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 |
|---|---|---|---|---|
386,649 | <p>If you were working in a number system where there was a one-to-one and onto mapping from each natural to a symbol in the system, what would it mean to have a representation in the system that involved more than one digit?</p>
<p>For example, if we let $a_0$ represent $0$, and $a_n$ represent the number $n$ for any... | N. S. | 9,176 | <p>The set</p>
<p>$${\mathbb N}[X]$$</p>
<p>is exactly a system as you describe. The constant polynomials are the whole numbers ${\mathbb N}$, and they represent the infinite "symbols" in your system, while $a_0a_1$ is actually the polynomials $a_0+a_1X$. </p>
<p>If you replace $\mathbb N$ by $\mathbb Z$ or $\mathbb... |
466,757 | <p>Suppose we have the following</p>
<p>$$ \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$$</p>
<p>where all the $a_{ij}$ are non-negative.</p>
<p>We know that we can interchange the order of summations here. My interpretation of why this is true is that both this iterated sums are rearrangements of the same series an... | Evan | 38,878 | <p>Well, I think that interpretation is a bit circular, as the reason absolute convergence allows for rearrangements without changing the limit is precisely because of this point. </p>
<p>The gist of it is that <strong>no cancellations due to sign are possible.</strong>
I think it is very illuminating to consider the... |
1,221,442 | <p>I have two uniform random variables $B$ and $C$ distributed between $(2,3)$ and $(0,1)$ respectively. I need to find the mean of $\sqrt{B^2-4C}$. Could I plug in the means for $B$ and $C$ and then solve or is it more complicated than that?<br>
The original question is here: <a href="https://math.stackexchange.com/qu... | André Nicolas | 6,312 | <p>You want to find the mean of $\sqrt{B^2-4C}$. This <em>cannot</em> be done by substituting $E(B)$ and $E(C)$ for $B$ and $C$ in the expression $\sqrt{B^2-4C}$. </p>
<p>Let $f_B(x)$ be the density function of $B$, and let $f_C(y)$ be the density function of $C$. So $f_B(x)=1$ on the interval $(2,3)$ and $0$ elsewher... |
3,118,298 | <p>So I have a formula for arc length <span class="math-container">$$s(t)=\int_{t_0}^t \vert\vert \dot\gamma(u)\vert\vert du$$</span>
I computed that <span class="math-container">$$\vert\vert \dot\gamma(t)\vert\vert=\sqrt{2(1-\cos t)}$$</span></p>
<p>Substituting this into the integral <span class="math-container">$$\... | J P | 551,142 | <p>It's not quite convincing enough for me. You didn't explicitly show why it was true. Let <span class="math-container">$L$</span> be the limit of the sequence <span class="math-container">$\{a_n\}$</span> We'll use the monotone convergence theorem as you said it was already proven to you. Clearly <span class="math-co... |
11,457 | <p>In their paper <em><a href="http://arxiv.org/abs/0904.3908">Computing Systems of Hecke Eigenvalues Associated to Hilbert Modular Forms</a></em>, Greenberg and Voight remark that</p>
<p>...it is a folklore conjecture that if one orders totally real fields by their discriminant, then a (substantial) positive proporti... | Dror Speiser | 2,024 | <p>I think the best way for one to become convinced that class numbers of real quadratic fields tend to be small, is to look at the continued fraction expansion of $\sqrt{D}$.</p>
<p>The period length of the continued fraction is about the regulator (up to a factor of $\log{D}$). One can easily compute some random con... |
107,171 | <p>I'm trying to find $$\lim\limits_{(x,y) \to (0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4} .$$
After I tried couple of algebraic manipulation, I decided to use the polaric method.
I choose $x=r\cos \theta $ , $y=r\sin \theta$, and $r= \sqrt{x^2+y^2}$, so I get </p>
<p>$$\lim\limits_{r \to 0} \frac{e^{-\frac{1}{r^2}... | GEdgar | 442 | <p>Some numbers are negative, so the title (not the actual statement, though) may refer to this one, easier to do:
$$
\sum_{k=-\infty}^\infty \frac{1}{(3k+1)^2} = \frac{4\pi^2}{27}
$$</p>
|
1,798,855 | <p>I'm trying to understand this proof that:</p>
<p>$M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time</p>
<p>Which is:</p>
<p>If $M=A\cup B$ is a separation, then $A$ and $B$ are open and closed. Recriprocally, if $A\subset M$ is open and closed, then $M = A\cup(M-... | Ross Millikan | 1,827 | <p>If <span class="math-container">$M=A \cup B$</span> is a separation, both <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are open as you say. Then <span class="math-container">$A = B^c$</span> is the complement of an open set, so is closed. The reciprically works the same way. ... |
75,925 | <p>I hope this question is focused enough – it's not about real problem I have, but to find out if anyone knows about a similar thing.</p>
<p>You probably know the <a href="https://en.wikipedia.org/wiki/Uncertainty_principle" rel="nofollow noreferrer">Heisenberg uncertainty principle</a>: For any function <span class="... | Piero D'Ancona | 7,294 | <p>There exists a plethora of inequalities relating weighted $L^p$ norms of a function and its derivatives. For instance you have the Caffarelli-Kohn-Nirenberg family of inequalities
$$\| |x|^{-\gamma}u\|_ {L^{r}}\le C \||x|^{-\alpha}\nabla u\|^{a}_ {L^{p}}\||x|^{-\beta}u\|^{1-a}_ {L^{q}}
$$
which hold for a quite larg... |
156,285 | <p>I have been working on this exercise for a while now. It's in B.L. van der Waerden's <em>Algebra (Volume I)</em>, page $19$. The exercise is as follows:</p>
<blockquote>
<p>The order of the symmetric group $S_n$ is $n!=\prod_{1}^{n}\nu$. (Mathematical induction on $n$.)</p>
</blockquote>
<p>I don't comprehend ho... | Dylan Moreland | 3,701 | <p>I start to get confused by the notation around halfway through. Here's what I'd do: identify $S_n$ with the subgroup of $S_{n + 1}$ consisting of those permutations that fix $n + 1$. Choose elements $\sigma_1, \ldots, \sigma_{n + 1}$ of $S_{n + 1}$ such that $\sigma_i(n + 1) = i$. For example, you could take $\sigma... |
7,761 | <p>Our undergraduate university department is looking to spruce up our rooms and hallways a bit and has been thinking about finding mathematical posters to put in various spots; hoping possibly to entice students to take more math classes. We've had decent success in finding "How is Math Used in the Real World"-type po... | Churning Butter | 4,988 | <p>The American Mathematical Society has a collection of beautiful posters in their Math Samplings. They are usually adding new ones every so often, and there and some new ones that are maybe a bit more what you'd call informative, that weren't there when you'd looked in the past:</p>
<p><a href="http://www.ams.org/sa... |
1,178,265 | <p>I'm supposed to be able to determine <strong><em>without calculations</em></strong> the determinant, inverse matrix, and n-th power matrix of the rotation matrix :</p>
<p>$\begin{pmatrix}
cos\theta & sin\theta \\
-sin\theta & cos\theta
\end{pmatrix} $</p>
<p>Can someone explain to me how I can do that ... | GFR | 64,803 | <p>The matrices you wrote is a rotation in the plane by an angle of $\theta$. Therefore its inverse can be obtained by replacing $\theta$ by $-\theta$ as a rotation can be undone by a rotation with the opposite angle. Similarly its $n$th power will be a rotation by the angle $n\theta$. Finally as the determinant measur... |
1,641,579 | <p>The conditional statement is:
If today is February 1, then tomorrow is Ground Hog's Day.
I need to negate this but I am confused. Would it just be If today is not February 1, then tomorrow is not Ground Hog's Day? I think that is an inverse statement though. Please help me negate this conditional statement.</p>
| p Groups | 301,282 | <p>Many (but not all) books of Group Theory / Algebra include this theorem with proof, but in my opinion, the book <em>Finite Group Theory</em> by Martin Isaacs gives very elegant, natural, motivational proof of it.</p>
<p>Although it is lengthy (p. 75-80), it is certainly written by author by keeping in mind the audi... |
1,641,579 | <p>The conditional statement is:
If today is February 1, then tomorrow is Ground Hog's Day.
I need to negate this but I am confused. Would it just be If today is not February 1, then tomorrow is not Ground Hog's Day? I think that is an inverse statement though. Please help me negate this conditional statement.</p>
| verret | 191,246 | <p>See Section 6.2 in The Theory of Finite Groups: An Introduction, by Kurzweil and Stellmacher</p>
|
1,641,579 | <p>The conditional statement is:
If today is February 1, then tomorrow is Ground Hog's Day.
I need to negate this but I am confused. Would it just be If today is not February 1, then tomorrow is not Ground Hog's Day? I think that is an inverse statement though. Please help me negate this conditional statement.</p>
| Rithvik Reddy | 737,143 | <p>You can find it in Section 6.6(pg 186) in Charles Weibel's book 'An Introduction to Homological Algebra'</p>
|
144,818 | <p>Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1<x_2<\cdots<x_n$.
Define \begin{equation*}
A=%
\begin{bmatrix}
0 & x_{2}-x_{1} & \cdots & x_{n-1}-x_{1} & x_{n}-x_{1} \\
x_{2}-x_{1} & 0 & \cdots & x_{n-1}-x_{2} & x_{n}-x_{2} \\
\vdots & \vdots & \ddots... | Dilawar | 1,674 | <p>Expanding Robert solution.</p>
<p>Let $det(A) = P(x)$. Let the polynomial on the right is a multi-variable polynomial $P(x)$.</p>
<p>If $x_1 = x_2$ then $det(A) = 0$ i.e $P(x) = 0$ i.e. $(x_1 - x_2)$ is a factor of $P(x)$.</p>
<p>If $x_2 = x_3$ then $det(A) = 0$ i.e $P(x) = 0$ i.e. $(x_2 - x_3)$ is a factor of $P... |
2,022,700 | <blockquote>
<p>In how many ways can the letters in WONDERING be arranged with exactly
two consecutive vowels</p>
</blockquote>
<p>I solved and got answer as $90720$. But other sites are giving different answers. Please help to understand which is the right answer and why I am going wrong.</p>
<p><strong>My Solut... | Community | -1 | <p>The number of arrangements with 3 consecutive vowels is correctly explained in the original post: the number is $15120$.</p>
<p>To find the number of arrangements with <em>at least</em> two consecutive vowels, we duct tape two of them together (as in the original post) and arrive at $120960$.</p>
<p>The problem wi... |
153,902 | <p>Let $A_i$ be open subsets of $\Omega$. Then $A_0 \cap A_1$ and $A_0 \cup A_1$ are open sets as well.</p>
<p>Thereby follows, that also $\bigcap_{i=1}^N A_i$ and $\bigcup_{i=1}^N A_i$ are open sets.</p>
<p>My question is, does thereby follow that $\bigcap_{i \in \mathbb{N}} A_i$ and $\bigcup_{i \in \mathbb{N}} A_i$... | talmid | 19,603 | <p>The union of <strong>any</strong> collection of open sets is open. Let $x \in \bigcup_{i \in I} A_i$, with $\{A_i\}_{i\in I}$ a collection of open sets. Then, $x$ is an interior point of some $A_k$ and there is an open ball with center $x$ contained in $A_k$, therefore contained in $\bigcup_{i \in I} A_i$, so this u... |
425,969 | <p>It seems striking that the cardinalities of <span class="math-container">$\aleph_0$</span> and <span class="math-container">$\mathfrak c = 2^{\aleph_0}$</span> each admit what I will call a "homogeneous cyclic order", via the examples of <span class="math-container">$ℚ/ℤ$</span> and <span class="math-conta... | Andreas Blass | 6,794 | <p>Adjoin to the theory of cyclic orders (as on the ncatlab page linked in the question) a 3-place function <span class="math-container">$f$</span> and axioms saying that, for each fixed <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, the function <span class="math-container">$z\map... |
3,256,646 | <p>I find it really hard to find the range. I usually substitute the x's with y and then solve for y, but it does not always work for me. Do you have any advice?</p>
<p>Function in question: </p>
<p><span class="math-container">$$f(x) = \frac{e^{-2x}}{x}$$</span></p>
| Community | -1 | <p>Let <span class="math-container">$x\in A,y\in B$</span>. Then if <span class="math-container">$x,y\in C'\subset C$</span>, I claim <span class="math-container">$C'$</span> is disconnected. Define <span class="math-container">$A'=A\cap C',B'=B\cap C'$</span>. Then <span class="math-container">$A'$</span> and <span... |
606,356 | <p>I would appreciate if somebody could help me with the following problem</p>
<p>Q: Quadratic Equation $x^4+ax^3+bx^2+ax+1=0$ have four real roots
$x=\frac{1}{\alpha^3},\frac{1}{\alpha},\alpha,\alpha^3(\alpha>0)$ and $2a+b=14$.</p>
<p>Find $a,b=?(a,b\in\mathbb{R})$</p>
| David Holden | 79,543 | <p>lab has a point. if we write $x=\alpha +\frac1{\alpha}$
then we get
$$
-ax + b = x^2+2
$$
or
$$
x= \frac12 \left((-a \pm \sqrt{a^2-4(2-b)}\right)
$$
but for a 'nice' problem the $2a+b=14$ should simplify the surd, and it doesn't quite seem to. maybe my arithmetic is wrong, or maybe i'm just a hopeless optimist with... |
365,631 | <p>Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, then show that the set
of things with the desired property has positive measure. Examples of this strategy
appear in... | user161212 | 161,212 | <p>A very famous and important theorem in the theory of metric embeddings is known as "Assouad's Embedding Theorem". It concerns <em>doubling</em> metric spaces: metric spaces for which there is a constant <span class="math-container">$D$</span> such that every ball can be covered by <span class="math-contain... |
97,449 | <p>I am trying to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using only elementary techniques from differential topology and this is proving to be trickier than I thought. I am aware of the usual proof for this result, which uses the cellular decomposition of $\mathbb{C}\mathrm{P}^2$ to get $\chi(\mathbb{C}\mathrm{P}^2) = ... | Igor Rivin | 11,142 | <p>By the way, a very cool (to my mind) way of computing the Euler characteristic of $\mathbb{C}P^n$ is to treat it as the $n$-fold symmetric product of $\mathbb{C}P^1 = \mathbb{S}^2$ with itself. Then, it is apparently a result of MacDonald (of Atiyah and M fame) that the Euler characteristic of an $n$-fold symmetric ... |
2,509,095 | <p>Is there a very simple test to check if a line <em>segment</em> in $3D$ space cuts a plane?
It is assumed we have the coordinates of the endpoints of the line segment, so $p_1,p_2$ and that we have the equation of the plane: $z = d$ (so for simplicity we're assuming it's a plane orthogonal to the z-axis).</p>
| Endre Moen | 413,323 | <p>So the plane is given by $ax + by + 0z -d = 0$ Using the parametric form - you must find the line equation in each direction. E.g $x=t, y=2+3t, z=t$</p>
<p>Then you plug that into the plain eq.: $a(t) + b(2+3t)0 + 0(t) - d = 0$. Take that value and plug it in for $t$ in the 3 equations given for the line.</p>
<p>E... |
438,336 | <p>This a two part question:</p>
<p>$1$: If three cards are selected at random without replacement. What is the probability that all three are Kings? In a deck of $52$ cards.</p>
<p>$2$: Can you please explain to me in lay man terms what is the difference between with and without replacement.</p>
<p>Thanks guys!</p>... | Community | -1 | <p>Hint: What's the probability that the first card would be a King? The second? And the third?</p>
<p>With replacement means that after you draw the card you put it back into the deck, then re-draw the next one completely random.</p>
|
3,068,031 | <blockquote>
<p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$G$</span>. Let also <span class="math-container">$a,~b\in G$</span> such that <span class="math-container">$ab\in H$</span>.</p>
<p>True or false? <span cla... | Lee Mosher | 26,501 | <p>Take <span class="math-container">$G$</span> to be the <a href="https://en.wikipedia.org/wiki/Free_group" rel="noreferrer">free group</a> on <span class="math-container">$a,b$</span>, whose elements are the reduced words in the alphabet <span class="math-container">$a,b,a^{-1},b^{-1}$</span>. </p>
<p>Take <span cla... |
2,933,375 | <p>I have a set of vectors, <span class="math-container">$M_1$</span> which is defined as the following:
<span class="math-container">$$M_1:=[\begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 1 \end{pmatrix}]$$</span>
I have to show that <span class="math-container">$M_1$</span> isn't a generating set ... | hmakholm left over Monica | 14,366 | <p>Some people even write things like <span class="math-container">$$\exists z \text{ s.t. } \forall y, y\notin z$$</span>
Personally I think this is a horrible practice. One can think what one wants about how it wastes space, but more importantly it reinforces the dangerous <strong>misconception</strong> that logical ... |
2,933,375 | <p>I have a set of vectors, <span class="math-container">$M_1$</span> which is defined as the following:
<span class="math-container">$$M_1:=[\begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 1 \end{pmatrix}]$$</span>
I have to show that <span class="math-container">$M_1$</span> isn't a generating set ... | Rob Arthan | 23,171 | <p>In my experience, mathematical logicians never use the form with ":", but split into two camps as regards the use of a ".": many people don't use a "." after quantifiers and take the quantifiers to have high precedence, so that <span class="math-container">$\forall x\forall y(x > y \implies x \ge y + 1)$</span> r... |
2,704,770 | <p>I need help in calculating this strange limit.</p>
<p>$$
\lim_{n \to \infty} n^2 \int_{0}^{\infty} \frac{sin(x)}{(1 + x)^n} dx
$$</p>
| vrugtehagel | 304,329 | <p>We can substitute $x=\frac{t}{n}$:</p>
<p>$$\lim_{n \to \infty} n^2 \int_{0}^{\infty} \frac{\sin(x)}{(1 + x)^n} dx=\lim_{n \to \infty} n^2 \int_{0}^{\infty} \frac{\sin(t/n)}{(1 + \frac tn)^n}\frac{1}{n}dt$$</p>
<p>Now in the denominator we get $(1+\frac{t}{n})^n$, and we're taking the limit $n\to\infty$; this beco... |
68,428 | <p>I am looking at the description of LTI systems in the time domain.</p>
<p>Intuitively, I'd have guessed it would be the composition of the input function and some "system function".
$$ y(t) = f(x(t)) = (f\circ x)(t)$$
Where $x(t)$ is the input, $y(t)$ output and $f(x)$ a "system function".</p>
<p>Why is it not tha... | Mike Stay | 756 | <p>Here's a partial answer: in the case of an endofunctor $F$ on a discrete category $C$ (i.e. $F$ is a function), the coend of $F$ gives the <em>set</em> of fixpoints rather than the <em>number</em>: A profunctor $F:C \not\to C$ adds extra morphisms to $C$ so that the result is still a category. I'll say these morphi... |
4,228,826 | <p>Consider the inequality
<span class="math-container">$$
1-\frac{x}{2}-\frac{x^2}{2} \le \sqrt{1-x} < 1-\frac{x}{2}
$$</span>
for <span class="math-container">$0 < x < 1$</span>. The upper bound can be read off the Taylor expansion for <span class="math-container">$\sqrt{1-x}$</span> around <span class="math... | Michael Rozenberg | 190,319 | <p>We need to prove that
<span class="math-container">$$(2-x-x^2)^2\leq4(1-x),$$</span> which is
<span class="math-container">$$x^2(3+x)(1-x)\geq0.$$</span></p>
|
1,134,145 | <p>A set S is bounded if every point in S lies inside some circle |z| = R other it is unbound. Without appealing to any limit laws, theorems, or tools from calculus, prove or disprove that the set {$\frac{z}{z^2 + 1}$; z in R} is bounded.</p>
<p>I imagine that it's simple, but I have no clue where to start due to the ... | Brian Rushton | 51,970 | <p>Can you prove that $\frac{z}{z^2+1}$ is less than some given number? For instance, it is not too hard to show that it is less than 1/2 (hint: start with $\frac{z}{z^2+1}<1/2$ and cross-multiply).</p>
|
393,712 | <p>I studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary density functions?</p>
| Chris Evans | 78,301 | <p>The standard answer is that measure theory is a more natural framework to work in. After all, in probability theory you are concerned with assigning probabilities to events (sets)... so you are dealing with functions whose inputs are sets and whose outputs are real numbers. This leads to sigma-algebras and measure t... |
3,188,298 | <blockquote>
<p>Let <span class="math-container">$f: X\to Y$</span> be bijective, and <span class="math-container">$f^{-1}: Y\to X$</span> be it's inverse. If
<span class="math-container">$V\subseteq Y$</span>, show that the forward image of <span class="math-container">$V$</span> under <span class="math-container"... | Martin Argerami | 22,857 | <p>No, what they want you to show is that the image of <span class="math-container">$f^{-1} $</span>:
<span class="math-container">$$\tag1
(f^{-1})(V)=\{f^{-1}(v):\ v\in V\}
$$</span>
is equal to the preimage of <span class="math-container">$V $</span> under <span class="math-container">$f $</span>:
<span class="math-c... |
78,443 | <p>Let $F$ be a number field and $A$ an abelian variety over $F$. It is known that if $A$ has complex multiplication, then it has potentially good reduction everywhere, namely there exists a finite extension $L$ of $F$ such that $A_L$ has good reduction over every prime of $L$.</p>
<p>And what about the inverse: if $A... | user2146 | 2,146 | <p>A key feature of the Nisnevich topology is that as a cd-structure (cf. [Voevodsky, Homotopy theory of simplicial sheaves in completely decomposable topologies]) it is complete and regular. This implies what Lurie calls Nisnevich excision in DAG XI. The proof of this "excision" relies on the existence of a "splitting... |
1,253,687 | <p>I don't know how to solve this one and the question is:</p>
<p>Find the values of a at which $y = x^3 + ax^2 + 3x + 1$.</p>
<p>My solution is:</p>
<p>$y'= 3x^2 + 2ax + 3$</p>
<p>I know that if $y' \ge 0$, $y$ should be always increasing. I don't know how to make it true. Please help and explain and thank you in ... | user156213 | 156,213 | <p>Try writing it in a different form:</p>
<p>$$y'=3(x^2+2ax/3+1)=3((x+a/3)^2-a^2/9+1)$$
Since $(x+a/3)^2$ is always non-negative, we need
$$-a^2/9+1 > 0$$
so
$$a^2 < 9\to -3<a<3$$
so the answer is (assuming strictly increasing)
$$a\in (-3,3)$$</p>
|
2,037,030 | <p>I am studying Distribution theory. But I am curious about that why we coin compact support. In what situation is it useful? Can any one give an intuitive way to explain this concept?</p>
| vidyarthi | 349,094 | <p>The functions with compact support are those that are zero outside of a compact set. This is quite useful in distributions as it tells us that the function dosent grow indefinitely. It is useful also in the theory of Differential Equations, Functional Analysis, Toplogy. Some examples include <a href="http://mathworl... |
898,755 | <p>The function $G_m(x)$ is what I encountered during my search for approximates of Riemann $\zeta$ function:</p>
<p>$$f_n(x)=n^2 x\left(2\pi n^2 x-3 \right)\exp\left(-\pi n^2 x\right)\text{, }x\ge1;n=1,2,3,\cdots,\tag{1}$$
$$F_m(x)=\sum_{n=1}^{m}f_n(x)\text{, }\tag{2}$$</p>
<p>$$G_m(x)=F_m(x)+F_m(1/x)\text{, }\t... | Daccache | 79,416 | <p>Well, I was able to prove <span class="math-container">$(3)$</span> in your answer above, meaning the bounds are proven! The 'proof' though is still much less formal than I'd like it to be, so it might need a little refinement. Here goes:</p>
<p>Intuitively, the LHS side of the inequality is the sum of all the posit... |
3,579,065 | <p>It is known that the quantity <span class="math-container">$\cos \frac{2π}{17}$</span> is a root of the <span class="math-container">$8$</span>'th degree equation,
<span class="math-container">$$x^8 + \frac{1}{2} x^7 - \frac{7}{4} x^6 - \frac{3}{4} x^5 + \frac{15}{16} x^4 + \frac{5}{16} x^3 - \frac{5}{32} x^2 -... | Community | -1 | <p>You probably know that <span class="math-container">$\frac{1}{1-x}=1+x+x^2+\dots$</span>. If not, observe that <span class="math-container">$$(1-x)(1+x+x^2+\dots)=(1+x+x^2+\dots)-(x+x^2+x^3+\dots)=1$$</span> Then <span class="math-container">$$\frac{1}{(1-x)^2}=\left(\frac{1}{1-x}\right)'=1+2x+3x^2+4x^3+\dots$$</spa... |
1,611,078 | <p>If we have the function $f : \mathbb{R}\rightarrow \mathbb{R} : x \mapsto x^2 + \frac{x}{3}$ and the sequence $(a_n)_{n \in \mathbb{N}}$ which is recursively specified for $n \in \mathbb{N_+}$:</p>
<p>$a_n =_{def} f(a_{n-1})$</p>
<p>(So the sequence is fixed by $a_0$) </p>
<p>How to determine all real numbers $x... | Asinomás | 33,907 | <p>for $x\in [-1,\frac{2}{3}]$ it converges. If $x=-1$ or $\frac{2}{3}$ it converges to $\frac{2}{3}$. Otherwise it converges to $0$.</p>
<p>For any other value of $x$ is goes to infinity.</p>
|
394,580 | <p>Let <span class="math-container">$S$</span> be a smooth compact closed surface embedded in <span class="math-container">$\mathbb{R}^3$</span> of genus <span class="math-container">$g$</span>.
Starting from a point <span class="math-container">$p$</span>, define a random walk as taking discrete steps
in a uniformly r... | Pierre PC | 129,074 | <p><strong>Edit:</strong> As far as I understand it, this is the approach of T. Sunada, as described in the paper linked in <a href="https://mathoverflow.net/a/394625/129074">R W's answer</a>.</p>
<p>Let us fix <span class="math-container">$\delta>0$</span> such that any pair of points can be joined by a finite sequ... |
1,716,656 | <p>I am having trouble solving this problem</p>
<blockquote>
<p>Julie bought a house with a 100,000 mortgage for 30 years being repaid with payments at the end of each month at an interest rate of 8% compounded monthly. If Julie pays an extra 100 each month, what is the outstanding balance at the end of 10 years im... | someguy | 326,526 | <p>Any result will do as long as the other die can score the same number plus two, that gets us with n-2 per die (n being number of sides). This gets us 2(n-2) posible results over n^2 (as we have two identical dice)</p>
<p>then the probability is: 2(n-2)/n^2 </p>
|
1,716,656 | <p>I am having trouble solving this problem</p>
<blockquote>
<p>Julie bought a house with a 100,000 mortgage for 30 years being repaid with payments at the end of each month at an interest rate of 8% compounded monthly. If Julie pays an extra 100 each month, what is the outstanding balance at the end of 10 years im... | AAAfarmclub | 326,781 | <p>Just for fun, I counted eight.<br>
<img src="https://i.stack.imgur.com/dG32E.png" alt="Dice image courtesy of Google[1]"></p>
|
449,631 | <p>Again a root problem..
$\sqrt{2x+5}+\sqrt{5x+6}=\sqrt{12x+25}$</p>
<p>Isn't there any standardized way to solve root problems..Can u plz help by giving some tips and stategies for root problems??</p>
| Clement C. | 75,808 | <p>It seems correct, up to a typo between $A$ and $E$, and braces missing in the union at the end (in $\bigcup\{x_{n_k}\}$ Also, you might want to define $n_{k+1}$ as
$$n_{k+1} = \inf\{ i > n_k \mid x_i \in E \}$$</p>
<p>Using the (equivalent, and pretty much identical up to notations sequence/function) definition ... |
559,194 | <p>$\mathscr{F}\{\delta(t)\}=1$, so this means inverse fourier transform of 1 is dirac delta function so I tried to prove it by solving the integral but I got something which doesn't converge.</p>
| meta_warrior | 73,032 | <p>$$\mathscr{F^{-1}}\{1\}=\int_{-\infty}^{\infty}e^{2\pi ixy}dy=\lim_{M\to\infty}\frac{\sin{2\pi Mx}}{\pi x}$$
Now we need to consider 2 cases:</p>
<p>1) $x=0$, then $\lim_{M\to\infty}\frac{\sin{2\pi Mx}}{\pi x}=\infty$</p>
<p>2) $x\ne0$, then $\lim_{M\to\infty}\frac{\sin{2\pi Mx}}{\pi x}=0$</p>
<p>Hence combining ... |
2,720,694 | <p>I am facing difficulty to calculate the second variation to the following functional.</p>
<p>Define $J: W_{0}^{1,p}(\Omega)\to\mathbb{R}$ by
$J(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^p\,dx$ where $p>1$.</p>
<p>I am able to calculate the first variation as follows:
$J'(u)\phi=\int_{\Omega}\,|\nabla u|^{p-2}\nabl... | Prasun Biswas | 215,900 | <p><strong>Hints:</strong></p>
<ul>
<li><p>If $f(a-x)=f(x)$ on $[0,a]$, then $\int_0^a f(x)~\mathrm dx=\int_0^a f(a-x)~\mathrm dx$</p></li>
<li><p>If $f(2a-x)=f(x)$ on $[0,2a]$, then $\int_0^{2a} f(x)~\mathrm dx=2\int_0^a f(x)~\mathrm dx$</p></li>
<li><p>Suppose $m=n=k=2^rs$ where $2\not\mid s$. Can you show that $$\i... |
435,298 | <p>Define
$$\langle X,Y \rangle := \operatorname{tr}XY^t,$$
where $X,Y$ are square matrices with real entries and $t$ denotes transpose.</p>
<p>I have some troubles in proving that
$$ \langle [X,Y],Z \rangle = - \langle Y,[X,Z] \rangle,$$
where square brackets denote commutator.</p>
<p>Let me update my <strong>questi... | S.B. | 35,778 | <p>$$\langle XY-YX,Z\rangle=\langle XY,Z\rangle-\langle YX,Z\rangle\\=\langle Y,X^tZ\rangle-\langle Y,ZX^t\rangle\\=\langle Y,[X^t,Z]\rangle$$
<strong>Counterexample for the OP's equation:</strong> Let $X=Z=\left[\array{0 & 1\\0 &0}\right]$ and $Y=\left[\array{1&0\\0&2}\right]$. We have $XZ-ZX=0\implies... |
473,508 | <p>Let $p > 2$ be a prime. Can someone prove that for for any $t \leq p-2$ the following identity holds</p>
<blockquote>
<p>$\displaystyle \sum_{x \in \mathbb{F}_p} x^t = 0$</p>
</blockquote>
| Servaes | 30,382 | <p>Let $t\in\{1,\ldots,p-2\}$ and let $g\in\Bbb{F}_p^{\times}$ be a primitive root. Then $g^t\neq1$ and $g^{p-1}=1$, so
$$\sum_{x\in\Bbb{F}_p}x^t=\sum_{k=0}^{p-2}g^{kt}=\frac{1-g^{(p-1)t}}{1-g^t}=0.$$</p>
|
4,353,203 | <p>While looking for an explanation to the first inequality, I spied another similar inequality. So, I will ask about both of them here.</p>
<p><span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span> are positive numbers.
<span class="math-container"... | Colin T Bowers | 65,280 | <p>This answer is here to flesh out the detail in the method provided by md5 in the accepted answer.</p>
<p>As md5 notes, given the setup:
<span class="math-container">\begin{equation}
\mathbb{P}(s_{2n} = -2n + 4k) = \frac{\binom{n}{k} \binom{n}{n-k}}{\binom{2n}{n}} , \quad k = 0, 1, ..., n .
\end{equation}</span>
A we... |
318,351 | <p>As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between two convex" functions ?</p>
| noobProgrammer | 61,722 | <p>Differential forms is a way of formulating a calculus on manifolds without a strict adherence to coordinates. Besides differential forms being necessary to learn some differential geometry, its absolutely necessary for understanding Stokes' Theorem (modern version), which generalizes the classical Kelvin Stokes into... |
122,471 | <p>Can anyone explain how I can prove that either $\phi(t) = \left|\cos (t)\right|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.</p>
| Did | 6,179 | <p><a href="http://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29#Properties" rel="noreferrer">Factoid 1</a>: If a characteristic function is infinitely differentiable at zero, all the moments of the corresponding random variable are finite.</p>
<p><a href="http://en.wikipedia.org/wiki/Characte... |
3,430,812 | <p>Consider the set of integers, <span class="math-container">$\Bbb{Z}$</span>. Now consider the sequence of sets which we get as we divide each of the integers by <span class="math-container">$2, 3, 4, \ldots$</span>.</p>
<p>Obviously, as we increase the divisor, the elements of the resulting sets will get closer and... | sanaris | 144,567 | <p>If you are physicist or applied mathematician, if you define <span class="math-container">$S_n = \{ z_i/n, z_i=i, i = 1..\infty \in \mathbb{Z}\}$</span>, then you can just say that <span class="math-container">$\lim_{n\rightarrow\infty} S = \{ \lim S_n \}$</span>, which will yield you <span class="math-container">$\... |
3,927,488 | <p>Given a random variable <span class="math-container">$X$</span> with finite expectation, I know that <span class="math-container">$$X_n\to X, a.s.\text{and} |X_n| \leq X\implies \mathbb{E}|X-X_n|\to 0 \text{ by DCT.}$$</span></p>
<p>I am wondering if it is possible to approximate <span class="math-container">$X$</s... | J. W. Tanner | 615,567 | <p>Start by setting up equations.</p>
<p>I prefer to write <span class="math-container">$A_0$</span> and <span class="math-container">$B_0$</span> for <span class="math-container">$C_{AO}$</span> and <span class="math-container">$C_{BO}$</span>, respectively, and <span class="math-container">$C$</span> for <span class... |
954,419 | <p>I am teaching myself mathematics, my objective being a thorough understanding of game theory and probability. In particular, I want to be able to go through A Course in Game Theory by Osborne and Probability Theory by Jaynes.</p>
<p>I understand I want to cover a lot of ground so I'm not expecting to learn it in le... | Trurl | 72,915 | <p>I'll second Shane's comments, but I think you should start studying game theory now. Schelling's The Strategy of Conflict is a classic, and very readable with very little math. A more formal book that I've read is Vorob'ev Game Theory (springer-verlag), about the level of an undergraduate math book. An undergradu... |
300,867 | <p>I am having a difficult time understanding where I went wrong with the following:
$$\begin{matrix}4x-y = 1 \\ 2x+3y = 3 \end{matrix} $$
$$\begin{matrix}4x-y = -3 \\ 2x+3y = 3 \end{matrix} $$</p>
<p>I found the inverse of the common coefficient matrix of the systems:
$$A^{-1} \begin{cases} \frac3{14}, \frac1{14} \\ ... | copper.hat | 27,978 | <p>Here is another approach. The idea is to enlarge $C$ so that it still remains in $U$, and is 'locally connected by straight lines'.</p>
<p>Let $\delta = \frac{1}{2}\sup \{ r | B(x, r) \subset U, \forall x \in C \}$. Since $C$ is compact, we have $\delta > 0$, and $\overline{B}(x,\delta) \subset U$ for all $x \in... |
3,316,730 | <p>I'm taking a course in Linear Algebra right now, and am having a hard time wrapping my head around bases, especially since my prof didn't really explain them fully. I would really appreciate any insight you could give me as to what bases are! Also, can there can be multiple different bases for a single subspace?</p>... | rschwieb | 29,335 | <p>They are subsets that “efficiently capture” the rest of the vector space. A sort of skeleton, if you will, or maybe like compressing a computer file.</p>
<p>This means that you can recover every other element in the space by using just the operations (scalar multiplication and addition) and furthermore there were a... |
909,741 | <blockquote>
<p><strong>ALREADY ANSWERED</strong></p>
</blockquote>
<p>I was trying to prove the result that the OP of <a href="https://math.stackexchange.com/questions/909712/evaluate-int-0-frac-pi2-ln1-cos-x-dx"><strong><em>this</em></strong></a> question is given as a hint.</p>
<p>That is to say: <em>imagine tha... | idm | 167,226 | <p>An other way:</p>
<p>Firstly $$\int_0^{\pi/2}\ln(\cos t)dt\underset{t=\frac{\pi}{2}-u}{=}\int_{0}^{\pi/2}\ln\left(\cos\left(\frac{\pi}{2}-u\right)\right)du=\int_0^{\pi/2}\ln(\sin u)du \tag 1 $$</p>
<p>Then,
$$\int_0^{\pi/2}\ln(\sin t)dt=\frac{1}{2}\left(\int_{0}^{\pi/2}\ln(\sin t)dt+\int_0^{\pi/2}\ln(\cos t)dt\rig... |
3,834,894 | <p>I understand a continuous function may not be differentiable. But does every continuous function have directional derivative at every point? Thanks!</p>
| Phil | 543,036 | <p>No. Consider the function <span class="math-container">$f(x,y) = e^{-\sqrt{x^2+y^2}}$</span>. Then <span class="math-container">$f$</span> is continuous everywhere, but <span class="math-container">$f(0,0)$</span> has no directional derivative at <span class="math-container">$(0,0)$</span>. I'll let you prove this r... |
147,994 | <p><strong>Preamble</strong></p>
<p>Consider a <a href="http://reference.wolfram.com/language/ref/SetterBar.html" rel="nofollow noreferrer"><code>SetterBar</code></a>:</p>
<pre><code>SetterBar[1, StringRepeat["q", #] & /@ Range@5]
</code></pre>
<blockquote>
<p><a href="https://i.stack.imgur.com/b4eqb.png" rel=... | garej | 24,604 | <p>To keep button size, one option would be to add <code>StringPadRight</code></p>
<pre><code>SetterBar[1, StringPadRight @ ( StringRepeat["q", #] & /@ Range@5 ),
Appearance -> {"Vertical", "Button"}]
</code></pre>
<p><a href="https://i.stack.imgur.com/OPnLx.png" rel="nofollow noreferrer"><img src="https://i.... |
706,980 | <p>If I know that $f(z)$ is differentiable at $z_0$, $z_0 = x_0 + iy_0$.
How do I prove that $g(z) = \overline{f(\overline{z})}$ is differentiable at $\overline z_0$?</p>
| Jonas Granholm | 134,205 | <p>The function $f(z)=u(x,y)+iv(x,y)$ is differentiable if the first partial derivatives of $u$ and $v$ exist, are continuous, and satisfy the Cauchy-Riemann equations $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\quad\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.$$</p>
<p>Since we know... |
3,291,549 | <p>How does one prove that the exponential and logarithmic functions are inverse using the definitions:</p>
<p><span class="math-container">$$e^x= \sum_{i=0}^{\infty} \frac{x^i}{i!}$$</span>
and
<span class="math-container">$$\log(x)=\int_{1}^{x}\frac{1}{t}dt$$</span></p>
<p>My naive approach (sort of ignoring issue... | Fractal | 688,603 | <p>We use the fact that <span class="math-container">$g(x)=\exp(x)$</span> is the unique function <span class="math-container">$g:\mathbb{R} \rightarrow (0,\infty)$</span> such that <span class="math-container">$g'(x)=g(x)$</span> and <span class="math-container">$g(0)=1.$</span></p>
<p>Since <span class="math-contain... |
490,802 | <p>Is $(x,3,5)$ a plane, for $x\in\mathbb{R}$?</p>
<p>I know that if two of the coordinates are "arbitrary", like $(x,y,4)$or $(3,y,z)$, then it creates a plane (for $x,y,z\in \mathbb{R}).$</p>
<p>Is there a way to tell if it would create a plane in $\mathbb{R}^3?$</p>
| Rebecca J. Stones | 91,818 | <p>The subset $$\{(x,3,5):x \in \mathbb{R}\} \subseteq \mathbb{R}^3$$ is not a plane. It is an affine space, a translation of the vector space $$\{(x,0,0):x \in \mathbb{R}\} \subseteq \mathbb{R}^3$$ which has dimension $1$. A plane has dimension $2$. So, by the <a href="http://en.wikipedia.org/wiki/Dimension_theorem... |
4,190,301 | <p>I found <span class="math-container">$\tilde{R}$</span> in a mathematical text, and I would like to know how this is pronounced. I tried to search on the internet but was not able to find anything related.</p>
| epi163sqrt | 132,007 | <p>The symbol <span class="math-container">$\tilde{R}$</span> is also pronounced <em>arr twiddle</em>. We can hear it for instance in these <em><a href="https://www.youtube.com/watch?v=mbv3T15nWq0&t=328s" rel="nofollow noreferrer">lecture notes</a></em> by F. P. Schuller in minute 27 where he defines a <em>linear m... |
4,344,571 | <p>In a previous exam assignment, there is a problem that asks for a proof that <span class="math-container">$\mathbb{Z}_{24}$</span> and <span class="math-container">$\mathbb{Z}_{4}\times\mathbb{Z}_6$</span> are <strong>not</strong> isomorphic.</p>
<p>We have <span class="math-container">$\mathbb{Z}_{24}$</span> is is... | A. Thomas Yerger | 112,357 | <p>You don't actually say what <span class="math-container">$f$</span> is somewhere. These are a bunch of manipulations about what would be true of <span class="math-container">$f$</span> if it existed. In particular, you take <span class="math-container">$f$</span> to be some bijection between these sets, but you are ... |
904,041 | <p>$$tx'(x'+2)=x$$
First I multiplied it:
$$t(x')^2+2tx'=x$$
Then differentiated both sides:
$$(x')^2+2tx'x''+2tx''+x'=0$$
substituted $p=x'$ and rewrote it as a multiplication
$$(2p't+p)(p+1)=0$$
So either $(2p't+p)=0$ or $p+1=0$</p>
<p>The first one gives $p=\frac{C}{\sqrt{T}}$
The second one gives $p=-1$. My questi... | atomteori | 156,639 | <p>Group theory gives a pretty direct answer. Use
$$
G(t,x)=(\lambda t,\lambda^\beta x)\lambda_o=1
$$Then apply the group to the DEQ:
$$
t\bigg(\frac{dx}{dt}\bigg)^2+2t\frac{dx}{dt}=x\rightarrow \lambda t\bigg(\frac{\lambda ^\beta dx}{\lambda dt}\bigg)^2+2\lambda t\frac{\lambda^\beta dx}{\lambda dt}=\lambda^\beta x
$... |
3,148,076 | <p>CONTEXT: Challenge question set by uni lecturer for discrete mathematics course</p>
<p>Question: Prove the following statement is true using proof by contradiction: </p>
<p>For all positive integers <span class="math-container">$x$</span>, if <span class="math-container">$x$</span>, <span class="math-container">$x... | Myunghyun Song | 609,441 | <p><strong>Hint:</strong> One of <span class="math-container">$x, x+2, x+4$</span> must be divisible by <span class="math-container">$3$</span>.</p>
|
3,031,460 | <blockquote>
<p>Give an example of an assertion which is not true for any positive
integer, yet for which the induction step holds.</p>
</blockquote>
<p>First of all, definition.</p>
<blockquote>
<p>In <strong>inductive step</strong>, we suppose that <span class="math-container">$P(k)$</span> is true for some p... | Badam Baplan | 164,860 | <p>There is no shortage of good answers to this question, so I'll give a couple examples in the hopes that thinking about them will help you to produce your own answers.</p>
<p>Two possibilities for <span class="math-container">$P(n)$</span> are</p>
<p>-the assertion that <span class="math-container">$n! = 0$</span> ... |
22,753 | <p>I've learned the process of orthogonal diagonalisation in an algebra course I'm taking...but I just realised I have no idea what the point of it is.</p>
<p>The definition is basically this: "A matrix <span class="math-container">$A$</span> is orthogonally diagonalisable if there exists a matrix <span class="math-co... | Qiaochu Yuan | 232 | <p>Mathematics is all about equivalence relations. Often we are not really interested in objects on-the-nose, we are interested in objects up to some natural <a href="http://en.wikipedia.org/wiki/Equivalence_relation">equivalence relation</a>. Matrices really define <em>linear transformations</em>, so in order to talk ... |
22,753 | <p>I've learned the process of orthogonal diagonalisation in an algebra course I'm taking...but I just realised I have no idea what the point of it is.</p>
<p>The definition is basically this: "A matrix <span class="math-container">$A$</span> is orthogonally diagonalisable if there exists a matrix <span class="math-co... | Tpofofn | 4,726 | <p>Matrices are complicated objects. At first glance they are rectangular arrays of numbers with a complicated multiplication rule. Diagonalization helps us reduce the the matrix multiplication operation to a sequence of simple steps which make sense. In your case you are asking about orthogonal diagonalization, so ... |
271,592 | <p>Let $A$ be a C$^*$-algebra. A pre-Hilbert $A$-module $H$ is a right $A$ module with a $A$-valued inner product (which is linear in the second variable and conjugate linear in the first variable) such that</p>
<p>1.$\langle \xi, \beta \cdot T \rangle_{A} = \langle \xi, \beta\rangle_{A}T$,</p>
<p>2.$\langle \xi, \xi... | Yemon Choi | 763 | <p>The following is based on a copy of some old handwritten notes; I haven't had time to refresh my memory on all the details. I will try to fill in these details in the next day or so.
$\newcommand{\Nat}{{\bf N}}\newcommand{\veps}{\varepsilon}$</p>
<hr>
<p>Using the notation of my comment:
take $V_0=\ell_1(\Nat)$ si... |
271,592 | <p>Let $A$ be a C$^*$-algebra. A pre-Hilbert $A$-module $H$ is a right $A$ module with a $A$-valued inner product (which is linear in the second variable and conjugate linear in the first variable) such that</p>
<p>1.$\langle \xi, \beta \cdot T \rangle_{A} = \langle \xi, \beta\rangle_{A}T$,</p>
<p>2.$\langle \xi, \xi... | heller | 62,272 | <p>Thanks Yemon. I fill in the details of you example. </p>
<p>Let $H=l^1(N)$. Then $\overline{H}=l^2(N)$.</p>
<p>Let $\xi_j = 1/\sqrt{2^n} \sum_{i=0}^{2^n-1} e_{2^n+i}$ where
$\{e_1, e_2, \ldots \}$ is the canonical orthonormal basis of $l^2(N)$.Define $V \in B(l^2(N))$ by $Ve_j = \xi_j$. Then it is clear that
... |
164,896 | <p>I want to create a list length <code>l</code> with the function $f(x_n)=x_{n-1}+r$ where $r$ is a random real number between -1 and 1 and $x_0=1$. It got it working like this:</p>
<pre><code>l = 50; a = Range[l]; a[[1]] = 0;
For[i = 2, i <= l, i++, a[[i]] = a[[i - 1]] + RandomReal[{-1, 1}]];
a
</code></pre>
<p>... | José Antonio Díaz Navas | 1,309 | <p>I am understanding you want to generate a sequence of real numbers recursively (with some randomness) starting with $x_0=1$, Maybe this is a solution:</p>
<pre><code>RandomSeed[2];
f[0] = 1;
f[n_Integer /; n > 0] := f[n] = f[n - 1] + RandomReal[{-1, 1}];
l = 50;
f[#] & /@ Range[0, l]
(*
{1, 1.44448, 0.6... |
1,702,616 | <p>I was working on a programming problem to find all 10-digit perfect squares when I started wondering if I could figure out how many perfects squares have exactly N-digits. I believe that I am close to finding a formula, but I am still off by one in some cases.</p>
<p>Current formula where $n$ is the number of digit... | Nigel Galloway | 63,512 | <p>The number of perfect squares between any two numbers a and b with a less than b is floor sqrt b - ceil sqrt a + 1. i.e. a=1000 b=2000. ceil sqrt 1000 = 32. floor sqrt 2000 = 44. So 32 33 34 35 36 37 38 39 40 41 42 43 and 44 when squared will be perfect squares between 1000 and 2000 and 44-32+1=13. If you are silly ... |
1,779,068 | <p>Let $H$ and $K$ be two subgroups of a group $G$ such that $[G : H]=2$ and $K$ is not a subgroup of $H$. Then show that $HK=G$.
Now, since $HK$ is a subset of $G$ we need only to show that $G$ is a subset of $HK$. But how can I show it? Please help me. Thank you in advance.</p>
| Community | -1 | <p>By Lagrange's,</p>
<p>$$[G : HK][HK:H] = [G:H] = 2$$</p>
<p>since $K \not\subset H$, $[HK:H] > 1 \implies [HK:H] =2 $, so $[G:HK] = 1 \implies G = HK$</p>
|
207,778 | <p>I want to save expressions as well as their names in a file.</p>
<pre><code> func[i_] := i;
Do[func[i] >>> out.m,{i,1,3}];
</code></pre>
<p>The output is </p>
<pre><code> cat out.m
1
2
3
</code></pre>
<p>However the desired output is</p>
<pre><code> cat out.m
func[1] = 1;
func[2... | Fortsaint | 10,101 | <pre><code>func[i_] := i
Do[
"func["<>ToString@i<>"] = "<>ToString@func@i >>> "out.m"
, {i,1,3}
]
</code></pre>
<p><strong>Edit 1</strong>: It's not clear to me what's your scope but, this does what you asked</p>
<pre><code>"func["<>ToString@#<>"] = "<>ToString@... |
1,660,289 | <p>I want to find the line that passes through $(3,1,-2)$ and intersects at a right angle the line $x=-1+t, y=-2+t, z=-1+t$. </p>
<p>The line that passes through $(3,1,-2)$ is of the form $l(t)=(3,1,-2)+ \lambda u, \lambda \in \mathbb{R}$ where $u$ is a parallel vector to the line. </p>
<p>There will be a $\lambda \i... | Erick Wong | 30,402 | <p>Julián Aguirre's answer is the definitive one, but I also want to give some insight on why this conjecture is kind of unreasonable (i.e. that one should expect it to become false by just doing a little computation).</p>
<p><strong>Caveat</strong>: none of the following is proven — for instance, it hasn't been prove... |
1,741,765 | <p>Problem description: Show that well-ordering is not a first-order notion. Suppose that $\Gamma$ axiomatizes the class of well-orderings. Add countably many constants $c_i$ and show that $\Gamma \cup \{c_{i+1} < c_i \mid i \in \mathcal{N}\}$ has a model. </p>
<p>So, I don’t quite get how to go about this. <a href... | Nick | 27,349 | <p>This implies the operation is commutative. First note that by choosing $b=a$, you get that $a^3 = a \# a \# a = a$ for any $a$. In particular, apply this observation to the element $a \# b$:</p>
<p>$$
a \# b = (a\# b) \# (a\# b) \# (a \#b) = (a \# b \# a)\#(b \#a \# b) = b \# a
$$</p>
|
134,937 | <p>Let $p \equiv q \equiv 3 \pmod 4$ for distinct odd primes $p$ and $q$. Show that $x^2 - qy^2 = p$ has no integer solutions $x,y$.</p>
<p>My solution is as follows.</p>
<p>Firstly we know that as $p \equiv q \pmod 4$ then $\big(\frac{p}{q}\big) = -\big(\frac{q}{p}\big)$</p>
<p>Assume that a solution $(x,y)$ does e... | Will Jagy | 10,400 | <p>You started out well enough. You got to: the assumption of a solution implies $(p|q) = 1$ and so $(q|p ) = -1.$</p>
<p>Next, we think we have $x^2 - q y^2 = p,$ in particular
$$ x^2 \equiv q y^2 \pmod p. $$ If $y \neq 0 \pmod p,$ then $y$ has a multiplicative inverse $\pmod p,$ and
$$ \left( \frac{x}{y} \right)^... |
3,706,332 | <p>Let <span class="math-container">$f: \mathbb{R} \to \mathbb{R} $</span> be a differentiable function. Is it true that <span class="math-container">$f$</span> is strictly increasing on <span class="math-container">$\mathbb R$</span> if and only if <span class="math-container">$f'(x) \geq 0$</span> on <span class="mat... | zhw. | 228,045 | <p>Suppose <span class="math-container">$f$</span> is differentiable and strictly increasing. Then yes, we must have <span class="math-container">$f'\ge 0$</span> everywhere. However, we can have <span class="math-container">$f'=0$</span> on an uncountable set, in fact a set of positive measure.</p>
<p>Example: Let <s... |
121,865 | <p>Can someone please help me clarify the notations/definitions below:</p>
<p>Does $\{0,1\}^n$ mean a $n$-length vector consisting of $0$s and/or $1$s?</p>
<p>Does $[0,1]^n$ ($(0,1)^n$) mean a $n$-length vector consisting of any number between $0$ and $1$ inclusive (exclusive)?</p>
<p>As a related question, is there... | diofanto | 27,163 | <p>The idea is simple… The abstract set of Topology</p>
<p>$$\displaystyle{\prod_{i\in I}X_i=\left\{x:I\rightarrow\cup_{i\in I}X_i \vert x(i)=x_i\in X_i,\;\forall i\in I \right\}}$$</p>
<p>where $x$ are continuous functions. (Also, you can see the continuous function as equivalence relation with $n$-length vector)</p... |
45,662 | <p>Does this undirected graph with 6 vertices and 9 undirected edges have a name?
<img src="https://i.stack.imgur.com/XwuUB.png" alt="enter image description here">
I know a few names that are not right. It is not a complete graph because all the vertices are not connected. It is close to K<sub>3,3</sub> the utility gr... | Alon Amit | 308 | <p>This is exactly $K_{3,3}$. What makes you say it's only "close" to it? Can you spot two independent sets of 3 vertices each here? Once you see that, and given that there are 9 edges, it <strong>must</strong> be the complete bipartite graph on two sets of 3 vertices each.</p>
|
45,662 | <p>Does this undirected graph with 6 vertices and 9 undirected edges have a name?
<img src="https://i.stack.imgur.com/XwuUB.png" alt="enter image description here">
I know a few names that are not right. It is not a complete graph because all the vertices are not connected. It is close to K<sub>3,3</sub> the utility gr... | Isaac Kleinman | 11,444 | <p>You can also think of it as the <a href="http://mathworld.wolfram.com/HararyGraph.html" rel="nofollow">Harary graph</a> $H_{3,6}$.</p>
|
3,055,208 | <p>I am trying to compute the below limit through Taylor series:
<span class="math-container">$\lim \limits_{x\to \infty} ((2x^3-2x^2+x)e^{1/x}-\sqrt{x^6+3})$</span></p>
<p>What I have already tried is first of all change the variable x to
<span class="math-container">$x=1/t$</span> and the limit to t limits to 0, so ... | DudeMan | 630,057 | <p>I think that you don't have to go that far. You know that (for large values of <span class="math-container">$x$</span>):
<span class="math-container">$$
e^{1/x}=\sum_{k=0}^{\infty}{\frac{1}{k!x^k}}\geq1
$$</span>
so that:
<span class="math-container">$$
(2x^3-2x^2+x)e^{1/x}-\sqrt{x^6+3} \geq (2x^3-2x^2+x)-\sqrt{x^6... |
3,055,208 | <p>I am trying to compute the below limit through Taylor series:
<span class="math-container">$\lim \limits_{x\to \infty} ((2x^3-2x^2+x)e^{1/x}-\sqrt{x^6+3})$</span></p>
<p>What I have already tried is first of all change the variable x to
<span class="math-container">$x=1/t$</span> and the limit to t limits to 0, so ... | TheSimpliFire | 471,884 | <p><strong>HINT:</strong></p>
<p><span class="math-container">$$\sqrt{x^6+3}=\frac1{t^3}\left(1+3t^6\right)^{1/2}=\frac1{t^3}+\frac{\frac12\cdot3t^6}{t^3}-\frac{\frac1{2\cdot4}\cdot(3t^6)^2}{t^3}+\frac{\frac{1\cdot3}{2\cdot4\cdot6}\cdot(3t^6)^3}{t^3}-\cdots\\(2x^3-2x^2+x)e^{1/x}=\left(\frac2{t^3}-\frac2{t^2}+\frac1t\r... |
4,136,082 | <p><span class="math-container">$$f(x) =
\begin{cases}
\cos(\frac{1}{x}) & \text{if $x\ne0$} \\
0 & \text{if $x=0$} \\
\end{cases}$$</span></p>
<p>How do I prove this function has Darboux's property? I know it has it because it has antid... | Community | -1 | <p>Well... it's obvious that it has Darboux's property on any interval <span class="math-container">$[a,b]$</span> with <span class="math-container">$0<a< b\lor a<b<0$</span>.</p>
<p>If <span class="math-container">$a\le 0<b$</span>, then <span class="math-container">$f(a),f(b)\in [-1,1]$</span> anyways ... |
2,467,327 | <p>How to prove that $441 \mid a^2 + b^2$ if it is known that $21 \mid a^2 + b^2$.<br>
I've tried to present $441$ as $21 \cdot 21$, but it is not sufficient.</p>
| Jack D'Aurizio | 44,121 | <p>If $3\mid(a^2+b^2)$ then $3$ divides both $a$ and $b$, since $-1$ is not a quadratic residue $\!\!\pmod{3}$.<br>
The same applies $\!\!\pmod{7}$. If $21\mid(a^2+b^2)$, from the CRT we get that $3$ and $7$ divide both $a$ and $b$, hence $3^2$ and $7^2$ divide both $a^2$ and $b^2$ and $3^2\cdot 7^2\mid (a^2+b^2)$ as w... |
3,884,098 | <p>Suppose <span class="math-container">$A$</span> is a <span class="math-container">$n \times n$</span> symmetric real matrix with eigenvalues <span class="math-container">$\lambda_1, \lambda_2, \ldots, \lambda_n$</span>, what are the eigenvalues of <span class="math-container">$(I - A)^{3}$</span>?</p>
<p>Are they <s... | Derek Luna | 567,882 | <p>I am not sure how much is going to go through without using a proof by contrapositive since any partition you make needs to have a finite number of elements and if you index a sum by the set <span class="math-container">$\{0,...,n\}$</span> we don't get information about every <span class="math-container">$f(x)$</sp... |
2,466,556 | <p><strong>Solution:</strong> </p>
<p>The vectors $\vec{AB}=(3,2,1)-(0,1,2)=3,1,-1$ and $\vec{AC}=(4,-1,0)-(0,1,2)=(4,-2,-2),$ are two direction vectors of the plane. A normal vector $\vec{n}$ to the plane is then given by $$\vec{n}=\vec{AB}\times\vec{AC}=(-4,2,-10).$$</p>
<p>Since $A$ is a point on the plane, we get... | Sir_Math_Cat | 330,367 | <p>The vector $(x-0,y-1,z-2)$ comes directly from the fact that $(0,1,2)$ is a point on the plane. For the second part of your question, if $\vec{v}$ is a vector and $\vec{n}$ is a normal vector to $\vec{v}$, then $\vec{v}\cdot\vec{n}=0$. We use the dot product between the normal vector and the cross product of the dis... |
881,831 | <p>It is trivial that a group $G$ is abelian if and only if every subgroup of $G$ with two generators is abelian (i.e., any two elements commute).</p>
<p>If $G$ is a nilpotent group, every subgroup with two generators must be nilpotent. Is the reciprocal true? More precisely:</p>
<blockquote>
<p>Let $G$ be a group ... | Geoff Robinson | 13,147 | <p>When $G$ is finite the answer is "yes", since any two elements of coprime order commute, so a Sylow $p$-subgroup is normal for each prime divisor $p$ of the group order. When $G$ is infinite, I don't know, but others might. Each element of $G$ is certainly an Engel element, but there are non-nilpotent groups in whic... |
223,385 | <p>I would like to recreate the following picture in Mathematica. I know how to draw a tree with GraphLayout. But I don't know how to create the shape of nodes as below. A bit hints about where to start will be appreciated!</p>
<p><a href="https://i.stack.imgur.com/PtK1F.png" rel="noreferrer"><img src="https://i.stack... | flinty | 72,682 | <pre><code>(* generate a random tree *)
edges = Table[i <-> RandomInteger[{0, i - 1}], {i, 1, 20}];
(* random circles appear on edges with degree 1 only *)
circles = If[# == 1, RandomInteger[{1, 4}], 0] & /@ VertexDegree[Graph[edges]];
gencircles[{x_, y_}, name_] :=
If[circles[[name + 1]] > 0,
Disk... |
1,903,235 | <p>According to Wikipedia, </p>
<blockquote>
<p>Hilbert space [...] extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions</p>
</blockquote>
<p>However, the article on Euclidean space states a... | user288972 | 288,972 | <p>A Hilbert space essentially is also a generalization of Euclidean spaces with infinite dimension.</p>
<hr>
<p><strong>Note</strong>: this answer is just to give an intuitive idea of this generalization, and to consider infinite-dimensional spaces with a scalar product that they are complete with respect to metric ... |
2,324,850 | <p>How to find the shortest distance from line to parabola?</p>
<p>parabola: $$2x^2-4xy+2y^2-x-y=0$$and the line is: $$9x-7y+16=0$$
Already tried use this formula for distance:
$$\frac{|ax_{0}+by_{0}+c|}{\sqrt{a^2+b^2}}$$</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Use <a href="http://www.sosmath.com/CBB/viewtopic.php?t=17029" rel="nofollow noreferrer">rotation of axes</a> to eliminate the $xy$ term from the equation of the parabola as the distance is invariant in rotation.</p>
<p>Now use parametric equation $P(h+at^2,k+2at)$ of the parabola $$(y-k)^2=4a(x-h)$$ ... |
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Qiaochu Yuan | 232 | <blockquote>
<p>What is the period of the Fibonacci sequence $F_n$ modulo a prime $p$? </p>
</blockquote>
<p>This is the <a href="http://en.wikipedia.org/wiki/Pisano_period" rel="nofollow">Pisano period</a>. It is difficult to say much about the exact period, but one can write down a number which is guaranteed to be... |
1,077,504 | <p>Evaluate:</p>
<p>$$\int_{0}^{\infty} \frac{1}{x^6 + 1} \,\mathrm dx$$</p>
<p>Without <strong>the use of complex-analysis.</strong></p>
<p>With complex analysis it is a very simple problem, how can this be done WITHOUT complex analysis?</p>
| Lucian | 93,448 | <blockquote>
<p><em>how can this be done WITHOUT complex analysis?</em></p>
</blockquote>
<p>$\quad$ All integrals of the form $~\displaystyle\int_0^\infty\frac{x^{k-1}}{(x^n+a^n)^m}dx~$ can be evaluated by substituting $x=at$ and $u=\dfrac1{t^n+1}$ , then recognizing the expression of the <a href="http://en.wikiped... |
1,570,044 | <p>How many arrangements of banana such that the "b" occurs before any of the "a's"?</p>
<p>This is more an inquiry into what I did wrong in my counting. I came up with a solution of: $$\binom{3}{1} \binom{5}{3}$$ where i did C(3,1) to account for the 3 possible places the "b" could go and C(5,3) to choose the posi... | grand_chat | 215,011 | <p>Another way to see this: If the
<em>b</em> must occur before any of the <em>a</em>'s, then our only decision is where to put the <em>n</em>'s. There are six slots available for the two <em>n</em>'s, giving ${6\choose2}=15$ possibilities. Once the <em>n</em>'s are placed we put <em>b,a,a,a</em> in the remaining open... |
3,407,489 | <p><span class="math-container">$\neg\left (\neg{\left (A\setminus A \right )}\setminus A \right )$</span></p>
<p><span class="math-container">$A\setminus A $</span> is simply empty set and <span class="math-container">$\neg$</span> of that is again empty set. Empty set <span class="math-container">$\setminus$</span... | fleablood | 280,126 | <p><span class="math-container">$A$</span> = everthing that is is <span class="math-container">$A$</span>.</p>
<p><span class="math-container">$A\setminus A$</span> = everything in <span class="math-container">$A$</span> that is not in <span class="math-container">$A$</span> = nothing = <span class="math-container">$\... |
241,612 | <blockquote>
<p>Find all eigenvalues and eigenvectors:</p>
<p>a.) $\pmatrix{i&1\\0&-1+i}$</p>
<p>b.) $\pmatrix{\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta}$</p>
</blockquote>
<p>For a I got:
$$\operatorname{det} \pmatrix{i-\lambda&1\\0&-1+i-\lambda}= \lambda^{2} - 2\lambda ... | Belgi | 21,335 | <p>For $a$ you can note that the matrix in case is upper triangular,
or use the fact the the quadratic formula is also valid over $\mathbb{C}$. </p>
<p>For $b$ the last equality you have is not true, how did the $\cos(\theta)$
coefficient of $\lambda$ disappeared ? you should apply the quadratic
formula in this case t... |
146,813 | <p>Is sigma-additivity (countable additivity) of Lebesgue measure (say on measurable subsets of the real line) deducible from the Zermelo-Fraenkel set theory (without the axiom of choice)?</p>
<p>Note 1. Follow-up question: Jech's 1973 book on the axiom of choice seems to be cited as the source for the Feferman-Levy m... | François G. Dorais | 2,000 | <p>This depends on exactly how you define Lebesgue measure since some definitions incorporate countable additivity. However, there is a model of ZF, the Feferman-Levy model, where $\mathbb{R}$ is a countable union of countable sets which ensures that any countably additive measure on $\mathbb{R}$ has to be trivial.</p>... |
107,525 | <p>Say I have two random variables X and Y from the same class of distributions, but with different means and variances (X and Y are parameterized differently). Say the variance converges to zero as a function of n, but the mean is not a function of n. Can it be formally proven, without giving the actual pdf of X and Y... | leonbloy | 312 | <p>The answer is yes. Let's assume the means verify $\mu_X < \mu_Y$, and let $c =(\mu_X +\mu_Y)/2$ the middle point. The "overlap area" (?) is</p>
<p>$$\int_{-\infty}^{\infty} \min(f_X(x),f_Y(x)) dx = \int_{-\infty}^c \cdots dx + \int_{c}^{\infty} \cdots dx$$ </p>
<p>The second term is: </p>
<p>$$\int_c^{\infty}... |
1,880,090 | <p>The solution states that the ball of radius $\epsilon >0$ around a real number $x$ always contains the non-real number $x+i\epsilon/2$. </p>
<p>I don't understand the answer, for every number $x \in \mathbb{R}$ there is an open ball, right? For every $x \in \mathbb{R}$ there is an $r>0$ such that I can form a... | quid | 85,306 | <p>You need to be careful what the definition of the sets is precisely. </p>
<p>You study a subset of the complex numbers thus your topological objects are those for the complex numbers. </p>
<p>Thus in this context $B_r (x) = \{z \in \mathbb{C} \colon |z-x| < r\}$. So it is the set of all <strong>complex numbers... |
1,832,080 | <p>The converse is pretty obvious. If G is a cycle, then it is isomorphic to it's line graph.
How to prove that if L(G) is isomorphic to G, then G is a cycle...?</p>
<p><strong>P.S.</strong>- Assume G is connected</p>
| joriki | 6,622 | <p>A vertex of $G$ of degree $d_i$ contributes $\binom{d_i}2$ edges to $L(G)$. Then with $n$ denoting the common number of vertices and edges of $G$ and $L(G)$,</p>
<p>$$
\sum_id_i=2n\;,
$$</p>
<p>$$
\sum_i\frac{d_i(d_i-1)}2=n\;,
$$</p>
<p>so</p>
<p>$$
\sum_id_i^2=4n
$$</p>
<p>and thus</p>
<p>\begin{align}
\opera... |
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