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 |
|---|---|---|---|---|
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | Paul Siegel | 4,362 | <p>I think there are two ways to approach a first course on manifolds: one can focus on either their geometry or their topology.</p>
<p>If you want to focus on geometry, then I think Anton Petrunin's suggestion is the end of the story. I'm a fourth year graduate student, and practically every time I find myself confu... |
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | Charlie Frohman | 4,304 | <p>The problem will be that the students do not have a firm grasp of multivariable calculus. </p>
<p>You should probably start with a rigorous review of multivariable calculus including the definition of the differentiable, C^1 implies differentiable on open sets, mixed partials are equal, inverse function theorem, l... |
294,519 | <p>The problem I am working on is:</p>
<p>Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.</p>
<p>a) $∀x(C(x)→F(x))$ </p>
<p>b)$∀x(C(x)∧F(x))$</p>
<p>c) $∃x(C(x)→F(x))$ </p>
<p>d)$∃x(C(x)∧F(x))$</p>
<h2>-----------------------... | Andreas Blass | 48,510 | <p>Your answers to parts a), b), and d) are OK; in part c) you've reversed the roles of "comedian" and "funny". The book's answers are correct and, in a couple of cases, closer to how people would ordinarily express these statements. One can make other equivalent statements that sound (to me) even more like ordinary ... |
1,877,558 | <p>For instance, let $(\mathbb{R}, \mathfrak{T})$ be $\mathbb{R}$ with the usual topology. </p>
<p>Why is that $\mathfrak{T} \times \mathfrak{T}$ is a basis on $\mathbb{R} \times \mathbb{R}$ instead of topology?</p>
<p>It seems that people just take $\mathfrak{T} \times \mathfrak{T}$ as a basis by definition. There m... | Arkady | 23,522 | <p>Hint: If I take the union of two elements of $\mathcal I\times\mathcal I$ (boxes), it is very unlikely that their union will be a box. So $I\times I$ will not be topology).</p>
<p>Since you want an explicit construction, take the union of the sets $A=(0,1)\times(0,1)$ and $B=(1,2)\times(1,2)$. This is not of the f... |
4,374,391 | <blockquote>
<p>Find prime number <span class="math-container">$p$</span> such that <span class="math-container">$19p+1$</span> is a square number.</p>
</blockquote>
<p>Now, I have found out, what I think is the correct answer using this method.<br />
Square numbers can end with - <span class="math-container">$1, 4, 9,... | angryavian | 43,949 | <ul>
<li>What is the cardinality of <span class="math-container">$\{0,1\}^n$</span> (in terms of <span class="math-container">$n$</span>)?</li>
<li>Given that the cardinality of <span class="math-container">$2^A$</span> is <span class="math-container">$2^{|A|}$</span>, does this suggest some necessary condition on <spa... |
947,358 | <p>Okay $g(x)= \sqrt{x^2-9}$</p>
<p>thus, $x^2 -9 \ge 0$</p>
<p>equals $x \ge +3$ and $x \ge -3$</p>
<p>thus the domains should be $[3,+\infty) \cup [-3,\infty)$ how come the answer key in my book is stating $(−\infty, −3] \cup[3,\infty)$. </p>
| taninamdar | 66,212 | <p>$$\begin{align}\sqrt{x^2 - 9} \text{ exists} &\implies x^2-9 \ge 0\\&\implies (x+3)(x-3) \ge 0 \\&\implies (x-3), (x+3) \text{ both are non-negative OR } (x-3),(x+3) \text{both are non-positive} \end{align}$$</p>
<p><strong>case 1</strong>
Now, $(x-3)(x+3)$ both are non-negative when $x-3 \ge 0$ and $x+... |
605,155 | <p>$\newcommand{\ker}{\operatorname{ker}}$</p>
<p>Proof that: $\ker AB\subseteq\ker A+\ker B$</p>
<p>my solution:</p>
<p>$x\in \ker AB\to ABx=0\to \begin{cases} Ax=0\to x\in \ker A\\Bx=0\to x\in \ker B\end{cases}$</p>
<p>$\to x\in \ker A+\ker B\to \ker AB\subseteq \ker A+\ker B$</p>
<p>Question: Do it right? if fa... | Ben Grossmann | 81,360 | <p>Note that we can have $x \in \ker(AB)$ with $x \not \in \ker(A)$ and $x \not \in \ker B$. For example, take
$$
A = \pmatrix{1&0\\0&0};\quad B=\pmatrix{0&1\\1&0}; \quad x=\pmatrix{1\\0}
$$
Note that $Ax \neq 0, Bx \neq 0,$ but $ABx=0$. Also, there are no vectors in the kernels of $A$ and $B$ whose s... |
1,910,983 | <p>What conditions are equivalent to <strong>singularity</strong> of matrix $A\in \mathbb{R}^{n,n}$.<br>
<strong>a.</strong> $\dim(ker A) \ge 0$<br>
<strong>b.</strong> There is exist vector $b$ such that $Ax=b$ is contradictory.<br>
<strong>c.</strong> $rank(A^T) < n$ </p>
<p><strong>a.</strong> is true for each ... | H. H. Rugh | 355,946 | <p>The sequence verifies $a_{10+k} = a_{10 + (k \; {\rm mod}\; 4)}$ for $k\geq 0$ (and there are no other relations). So look for $\ell=10+k$, $k\geq 0$ so that
$a_{10+k}=a_{20+2k}=a_{10+(10+2k)}$. And this is equivalent to $k\geq 0$ and
$$ k \equiv 10+2 k \ {\rm mod} \ 4 $$
or $k \equiv 2 \ {\rm mod} \ 4$.</p>
|
178,028 | <p>I am given $G = \{x + y \sqrt7 \mid x^2 - 7y^2 = 1; x,y \in \mathbb Q\}$ and the task is to determine the nature of $(G, \cdot)$, where $\cdot$ is multiplication. I'm having trouble finding the inverse element (I have found the neutral and proven the associative rule.</p>
| user29999 | 29,999 | <p>\begin{eqnarray}
\dfrac{1}{x+y\sqrt{7}} &=& \dfrac{x-y\sqrt{7}}{x^2-7y^2}\\
& =& x-y\sqrt{7}
\end{eqnarray}</p>
|
855,570 | <p>I am having trouble with what seems like it should be a simple problem. I am trying to find intersections of connections between multiple people but I want to include any intersection of connections found between any two of the sets.</p>
<p>For example, Let</p>
<p>$A$ = {January, February, March, April, May, Augus... | Ross Millikan | 1,827 | <p>It looks like you want any month that is in more than one subset. You can go through all the months and count how many subsets each one is in. If it is greater than one, put it in your final list.</p>
|
812,778 | <p>Prove that $(4/5)^{\frac{4}{5}}$ is irrational.</p>
<p><strong>My proof so far:</strong></p>
<p>Suppose for contradiction that $(4/5)^{\frac{4}{5}}$ is rational.</p>
<p>Then $(4/5)^{\frac{4}{5}}$=$\dfrac{p}{q}$, where $p$,$q$ are integers.</p>
<p>Then $\dfrac{4^4}{5^4}=\dfrac{p^5}{q^5}$</p>
<p>$\therefore$ $4^4... | Aryan | 153,656 | <p>A slightly modified approach.</p>
<p>$(\frac{4}{5})^\frac{4}{5} = (\frac{4}{5})^{1-\frac{1}{5}} = (\frac{4}{5})^1 \times(\frac{5}{4})^\frac{1}{5}$</p>
<p>Since the first factor is rational we need only show that the latter term $(\frac{5}{4})^\frac{1}{5}$ is irrational.</p>
<p>Let us assume that the latter is ra... |
1,304,529 | <p>I have come across these while studying the limsup & liminf of sequence of subset of a set. In order to understand that, I have to understand what least upper bound & greatest lower bound of a sequence of subset mean. I would be grateful if anyone helps me comprehend this concept intuitively as I am new &... | Mauro ALLEGRANZA | 108,274 | <p>Ref also to <a href="https://math.stackexchange.com/questions/1305004/what-is-meant-by-ordering-of-set-by-inclusion">your subsequent question</a>.</p>
<p>A <a href="http://en.wikipedia.org/wiki/Partially_ordered_set" rel="nofollow noreferrer">partially ordered set</a> is a very "simple" mathematical structure :</p>... |
2,565,802 | <p>Calculate the volume of the region bounded by $z=0, z=1,$, and $(z+1)\sqrt{x^2+y^2}=1$</p>
<p>The integral is $\int_{B}z\text{ dV}$</p>
<p>The area is like the thing between the top two green places. The first place is $z=1$, second is $z=0$</p>
<p>Clearly we have $0\leq z\leq 1$, but I'm not sure what to bound n... | N. F. Taussig | 173,070 | <p>A circle can be formed by joining the ends of a row together. We solve the problem for a row, then remove those solutions in which two of the selected people are at the ends of the row.</p>
<p>We will arrange $96$ blue balls and $4$ green balls in a row so that the green balls are separated. Line up $96$ blue bal... |
2,231,388 | <blockquote>
<p>Consider a ring map $B \rightarrow A$. Consider the map $f:A \otimes_{B}A \rightarrow A$, where $x \otimes y$ goes to $xy$. Let $I$ be the kernel of $f$. Why is it true that $I/I^2$ is isomorphic to $I \otimes_{A \otimes_{B}A} A$?</p>
</blockquote>
<p>This is what I've been able to prove till now:</p... | Andrew | 154,986 | <p>This can be made slightly more general: Let $R$ and $I$ be as you say, and let $M$ be any $R$-module. There is an isomorphism
$$
M\otimes_R R/I \cong M/IM.
$$
Here's how to see it: start with the exact sequence
$$
0 \to I \to R \overset{f}{\to} R/I \to 0
$$
(note that $f$ is the same as your $f$).
Then tensor wit... |
752,517 | <p>From Wikipedia</p>
<blockquote>
<p>...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different unless their equality follows from the group axioms (e.g. $st = suu^{−1}t$, but $s ≠ t$ for $s,t,u \in... | Community | -1 | <p>The unit circle is a subspace of the plane; thus if you already know the plane is Hausdorff, that carries over to the circle too.</p>
<p>Showing that</p>
<p>$$ \mathrm{R} \to (-\pi, \pi) \to \text{everything but $(-1,0)$} $$</p>
<p>is surjective and $f(\infty) = (-1,0)$ would be enough to show $f$ is surjective.<... |
412,642 | <p>Let $E=\mathcal{C}[0,1]$. How to prove that if $f_n\rightarrow f$ with the norm $\displaystyle{\|\cdot\|_\infty=\sup_{t\in[0, 1]}f(t)}$ then $f_n\rightarrow f$ with the norm $\displaystyle{\|\cdot\|_p=\left(\int_{0}^{1}|f(t)|^p\,dt\right)^{1/p}}$. </p>
<p>Give an example showing that the converse is not true</p>
| Mhenni Benghorbal | 35,472 | <p>Here is a start for the first part
$$ \displaystyle{\|f_n-f\|^p_p= \int_{0}^{1}|f_n(t)-f(t)|^p\,dt }\leq \int_{0}^{1}\sup|f_n(t)-f(t)|^p\,dt \leq \dots. $$</p>
|
1,538,496 | <p>I came across this riddle during a job interview and thought it was worth sharing with the community as I thought it was clever:</p>
<blockquote>
<p>Suppose you are sitting at a perfectly round table with an adversary about to play a game. Next to each of you is an infinitely large bag of pennies. The goal of the... | JMoravitz | 179,297 | <p>Yes, I've seen this one before. Assuming exactly one penny is allowed to be placed per turn:</p>
<blockquote class="spoiler">
<p> Go first and place a penny in the dead center of the table. From then on, any move your opponent makes, place a penny in the mirror opposite location (i.e. rotated 180 degrees). It ... |
1,335,842 | <p>The smallest solution to the above equation for various primes are:</p>
<p>$(p=2)$ $3^2 = 2*2^2 +1$</p>
<p>$(p=3)$ $2^2 = 2*1^2 +1$</p>
<p>$(p=5)$ $9^2 = 5*4^2 +1$</p>
<p>$(p=7)$ $8^2 = 7*3^2 +1$</p>
<p>Is there at least one solution for each prime?
If there is one solution, there are infinite.</p>
| Travis Willse | 155,629 | <p>If one permits a general positive integer $n$ in place of the prime $p$, the resulting equation (in integers $x, y$) is called <em><a href="https://en.wikipedia.org/wiki/Pell's_equation" rel="nofollow">Pell's equation</a></em>, and is usually rearranged to read
$$x^2 - n y^2 = 1.$$
Lagrange showed that there are... |
3,124,158 | <p>So what I want to prove is
<span class="math-container">$$ |xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3| $$</span>
for <span class="math-container">$x,y,z\in \mathbb{R}$</span>, and I'm aware that the RHS is just <span class="math-container">$|(x-1)^2+(y-1)^2+(z-1)^2|$</span>.</p>
<p>Now I'm able to prove ... | David Holden | 79,543 | <p>have you tried evaluating:
<span class="math-container">$$
(x-1)(y-1) + (y-1)(z-1) +(z-1)(x-1)
$$</span>
?</p>
|
2,007,373 | <p>At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object. </p>
<p>For me the first instance that comes to memory was in 7th grade in a inner city USA school district. </p>
<p>Getting to the point, my geometry teacher taught,</p>
<p>"a p... | hmakholm left over Monica | 14,366 | <p>The viewpoint you're groping towards is not crazy, and at least historically you're in excellent company -- e.g., Leibniz had similar ideas when he viewed an integral as a sum of the heights of infinitely many lines, weighted by their infinitesimal width, and this intuition is still the background for the notation $... |
2,755,733 | <p>Why in this <a href="https://math.stackexchange.com/questions/625112/if-tossing-a-coin-400-times-we-count-the-heads-what-is-the-probability-that-t">If, tossing a coin 400 times, we count the heads, what is the probability that the number of heads is [160,190]?</a> question heropup's asnwer is like that? </p>
<p>I ... | BruceET | 221,800 | <p>As mentioned in the linked post, if $X \sim \mathsf{Binom}(400, .5),$ then
$$P(160 \le X \le 200) = P(159 < X < 201) = P(159.5 < X < 200.5) = 0.5199$$
to four places. [The computation is from R statistical software, in which
<code>pbinom</code> is a binomial CDF.]</p>
<pre><code>pbinom(200, 400, .5) - ... |
189,014 | <p>Ok, I know the simple answer is to set some form of Hold attribute to the function but bear with me for a bit while I explain my motivation and why that is not quite what I want.</p>
<hr>
<p>I have a collection of data that is naturally grouped together and some functions that operate on that data. To me, the obvi... | Jason B. | 9,490 | <p>You should definitely go the route of using an inert wrapper for the association, <code>atomData[Association[...]]</code>. </p>
<p>You can make a general subvalue like what you have,</p>
<pre><code>atomData[a_Association][key_] := a[key];
</code></pre>
<p>and you can make more specific definitions like</p>
<pre... |
3,765,555 | <p>Let <span class="math-container">$\triangle ABC$</span> be an isosceles triangle with base <span class="math-container">$a$</span> and altitude to the base <span class="math-container">$b.$</span> I am trying to find the sides of the rectangle inscribed in <span class="math-container">$\triangle ABC$</span> if its d... | Narasimham | 95,860 | <p>Yes, any number of triangles can be constructed that way. Draw a line parallel to one leg. See where it cuts the altitude. Reflect this parallel parallel line about altitude to be parallel to the other leg. Draw the inscribed rectangle as shown including cutting points on both the slant sides of the isosceles triang... |
4,594 | <p>I would like to open an Excel file and manipulate it as a COM object. While I'm able to open an instance of excel with</p>
<pre><code>Needs["NETLink`"]
InstallNET[]
excel = CreateCOMObject["Excel.Application"]
</code></pre>
<p>This doesn't work for me:</p>
<pre><code> wb = excel@Workbooks@Open["D:\\prices.csv"]
<... | Szabolcs | 12 | <p>Per request, I'm posting this as an answer:</p>
<p>The same problem is mentioned in the following support article:</p>
<ul>
<li><a href="http://support.microsoft.com/kb/320369">http://support.microsoft.com/kb/320369</a></li>
</ul>
<p>The problem appears if the language of Excel differs from <a href="http://windo... |
3,545,250 | <p>Being new to calculus, I'm trying to understand Part 1 of the Fundamental Theorem of Calculus. </p>
<p>Ordinarily, this first part is stated using an " area function" <em>F</em> mapping every <em>x</em> in the domain of <em>f</em> to the number " integral from a to x of f(t)dt". </p>
<p>However, I encounter <stro... | José Carlos Santos | 446,262 | <p>Yes, <span class="math-container">$\int_a^bf(t)\,\mathrm dt$</span> is a number. But if you change <span class="math-container">$a$</span> or <span class="math-container">$b$</span> (or both), you usually get a different number. So, <span class="math-container">$(a,b)\mapsto\int_a^bf(t)\,\mathrm dt$</span> is a <em>... |
3,820,465 | <p>I'm working on the following problem but I'm having a hard time figuring out how to do it:</p>
<p>Q: Let A and B be two arbitrary events in a sample space S. Prove or provide a counterexample:</p>
<p>If <span class="math-container">$P(A^c) = P(B) - P(A \cap B)$</span> then <span class="math-container">$P(B) = 1$</sp... | Graham Kemp | 135,106 | <blockquote>
<p>Drawing Venn diagrams I can see how this is true, as <span class="math-container">$A \subset B$</span>, but I'm not sure how to formally prove this. Any help would be great!</p>
</blockquote>
<p>Nope. (Unless you have a typo.)</p>
<p><span class="math-container">$$\begin{align}\mathsf P(A^{\small\compl... |
1,121,205 | <p>Can we find an bijective continuous map $f:X\to Y$ from a disconnected topological space $X$ to a connected topological space $Y$?</p>
<p>It seems counter intuitive for me, but I am not able to prove that $f(X)$ will be disconnected. I cannot think of any counterexample either. Can someone help?</p>
| Seth | 31,659 | <p>Take the interval $[0,1)$ disjoint union with $[2,3]$ mapping in the obvious way to $[0,2]$.</p>
|
132,862 | <p>Is it true that given a matrix $A_{m\times n}$, $A$ is regular / invertible if and only if $m=n$ and $A$ is a basis in $\mathbb{R}^n$?</p>
<p>Seems so to me, but I haven't seen anything in my book yet that says it directly.</p>
| penartur | 25,016 | <p>You are trying to deduce something about the importance of 5th axiom only from the axioms itself, which is impossible.
The axioms by itself do not carry any significance. What is important is that the set defined by the axioms is (a) unique and (b) isomorphic to some real-world object (real-world natural numbers, as... |
3,366,064 | <p>I have a baking recipe that calls for 1/2 tsp of vanilla extract, but I only have a 1 tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere. </p>
<p>My question is, to what depth (as a percentage of hemisphere radius) must I fill my teaspoon with van... | Rafi | 307,853 | <p>Note about eyeballing: Your eye's reference is the surface of the spoon, so when you eyeball you may actually be measuring along the arc from the bottom of the spoon to its top edge. </p>
<p>That is, your eye may be watching the red curve, not the blue line:</p>
<p><a href="https://i.stack.imgur.com/H1lzK.png" rel... |
496,255 | <p>Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact? </p>
| triple_sec | 87,778 | <p><strong>Lemma 1</strong>: $a$ is odd $\Longrightarrow$ $a^2\equiv 1(\operatorname{mod} 4)$.</p>
<p><em>Proof</em>: $a^2-1=(a-1)(a+1)$. Since $a$ is odd, both $a-1$ and $a+1$ are even, so that $a^2-1$ is divisible by $4$. $\blacksquare$</p>
<p><strong>Lemma 2</strong>: $a$ is even $\Longrightarrow$ $a^2\equiv 0(\op... |
496,255 | <p>Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact? </p>
| user66733 | 66,733 | <p>I'll write another argument with more group theoretic flavor in my opinion. Suppose that $p=4k+3$ is a prime number and you can write $p=x^2+y^2$. then $x^2+y^2 \equiv 0 \pmod{p} \iff x^2 \equiv -y^2 \pmod{p} \iff (xy^{-1})^2 \equiv -1 \pmod{p}$. Therefore $t=xy^{-1}$ is a solution of $x^2 \equiv -1 \pmod{p}$.</p>
... |
1,012,895 | <p>I am stuck with my revision for the upcoming test.</p>
<p>The question asks"</p>
<p>An implementation of insertion sort spent 1 second to sort a list of ${10^6}$ records. How many seconds it will spend to sort ${10^7}$ records?</p>
<p>By using $\frac{T(x)}{T(1)}$ = $\frac{10^7}{10^6}$ I thought the answer was $10... | Irvan | 172,851 | <p><strong>Hint</strong>: what is the <a href="http://en.wikipedia.org/wiki/Time_complexity" rel="nofollow">Time Complexity</a> of insertion sort?</p>
|
1,634,725 | <p>It's bee a long time since I've worked with sums and series, so even simple examples like this one are giving me trouble:</p>
<p>$\sum_{i=4}^N \left(5\right)^i$</p>
<p>Can I get some guidance on series like this? I'm finding different methods online but not sure which to use. I know that starting at a non-zero num... | stackoverflowuser2010 | 9,177 | <p>The problem asks for a closed-form solution to:</p>
<p><span class="math-container">$$\sum_{i=4}^{N} 5^i = 5^4 + 5^5 + ... + 5^N$$</span></p>
<p>The OP's original intuition was correct:
<span class="math-container">$$\sum_{i=4}^{N} = \sum_{i=0}^{N} 5^i - \sum_{i=0}^{3} 5^i$$</span></p>
<p>More generally, for summing... |
3,892,246 | <p>I am dealing with sets of vectors <span class="math-container">$\big\{x_1, x_2, x_3, \dotsc, x_m \big\}$</span> from some abstract vector space <span class="math-container">$\mathcal{V}$</span>. <strong>Occasionally</strong>, <span class="math-container">$\mathcal{V}=\mathbb{R}^n$</span> an I need to address the ele... | Wuestenfux | 417,848 | <p>Well, there is no common notation here. The policy is to keep the notation as simple as possible.</p>
<p>In your case, I would write <span class="math-container">$x_{ij}$</span> for the <span class="math-container">$j$</span>th component of vector <span class="math-container">$x_i$</span>.</p>
|
4,486,594 | <p>Let <span class="math-container">$X$</span> be the Riemann surface of <span class="math-container">$w^{2} \ =\text{sin} \ z$</span> in <span class="math-container">$ \mathbb{C}^{2}$</span>, i.e. let <span class="math-container">$X = \{(z,w): w^2 = \text{sin} \ z\}$</span>.</p>
<p>The Riemann surface structure on <sp... | onriv | 195,865 | <p>I am trying to use some other way to prove that the Riemann surface <span class="math-container">$X=\{(z,w):w^2=\sin z\}$</span> has infinite genus, as shown in the answer of Kohan's. The main result I wanna to show is that:</p>
<blockquote>
<p>For any positive integer <span class="math-container">$n$</span>, there ... |
4,321,604 | <p>I have come across an expression like this,</p>
<p><span class="math-container">$$ \frac{f(x) + f(a)}{2\sqrt{f(x)f(a)}}\,\delta(x-a), $$</span></p>
<p>where I expected to find just <span class="math-container">$\delta(x-a)$</span>. When I thought about it, though, I realised maybe... they are identical? Because both... | Mostafa Ayaz | 518,023 | <p>The famous property of sampling of the delta function yields
<span class="math-container">$$
f(x)\delta(x-a)=f(a)\delta(x-a).
$$</span>
However, subtlety lies in this property. It should be mentioned that the validity of the above equation is the domain of <span class="math-container">$f(x)$</span>, which in this ca... |
1,617,698 | <p>While I was trying to find the formula of something by my own means I came across this sum which I need to solve, however I don't know if there is a solution for it, maybe it doesn't mean anything and I made a mistake. However if there's an equation which can replace this sum I will appreciate it a lot if you show m... | sinbadh | 277,566 | <p>Let $z=e^{i\theta}=\cos\theta+i\sin\theta$. Then $z^j=e^{ij\theta}=\cos j\theta+i\sin j\theta$ for all $j\in\mathbb{N}$.</p>
<p>Thus</p>
<p>$\begin{eqnarray}
1+\sin\theta+\sin2\theta+...+\sin n\theta&=&Im(z^0+z^1+z^2+...+z^n)\\
&=&Im\left(\frac{z^{m+1}-1}{z-1}\right)
\end{eqnarray}$</p>
<p>Putting... |
1,562,010 | <p>Let $f(x) \in \mathbb Z[x]$ be an irreducible monic polynomial such that $|f(0)|$ is not a perfect square . Then is $f(x^2)$ also irreducible in $\mathbb Z[x]$ ?</p>
<p>( It is supposed to have an elementary solution , without using any field-extension etc. )</p>
| Eric Wofsey | 86,856 | <p>Sophisticated answer: Let us suppose $f(x)\in \mathbb{Z}[x]$ is monic and irreducible of degree $n$ but $g(x)=f(x^2)$ is reducible. Let $K\supset\mathbb{Q}$ be a splitting field of $g$ and let $a_1,\dots,a_n\in K$ be the roots of $f$, so $\pm\sqrt{a_1}\dots,\pm\sqrt{a_n}$ are the roots of $g$. Since $f$ is irredu... |
313,030 | <p>I often find myself writing a definition which requires a proof. You are defining a term and, contextually, need to prove that the definition makes sense. </p>
<p>How can you express that? What about a definition with a proof?</p>
<p>Sometime one can write the definition and then the theorem. But often happens tha... | usul | 29,697 | <p>One approach always available is to decompose your problem into a series of definitions and theorems each of which is formally correct and relies only on the previous ones. It may require defining and naming sub-objects. For example: (1) Definition of ~. (2) Theorem: ~ is an equivalence relation. Proof. (3) Definiti... |
1,751,196 | <p>I have an interesting question that I certainly don't know how to solve it. I've already read many topics on probability, eg: <a href="https://math.stackexchange.com/questions/1041325/probability-that-someone-will-pick-a-red-ball-first">Probability that someone will pick a red ball first?</a> and <a href="https://ma... | barak manos | 131,263 | <p>If you choose with replacement:</p>
<p>$$\frac15$$</p>
<hr>
<p>If you choose without replacement:</p>
<p>$$\left(1-\frac15\right)\cdot\frac14$$</p>
|
1,751,196 | <p>I have an interesting question that I certainly don't know how to solve it. I've already read many topics on probability, eg: <a href="https://math.stackexchange.com/questions/1041325/probability-that-someone-will-pick-a-red-ball-first">Probability that someone will pick a red ball first?</a> and <a href="https://ma... | Fabich | 320,938 | <p>you need the blue ball to be in the bucket after first pick : probability $\frac{4}{5}$</p>
<p>you need to pick the blue ball among the 4 remaining balls : probability $\frac{1}{4}$</p>
<p>Global probability : $$\frac{1}{4}\times \frac{4}{5} = \frac{1}{5}$$</p>
<hr>
<p>You can also see it this way : The 5 balls... |
2,781,153 | <p>I've a right triangle that is inscribed in a circle with radius $r$ the hypotunese of the triangle is equal to the diameter of the circle and the two other sides of the triangle are equal to eachother.</p>
<blockquote>
<p>Prove that when you divide the area of the circle by the area of the triangle that you will ... | The Integrator | 538,397 | <p>Consider the triangle $\triangle ABC$ with angles $45^\circ-90^\circ -45^\circ$
and hypotenuse $BC = 2r$</p>
<p>$AB = BC\cdot \cos(45^\circ) = \sqrt2r$</p>
<p>$AC = BC\cdot \sin(45^\circ) = \sqrt2r$</p>
<p>Area of triangle = $\frac12\cdot AB\cdot AC = r^2$</p>
<p>ratio = $\frac{\pi r^2}{r^2} = \pi$</p>
|
227,833 | <p>In the documentation article for <code>Polygon</code> in Mathematica 12, there is an example with the input:</p>
<pre><code>pol = Polygon[{{1, 0}, {0, Sqrt[3]}, {-1, 0}}]
</code></pre>
<p>In the documentation article the output is displayed as:</p>
<blockquote>
<pre><code>Polygon[{{1, 0}, {0, Sqrt[3]}, {-1, 0}}]
</c... | m_goldberg | 3,066 | <p>The examples in the documentation article for <code>Polygon</code> should have been re-evaluated before Mathematica 12 was released, but it is evident that they weren't. This is true even for V12.1.1. If you manually evaluate the examples shown in the documentation they will show the new iconized argument form. This... |
1,137,079 | <p>I'm new to the concept of complex plane. I found this exercise:</p>
<blockquote>
<p>Let $z,z_1,z_2\in\mathbb C$ such that $z=z_1/z_2$. Show that the length of $z$ is the quotient of the length of $z_1$ and $z_2$.</p>
</blockquote>
<p>If $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ then $|z_1|=\sqrt{x_1^2+y_1^2}$ and $|z_2|... | Tim Raczkowski | 192,581 | <p>Since $|z|^2=z\overline{z}$,</p>
<p>$$\left|{z_1\over z_2}\right|^2={z_1\over z_2}{\overline{z_1}\over\overline{z_2}}={|z_1|^2\over|z_2|^2}.$$</p>
<p>Take square root of both sides to get your result.</p>
|
2,768,187 | <blockquote>
<p>Let $w$ and $z$ be complex numbers such that $w=\frac{1}{1-z}$, and
$|z|^2=1$. Find the real part of $w$.</p>
</blockquote>
<p>The answer is $\frac{1}{2}$ but I don't know how to get to it.</p>
<hr>
<p>My attempt</p>
<p>as $|z|^2=1$</p>
<p>$z\bar z = 1$</p>
<p>If $z = x+yi$</p>
<p>$z=\frac{1... | nonuser | 463,553 | <p>Pure geometric solution. </p>
<p>All points $z$ satisfying $|z|=1$ are on circle $\mathcal{C}$ with radius $r=1$ and center at $0$. </p>
<p>Now transformation $z\mapsto 1-z$ is reflection across $0$ followed by translation for $1$. So $\mathcal{C}$ ''moves'' to the right for $1$ and let this new circle be $\mathca... |
2,921,439 | <p>I got this summation from the book <a href="https://rads.stackoverflow.com/amzn/click/0201558025" rel="nofollow noreferrer">Concrete Mathematics</a> which I didn't exactly understand:</p>
<p>$$
\begin{align}
Sn &= \sum_{1 \leqslant k \leqslant n} \sum_{1 \leqslant j \lt k} {\frac{1}{k-j}} \\
&= \sum_{1 \leq... | user | 505,767 | <p>We have indeed that</p>
<p>$$\frac{x+1}{-x^3-x^2+4x+4}=\frac{x+1}{-x^2(x+1)+4(x+1)}=-\frac{1}{x^2-4}=\frac14 \frac1{x+2}-\frac14 \frac1{x-2}$$</p>
|
4,608,805 | <p>Suppose that I have a class of 35 students whose average grade is 90. I randomly picked 5 students whose average came out to be 85. Assume their grades are i.i.d and of normal <span class="math-container">$N(\mu, \sigma^2)$</span>. From the example I have seen, <span class="math-container">$\mu$</span> is usually ca... | JonathanZ supports MonicaC | 275,313 | <p>You have a valid point here. In fact, the population has 35 members, so its distribution <em>must</em> be discrete, while a normal distribution is continuous. The sentence starting "Assume ...." is telling you that we are going to approximate that discrete distribution with a continuous, normal one.</p>
<p... |
2,261,410 | <blockquote>
<p>The generating function for a Bessel equation is:</p>
<p>$$g(x,t) = e^{(x/2)(t-1/t))}$$</p>
<p>Using the product $g(x,t)\cdot g(x,-t)$ show that:</p>
<p>a) $$[J_0(x)]^2 + 2[J_1(x)]^2 + 2[J_2(x)]^2 + \cdots = 1$$</p>
<p>and consequently:</p>
<p>b)</p>
<p>$$|J_0(x)|\le 1, \... | Michael Seifert | 248,639 | <p>For part (a), note that you also have
$$
g(x,-t) = g(x,1/t) = \sum_{n = 0}^\infty J_n(x) \left( \frac{1}{t} \right)^n.
$$
You can then multiply this with the original series to get
$$
\left[ \sum_{n = -\infty}^\infty J_n(x) t^{-n} \right] \left[ \sum_{n = -\infty}^\infty J_n(x) t^n \right] = 1.
$$
By multiplying the... |
3,948,418 | <p>Ok so on doing a whole lot of Geometry Problems, since I am weak at Trigonometry, I am now focused on <span class="math-container">$2$</span> main questions :-</p>
<p><span class="math-container">$1)$</span> <strong>How to calculate the <span class="math-container">$\sin,\cos,\tan$</span> of any angle?</strong></p>
... | WindSoul | 715,008 | <p>For any value angle I would turn it into radians in order to use the power series of sine.</p>
<p><span class="math-container">$sin(x)=\sum_{k=0}^{\infty}{(-1)^k\frac{x^{2k+1}}{(2k+1)!}}$</span>, <span class="math-container">$x\epsilon\mathbb R$</span></p>
<p><span class="math-container">$\theta=\frac{143^o}{3}\Righ... |
3,948,418 | <p>Ok so on doing a whole lot of Geometry Problems, since I am weak at Trigonometry, I am now focused on <span class="math-container">$2$</span> main questions :-</p>
<p><span class="math-container">$1)$</span> <strong>How to calculate the <span class="math-container">$\sin,\cos,\tan$</span> of any angle?</strong></p>
... | G Cab | 317,234 | <p>I do not catch exactly what is your problem, so I am proposing some considerations which might be useful,
at least to substantiate what you need.</p>
<p>a) The sine of an angle will be rational when the angle corresponds to that of an integral triangle (i.e. a Pythagorean triple).<br />
Among the angles which are r... |
109,423 | <p>Let $f$ be an isometry (<em>i.e</em> a diffeomorphism which preserves the Riemannian metrics) between Riemannian manifolds $(M,g)$ and $(N,h).$ </p>
<p>One can argue that $f$ also preserves the induced metrics $d_1, d_2$ on $M, N$ from $g, h$ resp. that is, $d_1(x,y)=d_2(f(x),f(y))$ for $x,y \in M.$ Then, it's easy... | H-H | 528,586 | <p>It is actually an exercise in the Lee's book, I try to do it by following the hint. First, you have to understand the naturality of Riemannian connection, then everything will be clear. I like using $\nabla_{\frac{d}{dt}}$ instead of $D_t$ here.</p>
<p>First, Define an operator $\varphi^*\tilde{\nabla}_{\frac{d}{dt... |
30,718 | <p>As we all know, questions lacking context are strongly discouraged on this site. This includes mainly "homework questions" that look a bit like:</p>
<blockquote>
<p>Prove that <span class="math-container">$\lim_{x\to0}x^2=0$</span> using <span class="math-container">$\epsilon$</span>-<span class="math-container">... | quid | 85,306 | <p>For the specific question, the context seems completely clear. The question reads:</p>
<blockquote>
<p><strong>How deep is the liquid in a half-full hemisphere?</strong></p>
<p>I have a baking recipe that calls for 1/2 tsp of vanilla extract, but I only have a 1 tsp measuring spoon available, since the dishwasher is... |
4,376,076 | <p>i) the Matrix P has only real elements</p>
<p>ii) 2+i is an eigenvalue of Matrix P</p>
<p>I got that the zeros has to be <span class="math-container">$(x-(2+i))(x-(2-i))$</span> which is equal to <span class="math-container">$(x-2)^2 -i^2$</span> so the characteristical polynom is equal to <span class="math-containe... | Martin Argerami | 22,857 | <p>If you find <span class="math-container">$A$</span> with eigenvalues <span class="math-container">$\pm i$</span>, then <span class="math-container">$\alpha I+A$</span> has eigenvalues <span class="math-container">$\alpha \pm i$</span>. Indeed,
<span class="math-container">$$ \det\bigl(( \alpha I+A )-\lambda I \bigr)... |
603,986 | <p>Show that in a finite field $F$ there exists $p(x)\in F[X]$ s.t $p(f)\neq 0\;\;\forall f\in F$</p>
<p>Any ideas how to prove it?</p>
| Community | -1 | <p>Take some element $\alpha_1\in F$</p>
<p>Then consider $f_1(x)=(x-\alpha_1)+1$.. What would be $f_1(\alpha_1)$?</p>
<p>Soon you will see that $f(\alpha_1)$ is non zero but may probably for some $\alpha_2$ we have $f_1(\alpha_2)=0$</p>
<p>Because of this i would now try to include $(x-\alpha_2)$ in $f_1(x)$ to mak... |
2,265,782 | <p>Number of twenty one digit numbers such that Product of the digits is divisible by $21$</p>
<p>Since product is divisible by $21$ the number should contain the digits $3,6,7,9$ But i am unable to decide how to proceed...can i have any hint</p>
| Mark Fischler | 150,362 | <p>We will massage
$$(p\wedge q) \rightarrow p$$ into $\mbox{TRUE}$
showing justifications for each step.</p>
<p>By definition of the "implies" relation,
$$(p\wedge q) \rightarrow p = \lnot (p\wedge q) \lor q$$ </p>
<p>By negation of an and relationship
$$(p\wedge q) \rightarrow p = \lnot (p\wedge q) \lor p = ( (\... |
1,959,949 | <blockquote>
<p>We introduce new variables as
$\begin{cases}\xi:=x+ct\\\eta:=x-ct\end{cases}
$
which implies that
$
\begin{cases} \partial_ x=\partial_\xi+\partial_\eta\\\partial_t=c\partial_\xi+c\partial_\eta
\end{cases}
$</p>
</blockquote>
<p>This is from page 34 of <em>Partial Differential Equation --- an ... | Community | -1 | <p>There is a typo in Strauss's book. One should have
$$
\partial_t=c\partial_\xi-c\partial_\eta.
$$
Note that both $\xi$ and $\eta$ are functions of $x$ and $t$:
$$
\xi=h(x,t);\quad \eta=g(x,t).
$$</p>
<p>Now, suppose you have a function $u=u(\xi,\eta)$. Then you have
$$
u=u(h(x,t),g(x,t)).
$$
Can you write down by c... |
1,237,077 | <p>For a periodic function we have: $$\int_{b}^{b+a}f(t)dt = \int_{b}^{na}f(t)dt+\int_{na}^{b+a}f(t)dt = \int_{b+a}^{(n+1)a}f(t)dt+\int_{an}^{b+a}f(t)dt = \int_{na}^{(n+1)a}f(t)dt = \int_{0}^{a}f(t)dt.$$ , but I don't understand how we obtain $\int _{b+a}^{\left(n+1\right)a}\:f\left(t\right)\:dt=\int _b^{na}\:f\left(t\... | Idris Addou | 192,045 | <p>Let $F$ be a primitive of $f$. Then, by the fundamental theorem of calculus, one has</p>
<p>$ \frac{d}{db} \int_{b}^{b+a}f(t)dt= \frac{d}{db} [F(b+a)-F(b)]=F^ \prime (b+a)-F^ \prime (b)=f(b+a)-f(b)=0.$ </p>
<p>since $f$ is $a$-periodic. Then, the integral $\int_{b}^{b+a}f(t)dt$ is independent of $b$, so one can ... |
786,086 | <p>For my research I am working with approximations to functions which I then integrate or differentiate and I am wondering how this affects the order of approximation.</p>
<p>Consider as a minimal example the case of $e^x$ for which integration and differentiation doesn't change anything. Now if I would approximate t... | Claude Leibovici | 82,404 | <p>This is indeed a very interesting question about the use of truncated series for integration and differentiation.</p>
<p>Taking you example of $e^x$, if we write $$f(x) =1+x+\frac{x^2}{2}+O\left(x^3\right)$$ from a formal point of view (at least, in my opinion), the derivative should write $$f'(x) =1+x+O\left(x^2\r... |
737,835 | <p>Why is $[0,1]$ not homeomorphic to $[0,1]^2$? It seems that the easiest way to show this is to find some inconsistency between the open set structures of the two. It is clear that the two share the same cardinality. Both are compact. Both are normal since they are metric spaces. However, where to find the open set s... | user126154 | 126,154 | <p>As comments and other questions point out, this case is easily solved by a connectedness argument.</p>
<p>It may be worth to point out that the general proof that $\mathbb R^n$ is not homeomorphic to $\mathbb R^m$ for $m\neq n$ is not trivial.</p>
<p>There is a nice important result, called the invariance of domai... |
229,606 | <p>I need little help in proving the following result :</p>
<p>Consider the ring $R:=\mathbb{F}_q[X]/(X^n-1)$, where $\mathbb{F}_q$ is a finite field of cardinality $q$ and $n\in\mathbb{N}$. Then any ideal $I$ of $R$ is principle and can be written as $I=(g(X))$, such that $g(X)|(X^n-1)$.</p>
| Lior B-S | 26,713 | <p>If $\varphi\colon S\to T$ is an epimorphism of rings with $1$, and if $I$ is an ideal of $T$, then $J=\varphi^{-1}(I)$ is an ideal of $S$. </p>
<p>Now if $S$ is <a href="http://en.wikipedia.org/wiki/Principal_ideal_domain" rel="nofollow">PID</a>, that is, if every ideal is generated by a single element, then $J=xS$... |
1,109,552 | <p>So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) = 2$ and so $a^2+5b^2 = 2$ , however, since $b^2$ and $a^2$ are both positive integers then $b=0$ and $a=\sqrt{2}$ h... | Community | -1 | <p>I've seen these same arguments in quite a few books, it all looks pretty standard issue to me.</p>
<p>But what you seem to be missing is the <em>motivation</em> for all of this, the why do we care. That motivation is this famous fact: $$6 = 2 \times 3 = (1 - \sqrt{-5})(1 + \sqrt{-5}).$$ It's also true that $$6 = -2... |
526,820 | <p>How do I integrate the inner integral on 2nd line? </p>
<p><img src="https://i.stack.imgur.com/uIxQX.png" alt="enter image description here"></p>
<hr>
<p>$$\int^\infty_{-\infty} x \exp\{ -\frac{1}{2(1-\rho^2)} (x-y\rho)^2 \} \, dx$$</p>
<p>I know I can use integration by substitution, let $u = \frac{x-y\rho}{\sq... | martini | 15,379 | <p>Then, recalling that $u \mapsto (2\pi)^{-1/2}\exp(-u^2/2)$ is the density of a standard normal distribution, we have that
$$ \int_{\mathbb R} (2\pi)^{-1/2} \exp(-u^2/2)\, du = 1 \iff \int_{\mathbb R} \exp(-u^2/2)\, du = (2\pi)^{1/2} $$
and $\int_{\mathbb R} (2\pi)^{-1/2}u\exp(-u^2/2)\, du$ is the expectation of a s... |
3,058,139 | <p>Let us consider the statement <span class="math-container">$\exists x P(x)$</span> - translated into English, "there exists an <span class="math-container">$x$</span> in our universe of discourse such that <span class="math-container">$P(x)$</span> is true." In writing the negation of this, we are taught to switch q... | N. S. | 9,176 | <p>The two statements are equivalent, the reason why we write it that way it is because it is easier to deal/prove it. It is irrelevant which way it is easier to say it, the important thing is being able to use.</p>
<p>Just consider the statement <span class="math-container">$\exists x \in \mathbb Z, x^2+(x+1)^2 \mbox... |
2,117,420 | <p>We all know that particular solution of $A_{n} = A_{(n-1)} + f(n)$</p>
<p>where $f(n)=n^c$ , c is a random positive integer.</p>
<p>Can be set to $(n^c+n^{(c-1)}+.....+1)$</p>
<p>But what about when $c\lt0$?</p>
<p>How do we find a particular solution of the form:</p>
<blockquote>
<p>$A_{n} = A_{(n-1)} + f(n)... | MatheMagic | 397,530 | <p>We can write $M=I+N$ where </p>
<p>$$
I = \begin{pmatrix}1&0&0 \\ 0&1&0 \\0&0&1 \end{pmatrix}
\qquad
N = \begin{pmatrix}0&1&0 \\ 0&0&1 \\0&0&0 \end{pmatrix}
$$</p>
<p>Since $I$ is the identity, $N$ and $I$ commute; hence $e^{I+N} = e^I e^N$ and $N$ is nilpotent with ... |
921,893 | <p>If we have: $f(x)=\frac { 1+x }{ 1+{ e }^{ x } } $</p>
<p>I am told to determine if $f(x)=x$ has multiple roots on $\left[ 0;+\infty \right] $</p>
<p>I tried to manually solve this equation, but I don't understand the result:</p>
<p>$f(x)-x=0\rightarrow \frac { 1+x }{ 1+{ e }^{ x } } =0\rightarrow { e }^{ x }(x+... | Sheheryar Zaidi | 131,709 | <p>$$f(x) - x = \frac{1+x}{1+e^x} - x = \frac{1+x}{1+e^x} - \frac{x(1+e^x)}{1+e^x} = \frac{1-xe^x}{1+e^x}$$</p>
|
788,995 | <p>I need to prove that function $\mathbb R × \mathbb R → \mathbb R $ : $f(x,y) = \frac{|x-y|}{1 + |x-y|}$ is a metric on $\mathbb R$. First two axioms are trivial; it's the triangle inequality which is pain. $\frac{|x-y|}{1 + |x-y|}$ + $\frac{|y-z|}{1 + |y-z|} ≥ \frac{|x-z|}{1 + |x-z|} ⇒ \frac{|x-y| + |y-z| + 2|(x-y)... | Tom | 103,715 | <p><b>Hint</b>: For $a,b \geq 0$
$$\frac{a+b}{1+a+b} = \frac{a}{1+a+b} + \frac{b}{1+a+b} \leq \frac{a}{1+a}+\frac{b}{1+b}$$
Next, made judicious choices for $a$ and $b$. </p>
|
788,995 | <p>I need to prove that function $\mathbb R × \mathbb R → \mathbb R $ : $f(x,y) = \frac{|x-y|}{1 + |x-y|}$ is a metric on $\mathbb R$. First two axioms are trivial; it's the triangle inequality which is pain. $\frac{|x-y|}{1 + |x-y|}$ + $\frac{|y-z|}{1 + |y-z|} ≥ \frac{|x-z|}{1 + |x-z|} ⇒ \frac{|x-y| + |y-z| + 2|(x-y)... | Santosh Linkha | 2,199 | <p>Since $1 + |x-y| + |y -z| \ge 1 + |x-z|$, </p>
<p>$$1 - \frac{1}{1 + |x-z|} \le 1 - \frac{1}{1 + |x-y| + |y -z|} \\
\implies \frac{|x-z|}{1 +|x-z|} \le \frac{|x-y| + |y-z|}{1 + |x-y| + |y-z|} \le \frac{|x-y|}{1 + |x-y|} + \frac{|y-z|}{1 + |y-z|}$$</p>
|
1,586,286 | <p>There are $30$ red balls and $50$ white balls. Sam and Jane take turns drawing balls until they have drawn them all. Sam goes first. Let $N$ be the number of times Jane draws the same color ball as Sam. Find $E[N].$</p>
<p>I have been proceeding with indicators...</p>
<p>$$ I_{j} =
\begin{cases}
1, & \text{i... | MichaelChirico | 205,203 | <h1>Hint:</h1>
<p>Here's the simulated distribution (300,000 repetitions)</p>
<p><a href="https://i.stack.imgur.com/1GpDR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1GpDR.png" alt="enter image description here" /></a></p>
<p>(let me know if you want the code, it's pretty straightforward though)<... |
2,227,027 | <p>If $f(x)$ is defined everywhere except at $x=x_0$, would $f'(x_0)$ be undefined at $x=x_0$ as well?</p>
<p>One example is: $$f(x)=\ln(x)\rightarrow f'(x)=\frac{1}{x}$$</p>
<p>In this particular case, both $f(x)$ and $f'(x)$ are undefined at $x=0$. I wonder if this always holds true.</p>
<p>Thank you.</p>
| Ross Millikan | 1,827 | <p>The usual definition, <span class="math-container">$f'(x)=\lim_{h \to 0}\frac {f(x+h)-f(x)}h$</span> clearly requires that we be able to evaluate <span class="math-container">$f(x)$</span> in a neighborhood of <span class="math-container">$x$</span>. A challenging case is where there is a removable discontinuity. ... |
3,273,756 | <blockquote>
<p>I am supposed to give a 9-dimensional irreducible representation of <span class="math-container">$\mathfrak{so}(4)$</span>.</p>
</blockquote>
<p>I know that <span class="math-container">$\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$</span> and hence I have a 6-dimensional reducible rep... | Torsten Schoeneberg | 96,384 | <p>Almost every treatment of representations of real semisimple Lie algebras explicitly treats the basic case <span class="math-container">$\mathfrak{su}_2$</span>; the standard result is that for each positive integer <span class="math-container">$n$</span>, there is up to equivalence exactly one irreducible <span cla... |
1,358,735 | <p>I'm sorry to sound like a dummy, but I've had trouble with Algebra all my life. I'm studying online with Khan Academy and one of the questions is: </p>
<p>Point $E$'s $y$-coordinate is $0$, but its $x$-coordinate is not $0$.
Where could point $E$ be located on the coordinate plane?</p>
<p>There is not graph or not... | Tyler Hilton | 287 | <p>Each point is represented by its coordinates on the plane. So your point $E$ can be represented by coordinates $$(x, y)$$</p>
<p>You are already told that the $y$-coordinate is 0, so the point $E$ has the form $$(x, 0)$$</p>
<p>Since no information is given for $x$ (ie, no restrictions for $x$), plug in some value... |
1,666,615 | <blockquote>
<p>For the series $\sum_{k=1}^{\infty}a_k$, suppose that there is a number $r$ with $0\leq r<1$ and a natural number $N$ such that $$|a_k|^{1/k}<r\qquad\text{for all indices $k\geq N$}$$ Prove that $\sum_{k=1}^{\infty}a_k$ converges absolutely.</p>
</blockquote>
<p>Proof:</p>
<p>For a given $r\in... | choco_addicted | 310,026 | <p>If you know the comparison test, then you can prove the proposition easily. You know $|a_k| \le r^k$ for all $k \ge N$. Since $0\le r < 1$, the geometric series $\sum_{n=1}^{\infty} r^n$ converges. Therefore, by comparison test, $\sum_{n=1}^{\infty} |a_n|$ converges.</p>
<p>The proposition you have to prove is c... |
4,351,794 | <p>I am trying not exactly to solve equation, but just change it from what is on right side to what is on left side. But I didn't do any math for years and can't remember what to.</p>
<blockquote>
<p><span class="math-container">$$\frac{1}{2jw(1+jw)}=\frac{-j(1-jw)}{2w(1+w^2)}$$</span></p>
<p>Here, <span class="math-co... | Azur | 656,302 | <p><em>So, correct me if I'm wrong, but this equation seems to come from some electrical physics context? The reason I'm saying that is because it seems that <span class="math-container">$j$</span> is the imaginary unit (such that <span class="math-container">$j^2 = -1$</span>), which is usually called <span class="mat... |
947,730 | <p>I'm trying to do this for practice but I'm just going nowhere with it, I'd love to see some work and answers on it.</p>
<p>Thanks :)</p>
<p>Find a polynomial that passes through the points (-2,-1), (-1,7), (2,-5), (3,-1). Present the answer in standard form.</p>
<p>What I've tried:</p>
<p><img src="https://i.sta... | 0xfee1dead | 898,565 | <p><strong>Step 1 :</strong> List the equations that define the function according to ax<sup>3</sup> + bx<sup>2</sup> + cx + d:<br></p>
<pre>-8a+ 4b - 2c + d = -1 (equation1)<br>
-1a + 1b + -1c + d = 7 (equation2) <br>
8a + 4b + 2c + d = -5 (equation3) <br>
27a + 9b + 3c + d = -1 (equation4)</pre>
<p><strong>Step 2 :</... |
83,336 | <p>Sorry if this is a naive question-- I'm trying to learn this stuff (cross-posted from <a href="https://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups">https://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups</a>)</p>
<p>If $G$ is a group with... | Jesse Peterson | 6,460 | <p>If $G$ is compact this is the Frobenius Reciprocity Theorem, see e.g., Section 6.2 in Folland's A Course in Abstract Harmonic Analysis for a proof. When $G$ is not compact then this fails already for $H$ the trivial group and $U$, and $V$ trivial representations. Indeed, in this case Ind$_H^G(1_H) = L^2G$ the left... |
2,400,336 | <p>My first try was to set the whole expression equal to $a$ and square both sides. $$\sqrt{6-\sqrt{20}}=a \Longleftrightarrow a^2=6-\sqrt{20}=6-\sqrt{4\cdot5}=6-2\sqrt{5}.$$</p>
<p>Multiplying by conjugate I get $$a^2=\frac{(6-2\sqrt{5})(6+2\sqrt{5})}{6+2\sqrt{5}}=\frac{16}{2+\sqrt{5}}.$$</p>
<p>But I still end up w... | John Joy | 140,156 | <p>In this particular problem, you can pretty much guess the answer.</p>
<p>$$\sqrt{6-\sqrt{20}}=\sqrt{6-2\sqrt{5}}$$</p>
<p>Now, suppose that the $-2\sqrt{5}$ was the middle term of a perfect square trinomial, where $x = \sqrt{5}$. In other words, that middle term is $-2x$.</p>
<p>What would the first and last term... |
144,375 | <p>We know that every $2\times 2$ matrix in $PGL(2, \mathbb{Z})$ of order $3$ is conjugate to the matrix $$ \left( \begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) $$. </p>
<p>I am interested in finding out to what extent this holds for $3\times 3$ integer invertible matrices.</p>
<p>In other words how m... | Alex B. | 35,416 | <p>I will work in ${\rm GL}$ instead of ${\rm PGL}$.</p>
<p>The corresponding question over ${\rm GL}_3(\mathbb{Z})$ is essentially$^1$ equivalent to asking how many faithful $\mathbb{Z}[G]$-modules, free of rank 3 over $\mathbb{Z}$ there are up to isomorphism, where $G$ is the cyclic group of order 3. Any such module... |
3,375,375 | <p>I noticed this issue was throwing off a more sophisticated problem I'm working on. When computing the indefinite integral </p>
<p><span class="math-container">$$ I(x) = \int \frac{dx}{1-x} = \log | 1-x | + C,$$</span></p>
<p>I realized I could equivalently write</p>
<p><span class="math-container">$$ I(x) = - \... | Z Ahmed | 671,540 | <p><span class="math-container">$$I=\int \frac{dx}{1-x}=-\log (x-1), ~if~ x>1~~~(1)$$</span>
and
<span class="math-container">$$I=\int \frac{dx}{1-x}=-\log (1-x), ~if~ x<1~~~(2)$$</span></p>
<p>By the way <span class="math-container">$$I=\int \frac{dx}{1-x} = - \log |1-x|~~~~(3)$$</span>
will perform like (1) or... |
3,370,750 | <p>Show that the moment if inertia of an elliptic area of mass
M
and semi-axis
a
and
b
about a semi-diameter
of length
r
is <span class="math-container">$$\frac{Ma^2b^2}{4r^2}$$</span>.
My attempt.</p>
<ol>
<li>I know that MI about ox is <span class="math-container">${Mb^2 \over 4}$</span>.</li>
<li>MI about oy a... | achille hui | 59,379 | <p>Let <span class="math-container">$\theta$</span> be the angle between the axis you wish to compute MI and the <span class="math-container">$x$</span>-axis.</p>
<p>For any point <span class="math-container">$(x,y)$</span> in the plane, its distance to the axis equals to <span class="math-container">$|x\sin\theta - y... |
1,261,504 | <p>I am trying to proof $ab = \gcd(a,b)\mathrm{lcm}(a,b)$.</p>
<p>The definition of $\mathrm{lcm}(a,b)$ is as follows:</p>
<p>$t$ is the lowest common multiple of $a$ and $b$ if it satisfies the following:</p>
<p>i) $a | t$ and $b | t$ </p>
<p>ii) If $a | c$ and $b | c$, then $t | c$.</p>
<p>Similiarly for the $\g... | callculus42 | 144,421 | <blockquote>
<p>At the same time, my notes state this formula (with no explanation..)
that $z = \frac{x - \mu}{\sigma}$, where $z$ is a $z-score$. But what
is this and what does it have to even do with these problems?</p>
</blockquote>
<p>Suppose you have a random variable X, which is $ \mathcal N(\mu,\sigma^2 )... |
68,438 | <p>I recall reading somewhere that if a conformal class contains an Einstein metric then that metric is the unique metric with constant scalar curvature in its conformal class, with the exception of the case of the round sphere. Does this sound right? If it is true: where can I find the proof of this result? </p>
| Brian Clarke | 2,063 | <p>This is not the original reference, but the most general (i.e., applicable to the non-compact and pseudo-Riemannian cases) result I know is:</p>
<p><a href="http://www.ams.org/mathscinet-getitem?mr=1260173">Kühnel, W., & Rademacher, H. Conformal diffeomorphisms preserving the Ricci tensor. Proceedings of the A... |
3,401,260 | <p>I am supposed to find the intersection of :
<span class="math-container">$$\begin{cases} 2^{x}=y \\ 31x+8y-94=0 \end{cases}$$</span>
When I substitute the first equation into the second one:
<span class="math-container">$$\frac{94-31x}{8}=2^{x}$$</span> and I do not know how to continue. </p>
<p>Can anyone help me... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: <span class="math-container">$x$</span> can only be <span class="math-container">$2$</span>, <span class="math-container">$2$</span> solves the equation.</p>
|
3,401,260 | <p>I am supposed to find the intersection of :
<span class="math-container">$$\begin{cases} 2^{x}=y \\ 31x+8y-94=0 \end{cases}$$</span>
When I substitute the first equation into the second one:
<span class="math-container">$$\frac{94-31x}{8}=2^{x}$$</span> and I do not know how to continue. </p>
<p>Can anyone help me... | Cornman | 439,383 | <p><span class="math-container">$$\dfrac{94-31x}{8}=2^x$$</span></p>
<p>is correct, you might proceed like this: It is <span class="math-container">$8=2^3$</span> and we get:</p>
<p><span class="math-container">$$94-31x=2^{x+3}$$</span></p>
<p>Note that the RHS is always positive, while the LHS is negative when <spa... |
2,226,337 | <p>How many multiples of $5$ are greater than $60,000,$ and can be made from the digits:
$$0, 1, 2, 3, 4, 5, 6$$ </p>
<p>if <strong>all</strong> digits have to be used and each can only be used once with no repeats.</p>
<p>Am I looking at this in the wrong way too simplistically or is it simply $2 \times 6!$</... | Kanwaljit Singh | 401,635 | <p>First digit can't be 0. So we have two cases.</p>
<p>Case 1 -</p>
<p>Number start with 5. Then it should be end with 0.</p>
<p>So we have,</p>
<p>$1 \times 5 \times 4 \times 3 \times 2 \times 1 \times 1$</p>
<p>Case 2 - </p>
<p>Number start with any digit except 5 or 0. Then it should be end with either 5 or 0... |
850,390 | <p>Let $f(x)$ be differentiable function from $\mathbb R$ to $\mathbb R$, If $f(x)$ is even, then $f'(0)=0$. Is it always true?</p>
| gniourf_gniourf | 51,488 | <p>Let $f$ be an even function defined on a (symmetric) neighborhood of $0$ and differentiable at $0$. From the definition of differentiability:
$$f'(0)=\lim_{h\to0}\frac{f(h)-f(0)}h.$$
By composition, we also have:
$$f'(0)=\lim_{h\to0}\frac{f(-h)-f(0)}{-h}$$
and using the fact that $f$ is even, this equality can be wr... |
2,403,404 | <p>I would be thankful if anyone can answer my question. This is a very basic question. Let's say we wish to minimise the quantity</p>
<p>$$\hat{h}= \|h-h_i\|+\lambda\|h-u\|,$$</p>
<p>where:</p>
<p>$$h=[13,17,20, 17, 20, 14, 17, 18, 16, 15, 15, 12, 19, 13, 17, 13]^\top,\\ h_i=[18, 17, 14, 13, 17, ... | Michael Rozenberg | 190,319 | <p>I think your way is not true. Why the minimum occurs for right-angled triangle? </p>
<p>I think we can solve your problem by the following way.
Let $CD$ be an altitude of the triangle and $BD=x$.</p>
<p>Thus,
$$CD=\frac{2\cdot12}{6}=4$$ and by Minkowski (triangle inequality) we obtain:
$$AC+BC=\sqrt{(6-x)^2+4^2}+\... |
4,567,390 | <p>Let</p>
<ul>
<li><span class="math-container">$X$</span> be a metric space,</li>
<li><span class="math-container">$\mathcal C_b(X)$</span> the space of real-valued bounded continuous functions,</li>
<li><span class="math-container">$\mathcal C_0(X)$</span> the space of real-valued continuous functions that vanish at... | Oliver Díaz | 121,671 | <p>To show that <span class="math-container">$\Phi\circ f: X\rightarrow\Phi(f(X))$</span> is a homeomorphism it seems that is enough to show that for any sequence <span class="math-container">$(x_m:m\in\mathbb{N})\subset X$</span>, <span class="math-container">$\delta_{x_m}\stackrel{v}{\longrightarrow}\delta_x$</span> ... |
3,637,526 | <p>I'm trying to understand the proof of Theorem 16.10, Probability and Measure, Patrick Billingsley, I put part of it here exactly as presented in the book</p>
<p>Theorem: If <span class="math-container">$f,g$</span> are nonnegative and <span class="math-container">$\int_Afd\mu=\int_Agd\mu$</span> for all <span class... | Sam | 584,704 | <p>The set <span class="math-container">$B_n$</span> can be written as <span class="math-container">$B_n = \{g \gt 0\}\cap\{g \lt f\} \cap\{g \le n\}$</span>. The first and third sets are measurable by definition. For the second one, notice that <span class="math-container">$g < f \iff \exists r \in \mathbb Q$</span... |
4,427,651 | <p>In programming, we define an "array" (basically an ordered n-tuple) in the following way:</p>
<p><span class="math-container">$$a=[3,5].$$</span></p>
<p>Later on, if we want to refer to the first element of the predetermined array/pair/n-tuple, we write <span class="math-container">$a[0]$</span> (because i... | Joe | 623,665 | <p>If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are sets, the function <span class="math-container">$f:X\times Y\to X$</span> given by <span class="math-container">$f(x,y)=x$</span> is called the <em>projection from <span class="math-container">$X\times Y$</span> onto <span cla... |
512,591 | <p>It is always confusing to prove with $\not\equiv$. Should I try contrapositive?</p>
| Fly by Night | 38,495 | <p>If $a \not\equiv 0 \bmod 3$ then either $a \equiv 1 \bmod 3$ or $a \equiv 2 \bmod 3$.</p>
<ul>
<li>If $a \equiv 1 \bmod 3$ then $a^2 \equiv 1^2 = 1\equiv 1 \bmod 3$.</li>
<li>If $a \equiv 2 \bmod 3$ then $a^2 \equiv 2^2 = 4 \equiv 1 \bmod 3$.</li>
</ul>
<p>Hence, if $a \not\equiv 0 \bmod 3$ then $a^2 \equiv 1 \bmo... |
1,638,051 | <p>$$\int\frac{dx}{(x^{2}-36)^{3/2}}$$</p>
<p>My attempt:</p>
<p>the factor in the denominator implies</p>
<p>$$x^{2}-36=x^{2}-6^{2}$$</p>
<p>substituting $x=6\sec\theta$, noting that $dx=6\tan\theta \sec\theta$ </p>
<p>$$x^{2}-6^{2}=6^{2}\sec^{2}\theta-6^{2}=6^{2}\tan^{2}\theta$$</p>
<p>$$\int\frac{dx}{(x^{2}-36... | Nikunj | 287,774 | <p>Your solution is incorrect as $$\frac{\sec\theta}{\tan\theta}$$ is not the same as $$\ sin^{-1}\theta$$ It is however, equal to $\csc\theta$, a standard integral that you had got after a few steps</p>
|
2,683,326 | <p>I have a function $f(x)$ whose second order Taylor expansion is represented by $f_2(x)$. Is it true that $$f(x)>f_2(x)$$ for all $x$? Any help in this regard will be much appreciated. Thanks in advance.</p>
| NewGuy | 518,506 | <p>similar to what @hardmath said </p>
<p>you can break the line into predetermined equal segments for example (10,20,....,40,...100 etc)</p>
<p>Example
lets break the line into 20 segments, the total number of ways 6 line segments can be selected is </p>
<p>total number of positive integers in the equation</p>
<p>... |
1,261,067 | <p>$$\int \left(\frac15 x^3 - 2x + \frac3x + e^x \right ) \mathrm dx$$</p>
<p>I came up with
$$F=x^4-x^2+\frac{3x}{\frac12 x^2}+e^x$$
but that was wrong.</p>
| MonkeyKing | 225,981 | <p>The first term and the third terms are wrong.</p>
<p>First term: $\int \frac{1}{5}x^3 dx= \frac{x^4}{20}$. Try differentiating and check $\int x^p dx = \frac{1}{p+1} x^{p+1}$ if $p \neq -1$.</p>
<p>Third term: $\frac{3}{x} = 3x^{-1}$, and $\int x^{-1} = \ln x$, so $\int 3x^{-1} = 3\ln x$.</p>
<p>Finally, don't fo... |
1,261,067 | <p>$$\int \left(\frac15 x^3 - 2x + \frac3x + e^x \right ) \mathrm dx$$</p>
<p>I came up with
$$F=x^4-x^2+\frac{3x}{\frac12 x^2}+e^x$$
but that was wrong.</p>
| Hasan Saad | 62,977 | <p>$\int cx^n=c\int x^n=c\frac{x^{n+1}}{n+1}$ when $n\neq -1$</p>
<p>Thus, $\int \frac{1}{5}x^3=\frac{1}{20}x^4$</p>
<p>Also, by the same rule, $\int -2x=-x^2$</p>
<p>$\int \frac{1}{x}=\ln(x)$</p>
<p>Thus, $\int \frac{3}{x}=3\ln(x)$</p>
<p>$\int e^x=e^x$</p>
<p>Thus, by above, the required integral is </p>
<p>$F... |
1,176,958 | <p>I've been struggling with this for over an hour now and I still have no good results, the question is as follows:</p>
<blockquote>
<p>What's the probability of getting all the numbers from $1$ to $6$ by rolling $10$ dice simultaneously?</p>
</blockquote>
<p>Can you give any hints or solutions? This problem seems... | drhab | 75,923 | <p><strong>Hint</strong>:</p>
<p>If $X_{i}$ denotes the number of dice that show face $i$ then it
equals:</p>
<p>$1-P\left[X_{1}=0\vee X_{2}=0\vee X_{3}=0\vee X_{4}=0\vee X_{5}=0\vee X_{6}=0\right]$</p>
<p>Now apply <a href="http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow">inclusion... |
387,749 | <p>This comes from Guillemin and Pollack's book Differential Topology. The book claims that one cannot parametrize a unit circle by a single map. I thought we could (by a single angle $\theta$). </p>
<p>I think one possible answer might be the fact that if we let $\theta$ in [0, 2$\pi$), when $\theta$ approaches 2$\pi... | Henry T. Horton | 24,934 | <p>Let's look more closely at Guillemin and Pollack's definition of parametrization:</p>
<blockquote>
<p>Let $X$ be a subset of $\Bbb R^N$. A <strong>parametrization</strong> is a diffeomorphism $\phi: U \longrightarrow V$ from an open set $U \subset \Bbb R^k$ to an open subset $V \subset X$ (where we give $X$ the s... |
1,042,212 | <p>If $\mathbb{Z}_p \leq K$ an algebraic extension, then $K$ has the identity $$\forall a \in K, \exists b \in K \text{ with } a=b^p$$</p>
<p>The proof is the following:</p>
<p>Let $a \in K$.</p>
<p>We take $\mathbb{Z}_p \leq \mathbb{Z}_p(a)$, $a$ algebraic over $\mathbb{Z}_p$.</p>
<p>So, $[\mathbb{Z}_p(a) : \mathb... | Julián Aguirre | 4,791 | <p>Another example. Let $\Omega=(0,1)\subset\mathbb{R}$ and $f_n(x)=\sin(n\,\pi\,x)/n$. Then
$$
\|f_n\|_2\le\frac1n\to0\text{ as }n\to\infty.
$$
On the other hand
$$
\|f'_n\|_2^2=\pi^2\int_0^1\cos^2(n\,\pi\,x)\,dx=\frac{\pi^2}{2}.
$$</p>
|
1,042,212 | <p>If $\mathbb{Z}_p \leq K$ an algebraic extension, then $K$ has the identity $$\forall a \in K, \exists b \in K \text{ with } a=b^p$$</p>
<p>The proof is the following:</p>
<p>Let $a \in K$.</p>
<p>We take $\mathbb{Z}_p \leq \mathbb{Z}_p(a)$, $a$ algebraic over $\mathbb{Z}_p$.</p>
<p>So, $[\mathbb{Z}_p(a) : \mathb... | gerw | 58,577 | <p>You can take any orthonormal sequence $\{f_n\}$ in $H^1$. Then you have $f_n \rightharpoonup 0$, but $\|f_n\|_{H^1} = 1$. Using the compact embedding from $H^1$ to $L^2$, you have $f_n \to 0$ in $L^2$.</p>
<p>More generally, you can construct a counterexample by using any weakly but not strongly convergent sequence... |
2,154,608 | <blockquote>
<p>Let $a$, $b$ and $c$ be non-negative numbers such that $a^3+b^3+c^3=3$. Prove that:
$$a^4b+b^4c+c^4a\leq3$$</p>
</blockquote>
<p>This inequality similar to the following.</p>
<blockquote>
<p>Let $a$, $b$ and $c$ be non-negative numbers such that $a^2+b^2+c^2=3$. Prove that:
$$a^3b+b^3c+c^3a\le... | Oscar Cunningham | 1,149 | <p>In any regular $4$-polytope, three or more regular polyhedra meet at each edge. The angles between faces on these polyhedra have to add to less than $2\pi$ (in the same way that angles meeting at a vertex in a polyhedron must add to less than $2\pi$). By calculating the angles between faces in the regular polyhedra,... |
697,402 | <p>I have this limit:</p>
<p>$$ \lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2\sqrt{x^2+1}} $$ I tried to solve it by this:</p>
<p>$$ \lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2\sqrt{x^2+1}} = \lim_{x\to\infty}\frac{\frac{x^3}{x^3}+\frac{\cos x}{x^3}+\frac{e^{-2x}}{x^3}}{\frac{x^2\sqrt{x^2+1}}{x^3}} = \frac{0+0+0}... | Aloizio Macedo | 59,234 | <p>For sufficiently large $x$</p>
<p>$\displaystyle \frac{x^3-1}{x^2\sqrt{x^2+1}} \leq \frac{x^3+\cos x+e^{-2x}}{x^2\sqrt{x^2+1}} \leq \frac{x^3+2}{x^2\sqrt{x^2+1}}$</p>
<p>Now, </p>
<p>$\displaystyle \frac{2}{x^2\sqrt{x^2+1}} \rightarrow 0$, and $\displaystyle \frac{-1}{x^2\sqrt{x^2+1}} \rightarrow 0$</p>
<p>More... |
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