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 |
|---|---|---|---|---|
3,852,436 | <p>I'm struggling on the following problem:</p>
<blockquote>
<p>You are given N boxes indexed from 1 to N. Each box contains either
no coins or one coin. The number of empty boxes and the number of
boxes with one coin are denoted by n0 and n1, respectively. You take
a random subset of the boxes where each subset has... | lulu | 252,071 | <p>If <span class="math-container">$n_1=0$</span> then the answer is <span class="math-container">$1$</span> because all choices have no coins (and <span class="math-container">$0$</span> is even).</p>
<p>For <span class="math-container">$n_1>0$</span> the answer is <span class="math-container">$\frac 12$</span>.</p... |
1,531,154 | <p>A question from my calculus book states,</p>
<blockquote>
<p>Which points on the graph $y=4-x^2$ are the closest to the point (0,2)?</p>
</blockquote>
<p>Using some of my notes, I have a formula as follows (not sure what it's actually called):</p>
<p>$$d=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$</p>
<p>So I plugged... | Community | -1 | <p>If A=(0,2), and B=(a,b) a point of y=4-x^2, then AB is minimum when vector AB is normal to curve. dy/dx=-2a => m=1/2a, but (b-2)/a=1/2a => b=5/2.</p>
|
81,435 | <p>Normally we consider simple arithmetic to be related to the world of objects. So the sum $3+2=5$ means $3$ three apples and $2$ apples gives $5$ apples. But is there an alternative interpretation which does not have anything to do with discrete objects?</p>
| analysisj | 14,966 | <p>There are varying interpretations. One interpretation in set theory is that each number is the set of all numbers prior. For instance, one would be defined as the set containing zero, or the null set so {0} where 0 is the null set. 2 would be {0,1}, or {0,{0}}, etc. This is another interpretation of the numbers.... |
57,914 | <p>I have the following code to show a red area defined by inequalities:</p>
<pre><code>ClearAll["Global`*"];
p = Reduce[y <= 3/10 x + 18 && y > x^2/8, {x, y}]
r = RegionPlot[p, {x, -15, 18}, {y, -5, 25},
GridLines -> {Table[i, {i, -15, 18}], Table[j, {j, -5, 25}]},
PlotStyle -> Directive[{... | RunnyKine | 5,709 | <p>Here is another solution using <strong>V10</strong> functionalities:</p>
<pre><code>region = ImplicitRegion[y <= 3/10 x + 18 && y > x^2/8, {{x, -15, 18}, {y, -5, 25}}];
lis = Tuples[{Range[-15, 18], Range[-5, 25]}];
</code></pre>
<p>We create a <code>RegionMemberFunction</code> </p>
<pre><code>rm =... |
153,772 | <p>Is there any relation between the chromatic number of a graph $G$ and its complement $G'$ that are always true?</p>
<p>I saw these ones: $\chi(G)\chi(G')\geq n$ and $\chi(G)+\chi(G')\geq 2n$,</p>
<p>but I'm not pretty sure about them.</p>
| Holdsworth88 | 22,437 | <p>The following proof is taken from Graphs and Digraphs by Chartrand, Lesniak, and Zhang, who attribute proof to Hudson V. Kronk.</p>
<p>Let <span class="math-container">$G$</span> be a graph such that <span class="math-container">$V(G)=n$</span>. Suppose <span class="math-container">$\chi(G)=k$</span> and <span class... |
153,772 | <p>Is there any relation between the chromatic number of a graph $G$ and its complement $G'$ that are always true?</p>
<p>I saw these ones: $\chi(G)\chi(G')\geq n$ and $\chi(G)+\chi(G')\geq 2n$,</p>
<p>but I'm not pretty sure about them.</p>
| 1233dfv | 102,540 | <p>$(a)$ Prove that $\chi(G)\cdot \chi(G')\geq n$.</p>
<p>Proof: For every graph $G$ and $G'$ we know that $\chi(G)\geq {n\over \alpha(G)}$ and $\chi(G')\geq \omega(G')=\alpha(G)$ where $\alpha(G)$ and $\omega(G)$ denote the independence number and clique number of $G$. So $\chi(G)\cdot \chi(G')\geq {n\over \alpha(G)}... |
2,593,392 | <p>The corollary is given below:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/7R4vx.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7R4vx.png" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/cNDji.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | Martin Argerami | 22,857 | <p>The $r$ is $1/\|A^{-1}\|$. That's exactly what Corollary 8.2. says. </p>
|
1,276,206 | <p>I am writing a computer program that involves generating 4 random numbers, a, b, c, and d, the sum of which should equal 100. </p>
<p>Here is the method I first came up with to achieve that goal, in pseudocode:</p>
<pre><code>Generate a random number out of 100. (Let's say it generates 16).
Assign this value as th... | kasperd | 161,559 | <p>Your question mention an inefficient algorithm generating four independent and uniformly distributed numbers among the integers from 0 to 100 and repeating until their sum is 100. I'll assume you are satisfied with the distribution generated by that algorithm, but you are not satisfied with the performance.</p>
<p>... |
226,353 | <p>I want to solve the following equation</p>
<pre><code>y''[x] + a + b y[x] + c y[x]^2 == 0, y[∞] == 0, y'[∞] == 0
</code></pre>
<p>where <code>a</code>, <code>b</code> and <code>c</code> are constants.</p>
| Nasser | 70 | <p>Not an answer, but long comment and some analysis.</p>
<p>The problem can be more clearly seen as follows.</p>
<p>The solution to the ODE contains elliptic integrals, whose solution contain elliptic special functions.</p>
<p>The problem is evaluating this at <span class="math-container">$\infty$</span> and its deriv... |
26,313 | <p>In page 21 of <em>A Problem seminar</em>, D. J. Newman presents a novel way (at least for me) to determine the expectation of a discrete random variable. He refers to this expression as the <strong>failure probability formula</strong>. His formula goes like this</p>
<p>$f_{0}+f_{1}+f_{2}+\ldots$</p>
<p>where $f_{n... | Sasha Voronov | 17,470 | <p>From the point of view of super geometry, Manin and I introduced the notion of a super Weyl group in relation to the geometry of homogeneous superspaces. We constructed Schubert supercells which were labeled by elements of a super Weyl group. I am not sure whether this is a well-accepted definition; it was rather a ... |
880,437 | <p>If the numerator of a fraction is increased by $2$ and the denominator by $1$, it becomes $\displaystyle \frac{5}{8}$ and if the numerator and the denominator of the same fraction are each increased by $1$, the fraction becomes equal to $\displaystyle \frac{1}{2}$. Find the fraction.</p>
<p>I tried,
Let the numerat... | amWhy | 9,003 | <p>That's a fine start. Now you've got two equations in two unknowns. $$\frac{x+2}{y+1} = \frac 58\tag{1}$$</p>
<p>$$\frac{x+1}{y+1} = \frac 12 \tag{2}$$</p>
<p>Now, we have to assume that $y+1 \neq 0$, given the information that's impossible, so we can take $(1)$ for example, and "cross multiply" to get: $$ 8(x+2) ... |
597,845 | <p>I am almost embarrassed writing this. But can someone tell me why this may not be true (so, please give me a counter example) for a power series where $x \in [-1,1]$</p>
<p>$|\sum_{n \geq 0} a_n x^n| \leq \sum_{n \geq 0} a_n $</p>
<p>Where $\sum_{n\geq 0} a_n $ is known to be convergent. </p>
<p>What if $a_n \geq... | copper.hat | 27,978 | <p>Take $a_0=1, a_1=-1, x=-1$ and all other $a_k = 0$. Then
$|\sum_n a_nx^n| = |1.1+(-1)(-1)| = 2$, but $\sum_n a_n = 1+(-1) = 0$.</p>
<p>If $a_n \ge 0$, and $|x| \le 1$, then clearly $|\sum_a a_n x^n| \le \sum_n a_n |x^n| \le \sum_n a_n$.</p>
|
187,489 | <p>I have a long expression involving different complex numbers. I want to force <em>Mathematica</em> to replace <code>z Conjugate[z]</code> by <code>Abs[z]^2</code> without using polar coordinates in the whole expression. For example,</p>
<pre><code>z w Conjugate[z] Conjugate[w] + Conjugate[z] w + z^2
</code></pre>
... | Αλέξανδρος Ζεγγ | 12,924 | <p>One way is to use a repeatedly applied rule</p>
<pre><code>z w Conjugate[z] Conjugate[w] + Conjugate[z] w + z^2 //. a_ Conjugate[a_] :> Abs[a]^2
</code></pre>
<blockquote>
<pre><code>z^2 + Abs[w]^2 Abs[z]^2 + w Conjugate[z]
</code></pre>
</blockquote>
|
1,713,713 | <p>To my knowledge, the exponential function is the unique function satisfying</p>
<p>$f'=f$ and
$f(0)=1$</p>
<p>however, unless I've made a mistake, we have</p>
<p>$$\frac{\partial}{\partial x} (ax)^x = x (ax)^{x-1} a = ax (ax)^{x-1} = (ax)^x$$</p>
<p>and </p>
<p>$$(a0)^0 = 0^0 =1$$</p>
<p>so I feel like I must ... | Phillip Hamilton | 312,810 | <p>You've made a mistake. Distribute $x $</p>
<p>$(ax)^x = a^x x^x $</p>
<p>Now can what you've written for the derivative be true?</p>
|
2,604,372 | <p>The formula for adding two vectors, as defined in Kells' Analytical Geometry is </p>
<p>$AB+BC=AC$</p>
<p>This makes sense since we're concerned with both the direction and magnitude of the vectors. When I got to the first exercise though, I was given the question </p>
<blockquote>
<p>A man walks east 6 miles t... | zwim | 399,263 | <p>What is missing and confusing you are the arrows.</p>
<p>$\overrightarrow{AB}$ is a vector while $AB$ is a distance.</p>
<p>Vectors verify the additive relation: $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$</p>
<p>Vectors and distances are linked by the equality: ${AB}^{\ 2}=\overrightarrow{AB}\c... |
927,480 | <p>Consider a univariate function $f(x)$. I know the graphical intuition behind why $f'(x)=0$ at the extrema of $f$. But how do you prove it mathematically? </p>
<p>I start with the assumption of $x^*$ being a minimum (the maximum case can be proved likewise), then
$f(x^*+h) \geq f(x^*)$ where $h\in \mathcal{N}(x)$, $... | AlexR | 86,940 | <p><strong>Hint</strong><br>
Note that the binary representation of $2^n$ has $n+1$ bits.<br>
Find then that the binary representation of a sum of $2^{k_i}$ with <em>distinct</em> $k_i$ has $\max_i k_i + 1$ bits.<br>
Finally conclude that any integer in the interval $[2^n, 2^{n+1})$ has a binary representation of exact... |
269,474 | <p>If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that</p>
<p>$$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$</p>
<p>Also show when equality occurs.</p>
| Mikko Korhonen | 17,384 | <p>As noted in the other answer, it suffices to show $\dim\ \operatorname{Ker}(A)+\dim\ \operatorname{Ker}(B) \geq \dim\ \operatorname{Ker}(AB)$. This is equivalent to showing that $\dim\ \operatorname{Ker}(AB)/\operatorname{Ker}(B) \leq \dim\ \operatorname{Ker}(A)$. To do this, use the first isomorphism theorem for ve... |
2,214,287 | <p>My exam review states that I need to utilize the difference formula for sine to solve the equation on the interval $0 \leq \theta < 2\pi $</p>
<p>$$\sqrt3\sin \theta- \cos\theta = 1$$</p>
<p>I know that: $\sin \frac\pi3 = \frac{\sqrt3}{2}$
and $\cos\frac\pi3 = \frac12 $, so I divide each term by 2 and rewrite t... | PM. | 416,252 | <p>Approach this systematically as follows. Express your LHS (left-hand side) as
$R\sin(\theta-B)$ where you need to find $R$ and $B$. So
$$
R\sin(\theta-B)=R (\sin(\theta)\cos(B)-\cos(\theta)\sin(B))
$$
Comparing with $\sin(\theta)$ and $\cos(\theta)$ on your LHS we find
$$
R\cos(B)=\sqrt3
$$</p>
<p>$$
R\sin(B)=1
$$... |
1,123,694 | <p>Prove that $az^n+b\overline{z}^n=0$ when $|a|\ne|b|$ and $n\in\mathbb{N_1}$does not have any complex solutions except for $0$. What happens if $n\in\mathbb{C}$?</p>
<p>The first one seems very obvious, but is there any way to show it very formally? </p>
| Haha | 94,689 | <p>$z=|z|e^{i\theta}$. Then $az^n+b\overline {z}^n=|z|(ae^{in\theta}+be^{-in\theta})$. If $z\neq 0$ then $|z|\neq 0$ and so we must have $ae^{in\theta}+be^{-in\theta}=0$. Because $|a| \neq |b|$ we can suppose that $|a|\neq 0$ and thus $ae^{in\theta}+be^{-in\theta}=0\Leftrightarrow \frac {-b}{a}e^{-2in\theta} =1 \Right... |
2,426,244 | <p>Suppose $u:[0,1]\to\mathbf R$ is a bounded function, show that there exist $x,y\in[0,1]$ such that $|u(x)-u(y)|<|x-y|^{1/2}.$</p>
<p>I have some stupid trials but failed. It seems that we could argue via contradiction. </p>
<p>If for each $x,y\in[0,1]$ with $x\neq y,$ there holds $$|u(x)-u(y)|\geq|x-y|^{1/2}.$$... | Long | 41,744 | <p>Suppose without loss that $u : [0,1] \to [0,R]$ and consider the graph of
$u$ in $[0,1]\times[0,R]$. Note that the area of the big rectangle $[0,1]\times[0,R]$ is $R$.</p>
<p>Let $n$ be a positive integer.
For each integer $0 \leq k \leq n$ cover the point $(k/n,u(k/n))$ with the rectangle $[0,1]\times (u(k/n) + (... |
3,443,910 | <blockquote>
<p>Prove that <span class="math-container">$f:[0,1] \rightarrow\mathbb{R}:f(x)= \left \{\begin {array}{ll}
\sin \frac1x &, \textrm{if}~ x\in(0,1]\\
0 &, \textrm{if}~~x =0
\end{array}
\right.~~$</span> is Riemann integrable using Darboux sums.</p>
</blockquote>
<p><em>Attempt.</em> The proof... | zhw. | 228,045 | <p>Fix <span class="math-container">$n$</span> large. Then for <span class="math-container">$1/n\le x <y\le 1,$</span> the mean value theorem shows</p>
<p><span class="math-container">$$\tag 1 |f(y)-f(x)|\le n^2|y-x|.$$</span></p>
<p>Now let <span class="math-container">$m\in \mathbb N$</span> be greater than <spa... |
370,570 | <p>In the wikipedia page (<a href="http://en.wikipedia.org/wiki/Birthday_problem" rel="nofollow">http://en.wikipedia.org/wiki/Birthday_problem</a>) on birthday paradox the following statement has been said : "the probability that, in a set of $n$ "randomly chosen" people, some pair of them will have the same birthday. ... | Ross Millikan | 1,827 | <p>The assumption is that each person's birthday is chosen randomly from the $365$ days of the year.</p>
|
3,755,709 | <p>How do you prove that the derivative of <span class="math-container">$\tan^{-1}(x)$</span> is equal to <span class="math-container">$\frac{1}{1+x^2}$</span> geometrically?</p>
<p>I figured it out by working it out using implicit differentiation.</p>
<p>I also found how to plot a semi-circle using <span class="math-c... | Benjamin Wang | 463,578 | <p>Tried to do geometry, but ended up doing a hand-wave-y first-principles approach. It can be made rigorous though.</p>
<p>The angle addition formulae have <a href="https://en.m.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Angle_sum_identities" rel="nofollow noreferrer">geometrical proofs</a>.</p>
<p><span cl... |
4,318,497 | <p>I'm having trouble finding the line integral of this problem. I have been given a vector field</p>
<p><span class="math-container">$F=(2x\sin(\pi y)-e^z,\pi x^2\cos(\pi y)-3e^z,-xe^z)$</span></p>
<p>Where the curve <span class="math-container">$C$</span> intercepts between <span class="math-container">$z=\ln(1+x)$</... | Andrei | 331,661 | <p>Instead of
<span class="math-container">$$r(t)=(t)\hat{i}+(t)\hat{j}+(\ln(2)t)\hat{k}$$</span>
you have <span class="math-container">$$r(t)=(t)\hat{i}+(t)\hat{j}+\ln(1+t)\hat{k}$$</span>
Notice that when <span class="math-container">$t=1$</span>, <span class="math-container">$r(1)=(1,1,\ln 2)$</span>.</p>
|
2,584,862 | <p>I'm trying to evaluate the following integral $$\int_{0}^{\infty} \frac{(x+1)}{(1+(1+x)^2)x^{1/3}}dx$$
I'm using the branch cut from $[0, \infty)$ (positive real axis to be precise). For this cut, when I try to evaluate the residues ( they're $(+i-1)$ and $(-i-1)$ ) and I get the following values : $ \frac{1}{2(i-1... | Mark Viola | 218,419 | <p>Taking the branch cut along the positive real axis and integrating over the classical keyhole contour, $C$, we find that </p>
<p>$$\begin{align}
\oint_C \frac{(z+1)z^{-1/3}}{(z^2+2z+2)}\,dz&=\int_\epsilon^R \frac{(x+1)x^{-1/3}}{(x^2+2x+2)}\,dx-e^{-i2\pi/3}\int_\epsilon^R \frac{(x+1)x^{-1/3}}{(x^2+2x+2)}\,dx\\\\... |
2,269,874 | <p>I have been learning about the Fibonacci Sequence in school and came across the interesting property: $$\gcd(F_m, F_n) = F_{gcd(m, n)}$$ where $F_m$ and $F_n$ are the $mth$ and $nth$ number from the Fibonacci sequence.</p>
<p>Apparently, this is the defining property of a Strong Divisibility Sequence. I've been sea... | Somos | 438,089 | <p>Without loss of generality we can consider bags with only one kind of element. Let $b=|B|, c=|C|$. Then the formula gives $\min(b,c)/(b+c)$ but then $2\min(b,c)\leq(b+c)$. Note that the formula gives $1/2$ if the bags are equal.</p>
|
833,814 | <p>Can someone help me in this question : Let $z=(-1+i)^{11}+(-1-i)^{15}$ so </p>
<ol>
<li>$z=-96+160i$</li>
<li>$z=96-160 i$</li>
<li>$z=160-96i$</li>
<li>$z=-160+96i$</li>
</ol>
<p>what is the right answer ? Thanks in advance.</p>
| David Mitra | 18,986 | <p>Let $\epsilon>0$. </p>
<p>Choose $0<\delta<1$ so that $ f(x)<f(0)+\epsilon$ for all $0\le x<\delta$.</p>
<p>Write
$$
x\int_x^1{f(t)\over t^2}\,dt =
\underbrace{x\int_x^\delta{f(t)\over t^2}\,dt}_{A(x)}+
\underbrace{x\int_\delta^1{f(t)\over t^2}\,dt}_{B(x)}.
$$</p>
<p>Since $f(x)/x^2$ is bounded on... |
198 | <p>Here I mean the version with all but finitely many components zero.</p>
| Andrew Stacey | 45 | <p>This is the swindle, isn't it?</p>
<p>There's an elegant way to phrase this with lots of sines and cosines, but working it all out is too much like hard work. Here's the quick and dirty way.</p>
<p>Let $T: S^\infty \to S^\infty$ be the "shift everything down by 1" map.</p>
<p>Then for any point $x \in S^\infty$,... |
2,162,375 | <p>I want to prove that $\sqrt{a} + \sqrt{b} \le 2 \times \sqrt{a+b}$, I had the idea to draw it:<br>
<a href="https://i.stack.imgur.com/cyqj5.png" rel="noreferrer"><img src="https://i.stack.imgur.com/cyqj5.png" alt="enter image description here"></a></p>
<p>Would it be enough to prove what I want to prove? If not, i... | vadim123 | 73,324 | <p>Note that $a,b$ are both nonnegative.
$$\sqrt{a}\le \sqrt{a+b}$$
$$\sqrt{b}\le \sqrt{a+b}$$
Now add the two.</p>
|
4,157,630 | <p>Compute the factor group: <span class="math-container">$\mathbb{Z} \times \mathbb{Z} \big / \langle (1,2) \rangle$</span></p>
<p>I don't understand why the answer is <span class="math-container">$\mathbb{Z}$</span>.
Isn't <span class="math-container">$\mathbb{Z} \times \mathbb{Z} \big / \langle (1,2) \rangle \cong... | Richard Jensen | 658,583 | <p>Your proposed identity <span class="math-container">$\mathbb{Z} \times \mathbb{Z} \big / \langle (1,2) \rangle \cong \mathbb{Z} / \langle 1 \rangle \times \mathbb{Z} / \langle 2 \rangle$</span> is not true. The group on the right has order 2, while the group on the left has an infinite subgroup generated by <span cl... |
675,991 | <p>So here's the question I'm trying to answer:</p>
<blockquote>
<p>Suppose $p_n(x) = \sum_{k=1}^N a_k^{(n)} x^k$ is a sequence of polynomials such that $p_n \to f$ uniformly over $[0,1]$ for some function $f:[0,1] \to \mathbb{R}$. Prove that $f$ must itself be an $N^\text{th}$ degree polynomial.</p>
</blockquote>
... | Diego Fonseca | 165,639 | <p>A proof that for my criterion is elegant is using facts of functional analysis:</p>
<p>Let $P_N$ the set of all polynomials of degrre $N$, this is a vector space and isometrically isomorphic to $\mathbb{R}^{N+1}$ via
$$\begin{array}{rcl}\Phi: P_{N} &\rightarrow & \mathbb{R}^{N+1}\\ a_0+a_1 x +\cdots+a_N x^N... |
2,244,198 | <p>I am trying to prove that the </p>
<p>$A_n=n\int_{1}^{+\infty}\left(1-e^{-\frac{1}{x-1}}\right) e^{-x}\left(1-e^{-x}\right)^{n-1}$ $\Rightarrow$
$\lim_{n\to\infty}A_n=0$ .</p>
<p>In fact, by defining $f_n(x)=\color{red}{n}\left(1-e^{-\frac{1}{x-1}}\right) e^{-x}\left(1-e^{-x}\right)^{n-1}$, it is straightforward t... | zhw. | 228,045 | <p>As you were discovering, your problem is not really a DCT situation. Perhaps you have seen the following result, or something like it: Let $0<b\le 1.$ Suppose $f$ is continuous on $[0,b]$ with $f(0) = 0.$ Then</p>
<p>$$\lim_{n\to \infty} n\int_0^b f(x)(1-x)^{n-1}\,dx = 0.$$</p>
<p>Now in the given problem, mak... |
1,026,807 | <p>Let $R$ be a ring with unity and assume that $R$ has no nonzero zero-divisors. Let $a,b\in R$, and assume that $ab=1$. Show that $ba=1$, and therefore $a,b$ are units. </p>
<p>I think this question boils down to showing that $R$ is communitive under multiplication (of the ring R), but I don't know how to show it gi... | Matt Samuel | 187,867 | <p>With your assumptions, $R$ need not be commutative. We can prove the result as follows.</p>
<p>Since $ab=1$ we have that $bab=b$. If it were the case that $ba\neq 1$, then $bab-b=(ba-1)b=0$, and since $ba-1\neq 0$ this contradicts the hypothesis that $R$ has no nonzero zero divisors.</p>
|
2,734,257 | <p>I have tried to show that this limit :
$$\lim\limits_{n\to \infty }\frac{n}{n!^{\frac 1 n}}=e$$</p>
<p>using $ \lim (1+\frac 1 n)^{\frac 1 n} , n \to \infty $ , I don't find any equivalence , however wolfram alpha says that is $e$ as shown <a href="https://www.wolframalpha.com/input/?i=lim+(n+%2F+((n!)%5E(1%2Fn))... | user | 505,767 | <p>As an alternative without Stirling, note that</p>
<p><span class="math-container">$$\frac{n}{n!^{\frac 1 n}}=\left(\frac{n^n}{n!}\right)^{\frac 1 n}=(a_n)^\frac1n$$</span></p>
<p>and</p>
<p><span class="math-container">$$\frac{a_{n+1}}{a_n}=\frac{(n+1)^{n+1}}{(n+1)!}\frac{n!}{n^n}=\left(1+\frac1n\right)^n\to e$$</sp... |
1,066,921 | <p>Solve the system of equations $x^2=y^3, x^y=y^x$ in positive real numbers.</p>
<p>Taking $\ln$ of the second equation, we have $\ln x/x=\ln y/y$. This function is increasing in $(0,e)$ and decreasing in $(e,\infty)$. For any value of $x\neq e$, we can find a unique value of $y$ such that $x^y=y^x$. But how can we f... | Community | -1 | <p>Why don't you continue with the logarithms ?
$$\frac{\ln x}x=\frac{\ln y}y,\\\color{blue}{2\ln x=3\ln y},$$
then dividing memberwise,
$$2x=3y,$$
or
$$\color{blue}{\ln x+\ln 2=\ln y+\ln3}.$$</p>
<p>You now have two linear equations in $\ln x$ and $\ln y$.</p>
<blockquote class="spoiler">
<p> $$\ln x=3\ln\frac32,\... |
978,927 | <p>How would one prove the equality of the sum of squares of diagonals and twice the sum of squares of the two sides:</p>
<p>$$\left|\mathbf{p} + \mathbf{q}\right|^2 + \left|\mathbf{p} - \mathbf{q}\right|^2 = 2\left|\mathbf{p}\right|^2 + 2\left|\mathbf{q}\right|^2 $$</p>
<p>where $\mathbf{p}$ and $\mathbf{q}$ are vec... | mookid | 131,738 | <p><strong>Hint:</strong> with such a characteristic polynomial, the matrix is congruent to either
$$
\left( \begin{array}{ccc}
-3 & 1 & 0 &0\\
0 & -3 & 0 &0\\
0 & 0 & 1 &0\\
0 & 0 & 0 &5\end{array} \right)
$$
or
$$ \left( \begin{array}{ccc}
-3 & 0 & 0 &0\... |
1,026,066 | <p>Problem:
It is researched that 60% of people in city goes to cinema on daily basis, 40% of people goes to theater on daily basis. It is also known that 20% simultaneously goes to both theater and cinema.</p>
<p>What is</p>
<ol>
<li>
Probability that chosen person does not attend both.</li>
<li>Probability that ch... | ethanol | 193,076 | <p>Draw a venn diagram first. Makes life a lot easier.</p>
<p>Q 3 to 6 are of conditional probability.. Remember the basic formulae for conditional probability</p>
<p>P(A|B) = P(A ∩ B) / P(B)</p>
<p>Even better solve everything with the help of venn diagram</p>
<p>Say, if you've to calculate P(A given B) then</p>
... |
2,187,929 | <p>A speaks truth $3$ times out of $4$ and $B$ $7$ times out of $10$ . they both agree that a white ball has been drawn out from a bag containing $6$ balls of different color . find the probability that the statement is true . </p>
<p>my try .</p>
<p>probability when they say false and agree = $\left(\dfrac56\right)\... | Nathanael Skrepek | 423,961 | <h2>Correct Problem</h2>
<h3>Setting</h3>
<hr>
<p>There is a bag with six balls in it. Each ball has a different color. One ball is blindly drawn. Two people are being asked which color the ball has after they saw the ball.</p>
<p>Person $A$ tells to truth in $3$ out of $4$ cases. Person $B$ tells the truth in $7$ ... |
4,349,993 | <p>Suppose that <span class="math-container">$(X_i, \tau_{X_i})$</span> are path-connected topological spaces for all <span class="math-container">$i \in I$</span>. I know that the product <span class="math-container">$\Pi_{i \in I}X_i$</span> with its product topology is path-connected. But is the converse true ? If <... | Henno Brandsma | 4,280 | <p>If <span class="math-container">$X$</span> is path-connected and <span class="math-container">$f: X \to Y$</span> is continuous and onto then <span class="math-container">$Y$</span> is path-connected. Just like for connected spaces. (Proof: let <span class="math-container">$y_1, y_2 \in Y$</span>, find <span class="... |
1,721,584 | <p>The image attached below is a problem on induction, the proof has been included.
I am enquiring if anyone could explain line for line what the proof states with its notation ( the notation is new to me). (I have a bit of experience with proof by induction, but is stumped by this problem)</p>
<p><a href="https://i.s... | Santiago | 326,828 | <p>Since $(x^n)' = n x^{n-1}$, we have $(x^2)' = 2x$ and $x' = 1$, therefore $(x^2 -x)' = 2x-1$ - differentiation of functions is additive.</p>
|
1,510,693 | <p>Suppose $T\in(V)$ and $(T-2I)(T-3I)(T-4I) = 0$. Suppose $\lambda$ is an eigenvalue of $T$. Prove $\lambda = 2$ or $\lambda = 3$ or $\lambda = 4$</p>
<p>What properties of polynomials will prove this?</p>
| lab bhattacharjee | 33,337 | <p>$$0=-\sin x-\cos x\iff\sin x=-\cos x\iff\tan x=-1=-\tan\dfrac\pi4=\tan\left(-\dfrac\pi4\right)$$</p>
<p>$x=n\pi-\dfrac\pi4$ where $n$ is any integer</p>
|
4,027,927 | <p>I'm trying to find derivative of <span class="math-container">$\frac{\cos t-\sin t}{\cos t+\sin t}$</span> in a different way: there is a trick to find derivative of the form <span class="math-container">$\frac{ax+b}{cx+d}$</span>:
<span class="math-container">$$\left(\frac{ax+b}{cx+d}\right)'=\frac{ad-bc}{(cx+d)^2... | Tan | 814,070 | <p>You can also do:
<span class="math-container">$$\frac{\cos{t}-\sin{t}}{\cos{t}+\sin{t}} = \frac{(\cos{t}-\sin{t})(\cos{t}-\sin{t})}{(\cos{t}+\sin{t})(\cos{t}-\sin{t})} = \frac{1-\sin{2t}}{\cos^2{t}-\sin^2{t}} = \frac{1}{\cos{2t}}-\tan{2t} = \sec{2t}-\tan{2t}$$</span> Take the derivative to get <span class="math-cont... |
4,027,927 | <p>I'm trying to find derivative of <span class="math-container">$\frac{\cos t-\sin t}{\cos t+\sin t}$</span> in a different way: there is a trick to find derivative of the form <span class="math-container">$\frac{ax+b}{cx+d}$</span>:
<span class="math-container">$$\left(\frac{ax+b}{cx+d}\right)'=\frac{ad-bc}{(cx+d)^2... | Etemon | 717,650 | <p>Inspired by J.G.'s answer we can rewrite <span class="math-container">$\frac{\cos t-\sin t}{\cos t+ \sin t}$</span> as <span class="math-container">$\frac{1-\tan t}{1+\tan t}$</span>. Let's call it <span class="math-container">$f(t)$</span>:</p>
<p><span class="math-container">$$f(t)=\frac{-\tan t+1}{\tan t+1}$$</s... |
1,180,719 | <p>I'm having a presentation on Gauss-Seidel iterative method, and although it isn't mandatory
, I would like to have some practical examples for this method (a system of linear equations with n>=1000, preferrably in .txt form), as well as some implementation details (maybe block GS or something, since I haven't looked... | Community | -1 | <p>Since $x,y$ are positive integers, we have $x>n$ and $y>n$. Write $x=n+a$ and $y=n+b$. Substitute this and you get $n^2=a\cdot b$. Retracing the steps, we find that $n^2$ is written as a product of two numbers say, $p$ and $q$, then $$\frac{1}{n+p}+\frac{1}{n+q}=\frac{1}{n}$$ Thus the number of solutions of ou... |
1,279,105 | <p>how to prove convergence of $\{\sin(1/n)\}$ sequence when $n$ goes infinity, using limit definition?
I proved that $\{\sin(n)\}$ is divergent but i can not do this with that method.</p>
| Vim | 191,404 | <p>An extreme yet valid example: let $\{q_n\}$ be an enumeration of $\Bbb Q$, then the map $\Bbb N\ni n\mapsto q_n\in\Bbb Q$, both sides equipped with the Euclidean metric, is bijective and UC, but its inverse is <em>nowhere</em> continuous. </p>
|
717,882 | <p>The $x^{2/2}$ can be represented by these ways:
$$\begin{align}
x^{2\over2}=\sqrt{x^2} = |x|\\
\end{align}
$$
And<br>
$$\begin{align}
x^{2\over2}=x^{1} = x\\
\end{align}
$$
Which one is correct?
And what is the domain of $x^{2 \over 2}$?</p>
| Robert Israel | 8,508 | <p>$2/2 = 1$, so $x^{2/2} = x^1 = x$. Always. What is confusing you is that this may not be the same as $(x^2)^{1/2}$ or $(x^{1/2})^2$ when $x$ is not positive. In mathematics, an expression such as $x^p$ depends on the values of $x$ and $p$, not on the way they are represented.</p>
|
17,669 | <p>I just inherited two slide rules from my grandfather-in-law, one wood with smooth action despite nearly a century without
use.<sup>1</sup> (I used a K+E slide rule myself as an undergraduate in the 1970's.) It struck me that the conversion of multiplication/division to addition/subtraction via the
logarithmically ru... | guest | 13,241 | <p>(too-long comment) I think it's a nice adjunct, but I would not introduce the topic that way. Introduce it after teaching of rational (math meaning) exponents and roots in a rational (well thought out meaning) progression. I.e. on the board, with some discussion of concepts they learned in exponents that are also ... |
952,453 | <p>A fast food restaurant offers customer a choice of eight toppings that can be added to a hamburger. How many different hamburgers can be ordered?</p>
<p>Attempt: I don't know if this is correct 8!? I think there is no sufficient information. Can anyone please help me? Thank you.</p>
| Paul | 138,918 | <p>If a variable is present more than once (the $Y$ in the proposed eq.) then, as a general rule, you should start by removing any fractions and multiplying out any brackets. </p>
<p>Since $Y$ composes the denominator, start by multiplying both sides by $Y$ so that it cancels out on the left:
$$
Y\cdot\frac{(X-Y)}{Y}=... |
92,105 | <p>Trying to solve</p>
<blockquote>
<p>$f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges.</p>
</blockquote>
<p>I need to prove that:
$$\lim \limits_{x \to \infty} f(x) = 0$$</p>
<p>Would appreciate your help!</p>
| Jonas Meyer | 1,424 | <p>For all $\delta>0$, the fact that $\int_0^\infty f(x)dx=\sum\limits_{k=0}^\infty\int\limits_{k\delta}^{(k+1)\delta}f(x)dx$ converges implies that $\lim\limits_{k\to\infty}\int\limits_{k\delta}^{(k+1)\delta}f(x)dx=0$. Let $\varepsilon>0$ be given. Let $\delta>0$ be such that $|x-y|<2\delta$ implies $|f(... |
1,636,979 | <p>The question says:
<a href="https://i.stack.imgur.com/NEDTp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NEDTp.png" alt="enter image description here"></a></p>
<p>The solution set was posted and there are a few things I don't quite understand from it.</p>
<p><a href="https://i.stack.imgur.com... | Noah Schweber | 28,111 | <p>There appears to be a typo in the last bit. Here's what I would write:</p>
<blockquote>
<p>If $a^2=1$, then - since we know that $ab=1$ - we have $a^2=ab$. So multiplying both sides by $a^{-1}$ we get $a=b$, a contradiction.</p>
</blockquote>
<p>I don't see how the author deduces $b=1$ here; I think "$1$" should... |
1,374,676 | <p>The series is as follows:</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=0}^\infty \frac{(-1)^nx^n}{(n!)^2}$$</span></p>
</blockquote>
<p>I tried working on it. The square in the denominator is breaking me. Please If any one could help.
And I need to find the sum of this series.</p>
<p>Edit:</p>
<blockqu... | Jack D'Aurizio | 44,121 | <p>If $X_1,X_2,\ldots,X_{19}$ are i.i.d random variables, what it the probability that</p>
<p>$$ \min(X_i)\geq 120 $$
? Obviously, it is given by:
$$ \mathbb{P}[X_1 \geq 120]^{19}.$$
Since $e^{-x^2/2}$ is a fixed point of the Fourier transform, the arithmetic mean of $n$ i.i.d normally distributed $N(\mu,\sigma^2)$ ra... |
2,812,118 | <p>Followup to the accepted answer of this question <a href="https://math.stackexchange.com/questions/631042/direct-proof-of-empty-set-being-subset-of-every-set">Direct proof of empty set being subset of every set</a></p>
<p>I understood the answer based on the nature of vacous truth, however what if we verify a state... | Asaf Karagila | 622 | <p>If you mean that every member of the empty set is <em>not</em> a member of $A$, then yes, you are correct. This statement is vacuously true.</p>
<p>There are no members to the empty set.</p>
<p>The point is that the definition of $X\subseteq Y$ is universal, i.e. given by $\forall x$, whereas the definition of $X\... |
3,874,049 | <p>After writing an integral as a limit of a Riemann sum, how do we actually calculate the integral? It seems that generally, we're in some form that isn't simplified. For example, take</p>
<p><span class="math-container">$$\int_0^3e^xdx=e^x|_0^3=e^3-1.$$</span></p>
<p>But this is also <span class="math-container">$$\i... | José Carlos Santos | 446,262 | <p>It is unlikely that you can prove in a direct and simple way that<span class="math-container">$$\lim_{n\to\infty}\sum_{i=1}^n\frac{3e^{3i/n}}{n}=e^3-1.$$</span>However, before we are able to apply the Fundamental Theorem of Calculus to compute integrals, we must know what <span class="math-container">$\int_a^bf(t)\,... |
296,737 | <p>Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. I've shown previously that if the relationship $p^2=q^3$ holds for some $p$ and $q$ in one variable, there exists $r$ ... | Mariano Suárez-Álvarez | 274 | <p>Suppose there is an isomorphism $\phi:\def\CC{\mathbb C}\CC[X,Y]/(X^2-Y^3)\to\CC[T]$, and let $f=\phi(X)$ and $g=\phi(Y)$. Then $f^2=g^3$. It follows from this equality in $\CC[T]$ that $f$ and $g$ have exactly the same zeros. Moreover, if $a$ is one of those zeroes and $m$ and $n$ are the multiplicities of $a$ in $... |
296,737 | <p>Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. I've shown previously that if the relationship $p^2=q^3$ holds for some $p$ and $q$ in one variable, there exists $r$ ... | Mariano Suárez-Álvarez | 274 | <p>Recall that if $A$ is a $\def\CC{\mathbb C}\CC$-algebra and $\delta:A\to A$ is a $\CC$-linear map, we way that $\delta$ is a <em>derivation</em> of $A$ if for all $a$, $b\in A$ we have $$\delta(ab)=\delta(a)b+a\delta(b).$$</p>
<p>Let $d:\CC[t]\to\CC[T]$ be the usual derivative map, so that $d(f)=f'$ for all $f\in\C... |
296,737 | <p>Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. I've shown previously that if the relationship $p^2=q^3$ holds for some $p$ and $q$ in one variable, there exists $r$ ... | Mariano Suárez-Álvarez | 274 | <p>Let $A=\def\CC{\mathbb C}\CC[X,Y]/(X^2-Y^3)$. Since the polynomial $X^2-Y^3$ is prime, the algebra $A$ is a domain. A little computation shows that $X$ and $Y$ are irreducible (and non-units), so that the element $u=X^2$ has two different factorization as products of irreducible elements, so $A$ is not a unique fact... |
1,361,478 | <p>A function of a single variable is denoted $f(x)$, of two variables if denoted $f(x,y)$</p>
<p>What about infinite variables? How do we denote such thing and do such things exist?</p>
| user251257 | 251,257 | <p>For example: Let $f:\mathbb R^{\mathbb N}\to\mathbb R$ be defined by
$$ f((x_n)_{n\in\mathbb N}) = x_1, $$
where $\mathbb R^{\mathbb N}$ denotes the set of real valued sequences.</p>
<p>You may say that $f$ has <em>infinitely many</em> arguments, namely $x_n$, $n\in\mathbb N$. You may also say $f$ has <em>one</em> ... |
2,131,709 | <p><span class="math-container">$$F=\left(y\cos \left(xy\right)+e^{x+y}\right)i+\left(x\cos \left(xy\right)+e^{x+y}\right)j$$</span></p>
<p>also show that <span class="math-container">$$∫_cF ⋅ dr = e^2-e^{-2}$$</span>
where c is the straight line from <span class="math-container">$\left(-1,-1\right)$</span> to <span c... | Taufi | 325,684 | <p>This is quite easy when you know differential forms. </p>
<p>Let <span class="math-container">$∫_{C} F⋅ dr = ∫_{C} \,\left(y\cdot\cos \left(xy\right) + \exp \left(x+y\right)\right)dx + \left(x\cdot\cos \left(xy\right) + \exp \left(x+y\right)\right)dy, $</span></p>
<p>where <span class="math-container">$ω = \left(y... |
4,020,261 | <p>From my university notes:</p>
<blockquote>
<p><strong>Comment to slide 9</strong></p>
<p>By virtue of the result shown in this slide, we can talk about <em>the</em> least element of a set <span class="math-container">$D$</span>, if one exists, and we denote it with <span class="math-container">$\perp$</span>, pronou... | Rob Arthan | 23,171 | <p>In this CS context, I think <span class="math-container">$\sqsubseteq$</span> would be called the refinement relation (see, for example Morgan's <a href="http://www.cs.ox.ac.uk/publications/books/PfS/" rel="nofollow noreferrer"><em>Programming from Specifications</em></a>). The subscript <span class="math-container"... |
60,849 | <p>Given two free semicirculars X_1 and X_2 and a projection h in the von-Neumann algebra generated by X_1, how does one show that the von-Neumann algebra generated by {X_1, hX_2(1-h)} is a factor? It is easy to show that the two elements in the generating set are free. But I am unable to see what kind of an object hX_... | ght | 13,825 | <p>One way of thinking about the operator
$$
Y=hX_{2}(1-h)
$$
is to work with the random matrix models. More specifically, the operators $X_{1}$ and $X_{2}$ can be thought as the limit as $n\to\infty$ of two independent $n\times n$ Hermitian random matrices where the upper triangular parts are formed by i.i.d. Gaussi... |
60,849 | <p>Given two free semicirculars X_1 and X_2 and a projection h in the von-Neumann algebra generated by X_1, how does one show that the von-Neumann algebra generated by {X_1, hX_2(1-h)} is a factor? It is easy to show that the two elements in the generating set are free. But I am unable to see what kind of an object hX_... | Dima Shlyakhtenko | 12,660 | <p>Let me first point out that $X_1$ and $Y=h X_2 (1-h)$ are <em>not</em> freely independent. This is most easily seen if $h$ has trace 1/2, in which case $Y$ has range and support projections $h$ and $(1-h)$, respectively. But since the support and range projections of $Y$ belong to $W^*(Y)$, it would follow from th... |
1,359,624 | <p>How to find derivative of</p>
<p>$$f(x)=|\sin^{-1}(2x^2-1)|$$</p>
<p>Please provide stepwise mechanism.</p>
<p>The original question was to find domain of derivative of y=|arc sin(2x^2−1)|.</p>
<p>My METHOD- My attempt was to break y into intervals ,i.e., where \sin^{-1}(2x^2-1)>=0 and where \sin^{-1}(2x^2-1)<... | Emir Šemšić | 278,709 | <p><strong>HINT:</strong> $\left | x \right |=\sqrt{x^{2}}$</p>
|
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Gian Maria Dall'Ara | 1,049 | <p>Here is the proof of the equivalent statement "Every complex non-constant polynomial $p$ is surjective".<br>
1) Let $C$ be the finite set of critical points , i.e. $p'(z)=0$ for all $z\in C$. $C$ is finite by elementary algebra.</p>
<p>2) Remove $p(C)$ from the codomain and call the resulting open set $B$ and remo... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | KConrad | 3,272 | <p>Here is the proof by Pukhlikov (1997) at </p>
<p><a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mp&paperid=6&option_lang=eng">http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mp&paperid=6&option_lang=eng</a></p>
<p>which Ilya mentioned as being only in Russian s... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | gowers | 1,459 | <p>Here's another complex analysis proof that I heard about for the first time under a week ago (because it was set as a question on a course I am teaching for). Pick a circle large enough for the modulus of p(z) to be greater than |p(0)| everywhere in that circle. Inside that circle take a point w where the modulus of... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Yaakov Baruch | 2,480 | <p>At the risk of being highly downvoted, I can't resist reposting my comment
to Andrew L's answer (or rather, question) below:</p>
<p>is there a purely algebraic proof that for any non constant $P$ in $\mathbb{Q}[i][X]$
and $\epsilon>0$ in $\mathbb{Q}$, there is $q$ in $\mathbb{Q}[i]$ s.t. $|P(q)|<\epsilon$?</p... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Johannes Ebert | 9,928 | <p>There is one short proof on wikipedia, that shows the statement that any endomorphism $A$ of a finite-dimensional vector space $V$ of positive dimension has an eigenvalue. Look at the resolvent function $R_A:z \mapsto (A-z)^{-1}$. Outside the disc of radius $\|A\|$, it can be developped into a geometric series. Use ... |
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | Mike Jones | 2,344 | <p>I have some online mathematics exercises / tutorials (autocorrected), ranging from kindergarten to college, at: <a href="http://www.public-domain-materials.com/folder-student-exercise-tasks-for-mathematics-language-arts-etc---autocorrected.html" rel="nofollow">http://www.public-domain-materials.com/folder-student-ex... |
1,214,419 | <p>How do I show that any open interval is an half open set and use this to conclude that any open set is also half open?</p>
<p>I am in an introduction to proofs writing class. I have a feeling I need to use the fact that an open set is a union of open intervals but I don't know that I am even going in the right dir... | Reveillark | 122,262 | <p>Take any half open interval $(a,b)$, with $a<b$ (otherwise its empty and the conclusion is trivial). Let $x\in(a,b)$, this means $a<x<b$. </p>
<p>We can therefore find $c\in \mathbb{R}$ such that $a<c<x$. In particular, $x≥c$, which gives $x\in[c,b)\subset(a,b)$.</p>
|
2,494,160 | <p>I've been having trouble using the definition of a limit to prove limits, and at the moment I am trying to prove that
$$\lim_{n\to\infty} \frac{n^x}{n!}=0$$</p>
<p>for all $x$ which are elements of natural numbers.
I'm able to start the usual setup, namely let $0<\epsilon$ and attempt to obtain $\left\lvert\dfr... | ThePortakal | 137,487 | <p>We prove for $x=4$ and generalize it from there.</p>
<p>Let $\varepsilon >0$. By Archimedean property, there exists $N$ such that $\frac 1N < \frac \varepsilon 8$ and $\frac{N}{N-3}\leq 2$. </p>
<p>Now let $n-4 \geq N$.</p>
<p>$$\left| \frac{n^4}{n!}\right| = \left| \frac{1}{(n-4)!} \cdot \frac{n}{n-3} \cdo... |
1,066,311 | <p>Given a vector $\vec a\in\mathbb R^n$ and another $\alpha=(\|\vec a\|,0,\dots,0)$, how could I define an orthogonal matrix $M$ such that $M\vec a=\alpha$ and $M^{-1}=M^t$? For $\mathbb R^2$ I tried to use a generic matrix $\; Q=\left(\begin{array}{cc}a & c\\b & d\end{array}\right)$ and $M=Orthogonalize(Q)$ u... | TZakrevskiy | 77,314 | <p>Take any orthogonal basis $\{w_j\}$, $j=1\ldots n$ such that $w_1=\vec a/\|\vec a\|$.</p>
<p>Now take another orthogonal basis $\{v_j\}$, $j=1\ldots n$, such that $v_1=\vec \alpha/\|\vec\alpha\|$.</p>
<p>Now build a linear operator $M$ given by $$\forall j\quad Mw_j=v_j.$$</p>
<p>It is an orthogonal operator and... |
3,695,971 | <p>The way I understand it currently, saying "<em>only if <span class="math-container">$P$</span>, then <span class="math-container">$Q$</span></em>" is like saying that "<em>only if <span class="math-container">$P$</span> happens, <span class="math-container">$Q$</span> happens.</em>" To me, it seems to say the same t... | Bram28 | 256,001 | <p>There are certainly cases where the use of 'only if' is meant as 'if and only if'. For example, if a professor tells you that "Sorry, but you can take my course Calculus II only if you have taken Calculus I", the professor probably means an if and only if: taking Calculus I would be necessary ... but it would probab... |
2,401,252 | <p>Prove the identity
$$\frac{(2n)!}{2^n \cdot n!} = (2n-1)(2n-3)\cdots 3\cdot 1$$
using combinatorial proof.</p>
| Robert Z | 299,698 | <p>Hint. It counts the number of ways to choose $n$ disjoint pairs of objects from $2n$ objects.</p>
|
200,670 | <p>I use groupings as below: </p>
<pre><code>Join[
Groupings[IntegerPartitions[3], {A -> {2, Orderless}, B -> {2, Orderless}}],
{x}]
</code></pre>
<p>Which generates: </p>
<blockquote>
<pre><code>{A[2, 1], B[2, 1], A[A[1, 1], 1], A[B[1, 1], 1], B[A[1, 1], 1], B[B[1, 1], 1], 3}
</code></pre>
</blockquote>
... | Michael E2 | 4,999 | <p>As <a href="https://mathematica.stackexchange.com/questions/200642/ndsolve-problem-with-sign-friction#comment519037_200642">@Hugh points out</a>, the particle will remain at rest if it comes to rest at a point where the restoring force is less than or equal to the frictional force.
The interval where this occurs may... |
563,170 | <p>If $$\lim_ {x\to a}\ f(x)$$ exists and $$\lim_ {x\to a}\ (f(x)g(x))$$ exists, then $$\lim_ {x\to a}\ g(x)$$ exists. Please exdpain why the above statment is true or false.</p>
<p>EDIT: Edited question to be more specific.</p>
| eeeeeeeeee | 55,017 | <p>This yields the same result as @p.s. provided, but this is how I'd think to solve it.</p>
<p>Let $f_{Y_1}(t)$ denote the density of $Y_1$, and note that $$\{ Y_1 = min(Y_1,Y_2,Y_3) \} \iff \{ Y_2 \geq Y_1, Y_3 \geq Y_1\}.$$
Then, by independence,
\begin{align*}
P(Y_1 = min(Y_1,Y_2,Y_3)) &= \int_0^{\infty} P(Y_2... |
1,372,600 | <p>Prove that $\sqrt{47}$ is irrational number. </p>
<p>I know that a rational number is written as $\frac{p}{q}$ where $p$ & $q$ are co-prime numbers. But I do not have any idea to prove it irrational number. </p>
<p>Thank you very much!</p>
| Jack D'Aurizio | 44,121 | <p>$47$ is a prime, hence you may just mimic the proof of the irrationality of $\sqrt{2}$. Let: </p>
<p>$$\nu_{47}(m) = \max\{n\in\mathbb{N}: 47^n\mid m\}.$$
Assuming $\sqrt{47}=\frac{p}{q}$ with $\gcd(p,q)=1$, it follows that:
$$ p^2 = 47\, q^2 $$
but $\nu_{47}$ of the LHS is even while $\nu_{47}$ of the RHS is odd, ... |
1,372,600 | <p>Prove that $\sqrt{47}$ is irrational number. </p>
<p>I know that a rational number is written as $\frac{p}{q}$ where $p$ & $q$ are co-prime numbers. But I do not have any idea to prove it irrational number. </p>
<p>Thank you very much!</p>
| Michael Hardy | 11,667 | <p>If $\dfrac p q =\sqrt{47}$ then $p^2=q^2\cdot 47$. That means the number of $47$s in the prime factorization of $p^2$ is one more than the number of $47$s in the prime factorization of $q^2$. But that cannot happen because $p^2$ and $q^2$, both being squares, must both have an even number of $47$s in their factori... |
2,998,735 | <p>I am trying to solved this inequality for <span class="math-container">$k$</span>.</p>
<p><span class="math-container">$x^{2k}<\varepsilon\cdot k^k$</span></p>
<p>Here <span class="math-container">$k\in\mathbb{N}$</span> and <span class="math-container">$x,\varepsilon$</span> are fixed such that <span class="ma... | Mercy King | 23,304 | <p>Since <span class="math-container">$\ln$</span> is increasing, we have
<span class="math-container">$$
2k\ln(|x|)<k\varepsilon\ln(k)
$$</span>
i.e
<span class="math-container">$$
\frac{2k\ln(|x|)}{k\varepsilon}=2\varepsilon^{-1}\ln(|x|)<\ln(k)
$$</span>
Hence
<span class="math-container">$$
k>|x|^{2/\varep... |
2,980,446 | <p>I have this problem and I really don't know, how to edit it to get some solution.
<span class="math-container">$\lim_{n \to \infty} \sqrt{n} * [\sqrt{n+1} - \sqrt{n}]$</span></p>
<p>So my question Is what to do with <span class="math-container">$\sqrt{n} * [\sqrt{n+1} - \sqrt{n}]$</span>? </p>
| lab bhattacharjee | 33,337 | <p>Set <span class="math-container">$1/n=h$</span> to find</p>
<p><span class="math-container">$$\lim_{h\to0^+}\dfrac{\sqrt{1+h}-1}h$$</span></p>
<p>Now rationalize the numerator</p>
<p>Or</p>
<p>choose <span class="math-container">$\sqrt{1+h}=k\implies h=k^2-1$</span></p>
|
173,888 | <p>$$\frac{\mathrm{d}y}{\mathrm{d}x} = f(y)$$
where $f(y)$ is continuous on $|y-a|\leq \epsilon$,and $f(y)=0$ iff $y=a$.</p>
<p>To Proof : For the initial value point on $y=a$,the equation has local unique solution iff $\left|\int_a^{a+\epsilon}\frac{\mathrm{d}y}{f(y)}\right|= \infty$ </p>
<p>How to proof Initial val... | Robert Israel | 8,508 | <p>It's not true. Consider e.g. $$\dfrac{dy}{dx} = - y^{1/3}, y(0) = 0$$
Even though $\int_0^{\epsilon} - y^{-1/3}\ dy$ is finite, it's obvious that the only solution is $y = 0$. </p>
<p>What is true is this. Suppose $f(y) > 0$ for $a+\epsilon > y > a$ and $\displaystyle\int_{a}^{a+\epsilon} \dfrac{dy}{f(y)... |
79,041 | <p>This maybe a simple question, but I am just stuck with it.
I want to do some simulation, say with 0.9 probability, I get a 1, and 0.1 probability get a 0.</p>
<p>How would I do that? Where should I start?</p>
<p>Thanks!</p>
| Ajasja | 745 | <p>Take a uniform random distribution and check if it is above some threshold (0.9 in your case).</p>
<p>For example:</p>
<pre><code>dist[] := If[RandomReal[] > 0.9, 0, 1];
Table[dist[], {i, 100}]
</code></pre>
|
667,781 | <p>There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?</p>
| The_Sympathizer | 11,172 | <p>I believe there is only one function $f(x, y)$, $f: \mathbb{R}^{+} \times \mathbb{R} \rightarrow \mathbb{R}$ satisfying</p>
<ol>
<li>$f$ is continuous,</li>
<li>$f(x, 1) = x$,</li>
<li>$f(x, y + z) = f(x, y) f(x, z)$,</li>
</ol>
<p>and that is $f(x, y) = x^y = e^{\ln(x) y}$, where $e^x$ is defined by the exponenti... |
3,014,113 | <blockquote>
<p>A function <span class="math-container">$f$</span> defined on interval <span class="math-container">$(0,1)$</span> with a continuous twice derivation <span class="math-container">$(f\in{C^2(0,1)})$</span> satisfies <span class="math-container">$\lim_{x\to0^+}f(x)=0$</span> and <span class="math-contai... | RRL | 148,510 | <p>For <span class="math-container">$0 < y < x < 1$</span>, by Taylor's theorem there exists <span class="math-container">$\theta \in (0,1)$</span> such that</p>
<p><span class="math-container">$$f(y) = f(x) + f'(x)(y-x) + \frac{1}{2} f''(x - \theta(x-y)) (y-x)^2$$</span></p>
<p>Taking <span class="math-con... |
60,697 | <p>Let's see the following test case</p>
<pre><code>data = Table[{RandomReal[{-10, 10}], RandomReal[{-10, 10}]}, {i, 1, 50}];
l0 = ListPlot[data, PlotStyle -> {Blue, PointSize[0.01]}];
p0 = Plot[x*Sin[x], {x, -10, 10}, PlotStyle -> {Red, Thick}];
s0=Show[{l0, p0}, Frame -> True, FrameLabel -> {"x"... | ubpdqn | 1,997 | <p>Perhaps,</p>
<pre><code>s0 = Show[{l0, p0}, Frame -> True, FrameLabel -> {"x", "y"},
Axes -> False, GridLines -> {{1.5, 7}, {}},
GridLinesStyle -> Directive[Black, Thickness[0.004], Dashed],
Method -> {"GridLinesInFront" -> True},
PlotRange -> {{0, 10}, {-10, 10}}, ImageSize ->... |
221,500 | <p>I have a list of lists similar to this:</p>
<pre><code>L = {{"a", "b", "c"}, {"x", "c", "y"}, {"i", "j", "h"}, {"x", "b", "z"}}
</code></pre>
<p>Each list within <code>L</code> happens to be of length 3. Suppose I need to find the position of the lists that have a particular element (say, "b") at the <span class="... | kglr | 125 | <pre><code>Flatten @ Position[L, _List?(#[[2]] === "b" &)]
</code></pre>
<blockquote>
<pre><code>{1, 4}
</code></pre>
</blockquote>
<p>Making it a function:</p>
<pre><code>posF1 = Flatten @ Position[#, _List?(Function[x, x[[#2]] === #3])] &;
posF1[L, 2, "b"]
</code></pre>
<blockquote>
<pre><code> {1, 4}
</... |
1,628,839 | <p>For what values of $p$ does the following integral converge:</p>
<p>$\sum_{n=2}^{\infty} \frac{1}{n(\ln\ n)^p}.$</p>
<p>Ans. (Integral Test) $\int\limits_{n=2}^{n=\infty}\frac{1}{n(\ln n)^p} = \frac{1}{(-p+1)(ln\ n)^{p-1}}$</p>
<p>I know that $p \neq 1$, but I do not understand why the answer is $p > 1$ </p>
| lab bhattacharjee | 33,337 | <p>Integrate by parts,</p>
<p>$$\int(x-1)\sin4x\ dx=(x-1)\int\sin4x\ dx-\int\left(\dfrac{d(x-1)}{dx}\cdot\int\sin4x\ dx\right)dx$$</p>
|
2,781,867 | <p>$$
a+\frac{b}{2}+\frac{c}{3}=7 \left(1+\frac{1}{2}+\frac{1}{3} \right)
$$
Find the number of positive integral solution.</p>
| Michael Rozenberg | 190,319 | <p>By C-S
$$\sum_{k=1}^{2n+1}\frac{1}{n+k}\geq\frac{(2n+1)^2}{\sum\limits_{k=1}^{2n+1}(n+k)}=\frac{(2n+1)^2}{\frac{(2(n+1)+2n)(2n+1)}{2}}=1.$$</p>
|
3,103,160 | <p>I am dealing with some expressions containing combinatoric numbers. Does anybody know a formula for this?</p>
<p><span class="math-container">$$\displaystyle\sum_{k=0}^{\left\lfloor \dfrac{n}{2} \right\rfloor} \binom{n}{k}\binom{m}{k}$$</span></p>
| Ross Millikan | 1,827 | <p>You are intended to notice that <span class="math-container">$(2y-1)^2=4y^2-4y+1$</span> and that doubling your original equation will give the proper combination of <span class="math-container">$y^2$</span> and <span class="math-container">$y$</span>
<span class="math-container">$$4x^2+2y^2-2y=0\\8x^2+4y^2-4y=0\\8x... |
3,103,160 | <p>I am dealing with some expressions containing combinatoric numbers. Does anybody know a formula for this?</p>
<p><span class="math-container">$$\displaystyle\sum_{k=0}^{\left\lfloor \dfrac{n}{2} \right\rfloor} \binom{n}{k}\binom{m}{k}$$</span></p>
| Oscar Lanzi | 248,217 | <p>You double dipped. You were right to multiply by <span class="math-container">$2$</span>, which turns the <span class="math-container">$2(y-1/2)^2$</span> term to either <span class="math-container">$4(y-1/2)^2$</span> or <span class="math-container">$(2y-1)^2$</span>. Putting <span class="math-container">$2(2y-1)... |
3,294,237 | <blockquote>
<p><span class="math-container">$\begin{array}{l}{\text { if } \operatorname{gcd}(a, n)=1 \quad \& \operatorname{gcd}(b, m)=1} \\ {a^{x} \equiv s \bmod (n)} \\ {b^{x} \equiv s \bmod (m)} \\ {\text { is there result relating }(a b)^{x} \text { with } \bmod (m n) ? ?}\end{array}$</span></p>
</blockquot... | John Omielan | 602,049 | <p>Note that</p>
<p><span class="math-container">$$a^{x} \equiv s \pmod n \iff a^x - s = jn \; \text{, for some} \; j \in \mathbb{Z} \tag{1}\label{eq1}$$</span>
<span class="math-container">$$b^{x} \equiv s \pmod m \iff b^x - s = km \; \text{, for some} \; k \in \mathbb{Z} \tag{2}\label{eq2}$$</span></p>
<p>Multipl... |
45,429 | <p>I have two problems: </p>
<p><strong>1.-</strong> Let $X$ be a compact Hausdorff space, then $X$ has a basis with cardinality less than or equal to $|X|$.</p>
<p><strong>2.-</strong> Let $X$ be a Hausdorff space and $D$ a dense subset in $X$, then $|X|\leq|P(P(D))|$, where $P(D)$ is the power set of $D$. </p>
<p... | Henno Brandsma | 4,280 | <p>Problem 1. follows from the fact that the network weight of a compact Hausdorff space equals the weight of it, but this might be unfamiliar. A dedicated proof of this fact:</p>
<p>Let $\mathcal{T}$ be the original topology on $X$, the compact Hausdorff one. We can assume that $X$ is infinite, otherwise $X$ is discr... |
847,719 | <p>How is the derivative of $(5x-2)^3$ equal to $15(5x-2)^2$ and not $3(5x-2)^2$. According to $\frac{df}{dx} = nx^{n-1}$, it has to be $3(5x-2)^2$ right. Please explain.</p>
| Ellya | 135,305 | <p>We need to use the rule:</p>
<p>$\frac{d}{dx}(f(x)^n)=n f'(x)f(x)^{n-1}$, where $f(x)=5x-2$, and $n=3$, so $f'(x)=5$</p>
<p>Combining all of this we have:</p>
<p>$\frac{d}{dx}((5x-2)^3)=(3)(5)(5x-2^{3-1})=15(5x-2)^2$</p>
|
2,135,151 | <p>Prove that, If $r$ is a real number such that $r^2 = 2$, $r$ is irrational.</p>
<hr>
<p><strong>Proposition:</strong> If $r$ is a real number such that $r^2 = 2$, then $r$ is irrational.</p>
<p><strong>Hypothesis:</strong> If $r$ is a real number such that $r^2 = 2$.</p>
<p><strong>Conclusion:</strong> $r$ is ir... | Giulio | 401,660 | <p>To demonstrate it in this way you must know (or assume) that $\sqrt2\notin \Bbb Q$. So actually you didn't demonstrate anything. The point of the proposition is to show that you can't write $\sqrt2$ as $\frac pq$ where $p,q \in \Bbb Z$.</p>
<p>If you need the usual demonstration I can write it, but I think it's the... |
2,135,151 | <p>Prove that, If $r$ is a real number such that $r^2 = 2$, $r$ is irrational.</p>
<hr>
<p><strong>Proposition:</strong> If $r$ is a real number such that $r^2 = 2$, then $r$ is irrational.</p>
<p><strong>Hypothesis:</strong> If $r$ is a real number such that $r^2 = 2$.</p>
<p><strong>Conclusion:</strong> $r$ is ir... | Ennar | 122,131 | <p>The first thing you need to ask yourself is what $\sqrt 2$ is. This is not written neither in decimal form nor as a fraction, so we can't just say at a glance whether this is rational or not. Actually, we know nothing of it without precisely stating definition: </p>
<blockquote>
<p>$\sqrt 2$ is the positive root ... |
3,375,366 | <blockquote>
<p>How do we find the latus rectum of parabola when the equation is given in this polar form?
<span class="math-container">$$1/r = 1 + \cos t$$</span></p>
</blockquote>
<p>This curve cuts the <span class="math-container">$x$</span> axis on <span class="math-container">$1/2$</span> and <span class="mat... | Community | -1 | <p><span class="math-container">$$(1+x)^3+(1-x)^3$$</span> is an <em>even</em> polynomial of degree at most three. Hence the expansion has only even terms and must be of the form</p>
<p><span class="math-container">$$ax^2+b.$$</span></p>
<p>Setting <span class="math-container">$x=0$</span>, you find <span class="math... |
3,375,366 | <blockquote>
<p>How do we find the latus rectum of parabola when the equation is given in this polar form?
<span class="math-container">$$1/r = 1 + \cos t$$</span></p>
</blockquote>
<p>This curve cuts the <span class="math-container">$x$</span> axis on <span class="math-container">$1/2$</span> and <span class="mat... | Certainly not a dog | 691,550 | <p><span class="math-container">$\mathbf{Method\ 1}:$</span> <span class="math-container">$$S=(x+1)^3 - (x-1)^3 = 2((x+1)^2 + (x-1)^2 + (x+1)(x-1))$$</span><span class="math-container">$$=2(2(x^2+1) + x^2 -1)$$</span> <span class="math-container">$$=2(3x^2 + 1)$$</span> <span class="math-container">$$=6x^2+2$$</span> ... |
1,302,383 | <p>Can anyone help me find the solution to this integral:</p>
<p>$$\int\limits{(t-4)(t-2)^{4/5}}dt?$$</p>
<p>I think I need to expand the integrand but I do not know how. Thanks a lot!</p>
| Community | -1 | <p>You can expand and use a substitution. The integrand can be expanded to give
\begin{equation*}
(t-2)^{4/5}t-4(t-2)^{4/5}
\end{equation*}
and you can do an integration term-by-term. For the first integral, use the substitution $u=t-2$ and use the substitution $s=t-2$ for the second integral. This gives
$$\int u^{4/5... |
445,069 | <p>I am reading a textbook "Representation theory" by Fulton and Harris and I have a question.</p>
<p>They proved the following theorem on page 16.</p>
<p>With an Hermitian inner product on a set of class function, the characters of the irreducible representation of a finite group $G$ are orthonormal.</p>
<p>For a c... | mdp | 25,159 | <p>You should note that the dimension of the space of class functions is equal to the number of conjugacy classes, and that orthonormal vectors in a Hermitian inner product space are linearly independent.</p>
|
387,542 | <p>e.g. The function $e^x$ reflected through $y=x$ is $\ln x$. Is this always true OR just in some cases?</p>
| rurouniwallace | 35,878 | <p>Consider the reflection matrix:</p>
<p>$$\begin {bmatrix}
0 & 1 \\
1 & 0 \\
\end{bmatrix}$$</p>
<p>This matrix reflects a point over the y = x axis. Now consider a point on a function, (x,f(x)). With respect to the origin, it can be expressed as a vector:</p>
<p>$$\begin {bmatrix}
... |
33,430 | <p>Using <kbd>ctrl</kbd><kbd>/</kbd> you can make a fraction. If you have selected something it will appear in the numerator.</p>
<p>Does there exist a shortcut to make the selected text appear in the denominator instead?
If not, is it possible to create a shortcut that does this?</p>
| Romke Bontekoe | 1,178 | <p>None of the solutions so far makes use of the fact that the data are percentages and hence add op to (nearly) 100.</p>
<pre><code>(* Add the rows of the data list *)
Total[Rest /@ data, {2}]
(* out *)
{99.96, 99.98, 99.98, 99.99, 99.99, 99.99, 99.99, 99.97, 100., 99.99,100., 100.}
</code></pre>
<p>Borrowing from P... |
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