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 |
|---|---|---|---|---|
11,172 | <p>As a TA who led calculus* 1 and 2 discussion section and holds office hour** in the previous year, I heard the following (wrong) arguments several times.</p>
<blockquote>
<ol>
<li><p><span class="math-container">$\displaystyle \lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}=0$</span> because <span class="math-container">$\in... | leftaroundabout | 6,902 | <p>I find it in general a bit backwards if a teacher has to explain <em>why such and such claim isn't true</em>. For any given statement that that can't be proven, the default assumption should be that it's not true<sup>†</sup>.</p>
<p>The entire confusion could be avoided by never talking about “infinite... |
11,172 | <p>As a TA who led calculus* 1 and 2 discussion section and holds office hour** in the previous year, I heard the following (wrong) arguments several times.</p>
<blockquote>
<ol>
<li><p><span class="math-container">$\displaystyle \lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}=0$</span> because <span class="math-container">$\in... | Benoît Kloeckner | 187 | <p>I think your first strategy, giving counter-example, is a good start; you may have more success by <strong>giving counter-examples that carry meaning</strong> and <strong>relate to things the students already know</strong>. In the case at hand, you can also try to make the point that <strong>limits are not about ari... |
2,442,233 | <p>let $A= \{x^2 \mid 0 < x <1\}$ and $B =\{x^3 \mid 1 < x < 2 \}$.</p>
<p>Which of the statement is true?</p>
<p>1.there is a one to one, onto function from $A$ to $B$.</p>
<p>2.there is no onto function from $A$ to $B$</p>
<p>my attempt ; there will be no onto function from $A$ to $B$ because order ... | John Griffin | 466,397 | <p>"...because the order of $A$ is $2$ and the order of $B$ is $3$, so..."
What do you mean by the order of a set? How does this relate to the functions it admits?</p>
<p>Note that $A=(0,1)$ and $B=(1,8)$. Both of these sets have the same cardinality of $\mathbb{R}$. Thus there are bijections $f:A\to\mathbb{R}$ and $g... |
467,279 | <p>I'm reading Intro to Topology by Mendelson.</p>
<p>I'm in the section titled "Compact Metric Spaces".</p>
<p>The problem is in the title.</p>
<p>My attempt at the proof is as follows:</p>
<p>Let $\{a_n\}_{n=1}^\infty$ be a Cauchy sequence in $X$. We will show that $\{a_n\}_{n=1}^\infty$ converges to a point in $... | taylorlan | 226,447 | <p>Any sequence of points in a compact metric space has a convergent subsequence. We know that a sequence that has a convergent subsequence is in fact convergent. So this easily shows us that a compact metric space is complete. </p>
|
2,895,923 | <p>I am trying to solve this simple geometry problem but I am always tangled in so many equations it makes my head spin. I tried solving it via similar triangles but i cant seem to eliminate all the unwanted variables. Please help.</p>
<p>I have to prove $ r_1\times r_3=(r_2)^2$</p>
<p><a href="https://i.stack.imgur... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/6Lf4q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6Lf4q.png" alt="enter image description here"></a></p>
<p>We have
\begin{align}
\triangle AOB:\quad
|AO| &= \frac{r_1}{\sin\phi}
,\\
|AP|&=|AO|+|OP|=\frac{r_1}{\sin\phi}+r_1+r_2
,\\
\triangle APC:\qu... |
875,924 | <p>For what values of $x\in\mathbb{R}$ is $f$ continuous?</p>
<p>$f(x) = \left\{
\begin{array}{lr}
0 & \text{if}\, x \in \Bbb Q\\
1 & \text{if}\, x \notin \Bbb Q
\end{array}
\right.$</p>
<p>The solution I found:</p>
<p>$f(x)$ is continuous nowhere. For, given any number $a$ and any... | Shabbeh | 165,678 | <p>when ${x \to a}$ the $f(x)$ doesn't tend to any specific number because there are infinite irrational and rational numbers near the $a$.</p>
<p>but $f(a)$ is either $0$ or $1$ </p>
<p>and thus (as you noticed) $\lim_{x \to a}f(x) \neq f(a)$ </p>
<p>so $f(x)$ is not continuous.</p>
|
875,924 | <p>For what values of $x\in\mathbb{R}$ is $f$ continuous?</p>
<p>$f(x) = \left\{
\begin{array}{lr}
0 & \text{if}\, x \in \Bbb Q\\
1 & \text{if}\, x \notin \Bbb Q
\end{array}
\right.$</p>
<p>The solution I found:</p>
<p>$f(x)$ is continuous nowhere. For, given any number $a$ and any... | Anupam | 84,126 | <p>Let $x_0\in \mathbb Q$. Then $(x_n)$, where $x_n=x_0+\frac{\sqrt{2}}{n}$ is a sequence of irrationals such that $x_n\to x_0$. But $f(x_n)=1$ for all $n\in \mathbb N$ and $f(x_0)=0$. Thus $(f(x_n))$ can not converge to $f(x_0)$. Thus $f$ is not continuous at any point of $\mathbb Q$. </p>
<p>Similarly it can be show... |
951,914 | <p>Here's my question. We know that the absolute value of X looks like:
<img src="https://i.stack.imgur.com/ePQo9.png" alt="Absolute Value of x"></p>
<p>Clearly, we can see, since the absolute value of x is always greater than or equal to 0, the area under the curve is always positive. Why then does it integrate to th... | davidlowryduda | 9,754 | <p>Your intuition seems to be telling you that the antiderivative of an always-positive function should be always positive. But this is not correct. This is a counterexample. Integrating <span class="math-container">$x^2 + 1$</span> is another example: it's antiderivative is <span class="math-container">$\frac{x^3}{3} ... |
322,448 | <p>I'm supposed to show that: </p>
<p>$$y=\frac{5(x-1)(x+2)}{(x-2)(x+3)} = P + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$ </p>
<p>and the required answers are: $$ P=5, Q=4, R=-4 $$ </p>
<p>I tried to solve this with partial fractions like so: </p>
<p>$$5(x-1)(x+2) = A(x+3) + B(x-2)$$ </p>
<p>$\implies$ $A$=4, $B$=-4... | achille hui | 59,379 | <p>You don't need to do any long division. In the expression</p>
<p>$$y = \frac{N(x)}{D(x)} = \frac{5(x-1)(x+2)}{(x-2)(x+3)}$$</p>
<p>The numerator $N(x) = 5(x-1)(x+2)$ and denominator $D(x) = (x-2)(x+3)$
are polynomials of degree $2$. Since the roots of $D(x)$ are simple,
$y$ can be rewritten in the form:</p>
<p>$... |
13,274 | <p>I have searched stackoverflow (and comparable pages) for quite a while now (got redirected from there to this specialized stack), and I surrender. I am trying to evaluate an expression that is small in the end numerically. <br></p>
<p>Example:</p>
<pre>Log[Log[Log[6^5^4^3^2^1]]]=12.9525...<br></pre>
<p>WolframAlp... | Dr. belisarius | 193 | <p>For example you could do:</p>
<pre><code>rules = {Log[x_ y_] :> Log[x] + Log[y], Log[x_^k_] :> k Log[x]};
N[Defer@Log[Log[Log[6^5^4^3^2^1]]] //. rules]
(*
--> 12.9525
*)
</code></pre>
<p>But you should be aware that it doesn't work ad-infinitum because your expression stops transforming after you get to ... |
102,280 | <p>What are the usual tricks is proving that a group is not simple? (Perhaps a link to a list?)</p>
<p>Also, I may well be being stupid, but why if the number of Sylow p groups $n_p=1$ then we have a normal subgroup?</p>
| Norbert | 19,538 | <p>Consider the equality
$$
1+x+x^2+...+x^n=\frac{x^{n + 1} -1}{x-1}
$$
Differentiate it by $x$, then multiply by $x$:
$$
x+2x^2+3x^3+...+n x^n=\frac{nx^{n+2} - (n + 1)x^{n+1} + x}{(x-1)^2}
$$
Now we can substitute $x=\frac{1}{2}$ and obtain
$$
\sum\limits_{i=0}^n\frac{i}{2^i}=n\left(\frac{1}{2}\right)^{n} - (n + ... |
2,210,372 | <p>I need to identify which of the curves $A_1,...,A_5$ are related to algorithms whose run times are proportional to $n, \log(n), n^2, n^3$ and $1.1^n$:</p>
<p>(Mentioning that the first figure of the $A_5$ column should be $0.015$ (not $0.025$)</p>
<p><a href="https://i.stack.imgur.com/70I2d.png" rel="nofollow nore... | zwim | 399,263 | <p><a href="https://i.stack.imgur.com/5DXu3.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5DXu3.jpg" alt="enter image description here"></a></p>
<p>As Marcelo suggest, you can identify some of the curves if not all at sight.</p>
<h3>linear</h3>
<p>$A_4$ is a straight line, so this is a linear al... |
2,666,421 | <p>Consider a linear regression model, i.e., $Y = \beta_0 + \beta_1 x_i + \epsilon_i$, where $\epsilon_i$ satisfies the classical assumptions. The estimation method of the coefficients $(\beta_0 , \beta)$ is the least-squared method. What would be an intuitive explanation of why the sum of residuals is $0$? I know t... | Georges Elencwajg | 3,217 | <p>The simplest example is when $X$ is a smooth projective curve , say over $\mathbb C$.<br>
Then a prime divisor is just a single point $P\in X$ and the group morphism $$j:\mathbb Z\to \operatorname {Pic(X)}:n\mapsto [nP]$$ is injective.<br>
<strong>Reason:</strong> any principal divisor $$\operatorname {div}f=\sum ... |
114,340 | <p>Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've
$$\text{ rank of }\left(
\begin{array}{ccc}
0 &\frac{\partial F}{\partial z} &\frac{\partial F}{\partial w} \\
\frac{\partial F}{\partial z} &\frac{\partial^2 F}{\partial ^ 2z} &\frac{\... | John | 221,643 | <p>The dot product of the adjacency matrix with itself is a measure of similarity between nodes. For instance take the non-symmetric directed adjacency matrix A = </p>
<pre><code>1, 0, 1, 0
0, 1, 0, 1
1, 0, 0, 0
1, 0, 1, 0
</code></pre>
<p>then the dot of <span class="math-container">$A^T$</span>A (gram matrix) gives... |
103,084 | <p><strong>Bug introduced in 8.0 or earlier and fixed in 11.0.0</strong></p>
<p>Case number: 305932.</p>
<hr>
<p>In the front-end notebook (Mathematica 9.0 and 10.2):</p>
<pre><code>g[x: Optional[_,default]]:=x;
</code></pre>
<p>has no problems, and works correctly (<code>g[]</code> outputs <code>default</code>, a... | m_goldberg | 3,066 | <p>I rewrote your package like this</p>
<pre><code>BeginPackage["Dummy`"]
Clear[f,g,Thing]
Begin["`Private`"]
g[x_:default]:=x
f[x:(Thing->{_}):(Thing->{1})]:=x
End[]
EndPackage[]
</code></pre>
<p>and saved it to <code>/Users/oldmg/Desktop/Dummy.m</code> and ran the following code</p>
<pre><code>Quiet @ Rem... |
1,403,228 | <blockquote>
<p><strong>Question:</strong></p>
<p>Let <span class="math-container">$m, n, q, r \in \mathbb Z$</span>. If <span class="math-container">$m = qn + r$</span>, show that <span class="math-container">$\gcd(m, n) = \gcd(n, r)$</span>. Hence justify the Euclidean Algorithm.</p>
</blockquote>
<p>I found this que... | 727 | 126,929 | <p><strong>Hint:</strong> </p>
<p>Suppose $d$ divides both $m$ and $n$ (it needn't be the $\gcd$). Then $d$ divides $m+kn$ for any $k \in \mathbb{Z}$.</p>
|
2,845,001 | <p>If I know the length of a chord of a circle and the length of the corresponding arc, but do not know the circle's radius, is there a formula by which I can calculate the length of the sagitta?</p>
| paulwal222 | 1,040,297 | <p>These other answers are too complicated. Here is a simple way to approximate it, where s=sagitta, a=arc length, c=chord length:</p>
<p><span class="math-container">$$s = 0.42 \sqrt{ a^2 - c^2}$$</span></p>
<p>If the arc is more like a full semi-circle, then the constant will be a little closer to 0.41. If the arc is... |
1,870,675 | <blockquote>
<p>Let $R$ be a commutative ring with unity, and let $R^{\times}$ be the group of units of $R$. Then is it true that $(R,+)$ and $(R^{\times},\ \cdot)$ are not isomorphic as groups ? </p>
</blockquote>
<p>I know that the statement is true in general for fields. And it is trivially true for any finite ri... | Community | -1 | <p>The conjecture is false. Here is a counterexample.</p>
<p>Suppose $R$ is a ring with the property that every $r \in R^\times$ satisfies $r^2 = 1$.</p>
<p>Then both $(R,+)$ and $(R^\times,\cdot)$ are abelian groups with the property that every element has exponent $2$ — that is, they are vector spaces over $\... |
221,403 | <p>I have a series of functions <span class="math-container">$f_d$</span> in <span class="math-container">$d$</span> variables and would like compute the sum of each one evaluated at each lattice point within the <span class="math-container">$d$</span>-sphere of radius <span class="math-container">$R$</span>; that is, ... | David G. Stork | 9,735 | <pre><code>pts[dim_Integer, num_Integer] :=
Select[RandomVariate[UniformDistribution[Table[{-1, 1}, dim]], num],
Norm[#] <= 1 &]
</code></pre>
<p>Then</p>
<pre><code>pts[5,20]
</code></pre>
<p>(Note that this creates 20 sample points, but some subset are within the unit ball.)</p>
|
2,228,328 | <p>Is the function $ f(z)=\dfrac 1 {x^2+y^2} + i \dfrac 1 {x^2+y^2} $ is differentiable and holomorphic somewhere?</p>
<p>We have $z=x+iy$ and $f(z)= \dfrac{1+i}{|z|^2}$ . Now, $f(0)= \lim_{\delta z \rightarrow 0} \frac{f(0+\delta z) - f(0)}{\delta z}=$ undefined. So $f(z)$ is not differentiable at origin. Also sinc... | WB-man | 420,588 | <p>You can check for sure using the Cauchy-Riemann equations. If we have a complex function $f(x+yi) = u(x,y) + iv(x,y)$, then for $f$ to be differentiable at a point $z_0 = x_0 + y_0 i$, both
$$
\left\{
\begin{aligned}
\frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} \\
\frac{\partial u}{\partial y} ... |
85,622 | <p>I'm trying to understand how this works in terms of Galois theory and local class field theory. Assume we have an extension of local fields $E/L/K$ s.t. $E/L$ and $L/K$ are abelian. I'm interested in recognizing when $E/K$ is Galois. Clearly, $E/K$ is Galois if and only if $E$ is always fixed by an extension of an $... | Chandan Singh Dalawat | 2,821 | <p>When you have a finite abelian extension $E|L$ of a finite galoisian extension $L|K$ of local fields, the extension $E|K$ is going to be galoisian if and only if the subgroup $N_{E|L}(E^\times)\subset L^\times$ is $Gal(L|K)$-stable.</p>
<p>This is somewhat similar to the following purely algebraic fact : Let $K$ b... |
759,683 | <p>Given an undirected connected simple graph $G=(V,E)$ there are $2^{|E|}$ orientations. How many of these orientations are acyclic? </p>
| mjachi | 838,358 | <p>Very old, but for the sake of completeness, yes, it is well known that we can count</p>
<p><span class="math-container">$$ \text{AO}(G) = (-1)^{|V(G)|}\mathcal{X}_G(-1) = T_G(2,0)$$</span></p>
<p>where <span class="math-container">$\mathcal{X}_G$</span> denotes the chromatic polynomial of <span class="math-container... |
3,563,238 | <p>I try to understand the relation of likelihood to cross-entropy by reading <a href="https://en.wikipedia.org/wiki/Cross_entropy" rel="nofollow noreferrer">cross-entropy</a>.</p>
<p>The problem is I cannot understand the formula for the likelihood in the article. The likelihood is defined as follows</p>
<p><span cl... | Mini | 743,701 | <p>This is because in a block of length <span class="math-container">$N$</span>, outcome <span class="math-container">$i$</span> appears about <span class="math-container">$Np_i$</span> times and the probability of each of them is estimated as <span class="math-container">$q_i$</span>. This is why for each outcome it i... |
306,178 | <p>Given
$$
y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1.
$$
Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ? I consider the ratio $\frac{y_{n+1}}{y_n}$ but I got stuck.</p>
| Mikasa | 8,581 | <p>I think it needs just some basic algebraic manipulations: $$\frac{y_{n}}{y_{n+1}}=\frac{(1+\frac{1}n)^{n+1}}{(1+\frac{1}{n+1})^{n+2}}$$ You can show that the latter fraction is equal to $$(1+\frac{1}{n^2+2n})^{n+1}\times\frac{1}{1+1/(n+1)}$$ But $$(1+\frac{1}{n^2+2n})^{n+1}\ge {1+1/(n+1)}$$ Note that if $x\ge-1$ the... |
2,213,047 | <p>Let $F$ be the set of all functions $f : \mathbb{R} \to \mathbb{R}$. A relation $c$ is defined on $F$ by
$f c g$ if and only if $f(x) ≤ g(x)$ for all $x ∈ \mathbb{R} $.
Prove that '$c$' is a partial order.</p>
<p>I have proved that the relation is <em>reflexive</em>. Now I must prove that it is <em>antisymmetric</e... | phunfd | 361,692 | <p>Keep in mind that this is a relation on the functions, not the elements. You seem to be getting confused with this difference. If $f \prec g$ and $g \prec f$ then $f(x) \leq g(x)$ and $f(x) \geq g(x)$ for all $x \in R$. This means $f=g$. No need for injectivity :)</p>
|
2,916,168 | <p>I am a beginner in proofs and, unfortunately, I cannot wrap my mind around how to prove the simplest things, so I need a bit of help getting started. This is the proof that I am dealing with:</p>
<p>$\text{If }x< y< 0\text{, then }x^{2}> y^{2}\text{.}$</p>
<p>Thank you in advance.</p>
| mechanodroid | 144,766 | <p>If $x < y < 0$ then $|x| > |y|$ so $$x^2 = |x|^2 > |y|^2 = y^2$$</p>
|
226,488 | <p>I am currently working on a challenge problem where I need to show that there is a point $x \in \mathbb{R_+}$ such that $\cos(x) = 0$ using only a few properties of the cosine function. In particular, the only properties of the cosine function that I can use are:</p>
<ul>
<li><p>$\cos(x)$ is continuous</p></li>
<li... | EuYu | 9,246 | <p>If you have $n$ points uniformly distributed on a circle then the angle between the points is $\frac{2\pi}{n}$. If the distance between adjacent points is $m$ then the chord of the sector with angle $\frac{2\pi}{n}$ has length $m$. </p>
<p>Trigonometry then yields
$$r = \frac{m\sin\left(\frac{(n-2)\pi}{2n}\right)}{... |
2,673,078 | <p>Generalize the Monty Hall problem where there are $n \geq 3$ doors, of
which Monty opens $m$ goat doors, with $1 \leq m \leq n$.<br><strong>Original Monty Hall Problem:</strong> There are $3$ doors, behind one of which there is a car (which you want), and behind the other two of which there are goats (which you don’... | angryavian | 43,949 | <p>If your initial door is the car, the switching strategy fails.</p>
<p>$\frac{n-1}{n}$ of the time, your initial door is not the car.
Then excluding your initial door and the $m$ goat doors, the car must be in one of the $n-m-1$ remaining doors, and presumably the switching strategy picks uniformly at random. So, th... |
3,129,192 | <p>I've been trying to integrate
<span class="math-container">$$\int\frac{dx}{(x-a)(x-b)}$$</span></p>
<p>By using the substitution
<span class="math-container">$$x=a \cos^2 \theta + b \sin^2 \theta$$</span></p>
<p>The only problem here is I arrived at the result
<span class="math-container">$$\frac{2}{a-b} \ln |\cs... | Paul Frost | 349,785 | <p>Identifying <span class="math-container">$\mathbb{R}^2$</span> with <span class="math-container">$\mathbb{C}$</span>, we have <span class="math-container">$A = \{ z \in \mathbb{C} \mid 1 \le \lvert z \rvert \le 2 \}$</span>. Define
<span class="math-container">$$q : I^2 \to A, q(x,y) = 2(x+1)e^{2\pi iy} .$$</span>
T... |
1,326,525 | <p>Suppose I have some functions $f,g$ such that
$$f:\Bbb{R} \mapsto \Bbb{R}^2$$
$$g:\Bbb{R}^2 \mapsto \Bbb{R}^n$$</p>
<p><strong>My Question:</strong></p>
<p>For some $c \in \Bbb{R}$, is $g(f(c))$ a function of one variable? If so, why is the fact $g$ requires two arguments irrelevant? </p>
<p>EDIT: </p>
<p>Speci... | Mathxx | 221,836 | <p>As @AlexR said, $P(\text{dog} \cap \neg \text{cat}) = P(\text{dog}) - P(\text{dog} \cap \text{cat})$</p>
<p>So you'll get 0.1 for P(Dog and not cat).</p>
<p>$P(\text{Not cat} \mid \text{Dog})=\frac{0.1}{0.25}=0.4$</p>
|
1,212,177 | <p>Does $i\arg(e^{2z})=2iy?$ If it does I have solved my problem, and hence it seems like it must be the case, but I don't see it.</p>
<p>$$i\arg(e^{2z})=i\arg(e^{2x+2iy})=i\arg(e^{2x}e^{2iy})\implies \theta=2y(?)$$ Why does the $2x$ get 'ignored'?</p>
| dustin | 78,317 | <p>A property of the argument is $\arg(z_1z_2) = \arg(z_1)+\arg(z_2)$. Therefore,
$$
\arg(e^{2x}e^{2iy}) = \arg(e^{2x})+\arg(e^{2iy})
$$
For $\arg\in(-\pi,\pi)$, what is $ \arg(e^{2x})$?</p>
|
2,735,854 | <p>How many permutations of the word $STRESSLESSNESS$ begin OR end with an $E$? </p>
<p>Correct me if I'm wrong, but you would have to subtract the permutations where $E$ begins AND ends the permutation?</p>
| Leyla Alkan | 459,554 | <ul>
<li><p>$E\underbrace{------------}_{12 \text {letter}}E$<br>
Result: $\frac{12}{7!}$ , since there are seven $S$'s among the remaining letters(STRSSLSSNESS) we divide by $7!$ to eliminate overcounting</p></li>
<li><p>$E\underbrace{-------------}_{13 \text {letter}}$<br>
Result: $\frac{13}{2!7!}$ ,since there are s... |
186,310 | <p>I am trying to evaluate: </p>
<pre><code>For[i = 1, i < 100000, i++, h[0] = hs;
h[i]=hi /. Check[FindRoot[Xfnew[a, w, hi, mxs + i*step]*92*10^23) == 1198/10000, {hi, h[i - 1]}]];
If [h[i] == 25, Break[], AppendTo[hin, {mxs + i*step, h[i]}]]];
</code></pre>
<p>where mxs , hs, and step has some values, and hin ... | John Doty | 27,989 | <p>You're creating pairs <code>{m,h}</code> where each pair really only depends on the previous pair. That's perfect for <code>NestWhileList</code>. <code>i</code> is irrelevant, an artifact of procedural formulation: you don't need it. You need two functions. The first transforms an <code>{m,h}</code> pair into a new ... |
2,698,284 | <p>Here is the problem: </p>
<p>Let A be a set with n elements. Find an expression for S(n, 2), the number of partitions of A into exactly two subsets. You can either start with the general recurrence for S(n, k), or count S(n, 2) directly. </p>
<p>I'm having trouble understanding what exactly it wants me to do. So f... | Ashwin Trisal | 343,481 | <p>First, the problem isn't actually asking you to divide up $A$ into sets with exactly two elements -- for something like $|A|=3$, this is impossible. Instead, what they want is to consider all the ways that $A=B\cup C$, where $B\cap C=\emptyset$.</p>
<p>So how can we approach this? Generally, we do it by induction i... |
2,655,178 | <p>I am asked to find the equation of a cubic function that passes through the origin. It also passes through the points $(1, 3), (2, 6),$ and $(-1, 10)$. </p>
<p>I have walked through many answers for similar questions that suggest to use a substitution method by subbing in all the points and writing in terms of vari... | Ranjeev Grewal | 484,668 | <p>the general cubic equation is $$y=ax^3+bx^2+cx+d.$$Plug in the coordinates of the points for x and y, and you end up with a system of four equations in four variables, namely $a, b, c$ and $d$. Hope that helps!</p>
|
2,655,178 | <p>I am asked to find the equation of a cubic function that passes through the origin. It also passes through the points $(1, 3), (2, 6),$ and $(-1, 10)$. </p>
<p>I have walked through many answers for similar questions that suggest to use a substitution method by subbing in all the points and writing in terms of vari... | Mostafa Ayaz | 518,023 | <p>Hint:</p>
<p>It is proper to use the <strong>Lagrange function</strong> as following$$f(x)=\sum_{cyc}\dfrac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)}$$</p>
|
2,655,178 | <p>I am asked to find the equation of a cubic function that passes through the origin. It also passes through the points $(1, 3), (2, 6),$ and $(-1, 10)$. </p>
<p>I have walked through many answers for similar questions that suggest to use a substitution method by subbing in all the points and writing in terms of vari... | csar | 446,038 | <p>This problem is a special case in which all known values differ by fixed <span class="math-container">$\Delta x$</span> (here <span class="math-container">$\Delta x = 1$</span>). This is a very common problem when you have discrete equispaced values and you want to interpolate to find the "exact" maximum o... |
1,504,438 | <blockquote>
<p>the parallel sides of a trapezoid have lengths 7 cm and 15 cm. The two lower base angles are 30 and 60 degrees. The area of the trapezoid is..?</p>
</blockquote>
<p>Two 30-60-90 degree triangles form on each side of the trapezoid. If I determine the height of one triangle, I can easily calculate the ... | André Nicolas | 6,312 | <p>Hint: Note that $h\cot (30^\circ)+h\cot(60^\circ)$ is equal to the difference $15-7$ of the parallel side lengths.</p>
|
1,504,438 | <blockquote>
<p>the parallel sides of a trapezoid have lengths 7 cm and 15 cm. The two lower base angles are 30 and 60 degrees. The area of the trapezoid is..?</p>
</blockquote>
<p>Two 30-60-90 degree triangles form on each side of the trapezoid. If I determine the height of one triangle, I can easily calculate the ... | David | 119,775 | <p><strong>Hint</strong>. If the base angles are $30^\circ$ and $60^\circ$, then the trapezium can be visualised as a $30$–$60$–$90$ triangle with (horizontally placed) hypotenuse $15$, minus a similar triangle with hypotenuse $7$.</p>
|
1,098,587 | <blockquote>
<p>Prove that for a bipartite graph $G$ on $n$ vertices the number of edges in $G$ is at most $\frac{n^2}{4}$. </p>
</blockquote>
<p>I used induction on $n$. </p>
<p>Induction hypothesis: Suppose for a bipartite graph with less than $n$ vertices the result holds true. </p>
<p>Now take a bipartite g... | Asinomás | 33,907 | <p>The sum of the degrees of vertices $x$ and $y$ is indeed less than or equal to $n$. One reason for this is that if a vertex $v$ is adjacent to $x$ it cannot be adjacent to $y$ since $y$ and $v$ would be in the same part.</p>
<hr>
<p>I would like to share my proof, please tell me what you think of it: The bipartite... |
1,098,587 | <blockquote>
<p>Prove that for a bipartite graph $G$ on $n$ vertices the number of edges in $G$ is at most $\frac{n^2}{4}$. </p>
</blockquote>
<p>I used induction on $n$. </p>
<p>Induction hypothesis: Suppose for a bipartite graph with less than $n$ vertices the result holds true. </p>
<p>Now take a bipartite g... | bof | 111,012 | <p>Suppose $p,q$ are nonnegative integers with $p+q=n,$ and that $K_{p,q}$ has the maximum number of edges among all bipartite graphs with $n$ vertices.</p>
<p>If we move one vertex from the side with $p$ vertices to the side with $q$ vertices, we lose $q$ edges and gain $p-1$ new edges. Since the number of edges was ... |
17,819 | <p>$\mathfrak{sl}_2(\mathbb{C})$ is usually given a basis $H, X, Y$ satisfying $[H, X] = 2X, [H, Y] = -2Y, [X, Y] = H$. What is the origin of the use of the letter $H$? (It certainly doesn't stand for "Cartan.") My guess, based on similarities between these commutator relations and ones I have seen mentioned when pe... | Jim Humphreys | 4,231 | <p>Both terminology and notation in Lie theory have varied over time, but as far
as I know the letter H comes up naturally (in various fonts) as the next
letter after G in the early development of Lie groups. Lie algebras came
later, being viewed initially as "infinitesimal groups" and having labels like
those of th... |
2,885,441 | <p>Prove that</p>
<p>$$\sum_{n=-\infty}^{\infty} \frac{\cos n}{n^2+1} = \frac{\pi \cosh (\pi -1)}{\sinh \pi }$$</p>
<p>I already have a solution using Fourier expansion of the exponential function. I'm interested in a complex analysis approach. It would be natural to consider the function</p>
<p>$$f(z) = \frac{\pi \... | pisco | 257,943 | <p>Hint: try to integrate
$$f(z)=\frac{{\pi {e^{iz}}}}{{({z^2} + 1)({e^{2\pi iz}} - 1)}}$$
around the rectangular contour with vertices $\pm R \pm Ri$, with $R$ a half integer. You might want to check carefully that the integral indeed tends to $0$. </p>
<p>The sum of residues at $z=\pm i$ is $\frac{i}{2}\frac{{\pi \c... |
2,885,441 | <p>Prove that</p>
<p>$$\sum_{n=-\infty}^{\infty} \frac{\cos n}{n^2+1} = \frac{\pi \cosh (\pi -1)}{\sinh \pi }$$</p>
<p>I already have a solution using Fourier expansion of the exponential function. I'm interested in a complex analysis approach. It would be natural to consider the function</p>
<p>$$f(z) = \frac{\pi \... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
1,041,623 | <p>I have been suggested to read the <a href="http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf"><em>Advice to a Young Mathematician</em> section </a> of the <em>Princeton Companion to Mathematics</em>, the short paper <a href="http://alumni.media.mit.edu/~cahn/life/gian-carlo-rota-10-lessons.html"><em>Ten L... | Georges Elencwajg | 3,217 | <p>My advice would be:<br>
$\bullet $ Do many calculations<br>
$\bullet \bullet$ Ask yourself concrete questions whose answer is a number.<br>
$\bullet \bullet \bullet$ Learn a reasonable number of formulas by heart. (Yes, I know this is not fashionable advice!)<br>
$\bullet \bullet \bullet \bullet$ Beware the illus... |
1,041,623 | <p>I have been suggested to read the <a href="http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf"><em>Advice to a Young Mathematician</em> section </a> of the <em>Princeton Companion to Mathematics</em>, the short paper <a href="http://alumni.media.mit.edu/~cahn/life/gian-carlo-rota-10-lessons.html"><em>Ten L... | SQL Injection | 211,243 | <p>$\bullet$ Patience!</p>
<p>$\bullet\bullet$ Persistence.</p>
<p>$\bullet\bullet$ Work hard.</p>
<p>$\bullet\bullet\bullet$ Learn things very well. (in detail)</p>
<p>$\bullet\bullet\bullet\bullet$ Ask yourself lots of questions, even stupid ones! (when does this lemma work? when it doesn't? is there a generaliza... |
331,543 | <blockquote>
<p>Given a fair six-sided die. Find the probability generating functions for the number of the throw on which the rth six appears. Hence find the probability that the fifth six occurs on the 20th throw. </p>
</blockquote>
<p>For the rth six is it true that $X~Geo((1/6)^r)$, so PGF is $(1/6)^rt+(1/6)^r\t... | user65384 | 65,384 | <p>See "negative binomial". With parameters r and p, a negative binomial random variable represents the trial number of the rth success when each trial is independently a success with probability p.</p>
<p>r = 5<br>
p = 1/6<br>
i = 20</p>
<p>(sorry in advance for formatting, coding errors etc.)</p>
<p>$p(i=20)$ = ${... |
331,543 | <blockquote>
<p>Given a fair six-sided die. Find the probability generating functions for the number of the throw on which the rth six appears. Hence find the probability that the fifth six occurs on the 20th throw. </p>
</blockquote>
<p>For the rth six is it true that $X~Geo((1/6)^r)$, so PGF is $(1/6)^rt+(1/6)^r\t... | Martin Hansen | 646,413 | <p>A straight forward application of the Negative Binomial Distribution where the number of trials <span class="math-container">$n$</span> is fixed and made up of <span class="math-container">$k$</span> failures and <span class="math-container">$r$</span> successes, the <span class="math-container">$r$</span>th success... |
979,947 | <p>If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer?</p>
<p>UPDATE: I've done some more work and have come up with the following: Based on the equations I combined to arr... | Mastombord | 185,395 | <p>Your proof is right. In 4 out of 5 you are getting - 1/4 in 1 out of it is +1. This means mathematically speaking it does in the long term not make a difference if you guess or leave an answer blank. But that means it is exactly as good, not at least as good: on single test where you don't know any answers it migh... |
278,637 | <p>After spending more than 1 hr on this, and looking at many questions, I give up as I am not able to figure a solution.</p>
<p>I have my current package in the standard location given by <code>FileNameJoin[{$UserBaseDirectory, "Applications"}]</code> which on windows is</p>
<pre><code>C:\Users\Owner\AppData... | lericr | 84,894 | <p>There is a built-in function for checking whether an expression is atomic: <code>AtomQ</code>. So, if your definition of atomic matches Mathematica's, you could do something like:</p>
<pre><code>Scalarize[list_List] :=
With[
{flat = Flatten[list]},
If[
MatchQ[flat, {_?AtomQ}],
First@flat,
... |
3,133,831 | <p>I know that supremum means least upper bound. If I have a sequence of events, <span class="math-container">$\{A_n\}_{n=1}^\infty$</span></p>
<p>then <span class="math-container">$$\limsup_{n\rightarrow \infty} A_n = \lim_{n\rightarrow \infty} \sup_{j\geq n} A_j$$</span></p>
<p>I'm having trouble understanding this... | Berci | 41,488 | <p>Probability events are (the measurable) subsets of the probability space <span class="math-container">$\Omega$</span>, and the partial order among them is simply the relation 'being subset of'. So, for any countable set of events <span class="math-container">$B_i$</span>, their supremum is just <span class="math-con... |
245,789 | <p>Let $f$ be a function defined on the interval $(−1, 1)$ such that for all $x, y \in (−1, 1)$, $$f(x + y) = \frac{{f(x) + f(y)}}{{1 - f(x)f(y)}}.$$ Suppose that $f$ is differentiable at $x = 0$, show that $f$ is differentiable on $(−1, 1)$.</p>
<p>Need your kind guidance and help. Thanks in advance!</p>
| Pedro | 23,350 | <p>Assuming differentiability at $x=0$, we have that</p>
<p>$$\lim\limits_{h\to 0}\frac{f(h)-f(0)}{h}=\ell$$</p>
<p>exists.</p>
<p>Now, note that from $$f(x + y) = \frac{{f(x) + f(y)}}{{1 - f(x)f(y)}}.$$ we get that</p>
<p>$$f(x +h)-f(x) = \frac{{f(x) + f(h)}}{{1 - f(x)f(h)}}-f(x)$$</p>
<p>$$f(x +h)-f(x) = \frac{{... |
3,760,115 | <p>I have these constraints on a cost function</p>
<p><span class="math-container">$$
c = A+Bx=A+B\text{vec}\ (q^*q^\top),
$$</span>
where <span class="math-container">$(c,A)\in\mathbb{R}^{100}$</span>, <span class="math-container">$B\in\mathbb{C}^{100\times 81}$</span>, <span class="math-container">$x\in\mathbb{C}^{81... | Ben Grossmann | 81,360 | <p>So far, you have
<span class="math-container">$$
dc = B \operatorname{vec}(q^*\,dq^\top) + B\operatorname{vec}(dq^*\, q^\top).
$$</span>
To obtain the components of the gradient, it suffices to plug in <span class="math-container">$q = e_j$</span> (where <span class="math-container">$e_1,\dots,e_9$</span> denote the... |
947 | <p>I'm looking for the algorithm that efficiently locates the "loneliest person on the planet", where "loneliest" is defined as:</p>
<p>Maximum minimum distance to another person — that is, the person for whom the closest other person is farthest away.</p>
<p>Assume a (admittedly miraculous) input o... | user80022 | 80,022 | <p>just divide the points into a mesh of triangles . the entire sphere , including the surface points can be divided as such if the population call it P is divisible by 3 . (I have written a publication for the other cases In Deutschen Mathematica vol 127.6 )
then not to describe every kilobyte in that paper, you can n... |
2,198,293 | <p>Having a bit of trouble in what way I am suppose to go about solving this problem. Any guidance would be great.</p>
<p>Show that there is at least one real solution to:$$x^5 - x^2 - 4 = 0 $$</p>
<p>Thanks in advance.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>set $$f(x)=x^5-x^2-4$$ then we have $$f(1)=-4$$ and $$f(2)=2^5-4-4=24>0$$ and the function $$f(x)$$ is continuous, thus there is at least one solution in the interval</p>
|
1,842,781 | <p>Suppose we have an alphabet of the following allowed characters: </p>
<ul>
<li>the lowercase letters $a$ through $z$ (26)</li>
<li>the uppercase letters $A$ through $Z$ (26)</li>
<li>the numerals $0$ through $9$ (10)</li>
<li>the common punctuation marks: $.,;:'"()!?-$ ( 11)</li>
<li>the white space (1)</li>
</ul>... | heropup | 118,193 | <p>Let the numbers be arranged as follows: $$\begin{array}{ccccc} & a & & b & \\ 0 & & c & & 1000 \\ & d & & e & \end{array}$$ Then we have the relationships $$\begin{align*} 3a &= b + c \\ 3d &= c + e \\ 3b &= a + c + 1000 \\ 3e &= c + d + 1000 \\ 6c &am... |
2,912,401 | <p>I'm trying to prove that the sequence $\left(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\cdots\right)$ is not a Cauchy sequence. </p>
<p>I know that a sequence of real numbers is <em>not</em> Cauchy if there exists an $\epsilon>0$ such ... | ℋolo | 471,959 | <p>Cauchy sequence: $$\forall\varepsilon>0\exists N\forall n,m\quad n,m>N\implies|a_n-a_m|<\varepsilon$$</p>
<p>We want to show the negative:$$\exists\varepsilon>0\forall N\exists n,m\quad n,m>N\land |a_n-a_m|\ge\varepsilon$$</p>
<p>Take $\varepsilon=1/10$, and then, for all $N$ take $n=\sum_{i=1}^{N+1... |
1,025,588 | <p>I just started doing AM-GM inequalities for the first time about two hours ago. In those two hours, I have completed exactly two problems. I am stuck on this third one! Here is the problem:</p>
<p>If $a, b, c \gt 0$ prove that $$ a^3 +b^3 +c^3 \ge a^2b +b^2c+c^2a.$$</p>
<p>I am going crazy over this! A hint or pro... | Aditya Hase | 190,645 | <p>Using <a href="http://en.wikipedia.org/wiki/Rearrangement_inequality" rel="noreferrer">rearrangement inequality</a> with $a^2,b^2,c^2$ and $a,b,c$ we get
$$aa^2+bb^2+cc^2\ge a^2b +b^2c+c^2a$$</p>
|
4,014,022 | <p>In example <span class="math-container">$1.5$</span> of <em>Cracking the GRE Subject Test</em>, the authors make the following calculation in one step with no additional commentary:</p>
<blockquote>
<p>we interchange <span class="math-container">$x$</span> and <span class="math-container">$y$</span> and solve for <s... | 19aksh | 668,124 | <p>Using the quadratic formula for <span class="math-container">$ay^2 + by +c = 0$</span></p>
<p><span class="math-container">$$y = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$</span></p>
<p>Here <span class="math-container">$a = x$</span>, <span class="math-container">$b = 1$</span>, <span class="math-container">$c = -1$</span></... |
73,112 | <blockquote>
<p>Find <span class="math-container">$e^{At}$</span>, where <span class="math-container">$$A = \begin{bmatrix} 1 & -1 & 1 & 0\\ 1 & 1 & 0 & 1\\ 0 & 0 & 1 & -1\\ 0 & 0 & 1 & 1\\ \end{bmatrix}$$</span></p>
</blockquote>
<hr />
<p>So, let me just find <spa... | Sasha | 11,069 | <p>Consider $M(t) = \exp(t A)$, and as you noticed, it has block-diagonal form
$$
M(t) = \left(\begin{array}{cc} \exp(t B) & n(t) \\ 0_{2 \times 2} & \exp(t B) \end{array} \right).
$$
Notice that $M^\prime(t) = A \cdot M(t)$, and this results in a the following differential equation for $n(t)$ matrix:
$$
... |
73,112 | <blockquote>
<p>Find <span class="math-container">$e^{At}$</span>, where <span class="math-container">$$A = \begin{bmatrix} 1 & -1 & 1 & 0\\ 1 & 1 & 0 & 1\\ 0 & 0 & 1 & -1\\ 0 & 0 & 1 & 1\\ \end{bmatrix}$$</span></p>
</blockquote>
<hr />
<p>So, let me just find <spa... | Rodrigo de Azevedo | 339,790 | <p>Using induction, for <span class="math-container">$k \geq 1$</span>,</p>
<p><span class="math-container">$${\rm A}^k = \begin{bmatrix} {\rm B}^k & k \, {\rm B}^{k-1}\\ {\rm O} & {\rm B}^k\end{bmatrix}$$</span></p>
<p>and, thus,</p>
<p><span class="math-container">$$\exp\left( t \, {\rm A} \right) = \begin{bm... |
14,527 | <p>The <a href="https://en.wikipedia.org/wiki/Robinson%E2%80%93Schensted%E2%80%93Knuth_algorithm" rel="nofollow noreferrer">Robinson-Schensted correspondence</a> is a bijection between elements of the symmetric group and ordered pairs of standard tableaux of the same shape.</p>
<p>Some simple operations on tableaux cor... | Igor Pak | 4,040 | <p>When you conjugate diagrams and apply RSK or other Young tableau bijections, the answer is typically bad, with some rare exceptions. The right way to think of RSK is to think of row length being continuous while columns still integer (see e.g. my "Geometric proof of the hook-length formula" paper and refs therein).... |
434,685 | <p>Suppose that $X^*$ is the dual space of a normed space $X$. If we renorm the space $X^*$ with a new norm equivalent to the first one, is this new normed space the dual of $X$ as well?
(I think it suffices to prove that a functional $f$ is continuous with a norm1 if and only if it is continuous with norm2 where norm1... | fuglede | 10,816 | <p>For the last statement, suppose $f$ is continuous with respect to $\lVert \cdot \rVert_1$, and that $k,K > 0$ are so that $k \lVert v \rVert_1 < \lVert v \rVert_2 < K\lVert v \rVert_1$ for all $v \in X$. Let $\varepsilon > 0$ be given, and choose $\delta_0$ so that $\lvert f(v) - f(w) \rvert < \vareps... |
1,289,112 | <p>I am trying to prove $\exists n_0 > 0: \forall n > n_0: \log n < \sqrt n$. My attempt uses the series representation of the exponential function, but it does not seem to accomplish the proof:</p>
<p>$$
\log n < \sqrt n \\
\Leftrightarrow n < e^{\sqrt n} = \sum_{k=0}^\infty \frac{(\sqrt n)^k}{k!} = 1 ... | André Nicolas | 6,312 | <p>Hint: The series representation for the exponential will work. Use the fact that if $x$ is positive then
$$e^x\gt \frac{x^3}{3!}.$$</p>
|
1,289,112 | <p>I am trying to prove $\exists n_0 > 0: \forall n > n_0: \log n < \sqrt n$. My attempt uses the series representation of the exponential function, but it does not seem to accomplish the proof:</p>
<p>$$
\log n < \sqrt n \\
\Leftrightarrow n < e^{\sqrt n} = \sum_{k=0}^\infty \frac{(\sqrt n)^k}{k!} = 1 ... | zhw. | 228,045 | <p>In fact, $\ln x < \sqrt x$ for all $x>0.$ Let $f(x) = \sqrt x - \ln x.$ Then $f'(x) = 1/(2\sqrt x) - 1/x.$ This tells us $f'<0$ on $(0,4), f'(4)=0,$ and $f'>0$ on $(4,\infty).$ Thus $f(4)$ is the absolute minimum of $f$ on $[1,\infty).$ It is quite simple to check that $f(4) > 0,$ hence $f> 0$ on $... |
689,921 | <p>There are two teams.Two games were played.There are three possible outcomes which are win, lose or draw. how many permutations are there?</p>
| amWhy | 9,003 | <p>Teams A, B</p>
<p>In any game, if team A wins, team B loses. If team A loses, team B wins...etc. So the outcome for any one team determines the outcome for the other.</p>
<p>So we need only consider the outcomes for team A, given team A plays two games against B.</p>
<p>$1\quad2$ : Game<br>
$W \;\;W$<br>
$W \;\;... |
153,999 | <p>How to solve: $|\sqrt{x-1}-2| + |\sqrt{x-1}-3|=1$.</p>
<p>I would like to know how to solve an absolute value equation when there is a square root sign inside.</p>
| Cameron Buie | 28,900 | <p>I recommend the following approach, in this case. Let $y=\sqrt{x-1}$. Now you need only solve the equation $|y-2|+|y-3|=1$--which should have a closed interval's worth of solutions--then given any of the solutions, say $\alpha$, solve the equation $\sqrt{x-1}=\alpha$. In the end, you will obtain a closed interval's ... |
10,677 | <p>I have a question about a transformation of a matrix. Lets say we have the following matrix</p>
<p>$
M =
\left[ {\begin{array}{cc}
4 & 3 \\
4 & 3 \\
\end{array} } \right]
$
</p>
<p>Then I want to transform all numbers in the matrix with the following function:</p>
<p>$f(x) = (x - 4)^2 + 2x -4$ ... | Mitchell Watt | 3,564 | <p>Could use the "Hadamard Product"... I don't know whether there is a consistently accepted notation for such a product, however a few examples <a href="https://planetmath.org/HadamardProduct" rel="nofollow noreferrer">here</a> and on Wikipedia seem to use an open circle.</p>
<p>In that sense, your expressio... |
10,677 | <p>I have a question about a transformation of a matrix. Lets say we have the following matrix</p>
<p>$
M =
\left[ {\begin{array}{cc}
4 & 3 \\
4 & 3 \\
\end{array} } \right]
$
</p>
<p>Then I want to transform all numbers in the matrix with the following function:</p>
<p>$f(x) = (x - 4)^2 + 2x -4$ ... | Agustí Roig | 664 | <p>Maybe I don't understand your question, but: what's wrong with this?</p>
<p>$$
\begin{pmatrix}
f(4) & f(3) \\\
f(4) & f(3)
\end{pmatrix}
$$
</p>
<p>More generally, if you have a matrix $(a^i_j)$, you could write the resulting matrix as $(f(a^i_j))$.</p>
|
1,552,747 | <p>While we were introduced to integration, we were told about some basic concepts that, as we were told, could not be proved based on our level of sophistication. They are as follows:</p>
<ol>
<li>$$\int_a^b \! f(x) \, \mathrm{d}x=\phi(b)-\phi(a),$$ where $\phi$ is a primitive of $f$ in $[a,b]$</li>
<li>$$\int_b^a \!... | Siddharth Joshi | 288,487 | <p>You <strong>cannot associate any</strong> physical interpretation to $\int_b^a f(x) \ dx$</p>
<p>You're right, the sum would be the same irrespective of whether you sum it forward or backward. But $\int_b^a f(x) dx$ doesn't connote taking the sum in the opposite manner. If it were then $\int_b^a f(x) dx$ would have... |
1,552,747 | <p>While we were introduced to integration, we were told about some basic concepts that, as we were told, could not be proved based on our level of sophistication. They are as follows:</p>
<ol>
<li>$$\int_a^b \! f(x) \, \mathrm{d}x=\phi(b)-\phi(a),$$ where $\phi$ is a primitive of $f$ in $[a,b]$</li>
<li>$$\int_b^a \!... | Dorebell | 59,428 | <p>The first question is the fundamental theorem of calculus: it says that if there is a differentiable function $F$ with $\frac{dF}{dx} = f$, and $f$ is Riemann integrable on $[a,b]$, then $\int_{a}^b f(x) dx = F(b) - F(a)$. This can be proved from the definitions relatively easily, but not obviously. It will be cover... |
1,037,530 | <p>what is the basic difference between a mapping and a function? many say they are same but the opposite views are also seen. is mapping a restricted version of a function?</p>
| Cameron Williams | 22,551 | <p>I think a map and function are usually synonymous amongst mathematicians. Often you will see "let $f$ be a map" or "let $f$ be a function" used interchangeably.</p>
|
4,363,384 | <p>Let <span class="math-container">$A_i$</span>, <span class="math-container">$i \in I$</span> subsets of a space X given topology <span class="math-container">$\tau$</span>. Show that <span class="math-container">$\overline{\bigcup_i A_i }\subset \bigcup_i\overline{ A_i }$</span> does not hold necessarily :</p>
<p>... | José Carlos Santos | 446,262 | <p>Your example is fine. The equality <span class="math-container">$\overline{A\cup B}=\overline A\cup\overline B$</span> holds for pairs of subsets of <span class="math-container">$X$</span> and, more generally, for finite sets of subsets of <span class="math-container">$X$</span>. But you have provided an example wit... |
1,675,411 | <p>So far I have this:</p>
<p>First consider $n = 5$. In this case $(5)^2 < 2^5$, or $25 < 32$. So the inequality holds for $n = 5$.</p>
<p>Next, suppose that $n^2 < 2^n$ and $n \geq 5$. Now I have to prove that $(n+1)^2 < 2^{(n+1)}$.</p>
<p>So I started with $(n+1)^2 = n^2 + 2n + 1$. Because $n^2 < 2... | Adriano | 76,987 | <p>There are, of course, many ways to force $2^{n + 1}$ to show up on the RHS. I like to bound the lower order terms as multiples of the leading term by repeatedly applying the given inequality $n \geq 5$ so that I can apply the inductive hypothesis in one shot. Indeed, observe that:
\begin{align*}
(n + 1)^2
&= n^2... |
393,430 | <p><strong>Question:</strong> Is there a simple method for calculating the Fourier transform of a holomorphic complex function <span class="math-container">${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$</span>?</p>
<p>In order for my question to be well-posed I define a holomorphic function <span class="math-conta... | Sam Nead | 1,650 | <p>Suppose that <span class="math-container">$M$</span> is finite volume hyperbolic. Then the mapping class group is finite and there are algorithms to build its multiplication table. On the other hand, there is definitely no overall pattern to these groups - they are “just” the finite quotients of torsion free lattic... |
4,126,527 | <p>Suppose <span class="math-container">$w_1$</span>, <span class="math-container">$w_2$</span>, ..., <span class="math-container">$w_{n-1}$</span> are the complex roots not equal to <span class="math-container">$1$</span> of <span class="math-container">$z^n-1=0$</span>, where <span class="math-container">$n$</span> i... | FeedbackLooper | 423,711 | <p>The idea is that any root of unity complies <span class="math-container">$w\bar{w}=1$</span> and the main manipulation trick you need is:
<span class="math-container">$$
(w+\bar{w})^2 = w^2 + 2w\bar{w} + \bar{w}^2 = 2 + w^2+\bar{w}^2
$$</span>
so that <span class="math-container">$2-w^2-\bar{w}^2 = 2 - (w^2+\bar{w}^... |
1,928,892 | <p>I'm solving a graduate entrance examination problem.
We are required to establish the inequality using the following result:</p>
<p>for $x,y > 0$, $\frac{x}{y} + \frac{y}{x} > 2$ (1), which is easy to prove as it is equivalent to $(x - y)^2 > 0$.</p>
<p>But when it comes to an inequality combining $x, y,... | Bart Michels | 43,288 | <p>Using the <a href="https://en.wikipedia.org/wiki/Rearrangement_inequality" rel="nofollow">Rearrangement Inequality</a>:</p>
<p>$x,y,z$ and $\frac1{y+z},\frac1{z+x},\frac1{x+y}$ have the same ordering, so
$$(x,y,z)\left(\frac1{y+z},\frac1{z+x},\frac1{x+y}\right)\geq(x,y,z)\left(\frac1{x+y},\frac1{y+z},\frac1{z+x}\ri... |
2,487,044 | <p>Why are integral elements defined in terms of monic polynomials? Why do we wish to split the cases between non monic polynomials and monic polynomials? I.e, algebraic elements and integral elements. Motivation? </p>
| Ted | 15,012 | <p>Any answer might be a bit arguing from hindsight. One could say that the reason algebraic integers are important is because a lot of theory has been successfully developed using algebraic integers as a concept. </p>
<p>But one way to look at it is that the ring of integers of a number field is a ring which is a <em... |
1,435,718 | <blockquote>
<p>Find all pairs of prime numbers $p , q$ for which:
$$p^2 \mid q^3 + 1 \tag{A}$$
and
$$q^2 \mid p^6 − 1 \tag{B}$$</p>
</blockquote>
<p>The question is from the Bulgaria National Olympiad 2014.</p>
<p><em>I'm looking for any solution I may have missed, and generally any alternative method that m... | Batominovski | 72,152 | <p>Here is a shorter solution. If $p=2$, then $q^2\mid p^6-1=63$ implies that $q=3$. If $p=3$, then $q^2\mid 3^6-1=728$ leads to $q=2$. If $q=2$, then $p^2\mid q^3+1=9$ means $p=3$. If $q=3$, then $p^2\mid q^3+1=28$ gives $p=2$. That is, solutions with $p\leq 3$ or $q\leq 3$ are $(p,q)=(2,3)$ and $(p,q)=(3,2)$. F... |
76,474 | <p>Use Newton's method to apporzimate the indicated root of the equation correct to six decimal places. The negative root of $e^x = 4-x^2$</p>
<p>I do not know what a negative root is nor do I really know what I am supposed to do. I am guessing raise everything by loge.</p>
| ROSE | 71,470 | <p>e^x=4-x^2<br>
<strong>Subtract</strong> e^x<br>
4-x^2-e^x=0<br>
<strong>Take Derivative</strong><br>
-2x-e^x=0</p>
<p>plug this formula into the graphing calculator:</p>
<p>x-((4-x^2-e^x)/(-2x-e^x))<br><br> (NOTE - this formula can be used with any Newton's method problem when needing to find roots: x-(original fx... |
390,662 | <p>Let <span class="math-container">$X$</span> be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli spaces.</p>
<p>I've tried to read the original articles by Hitchin (1987) or Sim... | Sean Lawton | 12,218 | <p>I personally like the notes by Eugene Xia: <a href="https://arxiv.org/pdf/1404.5025.pdf" rel="noreferrer">Abelian and Non-Abelian Cohomology</a> to build intuition.</p>
<p>But for a definitive source, I would read Simpson: <a href="http://www.numdam.org/article/PMIHES_1994__79__47_0.pdf" rel="noreferrer">Moduli of r... |
27,865 | <p>Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a harmonic (or loop-based) decomposition.</p>
<p>Are there deep connections between these two transforms? <a href="http... | asmaier | 27,609 | <p>Laplace transform and Fourier transform are both special cases of the <a href="http://en.wikipedia.org/wiki/Linear_canonical_transformation" rel="nofollow">http://en.wikipedia.org/wiki/Linear_canonical_transformation</a>.</p>
|
12,047 | <p>How to prove the following trigonometric identities ?</p>
<p>1) If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
</p>
<p>I tried to prove it by using the the formula for $\tan(\alpha + \beta)$ but ain't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?</p>
... | Quixotic | 2,109 | <p>For (2)</p>
<p>$$\sec \theta + \tan \theta = \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta}= \frac{1 + \sin \theta}{\cos \theta}$$ $$ = \frac{(1 + \sin \theta)\cdot (1 - \sin \theta)}{\cos \theta \cdot(1 - \sin \theta)}\text{ [Multiplying both sides by } (1 - \sin \theta)\text{]} $$
$$ = \frac{\cos^2 \the... |
838,854 | <p>In page 4 of the following <a href="http://web.mat.bham.ac.uk/D.Kuehn/RamseyGreg.pdf" rel="nofollow">http://web.mat.bham.ac.uk/D.Kuehn/RamseyGreg.pdf</a> the text says</p>
<blockquote>
<p>In any graph the number of vertices with odd degree must be even. For this reason there cannot exist a red 5-regular subgraph ... | Vladimir | 154,757 | <p>Hint: (1) $Ax\cdot Ax=|Ax|^2$;</p>
<p>(2) One has
$$
Ax\cdot Ay=\frac14 (A(x+y)\cdot A(x+y)-A(x-y)\cdot A(x-y)).
$$</p>
|
838,854 | <p>In page 4 of the following <a href="http://web.mat.bham.ac.uk/D.Kuehn/RamseyGreg.pdf" rel="nofollow">http://web.mat.bham.ac.uk/D.Kuehn/RamseyGreg.pdf</a> the text says</p>
<blockquote>
<p>In any graph the number of vertices with odd degree must be even. For this reason there cannot exist a red 5-regular subgraph ... | goblin GONE | 42,339 | <p>Let $A$ denote an $n \times n$ real matrix.</p>
<blockquote>
<p><strong>Definitions.</strong> Call $A$ <em>dot-product preserving</em> iff $x \cdot y = Ax \cdot Ay$ for all relevant vectors $x$ and $y$. On the other hand, let
us call $A$ <em>length-preserving</em> iff $|x| = |Ax|$ for all relevant
vectors $x$... |
2,303,339 | <p>How does this proof work?</p>
<p><strong>Theorem.</strong>$\quad$Let $G$ be a group. Then $G$ has a unique identity. </p>
<p><strong>Proof.</strong>$\quad$Assume that $e$ and $f$ are two identities in $G$. Since $e$ is
an identity, $ef=f$; and since $f$ is an identity, $ef=e$. Thus $e=ef=f$. </p>
<p>I think ne... | user1551 | 1,551 | <p>Here is an answer that is light on computation but conceptually somewhat obscure.</p>
<p>If we perturb $A$ by some $\delta I_2$, both sides of the equality change by the same amount. The similar holds if we perturb $B$ by a scalar matrix. So, we may assume that both $A$ and $B$ are traceless. In this case, the equa... |
578,086 | <p>Hey I've been having problem finding the limit of the sequence below</p>
<p>$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}$</p>
<p>Thanks!</p>
| Claude Leibovici | 82,404 | <p>Use the simplest approximation for (n!) that is to say Log[n!] = n Log[n] - n<br>
<a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow">http://en.wikipedia.org/wiki/Stirling%27s_approximation</a>.<br>
Taking logarithms, you will easily show that the limit is 1/e as shown by Dedalus.</p>
|
2,963,560 | <blockquote>
<p>Given <span class="math-container">$a,b,c\in \Bbb Z$</span>, pairwise distinct, and <span class="math-container">$n\in \Bbb N\setminus\{0\}$</span> prove that
<span class="math-container">$$S(n)=\frac{a^n}{(a-b)(a-c)}+\frac{b^n}{(b-a)(b-c)}+\frac{c^n}{(c-a)(c-b)}\in \Bbb Z.$$</span></p>
<p>Sou... | Hw Chu | 507,264 | <p>Similar to your <span class="math-container">$S(1)$</span> case, write </p>
<p><span class="math-container">$$
S(n) = \frac{a^n(b-c) - b^n(a-c) + c^n(a-b)}{(a-b)(a-c)(b-c)}.
$$</span></p>
<p>Now look at the numerator.</p>
<p><span class="math-container">$$
a^n(b-c) - b^n(a-c) + c^n(a-b) = ab(a^{n-1}-b^{n-1}) - c(... |
1,179,959 | <p>taking $v$ as $x$ axis and $u$ as $y$ axis I would like to know whether ${1\over v}+{1\over u}={1\over f}$ has graph of the form $xy=c^2$?</p>
| Narasimham | 95,860 | <p>Yes, but note that u=v=0 at the origin and that there is a displacement of x- y- axes at origin (f,f).</p>
<p>In fact KF Gauss had got rid of this reciprocal stuff to neatly introduce the displaced axes $ x\cdot y $ form useful in Optics:</p>
<p>$$ \frac{1}{u} + \frac{1}{v}=\frac{1}{f} $$
$$ u\, v = ( u+v) f $$
$... |
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
3,556,334 | <p>Given a rectangle with dimensions <span class="math-container">$10$</span>cm and <span class="math-container">$6$</span>cm, show that for every <span class="math-container">$3$</span> points in the interior of the rectangle, the area of the triangle is less than <span class="math-container">$30$</span> cm<span clas... | Aniket Gupta | 751,375 | <p>The area of the rectangle is <span class="math-container">$60 cm^2$</span>. Consider a triangle with the base as one of the sides of the rectangle.The the 3rd vertex be located on the opposite side. Clearly the area of this triangle is <span class="math-container">$30cm^2$</span>. But since all vertices of the trian... |
479,671 | <p>Let $p$ and $q$ be two distinct primes. Prove that $$p^{q-1} + q^{p-1} =1 \mod pq$$ I try to used Fermat little theorem and I obtain the congruence $p^q +q^p=0 \mod pq$. From this I don know how to relate to the congruence on the question.</p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> It is congruent to $1$ modulo $p$, and also modulo $q$. Modulo $p$, the first part is congruent to $0$, and the scond part is congruent to $1$, by Fermat's Theorem.</p>
|
63,534 | <p>I found in an article <a href="http://dx.doi.org/10.1103/PhysRev.105.776" rel="nofollow">"Imperfect Bose Gas with Hard-Sphere Interaction"</a>, <em>Phys. Rev.</em> 105, 776–784 (1957) the following integral, but I don't know how to solve it. Any hints?</p>
<p>$$\int_0^\infty {\int_0^\infty {\mathrm dp\mathrm dq\fr... | Did | 6,179 | <p>Call $I(u,v,w)$ the integral to compute and note that this can be defined only when $4vw>u^2$. Using the definition of $\sinh$ and the parity of the function to be integrated one sees that
$$
4I(u,v,w)=\int_{-\infty}^\infty\int_{-\infty}^\infty{\mathrm dp\mathrm dq}\,\frac{e^{upq}}{q^2 - p^2}pq\, e^{-vq^2 - wp^2... |
375,025 | <p>How do I solve this simultaneous equation? I know it has multiple solutions, but would anyone be able to show me the exact steps in working these out for future reference? </p>
<p>$$
\begin{cases}
2x +y = 4\\
-6x - 3y = -12
\end{cases}
$$</p>
<p>Thank you...</p>
| lab bhattacharjee | 33,337 | <p>As $2x+y=4-6x-3y=-12$</p>
<p>So, $2x+y=-12,y=-12-2x$</p>
<p>Putting $y=-12-2x$ in $4-6x-3y=-12$</p>
<p>$4-6x-3\{-12-2x\}=-12$</p>
<p>or $52=12$ which is impossible</p>
<p>So, the simultaneous equation will not have any solution</p>
<p><strong>EDIT</strong> :
Due to the change in the question to $2x+y=4,-6x-3y=... |
1,784,469 | <p>I roll a biased dice 9000 times. the probability of seeing 1,2,3,4,5,6 are $\frac{1}{3}, \frac{1}{12}, \frac{1}{12}, \frac{1}{6}, \frac{1}{6}, \frac{1}{6}$ respectively. I need to use the Central limit theorem to estimate the probability that the total number of 1s that I see is within [2970,3040].</p>
<p>So far, I... | Steven | 315,088 | <p>Define $X_i$ as the number of 1's on the i's toss. $X_i=1$ if the $i^{th}$ toss is 1, and $X_i=0$ if the $i^{th}$ toss is not 1.</p>
<p>By central limit theorem, we can assume $X_i's$ have normal distribution with $E(X)=9000\times\frac{1}{3}=3000$.</p>
<p>$Var(X)=9000\times(\frac{1}{3}-(\frac{1}{3})^2)=2000$</p>
... |
1,784,469 | <p>I roll a biased dice 9000 times. the probability of seeing 1,2,3,4,5,6 are $\frac{1}{3}, \frac{1}{12}, \frac{1}{12}, \frac{1}{6}, \frac{1}{6}, \frac{1}{6}$ respectively. I need to use the Central limit theorem to estimate the probability that the total number of 1s that I see is within [2970,3040].</p>
<p>So far, I... | callculus42 | 144,421 | <p>The probability of seeing a one in a single roll is $\frac{1}{3}$. The probability of $\texttt{not}$ seeing a one in a single roll is $1-\frac{1}{3}=\frac{2}{3}$.</p>
<p>Each roll i is <a href="https://en.wikipedia.org/wiki/Bernoulli_distribution" rel="nofollow">bernoulli distributed</a>. Let´s denote this random v... |
805,902 | <p>I'm struggling to figure out what I'm exactly required to do. The problem states</p>
<p>"Compute which lines through the point $(1, 0)$ that are tangent to the parabola defined by $y = x^2$."</p>
<p>I believe it's a simple question however I've been going around this for quite a bit. </p>
<p>I'll appreciate any k... | Michael Hardy | 11,667 | <p>Certainly the $x$-axis is one such tangent line, since it passes through the point $(1,0)$ and is tangent to the parabola at its vertex. That much you get just from looking at the picture.</p>
<p>Classical geometry says that if you draw lines parallel to the axis of a parabola through the point of intersection of ... |
2,781,702 | <p>Honestly, I have no idea if I put the correct tag on this question, and I don't even know where to begin to solve an equation like this:
$$
f(d,n)=\sum_{i=1}^n\binom{d}{i}.
$$</p>
<p>Could someone explain what the "d" over the "i" inside the parenthesis means? I'm attempting to solve for when d is equal to 20, n is... | Ian | 83,396 | <p>${d \choose i}$ is read as "d choose i" and is called a binomial coefficient. It is given by the formula $\frac{d!}{i!(d-i)!}$.</p>
|
4,619,164 | <p>So one way to solve this ordinary differential equation is by computing the integral of both sides of <span class="math-container">$$\frac{y'}{y} = 1$$</span>
However, I did it another way and I think I made a mistake, but where is it? What I did is:
<span class="math-container">$$y' = y $$</span>
<span class="math-... | Átila Correia | 953,679 | <p>Just out of curiosity, you can also apply the integrating factor method:
<span class="math-container">\begin{align*}
y' = y & \Longleftrightarrow y' - y = 0\\\\
& \Longleftrightarrow e^{-x}y' - e^{-x}y = 0\\\\
& \Longleftrightarrow (e^{-x}y)' = 0\\\\
& \Longleftrightarrow e^{-x}y = k\\\\
& \Longl... |
1,148,674 | <p>I have always asked myself why this happens. </p>
<p>If $x = 4$, then $\sqrt{x} = 2$, but if I search for the $\sqrt{4}$, I get $2$ & $-2$.</p>
| layman | 131,740 | <p>The problem you are having is in how to interpret square roots. You have probably been taught that when we say $\sqrt{4}$, we are thinking about the "number that when you square it, you get $4$". If you are thinking about it this way, then of course $\sqrt{4}$ could be $2$ or $-2$ since when you square each of tho... |
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