qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,914,365 | <p>I'm pretty stuck with this exercise. Hope somebody can help me:</p>
<p>show that
<span class="math-container">$$
\sum_{n=2^{k}+1}^{2^{k+1}} \frac{1}{n} \geq\left(2^{k+1}-2^{k}\right) \frac{1}{2^{k+1}}=\frac{1}{2}
$$</span>
and use this to show that the harmonic series is divergent.</p>
<p>First I can't figure out ho... | gt6989b | 16,192 | <p><strong>HINT</strong>
<span class="math-container">$$
\begin{split}
\sum_{n=2^k+1}^{2^{k+1}} \frac1n
\ge \sum_{n=1}^{2^k} \frac{1}{2^k+n}
\ge \sum_{n=1}^{2^k} \frac{1}{2^{k+1}}
= \frac{1}{2^{k+1}} \sum_{n=1}^{2^k} 1
\end{split}
$$</span>
Can you finish?</p>
|
283,816 | <p>$ 2 \ln (5x) = 16$</p>
<p>$ \ln (5x) = 8 $</p>
<p>$ 5x = e^8 $</p>
<p>$ x = \dfrac {1}{5}e^8$</p>
<p>But why can't we do it like this:</p>
<p>$ \ln(5x)^2 = 16$</p>
<p>I thought that was a possibilty with logaritms?</p>
| Guest 86 | 58,804 | <p>$$\ln ((5x)^2) = 16,\qquad x>0$$</p>
<p>$$25x^2 = e^{16}$$</p>
<p>$$x^2 = \frac{e^{16}}{25}$$</p>
<p>$$x = \pm\sqrt{\frac{e^{16}}{25}} =\pm\frac{\sqrt{e^{16}}}{5}=\pm\frac{e^{16/2}}{5}=\pm \frac{e^8}{5}$$</p>
<p>But since you've introduced the square you have to go back and check the answers - The negative on... |
262,773 | <p>In the curve obtained with following</p>
<pre><code> Plot[{1/(3 x*Sqrt[1 - x^2])}, {x, 0, 1}, PlotRange -> {0, 1},
GridLines -> {{0.35, 0.94}, {}}]
</code></pre>
<p>how can one fill the top and bottom with different colors or patterns such that two regions are perfectly visible in a black-n-white printout?<... | cvgmt | 72,111 | <pre><code>Clear[plot, pts];
plot = Plot[{1/(3 x*Sqrt[1 - x^2])}, {x, 0, 1}, PlotRange -> {0, 1}];
pts = Cases[plot, Line[a_] :> a, All] // First;
Graphics[{{GrayLevel[.6], Polygon[pts]}, {EdgeForm[Dashed], LightGray,
Polygon[Join[{{pts[[1, 1]], 0}}, pts, {{pts[[-1, 1]], 0}}]]}},
PlotRange -> {{0, 1}, {0... |
2,901,971 | <p>The question is as follows: </p>
<blockquote>
<p>Prove that for every $\beta \in \mathbb{R}$, $\sup(-\infty, \beta) = \beta$.</p>
</blockquote>
<p>The goal of the problem is to prove this without using the $\epsilon-\delta$ method. My professor gave us an idea in class, but for some reason, it isn't really maki... | Misha Lavrov | 383,078 | <p>Here are a few conditions that are equivalent to having an $X$-saturating matching. (This is stronger than what the question asks for, but is still an answer.)</p>
<ol>
<li><p>$G$ has no vertex cover smaller than $X$. </p></li>
<li><p>$G$ has no independent set bigger than $Y$. (This follows from 1 because the comp... |
494,338 | <p>This is gre preparation question of data interpretation,
Distribution of test score among students(Score range -> total % of students)</p>
<blockquote>
<p>0-65 -> 16<br/>
65-69 -> 37<br/>
70-79 -> 25<br/>
80-89 -> 14 <br/>
90-100 -> 8<br/></p>
</blockquote>
<hr>
<p>Question is that</p>
<p>Which of t... | Daniel R | 83,553 | <p>The right column shows the percentage of students for each range. Adding the percentages, starting from the top (or bottom for that matter), the range where you cross the 50 % limit will be the your answer. </p>
<p>What you're doing wrong is that you use the percentages themselves as the dataset. The dataset is the... |
347,009 | <p>I am looking for a generalisation of a modular form that transforms as something like:</p>
<p><span class="math-container">$f(\frac{a \tau+b}{c \tau+d}) = (c \tau+d)^k c^k f(\tau)$</span></p>
<p>I understand this cannot be literally true, as the multiplier c^k is not a root of unity, but does something like this a... | Yoav Kallus | 20,186 | <p>Let A be the measure on R^2 invariant wrt rotations about the origin with total weight at each radius equal to the frequency of that distance in your histogram. This measure is obtained by rotationally averaging the autocorrelation measure. The Fourier transform of the autocorrelation measure is nonnegative. Since t... |
3,379,447 | <p>I am currently studying for an exam in Mathematical research tools. One concept that has come up again and again throughout the course is continuity and although the definition of Cauchy Continuity or Delta Epsilon continuity was discussed, we were never shown how to actually prove that a function is continuous. The... | fleablood | 280,126 | <p>To prove <span class="math-container">$f(x) $</span> is continuous at <span class="math-container">$x = c$</span> we must prove the limit of <span class="math-container">$\lim_{x\to c}f(x) = c$</span>.</p>
<p>So for any <span class="math-container">$\epsilon > 0$</span> there is a <span class="math-container">$... |
116,201 | <p>I am curious if the "<a href="http://neumann.math.tufts.edu/~mduchin/UCD/111/readings/architecture.pdf" rel="nofollow noreferrer">Bourbaki's approach</a>" to mathematics is still a viable point of view in modern mathematics, despite the fact that Bourbaki is vilified by many.</p>
<p>Even more specifically, does any... | AAK | 2,503 | <p>The following appears in "<a href="http://www.ams.org/notices/201009/rtx100901106p.pdf">Reminiscences of Grothendieck and his school</a>", published in Notices of the AMS:</p>
<blockquote>
<p>Bloch: I wonder whether today such a style of mathematics could exist.</p>
<p>Illusie: Voevodsky’s work is fairly gen... |
267,753 | <p>A number of seemingly unrelated elementary notions can be defined uniformly with help of (iterated) Quillen lifting property
(a category-theoretic construction I define below) "starting" to a single (counter)example or a simple class of morphisms,
for example a finite group being nilpotent, solvable, p-group, a top... | user108780 | 108,780 | <p>Properties of topological spaces. </p>
<p>Here I need to use some notation for finite topological spaces. I use the fact that a finite topological space
may be thought of as a category such that $card Hom(x,y)\leq 1$ for any objects $x,y$.
By convention, $ \{a{\rightarrow}b\}$ denote the space where $a$ is open and... |
1,972,252 | <blockquote>
<p>If $v^TAv = v^TBv$ for all constant vectors $v$ and $A,B$ are matrices of size $n$ by $n$, is it true that $A=B$? </p>
</blockquote>
<p>I have thought about using basis vectors but cannot get system of equations uniquely. Is there a trick here?</p>
| edm | 356,114 | <p>It is not true. Take $A$ to be the zero matrix, and $B$ to be any nonzero skew-symmetric matrix. You can prove that both sides are always zero, but $A$ and $B$ are not equal.</p>
<p>If you assume that both $A$ and $B$ are symmetric real matrices, then they are equal. You can see that $v^TAv = v^TBv$ is equivalent t... |
4,417,325 | <p>When I say "divisibility trick" I mean "a recursive algorithm designed to show that, after multiple iterations, if the final output is a multiple of the desired number, then the original was also a multiple of the same number." Here's an example for a divisibility trick for 17.</p>
<blockquote>
<... | Misha Lavrov | 383,078 | <p>We can invent "divisibility tricks" that don't have this property. For example:</p>
<p>"To test if <span class="math-container">$n$</span> is divisible by <span class="math-container">$3$</span>, write <span class="math-container">$n$</span> as <span class="math-container">$10q+r$</span>. Then, comput... |
2,375,023 | <p>So I have to find an interval (in the real numbers) such that it contains all roots of the following function:
$$f(x)=x^5+x^4+x^3+x^2+1$$</p>
<p>I've tried to work with the derivatives of the function but it doesn't give any information about the interval, only how many possible roots the function might have.</p>
| Michael Rozenberg | 190,319 | <p>Let $f(x)=x^5+x^4+x^3+x^2+1$.</p>
<p>Hence, $f'(x)=x(5x^3+4x^2+3x+2)$ and since $(5x^3+4x^2+3x+2)'=15x^2+8x+3>0$,
we see that the polynomial $5x^3+4x^2+3x+2$ has one real root $x_1$ and this root is negative.</p>
<p>Thus, $x_{min}=0$ and $x_{max}=x_1$. </p>
<p>But $f(0)>0$, which say that $f$ has an unique ... |
1,109,918 | <p>Is it always possible to add terms into limits, like in the following example? (Or must certain conditions be fulfilled first, such as for example the numerator by itself must converge etc)</p>
<p>$\lim_{h \to 0} {f(x)} = \lim_{h \to 0} \frac{e^xf(x)}{e^x}$</p>
| Vim | 191,404 | <p>It is very simple if you apply Taylor's series with Peano's residue:<br/>
When $h\to 0$, we have
$$f(x+h)=f(x)+hf'(x)+\frac{1}{2}h^2 f''(x)+o(h^2)$$
$$f(x-h)=f(x)-hf'(x)+\frac{1}{2}h^2 f''(x)+o(h^2)$$
where $o(h^2)$ denotes any infinitesimal amount that converges to zero faster than $h^2$.<br/>
Put them back to the ... |
1,423,456 | <p>I am completely stuck on this problem: $C[0,1] = \{f: f\text{ is continuous function on } [0,1] \}$ with metric $d_1$ defined as follows:</p>
<p>$d_1(f,g) = \int_{0}^{1} |f(x) - g(x)|dx $.</p>
<p>Let the sequence $\{f_n\}_{n =1}^{\infty}\subseteq C[0,1]$ be defined as follows:</p>
<p>$
f_n(x) = \left\{
\beg... | Squirtle | 29,507 | <p>Essentially the idea is that you start with functions that look like a flat line, a continuous elevation (or decline) and then a flat line. You then make the elevation (or decline) more and more steep until it disappears and you are left with a noncontinuous step function.</p>
<p>EDIT: The point is that in a comp... |
2,077,275 | <p>Let $U$ be the set $U$ of quaternions of unit length. I know that $U\times S^1$ is compact, connected and is a $2n$ manifold in a $2n+1$ dimensional vector space $V$.</p>
<blockquote>
<p>How can I construct a differentiable tangent vector field on $U\times S^1$ that has no zeroes?</p>
</blockquote>
<p>What I kno... | Community | -1 | <p>Assuming you're trying to solve for $(x, y, z)^{T}$ then</p>
<p>Hint: Rewrite
$$
\begin{pmatrix}
\sin a & \sin 2a & \sin 3a\\
\sin b & \sin 2b & \sin 3b\\
\sin c & \sin 2c & \sin 3c\\
\end{pmatrix}\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}=\begin{pmatrix}
\sin 4a \\\sin 4b \\ \sin 4c
\end{pmatr... |
1,879,440 | <p>I am reading the definition of associative <span class="math-container">$R$</span>-algebra, and am confused about the following definition from Wiki:</p>
<p><a href="https://en.wikipedia.org/wiki/Associative_algebra" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Associative_algebra</a></p>
<p><a href="http... | Berci | 41,488 | <p>There are two operations here: one is 'multiplication' of 'scalars' with algebra elements, denoted by dot, and the other one is the multiplication of the algebra, between two of its elements.</p>
<p>The axiom says:
$$r\cdot(xy)=(r\cdot x)\,y=x\,(r\cdot y),\\
r\in R,\quad x,y\in A\,.$$</p>
<p>Since $R$ is commutati... |
877,683 | <p><img src="https://www.anonimg.com/img/c0423915f5f5fae8f9790506c36393f8.jpg" alt="">
Based on my last question I learned that this is an envelope of a parabola</p>
<p><a href="https://math.stackexchange.com/questions/874478/what-is-this-geometric-pattern-called">What is this geometric pattern called?</a></p>
<p>But... | mvw | 86,776 | <p>We try to model the family of lines and then try to infer the envelope.</p>
<p>The guiding lines (left and right arms of the V shape) are
$$
g(t) = u_g \, (1-t) + v_g \, t \quad h(t) = u_h \, (1-t) + v_h \, t
$$
for $t \in [0, 1]$, where $u$ is the start point and $v$ the end point of that line.</p>
<p>A line $f_r... |
2,289,813 | <p>Let$\ f(x,y,z)=4xz -y^2 +z^2$ be a differentiable function, let$\ P=(0,1,1)$ and$\ Q=(1,3,2)$. Find$\ T$ such that $\ f(P)-f(Q) = \nabla f(T)(P-Q)$</p>
<p>What I did:</p>
<p>$\ f(P)=0$, $\ f(Q)=3$, $\ (P-Q)=(-1,-2,-1)$, $\nabla f(P-Q)=(-4z, 4y, -4x+2z)$</p>
<p>Let $\ T=(a,b,c)$ then$\ f(P)-f(Q) = \nabla f(T)(P-Q)... | Manuel Guillen | 416,103 | <p>The function $f$ takes a vector in $\mathbb{R}^3$ and returns a scalar in $\mathbb{R}$. The left hand side of the expression is a scalar, and so is the right hand side, because the product is a dot product:
\begin{align*}
f(P)-f(Q) &= \nabla f(T) \cdot (P-Q) \\
0 - 3 &= \langle 4z, -2y, 4x + 2z \rangle \cdot... |
1,783,837 | <p>I'm studying how to use Laplace transform to solve ODEs. <br/><br/>
I have thought to use this very simple example: $$y'(t)=t+1 \qquad y(0)=0$$</p>
<p>I can use integration to find $y(t)$: $$y(t)=\int (t+1) \ \ dt=\frac{1}{2} t^2+t+C$$</p>
<p>$C \in \mathbb{R} $, $C=0$ for the initial condition, so:</p>
<p>$$y(t... | kennytm | 171 | <p>Your inverse Laplace transform of $\frac1{s^3}$ is wrong, it should be $\frac{t^2}2$ as you expect.</p>
<p>$$\mathcal L^{-1} \left\{ \frac1{s^{n+1}} \right\} = \frac{t^n}{\color{red}{n!}} $$</p>
|
1,783,837 | <p>I'm studying how to use Laplace transform to solve ODEs. <br/><br/>
I have thought to use this very simple example: $$y'(t)=t+1 \qquad y(0)=0$$</p>
<p>I can use integration to find $y(t)$: $$y(t)=\int (t+1) \ \ dt=\frac{1}{2} t^2+t+C$$</p>
<p>$C \in \mathbb{R} $, $C=0$ for the initial condition, so:</p>
<p>$$y(t... | Ángel Mario Gallegos | 67,622 | <p>$$\mathscr{L}^{-1}\left\{\frac1{s^2}+\frac1{s^3}\right\}=\mathscr{L}^{-1}\left\{\frac1{s^2}\right\}+\mathscr{L}^{-1}\left\{\frac1{s^3}\right\}=\mathscr{L}^{-1}\left\{\frac1{s^2}\right\}+\color{blue}{\frac{1}{2}}\mathscr{L}^{-1}\left\{\frac{\color{blue}{2}}{s^3}\right\}=t+\frac12t^2$$</p>
|
3,371,694 | <p>Let <span class="math-container">$ \forall n \in \mathbf{N}, ~ u_{n+1} = \frac{1}{\tanh^2(u_n)} - \frac{1}{u^2_n} $</span> with <span class="math-container">$ u_0 = a > 0 $</span>.</p>
<p>What is the limit of <span class="math-container">$(u_n)$</span> ?</p>
<p>I tried to find a fixed point of this sequence but... | A.J. | 654,406 | <p>You just have to consider the symmetry of the situation. Since the circle has center <span class="math-container">$(0,0)$</span>, the fact that <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> have opposite <span class="math-container">$x$</span>-coordinates and are in opposite qua... |
2,108,352 | <p>It is a question in functional analysis by writer Erwin Kryzic </p>
| Franck Panche | 615,159 | <p>Ok. The problem is the triangle inequality, then: </p>
<p>Let <span class="math-container">$x,y,z \in \mathbb{R}$</span>, so <span class="math-container">$$d(x,z) = (x+y-y+z)^2$$</span> <span class="math-container">$$d(x,z) = ((x-y)+(y-z))^2$$</span> <span class="math-container">$$\qquad d(x,z) = d(x,y)+2(x-y)(y-z... |
183,100 | <p>Let $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ be the map
$$(B_1,B_2)\mapsto [B_1,B_2]$$ which takes two $2\times 2$ matrices to its Lie bracket. </p>
<p>Then why does $d\phi_{(B_1,B_2)}:M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ send
$$(D_1,D_2)\mapsto [B_1,D_... | Georges Elencwajg | 3,217 | <p>This has actually nothing to do with Lie brackets nor Lie algebras! </p>
<p>Given finite-dimensional complex (or real) vector spaces $E,F,G$ and a bilinear map $f:E\times F\to G$, the differential of $f$ at $(a,b)\in E\times F$ is given by the formula (whose proof follows directly from the definition)<br>
$$ df... |
539,457 | <p>Suppose we have a natural number $N$ with decimal representation $A_kA_{k-1}\ldots A_0$. How do I prove that if the $\sum\limits_{i=0}^kA_i$ is divisible by $9$ then $N$ is divisible by $9$ too?</p>
| Carlos Eugenio Thompson Pinzón | 99,344 | <p>If you are familiar with modularity you have:</p>
<ol>
<li>If $x \equiv a\mod{m}$ and $y \equiv b\mod{m}$ then $x+y\equiv a+b\mod{m}$.</li>
<li>If $x \equiv a\mod{m}$ and $y \equiv b\mod{m}$ then $x\cdot y\equiv a\cdot b\mod{m}$.</li>
<li>If $x \equiv 1\mod{m}$ then $x\cdot b\equiv b\mod{m}$.</li>
<li>If $x \equiv ... |
320,019 | <p>Assume $Y$ is non negative random variable. Prove that $X+Y$ is stochastically greater than $X$ for any random variable $X$.</p>
<p>We have to prove there that $\Pr(X+Y > x) \geq \Pr(X>x) $ for all $x$</p>
| Reid | 64,872 | <p>One approach:</p>
<p>Would be glad if anyone points out any mistake if there is one.(Or comment to say if there is no mistake).</p>
<p>$P(X+Y \leq x) = \int_{-\infty}^{\infty} F_X(x-y)dG_Y(y)dy$ where $F$ and $G$ are cdf of $X,Y$ respectively.</p>
<p>$\leq \int_{-\infty}^{\infty} F_X(x)dG_Y(y)dy$. As $F(x-t) ... |
2,591,976 | <p>If $f(x-1)=2x^2-10x+3$, find $f(x)$. I tried the problem and received the answer $f(x)=2x^2-14x-5$, is this right? </p>
| Enrico M. | 266,764 | <p>Use $x-1 = y$, hence $x = y+1$, thence</p>
<p>$$f(y) = 2(y+1)^2 - 10(y+1) + 3 = 2y^2 + 2 + 4y - 10y - 10 + 3$$</p>
<p>$$f(y) = 2y^2 - 6y - 5$$</p>
<p>Now just call $y = x$ and you're done.</p>
|
3,644,823 | <p>Let <span class="math-container">$f$</span> be a non constant holomorphic function on and inside of the unit circle <span class="math-container">$C:=\{z\in \mathbb C~:~|z|=1\}$</span>. Suppose <span class="math-container">$|f(z)|=1$</span> on <span class="math-container">$C$</span>, then for <span class="math-contai... | Conrad | 298,272 | <p><span class="math-container">$f$</span> has finitely many zeroes in <span class="math-container">$\mathbb D$</span> (why?); number them <span class="math-container">$a_1,...a_n$</span> repeating them if of higher multiplicities and take the finite Blaschke product <span class="math-container">$B(z)=\Pi_{k=1}^n \frac... |
17,270 | <p>I just joined MathSE and it's beautiful here, except for the fact that some unregistered users ask a question and never come back. Most of the time these questions are trivial, though they still consume answerers' (valuable) time which never gets rewarded. I thought it was okay until I saw someone's profile with the... | Anonymous Computer | 128,641 | <p>Seventy-two questions without any accepted answers? Guess I'm not answering any of your questions....</p>
<p>Being serious, accepting answers shows gratitude for the answerer that helped the person asking the question the most. If people don't accept answers, the answerers may be discouraged from answering because ... |
17,270 | <p>I just joined MathSE and it's beautiful here, except for the fact that some unregistered users ask a question and never come back. Most of the time these questions are trivial, though they still consume answerers' (valuable) time which never gets rewarded. I thought it was okay until I saw someone's profile with the... | Dunka | 159,628 | <p>It is okay to not accept any answers if they haven't really clarified what you were trying to solve. You're under no obligation to give people participation points. If the answers lead you to understanding the problem, you should always accept. In this particular case I think the fact that they have asked 72 questio... |
2,559,350 | <p>Consider a circumference centered in the origin and with radius $r$. Let $C$ be a point on the circumference, and let $A,B$ be its projections on the axes, respectively $x$-axis and $y$-axis. What is the length of $AB$?</p>
<p>I tried applying the laws of sines and cosines on $ABC$, but I only got tautologies....</... | heropup | 118,193 | <p>Think about what $|z-i|^4 = 1$ <em>means</em>. It means that the magnitude of the complex number $z-i$, when raised to the fourth power, equals $1$. Since magnitude is always a nonnegative real number, it follows that the magnitude itself must equal $1$, since the only nonnegative real solution to the equation $$x... |
132,879 | <p>This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section.</p>
<p>Let us recall Maxwell’s equations: Given a semi-riemannian 4-manifold and a 3-form $j$. We describe the field-strength differential form $F$ as a solution of th... | Igor Khavkine | 2,622 | <p>Willie's answer is of course correct. But the part that I think is most interesting in the physical context is only briefly mentioned in point 3. of the last paragraph. Let me expand on that.</p>
<p>The relevant condition that make everything work nicely is that the background Lorentzian spacetime be globally hyper... |
4,624,695 | <p>I have an exercise with text: With lines</p>
<p><span class="math-container">$$
p_1 \ldots \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-4}{1}, \quad p_2 \ldots \frac{x-5}{2}=\frac{y-1}{-1}=\frac{z-2}{1} .
$$</span>
Determine one plane <span class="math-container">$\pi$</span> with respect to which the lines <span class="mat... | Sammy Black | 6,509 | <p>Shift your coordinates so that <span class="math-container">$x = a$</span> is the origin. In other words, define
<span class="math-container">$u = x - a$</span> so that <span class="math-container">$x = a + u$</span>. Geometrically, this measures <span class="math-container">$u$</span> units to the right of <span cl... |
1,554,871 | <p>The mean value theorem tells</p>
<p>" If $f:[a,b]\rightarrow \mathbb R$ is continuous and $g$ is an integrable function that does not change sign on $[a, b]$, then there exists $c$ in $(a, b)$ such that</p>
<p>$\int_a^b f(x)g(x)dx=f(c)\int_a^b g(x)dx$."</p>
<p>If $g$ changes its sign, is this theorem still true?<... | npatrat | 220,440 | <p>Well, $f_n$ are continuous functions. If $(f_n(x))$ is uniformly convergent , then $\lim_{n \to \infty}f_n(x)$ is continuous, which is false.</p>
<p>So, our sequence is not uniformly convergent over $\mathbb{R}$. </p>
|
123,813 | <p>Let $E$ be an elliptic curve over $\mathbb{Q}$. It is known from Gross and Zagier that if $\textrm{rank}_{\textrm{an}}(E) \leq 1$, then</p>
<p>$$\textrm{rank}(E) \geq \textrm{rank}_{\textrm{an}}(E).$$ </p>
<p>Instead, suppose I know that $\textrm{rank}(E) \leq 1$, are there any (unconditional) inequalities that re... | user30035 | 30,035 | <p>This is not really the right question. I think it's easier to just list the two things that are known unconditionally and then you can figure out if this answers your question.</p>
<p>1) If analytic rank is zero then algebraic rank is zero.</p>
<p>2) If analytic rank is 1 then algebraic rank is 1.</p>
<p>Proof: s... |
2,040,175 | <p>The following diagram shows 9 distinct points chosen from the sides of a triangle.</p>
<p><a href="https://i.stack.imgur.com/3DUhm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3DUhm.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$(1)$</span> How many line seg... | Community | -1 | <p>With all vertices on different sides, you form $2\cdot3\cdot4$ triangles.</p>
<p>With two vertices on the same side, you form $1,3$ and $6$ pairs respectively; then you can match every pair with a vertex on another side, giving</p>
<p>$$1\cdot(3+4)+3\cdot(2+4)+6\cdot(2+3).$$</p>
<p>Total $79$.</p>
<p>(I assume t... |
216,473 | <p>Let me begin with some background: I used to enjoy mathematics immensely in school, and wanted to pursue higher studies. However, everyone around me at that time told me it was a stupid area (that I should focus on earning as soon as possible), and so instead I opted for an engineering degree. While in college, I us... | zyx | 14,120 | <blockquote>
<p>"I mean, if I can't even prove basic theorems related to Euclid's geometry, how can I ever hope to do some authentic research like the mathematicians I so admire? "</p>
</blockquote>
<p>These are different skills. There are many researchers who (as students) could solve all the problems in the calc... |
216,473 | <p>Let me begin with some background: I used to enjoy mathematics immensely in school, and wanted to pursue higher studies. However, everyone around me at that time told me it was a stupid area (that I should focus on earning as soon as possible), and so instead I opted for an engineering degree. While in college, I us... | Wouter Stekelenburg | 27,375 | <p>I don't have an answer, but I have a suggestion for finding one.</p>
<p>Most mathematics professors will allow you to visit lectures and exercise classes, and even welcome you if you show more enthusiasm than the average student (trust me: not hard to do). You won't get a degree that way, but you will get a better ... |
4,221,530 | <p>Assume terms <span class="math-container">$a_n$</span> and <span class="math-container">$a_{n+1}$</span> share some common factor <span class="math-container">$x$</span> so that <span class="math-container">$a_n = xm$</span> for some integer <span class="math-container">$m$</span> and <span class="math-container">$a... | Math Lover | 801,574 | <p><span class="math-container">$\triangle ABM$</span> is isosceles so <span class="math-container">$\angle AMB = \angle ABM = \cfrac{180^ \circ - 30^\circ}{2} = 75^ \circ$</span></p>
<p>Say side of hexagon is <span class="math-container">$a$</span> and perp from <span class="math-container">$T$</span> to <span class="... |
636,359 | <p>How do you proof that $\angle CTP = \angle CBP+ \angle BCN$ ?
Please check the image below:</p>
<p><img src="https://i.stack.imgur.com/SqeXa.png" alt="enter the great image description here"></p>
| MikhaelM | 247,415 | <p>$\angle CTP=\angle BTN=180^\circ-\angle PTN=180^\circ-\angle CTB=\angle CBT+\angle TCB = \angle CBP + \angle BCN$</p>
|
4,057,959 | <p>What is the number of N step random walks starting from a point (x0,y0) to a point (x1,y1) assuming each direction (right,left,up,down) has equal probability. I know the expression for the one dimension case is:</p>
<p><span class="math-container">$$ \binom{N}{\frac{N+(y1-y0)}{2}} $$</span></p>
<p>is there a similar... | grand_chat | 215,011 | <p>Here's another approach, less slick than that of @MikeEarnest (+1):</p>
<p>Wlog we are taking <span class="math-container">$N$</span> steps from <span class="math-container">$(0,0)$</span> to <span class="math-container">$(x,y)$</span>. Suppose <span class="math-container">$k$</span> of these steps are horizontal (e... |
3,612,916 | <p>Find a function <span class="math-container">$f$</span> such that <span class="math-container">$f(x)=0$</span> for <span class="math-container">$x\leq0$</span>, <span class="math-container">$f(x)=1$</span> for <span class="math-container">$x\geq1$</span>, and <span class="math-container">$f$</span> is infinitely dif... | David Kraemer | 224,759 | <p>Try the function</p>
<p><span class="math-container">$$
f(x)= \begin{cases}
0 & x < 0 \\
\frac{1}{1+\exp(-g(x))} & 0 \leq x \leq 1 \\
1 & 1 < x
\end{cases}
$$</span>
where <span class="math-container">$g(x)= (1-x)^{-1} - x^{-1}$</span>.</p>
<p>Here's a <a href="https://www.desmos.com/calculator/y... |
1,916,232 | <p>I have massive problems with questions like these:</p>
<p>Let $\{v_1, . . . , v_r\}$ be a set of linearly independent vectors in $\mathbb{R}^n$
(with
$r < n$), and let $w\in\mathbb{R}^n$ be a vector such that $w \in \mathrm{span}\{v_1, . . . , v_r\}$.
Prove that $\{v_1, . . . , v_r, w\}$ is a linearly independe... | Siong Thye Goh | 306,553 | <p>In general, there is no sure work formula to successfully prove something and there could be multiple ways to solve the same thing . The advices in the comments are very useful. </p>
<p>Reading mathematics does help if you can see the thought process.</p>
<p>Mathematical proofs is really like solving real life pr... |
1,438,512 | <p>I was given this problem at school to look at home as a challenge, after spending a good 2 hours on this I can't seem to get further than the last part of the equation. I'd love to see the way to get through 2) before tomorrow's lesson as a head start.</p>
<p>So the problem is as follows:</p>
<p>1) Quadratic Equat... | R.N | 253,742 | <p>1) Quadratic Equation $$2x^2 + 8x + 1 = 0$$ </p>
<p>i. $$\alpha + \beta=\frac{-b}{a}=\frac{-8}{2}=-4$$</p>
<p>ii. $$\alpha\beta=\frac{c}{a}=\frac{1}{2}$$
For remain just find roots and use this fact that if $x_1+x_2=s$ and $x_1 x_2=p$ equation will be $x^2 -sx+p=0$. </p>
|
175,535 | <p>I have <code>data1</code> and a target point <code>targetPts</code>, and want to find the closest point from the data. As you can see from below, the 29th point from the data is the closest point and thus the value of <code>data1[[29]]</code> is same as the desired value, which is <code>{0.67033, 0.84245}</code>.</p... | kglr | 125 | <pre><code>First @ Nearest[data1data2[[All, ;; 2]] -> "Index", targetPts]
</code></pre>
<blockquote>
<p>29</p>
</blockquote>
<pre><code>data1data2[[First @ Nearest[data1data2[[All, ;; 2]] -> "Index", targetPts]]]
</code></pre>
<blockquote>
<p> {0.67033, 0.84245, 0.177696, 0.49873} </p>
</blockquote>
<p... |
1,791,849 | <p>On the <a href="https://en.wikipedia.org/wiki/Pentagon" rel="nofollow">Wikipedia Page about Pentagons</a>, I noticed a statement in their work saying that $\sqrt{25+10\sqrt{5}}=5\tan(54^{\circ})$ and $\sqrt{5-2\sqrt{5}}=\tan(\frac {\pi}{5})$</p>
<p>My question is: How would you justify that? My goal is to simplify ... | egreg | 62,967 | <p>You surely know that $z=\cos\frac{2\pi}{5}+i\sin\frac{2\pi}{5}$ is the root in the first quadrant of $z^5-1=0$ so it satisfies
$$
z^4+z^3+z^2+z+1=0
$$
that can also be written as
$$
z^2+\frac{1}{z^2}+z+\frac{1}{z}+1=0
$$
or, by noting that $z^2+1/z^2=(z+1/z)^2-2$,
$$
\left(z+\frac{1}{z}\right)^2+\left(z+\frac{1}{z}\... |
393,280 | <p>I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar <a href="http://www.math.harvard.edu/%7Egaitsgde/grad_2009/SeminarNotes/Oct13(AffGr).pdf" rel="noreferrer">notes</a>
and there are some commutative algebra facts that I am not able to figure out by myself apparently, like th... | Dylan Wilson | 423 | <p>Here's an attempt (could be mistakes so be wary!).</p>
<p>We have
$$
M \otimes_{A[[t]]}^{\mathbb{L}} B[[t]] \cong (M \otimes_{A[[t]]}^{\mathbb{L}} A[[t]]/t^n) \otimes_{A[[t]]}^{\mathbb{L}} B[[t]].
$$</p>
<p>By associativity this is (quasi-isomorphic) to
$$
M \otimes_{A[[t]]}^{\mathbb{L}} B[[t]]/t^n.
$$
Since $A[... |
1,905,186 | <blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative Noetherian ring (with unity), and let <span class="math-container">$I$</span> be an ideal of <span class="math-container">$R$</span> such that <span class="math-container">$R/I \cong R$</span>. Then is it true that <span class="math-container"... | Community | -1 | <p>Assume $f:R/I\to R$ is an isomorphism and $I \ne (0)$. Let $\overline{J} = f^{-1}I\subset R/I$ and $J\subset R$ be the preimage of $\overline J$ in $R$. Now $I \ne (0)$ implies $J$ strictly contains $I$; but $R/J \cong (R/I)/\overline J$ which is isomorphic to $R/I$ via $f$, and so isomorphic to $R$ by hypothesis. N... |
496,178 | <p>Let $t$ be a positive real number. Differentiate the function</p>
<blockquote>
<p>$$g(x)=t^x x^t.$$</p>
</blockquote>
<p>Your answer should be an expression in $x$ and $t$.</p>
<p>came up with the answer </p>
<blockquote>
<p>$$(x/t)+(t/x)\ln(t^x)(x^t)=\ln(t^x)+\ln(x^t)=x\ln t+t\ln x .$$</p>
</blockquote>
<p... | user94529 | 94,529 | <p>$g(x)= t^xx^t = e^{ln(t^xx^t)}
=e^{xln(t) + tln(x)}$</p>
<p>Now,</p>
<p>$g'(x)= \frac{d}{dx}e^{ln(t^xx^t)}$</p>
<p>$=\frac{d}{dx}e^{xln(t) + tln(x)}$</p>
<p>$= e^{xln(t) + tln(x)}.\frac{d}{dx}({xln(t) + tln(x)})$</p>
<p>$= e^{xln(t) + tln(x)}.(ln(t)+\frac{t}{x})$</p>
<p>$=t^xx^t.(ln(t)+\frac{t}{x})$</p>
|
2,861,362 | <p>I have the following question:
- We proved that if $T$ is 1-1 and $\{v_1...v_n\}$ is linearly independent then $\{T(v_1)...T(v_n)\}$ is linearly independent! I understood the proof! But can’t $\{T(v_1)...T(v_n)\}$ be linearly independent without having $T$ 1-1?
The image I uploaded shows my work on proving that th... | Jack D'Aurizio | 44,121 | <p>In a elementary fashion we have
$$\begin{eqnarray*}\log\left(\frac{n+1}{n}\right)&=&\int_{n}^{n+1}\frac{dx}{x}=\int_{0}^{1}\frac{dx}{n+x}=\int_{-1/2}^{1/2}\frac{dx}{n+\frac{1}{2}+x}\\&=&\int_{0}^{1/2}\left(\frac{1}{n+\frac{1}{2}+x}+\frac{1}{n+\frac{1}{2}-x}\right)\,dx\\&=&\int_{0}^{1}\frac{dz... |
3,221,652 | <p><span class="math-container">$ a= \frac{2}{5}; f(x) = 10x^3 +6x^2 + x -2 $</span></p>
<p>Have difficulties with advanced polynomials of this type. I can not factor.To find the right root. Here is my solution:</p>
<p><span class="math-container">$f(x) = 10x^3 +6x^2 + x -2 = 10x^3 +6x^2 +1x^2 -1x^2 +x -2 = (5x^2+x)(... | Community | -1 | <p><span class="math-container">$f(\dfrac25)=\dfrac {80}{125}+\dfrac {24}{25}+\dfrac25-2=\dfrac{80}{125}+\dfrac{120}{125}+\dfrac{50}{125}-\dfrac{250}{125}=0$</span>. Thus it is a root.</p>
|
885,478 | <p>Let $f$ be a function defined on an interval $I$ differentiable at a point $x_o$ in the interior of $I$.</p>
<p>Prove that if $\exists a>0$ $ \ [x_o -a, x_o+a] \subset I$ and $ \ \forall x \in [x_o -a, x_o+a] \ \ f(x) \leq f(x_o)$, then $f'(x_o)=0$.</p>
<p>I did it as follows:</p>
<p>Let b>0.</p>
<p>Sin... | JC574 | 165,562 | <p>It seems you're pretty much there. There are a couple of things you can do to make the argument clearer, both in the early line using the property of differentiable:</p>
<p>1.Differentiation talks about a limit, make sure you say $\forall b > 0 $, as this is what gets you the final step.</p>
<p>2.You want to in... |
885,478 | <p>Let $f$ be a function defined on an interval $I$ differentiable at a point $x_o$ in the interior of $I$.</p>
<p>Prove that if $\exists a>0$ $ \ [x_o -a, x_o+a] \subset I$ and $ \ \forall x \in [x_o -a, x_o+a] \ \ f(x) \leq f(x_o)$, then $f'(x_o)=0$.</p>
<p>I did it as follows:</p>
<p>Let b>0.</p>
<p>Sin... | copper.hat | 27,978 | <p>This is not an answer, but is too long for the comments.</p>
<p>Another approach is to show that if $f'(x_0) > 0$, where $x_0$ is in the interior of $I$, then for some $\delta>0$, then if $x \in (x_0-\delta,x_0)$ we have $f(x) <f(x_0)$ and for $x \in (x_0, x_0+\delta)$ we have $f(x) > f(x_0)$.</p>
<p>A... |
2,657,632 | <p>My question involves part (b) of Chapter 11 problem 6.4 in Artin's Algebra textbook.</p>
<blockquote>
<p>In each case, describe the ring obtained from <span class="math-container">$\mathbb{F_2}$</span> by adjoining an element <span class="math-container">$α$</span> satisfying the given relation:</p>
<p>(a) <span cla... | Tsemo Aristide | 280,301 | <p>Yes, $x^2+1=(x+1)^2$ in $\mathbb{F_2}[x]$ you deduce that $\mathbb{F_2}[x]/(x^2+1)$ is a $2$ dimensiobal vector space over $\mathbb{F_2}$.</p>
<p>Adjoining an element (of the algebraic closure) which satisfies $\alpha^2+1=0$ is adjoining $1$ and the result is $\mathbb{F_2}$.</p>
|
2,983,370 | <p><strong>question:</strong></p>
<p>maximum value of <span class="math-container">$\theta$</span> untill which the approximation <span class="math-container">$\sin\theta\approx \theta$</span> holds to within <span class="math-container">$10\%$</span> error is </p>
<p><span class="math-container">$(a)10^{\circ}$</spa... | Jam | 161,490 | <p>This is similar to the other two answers but another way of thinking about it. By definition, the true <span class="math-container">$\theta$</span> is at <span class="math-container">$\left|\frac{\sin(\theta)-\theta}{\sin(\theta)}\right|=\frac{10\%}{100\%}$</span>. This would give us <span class="math-container">$\t... |
273,086 | <p>I'd like to deploy a website to display and increment counters that track daily activities.</p>
<p>Here's a minimal working example:</p>
<pre><code>bin=CreateDatabin[]
CloudDeploy[FormPage[{"Pushups" -> {0, 5, 10, 15, 20, 25, 30},
"Water" -> {0, .5, 1, 2}},
Data... | lericr | 84,894 | <ul>
<li><p><strong>Replace drop-downs with buttons.</strong> You can add your own controls for each form element. Not sure if SetterBar is what you want, and it seems to behave a bit strangely when it's deployed to the cloud, so you might have to experiment with other controls.</p>
<pre><code>exerciseForm =
FormPag... |
3,753,143 | <p>We spend a lot of time learning different theories (for instance, theory of differential forms, sobolev spaces, homology groups, distributions). Although (at least most parts of) these theories are very natural and understandable when we read them from books, they are very difficult to create at the first place: it ... | Martin Hansen | 646,413 | <p>Well, here is my h'penny's worth :</p>
<p>In trying and eventually succeeding in proving Fermat's last Theorem, Andrew Wiles developed vast swaths of new mathematics. (As had others before him even without managing a proof). The problems are all around; The Riemann Hypothesis - with the tantalising "video's&quo... |
3,267,216 | <p>An urn contains equal number of green and red balls. Suppose you are playing the following game. You draw one ball at random from the urn and note its colour. The ball is then placed back in the urn, and the selection process is repeated. Each time a green ball is picked you get 1 Rupee. The first time you pick a re... | Anurag A | 68,092 | <p>Assuming <span class="math-container">$ab>0$</span>.
<span class="math-container">\begin{align*}
\frac{a+b}{2\sqrt{ab}} & = \frac{m}{n}\\
n^2(a+b)^2& = 4m^2ab\\
n^2\left(\frac{a}{b}+\frac{b}{a}+2\right) &=4m^2.
\end{align*}</span>
Let <span class="math-container">$\frac{a}{b}=t$</span>, then you have ... |
3,120,090 | <blockquote>
<p>Find all the values of <span class="math-container">$\theta$</span> that satisfy the equation
<span class="math-container">$$\cos(x \theta ) + \cos( (x+2) \theta ) = \cos( \theta )$$</span></p>
</blockquote>
<p>I've tried simplifying with factor formulae and a combo of compound angle formulae, and... | Claude Leibovici | 82,404 | <p>Use
<span class="math-container">$$\cos(p)+\cos(q)=2 \cos \left(\frac{p+q}{2}\right)\cos \left(\frac{p-q}{2}\right)$$</span> Make <span class="math-container">$p=(x+2)\theta$</span> and <span class="math-container">$q=x\theta$</span> making your problem to be
<span class="math-container">$$2\cos((x+1)\theta) \cos(\t... |
227,296 | <p>I happened to ponder about the differentiation of the following function:
$$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$
Now, while I do know how to manipulate power towers to a certain extent, and know the general formula to differentiate $g(x)$ wrt $x$, where $$g(x)=f(x)^{f(x)^{f(x)^{f(x)^{f(x)^{f(x)^{f(x)... | Fedor Petrov | 4,312 | <p>Update: it answers a different question.</p>
<p>If $1/(N-1)>x> 1/N$, then $p_N:=(Nx)^{(N+1)x^{\dots}}=\infty$, $p_{N-1}=0$, $p_{N-2}=1$, and so we have a tower of height $N-3$. Thus your function has a jump in points $1/N$ and is smooth between consecutive jumps (values at points of jumps is difficult to defi... |
2,227,135 | <p>I have this information from my notes:<span class="math-container">$\def\rk{\operatorname{rank}}$</span></p>
<p>Let <span class="math-container">$A ∈ \mathbb{R}^{m\times n}$</span>. Then </p>
<ul>
<li><span class="math-container">$\rk(A) = n$</span></li>
<li><span class="math-container">$\rk(A^TA) = n$</span></li>... | Marc van Leeuwen | 18,880 | <p><span class="math-container">$\def\rk{\operatorname{rank}}$</span>No you cannot apply the same logic to <span class="math-container">$vv^T$</span>, in other words for <span class="math-container">$AA^T$</span> where <span class="math-container">$A$</span> is (still) a <span class="math-container">$1\times m$</span> ... |
699,933 | <p>There is a proof of the real case of Cauchy-Schwarz inequality that expands $\|\lambda v - w\|^2 \geq 0 $, gets a quadratic in $\lambda$, and takes the discriminant to get the Cauchy-Schwarz inequality. In trying to do the same thing in the complex case, I ran into some trouble. First, there are proofs <a href="http... | Eric Auld | 76,333 | <p>I just thought of $$|\lambda|^2\|v\|^2 - 2 \text{Re}(\lambda \langle v,w\rangle) + \|w\|^2 \leq |\lambda|^2\|v\|^2 + 2 | \lambda ||\langle v,w\rangle | + \|w\|^2,$$ which yields the same discriminant. This seems to work!</p>
|
3,960,189 | <p>An auto insurance company is implementing a new bonus system. In each month, if a policyholder does not have an accident, he or she will receive a $5 cashback bonus from the insurer. Among the 1000 policyholders, 400 are classified as low-risk drivers and 600 are classified as high-risk drivers. In each month, the p... | Ross Millikan | 1,827 | <p>Your first line computes that the average number of accident free drivers in a month is <span class="math-container">$840$</span>. Those people receive <span class="math-container">$840\cdot 5=4200$</span> for the month. Each month is the same, so the annual total is <span class="math-container">$12\cdot 4200=5040... |
2,727,974 | <blockquote>
<p>I am in need of this important inequality $$\log(x+1)\leqslant x$$.</p>
</blockquote>
<p>I understand that $\log(x)\leqslant x$. For $c\in\mathbb{R}$. However is it true that $\log(x+c)\leqslant x$?</p>
<p>It is hard to accept because it seems like $c$ cannot be arbitrary.
I have tried to prove thi... | fleablood | 280,126 | <p>"However is it true that log(x+c)⩽x?"</p>
<p>Let $x =1$ and $c = 500,000,000$ then is $\log 500,000,001 \le 1$?</p>
<p>It is true that if $\log (x+c) \le x\iff x+c \le e^x$ and it is true that $\frac {d(x+c)}{dx} = 1 \le \frac{d e^x}{dx} = e^x$ if $x > 0$. So, for positive $x$ we have that $e^x$ increases fast... |
2,029,538 | <blockquote>
<p>A function $f(x)$, where $x$ is a real number , is defined implicitly by the following formula:
$$f(x)=x-\int^{\frac{\pi}{2}}_0f(x)\sin(x)dx$$
Find the explicit function for $f(x)$ in its simplest form.</p>
</blockquote>
<p>This question appeared in the recent New Zealand Qualifications Authority... | David | 119,775 | <p><strong>Hint</strong>. The integral in your equation is a <strong>definite</strong> integral and evaluates to a constant. Therefore
$$f(x)=x-c$$
where $c$ is a constant satisfying
$$c=\int_0^{\pi/2} (x-c)\sin x\,dx\ .$$
Evaluate the integral, then solve the equation to find $c$.</p>
<p>Answer:</p>
<blockquote cl... |
4,233,619 | <p>Consider all natural numbers whose decimal expansion has only the even digits <span class="math-container">$0,2,4,6,8$</span>. Suppose these are arranged in increasing order. If <span class="math-container">$a_n$</span> denotes the <span class="math-container">$n$</span>-th number in this sequence then the value of ... | Lalit Tolani | 913,165 | <p>Note that <span class="math-container">$y^z$</span> is one number. To avoid confusion you can let it equal to <span class="math-container">$a$</span></p>
<p>Therefore <span class="math-container">$ \displaystyle\log x^{(y^{z})}=\log x^a=a\log x=y^z\log x$</span></p>
<p>Also <span class="math-container">$yz\log x=\lo... |
2,049,777 | <p>$2.$ Find the dimensions of </p>
<p>(a) the space of all vectors in $R^n$ whose components add to zero;</p>
<p>(c) the space of all solutions to $\frac{d^2y(t)}{dt^2} −3 \frac{dy(t)}{dt} +2y(t) = 0$. </p>
<p>for (a) Im pretty sure that the dimension is $n-1$ but people seem to differ, I thought it is $n-1$ since ... | Community | -1 | <p>For first choose (0,0,...,0,1,0,...0,-1) where 1 is in the i th place i=1 to n-1.
For the second the general solution is a.e^2x +b e^x. So the dimension is 2. Since e^2x and e^x are linearly independent functions.</p>
|
181,321 | <p>Assuming that <code>a.b=z</code>,where <code>b={b1,b2,b3}</code> and <code>z=x b1+y b2+z b3</code>, obviously <code>a={x,y,z}</code>, how to realize it by mathematica?</p>
| kglr | 125 | <p>If <code>a.{b1,b2,b3} == x b1+y b2+z b3</code> <em><strong>for all</strong></em> <code>{b1,b2,b3}</code>:</p>
<pre><code>Reduce[ForAll[{b1, b2, b3}, Dot[{a1, a2, a3}, {b1, b2, b3}] == x b1 + y b2 + z b3]]
</code></pre>
<blockquote>
<p>a3 == z && a2 == y && a1 == x </p>
</blockquote>
<p>or</p>
... |
2,348,811 | <p>Whenever I go through the big pile of socks that just went through the laundry, and have to find the matching pairs, I usually do this like I am a simple automaton:</p>
<p>I randomly pick a sock, and see if it matches any of the single socks I picked out earlier and that haven't found a match yet. If there is a mat... | lulu | 252,071 | <p>The expected number can be computed via Linearity of Expectation. Let $E[n,k]$ denote the answer and let $\{X_i\}_{i=1}^n$ denote the indicator variable for the $i^{th}$ pair. Thus $X_i=1$ if exactly one member of the $i^{th}$ pair has been chosen in your $k$ trials, and $X_i=0$ otherwise. It is easy to see that ... |
2,348,811 | <p>Whenever I go through the big pile of socks that just went through the laundry, and have to find the matching pairs, I usually do this like I am a simple automaton:</p>
<p>I randomly pick a sock, and see if it matches any of the single socks I picked out earlier and that haven't found a match yet. If there is a mat... | HelloGoodbye | 119,068 | <p>If you let your function that gives the expected number of socks in your 'no match yet' pile take $k$ and $2n-k$ as arguments, i.e.</p>
<p>$$f\,=\,f(k,\,2n-k),$$</p>
<p>it will be <a href="https://en.wikipedia.org/wiki/Symmetric_function" rel="nofollow noreferrer">symmetric</a>. The 'no match yet' pile is just the... |
3,946,974 | <blockquote>
<p>To prove:<span class="math-container">$$\text{If } A_1\subseteq A_2 \subseteq ... \subseteq A_n\text{ ,then } \bigcup_{i=1}^n A_i = A_n$$</span> using the axioms of ZFC Set Theory.</p>
</blockquote>
<p>Honestly, this statement is <strong>very obvious</strong>, but I do not want to take that for granted.... | Luiz Cordeiro | 58,818 | <p>To be completely formal, you'd could first prove by induction that <span class="math-container">$A_i\subseteq A_n$</span> for every <span class="math-container">$i\leq n$</span>. Then prove that <span class="math-container">$\bigcup_{i=1}^n A_i\subseteq A_n$</span> and that <span class="math-container">$A_n\subseteq... |
1,691,825 | <p>My textbook goes from</p>
<p>$$\frac{\left( \frac{6\ln^22x}{2x} \right)}{\left(\frac{3}{2\sqrt{x}}\right)}$$</p>
<p>to:</p>
<p>$$\frac{6\ln^22x}{3\sqrt{x}}$$</p>
<p>I don't see how this is right. Could anyone explain?</p>
| Martigan | 146,393 | <p>Well...</p>
<p>$\dfrac ab=a\times \dfrac 1b$</p>
<p>In your case $a=\frac{6\ln^22x}{2x}$ and $b=\frac{3}{2\sqrt{x}}$</p>
<p>Then... $\dfrac 1b=\dfrac{2\sqrt{x}}{3}$.</p>
<p>Can you go from here?</p>
|
2,033,790 | <p>How do you prove the sequence $x_n = (\frac{n}{2})^n$ diverges?</p>
<p>Here is my attempt: </p>
<p>Suppose $x_n \to L$. This means $(\forall \epsilon > 0) (\exists N \in \mathbb{N}) (\forall n>N)|x_n-L| < \epsilon$</p>
<p>Assume $n > N$</p>
<p>Then $|x_n-L| < \epsilon$</p>
<p>$|(\frac{n}{2})^n-L|... | Retired account | 372,313 | <p>Any sequence which is unbounded is divergent. Note that $(n/2)^{n}>n/2$ for $n>2$. Then if we pick any M, $n = 2M$ will have $(n/2)^{n}>n/2=M$. Therefore, the sequence is unbounded and divergent.</p>
|
33,303 | <p>Is it possible to attach certain pieces of code to certain controls in a Manipulate? For example, consider the following Manipulate</p>
<pre><code>Manipulate[
data = Table[function[x], {x, -Pi*10, Pi*10, Pi/1000}];
ListPlot[{x, data}, PlotRange -> {{start, stop}, Automatic}]
, {function, {Sin, Cos, Tan}}
, ... | Nasser | 70 | <p>This is good reason to use the second argument of dynamics.</p>
<p><img src="https://i.stack.imgur.com/ezhK3.gif" alt="enter image description here"></p>
<pre><code>Manipulate[
function; (*just to allow tracking, since not explicity in the command*)
ListLinePlot[data, PlotRange -> {{start, stop}, Au... |
2,082,815 | <p>Find the $100^{th}$ power of the matrix $\left( \begin{matrix} 1& 1\\ -2& 4\end{matrix} \right)$.</p>
<p>Can you give a hint/method?</p>
| kub0x | 309,863 | <p>Since $A=PDP^{-1}$ then $A^{100}=PD^{100}P^{-1}$</p>
<p>Then we know that $M=\begin{bmatrix} 1& 1\\ -2& 4\end{bmatrix}$ has eigenvalues $\lambda=(3,2)$ and eigenvector matrix P=$\begin{bmatrix} 1&1\\2&1\end{bmatrix}$ therefore $D=\begin{bmatrix} 3&0\\0&2\end{bmatrix}$</p>
<p>Now apply $A^{1... |
2,082,815 | <p>Find the $100^{th}$ power of the matrix $\left( \begin{matrix} 1& 1\\ -2& 4\end{matrix} \right)$.</p>
<p>Can you give a hint/method?</p>
| Community | -1 | <p>Let $$P= \begin {pmatrix} 1& 1\\ -2 & 4 \end {pmatrix} $$The key is to write the matrix $P $ in terms of a diagonal matrix $D $. Thus we wish to write $$P=ADA^{-1} $$ where $D $ is a diagonal matrix. Thus, $$P^n =AD^n A^{-1} $$ $D^n $ being very easy to compute, we find the values of $D $ and $A $ using eige... |
47,561 | <p>The Hilbert matrix is the square matrix given by</p>
<p>$$H_{ij}=\frac{1}{i+j-1}$$</p>
<p>Wikipedia states that its inverse is given by</p>
<p>$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-i}}{{i+j-2}\choose{i-1}}^2$$</p>
<p>It follows that the entries in the inverse matrix are all i... | Henricus V. | 84,447 | <p>(Note: The matrix elements are indexed from $0$. To avoid confusion, I will not index into a matrix without brackets)</p>
<p>Here is a naive approach to this problem, without hindsight of the already-derived formula.</p>
<blockquote>
<p>Lemma:
If $M \in \mathbb F^{n \times n}$ is an invertible matrix and $(\al... |
2,983,199 | <p>I am looking for an example demonstrating that <span class="math-container">$\lim\inf x_n+\lim \inf y_n<\lim \inf(x_n+y_n)$</span> but for the life of me i can't find one. any suggestions?</p>
| ZAF | 609,023 | <p>Let <span class="math-container">$(x_{n})_{n}$</span> such that <span class="math-container">$x_{2n} = 1$</span> and <span class="math-container">$x_{2n - 1} = -1$</span></p>
<p>Let <span class="math-container">$(y_{n})_{n}$</span> such that <span class="math-container">$y_{2n} = -1$</span> and <span class="math-co... |
914,440 | <p>By the definition of topology, I feel topology is just a principle to define "open sets" on a space(in other words, just a tool to expand the conception of open sets so that we can get some new forms of open sets.) But I think in the practical cases, we just considered Euclidean space most and the traditional form o... | Theon Alexander | 165,460 | <p>For starters, we have things defined abstractly (at least in principle) such as manifolds. These are defined through charts, which endow your manifold with a topology.</p>
<p>The important tool of partitions of unity (which reduces Stokes's theorem and others to a toy case) works due to the 2nd Countability Axiom (... |
142,507 | <p>Let $\psi(x) := \sum_{n\leq x} \Lambda(n)$ where $\Lambda(n)$ is the Von-Mangoldt function.
I want to show that if $$ \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} =1 $$ then also $$\lim_{x\rightarrow \infty} \frac{\pi(x) \log x }{x}=1.$$</p>
<p>I tried to play a little bit with $\psi$, what I want to show is that:... | Syd | 27,920 | <p>Consider each set as {0,1} Map the elements of the product onto the closed unit interval by mapping each to a binary number. For example, [0,1,0,1,0,1,..] maps to .01010101... [I'm assuming the Axiom of Choice.]</p>
<p>The image is uncountable with countably many duplications, hence B must have the same cardinality... |
2,947,911 | <p>I have a homework with this equation:
<span class="math-container">$$(x+2y)dx + ydy = 0$$</span></p>
<p>However I have no idea how to solve it.
I tried couple things:</p>
<ol>
<li>Is it linear equation, or does it have "standard form" : <span class="math-container">$\frac{dy}{dx}+\frac{x}{y} + 2 = 0$</span>. Well... | Joseph Vidal-Rosset | 62,171 | <p><a href="https://www.vidal-rosset.net/sequent_calculus_prover_with_antisequents_for_classical_propositional_logic.html" rel="nofollow noreferrer">This prover is helpful to reply to this question. See example 3</a></p>
|
1,038,152 | <p>Let $B \subseteq \mathbb{R}_{+}$ such that B is non-empty.
consider $B^{-1} = \left \{b^{-1} : b\in B \right \}$.<br>
Show that if $B^{-1}$ is unbounded from above, then $\inf\left(B\right)=0$</p>
<p>How can i prove that? tnx!</p>
| Matt Samuel | 187,867 | <p>The long exact sequence is not really helpful here. What we need to do is look at chain groups. Any two 0-chains in $A$ of the form $1f$ where $f$ is a constant map are homologous and map to a generator when we pass to homology. 0 chains of the form $1(i\cdot f)$ generate the direct summand in $H_0(X)$ corresponding... |
4,272,214 | <h2>The Equation</h2>
<p>How can I analytically show that there are <strong>no real solutions</strong> for <span class="math-container">$\sqrt[3]{x-3}+\sqrt[3]{1-x}=1$</span>?</p>
<h2>My attempt</h2>
<p>With <span class="math-container">$u = -x+2$</span></p>
<p><span class="math-container">$\sqrt[3]{u-1}-\sqrt[3]{u+1}=... | Rene Schipperus | 149,912 | <p>Use the fact that <span class="math-container">$a^3+b^3+c^3-3abc$</span> is divisible by <span class="math-container">$a+b+c$</span>.</p>
<p>Let</p>
<p><span class="math-container">$$a=\sqrt[3]{x-3}$$</span>
<span class="math-container">$$b=\sqrt[3]{1-x}$$</span>
<span class="math-container">$$c=-1$$</span></p>
<p>T... |
4,272,214 | <h2>The Equation</h2>
<p>How can I analytically show that there are <strong>no real solutions</strong> for <span class="math-container">$\sqrt[3]{x-3}+\sqrt[3]{1-x}=1$</span>?</p>
<h2>My attempt</h2>
<p>With <span class="math-container">$u = -x+2$</span></p>
<p><span class="math-container">$\sqrt[3]{u-1}-\sqrt[3]{u+1}=... | user | 505,767 | <p>We have that by <span class="math-container">$A^3-B^3=(A-B)(A^2+AB+B^2)$</span></p>
<p><span class="math-container">$$\sqrt[3]{x-3}+\sqrt[3]{1-x}=\sqrt[3]{x-3}-\sqrt[3]{x-1}=$$</span></p>
<p><span class="math-container">$$=\frac{-2}{\sqrt[3]{(x-3)^2}+\sqrt[3]{(x-3)(x-1)}+\sqrt[3]{(x-1)^2}}<0$$</span></p>
<p>indee... |
103,317 | <p>I have a 3D Plot with 3 separate functions for <code>z</code> in terms of <code>x</code> and <code>y</code>. I would like to convert this to a RegionPlot with the <code>max</code> of the three functions plotted for the variables <code>x</code> and <code>y</code>.</p>
<p>Essentially, this would be the top view of th... | Michael E2 | 4,999 | <p>Here's my answer <a href="https://mathematica.stackexchange.com/a/58257/4999">here</a> applied to this question:</p>
<pre><code>Manipulate[
With[{fns = {0.1,
Piecewise[{{α - ((2 α - 1) α)/y + (x*α^2)/(2 y), (1 - α) + α^2/(2 y) >= 0.35}}],
Piecewise[{{α + (1 - α)^2/(2 y),
(1 - α) + (((3 α ... |
2,491,881 | <p>I'm trying to factorise $(4 + 3i)z^2 + 26iz + (-4+3i)$.</p>
<p>I tried to use the quadratic formula to factorise $(4 + 3i)z^2 + 26iz + (-4+3i)$, where $b = 26i$, $a = (4 + 3i)$, $c = (-4+3i)$. This gives us the roots $z = \dfrac{-4i - 12}{25}$ and $z =-4i - 3$.</p>
<p>But $\left( z + \dfrac{4i + 12}{25} \right) \l... | Sarvesh Ravichandran Iyer | 316,409 | <p>See, the definition is very simple : given a sequence in the set, there exists a convergent subsequence, which converges in the set itself. Then, the set is compact.</p>
<p>Now, what you have to start with is a sequence $x_n \in X \cup Y$. There is nothing to assume about the nature of this sequence at all. It is t... |
2,971,980 | <p>Show that if <span class="math-container">$0<b<1$</span> it follows that
<span class="math-container">$$\lim_{n\to\infty}b^n=0$$</span>
I have no idea how to express <span class="math-container">$N$</span> in terms of <span class="math-container">$\varepsilon$</span>. I tried using logarithms but I don't see h... | Gâteau-Gallois | 393,828 | <p>Welcome to MSE. What you want is, given some <span class="math-container">$\epsilon > 0$</span>, <span class="math-container">$b^N \leq \epsilon$</span>. Note that, by properties of the logarithm, and since both <span class="math-container">$b$</span> and <span class="math-container">$\epsilon$</span> are positiv... |
843,763 | <blockquote>
<p>Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. </p>
</blockquote>
<p>Can anyone give some hint to solve this problem? I tried contradiction but could not reach a proof.</p>
<p>I spend part of the day studying this... | Adam Rubinson | 29,156 | <p>A pictoral representation of mm-aops's proof. I also avoid use of: "the distance from <span class="math-container">$a_n$</span> to <span class="math-container">$a_m$</span> is the same as between <span class="math-container">$a_{n-m}$</span> and <span class="math-container">$a_0 = 0$</span>."</p>
<p>Let <s... |
2,213,528 | <p>Combination formula is defined as:</p>
<p>$$\binom{n}{r}=\frac{n!}{(n-r)!\cdot r!}$$</p>
<p>We do $(n-r)!$ because we want the combinations of only $r$ objects given $n$. We do $r!$ again because in combinations, order does not matter so doing this gets rid of repeated results. But why? Why does $r!$ get rid of an... | jnez71 | 295,791 | <p>Like others have already pointed out, the conundrum you are facing is because you are confusing probability with probability <em>density</em>.</p>
<p>Recall that we have an underlying <a href="https://en.wikipedia.org/wiki/Probability_space" rel="nofollow noreferrer">probability space</a> $(\Omega, \mathscr{F}, P)$... |
1,104,163 | <p>I have come up with the equation in the form $${{dy}\over dx} = axe^{by}$$, where a and b are arbitrary real numbers, for a project I am working on. I want to be able to find its integral and differentiation if possible. Does anyone know of a possible solution for $y$ and/or ${d^2 y}\over {dx^2}$?</p>
| graydad | 166,967 | <p>I think there are some scenarios to consider if <span class="math-container">$a$</span> and or <span class="math-container">$b$</span> is equal to zero.</p>
<p><strong>Case 1:</strong> <span class="math-container">$a=0$</span></p>
<p>Then <span class="math-container">$$\frac{dy}{dx} = 0 \implies y = C$$</span></p>
<... |
3,220,135 | <p>How many integer numbers, <span class="math-container">$x$</span>, verify that the following</p>
<p><span class="math-container">\begin{equation*}
\frac{x^3+2x^2+9}{x^2+4x+5}
\end{equation*}</span></p>
<p>is an integer?</p>
<p>I managed to do:</p>
<p><span class="math-container">\begin{equation*}
\frac{x^3+2x^2+... | nonuser | 463,553 | <p>Let <span class="math-container">$t=x+2$</span>, then <span class="math-container">$$t^2+1\mid 3t+13$$</span> and thus <span class="math-container">$$t^2+1\mid t(3t+13)-3(t^2+1)= 13t-3$$</span>
so <span class="math-container">$$t^2+1\mid 13(3t+13)-3(13t-3) = 178$$</span></p>
<p>So <span class="math-container">$$t^... |
2,319,766 | <p>Lets say ,I have 100 numbers(1 to 100).I have to create various combinations of 10 numbers out of these 100 numbers such that no two combinations have more than 5 numbers in common given a particular number can be used max three times.
E.g.</p>
<ol>
<li>Combination 1: 1,2,3,4,5,6,7,8,9,10</li>
<li>Combination 2: ... | Community | -1 | <p>This is how I will model it: each o (of a total of 14) represents one animal and there are two / to divide them into three group; first group will be cats, second one dogs and the last one pigs.</p>
<p>An example demonstration:
ooo/oo/ooooooooo This is 3 cats, 2 dogs and 9 pigs.</p>
<p>You got the idea. So basical... |
496,479 | <p>This symbols are used to describe left recursion : </p>
<blockquote>
<p>$A\to B\,\alpha\,|\,C$<br>$B\to A\,\beta\,|\,D,$</p>
</blockquote>
<p>It is taken from :
<a href="http://en.wikipedia.org/wiki/Left_recursion" rel="nofollow">http://en.wikipedia.org/wiki/Left_recursion</a></p>
<p>How can these symbols be ... | Eike Schulte | 11,853 | <p>This notation is used in <a href="http://en.wikipedia.org/wiki/Formal_language" rel="nofollow" title="Wikipedia on formal languages">formal language theory</a> to describe <a href="http://en.wikipedia.org/wiki/Formal_grammar" rel="nofollow" title="Wikipedia on formal grammars">formal grammars</a>. A formal grammar c... |
4,350,781 | <p><a href="https://i.stack.imgur.com/57qXm.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/57qXm.jpg" alt="enter image description here" /></a></p>
<p>I was able to follow most of the proof but I don’t understand how the author concludes that <span class="math-container">$r=0$</span> at the final pa... | Miguel Mars | 226,969 | <p>First we can state that <span class="math-container">$z$</span> is the "free variable", so it doesn't depend on <span class="math-container">$x$</span> and <span class="math-container">$y$</span> but both of them will depend on <span class="math-container">$z$</span>. We can see in the figure that it goes ... |
1,258,623 | <p>Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.</p>
| Singh | 121,735 | <p>I used <a href="https://en.wikipedia.org/wiki/Beal%27s_conjecture" rel="nofollow">Beal's conjecture</a> to show that $ac+bd$ is composite.</p>
<p>We have $a^4+b^4=e^5$, Using Beal's conjecture we say that $a$ and $b$ will have common prime factor.</p>
<p>Let the common prime factor be $f$.</p>
<p>$a=a_1f$ and $b=... |
2,553,610 | <p>I come across this result:</p>
<blockquote>
<p>Any power is conditionally convergent for at most two values of $x$, the endpoints of its interval of convergence.</p>
</blockquote>
<p>If it is so then why?</p>
| Fred | 380,717 | <p>Let $\sum_{n=0}^\infty a_n(x-x_0)^n$ be a power series with radius of convergence $=r>0$.</p>
<p>If $r= \infty$, then the power series converges absolutely in each $x \in \mathbb R$.</p>
<p>Let $r< \infty$ and $I=(x_0-r,x_0+r)$. Then the power series converges absolutely in each $x \in I$.</p>
<p>Therefore ... |
2,985,256 | <blockquote>
<p>Let there are three points <span class="math-container">$(2,5,-3),(5,3,-3),(-2,-3,5)$</span> through which a plane passes. What is the equation of the plane in Cartesian form?</p>
</blockquote>
<p>I know how to find it in using vector form by computing the cross product to get the normal vector and p... | Josh B. | 611,981 | <p>You in fact can solve it by this method, even if it does seem like there will be infinite solutions. This is because the normal vector for a plane is not unique: if <span class="math-container">$\vec{n}$</span> is a normal vector to the plane, then <span class="math-container">$c\vec{n}$</span> is as well, provided ... |
3,860,623 | <p>I'm trying to prove
<span class="math-container">$$\forall z\in\mathbb C-\{-1\},\ \left|\frac{z-1}{z+1}\right|=\sqrt2\iff\left|z+3\right|=\sqrt8$$</span>
thus showing that the solutions to <span class="math-container">$\left|(z-1)/(z+1)\right|=\sqrt2$</span> form the circle of center <span class="math-container">$-3... | fgrieu | 35,016 | <p>Using thrice that <span class="math-container">$\forall c\in\mathbb C,\, \left|c\right|^2=c\,\bar c$</span>, I got it down to</p>
<p><span class="math-container">$$\begin{align}\left|\frac{z-1}{z+1}\right|=\sqrt2&\iff\left|z-1\right|=\sqrt2\,\left|z+1\right|\\
&\iff\left|z-1\right|^2=2\,\left|z+1\right|^2\\
... |
3,291,190 | <p>Compute</p>
<p><span class="math-container">$$\text{const} \left( \frac{1}{(1-z_1z_2)(1-z_1^2z_2)(1-z_1)(1-z_2)z_1^{3t}z_2^{2t}} \right) $$</span> </p>
<p>The answer should be a polynomial in <span class="math-container">$t$</span>. This is exercise 2.38 from Beck's Computing the Continuous Discretely, and the cor... | gt6989b | 16,192 | <p><strong>HINT</strong>
Denote the coefficient of <span class="math-container">$x^k$</span> in <span class="math-container">$f(x)$</span> as <span class="math-container">$\left[x^k\right]f(x)$</span>.</p>
<p>You are looking for
<span class="math-container">$$
\left[x^0 y^0 \right] \frac{F(x,y)}{x^{3t} y^{2t}}
= \left... |
4,402,613 | <p>I know that the limits of integration in spherical coordinates are these, but I can't find the reason why that 2 appears,but I can't find a way to go further, and evaluate r in a <span class="math-container">$a \sin \theta$</span>,</p>
<p><span class="math-container">$$V = \int_0^{2\pi}\int_0^{a\sin{\theta}}\int_{-\... | eyeballfrog | 395,748 | <p>The 2 comes from the limits of integration: <span class="math-container">$\sqrt{a^2-r^2} - (-\sqrt{a^2-r^2}) = 2\sqrt{a^2-r^2}$</span></p>
<p>As for proceeding, here's a hint:
<span class="math-container">$$
2\int_0^{2\pi}\int_0^{a\sin{\theta}}r\sqrt{a^2-r^2}\mathrm{d}r\mathrm{d}\theta = \int_0^{2\pi}\int_0^{a\sin{\... |
740,294 | <p>Let $y_n$ satisfy the nonlinear difference equation:</p>
<p>$$(n+1)y_n=(2n)y_{n-1}+n.$$</p>
<p>Let $u_n=(n+1) y_n$. Show that</p>
<p>$$u_n= 2u_{n-1}+n.$$</p>
<p>Solve the linear difference equation for $u_n$. Hence find $y_n$ subject to the initial condition $y_0=4$.</p>
<p>I have showed that $u_n=2u_{n-1}+n$, ... | vonbrand | 43,946 | <p>This is a linear recurrence of the first order, thus solvable. Write:
$$
y_{n + 1} - \frac{2 n + 2}{n + 2} y_n = \frac{n + 1}{n + 2}
$$
If this is written:
$$
y_{n + 1} - u_n y_n = f_n
$$
you can divide by the summing factor:
$$
s_n = \prod_{0 \le k \le n} u_n
$$
to get the nicely telescoping:
$$
\frac{y_{n + 1}}{s_... |
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