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 |
|---|---|---|---|---|
757,656 | <p>Find the inverse of the function $f(x)= \dfrac{2x-1}{x^2-1}.$</p>
<p>We switch the $x$ and $y$ letters and then solve the the equation, but it became kind of complicated while solving.</p>
| Haha | 94,689 | <p>$\frac {2x-1}{x^2-1}=y<=>yx^2-2x+(1-y)=0$ (1). We have that $D=(-2)^2-4y(1-y)=4y^2-4y+4=4(y^2-y+1)>0$ because for $y^2-y+1$ we have $D*=(-1)^2-4=-3<0$ and the coefficient of $y^2$ is positive. Thus $D>0$ and we have two real roots of (1). You can find them and use one of them by choosing it correctly!... |
1,350,062 | <p>In almost all of the physics textbooks I have ever read, the author will write the oscillating function as</p>
<blockquote>
<p>$$x(t)=\cos\left(\omega t+\phi\right)$$</p>
</blockquote>
<p>My question is that, is there any practical or historical reason why we should prefer $\cos$ to $\sin$ here? One possible e... | tomi | 215,986 | <p>I agree with you that one good reason for using $\cos$ is that it corresponds well with the initial conditions for harmonic motion.</p>
<p>Another good reason is experimental observation. If I come across a physical system that is oscillating, I will want to measure its amplitude and period. Examples include a swin... |
1,372,779 | <p>Given that $x$ is a positive integer, find $x$ in $(E)$.</p>
<p>$$\tag{E} j-n=x-n\cdot\left\lceil\frac{x}{n}\right\rceil$$
All $n, j, x$ are positive integers.</p>
| Ross Millikan | 1,827 | <p>Hint: $n\cdot\left\lceil\frac{x}{n}\right\rceil$ is $x$ rounded up to the next multiple of $n$, so the right side of your equation is just $x \bmod n-n$ unless $n|x$ in which case it is zero.</p>
|
832,206 | <p>I want to approximate the derivative of f(x)</p>
<h2>Finite difference</h2>
<p>$f'(x) \approx \frac{f(x+h)-f(x)}{h}$</p>
<p>I was taught that the error from the subtraction is blown up for small h. This I can verify with MATLAB.</p>
<h2>Smartass method</h2>
<p>So I though maybe the following would fix the probl... | Peter Franek | 62,009 | <p>Yes, you can see it by observing that the set of all solutions is a vector space of dimension $d$; this holds because if you choose $q_1,\ldots q_d$, the rest is clearly determined. The solutions $\{r_i^n\}$ are linearly independent (which can be shown by Vandermond determinant, for example), so they generate the wh... |
827,072 | <p>How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$
using the formula
$(x-q)^2 + (y-p)^2 = r^2$.</p>
<p>I know i need to use that formula but have no idea how to start, I have tried to start but don't think my answer is right.</p>
| Américo Tavares | 752 | <blockquote>
<p>I know i need to use that formula but have no idea how to start</p>
</blockquote>
<p>\begin{equation*}
\left( x-q\right) ^{2}+\left( y-p\right) ^{2}=r^{2}\tag{0}
\end{equation*}</p>
<p>A possible very elementary way is to use this formula thrice, one for each point. Since the circle passes through t... |
2,322,481 | <p>Look at this limit. I think, this equality is true.But I'm not sure.</p>
<p>$$\lim_{k\to\infty}\frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3^{k}}}=1$$
For example, $k=3$, the ratio is $1.000000000014$</p>
<blockquote>
<p>Is this limit <strong>mathematically correct</strong>?</p>
</blockquote>
| Mark Fischler | 150,362 | <p>$$1\leq \frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3^{k}}}=
1+\frac{\sum_{n=1}^{k-1} 2^{2\times3^{n}}}{2^{2\times3^{k}}} =
1+ \sum_{n=1}^{k-1} 2^{-2(3^k-3^{n})}
\leq 1+ \sum_{n=1}^{k-1} 2^{-2(3^k-3^{k-1})}\\=1+(k-1)2^{-4\cdot3^{k-1}}
$$
So for all $k$
$$
1\leq \frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3... |
2,322,481 | <p>Look at this limit. I think, this equality is true.But I'm not sure.</p>
<p>$$\lim_{k\to\infty}\frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3^{k}}}=1$$
For example, $k=3$, the ratio is $1.000000000014$</p>
<blockquote>
<p>Is this limit <strong>mathematically correct</strong>?</p>
</blockquote>
| Hitesh Seth | 460,801 | <p>I have taken a simpler approach to solve this.</p>
<p>$$\frac{\sum_{n=1}^k 2^{2X3^n}}{2^{2X3^k}}$$ can be written as:</p>
<p>$$\frac{\sum_{n=1}^k 4^{3^n}}{4^{3^k}}$$ which further can be written as :</p>
<p>$$\frac{4^{3^1}}{4^{3^k}} + \frac{4^{3^2}}{4^{3^k}} +\frac{4^{3^3}}{4^{3^k}}......... \frac{4^{3^{k-1}}}{4... |
3,407,474 | <p>I can understand that The set <span class="math-container">$M_2(2\mathbb{Z})$</span> of <span class="math-container">$2 \times 2$</span>
matrices with even integer entries is an infinite non commutative ring. But why It doesn't have any unity? I think <span class="math-container">$I(2\times2)$</span> is a unity for ... | Simon Fraser | 717,270 | <p><span class="math-container">$I(2\times 2)=\left[\begin{matrix}1&0\\0&1\end{matrix}\right]\not\in M_2(2\mathbb{Z})$</span> since <span class="math-container">$M_2(2\mathbb{Z})$</span> has only even integers as its entries. It can be shown using an arbitrary matrix <span class="math-container">$B$</span> that... |
1,424,297 | <p>Why is $a\overline bc + ab = ab + ac$? I think it has something to do with the rule $a + \overline a = 1$, right?</p>
| robjohn | 13,854 | <p>Start with the identity $1-\cos(2x)=2\sin^2(x)$ to get
$$
\begin{align}
\color{#C00000}{n^2}\left(\color{#00A000}{1-\cos\left(\frac1n\right)}\right)
&=\frac{\color{#00A000}{2\sin^2\left(\frac1{2n}\right)}}{\color{#C00000}{\frac1{n^2}}}\\
&=\frac12\left(\frac{\sin\left(\frac1{2n}\right)}{\frac1{2n}}\right)^2\... |
2,597,126 | <p>Given the following problem:</p>
<blockquote>
<p>Applying the division algorithm, prove that all whole numbers that are at the same time a square and a cube have the form $7k$ or $7k+1$.</p>
</blockquote>
<p>I am unable to interpret what it is asking of me and therefore I am unable to provide any solutions. Coul... | fleablood | 280,126 | <p>I'm wondering are we to assume the student knows anything. Even that $M =a^2 = b^3$ means that $M$ is 6-power. </p>
<p>We can use the remainder theorem to show that $a^2$ will have remainder $0,1,4,2,2,4,1$ if $a$ has remainder $0,1,2,3,4,5,6$</p>
<p>That is $a = 7m + k$ so $a^2 = 49m^2 + 14mk +k^2$ so $a^2$ wil... |
2,649,756 | <p>I was given the equation</p>
<p>$x=y^4$</p>
<p>and asked to find the points where the curvature was the biggest and the smallest. I know the curvature equation:</p>
<p>$κ(x)= \frac{|y″|}{\left(1+\left(y′\right)^2\right)^{\frac{3}{2}}}$</p>
<p>and I know $f(x)=y=\sqrt[4]{x}$ </p>
<p>but I am not sure how to mini... | Hw Chu | 507,264 | <p>Let $r_i = \frac{p_i}{q_i}$ and $q = \prod_{i=1}^n q_i$.</p>
<p>The group isomorphism $G \to G'$ given by $x \mapsto qx$ sends $G$ to a subgroup of $\mathbb Z$. Then use the fact that every subgroup of $\mathbb Z$ is cyclic.</p>
|
740,323 | <p>This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?</p>
<p><strong>PS:</strong> feel free to interpret the term <em>natural... | Lubin | 17,760 | <p>Here’s an example, far from the best, of prime numbers entering into a (relatively) geometric problem. Consider all the points on the unit circle, $X^2+Y^2=1$. Notice that by considering this as the set of complex numbers $a+bi$ of absolute value one, i.e. $a^2+b^2=1$, this has a natural group structure. Explicitly,... |
740,323 | <p>This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?</p>
<p><strong>PS:</strong> feel free to interpret the term <em>natural... | narsep | 425,570 | <p><a href="https://i.stack.imgur.com/j0g7j.png" rel="nofollow noreferrer">Events on the horizon of a 2D Universe</a></p>
<p>You may have a look to the graph; it represents all the numbers (red, up to 100) that "see" straight to the origin.
e.g. prime 7 is represented by the seventh vertical point-column, prime 13 by ... |
2,006,676 | <p>I'm trying to show that for a non-empty set $X$, the following statements are true for logical statements $P(x)$ and $Q(x)$:</p>
<ul>
<li>$∃x∈X$, ($P(x)$ or $Q(x)$) $\iff$ $(∃x∈X, P(x))$ or $(∃x∈X, Q(x))$</li>
<li>$∃x∈X$, ($P(x)$ and $Q(x)$) $\implies$ $(∃x∈X, P(x))$ and $(∃x∈X, Q(x))$</li>
</ul>
<p>Is it possib... | Daniel McLaury | 3,296 | <p>Not entirely rigorous as stated, but to get an idea of what's going on:</p>
<p>If $\frac{a}{b} \frac{c}{d} = 1$ then $\frac{c}{d} = \frac{b}{a}$. So this inverse $\frac{c}{d}$ exists provided that $a$ is something that's allowed in the denominator of an element of $A$ -- namely, if $a(0) \neq 0$. (Of course we're... |
4,126,124 | <p>I managed to come to the end of a proof regarding a determinant of a linear operator. However, I stuck at the end. I know it is kinda simple but I couldn't see it right away.</p>
<p>Here is my previous work:</p>
<p><span class="math-container">$F$</span> is a field. <span class="math-container">$V = M_{nxn}(F)$</spa... | José Carlos Santos | 446,262 | <p><em>By definition</em>, the open sets in a topological space <span class="math-container">$(X,\tau)$</span> <em>are</em> the elements of <span class="math-container">$\tau$</span>. This happens always, and not only when the topology is induced by a metric.</p>
<p>Concerning the last paragraph, I suppose that what yo... |
838,631 | <p>This question may be far too easy for this site but I always seem to get stuck when it comes to the triangle inequality.</p>
<p>For example, I am trying to prove that differentiability implies Lipschitz continuity. Ok, I don't really have a hard time with this except for the step when using the triangle inequality.... | Ross Millikan | 1,827 | <p>The triangle inequality gives you $|f(x)-f(x_0)|-|f'(x_0)(x-x_0) | \le |f(x)-f(x_0)-f'(x_0)(x-x_0) |$ so you have $|f(x)-f(x_0)|-|f'(x_0)(x-x_0) |\le |f(x)-f(x_0)-f'(x_0)(x-x_0) | < | x- x_0|\\|f(x)-f(x_0)|\le|f'(x_0)(x-x_0) |+ | x- x_0|$</p>
|
1,995,471 | <p>I'm trying to find all of the the roots to the following polynomial with a variable second coefficient:
$$P(x)=4x^3-px^2+5x+6$$
All of the roots are rational, and $p$ is too. It is also given that the difference of 2 roots equals the third, e.g. $r-s=t$. I would like to solve for the roots using relationships betwee... | Parcly Taxel | 357,390 | <p>Since $\frac p8$ is a root, we can perform long division on the polynomial and obtain
$$4x^3-px^2+5x+6=\left(x-\frac p8\right)\left(4x^2-\frac p2x+5-\frac{p^2}{16}\right)$$
where
$$\left(5-\frac{p^2}{16}\right)\left(-\frac p8\right)=-\frac{5p}8+\frac{p^3}{128}=6$$
$$p^3-80p-768=0$$
Since $p$ is rational, by the rati... |
1,981,553 | <p>When a person asks: "What is the smallest number (natural numbers) with two digits?"</p>
<p>You answer: "10".</p>
<p>But by which convention 04 is no valid 2 digit number?</p>
<p>Thanks alot in advance</p>
| Steven Alexis Gregory | 75,410 | <p>It depends on the context of the problem. For example. A number $n$ is called "magical" if $n^2$ ends with the digits of $n$. Below is a list of the first six magical numbers whose units digit is $5$. </p>
<pre><code> 5² = 25
25² = 625
625² = 390625
0625² = ... |
3,921,335 | <p>how can I formally proof that for a > 1: <span class="math-container">$$ a> \sqrt a > \sqrt[3]a > \sqrt[4]a ... $$</span>?
Can someone help? ;)</p>
| Olivier Moschetta | 369,174 | <p>Hint: Use logarithms to show that
<span class="math-container">$$a^{\frac{1}{n}}>a^{\frac{1}{n+1}}$$</span></p>
|
58,912 | <p>In the board game <a href="http://en.wikipedia.org/wiki/Hex_%28board_game%29" rel="nofollow">Hex</a>, players take turns coloring hexagons either red or blue. One player tries to connect the top and bottom edges of the board, colored red; the other tries to connect the left and right edges, colored blue. It is known... | Brian M. Scott | 12,042 | <p>If the path is required to be continuous, the answer is <em>no</em>. </p>
<p>Since $[0,1]$ is separable, there are only $2^\omega$ continuous functions from $[0,1]$ into $[0,1]^2$; enumerate those that yield paths connecting opposite sides of the square as $\{\varphi_\xi:\xi < 2^\omega\}$. For $\xi < 2^\omega... |
369,104 | <p>How do I take an algebraic expression and construct a tree out of it?</p>
<p>Sample equation:</p>
<p>((2 + x) - (x * 3)) - ((x - 2) * (3 + y))</p>
<p>If somebody can teach me in steps, that would be really helpful!</p>
| Brian M. Scott | 12,042 | <p>Each internal node (or vertex) of the tree will be labelled by one of the operators appearing in the expression, and each leaf by one of the operands. The algorithm is quite straightforward. Start with the operation that would be performed <strong>last</strong> if you were evaluating the expression. In the case of t... |
2,776,388 | <p>I tried this problem and I first found the lim of $x^x$ as $x$ approaches zero from right to be 1 (I did this by re-writing $x^x$ as an exponential) and when I repeated the same process to find lim of $x^{(x^x)}$, I found 1 again but the final answer should be ZERO. Could I have an explanation on why it's a zero?</p... | Martín-Blas Pérez Pinilla | 98,199 | <p>Using $\lim_{x\to 0^+}x^x = 1$, we have:
$$\lim_{x\to 0^+}x^{(x^x)} = 0^1 = 0.$$</p>
|
884,342 | <p>$N$ is a normal subgroup of $G$ if $aNa^{-1}$ is a subset of $N$ for all elements $a $ contained in $G$. Assume, $aNa^{-1} = \{ana^{-1}|n \in N\}$.</p>
<p>Prove that in that case $aNa^{-1}= N.$</p>
<p>If $x$ is in $N$ and $N$ is a normal subgroup of $G$, for any element $g$ in $G$, $gxg^{-1}$ is in $G$. Suppose $... | Bey | 2,313 | <p>By definition, we have $aNa^{-1}\subseteq N$ for all $a\in G$. This is equivalent to saying $aN\subseteq Na$ for all $a\in G$. (Why?) </p>
<p>Thus, we simply need to show $Na\subseteq aN$ for all $a\in G$.</p>
<p>To this end, take $x\in Na$. Then $x=n_1a$ for some $n_1\in N$, and so $a^{-1}x=a^{-1}n_1a=(an_2a^{-1}... |
2,753,504 | <p>Where $a,b$ and $c$ are positive real numbers.</p>
<p>So far I have shown that $$a^2+b^2+c^2 \ge ab+bc+ac$$ and that $$a^2+b^2+c^2 \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$$ but I am at a loss what to do next... I have tried adding various forms of the two inequalities but always end up with something extra on the ... | Dr. Sonnhard Graubner | 175,066 | <p>Use that $$\frac{ab+ac}{2}\geq a\sqrt{bc}$$
$$\frac{ab+bc}{2}\geq b\sqrt{ac}$$
$$\frac{ac+bc}{2}\geq c\sqrt{bc}$$</p>
|
2,647,555 | <p>In our probability class, we recently learned that the probability measure, $P$, is a set function that takes in a subset of some sample space, $\Omega$, and returns a numerical value that satisfies the probability axioms. We are now learning about conditional probabilities, namely, $P(A|B)$, for two events $A, B \s... | mordecai iwazuki | 167,818 | <p>Yes, we can see $A|B$ as a set. You can see it as a subset of the set $B$, it is precisely those elements in the set $B$ which are also in the set $A$. Since $B$ has already happened, we know we are "in" the set $B$. Then, we are looking for the elements in which $A$ also occurs, i.e. the elements which are in $B$ a... |
2,647,555 | <p>In our probability class, we recently learned that the probability measure, $P$, is a set function that takes in a subset of some sample space, $\Omega$, and returns a numerical value that satisfies the probability axioms. We are now learning about conditional probabilities, namely, $P(A|B)$, for two events $A, B \s... | Ethan Bolker | 72,858 | <p>Good question. The literal answer is "No, $A|B$ is not a set." But there is good intuition behind your question. $A|B$ is not a set, but $A \cap B$ is. Conditional probability defines a probability function on the subsets of the subset (event) $B$ of the universe $\Omega$. </p>
|
1,548,841 | <p>As you can see, In the image a rectangle gets translated to another position in the coordinates System.</p>
<p>The origin Coordinates are <code>A1(8,2) B1(9,3)</code> from the length <code>7</code> and the height <code>3</code> you can also guess the vertices of the rectangle.</p>
<p><b>Now the Rectangle gets move... | mvw | 86,776 | <p>Your transformation contains translation (2 parameter), rotation (1 parameter) and stretching, which I hope means scaling (1 parameter).</p>
<p>This in general is no linear but an affine transform, except for the case that the origin gets mapped to the origin, which I doubt here.</p>
<p>The transform would be like... |
871,730 | <p>Given x and y we define a function as follow : </p>
<pre><code>f(1)=x
f(2)=y
f(i)=f(i-1) + f(i+1) for i>2
</code></pre>
<p>Now given x and y, how to calculate f(n)</p>
<p>Example : If x=2 and y=3 and n=3 then answer is 1</p>
<p>as f(2) = f(1) + f(3), 3 = 2 + f(3), f(3) = 1.</p>
<p>Constraints are : x,y,n all... | evinda | 75,843 | <p>You can use this code:</p>
<pre><code>#include <stdio.h>
int fib(int n,int x,int y){
if (n==1) return x;
else if (n==2) return y;
else return fib(n-1,x,y) + fib(n-2,x,y);
}
int main()
{
int x,y,n,z;
printf("Give a value for x:");
scanf("%d",&x);
printf("Give a value for y:");
... |
871,730 | <p>Given x and y we define a function as follow : </p>
<pre><code>f(1)=x
f(2)=y
f(i)=f(i-1) + f(i+1) for i>2
</code></pre>
<p>Now given x and y, how to calculate f(n)</p>
<p>Example : If x=2 and y=3 and n=3 then answer is 1</p>
<p>as f(2) = f(1) + f(3), 3 = 2 + f(3), f(3) = 1.</p>
<p>Constraints are : x,y,n all... | Vishwa Iyer | 71,281 | <p>The general way to solve $$F(i+1)= F(i) - F(i - 1)$$
Is to first to see that if $F(i + 1)$ can be written as the sum of 2 geometric series, so that $$F(n) = ar^n + bs^n$$
So $$ar^{n+1} + bs^{n+1} = ar^{n} + bs^n - ar^{n-1} - bs^{n-1}$$
$$ar^{n-1}(1 - r + r^2) = -bs^{n-1}(1 - s + s^2)$$</p>
<p>Since $r\ne s$, it fo... |
244,489 | <p>Given:</p>
<p>${AA}\times{BC}=BDDB$</p>
<p>Find $BDDB$:</p>
<ol>
<li>$1221$</li>
<li>$3663$</li>
<li>$4884$</li>
<li>$2112$</li>
</ol>
<p>The way I solved it:</p>
<p>First step - expansion & dividing by constant ($11$):
$AA\times{BC}$=$11A\times{BC}$</p>
<ol>
<li>$1221$ => $1221\div11$ => $111$</li>
<li>$3... | Colm | 252,814 | <p>Another way of solving this problem is with bounding boxes. A bounding box is the minimum rectangle containing a shape orthogonal to two chosen axes. Consider the bounding boxes of the two squares A and B orthogonal to the main square Q. In each case, the remaining space makes four right angled triangles. Call them ... |
3,304,542 | <p>Let <span class="math-container">$X,Y$</span> be Banach Spaces, show that <span class="math-container">$x_{n} \xrightarrow{w} x$</span> and <span class="math-container">$T \in BL(X,Y)\Rightarrow Tx_{n} \xrightarrow{w} Tx$</span></p>
<p>Question: Does sequential continuity (which <span class="math-container">$T$</sp... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$y^{*} \in Y^{*}$</span> then <span class="math-container">$x^{*}(x)=y^{*}(Tx)$</span> defines a continuous linear functional on <span class="math-container">$X$</span>. Hence <span class="math-container">$y^{*}(Tx_n)\to y^{*}(Tx)$</span>. This implies that <span class="math-container... |
3,364,317 | <p>Points A and B fixed, and point C moves on circle such that ABC acute triangle. <span class="math-container">$AT = BT$</span> and <span class="math-container">$TM \perp AC, \, TN \perp BC$</span>. How can I proove that all the middle perpendiculars (perpendicular bissector) to <span class="math-container">$MN$</span... | Batominovski | 72,152 | <p>Let the tangent at <span class="math-container">$A$</span> and the tangent at <span class="math-container">$B$</span> of the circumscribing circle <span class="math-container">$\mathcal{C}$</span> of <span class="math-container">$\triangle ABC$</span> meet at <span class="math-container">$U$</span> and let <span cla... |
2,812,472 | <p>I am trying to show that</p>
<p>$$
\frac{d^n}{dx^n} (x^2-1)^n = 2^n \cdot n!,
$$ for $x = 1$. I tried to prove it by induction but I failed because I lack axioms and rules for this type of derivatives. </p>
<p>Can someone give me a hint?</p>
| cansomeonehelpmeout | 413,677 | <p><strong>Hint</strong>:</p>
<p>You could do this by induction, with the induction step:</p>
<p>$$\frac{d^{n+1}}{dx^{n+1}}(x^2-1)^{n+1}=\frac{d^n}{dx^{n}}\left (\frac{d}{dx}\left [(x^2-1)^n (x^2-1)\right ]\right )=2x(n+1)\frac{d^n}{dx^n}(x^2-1)^n+(2n+2)\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n$$</p>
<p>Now you only have to... |
16,247 | <p>Given a map $f:B^n \to S^n$, where $B^n$ is the unit ball and $S^n$ is the unit sphere, is it true that the degree of $f|_{S^n}$ is always 0, where $f_{S^n}$ is the restriction of $f$ to $S^n$? If so, why? </p>
<p>Thanks!</p>
| Sean Tilson | 627 | <p>This is just a bit more of an explanation of Mariano's answer:
The homotopy class of a map between spheres of the same dimension is completely determined by its degree, that is, two maps $f,g: S^n \to S^n$ are homotopic if and only if they have the same degree. Now you can compute, using which ever definition you li... |
19,672 | <p>The <a href="http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow noreferrer">Stirling number of the second kind</a> is the number of ways to partition a set of $n$ objects into $k$ non-empty subsets. In Mathematica, this is implemented as <code>StirlingS2</code>. How can I enumerate all t... | Mr.Wizard | 121 | <p>This is faster than the Combinatorica function:</p>
<pre><code>KSetP[{}, 0] = {{}};
KSetP[s_List, 0] = {};
KSetP[s_List, k_Integer] /; k > Length@s = {};
KSetP[s_List, k_Integer] /; k > 0 :=
Block[{ikf, s1 = s[[1]]},
ikf[set_] := Array[MapAt[#~Prepend~s1 &, set, #] &, Length@set];
Join[
Prepen... |
243,510 | <p>So this question is seriously flooring me.</p>
<p>Let $G$ be drawn in the plane so that it satisfies: </p>
<ol>
<li>The boundary of the infinite region is a cycle $C$</li>
<li>Every other region has boundary cycle of length $3$</li>
<li>Every vertex of $G$ not in $C$ has even degree</li>
</ol>
<p>Show that $\chi(... | Bumblebee | 51,408 | <p>We induct on $|E(G)|$. The base case is trivial.</p>
<p>Suppose first that there are two non-consecutive vertices $u, v$ of $C$ which are adjacent. Let $e$ be the edge joining them. Then, $e$ partitions the graph into two graphs $G_1$ and $G_2$ on either side of $e$ in the plane, both of which satisfy the induction... |
3,297,110 | <p>Evaluate <span class="math-container">$$\int\frac{1}{x^{\frac{25}{25} }\cdot x^{\frac{16}{25}}+x^{\frac{9}{25}}}dx$$</span></p>
<p>I start by factoring <span class="math-container">$$\int\frac{1}{x^{\frac{25}{25} }\cdot x^{\frac{16}{25}}+x^{\frac{9}{25}}}dx=\int\frac{1}{x^{\frac{9}{25}}\left(x^{\frac{32}{25}}+1\rig... | Mnifldz | 210,719 | <p>This is a cute <span class="math-container">$u$</span>-substitution problem. The crux of the problem is dividing out by the right factor of <span class="math-container">$x$</span>. Notice that if we factor out one copy of <span class="math-container">$x$</span> we obtain</p>
<p><span class="math-container">$$
\in... |
2,149,006 | <p>While learning the power rule, one thing popped up in my mind which is confusing me. We know what the power rule states :</p>
<p>$$\frac{\mathrm{d}}{\mathrm{d}x}(x^n) = nx^{n-1}$$ where $n$ is a real number.</p>
<blockquote>
<p>But instead of $n$, if we have a trig function like $\sin(x)$, <strong>will the powe... | Khosrotash | 104,171 | <p>I think you must start with $y=x^x$ then $y=f(x)^{g(x)}$
$$y=x^x \to \ln y=\ln x^x\\\ln y=x\ln x \to (\ln y=x\ln x')\\
\dfrac{y'}{y}=(1.\ln x+x .\dfrac1x) \\y'=y\times (\ln x+1) \\y'=x^x(\ln x+1) $$then for $$y=f(x)^{g(x)} \to \ln y=\ln f(x)^{g(x)} \\\ln y = g(x)\ln f(x) \to
(\ln y = g(x)\ln f(x))'\\\dfrac{y'}{y}=(... |
654,263 | <p>My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in probability/statistics -- mostly oriented toward the applications without much emphasis on mathematical rigor.</p>
<p>Now I am ve... | Geoff Pointer | 96,501 | <p>Get a copy of Elementary Probability by David Stirzaker from the library and take it from there. If you like it then buy a copy. I bought this book on a whim some years after majoring in pure maths and it rekindled my interest in probability.</p>
<p>Brief Review: It starts right from the beginning of Probability so... |
1,209,870 | <p>I am stuck in a question which says that:
A particle moves on the X- axis according to equation $x=A+Bsin(\omega t)$. The motion is simple harmonic. Find the amplitude of SHM.</p>
<p>The answer of the above problem is "<em>B</em>".<br>
My question is: how the above equation is simple harmonic? I do not know how t... | Thomas Olson | 68,485 | <p>I made a video on this. Are you the one person that seen it and hit the like button on it?</p>
<p><a href="https://www.youtube.com/watch?v=Y7utC53CNs4" rel="nofollow">https://www.youtube.com/watch?v=Y7utC53CNs4</a></p>
<p>I think these are nd rose curves. Assume x_value is from the spherical coordinates on the wik... |
1,209,870 | <p>I am stuck in a question which says that:
A particle moves on the X- axis according to equation $x=A+Bsin(\omega t)$. The motion is simple harmonic. Find the amplitude of SHM.</p>
<p>The answer of the above problem is "<em>B</em>".<br>
My question is: how the above equation is simple harmonic? I do not know how t... | Andrew D. Hwang | 86,418 | <p>This doesn't exactly answer the question, but with $k$, $m$, and $n$ positive integers, the parametric equations
\begin{alignat*}{3}
x(s, t) &= a\cos(mt) \cos^{k}(ns) &&\cos(t) &&\cos(s), \\
y(s, t) &= a\cos(mt) \cos^{k}(ns) &&\sin(t) &&\cos(s), \\
z(s, t) &= a\cos(mt) \co... |
1,353,184 | <p>Let $a$, $b$, and $c$ be positive real numbers. Prove that</p>
<p>$$\sqrt{a^2 - ab + b^2} + \sqrt{a^2 - ac + c^2} \ge \sqrt{b^2 + bc + c^2}$$</p>
<p>Under what conditions does equality occur? That is, for what values of $a$, $b$, and $c$ are the two sides equal?</p>
| Tae Hyung Kim | 94,401 | <p>The expressions $a^2 - ab + b^2$, $a^2 - ac + c^2$, and $b^2 + bc + c^2$ should remind you of the law of cosines. Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be vectors such that $\vec{a}$ and $\vec{b}$ are separated by 60 degrees, and $\vec{a}$ and $\vec{c}$ are separated by 60 degrees. Consider the triangle formed by the ... |
1,849,809 | <p>Let $n$ be a positive integer. Define $$\textbf{A}_n(x):= \left[\frac{1}{x+i+j-1}\right]_{i,j\in\{1,2,\ldots,n\}}$$ as a matrix over the field $\mathbb{Q}(x)$ of rational functions over $\mathbb{Q}$ in variable $x$. </p>
<p>(a) Prove that the <em>Hilbert matrix</em> $\textbf{A}_n(0)$ is an invertible matrix over ... | Sungjin Kim | 67,070 | <p>This is a solution to (d), and partially for (b). </p>
<p>We use <a href="https://en.wikipedia.org/wiki/Cauchy_matrix" rel="nofollow">Cauchy Matrix</a>, and its evaluation of inverse given by Schechter: </p>
<p>If $T$ is a $n\times n$ Cauchy matrix on the sequences $\{x_i\}$, $\{y_j\}$, then
$S=T^{-1}=[s_{ij}]$ is... |
3,382,561 | <p>I just encountered the random variable <span class="math-container">$Y = |X|$</span>, where <span class="math-container">$X \sim \text{N}(\mu, \sigma^2)$</span>. Now, based on what we know about the absolute value function, this random variable is still continuous; however, the absolute value function means that the... | GReyes | 633,848 | <p>New CDF:if <span class="math-container">$z\le 0$</span>, clearly <span class="math-container">$F_Y(z)=0$</span>.Otherwise,
<span class="math-container">$$
F_Y(z)=P\{|X|\le z\}=P\{-z\le X\le z\}=2\Phi(z)-1
$$</span>
where <span class="math-container">$\Phi$</span> is the CDF of the normal variable. Your PDF is zero f... |
2,063,038 | <p>Let <span class="math-container">$S$</span> be the region in the plane that is inside the circle <span class="math-container">$(x-1)^2 + y^2 = 1$</span> and outside the circle <span class="math-container">$x^2 + y^2 = 1 $</span>. I want to calculate the area of <span class="math-container">$S$</span>.</p>
<h3>Try:</... | Christian Blatter | 1,303 | <p>The shape $S$ is extremely unwieldy for integration, and has to be cut up in several parts for that purpose. Use elementary geometry instead:</p>
<p>The area in question is a full unit disk minus one third of such a disk, minus two small circular segments. The latter are a sixth of a disc with an equilateral trian... |
1,170,708 | <p>What functions satisfy $f(x)+f(x+1)=x$?</p>
<p>I tried but I do not know if my answer is correct.
$f(x)=y$</p>
<p>$y+f(x+1)=x$</p>
<p>$f(x+1)=x-y$</p>
<p>$f(x)=x-1-y$</p>
<p>$2y=x-1$</p>
<p>$f(x)=(x-1)/2$</p>
| BCLC | 140,308 | <p>$f(x) = x - 1 - y$ is wrong. $y$ is a function of $x$. In my opinion, it was such a good try. Wouldn't have thought of that approach. Anyway, is $f(x)$ supposed to be linear? Write out: $f(x) = ax + b$ if so.</p>
<p>Then plug in:</p>
<p>$a(x + 1) + b + ax + b = 1$</p>
<p>Hint: $1 = 0x + 1$.</p>
|
3,860,199 | <p>What is the integration of <span class="math-container">$\int x^{|x|} dx$</span>?<br />
Actually through several google search finally I have found a solution for the problem <span class="math-container">$\int x^{x} dx$</span> using Gamma function function and I am hardly sure that the same method can be applied to ... | Henry Lee | 541,220 | <p>assuming that this is for <span class="math-container">$x\in\mathbb{R}$</span> you will still run into problems with the output of the function being complex for <span class="math-container">$x<0$</span> since you will have a negative number to a non-integer power, so in that respect this function is very similar... |
174,075 | <p>What is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?</p>
| dtldarek | 26,306 | <p>There are different kind of contexts you use term <em>normal</em> in mathematics. You often use <em>perpendicular</em> in case of two or three dimensional geometry, and in hi-dimensional spaces (e.g. infinite-dimensional) a term <em>orthogonal</em> is more common. On the other hand in the context of vectors, <em>nor... |
174,075 | <p>What is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?</p>
| Dovydas Vaiksnys | 188,561 | <p>If normal is 90 degree to the surface, that means normal is used in 3D. Perpendicular in that case is more in 2D referring two same entities (line-line) with 90 degree angle between them. Normal in this case could refer 2 different entities (line-surface) making this term valid for 3D case (I definitely remember hea... |
4,566,926 | <p><span class="math-container">$a_n$</span> converges to <span class="math-container">$a \in \mathbb{R}$</span>, find <span class="math-container">$\lim\limits_{n \rightarrow \infty}\frac{1}{\sqrt{n}}(\frac{a_1}{\sqrt{1}} + \frac{a_2}{\sqrt{2}} +\frac{a_3}{\sqrt{3}} + \cdots + \frac{a_{n-1}}{\sqrt{n-1}} + \frac{a_n}{... | Thomas Andrews | 7,933 | <p>Define <span class="math-container">$$A_n=\sum_{k=1}^{n}\frac{a_k}{\sqrt{k}}$$</span> and <span class="math-container">$B_n=\sqrt n.$</span></p>
<p>The we want to find the limit of <span class="math-container">$\frac{A_n}{B_n}.$</span></p>
<p>Then Stolz-Cesàro requires us to find the limit:</p>
<p><span class="math-... |
27,951 | <p>Something I notice is when there's an advanced/specialized question, it often receives very few upvotes. Even if it is seemingly well written. I try to upvote advanced questions <strong>that I might not even understand</strong>, if they appear well written. </p>
<p>Is this good behaviour? Should we encourage upvoti... | Rob | 510,296 | <blockquote>
<p>Should we encourage upvotes on more advanced questions, that are seemingly well written, even if you don't understand it? ... Is this good behaviour? Should we encourage upvoting seemingly well-written questions even if you don't understand it? </p>
</blockquote>
<p>The lack of voters for some questi... |
524,073 | <p>Hey can some help me with this textbook question</p>
<p>Let $R^{2×2}$ denote the vector space of 2×2 matrices, and let</p>
<p>$S =\left\{
\left[\begin{matrix}
a \space b \\
b \space c \\
\end{matrix}\right]\mid a,b,c \in \mathbb{R}\right\}$</p>
<p>Find (with justication) a basis for $S$ and determine the dimensi... | Community | -1 | <p><strong>Hint</strong>: $\mathbb{R}^{2 \times 2}$ is a four-dimensional space, so the dimension of $S$ is at most $4$. In fact, it's at most $3$ since the set $S$ does not contain the matrix</p>
<p>$$\left[\begin{array}{c} 0 & 1 \\ -1 & 0 \end{array}\right]$$</p>
<p>So $S$ in fact has dimension at most $3$.... |
3,176,155 | <blockquote>
<p>Let <span class="math-container">$ABCD$</span> be a parallelogram. I proved that the angle bisectors of <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, <span class="math-container">$C$</span>, <span class="math-container">$D$</span> form a rectangle. How can I prove t... | Blue | 409 | <p>Let the angle bisectors at <span class="math-container">$A$</span> and <span class="math-container">$B$</span> meet at <span class="math-container">$P$</span>. Drop perpendiculars from <span class="math-container">$P$</span> to points <span class="math-container">$A^\prime$</span>, <span class="math-container">$B^\p... |
1,748,542 | <p>So a friend of mine has a little project going, and needs some help.</p>
<p>Basically, we want to create a function that takes two variables; One $X$, and one that we call $DC$ ("Difficulty Class, as this is for a pen-and-paper game).</p>
<p>The output should be $0$ if $X\leq (1/2)DC$, and it should be $100$ if $X... | bartgol | 33,868 | <p>As stated, there is no unique solution. You are looking for a curve of the form $f(x)=ax^3+bx^2+cx+d$. You have three conditions to enforce:</p>
<p>$$
f(DC/2)=0\\
f(2DC) = 100\\
f'(DC)=0
$$</p>
<p>If you plug those conditions in the expression for $f$ (and its derivative) you will get a system of three equations i... |
70,500 | <p>I am trying to find $\cosh^{-1}1$ I end up with something that looks like $e^y+e^{-y}=2x$. I followed the formula correctly so I believe that is correct up to this point. I then plug in $1$ for $x$ and I get $e^y+e^{-y}=2$ which, according to my mathematical knowledge, is still correct. From here I have absolutely n... | Gottfried Helms | 1,714 | <p>It may be more helpful to consider the significant hyperbolic identities first. We have in general: </p>
<p>$\small \begin{array} {rcllll}
1)& \exp(z) &=& \cosh(z) + \sinh(z) \\
2)& 1 &=& \cosh(z)^2 - \sinh(z)^2 \\
&&& \implies \\
3)&\sinh(z) &=& \pm \... |
2,459,169 | <p>Let $A$ be an $n \times n$ matrix. Show that if $A^2 = 0$, then $I − A$ is nonsingular and $(I − A)^{−1} = I + A$.</p>
<p>(Matrix Algebra)</p>
| ajotatxe | 132,456 | <p><strong>Hint</strong>:</p>
<p>Verify that $(I-A)(I+A)=I$.</p>
|
3,113,553 | <p>Let <span class="math-container">$X$</span> be a set, and <span class="math-container">$\mathscr{G}\subset \mathcal{P}(X)$</span> with <span class="math-container">$X\in\mathscr{G}$</span>. Then <span class="math-container">$\sigma(\mathscr{G})$</span> is defined as the “smallest” <span class="math-container">$\sigm... | Set | 26,920 | <p>It's actually not terribly difficult, the key observation is that the integral actually splits into a bunch of single variable integrals,</p>
<p><span class="math-container">\begin{align}
\int q_\phi({\bf z})\|{\bf z}\|_2^2d{\bf z}&=\int q_\phi({\bf z})(z_1^2+\ldots+z_n^2)d{\bf z}\notag\\
&=\int q_\phi({\bf... |
3,113,553 | <p>Let <span class="math-container">$X$</span> be a set, and <span class="math-container">$\mathscr{G}\subset \mathcal{P}(X)$</span> with <span class="math-container">$X\in\mathscr{G}$</span>. Then <span class="math-container">$\sigma(\mathscr{G})$</span> is defined as the “smallest” <span class="math-container">$\sigm... | MegaNightdude | 357,588 | <p>Perhaps a bit late, but may be useful for someone needing the full derivation of KL divergence:</p>
<h2>Worked derivation of KL-divergence</h2>
<p>By definition of KL divergence:
<span class="math-container">$$
\begin {equation}
D_{KL}(Q||P) = \int Q(X) \log \frac{Q(X)}{P(X)}dX
\end {equation}
$$</span></p>
<p>Or... |
1,090,658 | <p>I'm doing some previous exams sets whilst preparing for an exam in Algebra.</p>
<p>I'm stuck with doing the below question in a trial-and-error manner:</p>
<p>Find all $ x \in \mathbb{Z}$ where $ 0 \le x \lt 11$ that satisfy $2x^2 \equiv 7 \pmod{11}$</p>
<p>Since 11 is prime (and therefore not composite), the Ch... | Henry | 6,460 | <p>$2x^2 \equiv 7 \pmod {11} \iff x^2 \equiv 9 \pmod {11}$ </p>
<p>so you should not be surprised by the solutions $x \equiv \pm3 \pmod {11}$</p>
<p>i.e. $x \equiv 3 \pmod {11}$ or $x \equiv 8 \pmod {11}$</p>
|
203,138 | <p>I'm starting to take calculus in college (the first offered math class) and I was shown this problem. Find $f(x)$ when$$-\frac{df(x)}{dx}\cdot\frac{x}{f(x)}=1$$Rearranging this to make$$\frac{d\,f(x)}{dx}=-\frac{f(x)}{x}$$gives me an interesting relationship, and very strongly suggests to me a polynomial function, b... | bartgol | 33,868 | <p>Rearrange it like this</p>
<p>$$\frac{df}{f}=-\frac{dx}{x}$$</p>
<p>then integrate both sides each with respect to its variable. You're right, it is a polynomial...an easy one, I would say... =)</p>
<p>Edit: it is NOT a polynomial, as pointed out by Peter Tamaroff. My bad.</p>
|
46,446 | <p>I'm an highschool graduate who is currently waiting for college.</p>
<p>Meanwhile, I'm trying to do a little project by myself. (Computer stuff) And yesterday, I found that I needed to deal with something called "De Casteljau algorithm"</p>
<p>I know calculus (single-variable, but I'm thinking about learning multi... | lhf | 589 | <p>Try <a href="http://www.farinhansford.com/books/essentials-cagd/" rel="nofollow">The Essentials of CAGD</a> by Farin and Hansford. All books by Farin are very nice.</p>
|
88,880 | <p>In a short talk, I had to explain, to an audience with little knowledge in geometry or algebra, the three different ways one can define the tangent space $T_x M$ of a smooth manifold $M$ at a point $x \in M$ and more generally the tangent bundle $T M$:</p>
<ul>
<li>Using equivalent classes of smooth curves through ... | Steven Landsburg | 10,503 | <p>In each of the three cases, your definition is capturing the intuition of "directions near x"
--- an equivalence class of curves defines a "direction" in which the curves head out from x.
A derivation or a linear functional on the cotangent vectors is a directional derivative,
hence determined by a choice of direc... |
963,125 | <p>I'm not exactly sure where to start on this one. Any help would be greatly appreciated.</p>
<p>Show that $4$ does not divide $12x+3$ for any $x$ in the integers.</p>
<p>Here's what I have so far:</p>
<p>There exist c in the integers such that 4 | 12x+3. Then 12x+3 = 4y for some y in the integers. This is a contra... | Community | -1 | <p>A couple of hints:</p>
<ol>
<li><p>Any time you want to prove something for all integers a good thing to try would be induction (proving that the truth for $k<n$ implies the truth for $n$).</p></li>
<li><p>Remember that if a number divides $a$ and $b$ then it should divide $a+b$ as well.</p></li>
</ol>
|
330,065 | <p>What is an example of a manifold <span class="math-container">$M$</span> with <span class="math-container">$\dim(M)>1$</span> whose Lie algebra <span class="math-container">$\chi^{\infty}(M)$</span> of smooth vector fields admit an elliptic operator <span class="math-container">$D:\chi^{\infty}(M)\to \chi... | Ali Taghavi | 36,688 | <p>I would like to apply the hints presented in the answer by Bazin to give the following answer.</p>
<p><a href="http://www.numdam.org/article/CM_1981__43_2_239_0.pdf" rel="nofollow noreferrer">Every derivation of <span class="math-container">$\chi^{\infty}(M)$</span> is inner</a> and it is obvious that every inner ... |
293,234 | <p>I recently asked a question about <a href="https://math.stackexchange.com/questions/287116/proof-that-mutual-statistical-independence-implies-pairwise-independence">pairwise versus mutual independence</a> (also related to <a href="https://math.stackexchange.com/questions/281800/example-relations-pairwise-versus-mutu... | Ewan Delanoy | 15,381 | <p>For convenience I will sometimes denote the number $\sqrt[4]{2}$ by $\theta$.</p>
<p>What we have here is a tower of square roots : To $\mathbb Q$ we
can add successively $\sqrt{2},\sqrt[4]{2},i,\alpha,\beta,\gamma$. The main problem
is that in this sequence, some square root might already be contained in the
prec... |
981,748 | <p>We know a closed-form of the first two powers of the integral of <a href="http://mathworld.wolfram.com/Trilogarithm.html" rel="nofollow noreferrer">trilogarithm function</a> between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. From the result <a href="https://math.stackexchang... | Kirill | 11,268 | <p>Numerically,
$$ I_3 =
27 \zeta (5) \zeta (2)-3 \zeta (3)^2 \zeta (2)+156 \zeta (3) \zeta (2)-\tfrac{153}{8} \zeta (7)-90 \zeta (5)+\zeta (3)^3+21 \zeta (3)^2-660 \zeta (3)-420 \zeta (2)-315 \zeta (4)-\tfrac{477}{4} \zeta (6)+1680
$$
and $I_4$ doesn't seem to have a similar closed form.</p>
<p>The same approach that... |
146,075 | <p>Within my limited experience, I have only known free groups to occur through two mechanisms: as fundamental groups of trees (graphs) and ping-pong. And sometimes only through one way: the fact that sufficiently high-powers of hyperbolic elements in a Gromov-hyperbolic group generate a free group arises via ping-pong... | Ian Agol | 1,345 | <p>A couple of remarks: </p>
<p>If a subgroup is torsion-free, and intersects the pure braid group $P_n$ in a free group, then the group is also free. So the question can be interpreted in $P_n$. </p>
<p>There is the Birman exact sequence $F_{n-1} \to P_n \to P_{n-1}$, which is obtained as deleting a puncture, and ha... |
2,153,663 | <p>My discrete book says that the set $A = \{4,5,6\}$ and the relation $R = \{(4,4),(5,5),(4,5),(5,4)\}$ is symmetric and transitive but not reflexive.</p>
<p>I was wondering how this is possible, because if a set $A$ is symmetric, doesn't it also need to include $(5,6),(6,5),(4,6),(6,4)$? </p>
<p>Also if it's transi... | Patrick Stevens | 259,262 | <p>A <em>set</em> can't be symmetric; a <em>relation</em> can be. (By the way, it's possible for a set to be "transitive", but that doesn't mean the same thing as a transitive <em>relation</em>: a transitive set is a set $x$ such that if $z \in y$ and $y \in x$, then $z \in x$.)</p>
<p>A relation on a set need not inv... |
2,153,663 | <p>My discrete book says that the set $A = \{4,5,6\}$ and the relation $R = \{(4,4),(5,5),(4,5),(5,4)\}$ is symmetric and transitive but not reflexive.</p>
<p>I was wondering how this is possible, because if a set $A$ is symmetric, doesn't it also need to include $(5,6),(6,5),(4,6),(6,4)$? </p>
<p>Also if it's transi... | mle | 66,744 | <blockquote>
<p><strong>Def.</strong>: let be $A,R$ sets, we define $$ \operatorname{dom}(R):=\{x| \exists y : (x,y) \in R\} \\ \operatorname{cod}(R):=\{x| \exists y :(y,x)\in R\} \\ \operatorname{field}(R):=\operatorname{dom}(R)\cup \operatorname{cod}(R) \\ R \text{ is reflexive in }A \text{ if } \,\forall x \in A:(... |
3,717,465 | <p>I couldn't find a question answering this concept but they seem to be related.</p>
<blockquote>
<p>Extreme Value Theorem (two variables)</p>
<p>If f is a continuous function defined on a closed and bounded set
<span class="math-container">$A⊂\mathbb{R}^2$</span>, then f attains an absolute maximum and absolute
minim... | Henry Lee | 541,220 | <p>If you let:
<span class="math-container">$x=2^t$</span>
then remember we have:
<span class="math-container">$dx=\ln(2)2^tdt$</span> and our original equation is:
<span class="math-container">$$\frac{df(x)}{dx}-f(x/2)=0$$</span>
so plugging in we get:
<span class="math-container">$$\frac{1}{2^t\ln(t)}\frac{df(2^t)}{d... |
3,717,465 | <p>I couldn't find a question answering this concept but they seem to be related.</p>
<blockquote>
<p>Extreme Value Theorem (two variables)</p>
<p>If f is a continuous function defined on a closed and bounded set
<span class="math-container">$A⊂\mathbb{R}^2$</span>, then f attains an absolute maximum and absolute
minim... | Riemann'sPointyNose | 794,524 | <p>Notice</p>
<p><span class="math-container">$${f'(x)=f\left(\frac{x}{2}\right)\Leftrightarrow f''(x) = \frac{1}{2}f'\left(\frac{x}{2}\right), f'''(x)=\frac{1}{2^2}f''\left(\frac{x}{2}\right)}$$</span></p>
<p>Plugging in some numbers we get <span class="math-container">${f(0) = 10, f'(0) = 10, f''(0) = \frac{10}{2}, f... |
2,025,069 | <p>$$\sum_{n=0}^{\infty} (-1)^{n}(\frac{x^{2n}}{(2n)!}) * \sum_{n=0}^{\infty} C_{n}x^{n} = \sum_{n=0}^{\infty} (-1)^{n}(\frac{x^{2n+1}}{(2n+1)!})$$</p>
<p>How to find coefficients until the $x^{7} $ ? It is basically sin and written in form of sums but i'm not sure how should i use this form to calculate coefficients ... | Mike Earnest | 177,399 | <p>Take the equation
$$
(1-x^2/2+x^4/24-x^6/720+\dots)(C_0+C_1x+\dots+C_7x^7+\dots)=x-x^3/6+x^5/120-x^7/5040+\dots
$$
and expand out the left side. When you collect the powers of $x$, you get
$$
C_0+C_1x+(C_2-C_0/2)x^2+(C_3-C_1/2)x^3+\dots=x-x^3/6+x^5/120-x^7/5040+\dots
$$
On the left side, I only went up to $x^3$, yo... |
1,374,660 | <p>The IQ of actuarial science majors is assumed to be normally distributed with mean 112 and standard deviation of 14. In a class of 19 students, find the probability that the mean IQ of all 19 students is greater than 120. </p>
<p>I know this question is based of the CLT, but I don't know what to change when everyon... | 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... |
3,657,106 | <blockquote>
<p>Sam was adding the integers from <span class="math-container">$1$</span> to <span class="math-container">$20$</span>. In his rush, he skipped one of the numbers and forgot to add it. His final sum was a multiple of <span class="math-container">$20$</span>. What number did he forget to add?</p>
</block... | J. W. Tanner | 615,567 | <p><span class="math-container">$210$</span> minus the missing number (call it <span class="math-container">$m$</span>) <span class="math-container">$=20n$</span>.</p>
<p><span class="math-container">$210-m=20n$</span>, where <span class="math-container">$1\le m \le 20$</span>.</p>
<p>Can you take it from here?</p>
|
89,669 | <p>If we fix a locally profinite group $G$ , we note $R(G)$ the category of smooth representations of $G$, $\mathcal{E}$ the set of equivalence classes of $R(G)$, and finaly $Irr(G)$ the set of irreducible equivalence calasses. I recall that the theorem of Cantor-Bernstein says : If $E$ and $F$ two sets. If there is an... | Marc Palm | 10,400 | <p>Here are some observations, too long for a comment:</p>
<p>1) Note that cuspidal irreducible representation are compactly induced</p>
<p>$\sigma = c-ind_K^G \tau = Ind_K^G \tau$</p>
<p>2) You have the second adjointness theorem (proposition on pg. 20 Bushnell-Henniart "Local Langlands for GL(2)".)</p>
<p>$Hom_G(... |
1,602,392 | <p>I think I have a proof using Pythagoras for $\sqrt{a_1^2} + \sqrt{a_2^2} > \sqrt{a_1^2 + a_2^2}$.</p>
<p><em>I'm interested in whether there's a way to use that proof with Pythagoras to prove the general $a_n$ case (for this, hints are appreciated rather than complete proofs), and <strong>also</strong> in other ... | Community | -1 | <p>It's very easy using induction : </p>
<p>You proved the case when $n=2$ .</p>
<p>Now assume you know it for $n$ and want prove it for $n+1$ :</p>
<p>$$\sqrt{a_1^2}+\ldots+\sqrt{a_n^2}+\sqrt{a_{n+1}^2} >\sqrt{a_1^2+a_2^2+\ldots+a_n^2}+\sqrt{a_{n+1}^2}$$</p>
<p>Now use also the $n=2$ case to finnish it :</p>
<... |
3,107,858 | <p>I have to find the parametric equation for the line <span class="math-container">$M_1$</span> with the following info: </p>
<ul>
<li><span class="math-container">$M_1$</span> goes through the point <span class="math-container">$P(1,2,2)$</span></li>
<li>Is parallel with the plane <span class="math-container">$x + 3... | Steven Alexis Gregory | 75,410 | <p><span class="math-container">$M_1$</span> does not have to be <strong>on</strong> then plane <span class="math-container">$x + 3y + z = 1$</span>. If the line <span class="math-container">$M_1$</span> is parallel to the plane <span class="math-container">$x + 3y + z = 1$</span>, then it must be on the plane
<span c... |
181,839 | <p>Since the specific space $\mathbb{R}^k$ is given, this might be provable in ZF.</p>
<p>Let $\{F_n\}_{n\in \omega}$ be a family of closed subset of $\mathbb{R}^k$, of which the union is $\mathbb{R}^k$.</p>
<p>Suppose $\forall n\in \omega, {F_n}^o=\emptyset$ ($o$ denotes interior)</p>
<p>Fix $x_0 \in \mathbb{R}^k$
... | William | 13,579 | <p>The Baire Category Theorem implies that complete metric spaces are not first category (meager). That is, they are not a countable union of nowhere dense sets. A set $A$ is nowhere dense if $\text{Int}(\bar{A}) = \emptyset$. </p>
<p>Clearly $\mathbb{R}^n$ is a complete separable metric space. If $\mathbb{R}^n = \big... |
181,839 | <p>Since the specific space $\mathbb{R}^k$ is given, this might be provable in ZF.</p>
<p>Let $\{F_n\}_{n\in \omega}$ be a family of closed subset of $\mathbb{R}^k$, of which the union is $\mathbb{R}^k$.</p>
<p>Suppose $\forall n\in \omega, {F_n}^o=\emptyset$ ($o$ denotes interior)</p>
<p>Fix $x_0 \in \mathbb{R}^k$
... | Carl Mummert | 630 | <p>The answer follows from an application of some descriptive set theory and Shoenfield's absoluteness theorem. The rest of this answer works the same way in $\mathbb{R}^k$ as in $\mathbb{R}$, but I will write it up (shortly) in $\mathbb{R}$ for notational simplicity. </p>
<p>Actually, the principle in question is a v... |
181,839 | <p>Since the specific space $\mathbb{R}^k$ is given, this might be provable in ZF.</p>
<p>Let $\{F_n\}_{n\in \omega}$ be a family of closed subset of $\mathbb{R}^k$, of which the union is $\mathbb{R}^k$.</p>
<p>Suppose $\forall n\in \omega, {F_n}^o=\emptyset$ ($o$ denotes interior)</p>
<p>Fix $x_0 \in \mathbb{R}^k$
... | Brian M. Scott | 12,042 | <p>Although it’s essentially equivalent to Asaf’s, I prefer the following proof (in ZF) that every separable, completely metrizable space is Baire: I find it easier to work with dense open sets.</p>
<p>Let $\langle X,d\rangle$ be a complete, separable metric space, let $D=\{x_n:n\in\Bbb N\}$ be dense in $X$, let $\{U_... |
3,510,232 | <p>I am learning about limits and there is something that I cant quite understand:</p>
<p>If we have the function:</p>
<blockquote>
<p><span class="math-container">$$
f(x) =\frac{x^2-1}{x-1}
$$</span></p>
</blockquote>
<p>Let's say that we want to see which value for y (image) the function approaches as x (domain)... | Math1000 | 38,584 | <p>The rate at which the wasp population is growing is given by
<span class="math-container">$$
P'(t) = \frac{25000000 e^{-\frac{t}{2}}}{\left(1000 e^{-\frac{t}{2}}+1\right)^2}.
$$</span>
We want to find the value of <span class="math-container">$t$</span> that maximizes this function. Since <span class="math-container... |
3,510,232 | <p>I am learning about limits and there is something that I cant quite understand:</p>
<p>If we have the function:</p>
<blockquote>
<p><span class="math-container">$$
f(x) =\frac{x^2-1}{x-1}
$$</span></p>
</blockquote>
<p>Let's say that we want to see which value for y (image) the function approaches as x (domain)... | Quanto | 686,284 | <p>The growth is the fastest when the slope of <span class="math-container">$P(t)$</span> is the largest. Evaluate <span class="math-container">$
P'(t) = \frac{ke^{-t/2}}{\left(1000 e^{-t/2}+1\right)^2}$</span>, with <span class="math-container">$k=25mm$</span>. Then, cast it in the form</p>
<p><span class="math-conta... |
125,862 | <p>I have been given that I am working with the space of all 2x2 matrices. The basis $B$ for this space is given as a set of four 2x2 matrices, each with an entry of 1 in a unique position and zeroes everywhere else (sorry about the description in words - I don't know how to format matrices for this site).</p>
<p>I h... | Rasmus | 367 | <p>So let's look at the transformation map $T\colon\mathbb M_2\to\mathbb M_2$. Let's write the given basis as $\{e_{11},e_{12},e_{21},e_{22}\}$ and let's fix the order in which we have written it down.</p>
<p>We have $T(e_{ij})=e_{ji}$ for all $i,j\in\{1,2\}$. Hence, the matrix for $T$ in our chosen ordered basis look... |
125,862 | <p>I have been given that I am working with the space of all 2x2 matrices. The basis $B$ for this space is given as a set of four 2x2 matrices, each with an entry of 1 in a unique position and zeroes everywhere else (sorry about the description in words - I don't know how to format matrices for this site).</p>
<p>I h... | nrenga | 363,379 | <p>I thought I would clarify that the transpose is a linear operation by explicitly giving the set of linear operations that need to be performed on the original matrix to get its transpose. I will give the expression for the case of a square matrix $M_{n \times n}$ but this can be extended to arbitrary matrices too.</... |
689,591 | <p>Today at class, my teacher stated the following proposition saying it is obvious:</p>
<p>Let $x_0 \in U \subset \mathbb{R}^d$, $U$ open, and $f: U \to \mathbb{R}^m$ differentiable at $x_0$, then for any $v \in \mathbb{R}^d$ we have</p>
<p>$$ D_v f(x_0) = D_f (x_0)(v) $$</p>
<p>It is not obvious to me. Can someone... | copper.hat | 27,978 | <p>Let $\lambda(t) = x_0+t v$. Then $D_v f(x_0) = (f \circ \lambda)'(0)$.</p>
<p>Since $f$ is differentiable, we can apply the chain rule to get
$(f \circ \lambda)'(0) = D(f \circ \lambda)(0) = Df(\lambda(0)) D \lambda(0) = Df(x_0) v$.</p>
<p>(You can find this many texts, for example in look at the commentary after ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Gerry Myerson | 3,684 | <p>Instead of recommending some puzzles, I'll recommend some books containing many puzzles. Peter Winkler, Mathematical Puzzles; Peter Winkler, Mathematical Mind-Benders; Miodrag Petkovic, Famous Puzzles of Great Mathematicians. </p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Bugs Bunny | 5,301 | <p>A room contains 3 bulbs and 3 switches outside controlling the bulbs. Is it possible to determine which switch controls which bulb by entering the room only once and observing bulbs?</p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Kiochi | 4,241 | <p>You are blindfolded, then given a deck of cards in which 23 of the cards have been flipped up, then inserted into the deck randomly (you know this). Without taking the blindfold off, rearrange the deck into two stacks such that both stacks have the same number of up-flipped cards. (You are allowed to flip as many c... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | BlueRaja | 2,883 | <p>Assuming you have unlimited time and cash, is there a strategy that's guaranteed to win at roulette?</p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Christian Blatter | 6,958 | <p>A princess inhabits a flight of 17 rooms in a row. Each room has a door to the outside, and there is a door between adjacent rooms. The princess spends each day in a room that is adjacent to the room she was in the day before. One day a prince arrives from far away to woo for the princess. The guardian explains the ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Tony Huynh | 2,233 | <p>An evil sorceress is holding 100 princes captive. Right now they are all in the same prison cell and can discuss strategy. However, in a moment, they will each be taken to individual prison cells, where no communication is possible. After that, the sorceress will start randomly calling princes to her bedroom (one... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | user7361 | 7,361 | <p>First, I apologize that this puzzle is not "clean". It's more suitable for the bar rather than dinner (in fact I heard it at a party with mathematicians). I understand if it should be taken down or rephrased. I scanned the previous puzzles and I don't think this puzzle is a duplicate.</p>
<p>Suppose you are male(fe... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | jonderry | 8,013 | <p>Here's one I saw a while ago:</p>
<p>A prisoner is presented with the following challenge by one of the guards of the jail. The prisoner is to be blindfolded and then the guard will place $n$ coins on a circular turntable with any combination of heads and tails facing up (with at least one tails showing initially).... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Dave Futer | 8,183 | <p>Here's one of my favorites. There are 99 bags, each of which contains some number of apples and some number of oranges. Prove that there's a way to select 50 out of the 99 bags, such that these 50 <i>simultaneously</i> contain at least half the total number of apples and at least half the total number of oranges.</p... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | David Richter | 11,264 | <p>Can one partition the plane $\mathbb{R}^2$ by closed intervals of equal length? How? The answer to the first question is "yes". In other words, can one cover the plane with translates and rotations of a given closed line segment such that every point lies on exactly one segment?</p>
|
191,373 | <p>I have a usual mathematical background in vector and tensor calculus. I was trying to use the differential operators of Mathematica, namely <code>Grad</code>, <code>Div</code> and <code>Curl</code>. According to my knowledge, the definitions of Mathematica for <code>Grad</code> and <code>Div</code> coincides with th... | xzczd | 1,871 | <p>I fail to find a reference for the definition used by <code>Curl</code>, but manage to figure out how the <code>Curl</code> is defined.</p>
<p>I'll use <a href="http://www.dr-qubit.org/teaching/summation_delta.pdf" rel="noreferrer">Einstein summation convention</a> for simplicity. The defintion used by <code>Curl</... |
191,373 | <p>I have a usual mathematical background in vector and tensor calculus. I was trying to use the differential operators of Mathematica, namely <code>Grad</code>, <code>Div</code> and <code>Curl</code>. According to my knowledge, the definitions of Mathematica for <code>Grad</code> and <code>Div</code> coincides with th... | jose | 10,552 | <p>The definition used (motivated by exterior calculus) is as follows:</p>
<p>Given a rectangular array <span class="math-container">$a$</span> of depth <span class="math-container">$n$</span>, with dimensions <span class="math-container">$\{d, ..., d\}$</span> (so there are <span class="math-container">$n$</span> <sp... |
2,004,895 | <p>In my textbook there is a question like below:</p>
<p>If $$f:x \mapsto 2x-3,$$ then $$f^{-1}(7) = $$</p>
<p>As a multiple choice question, it allows for the answers: </p>
<p>A. $11$<br>
B. $5$<br>
C. $\frac{1}{11}$<br>
D. $9$</p>
<p>If what I think is correct and I read the equation as:</p>
<p>$$f(x)=2x-3$$
th... | lhf | 589 | <p>Your mistake is here:
$$
x-3=2y
$$
It should be
$$
x+3=2y
$$</p>
|
2,004,895 | <p>In my textbook there is a question like below:</p>
<p>If $$f:x \mapsto 2x-3,$$ then $$f^{-1}(7) = $$</p>
<p>As a multiple choice question, it allows for the answers: </p>
<p>A. $11$<br>
B. $5$<br>
C. $\frac{1}{11}$<br>
D. $9$</p>
<p>If what I think is correct and I read the equation as:</p>
<p>$$f(x)=2x-3$$
th... | Christian Blatter | 1,303 | <p>Interpret
$$f:\quad x\mapsto y:=2x-3$$
as a <em>flow diagram</em>: The operation $f$ takes a variable value $x$ as input and produces a variable value $y$ as output, whereby the exact formula for computing $y$ from $x$ is given.</p>
<p>I'm writing here because in your argument you replace without hesitation $y=2x-3... |
2,932,799 | <p><strong>Notations.</strong></p>
<p>Let <span class="math-container">$\xi=(\xi_1,\xi_2,\xi_3,\xi_4)\in\mathbb( R\setminus\mathbb Q)^4$</span> such that <span class="math-container">$\xi_1\xi_4-\xi_2\xi_3\ne 0$</span> and the <span class="math-container">$\xi_i$</span> are linearly independent over <span class="math-... | Lei Zhang | 361,807 | <p>Please see Theorem 8 in epubs.siam.org/doi/10.1137/0719030</p>
<p>The key point is r(A) is a simple eigenvalue. </p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.