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 |
|---|---|---|---|---|
916,963 | <p>I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. </p>
<p>$\textbf{Theorem:}$ If $P \rightarrow Q$ and $R \rightarrow \neg Q$, then $P \rightarrow \neg R$.</p>
<p>$\textbf{Proof:}$ (by contradiction)
Ass... | J Marcos | 65,869 | <p>Proofs by contradiction prove that your assumptions are <em>jointly incompatible</em>. In this case, from $P\to Q$ and $R\to\neg Q$ you may conclude that $P\uparrow R$, where $\uparrow$ denotes <a href="http://en.wikipedia.org/wiki/Sheffer_stroke" rel="nofollow">NAND</a>. Note that $P\uparrow R$ is true iff $P$ and... |
916,963 | <p>I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. </p>
<p>$\textbf{Theorem:}$ If $P \rightarrow Q$ and $R \rightarrow \neg Q$, then $P \rightarrow \neg R$.</p>
<p>$\textbf{Proof:}$ (by contradiction)
Ass... | Tanner Swett | 13,524 | <p>Great question. The answer is that, <em>at that point in the proof</em>, you're still assuming that $P$ is true. With the assumptions stated more prominently, the proof goes like this:</p>
<blockquote>
<p>We are given that $P \to Q$ and that $R \to \neg Q$. <strong>Now we begin imagining that $P$ is true.</strong... |
916,963 | <p>I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. </p>
<p>$\textbf{Theorem:}$ If $P \rightarrow Q$ and $R \rightarrow \neg Q$, then $P \rightarrow \neg R$.</p>
<p>$\textbf{Proof:}$ (by contradiction)
Ass... | Taemyr | 114,582 | <p>We are trying to prove that given $P \rightarrow Q$ and $R \rightarrow \neg Q$, we get $P \rightarrow \neg R$</p>
<p>The proof you describe derives a contradiction by assuming $P$ and $R$, so at least one of the assumptions have to be false.</p>
<p>You wonder why we have to select $R$ as the false assumption. We ... |
916,963 | <p>I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. </p>
<p>$\textbf{Theorem:}$ If $P \rightarrow Q$ and $R \rightarrow \neg Q$, then $P \rightarrow \neg R$.</p>
<p>$\textbf{Proof:}$ (by contradiction)
Ass... | Dan Christensen | 3,515 | <blockquote>
<p>So my question is: in general, when proving by contradiction, how do we know which assumption exactly is false? And how do we know that exactly one assumption must be wrong in order to proceed with the proof?</p>
</blockquote>
<p>If you look at a proof as an ordered (numbered) sequence of statements,... |
313,522 | <p>The problem I am currently working on is:</p>
<blockquote>
<p>Consider the following information: where <ul><li>$A$={Visa Card}</li> <li>$B$={MasterCard}</li></ul>
$P(A)=0.5$, $P(B)=0.4$, and $P(A \cap B) = 0.25$</p>
</blockquote>
<p>The part I am having difficulty with is part (e):</p>
<blockquote>
<p>Giv... | gelichor | 62,894 | <p>The probability of having at least one card is: $$P(A)+P(B)-P(A\cap B) = 0.65 $$</p>
<p>Denote $C$={ <em>at least one card</em> }.</p>
<p>The probability you need (definition of conditional probability):</p>
<p>$$ P(A\;|\;C) = \frac {P(A\cap C)}{P(C)}$$</p>
<p>If you have a Visa card, you have at least one card,... |
313,522 | <p>The problem I am currently working on is:</p>
<blockquote>
<p>Consider the following information: where <ul><li>$A$={Visa Card}</li> <li>$B$={MasterCard}</li></ul>
$P(A)=0.5$, $P(B)=0.4$, and $P(A \cap B) = 0.25$</p>
</blockquote>
<p>The part I am having difficulty with is part (e):</p>
<blockquote>
<p>Giv... | Gerry Myerson | 8,269 | <p>Suppose there are $100$ people. How many have a Visa card? how many a Mastercard? how many both? of those who have at least one, how many have a Visa card?</p>
|
4,039,424 | <p>I would like to calculate the elements of <span class="math-container">$\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})$</span>.
I know that the elements of <span class="math-container">$\mathbb{Q}(\sqrt[3]{2})$</span> have the form of <span class="math-container">${a+b\sqrt[3]{2}+c\sqrt[3]{4}}$</span>, where a,b,c <span class="ma... | TravorLZH | 748,964 | <p>Similar to how Gronwall deduced</p>
<p><span class="math-container">$$
\limsup_{n\to\infty}{\sigma(n)\over e^\gamma n\log\log n}=1
$$</span></p>
<p>We consider defining <span class="math-container">$a_n$</span> such that</p>
<p><span class="math-container">$$
\prod_{p\le p_{a_n}}p\le n\le\prod_{p\le p_{a_n+1}}p
$$</... |
3,613,180 | <p>Let <span class="math-container">$f:[0, \pi] \to \mathbb{R}$</span> with the <span class="math-container">$L^2$</span> inner product</p>
<p><span class="math-container">$$
\langle f,g \rangle = \int_0^{\pi} f(x)g(x) \mathrm{d}x
$$</span></p>
<p>I want to find a projection of <span class="math-container">$f(x)=1$</... | Robert Lewis | 67,071 | <p>Having inspected the cited problem and it's surroundings, I note that part (b) of problem (2) asks us to show that
<span class="math-container">$(y')^2 + y^2$</span> is constant; this is seen to be an easy consequence of</p>
<p><span class="math-container">$y'' + y = 0 \tag 1$</span></p>
<p>as follows: we have</p>... |
3,613,180 | <p>Let <span class="math-container">$f:[0, \pi] \to \mathbb{R}$</span> with the <span class="math-container">$L^2$</span> inner product</p>
<p><span class="math-container">$$
\langle f,g \rangle = \int_0^{\pi} f(x)g(x) \mathrm{d}x
$$</span></p>
<p>I want to find a projection of <span class="math-container">$f(x)=1$</... | marty cohen | 13,079 | <p>If <span class="math-container">$y+y''=0$</span>, then <span class="math-container">$yy'+y'y''=0$</span>
so <span class="math-container">$(y^2)'+((y')^2)' =0$</span>
so <span class="math-container">$y^2+(y')^2=c$</span>
for some real <span class="math-container">$c$</span>.</p>
<p>Since <span class="math-container"... |
86,536 | <p>Considering we have a an association:</p>
<pre><code>assc = <|"A" -> <|"a" -> 1, "aa" -> 2|>, "B" -> 0, "C" -> 5,"D" -> <|"d" -> 2, "dd" -> 12|>|>
</code></pre>
<p>Let's also consider we have 2 known lists and one list for nested keys
(**
- <em>how to create this list?</... | WReach | 142 | <p><code>MapIndexed</code> can be used to map a function to the inner-most values in nested associations, for example:</p>
<pre><code>MapIndexed[f, <| "A" -> <|"a" -> 1|>|>, {-1}]
(* <|"A" -> <|"a" -> f[1, {Key["A"], Key["a"]}]|>|> *)
</code></pre>
<p>The level specification <code... |
3,300,469 | <p>I have a problem counting all the possible ways of "pairing" two datasets of size n and m, including partial pairing. </p>
<p>Example:
Assume we have two sets <span class="math-container">$\{A,B\}$</span> and <span class="math-container">$\{1,2,3\}$</span>. My aim is to find all ways of pairing letters with numbers... | JMoravitz | 179,297 | <p>Assume our two sets are <span class="math-container">$A$</span> and <span class="math-container">$B$</span> where <span class="math-container">$A\cap B = \emptyset$</span>
Break into cases based on the number of pairs used. With <span class="math-container">$k$</span> pairs used, choose which <span class="math-cont... |
812,345 | <p><a href="https://math.stackexchange.com/a/87705/53259">This</a> proves: Similar matrices have the same characteristic polynomial. (Lay P277 Theorem 4)</p>
<p>I prefer <a href="https://math.stackexchange.com/a/8407/53259">https://math.stackexchange.com/a/8407/53259</a>, but this proves that they have the same eigenv... | Omran Kouba | 140,450 | <ol>
<li>If $A$ and $B$ have the same characteristic polynomial, then clearly the have the same eigenvalues, these are the zeros of the characteristic polynomial.</li>
<li>The converse is generally not true: for example
$$
A=\left[\matrix{1&0&0\cr
0&0&1\cr 0&0&0}\right],\quad
B=\left[\matrix{1&a... |
1,248,068 | <p>Let $S$ be a set of cardinality $\aleph_1$. Consider the directed family $\mathcal{C}$ (here <em>directed</em> means <em>directed with respect to the inclusion</em>) of all countably infinite subsets of $S$. Suppose that</p>
<p>$$\mathcal{C} = \bigcup_{n=1}^\infty \mathcal{C}_n$$</p>
<p>for some families $\mathcal... | hot_queen | 72,316 | <p>Yes. Letting $S = \omega_1$, one of the $C_n$'s must contain uncountably many ordinals and hence an uncountable linearly ordered subfamily. </p>
<p>Also not every uncountable family contains one. For example, an almost disjoint family.</p>
|
1,254,189 | <p>I know that I have to study the order of every element in $\mathbb{Q/Z}$.
But what do I do?
I've been struggling of what to do for this question</p>
| Dietrich Burde | 83,966 | <p>The abelian group $(\mathbb{Q},+)$ is torsion-free, but the abelian group
$(\mathbb{Q/Z},+)$ is not torsion-free. Hence these two groups cannot be isomorphic. Also, the $\mathbb{Z}$-module $\mathbb{Q}$ is flat, whereas $\mathbb{Q/Z}$ is not flat.</p>
|
59,846 | <p>In "The New Book of Prime Number Records", Ribenboim reviews the known results on the degree and number of variables of prime-representing polynomials (those are polynomials such that the set of positive values they obtain for nonnegative integral values of the variables coincides with the set of primes). For exampl... | Alon Amit | 308 | <p>Given all the evidence so far, I'm inclined to declare Ribenboim's parenthetic remark a typo. He probably meant either "it cannot be 1" or "it must be at least 2". It would be confusing and unusual for him to mention this off-hand with no reference as if it's a simple observation. <em>That</em> it certainly isn't. <... |
409,689 | <p>I have $(x_1, y_1), (x_2, y_2)$.</p>
<p>How do I find the point that's $d$ distance away from $(x_1, y_1)$ on a straight line to $(x_2, y_2)$?</p>
<p>I know I can get the length of the line with Pythagoras. I know if I drew a circle I could use the radius as distance and the point would be where the line and the c... | George V. Williams | 54,806 | <p>Consider a circle with radius $d$ and center $(x_1, y_1)$. This is the equation:</p>
<p>$$ (x - x_1)^2 + (y - y_1)^2 = d^2 $$</p>
<p>Let $m$ be the slope of the line from $(x_1, y_1)$ to $(x_2, y_2)$. ($m = \dfrac{y_2 - y_1}{x_2 - x_1}$). Our line must satisfy the equation:</p>
<p>$$ y - y_1 = m(x - x_1) $$
$$ y ... |
3,865,388 | <p>Let <span class="math-container">$A$</span> be the set of all <span class="math-container">$2\times2$</span> boolean matrices and <span class="math-container">$R$</span> be a relation defined on <span class="math-container">$A$</span> as <span class="math-container">$M \mathrel{R} N$</span> if and only if <span clas... | J.-E. Pin | 89,374 | <p>It looks like your structure is simply <span class="math-container">${\Bbb B}^4$</span>, the product of four copies of the Boolean lattice <span class="math-container">${\Bbb B} = \{0, 1\}$</span>, ordered by <span class="math-container">$0 \leqslant 1$</span>. Thus, yes, <span class="math-container">$(A, R)$</span>... |
3,865,388 | <p>Let <span class="math-container">$A$</span> be the set of all <span class="math-container">$2\times2$</span> boolean matrices and <span class="math-container">$R$</span> be a relation defined on <span class="math-container">$A$</span> as <span class="math-container">$M \mathrel{R} N$</span> if and only if <span clas... | Hans | 602,799 | <p>In general, if <span class="math-container">$\{ L_\alpha \}$</span> is a family of lattices, then the Cartesian product <span class="math-container">$\prod_{\alpha} L_{\alpha}$</span> also has the structure of a lattice given by <span class="math-container">$\mathbf{x} \wedge \mathbf{y} = (x_\alpha \wedge y_\alpha)_... |
1,398,956 | <p>I saw from literature that the expected value of a random variable $f(X)$ is either $E f(X)$, $E(f(X))$ or $E[f(X)]$. Is there a standard which one notation should one use? Is the expected value a function $f(X)\to\mathbb R$?</p>
| Karl | 203,893 | <p>I've seen $\langle X \rangle$ used as well for expected value. I quite like this as it makes moment generating functions look nice $\langle e^{tX} \rangle$ only has one 'e' compared with $\mathbb{E}\left[e^{tX}\right]$
I guess that's your answer, people choose notation to balance form and function. I suspect there ... |
3,100,957 | <p>A fair coin is tossed until one of the patterns show up: TTH or THT.
Let A be the event that TTH shows up before THT.</p>
<p>What is P(A)?</p>
<p>Here is my solution but I am not sure if it is correct or there is a better solution.</p>
<p>Let <span class="math-container">$p=P(A)$</span>. Define </p>
<p><span cl... | DanielWainfleet | 254,665 | <p>If <span class="math-container">$(X,d)$</span> is a metric space and <span class="math-container">$Y\subset X$</span> then for <span class="math-container">$Y$</span> to be compact it is necessary that <span class="math-container">$Y$</span> is bounded. That is, it is necessary that <span class="math-container">$B... |
2,995,327 | <p>Suppose a,b ∈ Z. If 4 | <span class="math-container">$(a^2 + b^2)$</span> then a and b are not both odd.</p>
<p>So, assuming that 4 | <span class="math-container">$(a^2 + b^2)$</span> and <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are odd</p>
<p>this gives <span class="mat... | Martund | 609,343 | <p>This is invalid. You have done a mistake in calculation.
RHS will be <span class="math-container">$4(l^2+u^2+l+u)+2$</span>, which is not divisible by 4.
Hope it helps:)</p>
|
1,199,304 | <p>Let $M\neq \{0\}$ be a semi-simple left $R$ module .Prove that it contains a simple sub-module.</p>
<p>An $R-$ module $M$ is said to be semi-simple if every submodule of $M$ is a direct summand of M
<strong>My solution</strong></p>
<p>Since $M\neq \{0\}$; $\exists m\in M$ such that $m\neq 0$.Then I can consider th... | Math137 | 60,099 | <p>Your definition is equivalent to "An $R$-module $M$ is semisimple if it can be written as a direct sum of family of simple modules" for example <a href="http://nptel.ac.in/courses/111102009/module2/lec4.pdf" rel="nofollow">see here page 2</a> for the proof, in which your method is subsidised by the proof of the theo... |
3,965,164 | <p>I know the standard and expanded forms of the equation of the circle in the simple 2d space,</p>
<p><span class="math-container">${(x-a)}^2+{(y-b)}^2=r^2$</span></p>
<p><span class="math-container">$x^2-2ax+y^2-2by=c$</span></p>
<p>So in 3d space what are the equations for a circle laying in an arbitrary plane,
and ... | Brian M. Scott | 12,042 | <p>In order to make the intersection equal to <span class="math-container">$\Bbb R\setminus\Bbb Q$</span>, you need to make sure that each <span class="math-container">$A_n$</span> contains every irrational number, and that each rational number is excluded from at least one <span class="math-container">$A_n$</span>. Th... |
2,545,516 | <p>So I have to assess the convergence of $$\displaystyle\sum_{n=1}^{\infty}\sin\left(\displaystyle\frac{1}{\sqrt{n}}\right).$$</p>
<p>I'm told that it diverges, but can't really see why.</p>
<p>The divergence test doesn't really help, because
$\lim\limits_{x\to\infty}\displaystyle\frac{1}{\sqrt{n}}=0$, so</p>
<p>... | Angel Moreno | 327,493 | <p>The function $\sin(x) \geq\dfrac{2}{\pi}x$ for $x\in[0,\pi/2]$</p>
<p>$\sum_{n} 1/n^{1/2}$ diverge</p>
<p>$\forall n: \sin(1/n^{1/2}) \geq \dfrac{2}{\pi n^{1/2}}$</p>
|
1,796,792 | <p>Is $\log_27$ a rational number?</p>
| DanielWainfleet | 254,665 | <p>If $a,b$ are positive integers and $2^{a/b}=7$ then $2^a=7^b$ which make an even number equal to an odd number.</p>
|
52,079 | <p>I'm in doubt about the topology of maps between fibres of vector bundles.</p>
<p>Consider $E$ and $F$ vector bundles and the set of all linear maps from a fibre of $E$ to a fibre of $F$, ie, the set of all linear maps $T:E_x \rightarrow F_y$, where $E_x$ is the fiber over $x$ and $F_y$ is the fiber over $y$.</p>
<... | HenrikRüping | 3,969 | <p>If both bundles were trivial, say $X\times \mathbb{R}^n\rightarrow X$ and $Y\times \mathbb{R}^n\rightarrow Y$ one could just take $X\times Y\times M(m,n,\mathbb{R})$.
Different choices of trivializations should give homeomorphic spaces, so this topology seems to be right.</p>
<p>In general the bundles are just loca... |
52,079 | <p>I'm in doubt about the topology of maps between fibres of vector bundles.</p>
<p>Consider $E$ and $F$ vector bundles and the set of all linear maps from a fibre of $E$ to a fibre of $F$, ie, the set of all linear maps $T:E_x \rightarrow F_y$, where $E_x$ is the fiber over $x$ and $F_y$ is the fiber over $y$.</p>
<... | Patrick I-Z | 11,885 | <p>I elaborate a little bit on what your question inspires me. I will treat a more general question, but you can reduce it to finite linear dimensional fiber bundles over manifolds. Let $\pi : E \to X$ and $\pi' : E' \to X$ be two projections (smooth in some sense, and you can assume that you deal with manifolds). You ... |
2,365,933 | <p>I'm aware of how we can simplify functions which have $Arc$ as an argument . For example $\sin(\cos^{-1}(x)) = \sqrt{1-x^2}$ but what about cases which $Arc$ is out of the parentheses ? For instance consider this : $\sin^{-1}(\tan x)$ . Is there any way for simplification ? </p>
| Vassilis Markos | 460,287 | <p>Well, let $f(x)=\arcsin(\tan(x))$, $x\in(-\frac{\pi}{4},\frac{\pi}{4})$. Now, since:
$$\arcsin(x)=\int_0^{x}\frac{1}{\sqrt{1-t^2}}dt$$
we have, that:
$$f(x)=\int_0^{\tan(x)}\frac{1}{\sqrt{1-t^2}}dt$$
So:
$$\begin{align*}f'(x)=&\frac{1}{\sqrt{1-\tan^2(x)}}\frac{1}{\cos^2(x)}=\frac{1}{\cos^2(x)\sqrt{1-\frac{\sin^2... |
376,796 | <p>This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures which have a reasonably simple mathematical formalization and even better if there is a related diagram that explains ... | Gabe K | 125,275 | <p>Mohammed Ghomi's <a href="https://mathoverflow.net/a/376809/2383">answer</a> reminds me of a related picture that Cedric Villani drew to depict Ricci curvature ([1] Chapter 14). Similar to the <span class="math-container">$\operatorname{CAT}(\kappa)$</span> inequality, this idea can be used to derive notions of Ricc... |
4,338,382 | <blockquote>
<p>Let <span class="math-container">$a,b>0$</span>. Prove that: <span class="math-container">$$\frac{1}{a^2}+b^2\ge\sqrt{2\left(\frac{1}{a^2}+a^2\right)}(b-a+1)$$</span></p>
</blockquote>
<p>Anyone can help me get a nice solution for this tough question?
My approach works for 2 cases:</p>
<p>Case 1: <sp... | Rushabh Mehta | 537,349 | <p>Well, let's run through the possibilities.</p>
<p><strong>Case 1</strong>: A knight</p>
<p>In this case, B is a spy and C is a knave.</p>
<p><strong>Case 2</strong>: A knave</p>
<p>In this case, B is not a spy, and thus a knight.</p>
<p><strong>Case 3</strong>: A spy</p>
<p>In this case, C is a knave, and B is a kni... |
9,918 | <p>I recently flagged as "rude or offensive" the comment </p>
<blockquote>
<p>As the tone should suggest, he’s a crank. It’s a hysterical screed with a few nuggets of fact surrounded by a great deal of nonsense. E.g., he may find that set theory ‘doesn’t make sense’, but a great many of us have no trouble making sen... | Stephen | 146,439 | <p>It seems that flagging a comment in which (you believe) a baseless accusation of crankdom has been made is the only recourse. If the moderators doesn't agree with you, it's probably best to shrug and move on. </p>
<p>At the moment, this question has 3 votes to close, so I am answering it even though it doesn't seem... |
275,539 | <p>Kind of leading on from my other question, how would I solve for $i$? Or how would I check that it is possible to have such an $i$?</p>
<p>First I had to check for all $2^i$ and clearly this doesn't happen as all $2^i$ are even and so I will just get even $x's$ such that $2^i \equiv x \mod 28$. So the next one I go... | Calvin Lin | 54,563 | <p>To solve $a^i \equiv 1 \pmod{n}$, first note that we must have $\gcd(a,n) = 1$ in order to get a positive (non-zero) integer solution. Like you realized, since $\gcd(2, 28)=2$, thus $2^i$ will always be a multiple of $2$, and hence cannot be of the from $28k+1$.</p>
<p>Given that condition, such an $i$ always exist... |
200,903 | <p>My teacher was explaining quadratics in my class and it was a little bit unclear to me. The problem was <br> <br>
Suppose $at^2 + 5t + 4 > 0$, show that $a > 25/16$ . <br> <br></p>
<p>My teacher said that there are no solutions for this function when it is greater than $0$ and used $b^2-4ac \lt 0$, and this ... | Thomas | 26,188 | <p>Edit: I will try to say something general first. Hopefully that will make my answer a bit easier to understand. But as a first thing, it might be helpful to take a look at for example the Wikipedia article on <a href="http://en.wikipedia.org/wiki/Quadratic_equation" rel="nofollow">quadratic equations</a>.</p>
<p>If... |
3,826,994 | <p>I would like to find <span class="math-container">$z$</span> which minimizes the below, when <span class="math-container">$x$</span> is held at a specific value.</p>
<p><span class="math-container">$f(x,z) =\sqrt{\sqrt{x^2 + z^2} - 0.25}$</span></p>
<p>For example; I would like to find the value of <span class="math... | Toby Mak | 285,313 | <p>You have not made a mistake. Observe that <span class="math-container">$x$</span> is a multiple of <span class="math-container">$y$</span>:</p>
<p><span class="math-container">$$x = \frac{-2b + 6b^2}{1 + b^2} = -b \cdot \frac{2 - 6b}{1 + b^2} = -by$$</span></p>
<p>which is what the answer states.</p>
|
2,512,294 | <p>So I've been given the following problem:</p>
<p>How many positive integers are there that can not be written as a sum of 5's and 7's? For example, 4 is one of those integers, but 19 is not because 19 = 5 + 7 + 7. How to solve this? </p>
| paw88789 | 147,810 | <p>Hints:</p>
<p>(1) Start at the beginning: Can you get $1$, $2$,...</p>
<p>(2) If you can write a number $n$ as a sum of $5$s and $7$s, you can write $n+5$ as a sum of $5$s and $7$s.</p>
<p>(3) If you ever achieve five consecutive numbers that you can write as a sum of $5$s and $7$s, what will hint (2) allow you ... |
38,659 | <p>I know how to use Matrix Exponentiation to solve problems having linear Recurrence relations (for example Fibonacci sequence). I would like to know, can we use it for linear recurrence in more than one variable too? For example can we use matrix exponentiation for calculating ${}_n C_r$ which follows the recurrence ... | Shahab | 10,575 | <p>The product rule is that if we have x ways of doing something and y ways of doing another thing, then there are xy ways of performing both actions. Here's how you can think of the product rule. The things in question are choosing a first chapter and choosing a second chapter.</p>
<p>First consider each of the ways ... |
3,490,329 | <blockquote>
<p>Show that a 2-dimensional subspace of the space of <span class="math-container">$2\times2$</span> matrices contains a non-zero symmetric matrix. </p>
</blockquote>
<p>I don't know if it should be written like the addition of two symmetric and skew-symmetric matrix or there is another way to show it. ... | almagest | 172,006 | <p>Take any two linearly independent matrices in the subspace:</p>
<p><span class="math-container">$\begin{pmatrix}a_1 & b_1\\ c_1 & d_1\end{pmatrix}$</span> and <span class="math-container">$\begin{pmatrix}a_2 & b_2\\ c_2 & d_2\end{pmatrix}$</span></p>
<p>If <span class="math-container">$b_1=c_1$</sp... |
2,996,920 | <p>Please recommend me a good book to study interpolation techniques such as polynomial interpolation, cubic, spline interpolations, if possible tell me the branch of mathematics that deals with this subject. I want to go in depth with this topic.</p>
| cqfd | 588,038 | <p>The topics mentioned above are usually dealt under numerical analysis. Some of the textbooks are:</p>
<ol>
<li>S.D.Conte and C.deBoor, <em>Elementary Numerical Analysis-an algorithmic approach</em>,
3rd Edn., McGraw Hill, 1980.</li>
<li><p>K. E. Atkinson, <em>An Introduction to Numerical Analysis</em>, 2nd Edn., Jo... |
1,270,042 | <p>$$(a+5)(b-1)=ab-a+5b-5=20-5=15.$$</p>
<p>So, both $a + 5$ and $b-1$ divide $15$. </p>
<p>Then, $a + 5$ is one of $15, -15, 3, -3, 5, -5, 1, -1$, so $a$ is one of $10, -20, -2, -8, 0, -10, -4, -6$ and $b – 1$ is one of $15, -15, 3, -3, 5, -5, 1, -1$, so $b = 14, -14, 4, -2, 6, -4, 2, 0$.</p>
<p>Could all possibili... | mathlove | 78,967 | <p>Having $$(a+5)(b-1)=15$$
gives you $$(a+5,b-1)=(1,15),(3,5),(5,3),(15,1),(-1,-15),(-3,-5),(-5,-3),(-15,-1),$$
i.e.
$$(a,b)=(-4,16),(-2,6),(0,4),(10,2),(-6,-14),(-8,-4),(-10,-2),(-20,0).$$
Since $a,b$ are positive integers, $(a,b)=(10,2)$ is the only solution.</p>
|
2,512,363 | <h2>Defining the barycentre and finding its variance</h2>
<p>I have a set of $N$ points at the locations $x_i$ which has weights $W_i$, $i=1,\ldots, N$ and want to find the barycenter (or center of gravity) </p>
<p>$$ B = {\sum_{i=1}^N W_i x_i\over \sum_{i=1}^N W_i}.$$</p>
<p>I want to find the variance of $B$.
I a... | Dean | 393,411 | <p>There are a couple of mistakes in your work here.</p>
<p>In your derivation you replace the denominator, $\sum W_i$ with its expectation value. You say you do this because it is a good approximation, but you are ignoring the variance it contributes. Would you still do that if the numerator was a constant? If so, yo... |
817,386 | <p>For a function $f$ I know that: $$\int{f'(r)dr}=f(r)$$ where $f(r)$ is known. knowing the result of this integral how can i calculate $$\int{(f'(r))^2dr}$$ Is there any relation between these integrals?</p>
| Gerry Myerson | 8,269 | <p>Let $f(x)=e^{x^2}$, so $(f'(x))^2=4x^2e^{2x^2}$. But $\int4x^2e^{2x^2}\,dx$ can't be evaluated in terms of elementary functions (exponentials, trig functions, polynomials, $n$th roots, etc.). </p>
|
226,551 | <p><strong>(1)</strong> Which graph classes are extremely tough to test for graph non-isomorphic pairs from isomorphic pairs?</p>
<p><strong>(2)</strong> Is there a repository of adjacencies from such classes?</p>
| joro | 12,481 | <p>Paper: <a href="http://mivia.unisa.it/wp-content/uploads/2013/05/desanto02.pdf" rel="noreferrer">A large database of graphs and its use for benchmarking
graph isomorphism algorithms</a></p>
<p>The graph instances and graph generator software are here:</p>
<p><a href="http://mivia.unisa.it/datasets/graph-database/... |
671,407 | <p>I have problem with equation: $4^x-3^x=1$. </p>
<p>So at once we can notice that $x=1$ is a solution to our equation. But is it the only solution to this problem? How to show that there aren't any other solutions? </p>
| imranfat | 64,546 | <p>You can make a graph of the function $y=4^x-3^x-1=0$. It has a y-intercept at (0,-1) and for $x<0$ the curve stays under the y-axis and for $X>0$ the curve is only increasing. </p>
|
129,293 | <p>I'm writing a survey that involves Levy processes and wanted to mention the different forms of the Levy-Khintchine formula found in literature.</p>
<p>The most common version seems to give the Levy symbol as</p>
<p>$$\Psi(u) = i\langle b,u \rangle - \frac{1}{2} \langle u,\Sigma u\rangle + \int_{\mathbb{R}^d} {(} e... | Community | -1 | <p>Here is part of <strong>Exercise 3.2.40</strong> from <em>Probability Theory: An Analytic View</em> by Daniel W. Stroock. He refers to
$$\int_{\{0<|y|<1\}} |y|\,K(dy)+K((-1,1)^c)<\infty\tag{3.2.2}$$ </p>
<blockquote>
<p>The difficulty of distinguishing between the drift and small jumps when (3.2.2) fai... |
1,689,523 | <p>I need help with this Laplace question.
<span class="math-container">$$f(t) = e^{-t} \sin(t) $$</span></p>
<hr />
<p>Answer should be <span class="math-container">$\dfrac{1}{s^2 + 2s + 2}$</span></p>
<hr />
<p>What I'm currently doing is as follows:</p>
<p><span class="math-container">$u = \sin(t)\qquad$</span> ... | Mark Viola | 218,419 | <p>You need not integrate by parts to evaluate this integral. In fact, one would need to integrate by parts twice. <strong>See the section following the highlighted SPOILER ALERT</strong></p>
<p>So, I thought it would be instructive to present a "trick" that we can use to quickly evaluate the integral of in... |
3,031,290 | <p>Can you choose <span class="math-container">$11$</span> different numbers among them so that the numbers <span class="math-container">$|a_1-a_2|, |a_2-a_3|,\ldots,|a_{10}-a_{11}|,|a_{11}-a_{1}|$</span> are all different. The smartest thing that my dumbest mind could accomplish is that all those differences are <span... | saulspatz | 235,128 | <p>I haven't been able to do this by any reasonable method. I worked out that the sum of the numbers that are bigger than both their neighbors (considering the numbers as arranged on a circle) minus the sum of the numbers that are smaller than both their neighbors must be <span class="math-container">$33,$</span> but ... |
4,177,639 | <p>I have an object with known coordinates in in 3D but on the ground (<code>z=0</code>). The object has a direction vector. My goal is to move this object on the ground (so <code>z</code> stays <code>0</code>) using its direction vector and via randomly-generated velocity vectors with one condition: I want to ensure t... | Claude Leibovici | 82,404 | <p>In the most general case, there is no analytical solution for the zero of function
<span class="math-container">$$f(x)=a^x+b^x-1$$</span> and numerical iterative methods would be required.</p>
<p>If wa assume <span class="math-container">$a>1$</span> and <span class="math-container">$b>1$</span>, <span class="... |
1,908,923 | <p>Let $X$ be a Riemannian manifold*, and $S$ a compact submanifold of $X$. </p>
<p>Assume there exists an <strong>open, dense</strong> subset $Y$ of $\,X$, such that for any element $y \in Y$, there exists a unique element in $S$ closest to $y$; i.e there is a function $\tilde s:Y \to S$ such that $$ d(y,\tilde s(y))... | HK Lee | 37,116 | <p>Yes </p>
<p>$c(t)$ is a shortest geodesic from $c(0)=s_0\in S$ to $c(1)=x$ For some $0<t_0<1$ assume that $$d(c(t_0),S)=d(c(t_0),s_1) \leq d(c(0),c(t_0)),\ s_1\in S$$</p>
<p>Then $$ d(x,S)\geq d(c(t_0),x) + d(c(t_0),s_1) \geq d(x,S) $$</p>
<p>It is a contradiction So $s_1=s_0$ If $y_n\in Y$ goes to $ c(t_0)... |
1,185,108 | <p>empty set is an subset of any sets maybe any collection of sets.</p>
<p>I wonder what about the case of the empty set being a member,not subset, of any collection (family) of sets.</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Counterexample: the only element of $\{\{\emptyset\}\}$ is $\{\emptyset\}\ne\emptyset$.</p>
|
1,257,598 | <p>Suppose A is a family of subsets of R with the property that the intersection of any two sets in A is finite. Show that $|A|\leq 2^{\aleph_0}$.</p>
<p>I was told that choosing a countable $D \subset B$ for all $B \in A$ would be helpful. I'm just really not sure where to go with this. Any hints would be appreciated... | Brian M. Scott | 12,042 | <p>Here’s an argument that’s a bit closer to the hint. Suppose that $|A|>2^{\aleph_0}$. $\Bbb R$ has only $2^{\aleph_0}$ finite subsets, so without loss of generality we may assume that every member of $A$ is infinite. For each $B\in A$ let $C_B$ be a countably infinite subset of $A$. $\Bbb R$ has $(2^{\aleph_0})^{\... |
1,297,690 | <p>If I have a program that creates, let's say, one billion integers, with each having a pure $50 - 50$ chance to be one or zero,</p>
<p>what is the chance of finding $x$ zeros in a row?</p>
<p>for brownie points, instead of the program creating a set billion numbers, what would the equation be with $z$ numbers?</p>
| Nick Peterson | 81,839 | <p><strong>Hint:</strong> If you never had $x$ zeros in a row, then (at least) one of the first $x$ numbers must be a one.</p>
<p>If $n_z$ is the number of arrangements, then you can partition $n_z$ based on which digit (of the first $x$) is the first to be a one. This will give you a recurrence relation in terms of ... |
942,030 | <p>Given a Hilbert space $\mathcal{H}$.</p>
<p>Consider spectral measures:
$$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad E(\mathbb{C})=1$$</p>
<p>Define its support:
$$\operatorname{supp}(E):=\bigg(\bigcup_{U=\mathring{U}:E(U)=0}U\bigg)^\complement=\bigcap_{C=\overline{C}:E(C)=1}C$$</p>
<p>By second c... | Community | -1 | <p>$\frac{d(\dot\theta^2)}{dt}$ is not obtained from $b'(\theta)\dot\theta$, but from $b(\theta)=\dot\theta^2$, by derivation.</p>
|
4,616,559 | <p>For each day we store a snapshot of data in a database. We want to balance the storage costs with the densitiy of snapshots.
The older a time frame is the fewer snapshots from this time frame we need.
For example: if we store 10 snapshots from last year then we would like to store only 1 snpashot from the time ten y... | joriki | 6,622 | <p>When you need to delete a snapshot, delete the one with minimal ratio <span class="math-container">$T/t$</span>, where <span class="math-container">$t$</span> is the time of the snapshot and <span class="math-container">$T$</span> is the length of the interval without snapshot that would result from deleting it.</p>... |
659,256 | <p>This might be a silly question to some, but I need some help in this topic. <br />
Iota, denoted as <em>'i'</em> is equal to the principal root of -1.
Therefore, </p>
<p>$\iota^2 = -1$</p>
<p>When studying Modulus, I was wondering..</p>
<p>$|\iota| = ?$</p>
<p>A Google search revealed that the value is <strong>... | Kindeep Singh Kargil | 422,502 | <p>Well, we can define modulus as <span class="math-container">$|x|$</span>= <span class="math-container">$\sqrt{x^2}$</span>.</p>
<p>So, since i=<span class="math-container">$\sqrt{(-1)}$</span></p>
<p>|i|=<span class="math-container">$\sqrt{(-1)^2}$</span></p>
<p>=<span class="math-container">$\sqrt{1}$</span></p>... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Henry | 6,460 | <p>Imagine a vertical sea cliff. Floating in the sea some distance away
from the cliff there is a boat, which is attached to a rope which goes
diagonally up to the top edge of the cliff and then continues on the
field at the top to a tractor. (Draw a picture.) The tractor then moves away from the
cliff edge, pulling ... |
3,779,589 | <p>Let the metric <span class="math-container">$d$</span> be defined as
<span class="math-container">$$
d(f,g) =\sup_{x\in[0,1]}|f(x)-g(x)|,
$$</span>
and let<br />
<span class="math-container">$$
H(x) = \begin{cases} 0 \text{ if } x \leq \frac{1}{2}\\ 1 \text { if } x > \frac{1}{2} \end{cases}.
$$</span>
Is <span c... | Àlex Rodríguez | 813,535 | <p>I think that is much easier to think in terms of the euler function. We know that when n=200, this function give us the value 80, so there are 80 numbers between 1 and 200 that do not have a 2 or 5 in its factorisation, and the problem is over.</p>
|
3,779,589 | <p>Let the metric <span class="math-container">$d$</span> be defined as
<span class="math-container">$$
d(f,g) =\sup_{x\in[0,1]}|f(x)-g(x)|,
$$</span>
and let<br />
<span class="math-container">$$
H(x) = \begin{cases} 0 \text{ if } x \leq \frac{1}{2}\\ 1 \text { if } x > \frac{1}{2} \end{cases}.
$$</span>
Is <span c... | Jan Eerland | 226,665 | <blockquote>
<p>Not a 'real' answer, but it was too big for a comment.</p>
</blockquote>
<p>I wrote and ran some Mathematica code:</p>
<pre><code>In[1]:=Length[ParallelTable[
If[TrueQ[If[IntegerQ[n/2] \[Or] IntegerQ[n/5], True, False]],
Nothing, n], {n, 1, 200}]]
</code></pre>
<p>Running the code gives:</p>
<pre>... |
2,469,798 | <p>Let $S = \left\{x \in \mathbb{Q} \mid 1 \leqslant {x}^2 \leqslant 29 \right\}$</p>
<p>What can we say about the supremum and infimum of this set? Would it be non-existent?</p>
<p>Would it be correct to say the following?</p>
<p>Suppose $ \sup S < \sqrt{29} $ then $ \exists x \in S $ such that $ x > \sup S$... | orangeskid | 168,051 | <p>Let's consider the set of rational numbers
$$\{ r \in \mathbb{Q} \mid r \ge 1 \text{ and } r^2 \le 29\}$$</p>
<p>The supremum of the set equals $\sqrt{29}$. Perhaps it is more interesting to show that there does not exist a supremum of this set in $\mathbb{Q}$. That is in some way obvious. But we may still play wi... |
34,215 | <p>How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues? </p>
| Gerry Myerson | 3,684 | <p>Talking to colleagues is good. Also, attending talks, reading papers and books, and teaching - I never knew much about differential equations until my department made me teach it (I still don't know much about differential equations, but at least I know enough to do a decent job of teaching it). </p>
|
188,087 | <p>Is there a function that can extract a list of variables in an expression?
For example, assume we have an expression</p>
<pre><code>x^2+y^3+z
</code></pre>
<p>This expression has variables x, y and z. The result should be</p>
<pre><code>{x, y, z}
</code></pre>
<p>. Is there a way to get this?</p>
| TimRias | 10,587 | <p>For polynomial expressions @Buddha_the_Scientist's suggestion <code>Variables</code> will work. For more general expressions</p>
<pre><code>expr = x^2 + y^3 + z
DeleteDuplicates@Cases[expr, _Symbol, ∞]
</code></pre>
<p>Should do the trick in most situations.</p>
|
188,087 | <p>Is there a function that can extract a list of variables in an expression?
For example, assume we have an expression</p>
<pre><code>x^2+y^3+z
</code></pre>
<p>This expression has variables x, y and z. The result should be</p>
<pre><code>{x, y, z}
</code></pre>
<p>. Is there a way to get this?</p>
| Michael E2 | 4,999 | <p>The undocumented <code>Integrate`getAllVariables</code> is a somewhat more robust version of <code>Variables</code>. It has a required second argument that specifies a variable to be excluded from the output. It just goes to show that internal functions are not always defined with the general user in mind.</p>
<pr... |
1,913,320 | <blockquote>
<p>Let <span class="math-container">$A=(a_{ij})_{n\times n}$</span> and <span class="math-container">$A=(a_{ij})_{n\times n}$</span> be two upper triangular matrices, i.e. <span class="math-container">$a_{ij}=b_{ij}=0$</span> whenever <span class="math-container">$i>j$</span>.</p>
<p><span class="math-c... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Suppose for contradiction that $\,2\,$ and $\,7\,$ are the only primes. Then $\,15 = 2*7+1\,$ has no smaller prime factors so is prime, contradiction. But $15$ is not prime in the <em>real</em> integers. Rather, it is prime only in the <em>hypothetical</em> integers having only the primes ... |
3,524,550 | <p>Lines in <span class="math-container">$\mathbb{R}^3$</span> are all congruent to one another, but circles in <span class="math-container">$\mathbb{R}^3$</span> are not all congruent to one another (because two different circles may have different radii). Visually, this is completely obvious. However, I would like ... | Jean Marie | 305,862 | <p>This is not a direct answer to your question, but an enlargment of its scope.</p>
<p>There exists a group which is transitive on the union of lines and circles : it is the <strong>anallagmatic group</strong>, existing in any dimension. I mention it in the third paragraph of <a href="https://math.stackexchange.com/q... |
1,272,499 | <p>Definite integral of $$\int_0^{2\pi} \frac{1}{2+\cos x}$$ without using improper integral, I want to solve this without having to use $-\infty$ and $\infty$ on the integrals limits. Is that possible?</p>
<p>The only way I can think of solving that is by using Weierstrass. $u = \tan \frac{x}{2}$, don't you have to m... | mickep | 97,236 | <p>Split the integral as $\int_0^{2\pi}=\int_0^{\pi}+\int_{\pi}^{2\pi}$. Do the substitution $u=x-\pi$ in the second one, and put the expressions under common denominator, and simplify. You should end up with
$$
4\int_0^{\pi}\frac{1}{4-\cos^2x}\,dx
$$
The integrand is now symmetric in $x=\pi/2$, so the integral equals
... |
1,272,499 | <p>Definite integral of $$\int_0^{2\pi} \frac{1}{2+\cos x}$$ without using improper integral, I want to solve this without having to use $-\infty$ and $\infty$ on the integrals limits. Is that possible?</p>
<p>The only way I can think of solving that is by using Weierstrass. $u = \tan \frac{x}{2}$, don't you have to m... | Jack Tiger Lam | 186,030 | <p>Make the substutition</p>
<p>$x \mapsto 2\theta$</p>
<p>$$\int_0^\pi \frac{2\text{d}\theta}{\sin^2{\theta}+3\cos^2{\theta}}$$</p>
<p>The polar integral for (half) the area of an ellipse is almost of this form, but some adjustments are required before we can proceed.</p>
<p>Factor out the 2 and insert a factor of... |
2,745,918 | <p>The numbers $1,2, \ldots, n$ are written in a board, with $n \in \mathbb{N}$. In every move, we can choose two numbers of the board, find their $\rm lcm$, and replace the two numbers with it. After $k$ moves, we find the sum of the numbers in the board, and we name it $S$. Find the minimum and the maximum value of $... | didgogns | 392,996 | <p>It is possible to make $S=2017$ starting from $n=11$. Merge $4, 5, 9, 11$ in any order to make the sequence $1, 2, 3, 6, 7, 8, 10, 1980$ which achieves the sum $2017$.</p>
<p>Now what is left is to prove that it is impossible to achieve the sum of $2017$ with $n=10$. Since $\gcd(1, \cdots, 10)=2520>2017$, there ... |
2,507,613 | <p>I am trying to teach myself group theory and I recently came across the topic of Isomorphisms.
I know that 2 groups are isomorphic if there is a one-on-one correspondence between their elements.
So if the groups have a different order, does that mean they are not isomorphic? Such as a group $S$ and its permutation g... | Community | -1 | <p>If two groups are isomorphic then their orders are the same. It follows from the fact that isomorphism between groups is a bijection. </p>
|
3,145,896 | <h1>Solve for <span class="math-container">$x$</span></h1>
<p>I have an equation that I have been working on solving; I know the solution, but I cannot get to it myself. Almost every simplification I do reverts back to a previous step. Can anyone show me how to solve for <span class="math-container">$x$</span> in this... | heropup | 118,193 | <p>If <span class="math-container">$x \in \mathbb R$</span>, the equation <span class="math-container">$$\log_6 (2x-3) + \log_6 (x+5) = \log_3 x$$</span> requires <span class="math-container">$x > 3/2$</span>. Under such an assumption, the LHS becomes <span class="math-container">$$\log_6 (2x-3)(x+5),$$</span> and ... |
312,145 | <p>OK, so the question says evaluate the integral
$$\int_{0}^{\pi}\frac{x}{(a^2\cos^2x+b^2\sin^2x)^2}dx$$
What I do is use the property that $\int_a^bf(x)dx=\int_a^bf(b+a-x)dx$ and this gives me ($I$ is the value of the integral)
$$\frac{2I}{\pi}=\int_{0}^{\pi}\frac{1}{(a^2\cos^2x+b^2\sin^2x)^2}dx$$
What should I do ah... | Sangchul Lee | 9,340 | <p>I prefer to the following method:</p>
<p>\begin{align*}
I
:= \int_{0}^{\pi} \frac{x}{(a^2 \cos^2 x + b^2 \sin^2 x )^2} \, dx
&= \frac{\pi}{2} \int_{0}^{\pi} \frac{dx}{(a^2 \cos^2 x + b^2 \sin^2 x )^2} \\
&= \pi \int_{0}^{\frac{\pi}{2}} \frac{dx}{(a^2 \cos^2 x + b^2 \sin^2 x )^2} \\
&= \pi \int_{0}^{\fra... |
312,145 | <p>OK, so the question says evaluate the integral
$$\int_{0}^{\pi}\frac{x}{(a^2\cos^2x+b^2\sin^2x)^2}dx$$
What I do is use the property that $\int_a^bf(x)dx=\int_a^bf(b+a-x)dx$ and this gives me ($I$ is the value of the integral)
$$\frac{2I}{\pi}=\int_{0}^{\pi}\frac{1}{(a^2\cos^2x+b^2\sin^2x)^2}dx$$
What should I do ah... | Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
\because I&=\int_0^\pi \frac{x}{\left(a^2 \cos ^2 x+b^2 \sin ^2 x\right)^2} \\& \stackrel{x\mapsto\pi-x}{=} \int_0^\pi \frac{\pi-x}{\left(a^2 \cos ^2 x+b^2 \sin ^2 x\right)^2} d x \\
&=\pi \int_0^\pi \frac{d x}{\left(a^2 \cos ^2 x+b^2 \sin ^2 x\right)^2}-I ... |
1,505,920 | <p>(I think) My textbook says something is a linear transformation if </p>
<ul>
<li>$L(ax) = aL(x)$</li>
<li>$L(x+y) = L(x) + L(y)$</li>
<li>$L(x) = A(x)$</li>
</ul>
<p>But a lot of sites I've been on haven't proved these 3 things, so I just wanted to make sure that this is the proper way to prove it.</p>
<p>For exa... | Red | 232,153 | <p>Yes your textbook is right, basically a function is a linear transformation if and only if scalar multiplicity is reserved meaning that letting <span class="math-container">$a $</span> be a real number then</p>
<p><span class="math-container">$L(a*x)=a*L(x)$</span></p>
<p>In your example if you wanted to show this... |
1,505,920 | <p>(I think) My textbook says something is a linear transformation if </p>
<ul>
<li>$L(ax) = aL(x)$</li>
<li>$L(x+y) = L(x) + L(y)$</li>
<li>$L(x) = A(x)$</li>
</ul>
<p>But a lot of sites I've been on haven't proved these 3 things, so I just wanted to make sure that this is the proper way to prove it.</p>
<p>For exa... | Eric Wofsey | 86,856 | <p>This is almost right, but that third condition shouldn't be there (in fact, I don't know what it means in general--what is "$A$"?). That is, a map $L$ between two vector spaces is a linear transformation if and only if it satisfies $L(ax)=aL(x)$ and $L(x+y)=L(x)+L(y)$ (for any scalar $a$ and any elements $x,y$ in t... |
1,501,595 | <blockquote>
<p>Let $A$ be an integral domain. Show that $\dim(A)=0 \iff A$ is a field.</p>
</blockquote>
<p>The backward implication is trivial.</p>
<p>For the forward implication, if we can show that $1 \in <a>$, where $a(\neq 0) \in A$. Then, we are done. However, I don't know how to show it. </p>
<p>Any ... | Bernard | 202,857 | <p>This means $(0)$ is the only prime ideal, since $A$ is a domain. As the set of non-invertible elements is the union of all prime/maximal ideals, it implies all non-zero elements are invertible.</p>
|
493,104 | <p>I'm finding the area of an ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. I know the answer should be $\pi ab$ (e.g. by Green's theorem). Since we can parameterize the ellipse as $\vec{r}(\theta) = (a\cos{\theta}, b\sin{\theta})$, we can write the polar equation of the ellipse as $r = \sqrt{a^2 \cos^2{\thet... | André Nicolas | 6,312 | <p>Your question has been answered, so now we look at how to find the area, using your parametrization, which is a perfectly good one. </p>
<p>The area is the integral of $|y\,dx|$ (or alternately of $|x\,dy|$. over the appropriate interval.</p>
<p>We have $y=b\sin\theta$ and $dx=-a\sin\theta\,d\theta$. So the area i... |
493,104 | <p>I'm finding the area of an ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. I know the answer should be $\pi ab$ (e.g. by Green's theorem). Since we can parameterize the ellipse as $\vec{r}(\theta) = (a\cos{\theta}, b\sin{\theta})$, we can write the polar equation of the ellipse as $r = \sqrt{a^2 \cos^2{\thet... | Rogelio Molina | 87,320 | <p>This is another way to do it when one know the area of a circle: Consider the area of a circle with radius 1 in coordinates $(\xi, \eta)$ this is:</p>
<p>$$
\int d\xi d \eta = \pi
$$</p>
<p>now if you define new coordinates in your ellipse equation $\xi = \frac{x}{a}, \quad \eta= \frac{y}{b}$ you obtain a circle o... |
2,868,595 | <p>A Vitali set is a subset $V$ of $[0,1]$ such that for every $r\in \mathbb R$ there exists one and only one $v\in V$ for which $v-r \in \mathbb Q$. Equivalently, $V$ contains a single representative of every element of $\mathbb R / \mathbb Q$.</p>
<p>The proof I read is in this short article on Wikipedia: <a href="h... | Alex R. | 22,064 | <p>The sets $V_k$ are disjoint and countable, hence the measure of the union is exactly equal to the sum of measures.</p>
|
164,152 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/33215/what-is-48293">What is 48÷2(9+3)?</a> </p>
</blockquote>
<p>Please, look at the picture?</p>
<p><img src="https://i.stack.imgur.com/3Tauh.jpg" alt="http://s16.radikal.ru/i190/1206/3b/7411739c6d... | Siminore | 29,672 | <p>Despite every personal belief, a notation like $a/b(c+d)$ <strong>is</strong> ambiguous. At school I learned that this should be
$$
\frac{a}{b}(c+d),
$$
but even in research papers somebody could read
$$
\frac{a}{b(c+d)}.
$$</p>
|
164,152 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/33215/what-is-48293">What is 48÷2(9+3)?</a> </p>
</blockquote>
<p>Please, look at the picture?</p>
<p><img src="https://i.stack.imgur.com/3Tauh.jpg" alt="http://s16.radikal.ru/i190/1206/3b/7411739c6d... | Aang | 33,989 | <p>when two operations have same precedence, then the operations are done from left to right.So here, division is done first and then multiplication.This is just a convention.Without this, this expression is ambiguous.</p>
|
2,197,790 | <h3>Question</h3>
<blockquote>
<p>A sequence $\{a_n\}$ of real numbers is said to be a Cauchy sequence of for
each $\epsilon$ > 0 there exists a number $N > 0$ such that m, $n > N$ implies
that $|a_n − a_m| <\epsilon$.</p>
<p>Prove that every convergent sequence is a Cauchy sequence</p>
</blockquot... | Yes | 155,328 | <p>The sequence $(a_{n})$ is convergent by assumption; let $l := \lim_{n}a_{n}$. Let $\varepsilon > 0$. Then there is some $N$ such that $|a_{n} - l| < \varepsilon/2$ for all $n \geq N$. Note that
$$
|a_{n} - a_{m}| \leq |a_{n}-l| + |l-a_{m}| < \varepsilon/2 + \varepsilon/2 = \varepsilon
$$
for all $n,m \geq ... |
25,260 | <h2>TL;DR:</h2>
<hr />
<p>Tell me which topics should i study the most, based on this three tests:</p>
<p>Mathematics (A):
<a href="https://www.studyinjapan.go.jp/ja/_mt/2021/06/2020_ga_math_a.pdf" rel="nofollow noreferrer">2020</a>
<a href="https://www.studyinjapan.go.jp/ja/_mt/2021/06/2019_ga_math_a.pdf" rel="nofollo... | James S. Cook | 128 | <p>It is discussed in all the introductory DEqns texts of which I've used. It's needed to complete the discussion of linear independence of solution sets. Together with Abel's formula it provides some rather general theorems for linear n-th order ODEs. The Wronskian lies at the heart of variation of parameters which is... |
670,522 | <p>In my very young mathematical career, I have worked a lot with modular forms. Recently, I worked as a teaching assistant in a course about geometry. At the end of the course, we dealt with hyperbolic geometry. It seems as if there is some relation between hyperbolic geometry and modular forms, for example, why is it... | Brian Fitzpatrick | 56,960 | <p>I think there's a much easier way to go about this. Let $W$ be a proper subspace of $V$. Then $W$ has a basis $\{w_1,\dotsc,w_k\}$. A standard theorem of linear algebra says this basis for $W$ can be extended to a basis $\{w_1,\dotsc,w_k,v_1,\dotsc,v_n\}$ for $V$. Now, let $f:W\rightarrow U$ be a linear map. Then le... |
291,957 | <p>Does there exist a simple expression for integrals of the form,</p>
<p>$I = \int_{-\infty}^0 H_n(u) H_m(u)\, \mathrm{e}^{-u^2}\,du$,</p>
<p>where $m$ and $n$ are nonnegative integers and $H_n$ is the $n$'th (physicists') Hermite polynomial?</p>
<p>When $n+m$ is even, the symmetry of the integrand and the orthogon... | Robert Israel | 8,508 | <p>It looks to me like we have exponential generating functions</p>
<p>$$\sum_{n=0}^\infty I(n,n+2k+1) t^n/n! = \dfrac{(-1)^{k+1}(2k)!}{k! (1-2t)^{k+3/2} (1+2t)^{k+1/2}}$$</p>
<p>EDIT: Hmm, these can be combined into a bivariate exponential generating function</p>
<p>$$ \sum_{n=0}^\infty \sum_{k=0}^\infty I(n,n+2k+1... |
291,957 | <p>Does there exist a simple expression for integrals of the form,</p>
<p>$I = \int_{-\infty}^0 H_n(u) H_m(u)\, \mathrm{e}^{-u^2}\,du$,</p>
<p>where $m$ and $n$ are nonnegative integers and $H_n$ is the $n$'th (physicists') Hermite polynomial?</p>
<p>When $n+m$ is even, the symmetry of the integrand and the orthogon... | user132949 | 132,949 | <p>This question is getting a little old now, but I feel I can add something here, for my own conscience, if nothing else.</p>
<p>My take on this problem is - simply define $u = \sqrt{v}$ in the integral and take advantage of the relation between the Hermite and Laguerre polynomials. i.e.</p>
<p>$$
H_{2n}(\sqrt{v}) =... |
2,359,700 | <p>Given the vector space, $ C(-\infty,\infty)$ as the set of all continuous functions that are always continuous, is the set of all exponential functions, $U=\{a^x\mid a \ge 1 \}$, a subspace of the given vector space?</p>
<p>As far as I'm aware, proving a subspace of a given vector space only requires you to prove c... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>$$f (x,0)=0$$
$$f (x,x^2)=\frac {1}{2} $$</p>
|
3,099,815 | <p>I need some help on how to approach this problem. I can't seem to find any examples that help me understand this, so if anyone has an approach example to post I would be very grateful:</p>
<p>"Consider a relation <span class="math-container">$R$</span> defined on the set of integers. Determine for the following if ... | Fred | 380,717 | <ol>
<li><p>is it true that for all <span class="math-container">$x \in \mathbb Z$</span> we have <span class="math-container">$x=2x$</span> ? If yes, then <span class="math-container">$R$</span> is reflexive, if no, then <span class="math-container">$R$</span> is not reflexive.</p></li>
<li><p>suppose that <span class... |
42,301 | <p>everyone! I am sorry, but I am an abcolute novice of Mathematica (to be more precise this is my first day of using it) and even after surfing the web and all documents I am not able to solve the following system: </p>
<pre><code>Solve[{y*(((y*x)/(beta*b))^(1/(beta - 1)) - v) - c*alpha ==
0, ((x/alpha))*(((y*x)... | halirutan | 187 | <p>How about using <code>SortBy</code> to sort your list by the last element and then take the first entry?</p>
<pre><code>First[SortBy[mya, Last]]
(* {0, 2, 5} *)
</code></pre>
<p>A simple iterative approach to go through your list exactly once and remember the minimum element can be written as</p>
<pre><code>Block... |
42,301 | <p>everyone! I am sorry, but I am an abcolute novice of Mathematica (to be more precise this is my first day of using it) and even after surfing the web and all documents I am not able to solve the following system: </p>
<pre><code>Solve[{y*(((y*x)/(beta*b))^(1/(beta - 1)) - v) - c*alpha ==
0, ((x/alpha))*(((y*x)... | Yi Wang | 7,253 | <p>I think @halirutan's answer is quite nice and clean. Nevertheless just give an alternative one:</p>
<pre><code>findLastMin[mat_] := Cases[mat, {__, Min@mat[[All, -1]]}]
findLastMin[mya]
</code></pre>
<blockquote>
<p>{{0, 2, 5}}</p>
</blockquote>
<p>There is additional {...} outside the desired output by the OP... |
212,949 | <p>A simple question:</p>
<p>I have this equation:</p>
<pre><code>eq1=Derivative[0, 1][T1][x, t] - Derivative[1, 0][T0][x, t]^2 -
T0[x, t]*Derivative[2, 0][T0][x, t] - Derivative[2, 0][T1][x, t] == 0;
</code></pre>
<p>I want only to select terms that contain T0 or its derivatives only, that is:</p>
<pre><code>-De... | NonDairyNeutrino | 46,490 | <pre><code>Block[{T1, Equal = Plus}, SetAttributes[T1, Constant]; eq1]
</code></pre>
<blockquote>
<pre><code>-Derivative[1, 0][T0][x, t]^2 - T0[x, t]*Derivative[2, 0][T0][x, t]
</code></pre>
</blockquote>
<p>Or (thanks to Mr. Wizard <a href="https://mathematica.stackexchange.com/a/75295/46490">here</a>)</p>
<pre><co... |
212,949 | <p>A simple question:</p>
<p>I have this equation:</p>
<pre><code>eq1=Derivative[0, 1][T1][x, t] - Derivative[1, 0][T0][x, t]^2 -
T0[x, t]*Derivative[2, 0][T0][x, t] - Derivative[2, 0][T1][x, t] == 0;
</code></pre>
<p>I want only to select terms that contain T0 or its derivatives only, that is:</p>
<pre><code>-De... | Michael E2 | 4,999 | <p>While the structural operation</p>
<pre><code>DeleteCases[eq1 /. Equal -> Subtract, _?(FreeQ[#,T0]&)]
</code></pre>
<p>works, I prefer using an algebraic approach on a algebraic problem.</p>
<pre><code>vars = Select[Not@*FreeQ[T0]]@Variables[eq1 /. Equal -> Subtract];
coeffs = CoefficientArrays[eq1 /. E... |
18,686 | <p>Suppose you have an arbitrary triangle with vertices $A$, $B$, and $C$. <a href="http://www.cs.princeton.edu/~funk/tog02.pdf">This paper (section 4.2)</a> says that you can generate a random point, $P$, uniformly from within triangle $ABC$ by the following convex combination of the vertices:</p>
<p>$P = (1 - \sqrt{... | mercio | 17,445 | <p>Pick $A,B,C = (0,0),(1,0),(1,1)$.
For any point $(x,y)$, we have that $(x,y)$ is in the triangle if and only if $0 < x < 1$ and $0 < y/x < 1$.</p>
<p>Now, we look for the distribution of $x$ and $y/x$. </p>
<p>Computing a few triangle areas, we can easily check that $P(0 < x < x_0) = x_0^2$.
Hen... |
1,942,578 | <p>Consider the following wedge</p>
<p><a href="https://i.stack.imgur.com/xiaPX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xiaPX.png" alt=""></a> cut from a cylinder of radius r.
The plane that cuts the wedge goes through the very bottom of the cylinder leading to an ellipse as the cross sectio... | user296113 | 296,113 | <p>First, notice that the functions $f_n$ are not defined on $0$ so there is no sense to say that the uniform convergence is actually on all $\Bbb R$. Secondly the point-wise convergence of $f_n$ is the null function on $(0,1]$ and we have</p>
<p>$$\Vert f_n\Vert_\infty=\sup_{x\in(0,1]}\vert f_n(x)\vert\ge f_n\left(\f... |
137,414 | <p>I am trying to evaluate the following expression numerically
$$\frac{d^2}{dt^2}e^{-2t^2}\int_0^\infty\frac{\xi/\sqrt{2}}{\xi^{3/2}}e^{(-\xi^2/2-2\xi t))}$$</p>
<p>My code is as follows</p>
<pre><code>f[t_]:=Exp[-2*t^2]*NIntegrate[Erf[\[Xi]/Sqrt[2]]/\[Xi]^(3/2)*Exp[-(\[Xi]^2/2)-2*\[Xi]*t],{\[Xi],0,Infinity}]
Der[t... | Dr. Wolfgang Hintze | 16,361 | <p>Notice: my formula contains a sign error (-t instaed of +t). But the derivation is still valid. Only the final numeric value must be taken at t = -5.</p>
<p>The integral can be written as</p>
<pre><code>f[t_] :=
NIntegrate[\[Xi]^(-3/2 )
Erf[\[Xi]/Sqrt[2]] Exp[-(1/2) (\[Xi] - 2 t)^2], {\[Xi],
0, \[Infinit... |
3,966 | <p>This type of answer is what I'm looking for:</p>
<pre><code>In[58]:= ArcTan @ 1
Out[58]= π/4
</code></pre>
<p>This is what mathematica gives me:</p>
<pre><code>In[59]:= ArcTan@2
Out[59]= ArcTan[2]
</code></pre>
<p>Is it possible to express <code>ArcTan</code> in terms of $\pi$? I understand some fractions woul... | Artes | 184 | <p>Maybe this </p>
<pre><code>HoldForm[Pi] (1/Pi ArcTan@2.)
</code></pre>
<p><img src="https://i.stack.imgur.com/6PKA9.gif" alt="enter image description here"></p>
<p>or if you want a nicer way </p>
<pre><code>Rationalize /@ (HoldForm[Pi] N@(1/Pi ArcTan@Range[5]))
</code></pre>
<p><img src="https://i.stack.imgur.... |
2,169,845 | <p>Basically it says given that s is a root of this polynomial: <span class="math-container">$(\sqrt3-\sqrt2)x^3 + \sqrt2x -\sqrt3 + 1$</span>, find another polynomial with integer coefficients that has the same root s as well.
I'm super stuck and am unsure on how to approach this problem.
I attempted to square some st... | zoli | 203,663 | <p>You can calculate the appropriate conditional density directly</p>
<p>$$f_{X\mid X>Y}(x)=\frac{dF_{X\mid X>Y}(x)}{d x}=$$
$$=\lim_{\Delta x\to 0}\frac{F_{X\mid X>Y}(x+\Delta x)-F_{X\mid X>Y}(x)}{\Delta x}=\lim_{\Delta x\to 0}\frac{P(X<x+\Delta x\mid X>Y)-P(X<x\mid X>Y)}{\Delta x}=$$
$$=\lim_... |
2,169,845 | <p>Basically it says given that s is a root of this polynomial: <span class="math-container">$(\sqrt3-\sqrt2)x^3 + \sqrt2x -\sqrt3 + 1$</span>, find another polynomial with integer coefficients that has the same root s as well.
I'm super stuck and am unsure on how to approach this problem.
I attempted to square some st... | Gordon | 169,372 | <p>Note that
\begin{align*}
E(X \mid X > Y) = \frac{E(X\mathbb{1}_{X>Y})}{P(X>Y)}= 2E(X\mathbb{1}_{X>Y}).
\end{align*}
Moreover,
\begin{align*}
E(X\mathbb{1}_{X>Y}) &= E\big(E(X\mathbb{1}_{X>Y} \mid X) \big)\\
&=E(X\Phi(X))\\
&=\int_{-\infty}^{\infty}x\Phi(x)\phi(x) dx,
\end{align*}
where ... |
4,570,329 | <p>In the textbook that I am working through, it is left as an exercise to prove the following claim</p>
<blockquote>
<p>Consider two linear maps <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> from <span class="math-container">$\mathbb{R}^n$</span> to <span class="math-container">$\... | belkacem abderrahmane | 660,639 | <p>Second proof
Remark that my first proof is more general in that it shows that the set of right invertible linear continous maps(linear maps <span class="math-container">$P:X\mapsto Y$</span> such that there's Some continous <span class="math-container">$T:Y\to X$</span> such that <span class="math-container">$PT=I$<... |
173,466 | <p>For a given matrix <code>M[n]</code> of size $ n\times n $ I want to define the following list of matrix-expressions:</p>
<pre><code>n=1
{Tr[M[1]]}
n=2
{Tr[M[2]]^2,Tr[M[2].M[2]]}
n=3
{Tr[M[3]]^3,Tr[M[3]]Tr[M[3].M[3]],Tr[M[3].M[3].M[3]]}
</code></pre>
<p>How could I generalize this relation for arbitrary $ n $? I... | kglr | 125 | <pre><code>f[k_] := Table[Tr[M @ k]^(k - i) Array[M @ k &, i, 1, Tr @ Dot @ ## &], {i, k}]
f /@ Range[5] // Column // TeXForm
</code></pre>
<blockquote>
<p>$\tiny\begin{array}{l}
\{\text{Tr}[M(1)]\} \\
\left\{\text{Tr}[M(2)]^2,\text{Tr}[M(2).M(2)]\right\} \\
\left\{\text{Tr}[M(3)]^3,\text{Tr}[M(3).M(3)] ... |
188,938 | <p>Hyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarely. (My frame of reference is that of someone with college freshman/sophomore, but not advanced math.)</p>
<p>Why is tha... | Fly by Night | 38,495 | <p>Well, yes, I would say so. After all, sine and cosine describe everything from lengths of sides of right-angled triangle to angular momentum. Rotation is so fundamental that we're bound to see sine and cosine very often. Of course $\sin$ and $\cos$ are just the opposite side of the exponential coin from $\sinh$ and ... |
188,938 | <p>Hyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarely. (My frame of reference is that of someone with college freshman/sophomore, but not advanced math.)</p>
<p>Why is tha... | Doc | 86,414 | <p>Tom,</p>
<p>Anon's answer is a good one, with one small excepton (or perhaps one wide misconception). What people commonly refer to as the complex plane is an unfortunate misnomer. It should be called the "complex number plane". By calling it the complex plane (which, by the way, is a quite widespread phenomenon)... |
230,204 | <p>Let $X$ be a compact, oriented Riemann manifold. Let $\pi_{P}: P \rightarrow X$ be a principal $G$-bundle over $X$, for a compact Lie group $G$. Let $(M, \omega)$ be a symplectic manifold endowed with a symplectic action of $G$. Denote by $\mathcal{N}:=C^{\infty}(P,M)^{U(1)}$ the space of smooth $G$-equivariant maps... | Tobias Diez | 17,047 | <p>Yes, the form $\Omega$ is closed and defines indeed a weak symplectic structure. This can be verified by a direct but a bit messy calculation. A cleaner way would be to generalize the ideas of <a href="http://arxiv.org/abs/1111.3889" rel="nofollow">Vizman: Induced differential forms on manifolds of functions</a> to ... |
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