)-f(x,0)=x\end{matrix}\right.$$ Any hints to solve it?</p>
| Sarvesh Ravichandran Iyer | 316,409 | <p>We can't solve this equation with the amount of information given, but yes:</p>
<p>If we let $f(x,y) = xy$, then certainly $f(x,1)-f(x,0) = x-0=x$ and $f(1,y)-f(0,y) = y - 0 = y$, so this is one solution. </p>
|
1,442,147 | <p>Could I quickly spot the inverse of a permutation from its 2-cycle composition?</p>
<p>For example, given that $\rho=(1 \ 9)(1 \ 4)(1 \ 5)(1 \ 8)(2 \ 10)(2 \ 3)(2 \ 6)(2 \ 7)$, how to find its inverse from this 2-cycle decomposition?</p>
| quid | 85,306 | <p>Yes, it is easy to write down the inverse of a permutation in this form.
It is also not much harder to do this for any permutation given in cycle form. </p>
<p>There are two main points to note: </p>
<ol>
<li><p>If you have $c_1 \dots c_n $ then the inverse is given by $c_n^{-1} \dots c_1^{-1}$. Where $c_i^{-1}$ i... |
938,429 | <p>The question is to evaluate $\displaystyle7\int\frac{dx}{x^2+x\sqrt{x}}$. My solution is attached.</p>
<p><img src="https://i.stack.imgur.com/79dEd.png" alt="This is the question."></p>
<p><img src="https://i.stack.imgur.com/GuZ7z.jpg" alt="This is the solution that I tried."></p>
<p>The problem of my solution i... | Mosk | 175,514 | <p>Try the substitution: $\sqrt{x}=t$</p>
|
43,650 | <p>Consider the following problem:</p>
<blockquote>
<p>Let $f:{\mathbb R}^3 \to{\mathbb R}$ be $$f(x,y,z)=x+4z$$ where $$x^2+y^2+z^2\leq 2.$$ Find the minimum of $f$. </p>
</blockquote>
<p>This is similar to the question <a href="https://math.stackexchange.com/q/41385/9464">here</a>. However, since this is not an a... | Jack Schmidt | 583 | <p>This answer is just meant to compare the solutions obtained by Lagrange multipliers versus elimination. Notice in the first that <em>x</em> and <em>z</em> are treated equally until we needed to solve { <em>u</em> = 2, <em>x</em> = 4<i>z</i> }, but in the second <em>x</em> was heavily favored the entire time. In th... |
762,483 | <p>I understand that exponents don't distribute over addition and have seen plenty of examples i.e. $$ (x + y)^2\neq x^2 + y^2 $$ but I'm wondering why that is. Multiplication distributes over addition e.g. $3(2+3) = 3(2) + 3(3)$ so if an exponent is just repeated multiplication why shouldn't the same be true for expon... | ajotatxe | 132,456 | <p>Perhaps this image answers your question:</p>
<p><img src="https://i.stack.imgur.com/4C132.gif" alt="enter image description here"></p>
|
1,040,846 | <p>I want to prove, that $a_n$ is a null sequence if $$\lim_{n \to \infty}|\frac{a_{n+1}}{a_n}|= c < 1$$</p>
<p>That means that $\forall \epsilon > 0: \exists N \in \mathbb{N}: n \ge N: |\frac{a_{n+1}}{a_n} - c| < \epsilon$</p>
<p>How can I get rid of the $a_{n+1}$ and the $c$, to show $\forall \epsilon >... | Felice Iandoli | 87,201 | <p>Maybe you mean that $a_n$ is vanishing at $\infty$. Anyway if $\lim\frac{a_n}{a_{+1}}=c<1$ then you have that $\sum_{n=1}^{\infty}a_n$ converges. By force the sequence $a_n$ must be vanishing at $\infty$. (See Ceraro theorem)</p>
|
1,907,758 | <p>In <a href="https://arxiv.org/pdf/1411.5057.pdf" rel="nofollow">this paper</a>, (section 2, page 2)$\mathcal{l}_1$ norm is replaced with reweighted $\mathcal{l}_2$ in an optimization problem. I don't understand how $\lVert x\rVert_1$ is replaced with $x^TWx$ and ow the solution has changed to its weighted version. <... | Robert Israel | 8,508 | <p>It appears that they are taking $W$ to be a diagonal matrix with diagonal entries $W_{ii} = |x_i|^{-1}$ (more precisely, they're taking a sequential approach where the matrix $W$ in the $k$'th iteration has diagonal entries
$|x^{k-1}_i|^{-1}$, where $x^{k-1}$ is the result of the $k-1$'th iteration).
Assuming every... |
1,947,630 | <p>Describe the set of relations on $\mathbb{Z}$ which are both symmetric and anti-symmetric. Hint: this set is infinite and contains one relation with which you are already familiar.</p>
<p>I know that this is clearly talking about the equality relation, but what I am confused about is what does a set of relations me... | Nick R | 320,894 | <p>You are correct that $S$ is the relation denoting equality. However note that any subset of $S$ is also a relation that is symmetric and antisymmetric. The idea here is that we do not need to define all the integers to be equal to each other.</p>
<p>Thus the answer you are looking for is the set of all subsets of $... |
1,947,630 | <p>Describe the set of relations on $\mathbb{Z}$ which are both symmetric and anti-symmetric. Hint: this set is infinite and contains one relation with which you are already familiar.</p>
<p>I know that this is clearly talking about the equality relation, but what I am confused about is what does a set of relations me... | Graham Kemp | 135,106 | <p>The identity/equality relation <em>is</em> both anti-symmetric and symmetric. However it is not the only such. The empty relation <em>is also</em> both anti-symmetric and symmetric. There are many others.</p>
<p>You wish to describe <em>all</em> relations (on the integers) which have the proper... |
291,676 | <p>I have a question about a problem I encountered:</p>
<p>$\exists$ a,b $\epsilon$ $\mathbb{R}$+ such that $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$</p>
<p>Any tips for going about solving this?</p>
<p>I tried:</p>
<p>$\sqrt{a+b}=\sqrt{a}+\sqrt{b}$</p>
<p>$a+b=a+b$</p>
<p>I have a feeling this isn't a legal operation...</p... | A Ricko Maulidar | 60,472 | <p>i think you just need to find the value of $a$ and $b$ so that the conditions are met
but we know that only $a=0$ or $b=0$ such that conditions met
is $0\in\mathbb{R^+}$? if yes then the statement is true</p>
|
142,879 | <p>I'm working through Dr. Pete Clark's convergence notes here: <a href="http://alpha.math.uga.edu/%7Epete/convergence.pdf" rel="nofollow noreferrer">http://alpha.math.uga.edu/~pete/convergence.pdf</a></p>
<p>and I've been thinking about Exercise 3.2.2 (b),</p>
<p>The question states that for the set <span class="math-... | Arturo Magidin | 742 | <p>You are misinterpreting 3.2.2(b). </p>
<p>In fact, $\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{n}$ does <em>not</em> converge absolutely, as you note; hence this series <em>also</em> does not converge unconditionally.</p>
<p>Indeed, for the series to converge unconditionally to $a$, it would have to be the case that ... |
1,665,931 | <p>Let $a,c \in \mathbb{R}$ and $b \in \mathbb{C}$. We consider the equation $$a \bar{z}z+ b\bar{z} + \bar{b}z+c=0.$$ What curve it represents in the complex plane?</p>
<p>I think it a circle, but I am not able to conclude. Any help?</p>
| Whiz_Geek | 230,983 | <p>It represents a conic where a=b and h=0, ie, a circle. It may also represent a pair of straight lines.</p>
|
176,055 | <p>I heard teachers say [cosh x] instead of saying "hyperbolic cosine of x".</p>
<p>I also heard [sinch x] for "hyperboic sine of x". Is this correct?</p>
<p>How would you pronounce tanh x? Instead of saying "hyperbolic tangent of x"?</p>
<p>Thank you very much in advance.</p>
| marty cohen | 13,079 | <p>I usually say "sine-h", "cos-h", and "tan-h" with the "h" pronounced "aich" like the letter.</p>
<p>Sometimes I pronounce "cosh" as a word with a long "o".</p>
<p>I guess this qualifies as an answer, instead of just a comment.</p>
|
176,055 | <p>I heard teachers say [cosh x] instead of saying "hyperbolic cosine of x".</p>
<p>I also heard [sinch x] for "hyperboic sine of x". Is this correct?</p>
<p>How would you pronounce tanh x? Instead of saying "hyperbolic tangent of x"?</p>
<p>Thank you very much in advance.</p>
| Community | -1 | <p>I believe that in UK and the Commonwealth countries, the accepted pronunciations are /ʃaɪn/ (like "shine"), /kɒʃ/ ("cosh") and /θæn/ (like "thank" without the k). American usage may differ.</p>
|
2,906,282 | <p>I want calculate the reflection from a straight <strong>R</strong> on the surface from the straight <strong>BC</strong>: </p>
<p><a href="https://i.stack.imgur.com/bOINy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bOINy.png" alt="enter image description here"></a></p>
<p>The triangle with th... | Jose Brox | 146,587 | <p>That $A$ is invertible means precisely that there is another matrix $A^{-1}$ such that
$$AA^{-1}=I=A^{-1}A.$$</p>
<p>It is easy to show that the inverse, if it exists, must be unique.</p>
<p>Now suppose $A,B$ to be invertible, and denote $C:=AB$. $C$ will be invertible if we can find $C^{-1}$ such that $CC^{-1}=I=... |
3,224,765 | <p>The following question was asked on a high school test, where the students were given a few minutes per question, at most:</p>
<blockquote>
<p>Given that,
<span class="math-container">$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$</span>
and,
<span class="math-container">$$Q(x)=x^4+x^3+x^2+x+1$$</span>
what is t... | DanLewis3264 | 480,329 | <p>While it may be a standard technique, as Bill's response details, I wouldn't say it's at all obvious at High School level. As a pre-Olympiad challenge problem, however, it's a good one. </p>
<p>My intuition is via cyclotomic polynomials -- <span class="math-container">$Q(x) = \Phi_5(x)$</span>, giving the idea to m... |
2,847,301 | <p>Find elements $a,b,$ and $c$ in the ring $\mathbb{Z}×\mathbb{Z}×\mathbb{Z}$ such that $ab, ac,$ and $bc$ are zero divisors but $abc$ is not a zero divisor.</p>
<p>Work:</p>
<ul>
<li><p>$a=(1,1,0)$</p></li>
<li><p>$b=(1,0,1)$</p></li>
<li><p>$c=(0,1,1)$</p></li>
</ul>
<p>Why this works: because $ab=(1,0,0)\neq(0,0... | Benjamin Dickman | 37,122 | <p>The $a, b,$ and $c$ that you suggest seem to work, although your reasoning is somewhat missing. Depending on where you are in your studies, it appears that you want to prove your choice of these three elements in $R = \mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}$ yield zero divisors $ab$, $ac$, and $bc$, but for which... |
785,441 | <p>Prepping for a comprehensive test in August and I am working on a problem from Royden 4th ed. (Chapter 2, #15): Show that if $E$ has finite measure and $\varepsilon>0$, then $E$ is the disjoint union of a finite number of measurable sets, each of which has measure at most $\varepsilon$.</p>
<p>Here is what I hav... | Laars Helenius | 112,790 | <p>I figured it out:</p>
<p>Since $E\subseteq\bigcup_{k=1}^\infty I_k$, then
$$
\begin{align*}
E&=E\cap\bigcup_{k=1}^\infty I_k\\
&=E\cap\left(\bigcup_{k=1}^N I_k\cup\bigcup_{k=N+1}^\infty I_k\right)\\
&=\left(E\cap\bigcup_{k=1}^N I_k\right)\cup \left(E\cap\bigcup_{k=N+1}^\infty I_k\right)\\
&=\left(E... |
990,418 | <p>What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. </p>
<p>I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin B+\sin C=\frac{3\sqrt3}2$. And also <a href="http://www.wolframalpha.com/input/?i=max+sin(a)%2Bsin(b)%2Bsin(c)... | Kim Jong Un | 136,641 | <p>For $x\in[0,\pi]$, the function $f(x)=\sin(x)$ is concave, so by Jensen's inequality, we have
$$
\frac{1}{3}f(A)+\frac{1}{3}f(B)+\frac{1}{3}f(C)\leq f\left[\frac{1}{3}(A+B+C)\right]=\sin(\pi/3)=\frac{\sqrt{3}}{2}.
$$
Equality is achieved when $A=B=C=\pi/3$.</p>
|
990,418 | <p>What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. </p>
<p>I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin B+\sin C=\frac{3\sqrt3}2$. And also <a href="http://www.wolframalpha.com/input/?i=max+sin(a)%2Bsin(b)%2Bsin(c)... | Community | -1 | <p>$$f(x,y,z)=\sin(x)+\sin(y)+\sin(z)$$
$$g(x,y,z)=x+y+z-\pi=0$$</p>
<p>$$\large\frac{\frac{\partial f}{\partial x}}{\frac{\partial g}{\partial x}}=
\frac{\frac{\partial f}{\partial y}}{\frac{\partial g}{\partial y}}=
\frac{\frac{\partial f}{\partial z}}{\frac{\partial g}{\partial z}}=k$$</p>
<p>$$\cos(x)=\cos(y)=\co... |
10,674 | <p>Let $p$ be a rational prime and $K$ a number field.
Dedekind's discriminant theorem tells us that
$p$ ramifies in $K$ $\iff$ $p$ divides the discriminant of $K$.
Hence if $p$ does not divide discriminant of $K$,
$(p)$ either splits, i.e., </p>
<p>(i) $(p)=P_1 \cdots P_g$ for $P_i \neq P_j$ and $g \geq 2$
or </p>... | David E Speyer | 297 | <p>Pete's answer is very good. Some self-promotion:</p>
<p>An <a href="http://sbseminar.wordpress.com/2007/08/24/factoring-polynomials-and-the-frobenius/" rel="nofollow noreferrer">expository post</a> on the relationship between factoring polynomials and factoring primes.</p>
<p>The <a href="https://mathoverflow.net/... |
2,434,812 | <p>I am only concerned with the integral of a measurable function $f : M \to \mathbb{R}^{\ast}_{+}$, where $\mathbb{R}^{\ast}_{+}$ contains all non-negative real numbers and positive infinity. As I understand it, this is defined as
$$\sup \left\{ \int s : \text{$s$ is simple and $s\le f$}\right\}.$$</p>
<p>My question... | barmanthewise | 453,518 | <p>If said function is measurable then yes, it is. Some textbooks say that a function $f$ is Lebesgue-integrable if and only if $\sup\{s \text{ simple} : s \leq f\} = \inf\{s \text{ simple} : s \geq f\}$.</p>
<p>It all comes down to showing that these definitions of measurable function $f:X \rightarrow \mathbb{R}$ are... |
29,530 | <p>On the questions tab can we have a filter feature (I mean like frequented votes etc) of watched tags. I am aware that the tags that I have watched are demarked in a different colour. But I would like to discuss the pros and cons of getting a dedicated watched tags filter because I feel most people are well versed in... | Martin Sleziak | 8,297 | <p>The search query <a href="https://math.stackexchange.com/search?q=intags:mine">intags:mine</a> returns only posts in your favorite tags. If you <a href="https://math.stackexchange.com/search?tab=active&q=intags:mine">sort by activity</a> and perhaps add <a href="https://math.stackexchange.com/search?tab=active&a... |
3,082,337 | <p>Forgive my ignorance.<br>
Is the condition <span class="math-container">$x\in\mathbb{R}$</span> necessary to the set statement <span class="math-container">$\{x \in\mathbb{R} \vert x> 0\}$</span>?<br>
In other words, if <span class="math-container">$x$</span> is greater than zero, then is it not, by definition, a... | Hagen von Eitzen | 39,174 | <p>One might say: For me, <span class="math-container">$\aleph_0>0$</span> and <span class="math-container">$\aleph_0\notin \Bbb R$</span>, so the restriction <span class="math-container">$x\in \Bbb R$</span> is necessary</p>
<hr>
<p>But we may also view this from a different perspective, namely that the <em>form<... |
4,090,609 | <p>Not any even rank real vector bundle over a smooth manifold has a complex structure, because there are even dimensional manifolds that have no almost complex structures. I am curious about the following special case: Suppose <span class="math-container">$M$</span> is a real compact, connected smooth manifold and <sp... | Moishe Kohan | 84,907 | <p>There are two obstructions, one of which is topological and second is geometric.</p>
<ol>
<li>The topological obstruction is that every complex vector bundle comes with a canonical orientation. Hence, you have to assume that your vector bundle <span class="math-container">$E\to M$</span> is orientable and its orient... |
2,685,822 | <p>How can we prove that $L = \lim_{n \to \infty}\frac{\log\left(\frac{n^n}{n!}\right)}{n} = 1$</p>
<p>This is part of a much bigger question however I have reduced my answer to this, I have to determine the limit of $\log(n^{n}/n!)/n$ when $n$ goes to infinity.</p>
<p>Apparently the answer is 1 by wolfram alpha but ... | Bernard | 202,857 | <p>$$n!\sim_\infty\sqrt{2\pi n\vphantom{h}}\,\Bigl(\frac ne\Bigr)^{\!n},\enspace\text{hence }\quad \frac{n^n}{n!} \sim_\infty\frac{\mathrm e^n}{\sqrt{2\pi n\vphantom{h}}} $$
so
$$\frac1n\,\log\biggl(\frac{n^n}{n!}\biggr)\sim_\infty\frac1n\Bigl(\frac12\log(2\pi n)+n\Bigr)=\underbrace{\frac{\log(2\pi n)}{2n}}_{\substack{... |
470,506 | <p>When is this true?
$$\lim_{r\to 0}\int_{-K}^K f(rx)dx=\int_{-K}^K \lim_{r\to 0} f(rx)dx$$
Is it true without the hypothesis of continuity of
$f$? </p>
<p>Thank you.</p>
| Gina | 102,040 | <p>Your method are doomed to fail. Here is why:</p>
<ol>
<li><p>Totally bounded is equivalent to the condition that the space have finite cover each with radius less than $\epsilon$ for any $\epsilon>0$.</p></li>
<li><p>Metric subspace of a totally bounded metric space is also totally bounded. Trivially prove from ... |
2,363,211 | <p>Can <span class="math-container">$406014677132263504491682$</span> be the sum of two fourth powers? It may be so. Can anyone use Wolfram Mathematica , SAGE , or some computer program, to check whether this number is the sum of two fourth powers ?</p>
<p>The complete factorization of this number is given by : <span... | Oscar Lanzi | 248,217 | <p>Hmmmm. $1459^1$. Any whole number where a $4n-1$ prime has an odd exponent can't be rendered as a sum of squares. Let alone fourth powers.</p>
|
2,887,858 | <p>I'm learning how to take surface integrals on the surface of spheres in $\mathbb{R}^n$. This question is related to <a href="https://math.stackexchange.com/questions/2887371/calculating-the-surface-integral-int-s-10y-j-d-sigmay">Calculating the surface integral $\int_{S_1(0)}y_j \ d\sigma(y)$</a> where I try to comp... | Batominovski | 72,152 | <p><strong>Solution Employing Symmetry</strong></p>
<p>Let $S^{n-1}$ denote the unit hypersphere in $\mathbb{R}^n$ (with coordinate vector $\mathbf{y}=(y_1,y_2,\ldots,y_n)$) centered at the origin $\boldsymbol{0}_n$. Write $\sigma_{n-1}$ for the hypersurface area measure of $S^{n-1}$. Declare the <em>northern</em> h... |
1,046,961 | <p>Find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x \in \mathbb{R}$, $f(x) + f(2x) = 0$ <br/>
I'm thinking; <br/>
Let $f(x)=-f(2x)$ <br/>
Use a substitution $x=y/2$ for $y \in \mathbb{R}$. <br/>
That way $f(y)=-f(y/2)=-f(y/4)=-f(y/8)=....$ <br/>
Im just not sure if this is a good approac... | JHalliday | 124,007 | <p>$f(x)=0$ is the only solution.</p>
<p>$$f(0)=-f(2\cdot 0) \iff f(0)=0$$</p>
<p>Suppose that for any $x$, $f(x) = \epsilon$. But this means that for any $\delta$ and for large $n$ we have:</p>
<p>$$\frac{x}{2^n} < \delta$$</p>
<p>$$\large f\bigg(\frac{x}{2^n}\bigg)= (-1)^n\cdot\epsilon$$</p>
<p>But supposing... |
2,411,568 | <p>I would like to find all $(a,b,c)\in\mathbb{N}^3$ such that $a\times 1111+b\times 111+c\times 11=9002$.</p>
<p>It is obvious that $a+b+c \equiv 2 \pmod{10}$. I made a little computer program, that basically looped over all possiblities and I found no answer, so I just have to prove it...<br>
I am thinking of an in... | N. S. | 9,176 | <p>$$a×1111+b×111+c×11=9002 \\
11 ( 101a +10b+c)= 9002-b$$</p>
<p>It follows that $b=11k+4$ for some $k \in \mathbb N$. The equation then becomes
$$101 a+10b +c=818-k \\
101 a+111k +c=778\\
$$</p>
<p>The problem boils down to solving
$$101a+111k \leq 778$$
as you can set $c=778-101a+111k$. </p>
<p>I claim that
$$1... |
248,658 | <p>Based on the next relation:</p>
<p>$$\det\begin{bmatrix}A & B \\ C & D\end{bmatrix} = \det(A)\det(D - CA^{-1}B),$$</p>
<p>I have that for computing the eigenvalues of the block matrix:</p>
<p>$$\det\begin{bmatrix}A-\lambda I & B \\ C & D-\lambda I\end{bmatrix} = \det(A-\lambda I)\det((D-\lambda I)... | Roberto Belotti | 215,178 | <p>I was about to ask the very same question when I found this one... I know that I'm two years late but this issue does intrigue me so I want to say that the formula in the question, although not valid in general, can be still useful in some situations to compute the eigenvalues of block matrix. In fact if $\lambda$ i... |
1,409,932 | <p>I am trying to make a random number generator that is sorta special. Basically it generates a number between 5, and 17. The twist that I need math help on is that I want to have a variable "P" that works like the following.</p>
<p>The higher p is the more likely numbers closer to 17 are to appear, the lower it is... | David K | 139,123 | <p>In principle, you can turn a uniform random number generator into a
generator of any probability distribution you want, using the following procedure. (I'll also suggest some simplifications and shortcuts later in the answer.)</p>
<p>Assuming the desired probability distribution has been specified by a
density func... |
353,630 | <p>Consider a uniform random tournament with <span class="math-container">$n$</span> vertices. (Between any two vertices <span class="math-container">$x,y$</span>, with probability <span class="math-container">$0.5$</span> draw an edge from <span class="math-container">$x$</span> to <span class="math-container">$y$</sp... | Iosif Pinelis | 36,721 | <p>You guess is correct, assuming that by <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> you meant <span class="math-container">$s_1$</span> and <span class="math-container">$s_2$</span>. </p>
<p>Indeed, the probability in question is <span class="math-container">$1-p_n$</span>... |
65,220 | <p>Si $f$ es una función continua y si $f(m/2^n)=0$ para todo entero $m$ y todo natural $n$, ¿cómo demuestro que $f(x)=0$ para todo numero real $x$?</p>
<hr/>
<p><em>[translation by mixedmath]</em></p>
<p>If $f$ is a continuous function and if $f\left(\dfrac{m}{2^n}\right) = 0$ for all integers $m$ and... | Arturo Magidin | 742 | <p>Let $x\in\mathbb{R}$, and let $\epsilon\gt 0$. Then there exists $\delta\gt 0$ such that for all $z$, if $|x-z|\lt\delta$ then $|f(x)-f(z)|\lt\epsilon$. Let $n$ be a positive integer such that $\frac{1}{2^{n}}\lt \delta$. Then there is an integer multiple of $\frac{1}{2^{n+1}}$ that lies in $(x-\frac{1}{2^n},x+\frac... |
89,987 | <p>In the sequence: </p>
<p>$$\lim _{n\rightarrow \infty }{\frac {n+1}{2\,n+3}}\neq 3$$</p>
<p>I know how to prove that the limit is actually $1/2$, but is there another way to prove that 1 is NOT the limit? </p>
<p>I tried to prove by negation showing that if 1 is the limit I can't find an $N$ that for every epsilo... | David Mitra | 18,986 | <p>Let $\epsilon=1/4$. Let $N$ be given. Then
$${n+1\over 2n+3}\le {n \over 2n}\le{1\over2}\quad\Rightarrow|{n+1\over 2n+3} -1|\ge 1/2.$$
This is true for all $n\ge N$. </p>
<p>So, there is no $N$ such that $|{n+1\over 2n+3} - 1|<\epsilon$
for all $n\ge N$. This shows the sequence does not converge to 1.</p>
|
4,491,385 | <p><a href="https://i.stack.imgur.com/ILqh9.png" rel="nofollow noreferrer">This is the graph plotted by Desmos for the inequality <span class="math-container">$~\log_{0.2}\left(x^{2}-x-2\right)>\log_{0.2}\left(-x^{2}+2x+3\right)$</span></a></p>
<p>Here, you can see that plotted interval is <span class="math-containe... | bobeyt6 | 1,017,316 | <p>The answer is on <span class="math-container">$(2, 2.5)$</span>. There aren't dash marks on <span class="math-container">$x=2$</span> because the function is undefined there and desmos doesn't know what to do with it. If it were really <span class="math-container">$[2, 2.5)$</span>, you would see a bolded line on <s... |
1,092,957 | <p>Given a set of natural numbers $S_1$ , $S$ and a number N .</p>
<p>Specification of sets are as follow .</p>
<p>$$S = \{1,\dots N\}$$
$$S_1\subset S \;and \;S_1=\{b_1,b_2,\dots,b_m\}$$</p>
<p>And $$S' = S\setminus S_1$$</p>
<p>Find the smallest natural n that can not be written as a sum of elements of S'.</p>
<... | Tom-Tom | 116,182 | <p>A possible rather stupid way to do it is to compute the sum $M$ of all elements of $S_1$. Form an array $a$ (filled with zeroes) with the numbers from $1$ to $M-1$. Then compute all the possible sums of elements of $S_1$ (there are $2^N-1$ such sums where $N=\text{Card}\, S_1$, the number of elements of $S_1$) and f... |
1,092,957 | <p>Given a set of natural numbers $S_1$ , $S$ and a number N .</p>
<p>Specification of sets are as follow .</p>
<p>$$S = \{1,\dots N\}$$
$$S_1\subset S \;and \;S_1=\{b_1,b_2,\dots,b_m\}$$</p>
<p>And $$S' = S\setminus S_1$$</p>
<p>Find the smallest natural n that can not be written as a sum of elements of S'.</p>
<... | Waffle | 55,527 | <p>EDIT: I just realized how simple this problem is. Here's the solution in Lua:</p>
<p>S={1,1,1,1,5,7,10,25,50,75,100,250,500,1000}
table.sort(S)</p>
<p>x=1
for i=1,#S do
if S[i]<=x then
x=x+S[i]
end
end</p>
<p>print(x)</p>
<p>Starting at x=1, go in ascending order increasing x if the number is less than or equ... |
4,610,313 | <p>I am working on AoPS Vol. 2 exercises in Chapter 1 and attempting to solve the below problem:</p>
<blockquote>
<p>Given that <span class="math-container">$\log_{4n} 40\sqrt{3} = \log_{3n} 45$</span>, find <span class="math-container">$n^3$</span> (MA<span class="math-container">$\Theta$</span> 1991).</p>
</blockquot... | boojum | 882,145 | <p>We could also take a "factorization" approach. Since
<span class="math-container">$$ \log_{4n} 40\sqrt{3} \ \ = \ \ \log_{3n} 45 \ \ = \ \ \alpha \ \ , $$</span></p>
<p>we can write
<span class="math-container">$$ (4n)^\alpha \ \ = \ \ 2^3·3^{1/2}·5 \ \ \ , \ \ \ (3n)^\alpha \ \ = \ \ 3^2·5 $$</span> <sp... |
268,308 | <p>In composing a proof that is reliant on proven theorems, does one simply assume the reader's familiarity with said theorems, or does one explicitly include their logic in the new logic? </p>
| amWhy | 9,003 | <p>It's a judgment call, <em>depending, in part, on the audience you are addressing</em>, and <em>your purpose</em> for writing a proof, or a paper with proofs, etc. And so, as Asaf answers, the answer to your question: "depends on the context."</p>
<p><em>If you are writing proofs for classes</em> (assigned or recomm... |
278,971 | <p>How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis? What are the connections between both conjectures if any?</p>
| Benjamin Dickman | 37,122 | <p><strong>Edit 2:</strong> For a more tenuous connection between these two theorems, one person whose name might crop up someday for both (please note I am just speculating here!) is that of Fields Medalist <a href="http://en.wikipedia.org/wiki/Atle_Selberg" rel="nofollow noreferrer">Atle Selberg</a>. His <a href="htt... |
737,054 | <p>Express the vector $\vec{u}$ below as a sum of two vectors $\vec{u}_1$ and $\vec{u}_2$, where $\vec{u}_1$ is parallel to the vector $\vec{v}$ given below, and $\vec{u}_2$ is perpendicular to $\vec{v}$. Make sure that the first vector in your sum is $\vec{u}_1$ and the second is $\vec{u}_2$.</p>
<p>$\vec{v} = [-3,-1... | user139388 | 139,388 | <p>Write
$$
u
= proj_v u + (u-proj_v u)
:= u_1 + u_2.
$$
So in your case you would get
$$
u_1
= \left(\frac{|u|\cdot |v|}{|v| \cdot |v|}\right)v
= \frac{22}{11}(-3,-1,1)
= (-6,-2,2)
$$
and
$$
u_2
= (-2,-10,6) - (-6,-2,2)
= (4,-8,4).
$$</p>
|
2,798,338 | <p>Let $G$ be a group of order 20, Prove that $G$ has a normal subgroup of order 5.</p>
<p>Obviously by Sylow theorem there is a subgroup of order 5, and since all Sylow p-subgroups are conjugate the only problem is to show that there is only one sungroup of order 5.</p>
<p>any help would be appreciated</p>
| N8tron | 32,820 | <p>As you said Sylow's thoerem gives there is a 5-sylow subgroup.</p>
<p>The number of sylow 5 subgroups $n_5$ divides 4 and $n_5$ is congruent to 1 modulo 5. </p>
<p>So $n_5=1+5k$ for some $k$ and $n_5=1+5k$ divides $4$. Since $n_k$ is a natural number $k\ge 0$ but if $k>0$ then $n_5>4$ contradicting that $n_... |
406,966 | <p>Define a graph with vertex set <span class="math-container">$\mathbb{R}^2$</span> and connect two vertices if they are unit distance apart. The famous <a href="http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem" rel="nofollow noreferrer">Hadwiger-Nelson problem</a> is to determine the <a href="http://en.wi... | MJD | 25,554 | <p>The first four perfect numbers have been known for at least 2300 years. Here are their prime factorizations:</p>
<p>$$\begin{align}
6 & = 2^{\hphantom{2}}\cdot 3 \\
28 & = 2^2\cdot 7 \\
496 & = 2^4\cdot 31 \\
8128 & = 2^6\cdot 127
\end{align}
$$</p>
<p>Any knucklehead, looking at these, or perhaps... |
63,763 | <p>For elliptic curve $y^{2}=x(x+a^{2})(x+(1-a)^{2})$,($a$ is a rational number and does not equal 0,1,1/2),is its rank always 0?</p>
| James Weigandt | 4,872 | <p>Your family of elliptic curves are exactly those elliptic curves with torsion subgroup containing $\Bbb Z / 2 \Bbb Z \times \Bbb Z / 4 \Bbb Z$. To see this make the linear change of variables $t = a/(1-a)$ and look at the parameterizations on page 101 of Husemöller's book "Elliptic Curves". </p>
<p>So in particular... |
599,232 | <p>Q)An object moving along a curve in the xy-plane has position ( x(t), y(t) ) at time t, with dx/dt = cos (3/2 t^2) and dy/dt = 3sin(t^2) FOR 0 <= t <=3 at t=2, the object is at (4,5). write the equation for the tangent line at this point.</p>
<p>Comments:
First and for most what should i understand from (x(t)... | user99680 | 99,680 | <p>The distance from $(2,0)$ to $(20,0)$ + the distance from $(2,0)$ to $(0,0)$ equals the radius.</p>
|
599,232 | <p>Q)An object moving along a curve in the xy-plane has position ( x(t), y(t) ) at time t, with dx/dt = cos (3/2 t^2) and dy/dt = 3sin(t^2) FOR 0 <= t <=3 at t=2, the object is at (4,5). write the equation for the tangent line at this point.</p>
<p>Comments:
First and for most what should i understand from (x(t)... | André Nicolas | 6,312 | <p>Let the centre of the circle be $O$, and let the point $(2,0)$ be $P$. Draw a line $PQ$ to the periphery of the circle, making an angle $\theta$ with the positive $x$-axis. We want to find the length of $PQ$.</p>
<p>Consider the triangle $OPQ$. We have $\angle OPQ=180^\circ-\theta$. By the Cosine Law, with $x=PQ$, ... |
599,232 | <p>Q)An object moving along a curve in the xy-plane has position ( x(t), y(t) ) at time t, with dx/dt = cos (3/2 t^2) and dy/dt = 3sin(t^2) FOR 0 <= t <=3 at t=2, the object is at (4,5). write the equation for the tangent line at this point.</p>
<p>Comments:
First and for most what should i understand from (x(t)... | Remco | 463,495 | <p>A different solution without having to solve an equation is by rotating the axis back and forth. (more suitable for mathematical programs)</p>
<p><strong><em>r</em></strong> is the radius of the circle.</p>
<p><strong><em>O</em></strong> is the origin at [0, 0].</p>
<p><strong><em>P</em></strong> is any point wit... |
325,890 | <p>To evaluate this integral I have got to use Parseval's Theorem and the fourier transform of</p>
<p>$$s(x)=\begin{cases}
0 & x\leq -a \\
1 & -a<x<a \\
0 & x \geq a.\end{cases}$$</p>
<p>This works out to be</p>
<p>$$\tilde{s}(k)=\frac{e^{ika}}{ik}-\frac{e^{-ika}}{ik}$$</p>
<p>I have then changed... | Ron Gordon | 53,268 | <p>The FT of $f(x) = \sin{x}/x$ is</p>
<p>$$\hat{f}(k) = \begin{cases} \\\pi & |k| < 1 \\ 0 & |k| > 1 \end{cases}$$</p>
<p>Parseval's theorem then implies that</p>
<p>$$\int_{-\infty}^{+\infty} dx \frac{\sin^2{x}}{x^2} = \frac{1}{2 \pi} \int_{-1}^1 dk \: \pi^2 = \pi $$</p>
|
983,923 | <p>Given $A$ and $B$ is the subset of $C$ and $f:C\mapsto D$,
$$f(A\cap B)\subseteq f(A) \cap f(B)$$ and the equality holds if the function is injective.</p>
<p>But why for the inverse, suppose that $E$ and $F$ is the subset of $D$,
$$f^{-1}(E \cap F) = f^{-1}(E) \cap f^{-1}(F)$$
without saying that the inverse functi... | Liealgebrabach | 185,117 | <p>They don't mean the inverse by $f^{-1}$ but the pre-image under f. Maybe once you have studied that definition, it will become more evident!</p>
|
2,900,112 | <p>Let $f:\mathbb{R}^n \to \mathbb{R}$. Let $x_0 \in \mathbb{R}^n$. Assume $n-1$ partials exist in some open ball containing $x_0$ and are continuous at $x_0$, and the remaining $1$ partial is assumed only to exist at $x_0$. A well known result states that this implies $f$ is differentiable at $x_0$. </p>
<p>My questi... | Paul Frost | 349,785 | <p>For $n = 2$ it is not sufficient to assume that the 2 partial derivatives merely exist.</p>
<p>This generalizes to abitrary $n$. Assume we could take $g(n) = n-2$. Consider any function $f : \mathbb{R}^2 \to \mathbb{R}$ and define $F : \mathbb{R}^n \to \mathbb{R}, F(x_1,\ldots,x_n) = f(x_1,x_2)$. We have $\frac{\pa... |
2,759,112 | <p>Are there cases where two events have high probability (i.e. each of them has high probability) but, at the same time, the probability that they both obtain is low? If yes, could you please provide a simple, daily-life example?</p>
| manofbear | 230,268 | <p>Coin flip: probability of heads is $1/2$, probability of tails is $1/2$, probability of both is $0$.</p>
|
4,447,329 | <p>There is an interesting inequality I've stumbled upon on the Internet. It has logarithms and trigonometry, but in contrast to something like <a href="https://math.stackexchange.com/questions/1029060/inequality-with-a-mixture-of-logs-and-trig">this</a> that uses trigonometry and logarithms separately, this one takes ... | richrow | 633,714 | <p>Let <span class="math-container">$O$</span> be an arbitrary point of the plane. Consider vectors <span class="math-container">$v_n:=\overrightarrow{OP_n}$</span>. Then, for all <span class="math-container">$n\ge 3$</span> we have
<span class="math-container">$$
v_{n}=\frac{1}{2}(v_{n-2}+v_{n-3})
$$</span>
or
<span c... |
4,447,329 | <p>There is an interesting inequality I've stumbled upon on the Internet. It has logarithms and trigonometry, but in contrast to something like <a href="https://math.stackexchange.com/questions/1029060/inequality-with-a-mixture-of-logs-and-trig">this</a> that uses trigonometry and logarithms separately, this one takes ... | Saaqib Mahmood | 59,734 | <p>For <span class="math-container">$n = 4, 5, 6, \ldots$</span>, since <span class="math-container">$P_n$</span> is the midpoint of the segment joining the points <span class="math-container">$P_{n-3}$</span> and <span class="math-container">$P_{n-2}$</span>, we can conclude that
<span class="math-container">$$
\vec{O... |
2,293,838 | <p>I have question(1) in the following picture:</p>
<p><a href="https://i.stack.imgur.com/ZwSLl.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZwSLl.png" alt="enter image description here"></a></p>
<p>and the solution to it is given in the following picture:</p>
<p><a href="https://i.stack.imgur.... | Alex R. | 22,064 | <p>To show that $a_n=3a_{n-2}$, note that if you have a palendrome of size $n$ and snip off the first and last letters, you are left with a palendrome of size $n-2$. This gives you a map from $n$ to $n-2$. To go from $n-2$ to $n$, convince yourself that it suffices to choose one of the 3 letters to then append to the b... |
654,089 | <p>The book "A First Course in Algebra" says</p>
<blockquote>
<p>In a finite dimensional vector space, every <strong>finite</strong> set of vectors spanning the space contains a subset that is a basis.</p>
</blockquote>
<p>All that is fine. But what about a span having an infinite number of vectors? Surely that too... | Christian Blatter | 1,303 | <p>Given that $V$ is finite-dimensional one can argue as follows: Take a basis $(e_i)_{1\leq i\leq n}$ of $V$. Each of the $e_i$ is a finite linear combination of vectors from the spanning set $S$. It follows that there is a finite subset $S_0\subset S$ that spans $V$. From your "First Course in Algebra" you can then ... |
2,021,734 | <p>Let $p$ be a prime number and $\mu_{p^{\infty}}$ denote $$\{ z\in \mathbb{C} : \exists k \ge 1 : z^{p^k}=1 \}$$ find all of its subgroups.</p>
<p>I was able to prove that its finite subgroups are of the form $\mu_{p^n}$, where $n$ is a positive integer (using Lagrange's Theorem and the fact that in, $\mu_{\infty}=\... | Andreas Caranti | 58,401 | <p>This group, the <a href="https://en.wikipedia.org/wiki/Pr%C3%BCfer_group" rel="nofollow noreferrer">Prüfer group</a>, has no proper infinite subgroup.</p>
<p>Suppose $H$ is a subgroup. Consider the set
$$
E_{H} = \{ k \in \mathbb{N} : \text{in $H$ there is an element of order $p^{k}$} \}.
$$
If $E_{H}$ is infinite... |
1,645,865 | <p>I am ultimately trying to prove, for an Exercise in Burton's Elementary Number Theory, that $x^4 - y^4 = 2z^2$ has no solution in the positive integers.</p>
<p>I can establish that if there is a solution, the solution with the smallest value of x has gcd(x,y)=1 </p>
<p>I see that $x^4 - y^4 = (x^2 + y^2)(x^2 - y^2... | πr8 | 302,863 | <p><strong>Hint:</strong> Try to sketch a graph of $y=x^2 e^x$, paying particular attention to any minima and maxima it may have. Once you've done this, superimpose $y=1$ onto this graph to see what any solutions might look like.</p>
<p>1) <strong>Existence</strong></p>
<p>If $f(x)=x^2e^x$, then</p>
<ul>
<li>$f$ is ... |
1,119,421 | <p>We know that p → q is not equivalent to q → p. But suppose we make a proof system that has all the rules of logical identities plus the rule (“commutativity of implies”) p → q ≃ q → p. (We are using the new symbol ≃ because these are not really equivalent.) </p>
<p>This new proof system is not sound. The point of t... | Aaron Maroja | 143,413 | <p>HInt: We have that $\mathbb{Z}_6 = \langle \overline{1}\rangle$ and $\mathbb{Z}_{18} = \{\overline{1}, \ldots, \overline{17}\}$. then there exists a homomorphism $f: \langle \overline{1}\rangle \to \mathbb{Z}_{8}$ such that $f(a) = b$, if and only if, $\mathcal{O}(b)$ divides $\mathcal{O}(a)$. </p>
<p>Note: $\mat... |
1,370 | <p>This is in reference to the recent <a href="https://mathematica.stackexchange.com/q/56829/3066">PlotLegends -> “Expressions”</a> question.</p>
<p>I filed a repot on this behavior with WRI tech support. A little later rcollyer posted an answer explaining it was new (undocumented) feature. This morning I received an... | Mr.Wizard | 121 | <p>I think it is appropriate to follow the logic that rm -rf explained here:</p>
<ul>
<li><a href="https://mathematica.meta.stackexchange.com/questions/1085/do-we-need-a-list-filtering-tag/1087#1087">Do we need a "list-filtering" tag?</a></li>
</ul>
<p>Simply adding <a href="https://mathematica.stackexchang... |
2,556,255 | <p>For work I have been doing a lot of calculations which look sort of like summing terms similar to $\frac{A}{1+x}$ and $\frac{B}{1+y}$ for some $A, B$ and small values of $0 \leq x, y \leq 0.1$. In my experience I have found that this is approximately equal to $\frac{A + B}{1 + \frac{A}{A+B}x + \frac{B}{A+B}y }$. The... | dxiv | 291,201 | <p>Hint: look at $\;\displaystyle \frac{a}{u} +\frac{b}{v} - \frac{(a+b)^2}{a u + b v } = \frac{ab(u-v)^2}{uv(a u + b v)}\,$ for $u=x+1, v=y+1$. The RHS gives the error of the proposed approximation.</p>
|
1,285,870 | <p>Is the collection of all cardinalities a set or a proper class?
Does anybody ever think about the problem?</p>
| Wojowu | 127,263 | <p>Collection of all cardinalities is indeed a proper class. To see this, note that there is at least as many cardinal numbers as ordinal numbers, because map $\alpha\rightarrow\aleph_\alpha$ is an injection.</p>
|
1,285,870 | <p>Is the collection of all cardinalities a set or a proper class?
Does anybody ever think about the problem?</p>
| Noah Schweber | 28,111 | <p>It is a proper class. There are several ways to see this. One is: suppose $X$ were the set of all cardinals (=initial ordinals). The ordinals - in particular, the cardinals - are well-ordered, so we may add them together (indexed by this well-order) to form a single "super-cardinal;" but it's easy to check that this... |
243,849 | <p>I was working on an examples from my textbook concerning transforming formulae into disjunctive-normal form (DNF) until I found an expression that I cannot solve. I hope somebody can help me transform the following statement into DNF:</p>
<p>$$ (\lnot q \lor r) \land ( q \lor \lnot r)$$</p>
| A.Schulz | 35,875 | <p>Make a table</p>
<pre><code> q r result
0 0 1
0 1 0
1 0 0
1 1 1
</code></pre>
<p>Now you can get the DNF formula, by "or-ing" all the rows with result=1:
$$ (\lnot q \land \lnot r ) \lor ( q \land r )$$</p>
|
430,142 | <p>How do I determine if $w$ is in the range of the linear operator $T$?</p>
<p>$$T:\Bbb R^3 \to \Bbb R^3, T(x,y,z)=(x-y,x+y+z,x+2z)\ ;\quad w=(1,2,-1)$$
I would appreciate the help.</p>
<p>Thanks</p>
| Mikasa | 8,581 | <p>Assuming you already showed that $T$ is a linear transformation, try to find $(x,y,z)\in\mathbb R^3$ such that $$\left\{
\begin{array}{ll}
x-y=1\\
x+y+z=2\\
x+2z=-1
\end{array}
\right.$$</p>
|
1,065,988 | <p>In <a href="https://www.youtube.com/watch?v=6Lm9EHhbJAY">this Numberphile video</a> it is stated that trisecting an angle is not possible with only a compass and a straight edge. Here's a way I came up with:</p>
<blockquote>
<p>Let the top line be A and bottom line be B, and the point intersecting P.<br>
1. Use... | Oleg567 | 47,993 | <p>If $\angle APB= \alpha$, then middle angle (middle part) will have measure
$$
\beta = 2\arctan \left(\frac{\tan(\alpha/2)}{3}\right) {\Large\color{red}{\ne}} \dfrac{\alpha}{3}.
$$</p>
<p>A few examples:
\begin{array}{|c|c|c|}
\hline
\alpha & \beta & error\\
\hline
3^\circ & 1.000203^\circ & 0.0203\%... |
461,268 | <p>A square hole of depth $h$ whose base is of length $a$ is given.
A dog is tied to the center of the square at the bottom of the hole by a rope of length
$L>\sqrt{2a^2+h^2}$ ,and walks on the ground around the hole.</p>
<p>The edges of the hole are smooth, so that the rope can freely slide along it. Find the sha... | WishingFish | 83,036 | <p>Consider the rope touches the edge of the hole at point $p$ with distance $x$ from the corner. Then the distance from the projection of the point $p$ on the edge to the point on the bottom $p^\prime$ is $a$.</p>
<p>The distance between $p^\prime$ to the center of the edge of bottom it lands on is $a/2$. Hence the d... |
432,262 | <p>Let <span class="math-container">$x_0=1$</span> and
<span class="math-container">$$x_{k+1} = (1-a_k)\left(\frac{3}{2}-\frac{1}{2}\frac{1}{x_k}\right)$$</span>
where <span class="math-container">$a_n$</span> is a known sequence satisfying that <span class="math-container">$a_k\in(0,1)$</span> for all <span class="mat... | Saúl RM | 172,802 | <p>If, as you say, <span class="math-container">$a_k<0.1$</span> for all <span class="math-container">$k$</span>, then we can prove by induction that <span class="math-container">$x_k>\frac{3}{4}$</span> for all <span class="math-container">$k$</span>, with induction step <span class="math-container">$x_{k+1}>... |
718,664 | <p>I'm trying to find a small primitive root modulo $p^k$, where $p$ is prime. My strategy is to test small numbers $g=2,3,\ldots$ until I find a primitive root modulo $p$. That is, until $ord_pg=\phi(p)=p-1$. There are results that suggest such a search won't take long.</p>
<p>To go from modulo $p$ to modulo $p^k$... | lhf | 589 | <p>Surprisingly, only two odd primes are known such that the smallest primitive root mod $p$ is not a primitive root mod $p^2$: $40487$ and $6692367337$. See <a href="https://oeis.org/A055578" rel="nofollow">OEIS:A055578</a>.</p>
|
4,589,828 | <p>I'm stuck in proving the function <span class="math-container">$$f(x) = \vert x + e^x\vert $$</span> has a minimum.</p>
<p>This is what I did:</p>
<p><span class="math-container">$$f'(x) = (1+e^x)\text{sgn}(1+e^x)$$</span></p>
<p>But this function is never zero, because the exponential is always positive. So when I ... | Anastasiya-Romanova 秀 | 133,248 | <p>You made mistake in the second integral. It should be
<span class="math-container">$$
I_2 = \frac{\theta}{2} + C_2 = \frac{\arcsin(x)}{2} + C_2
$$</span></p>
|
2,105,238 | <p>I need to prove that the function $f: \mathbb R_{>0}\times\mathbb R \to \mathbb R^2$ , $f(x,y)=(xy, x^2-y^2)$ is injective. I know I have to show that $f(x,y)=f(a,b)$ implies $x=a$ and $y=b$ but I have no idea how to prove it. Could you give me a hint?</p>
| C. Zhihao | 405,549 | <p>if $f(x,y) = f(a,b)$ you have two equations</p>
<p>$(1)\;\; xy = ab$</p>
<p>$(2)\;\; x^2 - y^2 = a^2 - b^2$.</p>
<p>Since $x > 0$, from (1) you get $y = ab/x$. Sub such $y$ on equation
(2) and simplify to get</p>
<p>$$x^4 - x^2 (a^2 - b^2) - (ab)^2 = 0$$</p>
<p>Use the quadratic equation and the fact that $... |
2,785,464 | <p>It's clear that every orthonormal basis is a frame, but is <strong>every</strong> basis a frame? In particular, is a non-orthogonal basis always a frame? If yes, why?</p>
<p>I'm asking about general Hilbert spaces, in particular infinite dimensional spaces.</p>
| user | 505,767 | <p>For finite dimensional vector spaces every basis is a frame in the sense that we can describe all the space in an unique manner by any basis.</p>
<p>For non orthogonal basis we lost the property that </p>
<p>$$ \langle\vec x,\vec v_i \rangle =\vec x\cdot \vec v_i=x_i$$</p>
<p>which holds for orthogonal basis, tha... |
2,687,375 | <p>Let <span class="math-container">$x_1, x_2, \ldots, x_n$</span> be a random sample from the Bernoulli (<span class="math-container">$\theta$</span>).</p>
<p>The question is to find the UMVUE of <span class="math-container">$\theta^k$</span>.</p>
<p>I know the <span class="math-container">$\sum_1^nx_i$</span> is the ... | Matteo Casarosa | 539,948 | <p>$ S$ is compact due to Heine-Borel theorem, as you mentioned, and it is also connected. You can prove this in several ways. One way is considering that $ S$ is the topological product of two connected spaces, another is to show that $S$ is path-connected (hint: it's also convex).</p>
<p>$T$ is not compact, because ... |
1,116,215 | <p>In C. Adam's Topology, it is written that $f(A) - f(B) \subset f(A - B)$ for any function, but $f(A) - f(B) = f(A - B)$ iff f is bijective. I can come the half way, i.e., $f(A) - f(B) \subset f(A - B)$; but for the $f(A - B) \subset f(A) - f(B)$, I don't know both of how to prove correctness of it for bijective and ... | Community | -1 | <p>For a counterexample when $f$ is not bijective : Let $f:\mathbb R \to \mathbb R : x \mapsto 1$. Take $A = \mathbb R$ and $B = \{0\}$. Then $f(A \setminus B) = \{1\}$ but $f(A) \setminus f(B) = \{1\} \setminus \{1\} = \emptyset$.</p>
|
1,131,158 | <p>I'm having an exam from linear algebra in a few days and I'm trying to understand some definitions from my workbook. Here's a definition I can't wrap my head around:</p>
<p>Let's assume we have two bases in vector space $X$: <strong>a</strong>$=(a_1,...,a_n)$ and <strong>b</strong>$=(b_1,...,b_n)$. Let $f$ be an e... | Muphrid | 45,296 | <p>$M_{f,a}$ is the matrix that <em>corresponds</em> to (or <em>represents</em>) the endomorphism $f$ for the choice of basis $a$.</p>
<p>$x$ is any vector. $\alpha$ corresponds to $x$, when you choose the basis $a$.</p>
<p>All this is doing is taking the abstract idea of an endomorphism--which is just a linear func... |
812,379 | <p>Is the above true? (I think it is!) if so, please can somebody explain why? I don't see it!</p>
| Nicky Hekster | 9,605 | <p>Perhaps you want to know the following fact (the ring $R$ is assumed to be commutative with identity). Let $$N= \{a \in R\mid a^n=0 \text{ for some integer } n\gt 0\}.$$ This set equals all the so-called <em>nilpotent elements</em>. One can show that actually $$N=\bigcap\{P \mid P \text{ is a prime ideal of }R\}.$$ ... |
3,502,551 | <blockquote>
<p>Does L' Hospital's rule pay-off at all in calculating:
<span class="math-container">$$\displaystyle\lim_{x\to 0}\frac{\sqrt{\cos2x}\cdot
e^{2x^2}-1}{\ln{(1+2x)\cdot\ln{(1+2\arcsin{x})}}}$$</span></p>
</blockquote>
<p>I posted <a href="https://qr.ae/TJIZ8z" rel="nofollow noreferrer">this question o... | Paramanand Singh | 72,031 | <p>My preferred approach is to use a set of well kwown limits combined with algebraic manipulation. This is a simple approach which requires familiarity with laws of limits. Techniques like L'Hospital's Rule and Taylor series are powerful but require some care (especially in case of L'Hospital's Rule one must first ens... |
3,794,418 | <p><strong>Background:</strong> I'm a math rookie, yet to enrol in university. I randomly started reading Mendelson's <em>Introduction to Mathematical Logic</em>, when I stumbled upon this paradox in the introductory section:</p>
<blockquote>
<p><strong>Grelling's Paradox:</strong> An adjective is called <em>autologica... | Noah Schweber | 28,111 | <p>If <span class="math-container">$A, A_a,$</span> and <span class="math-container">$A_h$</span> actually "make sense" - more on this below - then we clearly have that <span class="math-container">$A_a$</span> and <span class="math-container">$A_h$</span> partition <span class="math-container">$A$</span>: <s... |
1,045,444 | <p>I am having trouble understanding the role primary decomposition plays in ``interpreting'' the geometric picture of a scheme. Here are the examples I am struggling with from Eisenbud's <em>Commutative Algebra With a viewpoint towards Algebraic Geometry</em>. Assume $k$ is algebraically closed.</p>
<ol>
<li><p>Appar... | user84413 | 84,413 | <p>Hint: By the AM-GM inequality, we have that $\frac{\sqrt{a_n}}{n}\le\frac{1}{2}(a_n+\frac{1}{n^2})$</p>
|
1,045,444 | <p>I am having trouble understanding the role primary decomposition plays in ``interpreting'' the geometric picture of a scheme. Here are the examples I am struggling with from Eisenbud's <em>Commutative Algebra With a viewpoint towards Algebraic Geometry</em>. Assume $k$ is algebraically closed.</p>
<ol>
<li><p>Appar... | Jack D'Aurizio | 44,121 | <p>By the Cauchy-Schwarz inequality,
$$\sum \frac{\sqrt{a_n}}{n} \leq \sqrt{\zeta(2)\sum a_n}.$$</p>
|
669,696 | <p>How do I show by the Root Test that $$\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$$ converges or diverges? This is what I have done so far. Since we take $\sum\limits_{i=1}^\infty \sqrt[n]{|a_n|}$, we let $a_n = (2n^{1/n}+1)^n.$ This yields $\sum\limits_{n=1}^\infty\sqrt[n]{(2n^{1/n}+1)^n}$, which simplifies to $\sum\l... | user127.0.0.1 | 50,800 | <p>By the <em><a href="http://en.wikipedia.org/wiki/Root_test" rel="nofollow">root test</a></em> you get</p>
<p>$$\limsup\sqrt[n]{\left|2n^{1/n}+1\right|^n} = \limsup 2n^{1/n}+1 = 3 \gt 1$$</p>
<p>Thus your series diverges.</p>
|
960,880 | <p>Could you help me to explain how to find the solution of this equation
$$y ′ (t)=−y(t)-\frac1{2}*(1+e^{-2t})+1$$
Given $y(0)=0$
Thank all
This is my answer
$$y ′ (t)=−y(t)-\frac1{2}e^{-2t}+\frac1{2}$$
$$e^{2t}y ′ (t)=e^{2t}(−y(t)-\frac1{2}e^{-2t}+\frac1{2})$$
where
$$(e^{2t}y(t))′=e^{2t}y(t)′+2(e^{2t}y(t))=e^{2t}... | UserX | 148,432 | <p>Hints; Solve $$y'+y=\frac12 \left (1-e^{-2t} \right ) \stackrel{\cdot e^{\int 1 \mathrm{d}t}}{\iff} e^t y'+e^t y= e^t \frac12 \left (1-e^{-2t} \right )$$</p>
<p>Do you recognise the LHS as something familiar?</p>
|
3,232,319 | <p>I need to show that <span class="math-container">$e^{-it}=\frac{1}{e^{it}}$</span>.
but I don't understand what needs to be proven, it seems trivial to me. If anyone could help me.
Is the claim true even if t is not real?
Thank you</p>
| D.B. | 530,972 | <p>From Euler's formula,
<span class="math-container">$$e^{-it} = \cos(t)+i\sin(-t) = \cos(t)-i\sin(t).$$</span>
Also,
<span class="math-container">$$e^{it} = \cos(t)+i\sin(t).$$</span>
Now, compute the product
<span class="math-container">$$e^{it} \cdot e^{-it}.$$</span>
You find that the answer is <span class="math-c... |
3,232,319 | <p>I need to show that <span class="math-container">$e^{-it}=\frac{1}{e^{it}}$</span>.
but I don't understand what needs to be proven, it seems trivial to me. If anyone could help me.
Is the claim true even if t is not real?
Thank you</p>
| Elias Costa | 19,266 | <p>The inverse of the complex number <span class="math-container">$e^{t\cdot i}$</span> is, by definition, another complex number <span class="math-container">$e^{u+v\cdot i}$</span> such that
<span class="math-container">$$
e^{t\cdot i}\cdot e^{u+v\cdot i}=1
$$</span>
On the other hand we know that
<span class="math-... |
2,096,827 | <p><strong>Question</strong>: A sample size of 10 is taken with replacement from an urn that contains 100 balls, which are numbered 1, 2, ..., 100. (At each draw, each ball has the same probability of being selected).</p>
<p>There are 3 parts to the question, and I've included my work below. However, I'm not sure if i... | RGS | 329,832 | <p><strong>HINT:</strong></p>
<p>What happens if you sum those two results together? Or what if you subtract them?</p>
|
4,010,408 | <p>good day</p>
<p>Can anybody help me? thanks</p>
<p>Let <span class="math-container">${X_i}$</span> Independent and Identically Distributed Random Variables as normals with mean <span class="math-container">$μ$</span> and standard deviation 2, What is the minimum value of n such that <span class="math-container">$P(|... | herb steinberg | 501,262 | <p><span class="math-container">$\frac{S_n}{n}$</span> is normal with mean <span class="math-container">$\mu$</span> and standard devation <span class="math-container">$\frac{2}{\sqrt{n}}$</span>. To get your answer look up table of normal distribution.</p>
|
4,528,027 | <p>Im trying to prove that the function <span class="math-container">$$\begin{cases}f(x,y)=\dfrac{(2x^2y^4-3xy^5+x^6)}{(x^2+y^2)^2}, & (x,y)≠0\\ 0, & (x,y)=0\end{cases}$$</span> is continuous at point (0,0) using the rigorous defintion of a limit.</p>
<p>Attempting to find the upper limit of the function:
<span... | Theo Bendit | 248,286 | <p>Use the fact that <span class="math-container">$|xy| \le \frac{x^2 + y^2}{2}$</span>. You can then get,
<span class="math-container">$$3|xy^5| = 3|xy|y^4 \le \frac{3}{2}(x + y)^3$$</span>
and
<span class="math-container">$$2x^2y^4 = 2|xy|^2y^2 \le \frac{1}{2}(x + y)^3.$$</span>
You can then proceed as you were.</p>
|
516,280 | <blockquote>
<p>Given the equation <span class="math-container">$\displaystyle{\int_{-x}^x\exp({-t^2})dt}=-\ln(x)$</span>:</p>
<p>a. Simplify the integral using Gauss method with 3 points.</p>
<p>b. Solve given equation by Newton Raphson iterative method</p>
</blockquote>
<p>I succeeded simplifying the integral and got... | Amzoti | 38,839 | <p>I did not check the first part , but I think there is an issue as we should be able to validate the LHS and RHS when we find our $x$, and that is not checking out. Note, you formula did not specify which Gauss method to use and that normally means Gauss-Legendre which use Legendre Polynomials as $\displaystyle \int_... |
406,514 | <blockquote>
<p>Find the Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$.</p>
</blockquote>
<p>Any explanations during the demonstration, will be appreciated. Thanks!</p>
| DonAntonio | 31,254 | <p>Hints for you to understand and/or prove:</p>
<p>(1) If $\,\sigma\in\,\text{Gal}\,(f/\Bbb Q)\,$ , it <strong>must</strong> be that</p>
<p>$$\sigma(\sqrt 2)=\pm\sqrt 2\;,\;\;\sigma(i\sqrt 3)=\pm i\sqrt 3$$</p>
<p>(2) We have that $\,x^2+3\in\Bbb Q(\sqrt 2)[x]\,$ is irreducible (<strong>there</strong> , not only i... |
2,775,443 | <p>The statement $\displaystyle \sum_{n=1}^{\infty}a_n$ converges $\implies$ $\displaystyle \sum_{n=1}^{\infty}\cfrac{1}{a_n}$ looks natural but do we have this implication? I am checking alternating series as an counter-example but could not find one yet. What can we say about the implication?</p>
| farruhota | 425,072 | <p>If $\sum a_n$ and $\sum b_n$ are both positive and convergent, then $\sum (a_n+b_n)$ must be convergent, but:
$$a_n+b_n=a_n+\frac{1}{a_n}\ge 2.$$</p>
|
922,339 | <p>Let A={1,2,3,...,8,9} B={2,4,6,8} C={1,3,5,7,9} D={3,4,5} E={3,5}
Which of these sets can equal a set X under each of the following conditions?</p>
<p>a. X and B are disjoint</p>
<p>b. X is a subset of D but X is not a subset of B</p>
<p>c. X is a subset of A but X is not a subset of C</p>
<p>d. X is a subset o... | almagest | 172,006 | <p>You have found one solution to the equation. Now you want to find other solutions. Suppose you add $Dx$ to your solution. What happens. Well, if you differentiate $Dx$ (where $D$ is just a constant) twice wrt $x$ or twice wrt $y$ you get nil. So your solution $+Dx$ is also a solution. Similarly you can add $Ey$ and ... |
945,920 | <p>Let $X$ be a nonempty set and $d: X\times X\to R$ be a function such that for all $x,y\in X$ and all distinct $u, v\in X$ each of which is different from $x$ and $y$</p>
<p>(1) $ d(x,y)\geq 0$ ;</p>
<p>(2) $d(x,y)=0$ if and only if $x=y$;</p>
<p>(3) $d(x,y)=d(y,x)$;</p>
<p>(4) $d(x,y)\leq d(x,u)+d(u,v)+d(v,y)$.<... | Lucas Mann | 171,950 | <p>Your idea is correct. There is just one little mistake: You noticed $3 \cdot 3 \equiv 1 \bmod 4$. This does <em>not</em> imply $3^{2n+1} \equiv 1^{2n+1} \bmod 4$, but it tells you that
$$3^{2n+1} = 3 \cdot (3^2)^n \equiv 3 \cdot 1 = 3 \mod{4}$$
which is exactly what you want.</p>
<p>Now for $8$, try to do a similar... |
1,231,082 | <p>Let $\text{tr}A$ be the trace of the matrix $A \in M_n(\mathbb{R})$.</p>
<ul>
<li>I realize that $\text{tr}A: M_n(\mathbb{R}) \to \mathbb{R}$ is obviously linear (but how can I write down a <em>formal</em> proof?). However, I am confused about how I should calculate $\text{dim}(\text{Im(tr)})$ and $\text{dim}(\text... | Christian Blatter | 1,303 | <p>From $2^3=1\ (7)$ we get $2^{105}=1\ (7)$, and from $3^3=-1\ (7)$ we get $3^{105}=-1\ (7)$. It follows that $2^{105}+3^{105}=0\ (7)$.</p>
|
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