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 |
|---|---|---|---|---|
1,829,086 | <p>So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$
to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$</p>
<p>which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$
But I'm not sure if this is the right way to go, or if I try something else.</p>
<p>Any tips or methods would be very helpful.</p>
| mdcq | 454,513 | <p>Here is yet another alternative.</p>
<p>As already shown in other answers we can compute $\int \frac {1}{\cos(t)} \thinspace {\rm {d}} t$ by computing $\int \frac {1}{\sin(x)} \thinspace {\rm {d}} x$ upon making use of the substitution $x=t+\frac{\pi}{2}$ and ${\rm {d}} x = {\rm {d}} t$.</p>
<p>By using the trigon... |
3,332,927 | <p>An analytic function that maps the entire complex plane into the real axis must map the imaginary axis onto:</p>
<p>A) the entire real axis</p>
<p>B) a point</p>
<p>C) a ray</p>
<p>D) an open finite interval</p>
<p>E) the empty set</p>
<p>I was thinking that it might be a constant function.
Any help would be ... | achille hui | 59,379 | <p>If <span class="math-container">$f(\mathbb{C})\subset \mathbb{R}$</span>, then <span class="math-container">$|e^{if(z)}| = 1$</span> for all <span class="math-container">$z$</span>. By <a href="https://en.wikipedia.org/wiki/Liouville's_theorem_%28complex_analysis%29" rel="nofollow noreferrer">Liouville's theorem... |
728,916 | <p>I have been developing an RPG and I finally want to program a better development curve. I have decided to use the following equations</p>
<p>Linear, Quadratic and Cubic</p>
<p>The Linear one is easy. I have no problems with that. But I'm stuck with the other 2.</p>
<p><strong>Quadratic</strong></p>
<p>The formul... | lakshayg | 137,874 | <p>For the quadratic case all you need is two pairs of x and y. Let these be $(x_1, y_1) ,(x_2,y_2)$. On putting these values in your equation we get:
$$ax_1^2+bx_1=y_1-c$$
$$ax_2^2+bx_2=y_2-c$$
$$\implies \left[{\begin{array}{cc}
x_1^2 & x_1\\
x_2^2 & x_2\\
\end{array}}\right]
\left[{\begin{array}{... |
233,397 | <p>could anyone please clarify me the meaning of the term 'hypothesis'? </p>
<p>with relation to terms 'reasoning' and 'assumption' ?</p>
<p>Many thanks</p>
| Rekhaa Kale | 298,037 | <p>Hypothesis is a tentative supposition made to explain, understand, define or predict any event or situation.
This is further tested and verified and once proved, accepted as an explanation.
This means, hypothesis is not permanent in nature, but is temporary, till its truth and validity is proved. If it is not proved... |
898,002 | <p>Let's say I have a continuous piecewise function of a single variable, so that $y = f(x)$ if $x < c$ and $y = g(x)$ if $x>=c$. Is it right to say that the derivative of the function at $x=c$ exists iff $f'(c-)=g'(c+)$, where $f'$ and $g'$ are obtained using derivative rules?</p>
<p>This would seem reasonable ... | Vera | 169,789 | <p>In general a function $h$ is differentiable at $c$ iff $\lim_{x\rightarrow c+}\frac{h\left(x\right)-h\left(c\right)}{x-c}$
and $\lim_{x\rightarrow c-}\frac{h\left(x\right)-h\left(c\right)}{x-c}$
both exist and are equal. </p>
<p>In your case $\lim_{x\rightarrow c+}\frac{h\left(x\right)-h\left(c\right)}{x-c}=\lim_{x... |
2,649,907 | <p>As the titles states I have to determine weather or not polynomials of the form $a_{0}+a_{1}x$ is a subspace of $P_{3}$, the polynomials of the form $a_{0}+a_{1}x$ have $a_{0}$ and $a_{1}$ as real numbers.</p>
<p>So since the polynomial is of the third degree, the entire polynomial would look like:</p>
<p>$a_{0}+a... | Siong Thye Goh | 306,553 | <p>Step $1$: check that $Q=\{a_0+a_1x | a_0, a_1 \in \mathbb{R}\}$ is a subset of $P_3$.</p>
<p>This is true since we can write $a_0+a_1x=a_0+a_1x+0x^2+0x^3$.</p>
<p>Step $2$: check closure under addition: </p>
<p>$$(a_0 + a_1 x )+(b_0+b_1)x=(a_0+a_1)+(a_1+b_1)x$$ </p>
<p>Note that $a_0+a_1, b_0+b_1 \in \mathbb{R}$... |
4,346,765 | <p>While I was solving an integral using Feynman Integration, I came across the following differential equation:</p>
<p><span class="math-container">$$y’’’’-y’’+y=0$$</span></p>
<p>I tried substituting <span class="math-container">$y$</span> with an exponential function which failed. Can someone else show me how to sol... | reuns | 276,986 | <p>Take a solution to <span class="math-container">$a^2+b^2+1=0$</span>, there is always one for <span class="math-container">$k$</span> a finite field.</p>
<p>For <span class="math-container">$x\in k^4$</span>, let <span class="math-container">$P(x)=(x_1,x_2,ax_3+bx_4,-bx_3+ax_4)$</span>.</p>
<p>You'll get that <span ... |
4,488,740 | <p>It is known that <span class="math-container">$$\sin^{−1}x+\sin^{−1}y = \sin^{-1}\left[x\sqrt{1 – y^2} + y\sqrt{1 – x^2}\right] $$</span> if <span class="math-container">$x, y ≥ 0$</span> and <span class="math-container">$x^2+y^2 ≤ 1.$</span></p>
<p>I know that the given condition makes sure that <span class="math-c... | eyeballfrog | 395,748 | <p>Here's a general method for any rational <span class="math-container">$0 < m/n < 1$</span> that uses mean inequalities, which notably can be proven with no calculus.</p>
<p>By the arithmetic-geometric mean inequality, for any positive real numbers <span class="math-container">$a,b$</span> and integers <span cl... |
1,923,761 | <p>I have a problem in which I need to discover $f$ knowing that $$\left\{\begin{matrix}f(1,y)-f(0,y)=y\\f(x,1)-f(x,0)=x\end{matrix}\right.$$ Any hints to solve it?</p>
| Elias Costa | 19,266 | <p><strong>Hint.</strong> Assume $f:\mathbb{R}^2\to \mathbb{R}$. Take continuous $f$ (in fact you can prove it). Note that for all $\alpha,\beta \in\mathbb{R}$,
$$
\left\{\begin{matrix}
\beta f(1, y)-\beta f(0, y)=\beta y, \\
\alpha f( x,1)- \alpha f( x,0)=\alpha x
\end{matrix}\right.
\Leftrightarrow
\left\{\begin{mat... |
2,254,672 | <p>I'm trying to solve a differential equation for the modeling of an ultrasonic horn in wich his form is a catenary, and his differential equation for a wave involves a hyperbolic tangent. I have the solution for the differential equation, but I need to do the "steps by steps". I had transformed the trigonometric coef... | Alex Jones | 350,433 | <p>Starting from the second form:</p>
<p>Divide the whole equation by the constant coefficient of $v^{\prime\prime}$ to get $(1+u)^2v^{\prime\prime} + \left( \frac{rw}{kc} \right)^2v = (u^2+2u+1)v^{\prime\prime} + \omega^2v = 0$. Let $v = \sum_{i=0}^\infty a_iu^i$, and differentiation term-by-term gives $v^{\prime\pr... |
3,502,123 | <p>I've seen many proofs of this theorem. But, unable to think any example showing this.</p>
<p>Suppose, how to write (0,1) as a countable Union of disjoint open intervals. </p>
<p>No idea! I'm stuck </p>
<p>Plz help!</p>
| Severin Schraven | 331,816 | <p>I understood <span class="math-container">$\mu^k=\mu$</span> in a poinwise sense, i.e. for all measurable sets <span class="math-container">$A$</span> we should have <span class="math-container">$\mu(A)^k=\mu(A)$</span>. </p>
<p>For all measurable sets <span class="math-container">$A$</span> we have <span class="ma... |
3,371,516 | <p>The question is as follows:</p>
<p>Suppose that E, F, and G are events with P(A) = 19/100, P(B) = 7/25, P(C) = 3/10. Furthermore, suppose A andB are mutually exclusive, A and C are independent, and P(B | C) = 11/15. Find P(A ∪ B ∪ C).</p>
<p>My attempt is as follows:</p>
<p>P(A ∪ B ∪ C) = P(A) + P(B)... | azif00 | 680,927 | <p><span class="math-container">$$\textsf{P}(A\cup B\cup C)=\textsf{P}(A)+\textsf{P}(B)+\textsf{P}(C)-\textsf{P}(A\cap B)-\textsf{P}(A\cap C)-\textsf{P}(B\cap C)+\textsf{P}(A\cap B\cap C)$$</span>
Now, <span class="math-container">$\textsf{P}(A\cap B)=\textsf{P}(A\cap B\cap C)=0$</span> since <span class="math-containe... |
4,111,094 | <p>In a convex quadrilateral <span class="math-container">$PQRS$</span>, the areas of triangles <span class="math-container">$PQS$</span>, <span class="math-container">$QRS$</span> and <span class="math-container">$PQR$</span> are in the ratio <span class="math-container">$3 : 4 : 1$</span>. A line through <span class=... | Jean Marie | 305,862 | <p>Here is a solution using <a href="https://www.geogebra.org/m/c8DwbVTP" rel="nofollow noreferrer">barycentric coordinates</a> which is a way to work on ratios of areas. They are often useful in Mathematical Contests.</p>
<p>See the following picture.</p>
<p><a href="https://i.stack.imgur.com/xBjWF.jpg" rel="nofollow ... |
4,111,094 | <p>In a convex quadrilateral <span class="math-container">$PQRS$</span>, the areas of triangles <span class="math-container">$PQS$</span>, <span class="math-container">$QRS$</span> and <span class="math-container">$PQR$</span> are in the ratio <span class="math-container">$3 : 4 : 1$</span>. A line through <span class=... | Math Lover | 801,574 | <p><a href="https://i.stack.imgur.com/nuK5Q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nuK5Q.png" alt="enter image description here" /></a></p>
<p>As we know the ratios of the areas of internal triangles, we have</p>
<p><span class="math-container">$PO:OR = 3:4, QO:OS = 1:6$</span></p>
<p>We ass... |
271,886 | <p>I am interested in explicit generators of the cohomology $H^\bullet(SU(n),\mathbb{Z})$. Let $\omega = g^{-1} dg$ be the Maurer-Cartan form on $SU(n)$. The forms $\alpha_3,\alpha_5,\dots,\alpha_{2n-1}$, defined by
$$ \alpha_k := \text{Tr}(\omega^{k})$$
are bi-invariant and define classes in de Rham cohomology.</p>
<... | Matthias Wendt | 50,846 | <p>It seems the required coefficient is given in </p>
<ul>
<li>R. Bott and R. Seeley. Some remarks on the paper of Callias: "Axial anomalies and index theorems on open spaces". Comment. Math. Phys. 62 (1978), 235-245. <a href="https://projecteuclid.org/euclid.cmp/1103904396" rel="noreferrer">link to paper on Project E... |
2,544,544 | <p>There is a theorem stated in my textbook as following and my question is below the proof :</p>
<p>Let <span class="math-container">$0<p<q<\infty$</span> and let <span class="math-container">$f$</span> in <span class="math-container">$L^{p,\infty}(X,\mu) \cap L^{q,\infty}(X,\mu)$</span>, where <span class="... | Martin Argerami | 22,857 | <p>This is simply
$$
\frac{\|f\|_q^q}{\alpha^q}\geq\frac{\|f\|_p^p}{\alpha^p}
\iff
\alpha^{q-p}\leq \frac{\|f\|_q^q}{\|f\|_p^p}
\iff
\alpha\leq \left(\frac{\|f\|_q^q}{\|f\|_p^p}\right)^{\frac1{q-p}}=B.
$$
Thus
$$
\min\left\{\frac{\|f\|_q^q}{\alpha^q},\frac{\|f\|_p^p}{\alpha^p}\right\}
=\begin{cases}
\frac{\|f\|_p^p}{\a... |
2,041,499 | <blockquote>
<p>The solution to the BVP <span class="math-container">$\frac{d^2y}{dx^2}+y =\csc x$</span>, <span class="math-container">$0 < x < \frac{\pi}{2}$</span> <span class="math-container">$y(0)=0$</span>, <span class="math-container">$y(\pi/2)=0$</span> is</p>
<p><span class="math-container">$(A)$</span> ... | Kovalevskaya | 261,757 | <p>You have: </p>
<p>$$y''(x)+y(x)=\csc(x)$$</p>
<p>Solving an ODE like this can be solved by looking for the complementary and particular solution. For further information look at <a href="https://en.wikipedia.org/wiki/Variation_of_parameters" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Variation_of_para... |
1,328,738 | <p>Let $c$ be a close curve such that $c$ does not intersect itself, $c\in \mathbb{R}^2$ (in the plane), show that for all point $P$ that surrounded in $c$ there are two points $A,B$ on $c$ such that $P$ is in the middle of the interval of $A$ & $B$</p>
<p>As you can see English isn't my first language.</p>
| Sachchidanand Prasad | 249,258 | <p>Let $P(x)= x^4−2x^3+4x^2+6x−21=0$ Let $a,-a $ are the two roots given. So, $P(a)=P(-a)=0$
\begin{align}
a^4−2a^3+4a^2+6a−21 &= &a^4+2a^3+4a^2-6a−21\\
\implies 4a^3-12a =0\\
\implies a(a^2-3)=0\\
a=0,a=\pm\sqrt3
\end{align}
But $a=0$ is not a root of the polynomial. Hence $\pm \sqrt3$ are the two roots of $P... |
1,947,630 | <p>Describe the set of relations on $\mathbb{Z}$ which are both symmetric and anti-symmetric. Hint: this set is infinite and contains one relation with which you are already familiar.</p>
<p>I know that this is clearly talking about the equality relation, but what I am confused about is what does a set of relations me... | Adriano | 76,987 | <p>You're correct, it is the power set of $S$, namely:
$$
\mathcal P(S) = \{\{(x, x) \mid x \in A\} \mid A \subseteq \mathbb Z\}
$$</p>
|
4,403,249 | <p><span class="math-container">$$f(x,y)=(x^2+y^2)^{\left|x\right|} $$</span> find limit at <span class="math-container">$(0,0)$</span></p>
<p>moving into parametric form <span class="math-container">$f(r\cos\phi,r\sin\phi)=(r^2)^{r\left|\cos\phi\right|}= e^{2r\left|\cos\phi\right|\ln r}$</span></p>
<p><span class="mat... | dictatemetokcus | 868,849 | <p>Note that <span class="math-container">$\lim_{x \to 0} x \ln(x) = \lim_{x \to 0} \frac{\ln(x)}{\frac{1}{x}} = \lim_{x \to 0} \frac{\frac{1}{x}}{-\frac{1}{x^2}} = 0$</span>.</p>
<p>Now as <span class="math-container">$e^x$</span> is continuous and increasing over the reals, we have
<span class="math-container">$$e^{0... |
3,063,130 | <blockquote>
<p>Suppose function <span class="math-container">$f(z)$</span> is holomorphic on <span class="math-container">$\mathbb{D}(0,2)$</span> and <span class="math-container">$N>0$</span>
is an integer such that: <span class="math-container">$$ |f^{(N)}(0)| = N! \sup\{|f(z)|: |z|=1\} $$</span>
show that ... | Aphelli | 556,825 | <p>For every <span class="math-container">$0 \leq \theta \leq 1$</span>, <span class="math-container">$f(e^{2i\pi\theta})e^{-2iN\pi\theta}$</span> has an absolute value not greater than <span class="math-container">$a_N$</span>, but the integral over <span class="math-container">$[0;1]$</span> is exactly <span class="m... |
4,080,680 | <p>If <span class="math-container">$\sum a_n$</span> diverges does <span class="math-container">$\sum \frac{a_n}{\ln n}$</span> necessarily diverge for <span class="math-container">$a_n>0$</span>?</p>
<p>I tried <span class="math-container">$a_n=\frac{\ln n}{n^2}$</span> to try bag an easy counterexample but it turn... | Andrea Marino | 177,070 | <p>Other answers are enough, just an observation on how to derive a counterexample: using Abel summation formula, aka discrete integration by parts, the general term is</p>
<p><span class="math-container">$$\sim \sum A_n (1/ \log(x)) '_{x = n} =- \sum A_n (1/n \log(n) ^2 ) $$</span></p>
<p>Here <span class="math-conta... |
2,762,951 | <p>Let $u$ be a $W^{1,p}$-Sobolev function defined on the half disk $D^2_+ := \{(x,y)\in \mathbb{R}^2| x^2+y^2 < 1, y > 0\}$.</p>
<p>If I define a function $\tilde u$ by
$$ \tilde u\colon (x,y) \mapsto \begin{cases} u(x,y) & y > 0 \\ u(x,-y) & y < 0\end{cases}, $$
is this $\tilde u$ a Sobolev $W^{1... | Revzora | 485,453 | <p>Since no one answers, I write some ideas that can be useful.</p>
<p>I prove the analogous problem in $\mathbb{R}$ ( the same idea can be use in your problem ).</p>
<p>I want use that
\begin{equation}u\in W^{1,1}((a,b)) \text{ iff } u\in AC((a,b))\\ \text{
(In the sense that there is an absolutely continuous repre... |
3,096,351 | <blockquote>
<p><span class="math-container">$$x_1 + y_1 = 3$$</span>
<span class="math-container">$$y_1 - x_3 = -1$$</span>
<span class="math-container">$$x_3 + y_3 = 7$$</span>
<span class="math-container">$$x_1 - y_3 = -3$$</span></p>
<p>Find the values of <span class="math-container">$x_1$</span>, <spa... | Henno Brandsma | 4,280 | <p>Given: <span class="math-container">$A \subseteq B$</span>.</p>
<p><span class="math-container">$C \in \mathscr{P}(A)$</span> implies <span class="math-container">$C \subseteq A$</span> so <span class="math-container">$C \subseteq B$</span>, so <span class="math-container">$C \in \mathscr{P}(B)$</span>. Conclusion ... |
990,418 | <p>What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. </p>
<p>I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin B+\sin C=\frac{3\sqrt3}2$. And also <a href="http://www.wolframalpha.com/input/?i=max+sin(a)%2Bsin(b)%2Bsin(c)... | DuFong | 193,997 | <p>Observe that $\sin A+\sin B=2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2})\leq2\sin(\frac{A+B}{2})=2\cos\frac{C}{2}$</p>
<p>so it it sufficient to show that $\sin C+2\cos \frac{C}{2}\leq \frac{3\sqrt{3}}{2}$</p>
<p>In fact assume $t=\cos\frac{C}{2}$, above becomes $2t\sqrt{1-t^2}+2t$, so finally we only need to estimate ... |
101,566 | <p>How do I generalize the equation to be able to plug in any result for $\phi(n)=12$ and find any possible integer that works?</p>
| Angela Pretorius | 15,624 | <p>If there are more than 12 primes between n and n!, then $\phi(n!)>12$ and there is no $m>n!$ such that $\phi(m)>12$. </p>
<p>$n<120$ and a brute force search can be used. </p>
|
10,674 | <p>Let $p$ be a rational prime and $K$ a number field.
Dedekind's discriminant theorem tells us that
$p$ ramifies in $K$ $\iff$ $p$ divides the discriminant of $K$.
Hence if $p$ does not divide discriminant of $K$,
$(p)$ either splits, i.e., </p>
<p>(i) $(p)=P_1 \cdots P_g$ for $P_i \neq P_j$ and $g \geq 2$
or </p>... | Pete L. Clark | 1,149 | <p>The best explicit criterion that I know is the criterion of Kummer-Dedekind, which involves writing $K = \mathbb{Q}[t]/(P(t))$ and factoring $P(t)$ modulo the prime $p$. Then the factorization of $(p)$ in $\mathbb{Z}_K$ "has the same shape" as the factorization of $P(t)$ in $(\mathbb{Z}/p\mathbb{Z})[t]$: see e.g.</... |
1,089,949 | <p>I would like some comments on how I approach this problem. The part right before this problem in my homework asks for an existential formula that defines the set of even integers. Please let me know if my thoughts are not correct or could be better.</p>
<p><em>Edit: the language contains just one binary function sy... | Wojowu | 127,263 | <p>I'll show the following fact: if $P(x_1,x_2,...,x_k)$ is quantifier free in the language of $(\Bbb Z,+)$, then for all $v_1,...,v_k$ we have $P(v_1,...,v_k)\Leftrightarrow P(2v_1,...,2v_k)$. Indeed, by induction on construction of well-founded formulas: formula $t_1+...+t_i=t_{i+1}+...+t_j$ is true iff $2t_1+...+2t_... |
1,089,949 | <p>I would like some comments on how I approach this problem. The part right before this problem in my homework asks for an existential formula that defines the set of even integers. Please let me know if my thoughts are not correct or could be better.</p>
<p><em>Edit: the language contains just one binary function sy... | Primo Petri | 137,248 | <p>($\#$) Recall that embeddings preserve the truth of existential formulas.</p>
<p>Suppose for a contradiction that $\varphi(x)\in L$ defines the set of odd integers. The map $f:x\mapsto 2x$ is an embedding of $(\mathbb{Z},+)$ into itself. By ($\#$) above $\mathbb{Z}\models\phi(n)$ implies $\mathbb{Z}\models\phi(fn)$... |
3,082,337 | <p>Forgive my ignorance.<br>
Is the condition <span class="math-container">$x\in\mathbb{R}$</span> necessary to the set statement <span class="math-container">$\{x \in\mathbb{R} \vert x> 0\}$</span>?<br>
In other words, if <span class="math-container">$x$</span> is greater than zero, then is it not, by definition, a... | DanLewis3264 | 480,329 | <p>It's absolutely necessary. You could have for instance
<span class="math-container">$$\{x \in \mathbb{Q}: x>0\} $$</span>
which only includes positive rationals, not the irrationals.</p>
<p>It's common to abbreviate these to <span class="math-container">$\mathbb{R}_{>0}$</span> and <span class="math-container... |
4,173,559 | <p>The integral is</p>
<p><span class="math-container">$$\int\frac{1}{x\sqrt{1-x^2}}dx\tag{1}$$</span></p>
<p>I tried solving it by parts, but that didn't work out. I couldn't integrate the result of substituting <span class="math-container">$t=1-x^2$</span> either.</p>
<p>The answer is</p>
<p><span class="math-contain... | Quanto | 686,284 | <p>With the substitution <span class="math-container">$t=\sqrt{1-x^2}$</span>, the integral can be integrated as follows
<span class="math-container">$$\int\frac{1}{x\sqrt{1-x^2}}dx=-\int \frac1{1-t^2}dt= \frac12 \ln\frac{1-t}{1+t}= \frac12\ln \frac{1-\sqrt{1-x^2}}{1+\sqrt{1-x^2}}+C
$$</span>
which is the same as <span... |
3,688,829 | <p>We're currently analyzing the convergence of function sequences.</p>
<p>I need to prove
<span class="math-container">$$
\lim_{n \to \infty} \left(1 + \frac{z}{n}\right)^n
$$</span></p>
<p>is not uniformly convergent on <span class="math-container">$\mathbb{C}$</span>. Can I just use the equivalence <span class... | Aryaman Maithani | 427,810 | <p>Suppose that the convergence were uniform. Then, for <span class="math-container">$\epsilon = 1$</span>, there exists <span class="math-container">$N \in \Bbb N$</span> such that
<span class="math-container">$$\left|\left(1 + \dfrac{z}{n}\right)^n - e^z\right| < 1$$</span>
for all <span class="math-container">$n ... |
101,384 | <p>Calculate the Lebesgue integral of the function</p>
<p>$$ f(x,y)=\left\lbrace\begin{array}{ccl}[x+y]^{2} &\quad&|x|,|y| <12 ,\quad xy \leq 0\\
0 &\quad&\text{otherwise}\end{array} \right.$$</p>
<p>in $\mathbb{R}^2$.</p>
<p>Can anyone help with this? I can't find a way to make the expression of... | Norbert | 19,538 | <p>Denote
$$
A_{m,n}=\{(x,y):m\leq x<m+1,\quad n\leq y<n+1\}\qquad
a_{mn}=\int_{A_{m,n}}f(x,y)d\mu(x,y)
$$
then
$$
\int_{\mathbb{R}^2}f(x,y)d\mu(x,y)=\sum_{(m,n)\in\mathbb{Z}^2}a_{mn}
$$
From definition of $f$ it follows that $a_{mn}\neq 0$
only for pairs $(m,n)\in\mathbb{Z}^2$ such that $-N\leq m\leq N-1$, $-N\l... |
2,363,211 | <p>Can <span class="math-container">$406014677132263504491682$</span> be the sum of two fourth powers? It may be so. Can anyone use Wolfram Mathematica , SAGE , or some computer program, to check whether this number is the sum of two fourth powers ?</p>
<p>The complete factorization of this number is given by : <span... | Shuri2060 | 243,059 | <p>You may want to try some program like this in future for testing.</p>
<p>However, I cannot say anything myself about how the program handles precision, etc, with taking $4$th powers and roots of large numbers, but at least you can be sure it's possible if the program returns with a positive (as you can verify it yo... |
4,213,708 | <p><strong>Question</strong>
Is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? Give reasons for your answer.</p>
<p><strong>My attempt at the solution</strong>
Let us take a piece of un-stretched rubber ban... | Henry | 6,460 | <p>For me the translation of "choosing form <span class="math-container">$n+1$</span> a committee of <span class="math-container">$k$</span> people of which <span class="math-container">$1$</span> is president" would lead to the identity <span class="math-container">$${n+1 \choose k}{k \choose 1} = {n+1 \choo... |
4,213,708 | <p><strong>Question</strong>
Is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? Give reasons for your answer.</p>
<p><strong>My attempt at the solution</strong>
Let us take a piece of un-stretched rubber ban... | Saksham Sethi | 953,997 | <p>I'm not sure I understand your question, but I think you are not understanding how the <span class="math-container">$\binom{n}{k-1}$</span> is formed.</p>
<hr />
<p>Assume that the <span class="math-container">$(n-1)^{\text{th}}$</span> person is the president. We already counted the case when the president isn't in... |
291,102 | <p>I have a linear algebra question I need help with. </p>
<blockquote>
<p>Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the identity matrix. </p>
</blockquote>
<p>I know that $\|Ax\|_2 \leq C\|x\|_2$ for some constant $C$... | Zach L. | 43,128 | <p>Since $\|A\|_2 < 1$, the sequence of matrices $S_n = \sum_{i=0}^n A^n$ is Cauchy (as a sequence in $L^2$) and so converges in the $L^2$-norm to some matrix $S$. Now, $S_n(I - A) = I - A^n$, so taking the limit in $L^2$ gives $S(I-A) = I$. Something similar works for the other side.</p>
<p>There may be some other... |
291,102 | <p>I have a linear algebra question I need help with. </p>
<blockquote>
<p>Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the identity matrix. </p>
</blockquote>
<p>I know that $\|Ax\|_2 \leq C\|x\|_2$ for some constant $C$... | Mhenni Benghorbal | 35,472 | <p>A <a href="https://math.stackexchange.com/questions/325891/left-cdot-right-is-an-induced-norm-if-left-a-right-1-how/326098#326098">related problem</a>. Note this,</p>
<p>$$ (I-A)^{-1} = \sum_{n=0}^{\infty}A^n. $$</p>
<p>Now, for the above series to converge, you need to impose the condition $||A||_2<1$. Just co... |
1,780,352 | <p>Suppose we have a series</p>
<p>$$\sum^{\infty}_{n = 0} \frac{\sqrt{n}}{(n+1)^2 + |z|^2}$$</p>
<p>As mentioned in the title, how to find the radius of convergence?</p>
<p>In my opinion, $|z| \in \mathbb{C}$, $|z|^2$ is giving a real number, hence we can fix $|z|^2$ regardlessly. But then again, this is not a powe... | marty cohen | 13,079 | <p>I'll plod on and see
what happens.</p>
<p>We want to show that
$\int_{0}^{2\pi}\log|1-e^{it}|dt
= 0
$.</p>
<p>$\begin{array}\\
|1-e^{it}|
&=|1-\cos(t)-i\sin(t)|\\
&=\sqrt{(1-\cos(t))^2+\sin^2(t)}\\
&=\sqrt{1-2\cos(t)+\cos^2(t)+\sin^2(t)}\\
&=\sqrt{2-2\cos(t)}\\
&=\sqrt{2(1-\cos(t))}\\
&=\s... |
847,887 | <p>I'm having a very hard time resolving the system of equations after using the Lagrange Multipliers optimization method. For instance:</p>
<p>The plane $ x + y + 2z = 2 $ intersects the paraboloid $ z = x^2 + y^2 $ over an ellipse. Find the ellipse points that are nearer and farther from the origin.</p>
<p>I know t... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>As commented by David, some of your partial derivatives are wrong and I suppose that you have fixed them.</p>
<p>In any manner, the first four derivatives make a linear system. So, just solve it the manner you want and, as a result, $x,y,z,L_1$ will simply express as function of $L_2$.... |
4,374,739 | <p>Spivak's chapter 6, question 1 asks: for which of the following functions <span class="math-container">$f$</span> is there a continuous function <span class="math-container">$F$</span> with domain <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$F(x) = f(x)$</span> for all <sp... | evianpring | 333,842 | <p>This is roughly what the graph of <span class="math-container">$f$</span> looks like:</p>
<p><a href="https://i.stack.imgur.com/zJPLQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zJPLQ.png" alt="enter image description here" /></a></p>
<p>Note that I drew only a few blue points, representing ra... |
4,491,385 | <p><a href="https://i.stack.imgur.com/ILqh9.png" rel="nofollow noreferrer">This is the graph plotted by Desmos for the inequality <span class="math-container">$~\log_{0.2}\left(x^{2}-x-2\right)>\log_{0.2}\left(-x^{2}+2x+3\right)$</span></a></p>
<p>Here, you can see that plotted interval is <span class="math-containe... | Z Ahmed | 671,540 | <p><span class="math-container">$\log_{1/5}\left(x^{2}-x-2\right)>\log_{1/5}\left(-x^{2}+2x+3\right)$</span>
<span class="math-container">$\implies x^2-x-2>0 \& -x^2+2x+3>0 \implies -1> x ~\text{or}~ x>2 \& -1<x<3 \implies x \in (2,3)$</span></p>
<p>When <span class="math-container">$\log_{... |
2,897,904 | <p><a href="https://i.stack.imgur.com/ZwR5o.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZwR5o.png" alt="enter image description here" /></a></p>
<h2>Attempt:</h2>
<p>Let <span class="math-container">$x_{INi}$</span> be the number of people who travel from Ithaca to Newark in class <span class="ma... | orlp | 5,558 | <p>$$ \sum_{i=1}^3 (x_{INi}+x_{NBi}+x_{IBi}) \leq 30 $$</p>
<p>This constraint is incorrect. Consider that 30 passengers could enter at Ithaca and leave the airplane at Newark. Now your plane flies empty between Newark and Boston because this constraint disallows the empty seats to be filled.</p>
<p>Your other set of... |
254,236 | <p>If a matrix $A$ is diagonalizable, is $A$ invertible? </p>
<p>I know that $P^{-1}AP = \text{some diagonal matrix}$ and therefore $P$ is invertible, but not sure of $A$ itself.</p>
| Brett Frankel | 22,405 | <p>A square matrix is invertible if an only if its kernel is $0$, and an element of the kernel is the same thing as an eigenvector with eigenvalue $0$, since it is mapped to $0$ times itself, which is $0$.</p>
<p>When we diagonalize a matrix, we pick a basis so that the matrix's eigenvalues are on the diagonal, and al... |
1,635,548 | <p>Given two morphisms in some category, which is to say that you are told that $f$ and $g$ are in the cat $C$ and nothing more, how can you know if they are equal? Normally we appeal to the elements of their domain and codomain. Suppose you can't say anything about these sets, like you know nothing about the eleme... | Musa Al-hassy | 80,406 | <p>One answer is <strong>separating objects</strong>.</p>
<p>Say an object S is separating for iff
<em>for any</em> morphisms f, g : A ⟶ B in , we have
$$(∀ x : S ⟶ A • f x = g x) ⇔ f = g$$</p>
<p>(where the bullet • divides the declaration and body of the quantifier.)</p>
<p>That is, morphisms can be distinguished... |
1,635,548 | <p>Given two morphisms in some category, which is to say that you are told that $f$ and $g$ are in the cat $C$ and nothing more, how can you know if they are equal? Normally we appeal to the elements of their domain and codomain. Suppose you can't say anything about these sets, like you know nothing about the eleme... | Giorgio Mossa | 11,888 | <p>I assume you are interested in general techniques to prove when two morphisms are equal/different.</p>
<p>In general it is easier to prove when two morphisms are different. For instance, if the morphisms have different sources or targets they are different. Another way to distinguish two morphisms is by testing the... |
455,695 | <p>Let $V$ be the vector space of all real valued continuous functions. Prove that the linear operator $\displaystyle\int_{0}^{x}f(t)dt$ has no eigenvalues.</p>
| pitchounet | 61,409 | <p>Suppose that $\lambda \in \mathbb{R}^*$ is an eigenvalue of this operator associated to the eigenvector $f \neq 0$. Then, for all $x$, you have :</p>
<p>$$ \int_{0}^{x} f(t) \: dt = \lambda f(x) $$</p>
<p>with the condition that $f(0)=0$. If you differentiate the previous equality, you have $f(x) = \lambda f'(x)$ ... |
1,288,461 | <p>Here's the proof I was given:</p>
<p>Proposition.- If $A$ is countable then $\mathbb{R} \setminus A $ is dense.</p>
<p>Proof:
Suppose otherwise, then there exists real numbers $a$ and $b$, with $a < b$, such that there is no $a < x < b$ with $x \in \mathbb{R} \setminus A $. To put it in another way, if $a... | Kitegi | 120,267 | <p>$$\sum_{n=1}^\infty a_n=+\infty \tag{since $a_n\geq2/3$}$$
So the radius is $\leq1$.<br>
On the other hand, if $0<x<1$, then:
$$\sum_{n=1}^\infty a_nx^n\leq \sum_{n=1}^\infty\frac{5}{3}x^n<\infty$$
So the series converges for $|x|<1$</p>
<p>Therefore, the radius is $1$.</p>
|
278,971 | <p>How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis? What are the connections between both conjectures if any?</p>
| Sidharth Ghoshal | 58,294 | <p>Not too much, I don't belive that Twin PRime suggest much for Riemann (unless its some statistical stuff or basic guarantees).</p>
<p>The Twin Prime Conjecture is really part of a series of other conjectures which looks follows.</p>
<ol>
<li>Twin Prime: Are there infinite primes with difference 2?</li>
<li>De Poli... |
4,413,715 | <p>In trying to find a proof for the proposition below i came up with the following solution, which i haven't found on the internet. Can anyone tell me, if this proof is correct?</p>
<p>Proposition:
If <span class="math-container">$f$</span> is a strictly increasing function, then <span class="math-container">$f^{-1}$<... | Dr. Sundar | 1,040,807 | <p>I will establish this result as follows:</p>
<p>We start with <span class="math-container">$y < z$</span>, where <span class="math-container">$y, z$</span> are any two points in the range of <span class="math-container">$f$</span>.</p>
<p>Then <span class="math-container">$y = f(a)$</span> and <span class="math-c... |
4,413,715 | <p>In trying to find a proof for the proposition below i came up with the following solution, which i haven't found on the internet. Can anyone tell me, if this proof is correct?</p>
<p>Proposition:
If <span class="math-container">$f$</span> is a strictly increasing function, then <span class="math-container">$f^{-1}$<... | Incompl33t | 1,041,256 | <p>With the given answer and tips i make a second try. Hope this get's better. I included the existence of <span class="math-container">$f^{-1}$</span> in the proposition, since i didn‘t want to proof it here.</p>
<p>Proposition: If <span class="math-container">$f$</span> is a strictly increasing function and it’s inve... |
357,847 | <p>Let <span class="math-container">$n\ge 3$</span> be an integer. I would like to know if the following property <span class="math-container">$(P_n)$</span> holds: for all real numbers <span class="math-container">$a_i$</span> such that <span class="math-container">$\sum\limits_{i=1}^na_i\geq0 $</span> and <span clas... | Fedor Petrov | 4,312 | <p>Take <span class="math-container">$n=3k$</span>, <span class="math-container">$2k$</span> variables equal to <span class="math-container">$3$</span> and <span class="math-container">$k$</span> variables equal to <span class="math-container">$-5$</span> for large <span class="math-container">$k$</span>. Then <span cl... |
3,117,916 | <p><a href="https://i.stack.imgur.com/WZvzY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WZvzY.png" alt="enter image description here"></a></p>
<p>How can I tell the difference between the graphs?</p>
<p>I typically find the VA, HA and the x and y-intercepts but in this case, they are all the sa... | Myunghyun Song | 609,441 | <p>Let <span class="math-container">$n\ge 1$</span> be given. We may assume w.l.o.g. that <span class="math-container">$a<b$</span> and <span class="math-container">$f(a)<0<f(b)$</span>. By the continuity of <span class="math-container">$f$</span>, there exists <span class="math-container">$0<\epsilon<\f... |
3,117,916 | <p><a href="https://i.stack.imgur.com/WZvzY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WZvzY.png" alt="enter image description here"></a></p>
<p>How can I tell the difference between the graphs?</p>
<p>I typically find the VA, HA and the x and y-intercepts but in this case, they are all the sa... | user113102 | 620,412 | <p>Note that for fixed <span class="math-container">$n$</span> we can parameterize such sums by the step size <span class="math-container">$s$</span> and the initial value <span class="math-container">$x_1$</span>. Then we have a map <span class="math-container">$F: \mathbb{R}^2 \rightarrow \mathbb{R}$</span> given by ... |
63,763 | <p>For elliptic curve $y^{2}=x(x+a^{2})(x+(1-a)^{2})$,($a$ is a rational number and does not equal 0,1,1/2),is its rank always 0?</p>
| Siksek | 4,140 | <p>Although Junkie has answered the question, I'd like to point out that in the case of parametrized families of elliptic curves (such as this) it is often easy to find an explicit subfamily with positive rank. In the present case, let us take $x=2a^2$ and see what condition on $a$ forces this to give a point on the el... |
63,763 | <p>For elliptic curve $y^{2}=x(x+a^{2})(x+(1-a)^{2})$,($a$ is a rational number and does not equal 0,1,1/2),is its rank always 0?</p>
| Joe Silverman | 11,926 | <p>I'm probably going out on a limb here, but I think that the following conjecture is reasonable. </p>
<p><strong>Conjecture</strong>. Let $E:y^2=x^3+a(T)x+b(T)$ be an elliptic curve defined over $\mathbb{Q}(T)$ with the property that its $j$-invariant is not in $\mathbb{Q}$. Then there are infinitely many rational n... |
2,325,498 | <p>I have plotted a logit function and its derivative. My first question is that how can I interpret the derivative graph of the logit function and second, why in logit function, the second derivative becomes the logit function itself? </p>
<p><a href="https://i.stack.imgur.com/2jixx.png" rel="nofollow noreferrer"><i... | Franklin Pezzuti Dyer | 438,055 | <p>The second derivative of the logit function is not equal to itself. Look:
$$l(x)=\ln\bigg(\frac{x}{1-x}\bigg)$$
$$l(x)=\ln(x)-\ln(1-x)$$
Then differentiate:
$$l'(x)=\frac{1}{x}+\frac{1}{1-x}$$
Then differentiate again:
$$l''(x)=-\frac{1}{x^2}-\frac{1}{(1-x)^2}$$
The second derivative of the logit function is a compl... |
2,325,498 | <p>I have plotted a logit function and its derivative. My first question is that how can I interpret the derivative graph of the logit function and second, why in logit function, the second derivative becomes the logit function itself? </p>
<p><a href="https://i.stack.imgur.com/2jixx.png" rel="nofollow noreferrer"><i... | Bob Hanlon | 156,590 | <p>Since you appear to be using Mathematica</p>
<pre><code>logf[x_, a_, b_] := Log[a*x/(1 - b*x)]
With[{a = 1},
Plot3D[logf[x, a, b], {x, 0, 1}, {b, 0, 1},
ClippingStyle -> None,
PlotPoints -> 100]]
</code></pre>
<p><a href="https://i.stack.imgur.com/Op1mU.png" rel="nofollow noreferrer"><img src="https://... |
2,328,758 | <p>The Equation
$11^x + 13^x + 17^x =19^x $</p>
<p>Has </p>
<ol>
<li>No Real Roots</li>
<li>Only One Real Roots</li>
<li>Exactly Two Real Roots </li>
<li>More than Two Real Roots </li>
</ol>
<p>What I have done is </p>
<p>The function $f(x)=11^x + 13^x + 17^x -19^x $ is strictly increasing and being always positi... | Mathematical science | 603,336 | <p>Let y= f(x)= (11÷19)^x+(13÷19)^x+(17÷19)^x-1.
When we put x=0 then y=2 and when x->infinity then y->-1 so graph of f(x) cuts the x axis. Hence it has atleast one real root. So option 1 is wrong.
Also f'(x)<0 for all x
Let us suppose that f(x) has two real roots a,b such that f(a)=0 and f(b)=0. So f:[a,b]->R i... |
2,852,248 | <p><em>There are 5 classes with 30 students each. How many ways can a committee of 10 students be formed if each class has to have at least one student on the committee?</em></p>
<p>I figured that we first have to choose 5 people from each class, so there are $10^5$ options. There remain total of 29*5=145 students to ... | Boyku | 567,523 | <p>Here is how it works without inclusion-exclusion.</p>
<p>There are 7 ways of partitioning 10 into 5 positive parts. The number may be checked here <a href="https://oeis.org/A008284" rel="nofollow noreferrer">https://oeis.org/A008284</a></p>
<p>The problem is similar to a poker game hands problem but here we have 5... |
4,167,602 | <p>Does the probability that <span class="math-container">$\Pr \left[ {A < {R_3} \cup B < {R_2}} \right]$</span> with the information that <span class="math-container">${R_3} < {R_2}$</span> logically equivalent to <span class="math-container">$\Pr \left[ {\min \left[ {A,B} \right] < {R_3}} \right]$</span> ... | GReyes | 633,848 | <p>I do not think your question has anything to do with probabilities. It is about the relative strength or possible equivalency of the conditions</p>
<p><span class="math-container">$a)$</span> <span class="math-container">$A<R_3$</span> or <span class="math-container">$B<R_2$</span></p>
<p><span class="math-con... |
177,643 | <p>Let $I=\{0, 1, \ldots \}$ be the multiplicative semigroup of non-negative integers. It is possible to find a ring $R$ such that the multiplicative semigroup of $R$ is isomorphic (as a semigroup) to $I$?</p>
| Rick Decker | 36,993 | <p>Just use the sum formula twice:
$$
\begin{align*}
\tan(A+B+Y)=\tan((A+B)+Y) &= \frac{\tan(A+B)+\tan Y}{1-\tan(A+B)\tan Y}\\
&=\frac{\left(\frac{\tan A +\tan B}{1-\tan A\tan B}\right)+\tan Y}{1-\left(\frac{\tan A +\tan B}{1-\tan A\tan B}\right)\tan Y}\\
&= \frac{(\tan A + \tan B)+ \tan Y(1-\tan A\tan B)}{... |
1,595,297 | <p>Ordinary cohomology of topological space $X$ are known to be the cohomology of constant sheaf. </p>
<p><strong>Question</strong> Is there analogous description for equivariant cohomology?</p>
<p>More precisely. Consider category of $G$ equivariant sheaves $\mathcal{Sh}_G (X)$. Denote by $\Gamma_G := Inv \circ \Gam... | quinque | 167,762 | <p>This question does not actually makes sense.</p>
<p>Let me explain all this in the case of point.
If $G$ is discrete, then equivariant sheaves are just $G$ representations. If $G$ is a topological group, then one should consider topological vector spaces. The problem is that category of topological vector spaces is... |
2,738,679 | <p>I am having trouble to prove this exercise for a Real Analysis course. I tried two approaches. However, I am insecure about both of them.</p>
<p>This is the question:</p>
<p>If $X$ is a partition of ℕ* such that:</p>
<p>i) $1 ∈ X$;</p>
<p>ii) For all $n ∈ ℕ$*, if $n ∈ X$, then $2n ∈ X$;</p>
<p>iii) For all $n ∈... | Rob Arthan | 23,171 | <p>I like your second approach. We can formalise your second approach in easy stages like this:</p>
<p><strong>Lemma</strong> if $n \in X$, then so is every $m \in \Bbb{N}^*$ with $m < n$.</p>
<p><strong>Proof</strong> induction on $n - m$ using rule (iii).</p>
<p><strong>Lemma</strong> if $m \in \Bbb{N}^*$, then... |
6,155 | <p>I have a category $C$, which is equipped with a symmetric monoidal structure (tensor product $\otimes$, unit object $1$). My category also has finite coproducts (I'll write them using $\oplus$, and write $0$ for the initial object), and $\otimes$ distributes over $\oplus$.</p>
<p>By an <em>exponential monad</em>, ... | Tyler Lawson | 360 | <p>For a general complex oriented cohomology theory represented by a ring spectrum $R$, there is a "Hurewicz map" from $R$ to its smash product $H\mathbb{Z}\wedge R$ with the Eilenberg-Mac Lane object for the integers. $R$ has a formal group law associated to it as you stated. So does $H\mathbb{Z}\wedge R$; in fact, it... |
2,466,805 | <p>Hi maths peoples I have question how you show that function is greater or equal to zero because I want show that function is dense function and this is one of two condition for show it is dense function.</p>
<p>Example we have function </p>
<p>$$f(x) = \frac{1}{2} \cdot \sin(x) \text{ where } x \in \left[0, \pi\ri... | sam wolfe | 487,230 | <p>You can simply apply Chebyshev's inequality, which states that, for a random variable with mean $\mu$ and variance $\sigma^2$, we have for any real $k>0$ that$$
P(|X-\mu|\geq k\sigma)\leq \frac{1}{k^2}.
$$
In your case you have $\mu=0$ and $\sigma^2=1$. Hence, with $k=2$,$$
P(|X_i|\geq 2)\leq \frac{1}{2^2}=\frac{... |
105,583 | <p>this is a simple question, and excuse me if it's already been answered; I searched around and couldn't find anything.</p>
<p>I have two listplots, both along the same number of x data points, but with different y values. I want to find the difference between the two y values, while keeping the x values the same. I ... | Ali Hashmi | 27,331 | <p>try something like this:</p>
<pre><code>ListPlot@Thread[{Flatten[{#1[[1]][[All, 1]] - #1[[2]][[All,1]]}],
#1[[1]][[All, 2]]}] &@{{{1, 1}, {2, 2}, {3, 3}}, {{1,1}, {1, 2}, {1, 3}}}
</code></pre>
|
200,298 | <p>How to find the sum of the absolute values for the roots of this equation:</p>
<p>$$x^4-4x^3-4x^2+16x-8=0$$</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Suppose that $\rm\ g(x\!+\!1)\, =\, f(x^2),\:$ and that $\rm\:f(x)\:$ has roots $\rm\:0< s < 1 < r.\:$ </p>
<p>Then $\rm\:g\:$ has roots $\rm\:1\!-\!\sqrt{r}\, <\, 0\, <\, 1+\sqrt{r},\: 1\pm\sqrt{s},\:$ with absolute sum $\rm\ 2 + 2\,\sqrt{r}.$</p>
<p>In your case $\rm\:... |
1,702,885 | <blockquote>
<blockquote>
<p>Question: Prove $$ \frac{\sin^3(x)-\cos^3(x)}{\sin(x)+\cos(x)} = \frac{\csc^2(x) -\cot(x) -2\cos^2(x)}{1-\cot^2(x)} $$</p>
</blockquote>
</blockquote>
<p>RHS: $$ \frac{\csc^2(x) -\cot(x) -2\cos^2(x)}{1-\cot^2(x)} $$</p>
<p>$$ ⇔ \frac{\frac{1}{\sin^2(x)} +\frac{\cos(x)}{\sin(x)}-2\... | mathlove | 78,967 | <p>You are correct except one sign :
$$\frac{1\color{red}{-}\cos(x)\sin(x)-2\cos^2(x)\sin^2(x)}{(\sin(x)-\cos(x))(\sin(x)+\cos(x))}$$
(The error starts at the very beginning.)</p>
<p>Now using $1=\cos^2(x)+\sin^2(x)$,
$$\begin{align}&1-\cos(x)\sin(x)-2\cos^2(x)\sin^2(x)\\&=(1-2\cos(x)\sin(x))(1+\cos(x)\sin(x)... |
1,281,142 | <p>I am trying to find all the closed points of $\mathbb{A}_{\mathbb{R}}^2$.</p>
<p>After a quick google research, I found that $\mathbb{A}_{\mathbb{R}}^2 = \operatorname{Spec}(\mathbb{R}[x,y])$ and then all we need to find is the maximal ideals of $\mathbb{R}[x,y]$.</p>
<p>However, set-theoretically this equalty doe... | Slade | 33,433 | <p>Here is a more direct and canonical approach than the first one I posted:</p>
<p>First, an important fact: if $R=k[X,Y]$, and $\mathfrak{m} \subset R$ is a maximal ideal, then the field $R/\mathfrak{m}$ is a finite extension of $k$. This is known as <a href="http://en.wikipedia.org/wiki/Zariski%27s_lemma" rel="nof... |
1,166,382 | <blockquote>
<p>Evaluate the integral $\int_0^1 \cos(\ln(x)) \, dx$</p>
</blockquote>
<p>I was able to evaluate the improper integral which is:</p>
<p>$$\frac{x\left(\sin \ln x + \cos \ln x\right)}{2}$$</p>
<p>I was using the substitution $u = \ln x$, and afterward I did integration by parts twice and got the resu... | Jean-Sébastien | 31,493 | <p><strong>Hint</strong></p>
<p>$$
\lim_{x\to 0^+} -\frac{x}{2}\leq \lim_{x\to 0^+} \frac{x\sin(\ln(x))}{2}\leq \lim_{x\to 0^+} \frac{x}{2}
$$</p>
<p>Works also for $x\cos(\ln(x))$.</p>
|
430,142 | <p>How do I determine if $w$ is in the range of the linear operator $T$?</p>
<p>$$T:\Bbb R^3 \to \Bbb R^3, T(x,y,z)=(x-y,x+y+z,x+2z)\ ;\quad w=(1,2,-1)$$
I would appreciate the help.</p>
<p>Thanks</p>
| copper.hat | 27,978 | <p>The first and last equations allow you to write $y,z$ in terms of a formula in $x$. Substitute these expressions into the second equation and solve for $x$. Then compute $y,z$ using the initial formulae.</p>
|
3,443,711 | <p>I'm not sure I get this.</p>
<p>Because the sine function is an odd function, for
a negative number u, sin2u= -2sinucosu</p>
<p>Is it true or false and why?</p>
| Vasili | 469,083 | <p>Using diameter of circumcircle: <span class="math-container">$$\frac{\sin (2B)}{a}=\frac{\sin B}{b}=\frac{\sin (\pi-3B)}{c}=\frac{1}{D}$$</span> From here we get the function to maximize: <span class="math-container">$p(B)=D(\sin B + \sin 2B + \sin 3B)$</span><p>
Since sine is concave on the interval of interest, we... |
1,030,128 | <p>If <span class="math-container">$\alpha$</span> is of bounded variation on <span class="math-container">$[a,b]$</span>, then it is continuous almost everywhere on <span class="math-container">$[a,b]$</span>.</p>
<p>I know that a function is of bounded variation iff it is the difference of two monotone functions. Mon... | Thomas Andrews | 7,933 | <p>Any class that would reject the above proof is probably emphasizing formality over clarity.</p>
<p>Still, you can say more explicitly:</p>
<ol>
<li><p>If $f=g-h$ where $f,g,h$ are functions on $[a,b]$, and the set of discontinuities of $g$ is $U$ and the set of discontinuities of $h$ is $V$, then the set of discon... |
1,030,128 | <p>If <span class="math-container">$\alpha$</span> is of bounded variation on <span class="math-container">$[a,b]$</span>, then it is continuous almost everywhere on <span class="math-container">$[a,b]$</span>.</p>
<p>I know that a function is of bounded variation iff it is the difference of two monotone functions. Mon... | Oliver Díaz | 121,671 | <p>Just to add that if a function <span class="math-container">$\alpha$</span> is of bounded variation over a finite interval <span class="math-container">$[a,b]$</span>, then any point of continuity of <span class="math-container">$\alpha$</span> is a point of continuity of the variation function defined as <span cla... |
2,105,238 | <p>I need to prove that the function $f: \mathbb R_{>0}\times\mathbb R \to \mathbb R^2$ , $f(x,y)=(xy, x^2-y^2)$ is injective. I know I have to show that $f(x,y)=f(a,b)$ implies $x=a$ and $y=b$ but I have no idea how to prove it. Could you give me a hint?</p>
| DeepSea | 101,504 | <p>So: $xy = ab, x^2-y^2 = a^2-b^2\implies x^2-a^2=y^2-b^2$. We should first make the added condition that $x,y,a,b$ be positive reals to make the result valid. Thus with this in mind, if $x > a > 0 \implies y > b\implies xy > ab$. Thus $x = a$, and then $y = b$.</p>
|
2,835,038 | <p>Assuming that sum of probabilities for all possible events that can occur should sum to 1, how does one denote this for a conditional probability? Is it $P(A|E_1) + P(A|E_2) + ... = 1$, where $E_i$ is a specific event to be conditioned on? Or is the answer something else entirely? </p>
| BallBoy | 512,865 | <p>This is a good example where intuition on conditional probability can be a good check of the formula. Suppose $P(A)=0$ (which is clearly possible); then we know intuitively that $P(A|E_i)=0$ for any $i$ (because event $A$ never happens, so how can it happen conditioned on something?), so the sum you gave will be $0$... |
2,835,038 | <p>Assuming that sum of probabilities for all possible events that can occur should sum to 1, how does one denote this for a conditional probability? Is it $P(A|E_1) + P(A|E_2) + ... = 1$, where $E_i$ is a specific event to be conditioned on? Or is the answer something else entirely? </p>
| rmdmc89 | 200,513 | <p>The previous answers are more than enough to understand what is going on.</p>
<p>I just want to state the general proposition (implicit in the answers) with a formal proof.</p>
<blockquote>
<p>Proposition: Let <span class="math-container">$E_1,...,E_n,...$</span> be a countable collection of sets such that <span cla... |
85,984 | <p>I was doing good at school in plane geometry and trigonometry - especially in geometric proofs like proving the equality of two line segments or two angles - more than I was doing in analytic geometry.</p>
<p>I am considering doing research in mathematics to be my career (and my life) someday. and I am wondering ab... | Levon Haykazyan | 11,753 | <p>Actually there is a theorem of Tarski that elementary Euclidean geometry is decidable. Roughly speaking this means that there is a computer program that can decide if a given statement of elementary Euclidean geometry is true (given enough time). Generally mathematicians do not like to do things that can be done on ... |
855,686 | <p>In a triangle, the sum of two sides is $x$ and the product of the same two sides is $y$.
If $x^2 - c^2=y$ where c is the third side, then what is the ratio of the inradius to the circumradius of the triangle?</p>
<p>I guess I have found half of it: if the two sides of the triangle are $a$ and $b$, then $x=a+b$ and... | shabo | 130,786 | <p>$a+b=x$, $a.b=y$ and it is found $m(\widehat{ACB})=\frac{2\pi }{3}$ since $%
x^{2}-c^{2}=y$. Then area of the triangle is $S(ABC)=\frac{1}{2}ab\sin (%
\frac{2\pi }{3})=\frac{abc}{4R}=\frac{a+b+c}{2}r$, where $R$ and r are
circumradius and inradius respectively. Then you can easily calculate $R=%
\frac{c}{\sqrt{3}}$ ... |
706,250 | <p>For all $n\ge3\in \mathbb N$, if $n \equiv 3 \pmod{4}$ then ${3^n} \equiv 2 \pmod{5}$.</p>
<p>I tried to set $n = 3+4k$ but it doesn't help.</p>
<p>Any hints first please?</p>
| Flowers | 93,842 | <p>For all $n\geq 3$, if $n = 3 \pmod 4$ then $3^n = 2 \pmod 5$.</p>
<p>$$3^{3+4k} = 2 \pmod 5\\
2\cdot3^{4k} = 2 \pmod 5\\
3^{4k} = 1 \pmod 5\\
\left((-2)^4\right)^{k} = 1 \pmod 5\\
1^k = 1 \pmod 5$$</p>
|
1,116,215 | <p>In C. Adam's Topology, it is written that $f(A) - f(B) \subset f(A - B)$ for any function, but $f(A) - f(B) = f(A - B)$ iff f is bijective. I can come the half way, i.e., $f(A) - f(B) \subset f(A - B)$; but for the $f(A - B) \subset f(A) - f(B)$, I don't know both of how to prove correctness of it for bijective and ... | drhab | 75,923 | <p>Let $y\in f(A-B)$ and note that $f(A-B)\subseteq f(A)$ as a consequence of $A-B\subseteq A$. So automatically we have $y\in f(A)$. Some $a\in A-B$ must exist with $y=f(a)$. Now suppose that also $y=f(b)$ for some $b\in B$. Then injectivity of $f$ provides a contradiction and we can conclude that it implies that $y\n... |
4,345,182 | <p>Let <span class="math-container">$f \in L^1(\mathbb{R})$</span> be an integrable function on the real line.</p>
<p>Let <span class="math-container">$ p = x^m + a_1 x^{m-1} + \cdots + a_m \in \mathbb{R}[x] $</span> be a real polynomial of degree <span class="math-container">$m$</span>.</p>
<p>Consider the function <s... | xpaul | 66,420 | <p>You don't need advance technique. In fact,
<span class="math-container">\begin{eqnarray}
\int_{x}^{x+T}g(t)\,dt&=&\int_0^{x+T} g(t)\,dt-\int_0^{x} g(t)\,dt\\
&=&\int_{-T}^{x} g(u+T)\,dt-\int_0^{x} g(t)\,dt\\
&=&\int_{-T}^0g(t)\,dt=\int_{0}^Tg(T+t)\,dt\\
&=&\int_{0}^Tg(t)\,dt.
\end{eqn... |
2,529,990 | <p>Find the area of the shape surrounded by $y = \sin(x)$, $y = -\cos(x)$, $x = 0$, $x = π/4$.
Do I subtract $S_2$ from the $S_1$? How do I find the shape's area?
<a href="https://i.stack.imgur.com/67G7B.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/67G7B.png" alt="Picture of the graph: "></a></p>
| gen-ℤ ready to perish | 347,062 | <p>The idea is that area is height integrated over the width. The “height” is $$h(x) = (\sin x)-(-\cos x)$$ and the “width” is $dx$. Then the area is </p>
<p>$$\begin{align}
\text{area} &= \int_0^{\pi/4} h(x) \, dx \\
&= \int_0^{\pi/4} \left( \sin x + \cos x \right) \, dx \\
&= \int_0^{\pi/4} \sin(x) \, dx... |
101,828 | <p>In what sense is the limit of discrete series representation of $SL(2, \mathbb{R})$ a limit of discrete series representations? Where does the name origin from?</p>
| Asaf | 8,857 | <p>Here is the explanation I know, just for $SL_2$.</p>
<p>The discrete series rep. have realizations in the Hardy spaces $H_n$ which have the norm -
$$\|f\|_ n ^2 = n\int_{D}|f(z)|^2(1-|z|^{2})^{(n-1)}dxdy$$
notice this norm is scaled a bit differently than usual.
The limit of discrete series is realized inside $H_... |
4,518,339 | <p>Before I start, please excuse my jargon and any preconceived notions. I am trying to fix how I think about this. It feels as if I am missing something.</p>
<p>So I understand that matrix multiplication linearly maps a vector or matrix to that basis. It describes a linear transformation, as if the coordinate plane sc... | Ross Millikan | 1,827 | <p>Let the first number be <span class="math-container">$a$</span>, the difference between the first and second <span class="math-container">$d$</span>, and the increase in difference between (third-second) and (second-first) be <span class="math-container">$k$</span>. Then the numbers are
<span class="math-container"... |
3,291,975 | <p>Transpose this formula to make <span class="math-container">$y$</span> the subject.</p>
<p><span class="math-container">$$x=\sqrt{x^2y^2+1-y}$$</span></p>
<p>My try:</p>
<p><span class="math-container">$$x^2=x^2y^2+1-y$$</span></p>
<p><span class="math-container">$$x^2-x^2y^2=1-y$$</span></p>
<p><span class="ma... | Amit Rajaraman | 447,210 | <p><strong>Hint:</strong> </p>
<p><span class="math-container">$(1-y^2)=(1-y)(1+y)$</span></p>
<p>For <span class="math-container">$y\neq1$</span>, ...</p>
|
452,889 | <p>My question relates to the conditions under which the spectral decomposition of a nonnegative definite symmetric matrix can be performed. That is if $A$ is a real $n\times n$ symmetric matrix with eigenvalues $\lambda_{1},...,\lambda_{n}$, $X=(x_{1},...,x_{n})$ where $x_{1},...,x_{n}$ are a set of orthonormal eigenv... | Kendra Lynne | 83,385 | <p>All symmetric matrices are diagonalizable, therefore they have $n$ eigenvalues (which don't have to be distinct, by the way), all of which are real. The spectral theorem says:</p>
<blockquote>
<p>We can decompose any symmetric matrix $A\in S^n$ using <strong>symmetric eigendecomposition</strong>:
$$
A = \sum_{i... |
103,948 | <p>I have asked this sort of <a href="https://math.stackexchange.com/questions/69178/what-are-the-odds-of-rolling-a-3-number-straight-throwing-6d6">question before</a>, and I have a new similar question <a href="https://boardgames.stackexchange.com/questions/6375/is-there-a-best-character-in-button-men-is-there-a-worst... | Robert Israel | 8,508 | <p>Let $X$ be the number of rolls of 4 or less, $Y$ the number of rolls of exactly 5. You want the probability that $X+Y \ge 3$ and $X \ge 2$. This is $\sum_{x=2}^5 \sum_{y=\min(3-x,0)}^{5-x} P(X=x, Y=y) = P(X=2,Y=1) + P(X=2,Y=2) + P(X=2,Y=3) + P(X=3,Y=0)+P(X=3,Y=1)+P(X=3,Y=2)+P(X=4,Y=0)+P(X=4,Y=1)+P(X=5,Y=0$
where (... |
647 | <p>For example, I gave an exam earlier today with a problem that ended in the sentence</p>
<blockquote>
<p>Use the chain rule to find $(f\circ g)'(3)$.</p>
</blockquote>
<p>During the exam, one of the students asked me what the circle between the $f$ and $g$ means, and I answered that it represents the composition ... | dtldarek | 42 | <p>In some cases (e.g., during an open-book exam), you can opt for no-questions policy.</p>
<p>However, I think that case-by-case is the best option. Each time a student asks a question, what I ask myself is (with some stretch, you could call this a “general rule”),</p>
<p>$$\text{Would the answer give the student un... |
647 | <p>For example, I gave an exam earlier today with a problem that ended in the sentence</p>
<blockquote>
<p>Use the chain rule to find $(f\circ g)'(3)$.</p>
</blockquote>
<p>During the exam, one of the students asked me what the circle between the $f$ and $g$ means, and I answered that it represents the composition ... | Brendan W. Sullivan | 80 | <p>Since <a href="https://matheducators.stackexchange.com/a/648">dtldarek's answer</a> addresses well the issues of fairness to students, I'll mention another consideration. When writing exam questions, I try to make sure each question has a certain <em>intent</em>, that it probes the student's knowledge of a certain c... |
27,509 | <p>I have a set of data of the kind {{$x_i$, $y_i$, $z_i$}, ...} at randomly chosen points $x_i, y_i$. </p>
<p>The $z_i$ are supposed to be a smooth function of the $x_i$ and $y_i$. Unfortunately they turn out not to be. A few points stick out. These are local extrema - I want to find them and remove them from the dat... | Kuba | 5,478 | <p>An alternative way (notice usage of <code>Nearest</code> in case of random distributed points):</p>
<pre><code>pts = Table[RandomReal[{0, 2 Pi}], {1000}, {2}];
data = {##, Sin[#1] Cos[#2]} & @@@ pts;
(*and some noise*)
dev = Table[{RandomReal[{0, 2 Pi}], RandomReal[{0, 2 Pi}], RandomReal[{2, 5}]}, {20}];
data =... |
960,880 | <p>Could you help me to explain how to find the solution of this equation
$$y ′ (t)=−y(t)-\frac1{2}*(1+e^{-2t})+1$$
Given $y(0)=0$
Thank all
This is my answer
$$y ′ (t)=−y(t)-\frac1{2}e^{-2t}+\frac1{2}$$
$$e^{2t}y ′ (t)=e^{2t}(−y(t)-\frac1{2}e^{-2t}+\frac1{2})$$
where
$$(e^{2t}y(t))′=e^{2t}y(t)′+2(e^{2t}y(t))=e^{2t}... | Dr. Sonnhard Graubner | 175,066 | <p>one solution of the equation $y'(t)+y(t)=0$ is $y=C_1e^{-t}$ </p>
|
960,880 | <p>Could you help me to explain how to find the solution of this equation
$$y ′ (t)=−y(t)-\frac1{2}*(1+e^{-2t})+1$$
Given $y(0)=0$
Thank all
This is my answer
$$y ′ (t)=−y(t)-\frac1{2}e^{-2t}+\frac1{2}$$
$$e^{2t}y ′ (t)=e^{2t}(−y(t)-\frac1{2}e^{-2t}+\frac1{2})$$
where
$$(e^{2t}y(t))′=e^{2t}y(t)′+2(e^{2t}y(t))=e^{2t}... | Yiorgos S. Smyrlis | 57,021 | <p>Multiply the equation by $e^t$, then
$$
e^t(y'+y)=\frac{e^t}{2}-\frac{e^{-t}}{2}
$$
or
$$
\big(e^t y\big)'=\frac{1}{2}\big(e^t+e^{-t}\big)'
$$
or
$$
e^t y=\frac{1}{2}\big(e^t+e^{-t}\big)+c,
$$
for some $c$ constant, or
$$
y=\frac{1}{2}\big(1+e^{-2t}\big)+c\,e^{-t}.
$$</p>
|
3,609,214 | <p>What I want to do is to first calculate all the possible permutations of the letters of the given word. Once I do that, I plan to keep an S in the 5th position and calculate possible permutations. But the question is do I have to multiply it by 2 and THEN deduct it from the total number of perms? Or will I get the c... | Jonas Linssen | 598,157 | <p>I think we need maximality of the blocks rather than maximality of the graph here.</p>
<p>Assume there is a graph, which cannot be written as edge disjoint union of blocks, that is to say there are two <em>distinct</em> blocks <span class="math-container">$A,B$</span>, which share a common edge <span class="math-co... |
1,671,572 | <p>$\|A\vec{x}\|\leq\|A\|\space\|\vec{x}\|$ where $A$ is a $m\times n$ matrix and $\vec{x}$ is a n-dimensional column vector. Assume that $\|A\|=\sqrt{\Sigma_{i}\Sigma_{j}a_{ij}^{2}}$</p>
| C. Dubussy | 310,801 | <p>Note that $$\|A\| = \sup_{\|x\|=1} \|Ax\|.$$ Your inequality is obviously true if $x =0$. Then, for $x \neq 0$ one has $$\|A\frac{x}{\|x\|}\| \leq \|A\|,$$ because of the sup.</p>
|
516,280 | <blockquote>
<p>Given the equation <span class="math-container">$\displaystyle{\int_{-x}^x\exp({-t^2})dt}=-\ln(x)$</span>:</p>
<p>a. Simplify the integral using Gauss method with 3 points.</p>
<p>b. Solve given equation by Newton Raphson iterative method</p>
</blockquote>
<p>I succeeded simplifying the integral and got... | Barry Cipra | 86,747 | <p>I think what you need to do is rewrite the integral using the change of variable $t=ux$ and <em>then</em> using <a href="http://en.wikipedia.org/wiki/Gaussian_quadrature" rel="nofollow">Gauss quadrature</a> to get</p>
<p>$$\int_{-x}^x \exp(-t^2)dt= x\int_{-1}^1 \exp(-x^2u^2)du = x\left({8\over9}+{10\over9}e^{-3x^2/... |
406,514 | <blockquote>
<p>Find the Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$.</p>
</blockquote>
<p>Any explanations during the demonstration, will be appreciated. Thanks!</p>
| Islands | 40,177 | <p>By tower law we know that $[\mathbb Q(i\sqrt[]{3}, \sqrt[]{2}):\mathbb Q]= 4$, since $[\mathbb Q(i\sqrt[]{3}, \sqrt[]{2}):\mathbb Q(\sqrt[]{2})]=2$ and $[\mathbb Q(\sqrt[]{2}):\mathbb Q]=2$.</p>
<p>Let $i$ be the inclusion map $i:\mathbb Q \hookrightarrow \mathbb Q(i\sqrt[]{3}, \sqrt[]{2})$. Then $i(x^2 - 2) =x^2 -... |
214,832 | <p>Say I have the function: </p>
<pre><code>dep = TextStructure["He wrote a book. I read the book he wrote.",
"DependencyStrings", PerformanceGoal -> "Speed"]
</code></pre>
<p>Which outputs:</p>
<pre><code>{"(wrote, 2)((nsubj, (He, 1)), (dobj, (book, 4)((det, (a, 3)))))", \
"(read, 2)((nsubj, (I, 1)), (dobj, (... | Mr.Wizard | 121 | <p>Try these:</p>
<pre><code>StringCases[dep,
"((nsubj, (" ~~ w : WordCharacter .. ~~ ", " ~~ DigitCharacter .. ~~ "))" :> w]
StringCases[dep, "(" ~~ w : WordCharacter .. ~~ ___ :> w]
</code></pre>
<blockquote>
<pre><code>{{"He"}, {"I", "he"}}
{{"wrote"}, {"read"}}
</code></pre>
</blockquote>
|
2,775,443 | <p>The statement $\displaystyle \sum_{n=1}^{\infty}a_n$ converges $\implies$ $\displaystyle \sum_{n=1}^{\infty}\cfrac{1}{a_n}$ looks natural but do we have this implication? I am checking alternating series as an counter-example but could not find one yet. What can we say about the implication?</p>
| Hagen von Eitzen | 39,174 | <p>The convergence of $\sum a_n$ implies that $a_n\to 0$. The convergence of $\sum\frac1{a_n}$ would imply $\frac1{a_n}\to 0$.</p>
|
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