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 |
|---|---|---|---|---|
1,671,357 | <p>I'm trying to solve a minimization problem whose purpose is to optimize a matrix whose square is close to another given matrix. But I can't find an effective tool to solve it.</p>
<p>Here is my problem:</p>
<blockquote>
<p>Assume we have an unknown Q with parameter $q11, q12,q14,q21,q22,q23,q32,q33,q34,q41,q43... | Samrat Mukhopadhyay | 83,973 | <p>If $Q\ne \pm G$, then a minima cannot be achieved. You could try to write down $Q$ as $Q=G+\delta$ where $$\delta=\begin{pmatrix}
\delta_1 & -\delta_1\\
\delta_2 &-\delta_2
\end{pmatrix}$$ where the delta's have to be chosen in a way such that they satisfy the constraints, i.e. $$-0.4\le \delta_1,\delta_2\le... |
2,038,520 | <p>I know that the series b. converges as $\sum \frac{1}{n^p}$ converges for $p>1$, So a. also converges. I want to know the sum.</p>
<blockquote>
<blockquote>
<p>a.$1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+.....$</p>
<p>$b.1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+.....$</p>
</blockquote>
</blockquote... | Macavity | 58,320 | <p>If you know <a href="https://en.wikipedia.org/wiki/Basel_problem" rel="nofollow noreferrer">$\displaystyle \sum_1^\infty \frac1{n^2} = \frac{\pi^2}6$</a>, then this is easy, as the even series is just $\displaystyle \sum_1^\infty \frac1{4n^2} = \frac{\pi^2}{24}$.</p>
<p>I leave the odd series for you to separate ou... |
2,137,591 | <blockquote>
<p>$$\int \frac{1}{x+x\log x}\,dx$$</p>
</blockquote>
<p>I couldn't use any of the integration techniques to solve this, any help will be appreciated!</p>
| Community | -1 | <p><strong>"HINT":</strong> (Assuming of course that $\log=\ln$) It is $$\int\frac{1}{x}\frac{1}{1+\ln x}\,dx$$ so you can make the substitution $y=\ln x$.</p>
|
3,710,377 | <p>In <em>Postmodern Analysis</em> by Jurgen Jost, the Lebesgue integral of a step function is defined as follows:</p>
<p>Suppose we have a step function <span class="math-container">$t:W\subset\mathbb{R}^d\to \mathbb{R}$</span> defined on a cube <span class="math-container">$W\subset\mathbb{R}^d$</span> given by
<sp... | Ross Millikan | 1,827 | <p>Hint: Suppose <span class="math-container">$g(x)=\frac x2$</span>. Of course, <span class="math-container">$f(x)=2x$</span> makes <span class="math-container">$f(g(x))=x$</span>, but there are other possibilities. Can you find one?</p>
|
51,509 | <p>Here is a problem due to Feynman. If you take 1 divided by 243 you get 0.004115226337 .... It goes a little cockeyed after 559 when you're carrying out the decimal expansion, but it soon straightens itself out and repreats itself nicely. Now I want to see how many times it repeats itself. Does it do this indefinitel... | Mr.Wizard | 121 | <p>Since belisarius specifically refused to expound on his answer, which arguably would make my editing it for such purpose tantamount to vandalism, I shall post my own.</p>
<p>Regarding <a href="http://reference.wolfram.com/mathematica/ref/RealDigits.html" rel="noreferrer"><code>RealDigits</code></a>:</p>
<blockquot... |
51,509 | <p>Here is a problem due to Feynman. If you take 1 divided by 243 you get 0.004115226337 .... It goes a little cockeyed after 559 when you're carrying out the decimal expansion, but it soon straightens itself out and repreats itself nicely. Now I want to see how many times it repeats itself. Does it do this indefinitel... | Greg Hurst | 4,346 | <p>You can also call <code>WolframAlpha["1/243"]</code>.</p>
|
2,027,337 | <p>My homework sets up the problem accordingly:</p>
<blockquote>
<p>An object moves horizontally in one dimension with a velocity given by
v(t) = $8\cos\left(\frac{\pi \cdot t}{6}\right)$ m/s.</p>
<p>Find the The position of the object is given by s(t) =
$s\left(t\right)=\int _0^t\:v\left(y\right)\:dy\:$ ... | 5xum | 112,884 | <p>Well, the task tells you two things:</p>
<ol>
<li>The velocity, $v(t)$, is given as $v(t)=8\cos\frac{\pi-t}{6}$</li>
<li>The location is $\int_0^t v(y) dy$.</li>
</ol>
<p>From $1$, you know that $$v(t)=8\cos\frac{\pi-t}{6}$$ meaning that $$v(y)=8\cos\frac{\pi-y}{6}$$</p>
<p>From the second, you then get </p>
<p>... |
2,384,422 | <p>I'm really stuck on how to go about solving the following first order ODE; I've got little idea on how to approach it, and I'd really appreciate if someone could give me some hints and/or working for a solution so I can have a reference point on how to approach these sorts of problems.</p>
<p>The following is one o... | Hasek | 395,670 | <p>This kind of ODE should be solved as follows:</p>
<ol>
<li>Solve the corresponding <strong>homogeneous</strong> equation.</li>
</ol>
<p>In your case it is $y'+y\cos(x)=0$ which has solution $y=c\cdot e^{-\sin(x)}$.</p>
<ol start="2">
<li><strong>Consider constant</strong> in previous solution <strong>as a functio... |
1,630,480 | <p>Assume $U\subset\mathbb{R}\times\mathbb{R}^{n}=\mathbb{R}^{n+1}$, $U$ is open and $(t_0, \bf{x}$$_0)\in U$. Assume ${\bf f} (= {\bf f}(t,{\bf x})) : U \to \mathbb{R}$ is <em>continuous</em>. Then the following is called an <em>initial value problem</em>, with <em>initial condition</em>:</p>
<p>\begin{align*}
\frac{... | epi163sqrt | 132,007 | <p>We see in OPs example all $5$ different ways to multiply four matrices according to the associative law. This corresponds to the <em><a href="https://en.wikipedia.org/wiki/Catalan_number" rel="noreferrer">Catalan number</a></em></p>
<p>$$C_3=\frac{1}{4}\binom{6}{3}=5$$.</p>
<p>We write these $5$ variants explicite... |
2,917,535 | <p>I have found this problem in a 10th grade textbook and it's given me headaches trying to solve it. It says, determine the set:</p>
<p>$$ A = \left \{ x \in \mathbb Z| \root3\of{\frac{7x+2}{x+5}} \in \mathbb Z\right \} $$</p>
<p>So I have to find a condition for x so that the expression under the radical is a perfe... | ajotatxe | 132,456 | <p>Hint: for "large" $x$, the radical is between $6$ and $8$</p>
|
2,917,535 | <p>I have found this problem in a 10th grade textbook and it's given me headaches trying to solve it. It says, determine the set:</p>
<p>$$ A = \left \{ x \in \mathbb Z| \root3\of{\frac{7x+2}{x+5}} \in \mathbb Z\right \} $$</p>
<p>So I have to find a condition for x so that the expression under the radical is a perfe... | Sarvesh Ravichandran Iyer | 316,409 | <p>The key point is that if $\sqrt[3]{a}$ is an integer, then $a$ is an integer, because $a$ is the cube of some integer, and the cube of an integer is always an integer.</p>
<p>Therefore, we conclude that $\frac{7x+2}{x+5}$ is an integer. Write this as $\frac{(7x+35) - 33}{x+5} = 7 - \frac{33}{x+5}$.</p>
<p>Now, if ... |
1,115,645 | <p>I understand that a primitive polynomial is a polynomial that generates all elements of an extension field from a base field. However I am not sure how to apply this definition to answer my question. Can someone explain to me how I need to start please?</p>
| Will Brooks | 206,592 | <p>If you are happy to be vulgar, you can simply evaluate the expression at all 7 integer values, and show none of them is $0(7)$</p>
<p>Alternatively, $(x+4)^2 + 1\equiv x^2 + x + 3(7)$ and $-1$ is not a quadratic residue of $7$</p>
|
602,286 | <p>I'm reading a paper which uses the following fact; it appears to be standard but I am not sure where to look for a proof.</p>
<blockquote>
<p><strong>Claim.</strong> Let $M$ be a complete Riemannian manifold (assumed to be second countable, so no long lines). There is an increasing sequence of open sets $U_n$ wi... | Community | -1 | <p>I'll get you started: Factoring the right side, we find that</p>
<p>$$\frac{dy}{dx} = x^2 y + x^2 - (y + 1) = (y + 1)(x^2 - 1)$$</p>
<p>Upon rearrangement,</p>
<p>$$\frac{dy}{y + 1} = (x^2 - 1) dx$$</p>
|
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | Will Jagy | 10,400 | <p>I can only imagine this was intended to be about
$$ (10a + 5)^2 = 100 a (a+1) + 25, $$
$$ 15^2 = 225, $$
$$ 25^2 = 625, $$
$$ 35^2 = 1225, $$
$$ 45^2 = 2025. $$
Then
$$ 44^2 = 2025 - 2 \cdot 45 + 1 = 2025 - 90 + 1 < 2017. $$</p>
<p>EXAMPLE: factor $10001 = 10^4 + 1$</p>
<p>$$ 105^2 = 11025 $$
$$ 105^2 ... |
3,575,334 | <p>I am trying to show that <span class="math-container">$\int_{-b}^{b} \frac{f(N+\frac{1}{2} + it)}{e^{2\pi i(N+\frac{1}{2} + it)}-1} dt \to 0$</span> as <span class="math-container">$N \to \infty$</span> where <span class="math-container">$|f(N+1/2+it)| \le A/(1+(N+1/2)^2)$</span> for some constant <span class="math-... | roundsquare | 706,295 | <p>J. W. Tanner's answer is correct, but just to do it via a Venn diagram:</p>
<p><span class="math-container">$A - C$</span> is region I and region IV</p>
<p><span class="math-container">$C - B$</span> is region III and region V</p>
<p>Since there are no regions in both, <span class="math-container">$(A - C)\cap(C ... |
930,949 | <p>Given that the circle C has center $(a,b)$ where $a$ and $b$ are positive constants and that C touches the $x$-axis and that the line $y=x$ is a tangent to C show that $a = (1 + \sqrt{2})b$</p>
| Dan Christensen | 3,515 | <blockquote>
<p>Why are there so many different definitions of predicate??</p>
</blockquote>
<p>I would be hard to go wrong just memorizing the definition given by your prof. It's enough to know that there are many subtle variations. At this point in your career, it would be a waste of time to try to determined which d... |
2,674,102 | <p>Is the following Proof Correct?</p>
<blockquote>
<p>Given that $T\in\mathcal{L}(\mathbf{R}^2)$ defined by $T(x,y) =
(-3y,x)$. $T$ has no eigenvalues.</p>
</blockquote>
<p><em>Proof.</em> Let $\sigma_T$ denote the set of all eigenvalues of $T$ and assume that $\sigma_T\neq\varnothing$ then for some $\lambda\in\s... | SAHEB PAL | 309,736 | <p><strong>Another approach:</strong></p>
<p>The matrix representation of $T$ w.r.t. standard basis $\{(1,0),(0,1)\}$ of $\mathbb{R}^2$ is $A=\begin{pmatrix}0&-3\\1&0\end{pmatrix}$. So the characteristic equation $|A-\lambda I|=0 $ gives $\lambda^2+3=0$. Thus $T$ has no eigenvalues.</p>
|
1,017,411 | <blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative Ring with <span class="math-container">$1$</span> and <span class="math-container">$M$</span> a <span class="math-container">$R$</span>-Module. <span class="math-container">$$\varphi: \begin{cases}R & \longrightarrow \text{end}_R(M) \\ a &... | Timbuc | 118,527 | <p>I think you're forgetting a tiny thing here: observe $\;\delta\;$ is an $\;R$- homomorphism, and thus</p>
<p>$$\delta(x)=\delta(x\cdot1)=x\cdot\delta(1)\;,\;\;\forall\,x\in R$$</p>
<p>I think this solves the whole conundrum.</p>
|
1,949,966 | <h2>Q 1a</h2>
<p>Is it possible to define a number $x$ such that $|x|=-1$, where $|\cdot|$ means absolute value, in the same manner that we define $i^2=-1$?</p>
<p>I have no idea if it makes sense, but then again, $\sqrt{-1}$ used to not be a thing either.</p>
<p>To be more explicit, I want as many properties to hol... | Arthur | 15,500 | <p>First of all, you can define $|\cdot|$ to mean whatever you want in any given context, as long as you're clear and upfront about it.</p>
<p>That being said, one usually wants $|\cdot|$ to be a <em>norm</em>, which means it fulfills a certain list of criteria. Among them is $|x|\geq 0$. If you break these rules, doe... |
1,706,939 | <p>Can anyone share an easy way to approximate $\log_2(x)$, given $x$ is between $0$ and 1?</p>
<p>I'm trying to solve this using an old fashioned calculator (i.e. no logs)</p>
<p>Thanks!</p>
<p>EDIT: I realize that I stepped a bit ahead. The x comes in the form of a fraction, e.g. 3/8, which is indeed between 0 and... | CiaPan | 152,299 | <p>If your input involves just multiplication or division of small natural numbers and you don't need accuracy exceeding 4 decimal numbers, then logarithmic tables could be the simplest solution.</p>
<p>See for example</p>
<ul>
<li><a href="http://www.rapidtables.com/math/algebra/logarithm/Logarithm_Table.htm" rel="n... |
1,706,939 | <p>Can anyone share an easy way to approximate $\log_2(x)$, given $x$ is between $0$ and 1?</p>
<p>I'm trying to solve this using an old fashioned calculator (i.e. no logs)</p>
<p>Thanks!</p>
<p>EDIT: I realize that I stepped a bit ahead. The x comes in the form of a fraction, e.g. 3/8, which is indeed between 0 and... | N74 | 288,459 | <p>When $x$ is in the range $]0.5, 1[$ you can easily find a binary representation of the $\log_2(x)$ using this algorithm:</p>
<ol>
<li>Start writing $-0.$ (with the dot as the result will be fractional) and evaluate $z=1/x$.</li>
<li>if $z^2>2$ append $1$ and let $z=z/2$, else append $0$, to the result.</li>
<li>... |
1,679,920 | <p>I'm working for a firm, who can only use straight lines and (parts of) circles.</p>
<p>Now I would like to do the following: imagine a square of size $5\times5$. I would like to expand it with $2$ in the $x$-direction and $1$ in the $y$-direction. The expected result is a rectangle of size $7\times9$. Until here, e... | Paul H. | 278,067 | <p>Here is an article that I used to implement a routine for converting Bézier curves to tangent arcs that may help you. It includes C++ code.
<a href="http://www.ryanjuckett.com/programming/biarc-interpolation/" rel="nofollow">Biarc Interpolation</a></p>
<p>If this doesn't qualify as an answer on this site, maybe som... |
242,636 | <p>I am interested in the proof of the following result:
Suppose that $A > 1$, $\lambda \in \mathbb{R}$, and for $0 < Z \leq 1$, let $U(Z)$ be the number of integer solutions $v$ of
\begin{eqnarray}
|v| < ZA \ \ \ \text{ and } \ \ \ \| \lambda v \| < Z A^{-1}.
\end{eqnarray}
Then, if $0 < Z_1 < Z... | user94168 | 94,168 | <p>Is it a reference you want? Check chapter 12 (if I remember correctly) in Davenport's book on diophantine equations and inequalities.</p>
|
242,636 | <p>I am interested in the proof of the following result:
Suppose that $A > 1$, $\lambda \in \mathbb{R}$, and for $0 < Z \leq 1$, let $U(Z)$ be the number of integer solutions $v$ of
\begin{eqnarray}
|v| < ZA \ \ \ \text{ and } \ \ \ \| \lambda v \| < Z A^{-1}.
\end{eqnarray}
Then, if $0 < Z_1 < Z... | js21 | 21,724 | <p>I can show the (exact) inequality
$$(*) \ \ \ \ \ \ V(Z_1) \geq \left(\frac{Z_1}{Z_2} \right)^2 V(Z_2) \ \ \quad (\frac{2}{A} \leq Z_1 \leq Z_2 \leq \frac{A}{2}), $$
for a smoothed version of $U$ defined by
$$
V(Z) = \sum_{\nu \in \mathbb{Z}} \mathrm{sinc}^2\left( \frac{\nu}{2ZA} \right) \left( 1 - \frac{A ||\lambd... |
1,940,448 | <p>I am stuck on this question. Could someone help me?</p>
<p>$$ \text{Find value of } S = \displaystyle\sum_{n=0}^{\infty} \cfrac{1}{n!(n+2)} $$</p>
<p>I am supposed to show that $ S = 1 $ in two ways: <br /><br />
1) Integrate the taylor series of $ xe^x $ <br />
2) Differentiate the taylor series of $ \frac{e^{x-1... | Carl Schildkraut | 253,966 | <p>For (1), you need to be careful with using $x$ as your dummy variable: try</p>
<p>$$\int_0^t xe^x\ dx = \int_0^t \sum_{n=0}^{\infty} \frac{x^{n+1}}{n!}\ dx$$</p>
<p>$$ = \sum_{n=0}^{\infty} \int_0^t\frac{x^{n+1}}{n!}\ dx$$</p>
<p>$$ = \sum_{n=0}^{\infty} \frac{x^{n+2}}{n!(n+2)}\bigg|_0^t$$</p>
<p>$$ = \sum_{n=0}... |
1,940,448 | <p>I am stuck on this question. Could someone help me?</p>
<p>$$ \text{Find value of } S = \displaystyle\sum_{n=0}^{\infty} \cfrac{1}{n!(n+2)} $$</p>
<p>I am supposed to show that $ S = 1 $ in two ways: <br /><br />
1) Integrate the taylor series of $ xe^x $ <br />
2) Differentiate the taylor series of $ \frac{e^{x-1... | Khosrotash | 104,171 | <p>Another way $$S = \sum_{n=0}^{\infty} \cfrac{1}{n!(n+2)}\\S = \displaystyle\sum_{n=0}^{\infty} \cfrac{1}{n!(n+2)}\times \frac{n+1}{n+1}\\=\sum_{n=0}^{\infty} \cfrac{n+1}{(n+2)!}\\=
\sum_{n=0}^{\infty} \cfrac{(n+2)-1}{(n+2)!}\\=
\sum_{n=0}^{\infty} \cfrac{n+2}{(n+2)!}-\cfrac{1}{(n+2)!}\\=
\sum_{n=0}^{\infty} \cfrac{1... |
4,263,629 | <blockquote>
<p>Let <span class="math-container">$A=\{(x,y) \in \Bbb R^2 \mid x \ge 1, 0<y<\frac{1}{x^2}\}$</span>. Show that <span class="math-container">$m_2(A) < \infty$</span> where <span class="math-container">$m$</span> is the Lebesgue measure.</p>
</blockquote>
<p>I now that the integral <span class="ma... | Mark | 470,733 | <p>You are close. Measure of a set is equal to the Lebesgue integral of the constant function <span class="math-container">$1$</span> on that set. So the measure of your set is equal to the double integral <span class="math-container">$\iint\limits_A 1 dxdy$</span>. Since the function has an absolutely convergent impro... |
1,855,824 | <blockquote>
<p>Given $a_1=1$ and $a_n=a_{n-1}+4$ where $n\geq2$ calculate,
$$\lim_{n\to \infty }\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\cdots+\frac{1}{a_na_{n-1}}$$</p>
</blockquote>
<p>First I calculated few terms $a_1=1$, $a_2=5$, $a_3=9,a_4=13$ etc. So
$$\lim_{n\to \infty }\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\cdots+... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\dfrac4{a_ma_{m-1}}=\dfrac{a_m-a_{m-1}}{a_ma_{m-1}}=?$$</p>
<p>$a_m=1+4\cdot(m-1)=?$</p>
<p>Do you recognize the <a href="https://en.wikipedia.org/wiki/Telescoping_series">Telescoping series</a>?</p>
|
3,725,385 | <p><span class="math-container">$\{(x, y, z)\} \space$</span> with <span class="math-container">$\space x + y + z = 0$</span></p>
<p>Working through some problems in a textbook and I'm not very confident about checking if subsets are subspaces. I know that for a subset to be a subspace of <span class="math-container">$... | Maryam | 626,408 | <p>It is because the subset is the inverse image of the subspace <span class="math-container">$\{0\}$</span> under the linear map <span class="math-container">$(x,y,z)\mapsto x+y+z$</span>.</p>
|
3,245,796 | <p>Let <span class="math-container">$f$</span> have a continuous second derivative. Prove that</p>
<p><span class="math-container">$$f(x) = f(a) + (x - a)f'(a) + \int_a^x(x - t)f''(t) dt.$$</span></p>
<p>This is a modification of exercise 6.6.4 from Advanced Calculus by Fitzpatrick. I have seen that this question has... | peek-a-boo | 568,204 | <p>Let <span class="math-container">$u(t) = f'(t)$</span> and let <span class="math-container">$v(t) = t-x$</span> (don't be confused by the fact that there's an <span class="math-container">$x$</span> in the definition of <span class="math-container">$v(t)$</span>; right now we are keeping <span class="math-container"... |
833,143 | <p>Wolfram alpha solves $\sqrt{x+1}\ge\sqrt{x+2}+\sqrt{x+3}$ for $x$, and answers $x=-2/3(3+\sqrt{3})$. How did it do it? Thanks!</p>
| Hagen von Eitzen | 39,174 | <p>We need $x\ge -1$ in order for all roots to be defined. Then the right hand side is positive, hence in lets divide by the positive number $\sqrt{x+1}$ (and note that $x+1>0$):
$$ 1\ge\sqrt{1+\frac1{1+x}}+\sqrt {1+\frac{2}{1+x}}\ge 2$$
contradiction!</p>
|
1,685 | <p>Are there books or article that develop (or sketch the main points) of Euclidean geometry without fudging the hard parts such as angle measure, but might at times use coordinates, calculus or other means so as to maintain rigor or avoid the detail involved in Hilbert-type axiomatizations?</p>
<p>I am aware of Hilbe... | Robin Chapman | 226 | <p>You might look at Hartshorne's
<a href="http://books.google.co.uk/books?id=EJCSL9S6la0C&lpg=PP1&dq=hartshorne&pg=PP1#v=onepage&q&f=false" rel="noreferrer"><em>Geometry: Euclid and Beyond</em></a>.</p>
|
1,685 | <p>Are there books or article that develop (or sketch the main points) of Euclidean geometry without fudging the hard parts such as angle measure, but might at times use coordinates, calculus or other means so as to maintain rigor or avoid the detail involved in Hilbert-type axiomatizations?</p>
<p>I am aware of Hilbe... | Anton Petrunin | 12,434 | <p>Try <a href="http://www.jstor.org/discover/10.2307/1968336?uid=3739864&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=21102247102467" rel="nofollow"><em>A Set of Postulates for Plane Geometry</em> by Birkhoff</a>.</p>
|
2,728,248 | <blockquote>
<p>Let $K=\mathbb{Q}(\sqrt{-2})$. Show that $\mathcal{O}(K)$ is a principal ideal domain. Deduce that every prime $p\equiv 1, 3$ (mod 8) can be written as $p = x^2 + 2y^2$ with $x, y \in \mathbb{Z}$.</p>
</blockquote>
<p>As $−2$ is squarefree $6\equiv 1$ (mod 4) we have $\mathcal{O}(K) = \mathbb{Z}[
\s... | Mathmo123 | 154,802 | <p>The theorem you're after is the Kummer-Dedekind theorem:</p>
<blockquote>
<p><strong>Theorem</strong>: Let $p$ be a prime, and let $\beta\in \mathcal O_K$ be such that $K=\mathbb Q(\beta)$ and $p\nmid (\mathcal O_K:\mathbb Z[\beta])$. Let $f(X)$ be the minimal polynomial of $\beta$ over $\mathbb Q$. Suppose that... |
2,065,254 | <p>Let $f: \mathbb{R} \to \mathbb{R}$ be a function that is twice differentiable.</p>
<p>We know that:
$$\lim_{x\to-\infty}\ f(x) = 1$$</p>
<p>$$\lim_{x\to\infty}\ f(x) = 0$$</p>
<p>$$f(0) = \pi$$</p>
<p>We have to prove that there exist at least two points of the function in which $f''(x) = 0$.</p>
<p>How could w... | Doug M | 317,162 | <p>since f(0) > 1 $\lim_\limits {x\to -\infty} f(x)$ and $\lim_\limits {x\to \infty} f(x)$ and $f(x)$ is not monotonic.</p>
<p>Since $f(x)$ is continuous and differentiable, there it takes on a maximum value somewhere. $f'(c) = 0$</p>
<p>There exists an $x$ on each tail such that f'(x) = 0$</p>
<p>By the mean val... |
4,498,199 | <p>Exercise 1.2.1(vii) from Page 5 of Keith Devlin's "The Joy of Sets":</p>
<blockquote>
<p>Prove the following assertion directly from the definitions. The
drawing of "Venn diagrams" is forbidden; this is an exercise in the
manipulation of logical formalisms.
<span class="math-container">$$(x\subse... | Dan Velleman | 414,884 | <p>You could try doing these proofs using Proof Designer:</p>
<p><a href="https://djvelleman.people.amherst.edu/pd.html" rel="nofollow noreferrer">https://djvelleman.people.amherst.edu/pd.html</a></p>
<p>I think the result would be the kind of proof that Devlin has in mind.</p>
|
2,134,653 | <blockquote>
<p>There is a Vessel holding 40 litres of milk. 4 litres of Milk is initially taken out from the Vessel and 4 litres of water is then poured in. After this 5 litres of mixtures of Mixture is replaced with six litres of water and finally six litres of Mixture is Replaced with the six litres of water. How ... | Peter Phipps | 15,984 | <p>Initial congifuration is $40$ litres of milk and no water, say $m=40$ and $w=0$.</p>
<p>Remove $4$ litres of milk and add $4$ litres of water: $m=36, w=4$.</p>
<p>The mixture is $36:4$ or $9:1$. $5$ litres of mixture contains $5\times \frac{9}{10}=\frac 92$ litres of milk and $5\times \frac{1}{10}=\frac 12$ litres... |
1,795,836 | <p>Let's say that $A \subset X$ is a deformation retract. It follows that $A$ is both a retract and a space homotopically equivalent to $X$. Is the converse true? Probably not, but I couldn't find any example yet.</p>
<p>More specifically the converse would be:</p>
<p>If $A \subset X$ is a retract which is homotopic ... | Stefan Hamcke | 41,672 | <p>No. Let $X = \{0,1,2,3,\dots\}$ and $A = \{1,2,3,\dots\}$, both with the discrete topology, and let $i: A \to X$ be the inclusion. Then $i$ has a retraction $r: X \to A, n\mapsto\max\{n,1\}$, and is even a cofibration. $X$ and $A$ are clearly isomorphic. The inclusion, however, is not a homotopy equivalence.</p>
|
39,476 | <p>Fold is an extension of Nest for 2 arguments. How does one extend this concept to multiple arguments. Here is a trivial example:</p>
<pre><code>FoldList[#1 (1 + #2) &, 1000, {.01, .02, .03}]
</code></pre>
<p>Say I want do something like:</p>
<pre><code>FoldList[#1(1+#2)-#3&,1000,{.01,.02,.03},{100,200,3... | Leonid Shifrin | 81 | <p>Yes, there is. Group your extra arguments in a list, and address them by their positions in the function under <code>Fold</code>. For your particular example:</p>
<pre><code>FoldList[#1 (1 + First@#2) - Last@#2 &, 1000, Transpose@{{.01, .02, .03}, {100, 200, 300}}]
(* {1000, 910., 728.2, 450.046} *)
</code></p... |
21,752 | <blockquote>
<p>"Let $P$ be the change-of-basis matrix
from a basis $S$ to a basis $S'$ in a
vector space $V$. Then, for any vector
$v \in V$, we have $$P[v]_{S'}=[v]_{S}
\text{ and hence, } P^{-1}[v]_{S} =
[v]_{S'}$$</p>
<p>Namely, if we multiply the coordinates
of $v$ in the original b... | Qiaochu Yuan | 232 | <p>The situation here is closely related to the following situation: say you have some real function $f(x)$ and you want to shift its graph to the right by a positive constant $a$. Then the correct thing to do to the function is to shift $x$ over to the <em>left</em>; that is, the new function is $f(x - a)$. In essence... |
21,752 | <blockquote>
<p>"Let $P$ be the change-of-basis matrix
from a basis $S$ to a basis $S'$ in a
vector space $V$. Then, for any vector
$v \in V$, we have $$P[v]_{S'}=[v]_{S}
\text{ and hence, } P^{-1}[v]_{S} =
[v]_{S'}$$</p>
<p>Namely, if we multiply the coordinates
of $v$ in the original b... | Nick Alger | 3,060 | <p>One major reason is practical. The matrix that converts vectors in the new coordinates into the old coordinates is easy to come by: you just put your new basis vectors as columns of the matrix.</p>
<p>Then to find the matrix going the other way around, you have to compute the inverse of this matrix.</p>
<p>Thus, i... |
1,651,991 | <p>Let $p(x)$ be an odd degree polynomial and let $q(x)=(p(x))^2+ 2p(x)-2$ </p>
<p>a) The equation $q(x)=p(x)$ admits atleast two distinct real solutions.</p>
<p>b) The equation $q(x)=0$ admits atleast two distinct real solutions.</p>
<p>c) The equation $p(x)q(x)=4$ admits atleast two distinct real solutions.</p>
<... | MPW | 113,214 | <p><strong>Hint:</strong> First, determine the slope $m$ of the line through the given points.</p>
<p>Then, pick one of the points and determine how much $x$ changes when moving from that point to your new value of $x$. We write this as $\Delta x = x_{new}-x_{old}$.</p>
<p>Use the fact that $m = \frac{\Delta y}{\Delt... |
2,418,440 | <p><strong>Defn:</strong> Let $f$ be a function from $\mathbb{R}$ into a set $X$. We say that $f$ is <em>periodic</em> if there exists $p>0$ such that for all $x\in \mathbb{R}$, we have $f(x+p)=f(x)$. </p>
<p><strong>Prove</strong>: If $f$ is a continuous periodic function from $\mathbb{R}$ into a metric space $M$,... | orangeskid | 168,051 | <p>HINT:</p>
<p>your idea is to translate each point to the interval $[0,p]$ and hope that the translates are also close by. Better to translate both by the same multiple of p, so that the smaller one lands inside, and the larger one is still guaranteed not too far. So you consider the restriction of your function to ... |
1,731,978 | <p>For two complex numbers $z_1$ and $z_2$, it is given that: </p>
<blockquote>
<p>$$|z_1+z_2|>|z_1-z_2|$$</p>
</blockquote>
<p>How could we prove that $-\frac{\pi}{2}<arg\big(\frac{z_1}{z_2}\big)<\frac{\pi}{2}$</p>
<p>If I take $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ I get $x_1x_2+i y_1y_2=0$ but it does help... | Arthur | 15,500 | <p>Hint 1: $|z_1 + z_2| > |z_1 - z_2|$ means that $z_1$ and $z_2$ are closer to one another in the complex plane than $z_1$ and $-z_2$ are.</p>
<p>Hint 2: What is the geometric significance of $\operatorname{arg}(z_1/z_2)$?</p>
|
1,731,978 | <p>For two complex numbers $z_1$ and $z_2$, it is given that: </p>
<blockquote>
<p>$$|z_1+z_2|>|z_1-z_2|$$</p>
</blockquote>
<p>How could we prove that $-\frac{\pi}{2}<arg\big(\frac{z_1}{z_2}\big)<\frac{\pi}{2}$</p>
<p>If I take $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ I get $x_1x_2+i y_1y_2=0$ but it does help... | K.K.McDonald | 302,349 | <p>factor $z_2$ in each side and get:
$$|z_1+z_2|>|z_1-z_2|=|z_2(\frac{z_1}{z_2}+1)|>|z_2(\frac{z_1}{z_2}-1)|=|z_2|\frac{z_1}{z_2}+1|>|z_2||\frac{z_1}{z_2}+1|=|\frac{z_1}{z_2}+1|>|\frac{z_1}{z_2}-1|$$
this statement means that $|\text{Real}\{\frac{z_1}{z_2}+1\}|>|\text{Real}\{\frac{z_1}{z_2}-1\}|$ which ... |
1,673,854 | <p>Taylor Series of $f(x) = \sqrt{x}$ about $c = 1$</p>
<p>I've tried doing this problem but stuck at finding a pattern..</p>
<p>Work:</p>
<p>$$T_n = \sum^\infty_{n=0}\frac{f^n(c)}{n!}(x-c)^n = f(a) + \frac{f'(c)}{1!}(x-a)^1 + \frac{f''(c)}{2!}(x-a)^2+... $$</p>
<p>So $f(x) = \sqrt{x}$</p>
<p>$$ f'(x)=\frac12x^{-\... | Domenico Vuono | 227,073 | <p>Hint: note that if the series $$\sum_{n=1}^{+ \infty}\frac{(2n-1)!}{(2n+1)!}$$ converges $$\lim_{n\to \infty}\frac{(2n-1)!}{(2n+1)!}=0$$ you can use the ratio test to show that the series converges.</p>
|
198,945 | <p>Why and how publishing a paper in proceedings?<br>
What are the difference with a "classical" journal?<br>
What's the list of the main proceedings in which one can publish?<br>
Do proceedings papers (never, sometimes, often or always) appear on mathscinet?</p>
| Pace Nielsen | 3,199 | <p>I agree completely with Andreas' answer. One further consideration is publicity. It is easy for papers published in conference proceedings to become lost to general knowledge, or known only to very specialized groups. By publishing in a regular and reputable journal, the chances others will read your paper goes u... |
2,281,894 | <blockquote>
<p>The Hardy space <span class="math-container">$H^2(\mathbb{D})$</span> is defined to be the space of all functions <span class="math-container">$f$</span> >holomorphic on the unit disk <span class="math-container">$\mathbb{D}$</span> with the norm <span class="math-container">$\lVert \cdot \rVert_H$</... | Fred | 380,717 | <p>Inner product: $(f|g)=\sup_{0<r<1}\int_0^{2\pi}f(re^{i\theta}) \overline{g(re^{i\theta})}d\theta$</p>
|
3,621,223 | <p>I use a software called Substance Designer which has a Pixel Processor where I can assign to every pixel of a image a gray-scale value defined by a series of operations.</p>
<p>I am basically trying to generate a <a href="https://i.stack.imgur.com/6jhYa.jpg" rel="nofollow noreferrer">"normal gradient"</a> generated... | David G. Stork | 210,401 | <p>This figure may help those who wish to solve this problem:</p>
<p><a href="https://i.stack.imgur.com/hIeFH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hIeFH.png" alt="enter image description here"></a></p>
<p>By the way, here are three origin-centered ellipses with the same eccentricity <spa... |
2,325,436 | <p>I was reading <em>Introduction to quantum mechanics</em> by David J. Griffiths and came across following paragraph:</p>
<blockquote>
<p><span class="math-container">$3$</span>. The eigenvectors of a hermitian transformation span the space.</p>
<p>As we have seen, this is equivalent to the statement that any hermitia... | Cameron Williams | 22,551 | <p>Turning my comments into an answer and adding details since this is an important question in mathematical physics:</p>
<p>Many potentials will cause serious issues regarding completeness of eigenfunctions of Schrodinger operators. Really, we don't <em>need</em> completeness of the eigenfunctions, though it is desir... |
3,460,843 | <p>I understand that the way to calculate the cube root of <span class="math-container">$i$</span> is to use Euler's formula and divide <span class="math-container">$\frac{\pi}{2}$</span> by <span class="math-container">$3$</span> and find <span class="math-container">$\frac{\pi}{6}$</span> on the complex plane; howeve... | Canardini | 341,007 | <p><span class="math-container">$$\int{\frac{-3dx}{x^2+4}}=-3\int{\frac{dx}{x^2+4}}=-3\int{\frac{dx}{4\left(\frac{x^2}{4}+1\right)}}$$</span></p>
|
95,741 | <p>I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to $\mathbb R$. But I am not ok with this answer. I need a simple way to explain the differences between mapping and func... | Arimakat | 151,617 | <p><a href="http://books.google.rs/books/about/Introduction_to_Smooth_Manifolds.html?id=xygVcKGPsNwC&redir_esc=y">John M. Lee, Introduction to Smooth Manifolds, 2002</a>:</p>
<p>Although the terms <strong>function</strong> and <strong>map</strong> are technically synonymous, in studying
smooth manifolds it is ofte... |
4,329,888 | <p>I have a problem of understanding how to find shaded regions in Complex Plane.</p>
<p><span class="math-container">\begin{array}{l}
|z-2i|\ \geqslant \ |z+6+4i|\\
\\
\sqrt{x^{2} +( y-2)^{2}} =\sqrt{( x+6)^{2} +( y+4)^{2}}\\
x^{2} +( y-2)^{2} =( x+6)^{2} +( y+4)^{2}\\
x^{2} +y^{2} -4y+4=x^{2} +12x+36+y^{2} +8y+16\\
1... | JMP | 210,189 | <p>Start from</p>
<p><span class="math-container">$$\sin\pi z = \sum_{n=0}^\infty \frac{(-1)^n\pi^{2n+1}}{(2n+1)!}z^{2n+1}$$</span></p>
<p>and</p>
<p><span class="math-container">$$-1 = \cos\pi = \sum_{n=0}^\infty \frac{(-1)^n\pi^{2n}}{(2n)!}$$</span></p>
<p>Consider</p>
<p><span class="math-container">$$\sin\pi z \cos... |
3,987,718 | <p>Let <span class="math-container">$L \in \mathbb{R}$</span> and let <span class="math-container">$f$</span> be a function that is differentiable on a deleted neighborhood of <span class="math-container">$x_{0} \in \mathbb{R}$</span> such that <span class="math-container">$\lim_{x \to x_{0}}f'(x)=L$</span>.</p>
<p>Fin... | Wuestenfux | 417,848 | <p>Take the function <span class="math-container">$f(x) = |x-x_0|$</span>.</p>
<p><span class="math-container">$\lim_{x\rightarrow x_0^-} \frac{|(x_0-h)+x_0|}{-h} = \frac{|h|}{-h}$</span> and</p>
<p><span class="math-container">$\lim_{x\rightarrow x_0^+} \frac{|(x_0+h)+x_0|}{h} = \frac{|h|}{h}$</span>.</p>
|
2,662,033 | <p>My question is how to prove that $(X,d)$ is complete if and only if $(X,d')$ is complete.</p>
<p>I have that $d$ and $d'$ are strongly equivalent metrics and I have used this to show that a sequence $x_{n}$ is Cauchy in $(X,d)$ if and only if it is Cauchy in $(X,d')$.</p>
<p>I have the definition of complete as:
"... | user284331 | 284,331 | <p>So $\alpha d'(x,y)\leq d(x,y)\leq\beta d'(x,y)$ for some $\alpha,\beta>0$.</p>
<p>Given $d(x_{n},x_{m})\rightarrow 0$ and $d'$ is complete, then $\alpha d'(x_{n},x_{m})\rightarrow 0$, so $d'(x_{n},x_{m})\rightarrow 0$, then for some $x\in X$, $d'(x_{n},x)\rightarrow 0$, and hence $\dfrac{1}{\beta}d(x_{n},x)\righ... |
270,641 | <p>I want to find the inverse triple Laplace transform of <span class="math-container">$L^{-1}_{x_{3}} L^{-1}_{x_{2}} L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}} \right]$</span>. I did
<span class="math-container">\begin{align*}
L^{-1}_{x_{3}} L^{-1}_{x_{2}} L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s... | Abdulhameed Qahtan Abbood Alta | 87,352 | <p>I have another way to solve the problem by using the Taylor series of three variables and I am wondering whether it is correct or not:
<span class="math-container">\begin{align*}
L^{-1}_{x_{3}} L^{-1}_{x_{2}} L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}} \right] &= L^{-1}_{x_{3}} \left[ L^{-1}_{x... |
654,968 | <p>Let $g(x)=x+6$ and $h(x)=\frac{4}{x}$. Compute $\displaystyle\left(\frac{h}{g}\right)(5)$.</p>
<p>I've plugged $5$ in for $x$ but I keep coming up with $.07$ and thanks to webassign I know that is wrong. I'm sure I'm missing something basic but what is it?</p>
| TZakrevskiy | 77,314 | <p>The space is a countable union of balls centered in zero:
$$A = \bigcup_{n\in N}B(0,n).$$</p>
<p>The image of $B(0,n)$ is precompact, therefore, separable. Countable union of separable sets is separable.</p>
|
203,378 | <p>I am trying to solve this equation where I need the solution of K in term of v</p>
<pre><code> Solve[1 -
K - (54 (20 -
K) v (2 (-10 +
K) (5 (1300 - 10 K + 3 K^2 - 100 (2 + K)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
3 + (Sqrt[3] (-10 + K))/Sqrt[... | Mariusz Iwaniuk | 26,828 | <p>Massages from <code>Solve</code> says: <a href="https://en.wikipedia.org/wiki/Transcendental_equation" rel="nofollow noreferrer">transcendental equation</a> can't be solved
analytically,but we can plot solution of function <code>k[v]</code>.</p>
<pre><code>eq = 1 - k - (108 (20 - k) (-20 + k)^3 v Hypergeometric2F... |
4,084,517 | <p>In geometry of 2D and 3D, it's not uncommon for people to call a square or rectangle a <code>Box</code> in the field I work in. This makes naming things easier since it's clear what's in a folder of 'boxes'.</p>
<p>Does the similar name exist for a circle and a sphere? Interestingly we have circles in 3D that could ... | RobertTheTutor | 883,326 | <p>We often refer to an "<span class="math-container">$n$</span>-sphere", usually to say <span class="math-container">$S^{1}$</span> is a circle and <span class="math-container">$S^2$</span> is the surface of an ordinary sphere. An <span class="math-container">$n$</span>-sphere sits in <span class="math-cont... |
2,285,299 | <p>For $ c> b>a>0 $
Is this inequality true?
$$ c^2+ab> ac+bc $$</p>
<p>If yes can anybody please provide hint so I can solve it? </p>
| Gautam Shenoy | 35,983 | <p>Hint: </p>
<p>$$(c-b)c > (c-b)a$$</p>
<p>This is because $c-b > 0$ and $c > a$</p>
|
3,841,806 | <p>Using spherical coordinates I have to find the volume of a cone <span class="math-container">$z=\sqrt{x^2+y^2}$</span> inscribed in a sphere <span class="math-container">$(x-1)^2+y^2+z^2=4.$</span></p>
<p>I can`t find <span class="math-container">$\rho$</span> because the center of sphere is displaced from the origi... | Narasimham | 95,860 | <p>HINT</p>
<p>Eliminating <span class="math-container">$z$</span> between the two given equations and simplifying one obtains</p>
<p><span class="math-container">$$ x^2-x+\frac12+y^2-2=0 $$</span></p>
<p>the cylinder radius <span class="math-container">$ R=\frac{\sqrt 7}{2}$</span></p>
<p><span class="math-container">... |
1,534,694 | <p>I tried to solve for the following limit: </p>
<p>$$\lim_{x\rightarrow \infty} (e^{2x}+x)^{1/x}$$
and I reached to the indeterminate form:
$${4e^{2x}}\over {4e^{2x}}$$
if I plug in, I will get another indeterminate form! </p>
| Thomas Andrews | 7,933 | <p>Tricky solution:
$$\begin{align}
(e^{2x}+x)^{1/x}&=e^{2x/x}\left(1+\frac{1}{e^{2x}/x}\right)^{1/x}\\
&=e^2\left(\left(1+\frac{1}{e^{2x}/x}\right)^{e^{2x}/x}\right)^{e^{-2x}}
\end{align}$$</p>
<p>Since $e^{2x}/x\to\infty$ and $(1+1/y)^y\to e$ as $y\to\infty$, and $e^{-2x}\to 0$, you get the limit is $e^2$.</... |
2,250,339 | <p>The Mad Hatter sets up what he believes is a zero-knowledge protocol. The integer n is the product of two large primes p and q and he wants to prove to the March Hare that he knows the factorization of n without revealing to anyone the actual factors $p$, $q$. He devises the following procedure: </p>
<p>March Hare ... | hmakholm left over Monica | 14,366 | <p>Unless $x$ happens to be a multiple of $p$ or $q$, the number $y=x^2$ has four square roots modulo $pq$, namely the solutions for $z$ of the Chinese remainder systems
$$ z \equiv \pm x \pmod p \\ z \equiv \pm x \pmod q $$
for each of the four combinations of $\pm$.</p>
<p>If the March Hare chooses $x$s at random, t... |
194,671 | <p>I'm searching for two symbols - considering they exist - (1) unknown value; (2) unknown probability.</p>
<p><strong>Note</strong>: I thought that $x$ was used in a temporary context, whenever I see it, it remains unknown until an evaluation is made. I was thinking in a "unknown and impossible to be known" context. ... | Seyhmus Güngören | 29,940 | <p>As the others already mentioned. There are conventional variables to the unknowns such as $x$ for an unknown value or $P(A)$ for an unknown probability. We take them as variables and assume that they are unknown however there is a certain probability that they take some reasonable values. That is actually the reason... |
194,671 | <p>I'm searching for two symbols - considering they exist - (1) unknown value; (2) unknown probability.</p>
<p><strong>Note</strong>: I thought that $x$ was used in a temporary context, whenever I see it, it remains unknown until an evaluation is made. I was thinking in a "unknown and impossible to be known" context. ... | Vlad Atanasiu | 157,761 | <p>You might want to borrow notations from outside mathematics if there is none which suits you. Here are some suggestions: A practical notation for the unknown is the single or triple <strong>interrogation</strong> mark "<code>?</code>" / "???", as you might use in a table or a database. Sometimes the <strong>asterisk... |
3,637,785 | <p>Prove that
<span class="math-container">$${2n \choose n} 2^{-2n} = (-1)^n {-\frac12 \choose n},$$</span></p>
<p><span class="math-container">$$\frac{1}n {2n -2 \choose n-1} 2^{-2n +1} = (-1)^{n-1} {\frac12 \choose n}.$$</span></p>
<p>The second part can be proved by replacing <span class="math-container">$n$</spa... | jamie | 737,935 | <p>I was struggling to see how the last digit of <span class="math-container">$b^2$</span> must be 0. I believe the answer is that it is not necessary, it could be 0 or 5. It can be seen that the last digits of <span class="math-container">$a^2,b^2$</span> must be in 0,1,4,9,6,5 , so cannot be 2 or 8,</p>
<p><span cla... |
3,684,799 | <p>When the theory of groups is built up from its axioms, it is often necessary to establish very simple results such as</p>
<p><span class="math-container">$ax = xa \Longrightarrow a^{-1}x = xa^{-1}. \tag 1$</span></p>
<p>Thus we ask how the title question might be proved.</p>
| Martin Argerami | 22,857 | <p>Your argument is correct, but showing that the spectrum is <span class="math-container">$\{0\}$</span> does not in general imply that <span class="math-container">$T=0$</span>; it does, though, when <span class="math-container">$T$</span> is selfadjoint but you need to include that argument. </p>
<p>The two usual w... |
232,424 | <p>Are there any claims and counterclaims to mathematics being in some certain cases a result of common sense thinking? Or can some mathematical results be figured out using just pure common sense i.e. no mathematical methods? </p>
<p>I'd also appreciate any mentions relating to sciences, social sciences or ordinary l... | Martin Argerami | 22,857 | <p>There is this saying among mathematicians, that you don't really understand something until it becomes obviously trivial. So, in that sense, all of mathematics is "common sense thinking". </p>
|
2,321,667 | <p>Patrick Suppes in his book <a href="http://rads.stackoverflow.com/amzn/click/0486406873" rel="nofollow noreferrer">Introduction to Logic</a> on page 63 asks a reader to proof a statement
$$\forall x\forall y\forall z(xPy\land yPz\to xPz)$$ from the theory which he calls "Theory of rational behavior". The statement i... | Bram28 | 256,001 | <p>Do a proof by contradiction, i.e. continue with:</p>
<p>$$\begin{array}{p}
\{9\}&(9)&\neg xPz&\text{Assumption} \\
\{3,9\}&(2)&zQx&\text{from (3)(9)}\\
\{1,2,3,4,9\}&(10)&zQy&\text{from (1)(6)(10)}\\
\{3,4\}&(11) & \neg zQy & \text{from (5)} \\
\{1,2,3,4,9\}&(12) ... |
854,438 | <p>I've read in a lot of places how there was a "foundational crisis" in defining the "foundations of mathematics" in the 20th century. Now, I understand that mathematics was very different then, I suppose the ideas of Church, Godel, and Turing were either in their infancy, or not well-known, but I still hear this kind... | Trevor Wilson | 39,378 | <p>The question is broad, so I'll just try to address one of the sub-questions. I'll change it a little bit to allow for the possibility of pluralism:</p>
<blockquote>
<p>What is the meaning of finding a "foundation of mathematics”?</p>
</blockquote>
<p>As an example, I'll try to explain why set theory is a founda... |
4,249,794 | <p>i have to determine whether this graph is bipartite or not:</p>
<p><img src="https://i.stack.imgur.com/AjxQzl.png" alt="" /></p>
<p>I have found an answer but i am not sure about it. If we divide the vertices set into <span class="math-container">$\{a,d,c,h\}$</span> and <span class="math-container">$\{b,f,e,g\}$</s... | David G. Stork | 210,401 | <p>If you want to test large graphs, use software, such as <em>Mathematica</em>:</p>
<pre><code>myGraph =
Graph[{a \[UndirectedEdge] b, a \[UndirectedEdge] e,
a \[UndirectedEdge] f, b \[UndirectedEdge] h,
c \[UndirectedEdge] f, c \[UndirectedEdge] g,
d \[UndirectedEdge] e, d \[UndirectedEdge] f,
d \[U... |
2,089,502 | <blockquote>
<p>How many numbers are there from $1$ to $1400$ which maintain these conditions:
when divided by $5$ the remainder is $3$ and when divided by $7$ the remainder is $2$?</p>
</blockquote>
<p>How can I start? I am newbie in modular arithmetics. I can just figure out that the number $= 5k_1+3 = 7k_2+2$. ... | Glitch | 74,045 | <p>Assume initially that $x \le y$. From the FTC we have
$$
\cos(y) - \cos(x) = \int_x^y \cos'(z) dz = -\int_x^y \sin(z) dz
$$
and so
$$
|\cos(y) - \cos(x)| \le \left\vert -\int_x^y \sin(z) dz \right\vert \le \int_x^y |\sin(z)| dz \le \int_x^y dz = y-x = |y-x|.
$$
If $y \le x$ a similar argument works, which I'll lea... |
2,089,502 | <blockquote>
<p>How many numbers are there from $1$ to $1400$ which maintain these conditions:
when divided by $5$ the remainder is $3$ and when divided by $7$ the remainder is $2$?</p>
</blockquote>
<p>How can I start? I am newbie in modular arithmetics. I can just figure out that the number $= 5k_1+3 = 7k_2+2$. ... | Fernando Revilla | 401,424 | <p>$$\left|\cos x -\cos y\right|=\left|−2\sin\frac{x+y}{2}\cdot\sin\frac{x-y}{2}\right|$$ $$=2\left|\sin\frac{x+y}{2}\right|\left|\sin\frac{x-y}{2}\right|\le 2\cdot 1\cdot\left|\frac{x-y}{2}\right|=\left|x - y\right|.$$</p>
|
1,518,393 | <p>So I honestly don't even know where to start. Does it mean that a number k within the permutations is what we are looking for or something like that?</p>
| user21820 | 21,820 | <p>Every permutation can be decomposed into disjoint cycles. For example $(1,2,3) (4,5)$ can be used to denote the permutation that maps $1 \to 2 \to 3 \to 1$ and $4 \to 5 \to 4$. The question asks how many permutations of $n$ objects have $1,2,\cdots,k$ all in different cycles.</p>
<p><strong>Hint</strong>: The easie... |
1,518,393 | <p>So I honestly don't even know where to start. Does it mean that a number k within the permutations is what we are looking for or something like that?</p>
| Yuval Filmus | 1,277 | <p>Take a random permutation in $S_n$, decompose it as a product of disjoint cycles, and erase all numbers larger than $k$. Show that the resulting permutation is a random permutation in $S_k$. Since the probability that a random permutation in $S_k$ satisfies your condition is $1/k!$ (only the identity permutation), t... |
1,612,808 | <p>Suppose that $X$ is a finite $G$-set. A group $G$ is of prime power if $|G|=p^n$ for $p$ prime.</p>
<p>The fixed point set $X_G=\{x\in X : gx=x$ $\forall g\in G\}$.</p>
<p>I'm asked to prove that $|X|=|X_G|$ (mod $p$), but I'm unsure of how I should start.</p>
| Francis Begbie | 300,218 | <p>Note that $G$ acts on $X-X_G$, and $p$ divides the size of every orbit in $X-X_G$ by the Orbit Stabilizer Theorem, hence $p$ divides $|X-X_G|$, i.e. $|X|$ and $|X_G|$ are congruent modulo $p$.</p>
|
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Bill Dubuque | 6,716 | <p>One striking example that comes to mind is Nathan Jacobson's proof that rings satisfying the identity $X^m = X$ are commutative. This is model-theoretic and proceeds by a certain type of factorization which reduces the problem to the (subdirectly) irreducible factors of the variety. These turn out to be certain fini... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Pietro Majer | 6,101 | <p>A beautiful classical example from Functional Analysis is the Hausdorff moment problem: characterize the sequences
<span class="math-container">$m:=(m_0,m_1,\dots)$</span> of real numbers that are moments of some positive, finite Borel measure on the unit interval <span class="math-container">$I:=[0,1]$</span>:
<spa... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Peter LeFanu Lumsdaine | 2,273 | <p>A toy example, using the Yoneda lemma:</p>
<p><strong>Claim:</strong> There are two canonical bialgebra structures (the “additive” and “multiplicative” structures) on $k[x]$, and one of them (the additive one) in fact makes it a Hopf algebra.</p>
<p><strong>Proof 1:</strong> (Calculation.) Write down the formula... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Qiaochu Yuan | 290 | <p>The first proof I ever saw of the orthogonality relations for characters of finite groups was computational: it did a lot of matrix computations and manipulations of sums, which I didn't like at all. There is a much more conceptual proof which begins by observing that Schur's lemma is equivalent to the claim that</... |
1,336,381 | <p>Find the sum of solutions to:</p>
<p>$$ 2\log^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0 $$</p>
<p>I'm not sure about what to do with the absolute values, how can I get rid of them?</p>
<p>Should I solve for all various cases depending on the sign of $x+1$ and $x-1$?</p>
| Ben Grossmann | 81,360 | <p>The answer is no. As a counterexample, consider the sequence
$$
f_n(x) = \frac{n}{n+1}\cdot \frac{1}{x}
$$</p>
|
4,275,780 | <blockquote>
<p>If <span class="math-container">$0<a<b$</span> and <span class="math-container">$0<c<d$</span> then <span class="math-container">$\frac{c+a}{d+a} <\frac{c+b}{d+b}.$</span></p>
</blockquote>
<p>I get to <span class="math-container">$$d+a<d+b \Longrightarrow \frac{1}{d+b} < \frac{1}{d... | John Joy | 140,156 | <p>It may be helpful to just rename some of the quantities.</p>
<p>Suppose
<span class="math-container">$$\alpha = c+a\\
\beta=d+a\\
\gamma=b-a$$</span>
then
<span class="math-container">$$\begin{align}
\frac{\alpha}{\beta}&\lt\frac{\alpha+\gamma}{\beta+\gamma}\\
\alpha\beta + \alpha\gamma&\lt \alpha\beta +\bet... |
2,910,101 | <p>A positive integer X is said to be a cube-loving number if it can be written as $(a^3) \cdot b$, for some positive integers $a$ and $b$ ($a>1$,$b \ge 1$). Given a positive integer $n$, determine the number of Cube-loving numbers less than or equal to $n$.</p>
| Yanlong LIU | 1,057,860 | <p>Just use the theorem 2.3.1, which is the usual Chernoff's inequality about upper bound can work, we can get almost the same result like the first answer, which is <span class="math-container">$t \mathrm{log}(t) + O(t) \geq \mathrm{log}(n) - \epsilon$</span>, just show <span class="math-container">$t = O(\mathrm{log}... |
3,257,799 | <blockquote>
<p>Find all values of <span class="math-container">$a$</span> for which the equation
<span class="math-container">$$ (a-1)4^x + (2a-3)6^x = (3a-4)9^x $$</span>
has only one solution.</p>
</blockquote>
<p><br>
I have two cases, one when <span class="math-container">$a = 1$</span> and other when Discr... | nonuser | 463,553 | <p>Write <span class="math-container">$t=3^x/2^x>0$</span>, then we have <span class="math-container">$$(a-1)+(2a-3)t=(3a-4)t^2$$</span></p>
<p>Case <span class="math-container">$a={4\over 3}$</span> then the equation is linear so <span class="math-container">$$t= {1-a\over 2a-3}=1\implies \Big({3\over 2}\Big)^x= 1... |
69,448 | <p>What <code>Method</code> options are allowed for <code>DensityPlot</code> and <code>ContourPlot</code>? I am unable to find this information either in MMA documentation or in SE. Thanks.</p>
| Simon Woods | 862 | <p>As far as I know there is no documented list of <code>Method</code> options for <code>ContourPlot</code> and <code>DensityPlot</code>. If you want to experiment there is a large list of strings in <code>Charting`CommonDump`$VisualizationMethodOptions</code> to have a look at. Some of these are option settings, some ... |
4,549,340 | <p>I have heard people say that the flight time from Fort Lauderdale to Seattle is the longest possible flight time within the continental United States. However, upon further consideration, I realized that the curvature of the Earth may cause the visible distance on a map to decrease when traveling north (the circumfe... | Rohit Yadav | 910,446 | <p>For all <span class="math-container">$n$</span>, <span class="math-container">$m$$>$$N_ε$</span>, without loss of genarality, let m>n,</p>
<p>|<span class="math-container">$x_n$</span>−<span class="math-container">$x_m$</span>|<span class="math-container">$<=$</span>|<span class="math-container">$x_n$</spa... |
1,653,106 | <p>I was following a calculus tutorial that factored the equation $x^4-16$ into $(x^2 +4) (x+2)(x-2)$.</p>
<p>Why is the factorization of $x^4-16 = (x^2 + 4)(x+2)(x-2)$ rather than $(x^2 - 4)(x^2 +4)$? </p>
| Cameron Buie | 28,900 | <p>They are <em>both</em> factorizations of $x^4-16,$ but $(x^2+4)(x^2-4)$ is a less complete factorization.</p>
|
2,990,947 | <p>If r stands for counter-clockwise 90 degree rotation, s stands for horizontal flip.
<span class="math-container">$D_4= \{1, r, r^2, r^3, s, rs, r^2s, r^3s\}$</span>. What rule should I apply to find the subgroups of <span class="math-container">$D_4$</span>? Should I just put elements with same order in the same sub... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Pick elements one by one and see what happens to their generated subgroups (i.e. orbits under the operation <span class="math-container">$\cdot$</span>). Then try to mix them with each other. E.g.</p>
<ul>
<li><span class="math-container">$O(s): 1, s, s^2=1$</span></li>
<li><span class... |
2,662,554 | <p>I have to use Proof by contradiction to show what if $n^2 - 2n + 7$ is even then $n + 1$ is even. </p>
<p>Assume $n^2 - 2n + 7$ is even then $n + 1$ is odd. By definition of odd integers, we have $n = 2k+1$. </p>
<p>What I have done so far:</p>
<p>\begin{align}
& n + 1 = (2k+1)^2 - 2(2k+1) + 7 \\
\implies &am... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Note that</p>
<ul>
<li>$n^2-2n+7=n^2-2n+1+6=(n-1)^2+6$ is even $\iff$ $n-1$ is even</li>
</ul>
<p>and</p>
<ul>
<li>$n+1=(n-1)+2$</li>
</ul>
|
81,257 | <p>The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,</p>
<p>$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta A}$</p>
<p>where $YM(A)$ is the Yang Mills 'action' the integral of the curvature square,</p>
<p>$YM(A) = \... | Orbicular | 3,509 | <p>I think you should just take a look at the following paper:</p>
<p><a href="http://arxiv.org/PS_cache/arxiv/pdf/1103/1103.0845v1.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/1103/1103.0845v1.pdf</a></p>
<p>(In particular it should exponential convergence.) Should there still be question, you might post ... |
81,257 | <p>The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,</p>
<p>$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta A}$</p>
<p>where $YM(A)$ is the Yang Mills 'action' the integral of the curvature square,</p>
<p>$YM(A) = \... | Willie Wong | 3,948 | <p>Donaldson and Kronheimer wrote their book by 1990. There were some further developments about long time behaviour of Yang-Mills flow on four manifolds by, among others, Struwe and collaborators. You may try starting with <a href="http://www.ams.org/mathscinet-getitem?mr=1443269" rel="nofollow">Schlatter's dissertati... |
3,806,122 | <p>I tried using Chinese remainder theorem but I kept getting 19 instead of 9.</p>
<p>Here are my steps</p>
<p><span class="math-container">$$
\begin{split}
M &= 88 = 8 \times 11 \\
x_1 &= 123^{456}\equiv 2^{456} \equiv 2^{6} \equiv 64 \equiv 9 \pmod{11} \\
y_1 &= 9^{-1} \equiv 9^9 \equiv (-2)^9 \equiv -512... | Community | -1 | <p>By Euler's theorem, we first get <span class="math-container">$123^{40}\cong1\pmod{88}$</span>, since <span class="math-container">$\varphi(88)=40$</span>. This results in <span class="math-container">$35^{16}\pmod{88}$</span>, easily.</p>
<p>Now we use CRT: <span class="math-container">$\begin{cases}x\cong 35^{16}... |
3,525,621 | <p>Find all integral solutions to the equation <span class="math-container">$x^2 + 4xy - y^2 = m$</span> with <span class="math-container">$-5 \leq m \leq 10$</span>.</p>
<p>I know that I can set <span class="math-container">$m = -5$</span> to <span class="math-container">$m = 10$</span> and solve all of the equations... | Ali Shadhar | 432,085 | <p><span class="math-container">$$\int_0^1\frac{\ln(1+x)-\ln(1-x)}{x}\ dx$$</span>
<span class="math-container">$$=\int_0^1\frac{\ln(1+x)}{x}\ dx-\int_0^1\frac{\ln(1-x)}{x}\ dx$$</span></p>
<p><span class="math-container">$$=-\operatorname{Li}_2(-x)|_0^1+\operatorname{Li}_2(x)|_0^1$$</span></p>
<p><span class="math-c... |
1,234,726 | <p>How many lattice paths are there from $(0, 0)$ to $(10, 10)$ that do not pass to the point $(5, 5)$ but do pass to $(3, 3)$?</p>
<p>What I have so far:</p>
<p>The number of lattice paths from $(0,0)$ to $(n,k)$ is equal to the binomial coefficient $\binom{n+k}n$ (according to Wikipedia). So the number of lattic... | MBW | 6,884 | <p>Decompose your lattice paths in two parts: the one up until reaching $(3,3)$ and from $(3,3)$ to $(10, 10)$ not passing by $(5,5)$. We can translate this second part to be actually the number of paths from $(0, 0)$ to $(7,7)$ not passing by $(2,2)$. The fact that your paths must pass through $(3,3)$ make these probl... |
348,532 | <p>Consider the following integral
<span class="math-container">$$
I_\delta(\lambda)=\int_0^\delta e^{i\lambda \exp(-x^{-2})}dx.
$$</span>
Here, <span class="math-container">$\phi(x)=\exp(-x^{-2})$</span> is the phase function. I would like to study the rate of decay of <span class="math-container">$I(\lambda)$</span> ... | Bazin | 21,907 | <p>Let me fix <span class="math-container">$\delta=1$</span> for simplicity. Let us use a Van der Corput method. We have for <span class="math-container">$\epsilon\in (0,1)$</span> to be chosen later, with <span class="math-container">$\phi(x)= e^{-x^{-2}}, $</span> noting that <span class="math-container">$\phi'(x)=\p... |
163,589 | <p>The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case of 3 vector spaces. For one vector space there are two orbits: 0 vector, and non-zero vector. For two vector spaces, ... | Nathaniel Johnston | 11,236 | <p>For what it's worth, in the case when $U,V$, and $W$ all have dimension $2$ (i.e., a case that is much simpler than the $4$-dimensional one you're interested in), it is known that there are exactly six orbits. In particular, every vector is in the orbit of exactly one of these six vectors (where $\{\mathbf{e}_1,\mat... |
4,234,095 | <p>I need to show that <span class="math-container">$[\mathbb{Q}(2^{1/4},2^{1/6}):\mathbb{Q}]$</span> is a field extension of degree <span class="math-container">$12$</span>. It is possible to show that the degree is at least <span class="math-container">$12$</span> because it is divisible by <span class="math-containe... | Maths Rahul | 865,134 | <p>(1) <span class="math-container">$[\mathbb{Q}(2^{1/4}):\mathbb{Q}]=4$</span> since <span class="math-container">$2^{1/4}$</span> satisfies <em>irreducible</em>
polynomial <span class="math-container">$x^4-2$</span> of degree <span class="math-container">$4$</span> over <span class="math-container">$\mathbb{Q}$</span... |
2,358,490 | <p>Let $V$ be a finite dimensional vector space over the field $K$, and let $W_1$ and $W_2$ be subspaces. Express $(W_1+W_2)^{\perp}$ in terms of $W_1^{\perp}$ and $W_2^{\perp}$. Also, express $(W_1\cap W_2)^{\perp}$ in terms of $W_1^{\perp}$ and $W_2^{\perp}$.</p>
<p>I have no idea what this exercise is asking. Remar... | C. Ding | 320,080 | <p>Hint: $(W_1+W_2)^{\perp}=W_1^\perp\cap W_2^\perp$;
$(W_1\bigcap W_2)^{\perp}=W_1^\perp +W_2^\perp$.</p>
|
550,230 | <p>If 2 vectors form a basis for $\mathbb{R}^2$, must these 2 vectors always be orthogonal to each other?</p>
<p>For instance, the standard bases in $\mathbb{R}^2$ are definitely orthogonal (easily drawn). How about other bases?</p>
| GAM | 58,916 | <p>No, consider $A=\left\{\left[
\begin{array}{c}
1\\
2\\
\end{array}
\right],\left[
\begin{array}{c}
0\\
2\\
\end{array}
\right]\right\}\subset\mathbb{R^2}$. $A$ is linearly independent and spans $\mathbb{R^2}$, so $A$ forms a basis for $\mathbb{R^2}$. However, $\left[
\begin{array}{c}
1\\
2\\
\end{array}
\right]$ and... |
3,491,978 | <blockquote>
<p>Let (X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that for every x ∈ X, B(x,ε) is contained in some member of the cover.</p>
</blockquote>
<p>My attempt:</p>
<p>(X,d) is compact. Therefore there exists a finite subcover of X.</p>
<p>Any element x in X must lie ... | Matematleta | 138,929 | <p>The Wiki proof linked in the comments uses the fact that a continuous function on a compact set reaches its extrema. If you want a proof from scratch and closer to what you are trying to do, here are a few hints:</p>
<p><span class="math-container">$1).\ $</span> Let <span class="math-container">$\mathcal A$</span>... |
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