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 |
|---|---|---|---|---|
2,605,626 | <p>$\sin(t)$ is continuous on $[0,x]$ and $\frac{1}{1+t}$ is continuous on $[0,x]$ so $\frac{\sin(t)}{1+t}$ is continuous on $[0,x]$ so the function is integrable.</p>
<p>How do I proceed? What partition should I consider ? </p>
<p>Edit : We haven't done any properties of the integral so far except the basic definiti... | DIEGO R. | 297,483 | <p>I accidentally worked on that problem today. I'm surprised by the coincidence. Here I share my notes.
<a href="https://i.stack.imgur.com/RmvqA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RmvqA.png" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/VDWL8.png" re... |
1,394,490 | <p>I stumbled upon "the God proof" which goes:</p>
<p>$0 = 0 + 0 + 0...$</p>
<p>$ = (1-1) + (1-1) + (1-1) + ...$</p>
<p>$= 1 - 1 + 1 - 1 + 1 - 1 + ...$ </p>
<p>$= 1 + (-1+1) + (-1+1) + (-1+1) + ...$ </p>
<p>$= 1$</p>
<p>Even though this result is obviously wrong, I can't quite pinpoint exactly what the 'illegal'... | N. S. | 9,176 | <p>The equality
$$(1-1) + (1-1) + (1-1)...=1-1+1-1+1-1+...$$
is wrong, as the first series converges, while the second doesn't.</p>
|
630,614 | <p>Let $n\ge 2$ be an integer. The symmetric group $S_3$ acts on the set $M_n$ of polynomials in $\mathbb{C}[x_1,x_2,x_3]$ whose monomials are of the form $x_1^{a_1}x_2^{a_2}x_3^{a_3}$ with $0\le a_i\le n$ in the obvious way.</p>
<p>Is there a simple way to describe how $M_n$ is decomposed as a sum of the three irredu... | Mariano Suárez-Álvarez | 274 | <p>Let us compute the character of the representation of $M_n$. We need to compute the trace of the action of the identity element, of $(12)$ and of $(123)$. These permute the monomials in $M_n$, so we are just counting how many monomials each of these permutations fix.</p>
<p>First: the identity element of course fix... |
1,400,109 | <p>I am wondering if anyone can help me find an analytical solution to the roots of the following function:
<span class="math-container">$$f(b) = c\log \left( \frac{b}{a} \right) + (n-c)\log \left( \frac{1-b}{1-a} \right),$$</span>
<span class="math-container">$a,b \in (0,1)$</span> and <span class="math-container">$n... | Claude Leibovici | 82,404 | <p>I prefer to add another answer to this problem rather than to edit the previous one.</p>
<p>Setting $n=tc$ in the previous equation, we then have $$f(b)=\log \left(\frac{b}{a}\right)+(t-1) \log \left(\frac{1-b}{1-a}\right)$$ $$f'(b)=\frac{1-b t}{b(1-b)}$$ $$f''(b)=-\frac{tb^2 -2 b+1}{b^2(1-b)^2}$$ The maximum of th... |
8,193 | <p><strong>NB. Some answers appear to be for a question I did not ask, namely, "Why is standardized testing bad?" Indeed, these answers tend to underscore the premise of my actual question, which can be found above.</strong></p>
<p>As a foreigner who has spent some time in the US, it seems to me that in the U... | Aeryk | 401 | <p>In addition to the other great answers already posted, here's another thing that should be mention: Americans strongly buy into the rags-to-riches story; the idea that anyone can become successful. There are numerous examples of individuals in U.S. history that come from background with little formal education, but ... |
8,193 | <p><strong>NB. Some answers appear to be for a question I did not ask, namely, "Why is standardized testing bad?" Indeed, these answers tend to underscore the premise of my actual question, which can be found above.</strong></p>
<p>As a foreigner who has spent some time in the US, it seems to me that in the U... | Amy B | 5,321 | <p>I read through the answers and am surprised no one mentioned that issue of bias against race and lower class. There is a history of tests being designed for white middle class students. Students from other backgrounds did poorly because they didn't have the background or context to understand the questions. The r... |
3,057,874 | <blockquote>
<p>The following formula shall be proved by induction:
<span class="math-container">$$F(m+n) = F(m-1) \cdot F(n) + F(m) \cdot F(n+1)$$</span>
Where <span class="math-container">$F(i), i \in \mathbb{N}_0$</span> is the Fibonacci sequence defined as:
<span class="math-container">$F(0) = 0$</span>, <s... | Robert Z | 299,698 | <p>A combinatorial proof (no induction).</p>
<p>Using the representation of the Fibonacci numbers as the numbers of domino coverings the given identity can be shown in a quick and elegant way. </p>
<p>It is known that the <a href="https://math.stackexchange.com/questions/1317885/dominos-2-times-1-on-2-times-n-and-on-... |
1,432,341 | <p>I am tasked with finding the region of the complex plane under condition:
$$\left|\frac{z-2i}{z+2}\right|\ge 1$$
I can then calculate that
$|z-2i|\ge|z+2|$. Thus, I can say I'm looking for the region where the distance from $z$ to $2i$ is greater than the distance from $z$ to $-2$. Imagining the plane, I feel as t... | Michael Joyce | 17,673 | <p>Your geometric interpretation and your corrected algebraic calculation are correct.</p>
|
2,417,506 | <blockquote>
<p>Find the radius of convergence of $\displaystyle{\sum_{n=0}^{\infty}} {(n!)^3 \over (3n)!}z^{3n} \ ?$</p>
</blockquote>
<p>I applied Cauchy-Hadamard test and the result is coming $0$ (radius of convergence). To obtain the limit I also used <a href="http://priti2212.blogspot.in/2013/05/cauchys-theorem... | R. J. Mathar | 477,636 | <p>If we represent the factorials everywhere as gamma-functions and Pochhammer symbols, this is actually a generalized hypergeometric function 4F3(...;;;z/3). According to <a href="https://en.wikipedia.org/wiki/Generalized_hypergeometric_function" rel="nofollow noreferrer">wikipedia</a> the radius of convergence is whe... |
2,956,158 | <blockquote>
<p>Given <span class="math-container">$z = \cos (\theta) + i \sin (\theta)$</span>,
prove <span class="math-container">$\dfrac{z^{2}-1}{z^{2}+1} = i \tan(\theta)$</span></p>
</blockquote>
<p>I know <span class="math-container">$|z|=1$</span> so its locus is a circle of radius <span class="math-contai... | lab bhattacharjee | 33,337 | <p>As <span class="math-container">$z\ne0,$</span> with <span class="math-container">$|z|=1$</span></p>
<p><span class="math-container">$$\dfrac{z^2-1}{z^2+1}=\dfrac{z-\dfrac1z}{z+\dfrac1z}$$</span></p>
<p>Now <span class="math-container">$\dfrac1z=\dfrac1{\cos\theta+i\sin\theta}=\cos\theta-i\sin\theta$</span></p>
|
3,421,455 | <p>I'm not sure if I'm supposed to use integration by substitution here, but here's the question:</p>
<p><span class="math-container">$$\int ^{10}_{0}f\left( x\right) dx=25$$</span>
Find the value of
<span class="math-container">$$\int ^{e^{2}}_{1}\dfrac {f\left( 5\times \ln \left( x\right) \right) }{x}dx$$</span>
Do ... | Certainly not a dog | 691,550 | <p>The appearance of both <span class="math-container">$\ln x$</span> and <span class="math-container">$\frac 1x$</span> indicates that it is indeed a good substitution. And you have done it correctly in your edit. Note that the bounds of definite integration are the lower and upper limits of the values of <span class=... |
2,033,832 | <p>I am a math enthusiast in electrical engineering and I am planning on learning Differential Geometry for applications in Control Theory. I want to teach myself this beautiful branch of mathematics in a rigorous way.</p>
<p>I am currently going through Chapman Pugh's Real Analysis, I am then planning on studying Mun... | Dac0 | 291,786 | <p>I taught myself Differential Geometry so I can tell you everything it's needed. First of all you will have to decide if go for classic differential geometry or calculus on manifold. I would suggest Calculus on Manifold since with a little bit of effort you will gain a lot.
Having said that there's one secret to lea... |
2,138,663 | <p>Suppose a function $f$ is Riemann integrable over any interval $[0,b]$. By definition the improper integral is convergent if there is a real number $I$ such that</p>
<p>$$\lim_{b \to \infty}\int_0^b f(x) dx= I := \int_0^\infty f(x)dx.$$</p>
<p>I have shown that if $f$ is nonnegative then this is equivalent for $n ... | Fimpellizzeri | 173,410 | <p>Not true. You can choose the $A_n$ in a biased way. Consider $f(x)=\sin(x)$ and $A_n=[0,n\pi/2]$.</p>
|
330,710 | <p>Input: a set of $n$ points in general position in $\mathbb{R}^2$.</p>
<p>Output: the pair of points whose slope has the largest magnitude.</p>
<p>Time constraint: $O(n \log n)$ or better.</p>
<p>Please don't spoil the answer for me - I'm just stuck and looking for a nudge in the right direction. Thanks!</p>
| dtldarek | 26,306 | <p><strong>Hint:</strong></p>
<p>Consider points $A$, $B$, $C$ such that $A_x < B_x < C_x$, then the slope of $AC$ is smaller or equal than the maximum of magnitudes of slopes of $AB$ and $BC$.</p>
<p>Good luck!</p>
|
1,000,025 | <p>Well the question is the title. I tried to grab at some straws and computed the Minkowski bound. I found 19,01... It gives me 8 primes to look at. I get</p>
<p>$2R = (2, 1 + \sqrt{223})^2 = P_{2}^{2}$</p>
<p>$3R = (3, 1 + \sqrt{223})(3, 1 - \sqrt{223}) = P_{3}Q_{3}$</p>
<p>$11R = (11, 5 + \sqrt{223})(11, 5 - \sqr... | Jyrki Lahtonen | 11,619 | <p>Cam would know this better, but let me give a few pointers for pencil & paper calculations. IIRC calculations like this use heavily the following rules. In what follows $(a_1,a_2,\ldots,a_n)$ is the ideal generated by the listed elements.</p>
<p>We have the rules</p>
<ul>
<li>$(a_1,a_2,\ldots,a_n)=(a_1,a_2,\ld... |
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| Warren Hill | 86,986 | <blockquote>
<p>Radians make it possible to relate a linear measure and an angle
measure. A unit circle is a circle whose radius is one unit. The one
unit radius is the same as one unit along the circumference. Wrap a
number line counter-clockwise around a unit circle starting with zero
at (1, 0). The length ... |
2,143,227 | <p>I wish to show whether or not the sequences $f_n(x)=\chi_{[n,n+1]}$ is tight</p>
<p>By definition, Let $(X,m,\mu)$be a measure space, the sequence ${f_n}$ is called tight over $X$ if $\forall \epsilon>0$ $ \exists X_0 \subset X $,with $\mu (X_0)<\infty$ , such that $\forall n$ $\int_{X-X_0}|f_n|d\mu <\e... | Hermès | 127,149 | <p>$X_0$ must be independent of $n$, which is not the case in your example. I assume in the following that $\mu$ is the Lebesgue measure on the real line $\mathbb{R}$.</p>
<p>Let $X_0 \subset X$ such that $\mu(X_0)$ is finite. </p>
<blockquote>
<p>Given any $\varepsilon > 0$, there exists $N_\varepsilon \in \mat... |
63,589 | <p>A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the corresponding degree sequences $d_U$ and $d_V$, then it is easy to see that the largest eigenvalue $\lambda_{\max}$ ... | ght | 13,825 | <p>Please find <a href="https://mathoverflow.net/questions/59450/boundaries-of-the-eigenvalues-of-a-symmetric-matrix-or-of-its-lapacian/59556#59556">here</a> some relations between the eigenvalues of the Laplacian and properties of the graph. You can also take a look at Fan Chung's book <em>"Spectral Graph Theory"</em>... |
63,589 | <p>A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the corresponding degree sequences $d_U$ and $d_V$, then it is easy to see that the largest eigenvalue $\lambda_{\max}$ ... | Dan Stahlke | 18,969 | <p>Assume the graph is connected.</p>
<p>Let $\left| 1 \right>$ be the vector of all 1's (in Dirac notation), and let $A$ be the adjacency matrix. Then $\left<1|A|1\right>$ is the number of edges of the graph (well, actually twice the number of edges). Similarly, $\left<1|A^n|1\right>$ is the number o... |
155,359 | <p>I tried to compute the 40-th iteration by FindRoot but without result.
Please If any body solve that, I would to thank him. This question has a
nonlinear system. And the solution for n=1 is solved. But there is problem
when n=2,3, ...40.</p>
<p>The following code for computing the four unknowns for the n... | partida | 15,961 | <p>First method:</p>
<pre><code>run[cmdLine_] := Module[{out},
RunProcess[{$SystemShell, "/c", cmdLine <> ">test.txt"}]["StandardOutput"];
out = Import["test.txt", CharacterEncoding -> "CP936"];
DeleteFile["test.txt"]; out]
run["echo 你好"]
</code></pre>
<bl... |
1,048,045 | <blockquote>
<p>$$\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$$</p>
</blockquote>
<p>I try to solve it, but failed. Who can help me to find it?</p>
<p>I encountered this integral when trying to solve $\displaystyle{\int_0^\pi\frac{x\cos(x)}{1+\sin^2(x)}\,dx}$.</p>
| Lucian | 93,448 | <p><strong>Hint:</strong> Letting $t=\sin x$, the integral becomes $F(1)$, where $F(a)=\displaystyle\int_0^1\frac{\arctan(at)}{\sqrt{1-t^2}}dx$. Evaluate $F'(a)$.</p>
|
1,379,456 | <p>i am help</p>
<p>Calculate:</p>
<p>$$(C^{16}_0)-(C^{16}_2)+(C^{16}_4)-(C^{16}_6)+(C^{16}_8)-(C^{16}_{10})+(C^{16}_{12})-(C^{16}_{14})+(C^{16}_{16})$$</p>
<p>PD : use $(1+x)^{16}$ and binomio newton</p>
| Michael Hardy | 11,667 | <p>$$
(C^{16}_0)-(C^{16}_2)+(C^{16}_4)-(C^{16}_6)+(C^{16}_8)-(C^{16}_{10})+(C^{16}_{12})-(C^{16}_{14})+(C^{16}_{16})
$$</p>
<p>$$
1,0,-1,0,\overbrace{1,0,-1,0}^{\text{This repeats}\ldots},\ldots
$$</p>
<p>Powers of the imaginary number $i$ are:
$$
1,i,-1,-i,\overbrace{1,i,-1,-i}^{\text{This repeats}\ldots},\ldots
$$
... |
1,379,456 | <p>i am help</p>
<p>Calculate:</p>
<p>$$(C^{16}_0)-(C^{16}_2)+(C^{16}_4)-(C^{16}_6)+(C^{16}_8)-(C^{16}_{10})+(C^{16}_{12})-(C^{16}_{14})+(C^{16}_{16})$$</p>
<p>PD : use $(1+x)^{16}$ and binomio newton</p>
| haqnatural | 247,767 | <p>you can calculate it as follows instead of calculating combinatorics</p>
<p>$${ \left( 1+i \right) }^{ 16 }={ \left( { \left( 1+i \right) }^{ 2 } \right) }^{ 8 }=\left( 1+2i-1 \right) ^{ 8 }={ \left( 2i \right) }^{ 8 }=256$$</p>
|
916,120 | <p>What is negation of <strong>All birds can fly.</strong></p>
<p>The question seems bit funny but i don't know which of the following two sentences is correct:</p>
<ol>
<li>Some birds can not fly</li>
<li>There is at least one bird which can not fly.</li>
</ol>
<p>Both the sentence seems almost logically same. But ... | RutvikSutaria | 169,774 | <p>Both the sentence are correct. </p>
<p>I have come through many questions like these in which two answer are correct, when i asked to my teacher about it teacher he told me to mark first correct option.</p>
<p>So if it is asked as multiple choice question then you should mark which ever comes first.</p>
|
1,432,429 | <p>Problem to finish the question: If $n > 4$ is compound then $(n-1)!\equiv 0\pmod n$.
If $n = a\cdot b$ there is no problem, once $a, b$ are factors of $(n-1)!$. The problem is when $ n = p^2$. I know that once $p > 4$ then $p^2 \ge 3$. But, how can I justify that $p^2$ is a factor of $(n-1)!$?</p>
<p>Thanks ... | Bernard | 202,857 | <p>If $n=p^2$, $n-1=p^2-1$, so $\;p, 2p,\dots ,(p-1)p$ are distinct factors of $\;(n-1)!$, hence $p^{p-1}$ is a factor thereof. As $n=p^2>4$, $p\ge 3$, and $p-1\ge 2$.</p>
|
648,809 | <p>I have a matrix $A$ given and I want to find the matrix $B$ which is closest to $A$ in the frobenius norm and is positiv definite. $B$ does not need to be symmetric.</p>
<p>I found a lot of solutions if the input matrix $A$ is symmetric. Are they any for a non-symmetric matrix $A$? Is it possible to rewrite the pro... | Skeptical Khan | 475,966 | <p>Let suppose C is non positive definite correlation matrix $$C=Q\Lambda Q^*=Q (\Lambda_+ -\Lambda_-)Q^*$$ Where $\Lambda$ is diagonal matrix of Eigen values. Replace all negative eigen values with zero.
Then we use the Symmetric , non negative definite matrix $\rho^2C$ with suitable value of $\rho$. Two choices of ... |
4,508,474 | <p>I would like to solve the ODE
<span class="math-container">$$\left(x-k_1\right)\frac{dY}{dx} + \frac{1}{a}x^2\frac{d^2 Y}{d x^2}-Y+k_2=0$$</span>
with boundary conditions <span class="math-container">$Y\left(k_3\right)=0$</span> and <span class="math-container">$\frac{d Y}{d x}\left(\infty\right)=1-b$</span> (meanin... | Mockingbird | 716,126 | <p>According to your statement of Bézout's Theorem, (the curves defined by) two polynomials of degree <span class="math-container">$m,n$</span> intersect <em>at most</em> at <span class="math-container">$mn$</span> points.
Since <span class="math-container">$2 \le 4$</span> everything's fine.</p>
<p>There is however a ... |
4,508,474 | <p>I would like to solve the ODE
<span class="math-container">$$\left(x-k_1\right)\frac{dY}{dx} + \frac{1}{a}x^2\frac{d^2 Y}{d x^2}-Y+k_2=0$$</span>
with boundary conditions <span class="math-container">$Y\left(k_3\right)=0$</span> and <span class="math-container">$\frac{d Y}{d x}\left(\infty\right)=1-b$</span> (meanin... | quarague | 169,704 | <p><em>Mockingbird</em> gave a good answer providing the general reasoning and <em>emacs drives me nuts</em> computed an example. However in their example the two circles do not interset in the real plane at all so I thought it would be useful to also look at an example where there are two intersections in the real pla... |
2,853,673 | <p>I came across this as one of the shortcuts in my textbook without any proof.<br>
When $b\gt a$, </p>
<blockquote>
<p>$$\int\limits_a^b \dfrac{dx}{\sqrt{(x-a)(b-x)}}=\pi$$</p>
</blockquote>
<hr>
<p><strong>My attempt :</strong></p>
<p>I notice that the the denominator is $0$ at both the bounds. I thought of su... | Sangchul Lee | 9,340 | <p>Let $m = \frac{b+a}{2}$ and $r = \frac{b-a}{2}$. Consider the circle</p>
<p>$$ (x - m)^2 + y^2 = r^2. $$</p>
<p>Part of this locus with $y \geq 0$ is given by $y = \sqrt{r^2 - (x-m)^2} = \sqrt{(x-a)(b-x)}$ for $a \leq x \leq b$. By the implicit differentiation, this function satisfies $ 2(x - m) dx + 2ydy = 0 $ an... |
2,885,374 | <blockquote>
<p>Suppose, to build a box (a rectangular solid) of fixed volume and square base, the cost per square inch of the base and top is twice that of the four sides. Choose dimensions for the box that minimize the cost of building it, if the volume is (say) $2000$ cubic inches.</p>
</blockquote>
<p>Here is w... | mechanodroid | 144,766 | <p>You got $C(x,h) = 4n(x^2 + xh)$ and we know that $V = x^2h$ is fixed. Eliminating $h$ gives:</p>
<p>$$C(x) = 4n\left(x^2 + \frac{V}x\right)$$</p>
<p>Now we can use the AM-GM inequality:</p>
<p>$$C(x) = 4n\left(x^2 + \frac{V}x\right) = 4n\left(x^2 + \frac{V}{2x} + \frac{V}{2x}\right) \ge 4n \cdot 3\sqrt[3]{x^2 \cd... |
2,281,184 | <p>I was going through theorem 1.21 from Rudin (without looking at the proof) and I wanted to show that the supremum of $E = \{ y : y > 0, y^n < x \}$ must be the element that satisfies $y^n = x$. But for me to even attempt that next step, I need to show that $E$ is non-empty (then I need to show its bounded) so ... | Red shoes | 219,176 | <p>If $ 1 < x $ then the point $y=1$ is in set. If $ 0< x \leq 1 $ then the point $\frac{x}{2}$ is in set. </p>
|
2,281,184 | <p>I was going through theorem 1.21 from Rudin (without looking at the proof) and I wanted to show that the supremum of $E = \{ y : y > 0, y^n < x \}$ must be the element that satisfies $y^n = x$. But for me to even attempt that next step, I need to show that $E$ is non-empty (then I need to show its bounded) so ... | Alex Provost | 59,556 | <p>Note for instance that if $a < b$ ($a,b$ any two real numbers), then their <em>average</em> $\frac{a+b}{2}$ lies between them: $a < \frac{a+b}{2} < b$. Indeed, it follows from adding $a$ (resp. $b$) to both sides of the original inequality to get $2a < a + b$ (resp. $a + b < 2b$) and multiplying both ... |
3,944,727 | <p>Suppose we have <span class="math-container">$81$</span> positive integers that form a <span class="math-container">$9 \times 9$</span> matrix so that the first row forms an arithmetic sequence and every column forms a geometric sequence with the same common ratio. If <span class="math-container">$a_{2,4} = 1, a_{4,... | PTDS | 277,299 | <p>Let <span class="math-container">$a_{1,5} = x$</span>, the common difference of the arithmetic sequence be <span class="math-container">$d$</span> and the common ratio of the geometric sequence be <span class="math-container">$r$</span></p>
<p>Given: <span class="math-container">$a_{2,4} = 1, a_{4,2} = \frac{1}{8}, ... |
3,944,727 | <p>Suppose we have <span class="math-container">$81$</span> positive integers that form a <span class="math-container">$9 \times 9$</span> matrix so that the first row forms an arithmetic sequence and every column forms a geometric sequence with the same common ratio. If <span class="math-container">$a_{2,4} = 1, a_{4,... | Itachi | 655,021 | <p>Let, the first entry <span class="math-container">$a_{11}=a$</span>. Then the first row is <span class="math-container">$a, a+d, a+2d,...a+8d$</span>.</p>
<p>Now, let <span class="math-container">$r$</span> be the common ratio for each column.</p>
<p>Then, the <span class="math-container">$i^{th}$</span> column look... |
1,270,802 | <p>I just finished an exam, it has the following question: Where is the point on the plane $3x + 5y + z = 18$ has the shortest distance to $(0,0,0)$?</p>
<p>I found this question similar: <a href="https://math.stackexchange.com/questions/355460/find-the-point-on-the-plane-2x-y-2z-20-nearest-the-origin">Find the point ... | Mitchell Spanheimer | 238,684 | <p>Subtracting this way will give you a negative number, the easiest way to solve a problem like this is to just switch the two numbers around, so that the big number is on top, then subtract, and add a negative sign in front of the result, or like some people suggested, multiply by -1. It is the same answer either way... |
1,690 | <p>I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations. In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how ... | user839 | 839 | <p>You have some really good pros and cons for explanations on a true/false test. If that is the method that you would like to keep using, you could always try having students explain why the find a statement/problem true or false. For example, have the directions read something like: Mark the statement either true or ... |
1,690 | <p>I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations. In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how ... | Jyrki Lahtonen | 282 | <p>I use such questions frequently, but in a limited fashion. Typically my exam contains five or six question, and one of those consists of, say, four true/false questions. The way I do it is that I make four claims, and students are to either give a quick proof or give a counterexample. I design them in a way that eit... |
4,631,283 | <p>A question in my textbook says to evaluate <span class="math-container">$\displaystyle \int \frac{1}{\sqrt{x^2-a^2}}~dx$</span> where <span class="math-container">$a \gt 0$</span>. I know how to solve the integral using trig substitution but what i do not understand is why is <span class="math-container">$a \gt 0$<... | Jap88 | 543,634 | <p>Another way is to use a computer algebra package and eliminate the system of equations. This gives the following 8th degree polynomial:
<span class="math-container">$$\Phi(x,y)=-R^2x^6 - 3R^2x^4y^2 - 3R^2x^2y^4 - R^2y^6 + r^2x^6 - 6r^2x^4y^2 + 9r^2x^2y^4 - 2rx^7 + 2rx^5y^2 + 10rx^3y^4 + 6rxy^6 + x^8 + 4x^6y^2 + 6x^4... |
44,623 | <p>For a bilinear function $T$, it can be shown that $\lVert T(x,y)\rVert\leq C \lVert x\rVert \lVert y \rVert$</p>
<p>I saw some books say a bilinear function T is Lipschitz with Lipschitz constant $C$ given the above inequality holds. </p>
<p>Now I'm confused because a function T is Lipschitz if $\lVert T(\alpha)-T... | patricktokeeffe | 367,616 | <p>Assuming you use the standard meterological convention that wind direction is the <em>source</em> direction of winds (i.e. 270º means blowing west → "here"):</p>
<p>Given two arrays containing wind speed (<code>WS</code>) and wind direction (<code>WD</code>, in degrees) observations, the vector mean wind direc... |
415,795 | <p>I am currently doing exercises from graph theory and i came across this one that i can't solve. Could anyone give me some hints how to do it?</p>
<p>Prove that for every graph G of order $n$ these inequalities are true:
$$2 \sqrt{n} \le \chi(G)+\chi(\overline G) \le n+1$$</p>
| Hagen von Eitzen | 39,174 | <p>For the right hand inequality, use induction:
The case $n=1$ is clear as $\chi(G)=\chi(\overline G)=1$.
If $n>1$, select a vertex $v\in G$.
By induction, we may assume that $\chi(G-v)+\chi(\overline {G-v})\le n$.
Clearly, $\chi(G)\le \chi(G-v)+1$ and $\chi(\overline G)\le\chi(\overline{G-v})+1$.
If the degree $\r... |
1,139,847 | <p>I am trying to solve a differential equation, but I can't solve an integral, because I forgot which rule to apply. What rule do I use to do $$\int \frac{1}{3y-y^2}\mathrm dy\ ?$$</p>
| Olivier Oloa | 118,798 | <p><strong>Hint.</strong> By partial fraction decomposition, you have
$$
\frac{3}{3y-y^2}=\frac{1}{y}-\frac{1}{y-3}
$$
then integrate each part.</p>
|
3,996,442 | <blockquote>
<p>If <span class="math-container">$P(x)= \sum_{n=0}^{\infty} a_{n} x^{n}$</span>. It is known that <span class="math-container">$P$</span> satisfies: <span class="math-container">$$P^{\prime}(x)=2xP(x)$$</span> <span class="math-container">$\\$</span> for all <span class="math-container">$ x\in \mathbb{R}... | Claude Leibovici | 82,404 | <p>If <span class="math-container">$P(x)$</span> is of degree <span class="math-container">$n$</span>, then the lhs is of degree <span class="math-container">$(n-1)$</span> and the rhs is of degree <span class="math-container">$(n+1)$</span>.</p>
<p>Since we compare the coefficients of same power, we have only odd or ... |
278,528 | <p>Let's suppose we have the following structure of data</p>
<pre><code>data = {{1}, {0.0109, -12.7758, -0.00980164, 0.00032368},
{1.0218, -12.7764, -0.00948724, 0.00064337},
{2.0327, -12.7772, -0.00905215, 0.00095516},
{2}, {0.0109, -12.7758, -0.00980164, 0.00032368},
... | Najib Idrissi | 85,027 | <p>Maybe there is something more elegant, but there you go:</p>
<pre><code>Cases[data, {n_Integer} | {_?(LessThan[2]), b__} :> {n, b}]
</code></pre>
|
278,528 | <p>Let's suppose we have the following structure of data</p>
<pre><code>data = {{1}, {0.0109, -12.7758, -0.00980164, 0.00032368},
{1.0218, -12.7764, -0.00948724, 0.00064337},
{2.0327, -12.7772, -0.00905215, 0.00095516},
{2}, {0.0109, -12.7758, -0.00980164, 0.00032368},
... | kglr | 125 | <pre><code>SequenceReplace[{ a : {_, _, _, _} ...} :>
Sequence @@ (Rest /@ Select[First@# <= 2 &]@{a})] @ data
</code></pre>
<blockquote>
<p>{{1}, {-12.7758, -0.00980164, 0.00032368}, {-12.7764, -0.00948724, 0.00064337},<br />
{2}, {-12.7758, -0.00980164, 0.00032368}, {-12.7764, -0.00948724,
0.00064337},... |
48,571 | <p>Hi,</p>
<p>Let $R$ be equipped with the usual Borel structure. Let $F$ be a Borel subset and $E$ be a closed subset of $R$. Then $F+E=(f+e: f\in F, e \in E \)$ is Borel? If yes, is it true for any locally compact topological group? Thanks in advance.</p>
| Andrey Rekalo | 5,371 | <p>No. Erdös and Stone showed that the sum of two subsets $E$, $F\subset\mathbb R$ may not be Borel even if one of them is compact and the other is $G_\delta$ (see <a href="http://www.jstor.org/pss/2037209" rel="noreferrer"><em>"On the Sum of Two Borel Sets"</em></a>, Proc. Am. Math. Soc., Vol. 25, (1970), pp. 304-306)... |
604,359 | <p>My experience with non commutative rings is limited to 2 by 2 matrices and the quaternions. The first of which is not a domain, and the latter is a division ring. I'm looking for an example of a domain that is not a division ring. </p>
<p>Invertible matrices do not produce an example, as they must be division rings... | Jose Brox | 146,587 | <p>I would say the simplest example, in the sense of most elementary from some perspective, is the unital free ring in two noncommuting variables, $\mathbb{Z}\langle X,Y\rangle$. You can think about it as the ring of "polynomials" in two variables which do not commute, with integer coefficients. It is clearly noncommut... |
2,526,932 | <p>So, I started out with $$f(x)=e^{-x^2}cos(x) \;\;\;at\;\;\; a=0$$ And after finding the Taylor Polynomial $T_3(x)$ for that function, I have $$T_3(x)=1-{{3x^2}\over 2}$$ Now hopefully that is correct. Next, I need to use the Taylor inequality to estimate the accuracy of the approximation $f(x) \approx T_n(x)$ where ... | Andreas | 317,854 | <p><a href="https://math.stackexchange.com/questions/193702/">Here</a> you find the following formula:</p>
<p>$$
\frac{d^n}{dx^n} e^{x^2} = \left( \sum_{j=0}^{\lfloor n/2 \rfloor} \frac{n!}{j!(n-2j)!}(2x)^{n-2j} \right) e^{x^2}
$$</p>
<p>Now taking the derivative at the argument $ i x$ gives
$$
\frac{d^n}{dx^n} e^{... |
3,249,064 | <p>I've read the question: "Why does the derivative of sine only work for radians?" and I can follow the derivation for the derivative of sine when measured in degrees, but the result confuses me. </p>
<p>Does this mean the derivative of the sine changes values when measured in different units? </p>
<p>For example, w... | heropup | 118,193 | <blockquote>
<p>What's so special about radians, anyway?</p>
</blockquote>
<p>That is the crux of your question. And the answer is simple: radian measure is unique in the sense that <strong>the radian measure of an angle equals the length of the unit circle arc that it subtends</strong>. So in the unit circle, a ... |
3,249,064 | <p>I've read the question: "Why does the derivative of sine only work for radians?" and I can follow the derivation for the derivative of sine when measured in degrees, but the result confuses me. </p>
<p>Does this mean the derivative of the sine changes values when measured in different units? </p>
<p>For example, w... | Lee Mosher | 26,501 | <p>Here's a point to think about. </p>
<p>Let me use the variable <span class="math-container">$x$</span> for angle measured in radians, and the variable <span class="math-container">$\theta$</span> for angle in degrees. So, <span class="math-container">$\theta = \frac{180 x}{\pi} x$</span> and <span class="math-conta... |
3,249,064 | <p>I've read the question: "Why does the derivative of sine only work for radians?" and I can follow the derivation for the derivative of sine when measured in degrees, but the result confuses me. </p>
<p>Does this mean the derivative of the sine changes values when measured in different units? </p>
<p>For example, w... | ryang | 21,813 | <p>That's right: the slope of sine in the degrees world is not merely the cosine of the equivalent radian angle.</p>
<p>It is a fallacy that having the same value at equivalent inputs guarantees having the same slope at equivalent inputs, and the crux of the matter is that the <strong>sine function that takes in degree... |
755,989 | <p>I am looking for an intuitive explanation for the identity:</p>
<p>$$\binom{n}{h}\binom{n-h}{k} = \binom{n}{k}\binom{n-k}{h}$$</p>
<p>Thanks!</p>
| Ittay Weiss | 30,953 | <p>both count the number of possibilities for choosing two disjoint subsets from a set of size $n$, one with $h$ elements the other with $k$ elements. </p>
|
947,622 | <p><img src="https://i.stack.imgur.com/TBGYx.png" alt="">Find variance from the graph given. I know the mean is 6 but I have no idea how to find the variance using this graph</p>
| Hinrik Ingolfsson | 178,619 | <p>Let $f(x) = \frac{1}{\sigma \sqrt{2\pi} } e^{-\frac{(x-6)^2}{2\sigma^2} }$ be the pdf. From the graph we see that $f(6) = \frac{1}{\sigma \sqrt{2\pi} } \approx .08$, and you can use that equation to solve for $\sigma^2.$</p>
|
292,594 | <p>Today I have encounter an integral:</p>
<p>$$\int_0^{\infty}\left[\frac{1}{3}\frac{\sin x}{x}+\cdots+\frac{1\times4\times\cdots\times(3n-2)}{3^nn!}\left(\frac{\sin x}{x}\right)^n+\cdots\right]\text{d}x$$</p>
<p>since $$\int_0^{\infty}\frac{\sin x}{x}=\frac{\pi}{2}$$</p>
<p>so I want to estimate $$\sum_{n=1}^{\inf... | Community | -1 | <p>Since $$\prod_{k=1}^{n}\left(1-\frac{2}{3k}\right) = \left|{-1/3\choose n}\right|\sim {n^{-2/3}\over\Gamma(1/3)},$$ the sum diverges. </p>
|
466,271 | <blockquote>
<p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, and <span class="math-container">$d$</span> be the lengths of the sides of a quadrilateral. Show that
<span class="math-container">$$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\... | Taha Direk | 676,723 | <p>Let <span class="math-container">$\;$</span> <span class="math-container">$LHS=f(a,b,c,d)$</span>. Note that <span class="math-container">$f(a,b,c,d)>f(a-k,b-k,c-k,d-k)$</span> for <span class="math-container">$0<k\le\min\{a,b,c,d\}$</span>. So, WLOG we can take <span class="math-container">$d=0$</span> to pro... |
4,016,289 | <p>Let's consider the parametric integral:</p>
<p><span class="math-container">$F:\mathbb{R}\to\mathbb{R}$</span>, where <span class="math-container">$F(x):=\int\limits_0^1 \frac{1}{x}\left(e^{-x^2(1+t)}(1+t)^{-1}-(1+t)^{-1}\right)dt$</span>.</p>
<p>What is the value of the limit: <span class="math-container">$$\lim\li... | Community | -1 | <p>The error is in the step "Then the original problem is equivalent to evaluating ...". It is not. To be precise,
<span class="math-container">$$
\lim_{x\to 0}\frac{x^2+x+1}{2x^2-3}
=\lim_{x\to \color{red}{0}}\frac{1+\frac1x+\frac1{x^2}}{2-\frac3{x^2}}
\color{red}{\ne}
\lim_{x\to \color{red}{\infty}}\frac{1+... |
1,192,434 | <p>I have a problem with integrating of fraction
$$
\int \frac{x}{x^2 + 7 + \sqrt{x^2 + 7}}
$$
I have tried to rewrite it as $\int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{x^4 + 13x^2 + 42} = \int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{(x^2 + 6)(x^2 + 7)}$ and then find some partial fractions, but it wasn't succesful.</p>
| Joel | 85,072 | <p>If you expand each of the geometric series in the $A_n$, and combine each of the series together as one sum (you can since these are each absolutely convergent), then you can demonstrate that this is tending to $x(1-x)^{-1}$.</p>
<p>What you need to make sure of though, is that $x^k$ can appear only once between ea... |
1,192,434 | <p>I have a problem with integrating of fraction
$$
\int \frac{x}{x^2 + 7 + \sqrt{x^2 + 7}}
$$
I have tried to rewrite it as $\int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{x^4 + 13x^2 + 42} = \int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{(x^2 + 6)(x^2 + 7)}$ and then find some partial fractions, but it wasn't succesful.</p>
| Barry Cipra | 86,747 | <p>Treating this strictly as a multiple-choice question, some simple considerations narrow it down to a) as the only reasonable possibility: c) and d) are clearly less than $1$ for all $x$, but if $x$ is close to $1$, then $A_n$ is clearly (much) greater than $1$, while if $x\approx0$, then $A_n\approx0$ is not too ha... |
4,127,553 | <p>So for this problem I need to use the fact that <span class="math-container">$\frac {1-r^2}{1-2r\cos\theta+r^2}$</span>=<span class="math-container">$1+2\sum_{n=1}^{\infty} r^n\cos n\theta$</span>.</p>
<p>I replaced the term in the integral but i ended up getting <span class="math-container">$\sum_{n=1}^{\infty} r^n... | Disintegrating By Parts | 112,478 | <p>You have reduced the problem to evaluating
<span class="math-container">$$
\frac{1}{\pi}\int_{-\pi}^{\pi}\left(1+2\sum_{n=1}^{\infty}r^n\cos(n\theta)\right)^2d\theta.
$$</span>
The terms <span class="math-container">$1,\cos(\theta),\cos(2\theta),\cdots$</span> are mutually orthogonal on <span class="math-contai... |
1,793,375 | <p>Question: </p>
<p><a href="https://i.stack.imgur.com/TbjfY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TbjfY.png" alt="enter image description here"></a></p>
<p>My work for parts a and b:
<a href="https://i.stack.imgur.com/C3CgR.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.... | BruceET | 221,800 | <p>Here is a simulation using R statistical ssoftware of a million performances of this experiment,
where $P(Heads) = .3$ for the biased coin. Results should be
accurate to a couple of decimal places. You can use them as a
'reality check' for your work.</p>
<pre><code> m = 10^6; x = y = numeric(m)
for(i in 1:m) {
... |
14,486 | <h2>Speculation and background</h2>
<p>Let <span class="math-container">$\mathcal{C}:=\mathrm{CRing}^\text{op}_\text{Zariski}$</span>, the affine Zariski site. Consider the category of sheaves, <span class="math-container">$\operatorname{Sh}(\mathcal{C})$</span>.</p>
<p>According to <a href="https://ncatlab.org/nlab/s... | Jonathan Wise | 32 | <p>I'm having a little trouble teasing out exactly what your question is, so I'll just write some things about sheaves that seem related and hope they are helpful.</p>
<p>Suppose $C$ is a site. Let $\hat{C}$ be its category of presheaves and $\tilde{C}$ its category of sheaves. The topology defined in SGA 4, II.5 is... |
14,486 | <h2>Speculation and background</h2>
<p>Let <span class="math-container">$\mathcal{C}:=\mathrm{CRing}^\text{op}_\text{Zariski}$</span>, the affine Zariski site. Consider the category of sheaves, <span class="math-container">$\operatorname{Sh}(\mathcal{C})$</span>.</p>
<p>According to <a href="https://ncatlab.org/nlab/s... | Shizhuo Zhang | 1,851 | <p>Check out the paper of Kontsevich-Rosenberg <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=2331">noncommutative space.</a>, they defined formally open immersion and open immersion completely functorial way. This definition is nothing to do with "noncommutative" </p>
<p>Definition:
Formally open immersion i... |
14,486 | <h2>Speculation and background</h2>
<p>Let <span class="math-container">$\mathcal{C}:=\mathrm{CRing}^\text{op}_\text{Zariski}$</span>, the affine Zariski site. Consider the category of sheaves, <span class="math-container">$\operatorname{Sh}(\mathcal{C})$</span>.</p>
<p>According to <a href="https://ncatlab.org/nlab/s... | JBorger | 1,114 | <p>I'm not sure if this will satisfy you, but a map of schemes is an open immersion if and only if it is an etale monomorphism. Etale means, by definition, formally etale and locally of finite presentation, both of which conditions have simple formulations in terms of functors of points, from Rings to Sets. Likewise, a... |
65,923 | <p>The <a href="http://en.wikipedia.org/wiki/Parity_of_a_permutation">sign of a permutation</a> $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula</p>
<p>$${\rm sgn}(\sigma) = (-1)^m$$</p>
<p>where $m$ is the number ... | Niels J. Diepeveen | 3,457 | <p>So you have a permutation $f: X \to X$ for which you can efficiently compute $f(x)$ from $x$.</p>
<p>I think a good way to do any of the things you mentioned is to make a checklist for the elements of $X$.
Then you can start with the first unchecked element and follow the chain $x, f(x), f(f(x))$, etc. checking off... |
1,585,061 | <p>Starting from the formula for work given a constant force $W = f s$, if you take the differential of both sides you would expect to get:</p>
<p>$$\mathrm{d}W = f \mathrm{d}s + s \space \mathrm{d}f$$</p>
<p>by an application of the product rule on the right hand side when taking the differential.</p>
<p>However, c... | Mavericktoday | 300,090 | <p>The complete differential is actually the correct definition of Work. The usage usually depends on the what kind of force is exerted. For example, in a system with impulse force, you will account for the change in force as well.
Additionally, the formula changes depending on whether the final energy stored is potent... |
3,223,371 | <p>In order to prove if a relation is an equivalence relation, it needs to be show that is all of: </p>
<ul>
<li>Reflexive </li>
<li>Symmetric </li>
<li>Transitive </li>
</ul>
<p>Whilst I am familiar with this, I am unsure how to approach the following set of questions: </p>
<p>State and explain whether each of... | Maximilian Janisch | 631,742 | <p>How to deal with cartesian products of cartesian products:</p>
<p>Let <span class="math-container">$\Omega := \mathbb{Z}\times\mathbb{Z}$</span> (or, respectively, <span class="math-container">$\mathbb{Z}\times\mathbb{Z}\setminus \{0\}$</span>).</p>
<p><strong>For reflexivity:</strong> You are supposed to prove (o... |
3,790,726 | <p><span class="math-container">$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$</span></p>
<hr />
<p>By maths calculator it results 1.
I calculate and results <span class="math-container">$\sqrt{-\frac{1}{2}}$</span>.</p>
<p><span class="math-container">$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$</span>... | Mike | 792,125 | <p>The square of a negative number is positive.</p>
<p><span class="math-container">$$\left(-\frac{\sqrt{3}}{2}\right)^2=\frac{3}{4}$$</span></p>
|
1,115,732 | <p>I'm trying to do the following probability question involving, I think, the ''amended'' multiplication rule:</p>
<p>A Jar contains 3 red and 5 black balls. What is the probability of drawing
2 red balls simultaneously ?</p>
<p>I used the formula - $P(A \cap B) = P(A)\cdot P(B\mid A)$</p>
<p>I.E. P(Red ball and th... | Jeffrey | 200,703 | <p>Correct.</p>
<p>One way to confirm this works is to look at all the possible outcomes:</p>
<p>There's $8 \times 7 \over 2$ combination of balls to pick.</p>
<p>There's $3 \times 2 \over 2$ combination of red balls you can pick.</p>
<p>${3 \times 2 \over 2} \over {8 \times 7 \over 2}$ is ${3 \over 8} \times {2 \o... |
543,599 | <p>I have a question that is:</p>
<p>$a_1 = 1, a_{k+1} = (k+1)+a_k$</p>
<p>compute $a_8$</p>
<p>I suspect $a_8 = 46$</p>
<blockquote>
<p>from:
$${a_2 = 1 + 1 + 1 = 3}$$
$${a_3 = 2 + 1 + 3 = 6}$$
$$...$$</p>
</blockquote>
<p>Am I computing this correctly?</p>
<p>Thanks</p>
| Penguino | 90,137 | <p>These are the triangular numbers. It is fairly simple to show that A(k) = k(K+1)/2</p>
|
856,237 | <p>How can be proven the following inequality?
$$\forall{x\in\mathbb{R}},\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$
Thanks</p>
| agha | 118,032 | <p>For $x<0$ inequality is obvious.</p>
<p>$\displaystyle(\frac{x}{2}-1)^2=\frac{x^2}{4}-x+1 \geq 0$, so</p>
<p>$\displaystyle \frac{x^2}{4}+2 \geq x+1$, but for $x \geq 0$</p>
<p>$x \geq x\ (\sin x)^2$ and $1 \geq \sin 2x$</p>
<p>If you want to have $\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$ instead... |
856,237 | <p>How can be proven the following inequality?
$$\forall{x\in\mathbb{R}},\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$
Thanks</p>
| Community | -1 | <p>$$\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$
$$\sin(x)(2\cos(x)+x\sin(x))\lt\dfrac{1}{4}x^2+2$$
while the right side is positive for all $x \in \mathbb{R}$ it is enoght that we show:
$$|\sin(x)|\cdot|(2\cos(x)+x\sin(x))|\lt\dfrac{1}{4}x^2+2$$
by the Cauchy–Schwarz inequality we have:
$$|\sin(x)|\cdot\sqr... |
1,853,306 | <blockquote>
<p>Consider the relations R and S on <span class="math-container">$\Bbb N$</span> defined by <span class="math-container">$x\; R\; y$</span> iff</p>
<p><span class="math-container">$2 \;$</span>divides <span class="math-container">$x + y$</span>
and <span class="math-container">$x \;S \;y$</span> iff <sp... | Doug M | 317,162 | <p>suppse L = $\begin{pmatrix} \cos\theta & \sin\theta\\ -\sin\theta& \cos\theta\end{pmatrix}$
i.e. a rotation of $\theta$ degrees. L is ortho-normal. Now we rotate it $\phi$ more degrees. And we get a rotation of $\theta + \phi.$ All of those matrices are ortho-normal.</p>
|
1,292,500 | <p>A very, very basic question.</p>
<p>We know
$$-1 \leq \cos x \leq 1$$
However, if we square all sides we obtain
$$1 \leq \cos^2(x) \leq 1$$
which is only true for some $x$.</p>
<p>The result desired is
$$0 \leq \cos^2(x) \leq 1$$
Which is quite easily obvious anyway. </p>
<p>So, what rule of inequalities am I for... | Ahmed | 55,273 | <p>since $a>b \ \space \forall a,b \in R$</p>
<p>It is not necessary
...
$a^2>b^2 \ \ \forall a , b \in R$
let $ a=-1 b=-4 $ </p>
<p>$ a>b \ But \ b^2>a^2 $</p>
|
1,292,500 | <p>A very, very basic question.</p>
<p>We know
$$-1 \leq \cos x \leq 1$$
However, if we square all sides we obtain
$$1 \leq \cos^2(x) \leq 1$$
which is only true for some $x$.</p>
<p>The result desired is
$$0 \leq \cos^2(x) \leq 1$$
Which is quite easily obvious anyway. </p>
<p>So, what rule of inequalities am I for... | Miz | 205,000 | <p>There is a small distinction to the statement $a \leq b \; \implies \; a^2 \leq b^2$ and that is that this statement holds only when $a, b \geq 0$. However if $a, b \leq 0$ the opposite is true $a \leq b \; \implies \; a^2 \geq b^2$. So you need to check the signs of the terms you are squaring in an inequality. </p>... |
1,794,459 | <p>We have set $S=\{ (a,b) | a,b \in \mathbb{Q}\}$</p>
<p>And we know that $(a,b)\sim \mathbb{R}$ , so $k((a,b))=c$. And $\mathbb Q \sim \mathbb N$, so $k(\mathbb Q)=\aleph_0$.</p>
<p>I don't know how to put all informations together.</p>
<p>I was also thinking that $S \subset \mathbb R$ and than we know $k(S) \le k... | Asaf Karagila | 622 | <p>You're mixing the cardinality of the interval, with the cardinality of the <em>set</em> of intervals.</p>
<p>Note that while $\Bbb R$ has cardinality $2^{\aleph_0}$, $\{\Bbb R\}$ has cardinality $1$. It is a singleton.</p>
<p>So to see that $\{(a,b)\subseteq\Bbb R\mid a,b\in\Bbb Q\}$ is countable, note that each s... |
2,148,389 | <p>Consider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 = 0$, I get a parabola, not a pair of straight lines. Why?</p>
<p>Plotting: <code>x^2 - 5x + 6 = 0</code> </p>
<p>at <... | JJacquelin | 108,514 | <p>$x^2-5x+6=0\quad$ is not the equation of the parabola. </p>
<p>The equation of the parabola is $\quad x^2-5x+6=y(x)$</p>
<p>$(x-3)(x-2)=0\quad$ is not the equation of two straight lines. </p>
<p>The equations of the two straight lines is $\quad x-3=y(x)\quad\text{and}\quad x-6=y(x)\quad$ or :
$$(x-2-y)(x-3-y)=0$$... |
2,148,389 | <p>Consider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 = 0$, I get a parabola, not a pair of straight lines. Why?</p>
<p>Plotting: <code>x^2 - 5x + 6 = 0</code> </p>
<p>at <... | Paul Sinclair | 258,282 | <p>I believe the other answers are correct in the interpretation of the question and in the way that they have addressed the issue. But I would like to look at the question a bit differently. Whether this is of any use or interest is up to the OP.</p>
<p>$y = (x -2)$ and $y = (x-3)$ are lines. So why is their product ... |
2,960,132 | <p>Prove or disprove each of the follow function has limits <span class="math-container">$x \to a$</span> by the definition </p>
<p><span class="math-container">$\lim_{(x, y) \to (0, 0)} \frac{x^2y}{x^2 + y^2}$</span> </p>
<p>Let <span class="math-container">$y = x^2$</span></p>
<p><span class="math-container">$... | HK Lee | 37,116 | <p>(1) Note that <span class="math-container">$X=(\mathbb{R}^2, d)$</span> is a metric space where
<span class="math-container">$d(x,y)=\|x-y\|$</span>. Hence <span class="math-container">$T$</span> is an isometry. </p>
<p>(2) Assume that
<span class="math-container">$$ \| v-u\| +\|u-x\|=\|v-x\|$$</span></p>
<p>Then ... |
3,965,967 | <p>Assume that customers arrive to a bus stop according to a homogenous Poisson process with rate <span class="math-container">$\alpha$</span> and that the arrival process of buses is an independent renewal process with interarrival distribution <span class="math-container">$F$</span>. <strong>What is the long run perc... | user8675309 | 735,806 | <p>I ended up writing my comment as a full solution. Main ideas:</p>
<p><strong>1.)</strong> Poisson process has zero probability of arrival at a particular time -- i.e. there is zero probability of a 'collision'<br />
<strong>2.)</strong> Define a renewal reward process, where a new epoch begins each time a bus arriv... |
1,817,300 | <p>How to integrate $\frac {\cos (7x)-\cos (8x)}{1+2\cos (5x)} $ ? </p>
<p>All I could do is apply difference of cosines formula in numerator.After that I'm stuck.Can somebody please help me?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\dfrac{\cos y-\cos(6A+y)}{1+2\cos2A}=\dfrac{2\sin(3A+y)\sin3A}{1+2(1-2\sin^2A)}$$
$$=\dfrac{2\sin(3A+y)\sin A(3-4\sin^2A)}{3-4\sin^2A}=\cos(2A+y)-\cos(4A+y)$$</p>
<p>Here $2A=5x,6A+y=8x\implies y=-7x$</p>
|
546,505 | <p>About Goldbach conjecture, (that Every even integer greater than 2 can be expressed as the sum of two primes) and if algorithms exist to solve the Halting Problem, then algorithm that determine Goldbach conjecture is true or false is exist? please explain</p>
| William Ballinger | 79,615 | <p>Although there is not algorithm that can solve the halting problem, it is still meaningful to talk about an imaginary computer that has some procedure that decides whether a given program halts or not, called an oracle for the halting problem. </p>
<p>If we had an oracle for the halting problem, then we could immed... |
2,759,306 | <p>I have been struggling to find out a solution for a case where i have a cubic Bezier curve where two arbitrary control points of the one are equal, therefore i should show that this curve can be the quadratic curve. Do you have any thoughts how to prove it?</p>
| Dr. Richard Klitzing | 518,676 | <p>Scroll down in <a href="http://caffeineowl.com/graphics/2d/vectorial/bezierintro.html" rel="nofollow noreferrer">http://caffeineowl.com/graphics/2d/vectorial/bezierintro.html</a> to the title "Transforming a quadratic Bezier in a cubic Bezier". There it is mentioned how the handles $C_1$ and $C_2$ of a cubic curve a... |
228,233 | <p>I need a continuous function $f:\mathbb{Q}\rightarrow \mathbb{R}$ and discontinuous $g:\mathbb{R}\rightarrow \mathbb{R}$
s.t $f(x)=g(x)$ for all rational $x$ s. So if I say $f(x)=0$ and $g(x)=0$ for $x \in \mathbb{R}\setminus\{\sqrt2\}$ and $g(x)=1$ at $x=\sqrt 2$ .would I be right?</p>
| hmakholm left over Monica | 14,366 | <p>Yes, that will work.</p>
<p>More interesting examples would be</p>
<p>$$ g(x) =\begin{cases} 1 & x\ge \sqrt 2 \\ 0 & x < \sqrt 2\end{cases} $$</p>
<p>or</p>
<p>$$ g(x) = \begin{cases} 1 & x\notin\mathbb Q \\ 0 & x \in \mathbb Q \end{cases} $$</p>
<p>with $f$ in each case being the restriction... |
228,233 | <p>I need a continuous function $f:\mathbb{Q}\rightarrow \mathbb{R}$ and discontinuous $g:\mathbb{R}\rightarrow \mathbb{R}$
s.t $f(x)=g(x)$ for all rational $x$ s. So if I say $f(x)=0$ and $g(x)=0$ for $x \in \mathbb{R}\setminus\{\sqrt2\}$ and $g(x)=1$ at $x=\sqrt 2$ .would I be right?</p>
| Mike | 25,748 | <p>You're right. But if that wasn't obvious to you, then a more interesting question is: why are you right? Can you prove it from the definition of continuity?</p>
|
3,847,358 | <p>If <span class="math-container">$\lim \limits_{n \to \infty} x_n + x_{n+1} =0 $</span> is <span class="math-container">$\lim \limits_{n \to \infty} \frac{x_n}{n}=0$</span>?</p>
<p>If <span class="math-container">$\lim \limits_{n \to \infty} x_n - x_{n+1} =0 $</span> then I would say <span class="math-container">$x_n... | user823011 | 823,011 | <p>If you have the basic knowledge about differential form and exterior differentiation, then <span class="math-container">$\mathrm{d}\mathrm{d}U$</span> do have its mathematical meaning.
<span class="math-container">$$
\mathrm{d}\mathrm{d}U=\mathrm{d}T\wedge\mathrm{d}S-\mathrm{d}P\wedge\mathrm{d}V=(\dfrac{\partial T}{... |
2,217,630 | <p>I need to find how many <em>real</em> roots this polynomial has and prove there existence. I was wondering if my logic and thought process was correct.</p>
<blockquote>
<p>Determine the number of <em>real</em> roots and prove it for $x^3 - 3x + 2$</p>
</blockquote>
<p>First, note that $f'(x) = 3x^2 - 3$ and so <... | Jack Smith | 841,375 | <p><span class="math-container">$x^3 - 3x + 2$</span></p>
<p>Possible roots are ±1 and ±2:</p>
<p><span class="math-container">$f(1) = 1 - 3 + 2 = 0$</span></p>
<p><span class="math-container">$f(2) = 8 - 6 + 2 = 4$</span></p>
<p><span class="math-container">$f(-1) = (-1) - (-3) + 2 = 6$</span></p>
<p><span class="math... |
618,665 | <p>Show that $\sqrt{13}$ is an irrational number.</p>
<p>How to direct proof that number is irrational number. So what is the first step..... </p>
| Lubin | 17,760 | <p>The equation $m^2=13n^2$ is a direct contradiction to the Uniqueness part of the Fundamental Theorem of Arithmetic, since the left side has evenly many $13$’s, while the right side has oddly many.</p>
|
1,102,216 | <p>Let $\mathcal{F}\left[f(t)\right](x)$ be the Fourier Transform of $f$, defined regularly as</p>
<p>$$\mathcal{F}\left[f(t)\right](x)=\int_{-\infty}^{\infty}f(t)e^{-itx}\,dt$$</p>
<p>And let $\mathcal{F}^{-1}\left[g(x)\right](t)$ be the Inverse Fourier Transform of $g$, defined regularly as $$ \mathcal{F}^{-1}\left... | Nya | 281,178 | <p>There is no $e^{ixt}e^{-ixt}$ but $e^{ixt}e^{-ixw}$ in Fourier transform.<br>
The key is $ \int_{-\infty}^{\infty} e^{-jwt} \, dw = 2\pi \delta(t) $.<br>
I'm going to show: <br>
$ f(s) = \frac{1}{2\pi} \int_{-\infty}^{\infty} [ \int_{-\infty}^{\infty} f(t)e^{-jwt} \, dt ] e^{jws} \, dw \\
=\frac{1}{2\pi} \int_{-\inf... |
2,203,995 | <blockquote>
<p>Show that the number of nonisomorphic groups of order $8181=3^4\cdot 101$ is equal to the number of nonisomorphic groups of order $81$. Find all the abelian groups of order $8181$ and at least one that is not abelian.</p>
</blockquote>
<p>the second part (finding all abelian groups) doesn't seem very... | Matt Samuel | 187,867 | <p>Both the Sylow $3$-subgroup and the Sylow $101$-subgroup are normal, so a group $G$ of order $8181$ is a direct product $A\times B$ where $A$ is of order $81$ and $B$ is of order $101$. There is a unique isomorphism class of groups of order $101$, so a group of order $8181$ is just the direct product of a group of o... |
3,305,140 | <p>So, I'm going to take an enumerative combinatorics class this upcoming semester. I began reading about it and came across and interesting example, but I am not sure how they arrive at their final answer. The example is in in the image I included. I don't know how they determined the equations for <span class="math... | Lutz Lehmann | 115,115 | <p>The absolute values in your calculation were applied properly where they are needed. However, in the next step you can remove them as follows:</p>
<p>As you observed, there are constant solutions at <span class="math-container">$u=0$</span> and <span class="math-container">$u=1$</span>. Due to uniqueness of solutio... |
1,006,445 | <p>Prove $(0,1)$ and $[0,1]$ have the same cardinality. </p>
<p>I've seen questions similar to this but I'm still having trouble. I know that for $2$ sets to have the same cardinality there must exist a bijection function from one set to the other. I think I can create a bijection function from $(0,1)$ to $[0,1]$, ... | Jean-Claude Arbaut | 43,608 | <p>Use <a href="http://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" rel="nofollow">Cantor-Bernstein theorem</a>:</p>
<p>You can trivially find a bijection between $(0,1)$ and $(1/4,3/4)\subset[0,1]$, hence $\mathrm{Card} (0,1) \leq \mathrm{Card} [0,1]$.</p>
<p>Likewise, there is a trivial bijection ... |
3,300,954 | <p>How to prove P(A ∩ B) ≤ P(A) using probability theory?</p>
<p>I understand this when drawn on a Venn Diagram but am unsure how it translates to a formal proof. </p>
| Claude Leibovici | 82,404 | <p>You need to find the zero of function
<span class="math-container">$$f(t)=30\,t \,e^{30t}-\frac 3 {10}$$</span> For the time being, let <span class="math-container">$x=30t$</span> and the problem becomes
<span class="math-container">$$g(x)=xe^x-\frac 3 {10}\implies g'(x)=(x+1)e^x > 0 \quad \forall x >0$$</span... |
1,604,204 | <blockquote>
<p><span class="math-container">$a,b,c$</span> are in A.P ; <span class="math-container">$p,q,r$</span> are in H.P. And <span class="math-container">$ap,bq,cr $</span> are in G.P. Then what is the value of <span class="math-container">${p\over r} + {r\over p}\ \ ?$</span></p>
<p><span class="math-container... | Adriano | 76,987 | <p>Let's eliminate $d, k, m$:
\begin{align}
c - b &= b - a \tag 1\\
\frac{1}{r} - \frac{1}{q} &= \frac{1}{q} - \frac{1}{p} \tag 2 \\
\frac{cr}{bq} &= \frac{bq}{ap} \tag 3
\end{align}
We're aiming for $\frac{p}{r} + \frac{r}{p} = \frac{p^2 + r^2}{rp}$. That sort of looks like $(2)$. Let's try manipulating $(... |
4,105,190 | <p>Just been trying to prove the following by mathematical deduction for research but having some issues. Mind helping out?</p>
<p>Prove that A = ∅ if and only if B = A∆B.</p>
<p>What I have so far...
A∆B = (A-B)∪(B-A)
= (A∩B^Compliment)∪(B∩A^Compliment)</p>
<p>But not too sure how to explain or go from here...</p>
| Igor Rivin | 109,865 | <p>Hint: can you produce upper and lower bounds for the numerator?</p>
|
194,179 | <p>I want to work with linear expressions involving the formal symbol <span class="math-container">$w[a_1,...,a_n]$</span>, and I would like Mathematica to know that expressions such as</p>
<pre><code>w[a,b,d] + w[a,d,b] = 0
</code></pre>
<p>i.e. that the symbol <code>w</code> is antisymmetric with respect to the swa... | Carl Woll | 45,431 | <p>You just need to use the 3-arg syntax for Subsets. Here is a function that does this:</p>
<pre><code>MultipleRandomSubsets[list_, length_, count_] := Module[{total},
total = Binomial[Length @ list, length];
Join @@ Map[Subsets[list, {length}, {#}]&] @ RandomSample[1 ;; total, count] /; count <= total... |
2,522,291 | <p>I have two matrices where $A = $ \begin{bmatrix}1&-2\\-1&3\end{bmatrix} and $B =$ \begin{bmatrix}0&-1\\2&-2\end{bmatrix}. The question asks to Solve for the matrix C (i.e. find matrix C):
$$(((AB^T)^{-1})^TB)^T= 2C + B$$</p>
<p>I have done the calculations by following the formula and get \begin{bma... | gt6989b | 16,192 | <p>Note that
$$
(((AB^T)^{-1})^TB)^T
= B^T ((AB^T)^{-1})
= B^T \left(B^T\right)^{-1} A^{-1}
= A^{-1}.
$$
So you have
$$
A^{-1} = 2C+B
$$
and hence
$$
C = \frac{A^{-1}-B}{2}
$$</p>
|
3,831,510 | <p>If <span class="math-container">$a$</span> is a constant, what is the name of a curve of the form <span class="math-container">$a*(x+y) = x*y$</span>? And how is this equation related to more this curve's more general equations/characteristics? Plotting this curve, I would risk calling it a hyperbola, but I'm not su... | Soumyadwip Chanda | 823,370 | <p>Every 2-degree equation in x and y represents a conic. So, if you see two branches in its graph, it is always going to be a hyperbola.</p>
|
112,333 | <p>Consider a set of rules, e.g.</p>
<pre><code>{a -> aa, b -> bb, c -> cc, d -> dd, e -> ee}
</code></pre>
<p>I want to remove from this list all patterns of the form e->_ and to do so, I would like to form the Complement of matching patterns in my original set.</p>
<p>However if I use Cases, the des... | Adalbert Hanßen | 21,390 | <pre><code>Select[{a -> aa, b -> bb, c -> cc, d -> dd, e -> ee} ,
MatchQ[#, Rule[e, Blank[]]] &]
</code></pre>
<p>returns</p>
<pre><code>{e->ee}
</code></pre>
<p>To remove all patterns involving e->_ use</p>
<pre><code>Complement[#, Select[#, MatchQ[#, Rule[e, Blank[]]] &]] & @ {a ... |
2,758,434 | <p>I have an Mercator Projection Map:
<a href="https://i.stack.imgur.com/RDpHU.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RDpHU.jpg" alt="enter image description here"></a></p>
<p>And I try to calculate the Latitude from a specific Y-Point on the map.
According to this article : <a href="http:/... | Crostul | 160,300 | <p>Counterexample:</p>
<p>$R=\Bbb Q [X_1, X_2, \dots , X_n, \dots]$. Define the ring morphism $f: R \to R$ by
$$\begin{cases}X_i \mapsto X_{i-1} & i>1 \\ X_1 \mapsto 0
\end{cases}
$$</p>
<p>and call $I= \ker f$.</p>
<p>Then $$R/I \cong \mathrm{Im} f= R$$</p>
|
2,758,434 | <p>I have an Mercator Projection Map:
<a href="https://i.stack.imgur.com/RDpHU.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RDpHU.jpg" alt="enter image description here"></a></p>
<p>And I try to calculate the Latitude from a specific Y-Point on the map.
According to this article : <a href="http:/... | Community | -1 | <p>There's a subtlety here: what do you mean by "is isomorphic to"?</p>
<p>In a setting where the constructions of <span class="math-container">$A$</span> and/or <span class="math-container">$B$</span> yield a canonical map <span class="math-container">$f:A \to B$</span>, the phrase "<span class="math-container">$A$</... |
159,775 | <p>Anybody can help me to have an idea about an example showing the difference of a Physical measur with compare to an SRB measure?</p>
<p>By a Physical measure i mean in the sense of $\nu$ a hyperbolic(non-atomic) measure then having positive Lebesgue Basin of attraction.
&
By an SRB measure i mean in the sense ... | Martin Andersson | 50,157 | <p>Physical measure always means that the basin has positive Lebesgue measure. </p>
<p>SRB is simetimes synonymous to physical measure, and simetimes used to mean that it is hyperbolic and has a desintegration along unstable manifolds which is absolutely continuous wrt leaf volume. If you further suppose that such a m... |
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