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 |
|---|---|---|---|---|
1,373,103 | <p>I was wondering if $|f(x)g(x)| = |f(x)| |(g(x)|$ is true all the time as in the case of real numbers.</p>
<p>I was not convinced enough that that was true.</p>
<p>But I can't think of any counterexample.</p>
<p>Thank you.</p>
| Tucker | 256,305 | <p>For $|x|\leq1$, the Maclaurin series (of cos(x)) is an decreasing alternating series, so the error in your approximation will be no worse then the next non-zero term in the series, namely $|-\frac{x^{6}}{6!}|$. However you will want to review estimates for the remainder in a Taylor series. You will not always have t... |
2,441,894 | <p>The matrix $$\pmatrix{100\sqrt{2}&x&0\\-x&0&-x\\0&x&100\sqrt{2}},\quad x>0$$ have two equal eigenvalues. How can I find $x$?
What I tried is this. If $\lambda_1$ is doubly degenerate and $\lambda_2$ the third eigenvalue, then the characteristic equation is $(\lambda-\lambda_1)^2(\lambda-\... | euraad | 443,999 | <p>Here is the answer.</p>
<p>$$A = [A;A](:)$$</p>
<p>Easy!</p>
|
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">$$\... | MAHI | 821,955 | <p><em>My Approach-</em></p>
<p>Let M be the LUB of {<span class="math-container">$x_n$</span>}.
Then <span class="math-container">$x_n \in N_ε(M)$</span> for some <span class="math-container">$n\in S\forall ε>0$</span> ,(i. e.<span class="math-container">$x_n \in N_ε(M)$</span> when <span class="math-container">$K... |
11,457 | <p>In their paper <em><a href="http://arxiv.org/abs/0904.3908">Computing Systems of Hecke Eigenvalues Associated to Hilbert Modular Forms</a></em>, Greenberg and Voight remark that</p>
<p>...it is a folklore conjecture that if one orders totally real fields by their discriminant, then a (substantial) positive proporti... | JSE | 431 | <p>Maybe it's worth a word about <em>why</em> Cohen-Lenstra predicts this behavior. Suppose K is a field with r archimedean places. Then Spec O_K can be thought of as analogous to a curve over a finite field k with r punctures, which is an affine scheme Spec R. Write C for the (unpunctured) curve. Then the class gr... |
107,171 | <p>I'm trying to find $$\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}... | Peđa | 15,660 | <p>According to Maple solution is given by :</p>
<p>$$\displaystyle\sum_{n=0}^{\infty} \frac{1}{(3n+1)^2} = \frac{1}{9} \Psi\left(1,\frac{1}{3}\right)$$</p>
<p>where $\Psi\left(1,\frac{1}{3}\right)$ is <a href="http://mathworld.wolfram.com/PolygammaFunction.html" rel="nofollow">polygamma function</a> .</p>
|
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>$$\lim\limits_{r \to 0} \frac{e^{-\frac{1}{r^2}... | Riccardo.Alestra | 24,089 | <p>The sum is $$\frac19{\Psi(1,\frac13)}$$</p>
<p>where $\Psi$ is the Poligamma function</p>
|
107,171 | <p>I'm trying to find $$\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}... | Daoyi Peng | 29,776 | <p>The sum can express
$$\begin{align}
& \sum\limits_{n=0}^{\infty }{\frac{1}{{{(3n+1)}^{2}}}} \\
& =\frac{1}{9}\left[ \frac{{{\Gamma }''}(1/3)}{\Gamma (1/3)}-{{\left( \frac{{\Gamma }'(1/3)}{\Gamma (1/3)} \right)}^{2}} \right] \\
& =1+\frac{1}{9}\int_{0}^{\infty }{\frac{t{{\mathrm{e}}^{-t/3}}}{{{\mat... |
3,005,842 | <blockquote>
<p>Let <span class="math-container">$(X,Y)$</span> be the coordinates of a point uniformly chosen from a
quadrilateral with vertices <span class="math-container">$(0,0)$</span>, <span class="math-container">$(1,0)$</span>, <span class="math-container">$(1,1)$</span>, <span class="math-container">$(0,2)$</s... | Graham Kemp | 135,106 | <p>To be correct you need to include the support. This can be expressed in two equivalent ways: <span class="math-container">$$f_{X,Y}(x,y)~{=\tfrac 32\mathbf 1_{0\le x\leq 1, 0\leq y\leq2-x}\\=\tfrac 32\mathbf 1_{0\le y\leq 2, 0\leq x\leq \min(1,2-y)}}$$</span></p>
<p>Then integrate over this support with respect to ... |
1,576,713 | <p>$X$ and $Y$ are two sets and $f:X\to Y$. If $f(C)=\{f(x):x\in C\}$ for $C\subseteq X$ and $f^{-1}(D)=\{x:f(x)\in D\}$ for $D\subseteq Y$, then the true statement is </p>
<p>(A) $f(f^{-1}(B))=B$</p>
<p>(B) $f^{-1}(f(A))=A$</p>
<p>(C) $f(f^{-1}(B))=B$ only if $B\subseteq f(X)$</p>
<p>(D) $f^{-1}(f(A))=A$ only if $... | user 1 | 133,030 | <p><code>(A) is not true</code>.<br>
Let $X=A=\{a\}$, and $Y=B=\{a,b\}$. Define $f:X\to Y; \quad f(a)=a.$ Then $f(f^{-1}(B))=\{a\}\neq B.$ </p>
<hr>
<p><code>(B) is not true</code>.<br>
Let $X=Y=\{a,b\}$, and $A=B=\{a\}$ . Define $f:X\to Y; \quad f(x)=a, \forall x\in X.$ Then $f^{-1}(f(A))=X\neq A.$ </p>
<hr>
<p... |
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... | celeriko | 3,237 | <p>There is a series of posters made by Key Curriculum Press that each detail a different culture and their contribution to mathematics. They are colorful, informative, and have, in my experience, been very engaging for students to look at and read.</p>
<p>Unfortunately, with the McGraw-Hill purchase of Key Curriculu... |
18,459 | <p>Does anyone know of any studies or have personal experience dealing with difficulties (if any) faced by students studying mathematics if they come from countries which use languages written from right-to-left or top-down? </p>
<p>I have been wondering about this recently because I have been working on supplemental ... | Amy B | 5,321 | <p>I have some personal experience.</p>
<p>I taught in a school that has Israeli students whose families had moved to the US. These children had Hebrew as their native language. Hebrew is written right to left. I never saw these students struggle with math because they learned to write from right to left.</p>
<p>Furth... |
3,346,775 | <p>Do there exist non-zero expectation, dependent, uncorrelated random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>? The examples that I have found have at least one of the variables have zero expectation.</p>
| Robert Israel | 8,508 | <p>Take any example and add nonzero constants to <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. This changes the expectations but does not affect dependency or correlation.</p>
|
2,185,585 | <p>Triangular numbers (See <a href="https://en.wikipedia.org/wiki/Triangular_number" rel="noreferrer">https://en.wikipedia.org/wiki/Triangular_number</a> )</p>
<p>are numbers of the form $$\frac{n(n+1)}{2}$$</p>
<p>In ProofWiki I found three claims about triangular numbers. The three claims are that a triangular num... | Qiaochu Yuan | 232 | <p>Yes, the real $1$-dimensional projective line $\mathbb{RP}^1$ is homeomorphic (in fact diffeomorphic) to the circle $S^1$. It can be thought of as the circle $S^1$ with antipodal points identified, which reflects the fact that the circle is its own double cover: the double cover map is given explicitly by</p>
<p>$$... |
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 total number of ways of arranging the letters is $\frac{9!}{2!} = 181440$. Of these, let us count the cases where no two vowels are together. This is $$\frac{6!}{2!} \times \binom{7}{3}\times 3! = 75600$$
Again, the number of ways in which all vowels are together is 15120. Thus the number of ways in which exactl... |
287,947 | <p>For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of the cosines of rational multiples of $\pi$?</p>
<p>Products of linear combinations of cosines of rational multiples... | Johannes Hahn | 3,041 | <p>Yes, that is true. The general case is the Kronecker-Weber theorem as Lucia mentioned in the comments. For square roots one can be more explicit and prove $\mathbb{Q}(\zeta_p) \cap \mathbb{R} = \mathbb{Q}[\sqrt{ (-1)^{(p-1)/2} p }]$ for odd prime numbers $p$ using properties of the Legendre symbol. Therefore you can... |
521,589 | <p>In a rectangle $ABCD$, the coordinates of $A$ and $B$ are $(1,2)$ and $(3,6)$ respectively and some diameter of the circumscribing circle of $ABCD$ has equation $2x-y+4=0$. Then the area of the rectangle is:</p>
<p>My work: I found the equations of $AD$ and $BC$ of the rectangle. Taking the points $C$ and $D$ as $(... | copper.hat | 27,978 | <p>Here is one ugly way.</p>
<p>Since the diameter lies on the given line, we know the centre of the circle $P=(x,y)$ lies on this line. The distance from $P$ to $A=(1,2)$ and $B=(3,6)$ must be the same so we have the equation:
$(x-1)^2+(y-2)^2 = (x-3)^2+(y-6)^2$. Simplifying gives the equation $2y+x = 10$.
Since $P$ ... |
50,521 | <p>I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. </p>
<p>Is there something that still needs to be justified mathematically in order to have solid foundations? </p>
| Willie Wong | 1,543 | <p>Let me sneak in an answer before this gets closed, because the answer to this question is likely to be very very different depending if you ask a mathematician or a physicist. <code>:)</code> So here is a long list of questions that a <em>mathematician</em> considers to be important open problems in general relativ... |
394,085 | <p>How is it possible to establish proof for the following statement?</p>
<p>$$n = \frac{1}{2}(5x+4),\;2<x,\;\text{isPrime}(n)\;\Rightarrow\;n=10k+7$$</p>
<p>Where $n,x,k$ are $\text{integers}$.</p>
<hr>
<p>To be more verbose:</p>
<p>I conjecture that;</p>
<p>If $\frac{1}{2}(5x+4),\;2<x$ is a prime number, ... | Warren Moore | 63,412 | <p>What is $x$? I assume that it is some natural number: then the conjecture is false. If you let $x=0$, then $n=2$, but $n\equiv 2\text{ (mod }10)\not\equiv 7\text{ (mod 10})$.
<hr />
For $n$ to be an integer, we must have $x$ even, and since $n\ne 2$, we must have $n$ odd. So in particular, we cannot have $5x+4$ even... |
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>
| cmk | 671,645 | <p>First, <span class="math-container">$A$</span> and <span class="math-container">$B$</span> should be open.</p>
<p>Suppose that we can take <span class="math-container">$x\in A,\ y\in B.$</span> Then, by assumption, there exists a connected subset <span class="math-container">$C'$</span> of <span class="math-contain... |
227,562 | <p>Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$. </p>
<p><strong>Question 1.</strong> Is it true that there exists $\varepsilon_0>0$ such that for any $0<\varepsilon <\varepsilon_0$ the intersection $K\cap \varepsilon S^{n-1}$ is contractible? Here $\var... | Mohammad Ghomi | 68,969 | <p>The answer to Question 1 is yes, which is precisely Lemma 3.6 in the paper:</p>
<p><a href="http://people.math.gatech.edu/~ghomi/Papers/torsionrq.pdf" rel="nofollow noreferrer">Boundary torsion and convex caps of locally convex surfaces,
J. Differential Geom., 105 (2017), 427-486.</a></p>
<p>Although the lemma is... |
1,084,041 | <p>Introduction: I've been studying integrals of the form $$\int_0^\infty \frac{x^a}{(e^x-1)^b}dx$$ where a and b are real parameters. I've been able to find closed forms for the integral in terms of the Riemann Zeta function, the Gamma function and the Polygamma functions provided the integral converges when at least ... | Olivier Oloa | 118,798 | <p>A partial answer.</p>
<p>I'm sure you already know the Lerch transcendent, a <strong>special function</strong> which may initially be defined as
$$\Phi(z,s,a):=\sum_{k=0}^\infty\frac{z^k}{(a+k)^s}, \quad a>0,\Re s>1,|z|<1.$$
It admits the following integral representation, which you obtain by expanding the... |
1,084,041 | <p>Introduction: I've been studying integrals of the form $$\int_0^\infty \frac{x^a}{(e^x-1)^b}dx$$ where a and b are real parameters. I've been able to find closed forms for the integral in terms of the Riemann Zeta function, the Gamma function and the Polygamma functions provided the integral converges when at least ... | Lucian | 93,448 | <p>Starting from the general <span class="math-container">$~\displaystyle\int_0^\infty\frac{x^a}{e^x-u}dx~=~\frac{\Gamma(a+1)\cdot\text{Li}_{a+1}(u)}u,~$</span> we can then </p>
<p>deduce, by way of repeated differentiation under the integral sign with regard to <em>u</em>, </p>
<p>that <span class="math-container">$... |
1,181,631 | <p>Let $f : \mathbb R \to \mathbb R$ continuous. Prove that graph $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed.</p>
<p>I'm a little confused on how to prove $G$ is closed. I get the general strategy is to show that every arbitrary convergent sequence in $G$ converges to a point in $G$.</p>
<p>Here is what I tri... | Marm | 159,661 | <p>Define at first $F:\mathbb R^2 \rightarrow \mathbb R$,$F(x,y)=f(x)-y$.</p>
<p>Next note that $F$ is continuous (because $f$ and "+" are continuous)</p>
<p>Then the graph is exactly the inverse image of $\{0\}\in \mathbb R$, hence</p>
<p>the graph is closed as an inverse image of a closed set under a continuous fu... |
1,887,536 | <p>Howdy just a simple question,</p>
<p>I know when A is diagonalizable, the eigenvalues of F(A) are just simply $F(\lambda_i)$ where $\lambda_i \exists \sigma (A)$</p>
<p>I'm interested in the case of when A is not diagonalizable. I look at A as a Jordan form, but I cannot seem to show that when $A$ is not diagonali... | imranfat | 64,546 | <p>As hinted in a comment, we assume here that the coefficients $a,b,c,d$ are real.
If $x=1+4i $ is a root, then also its conjugate $x=1-4i$. Now here is where I differ.
It is better to write $x-1=4i$, then square which gives $x^2-2x+1=-16. $ Do you recognize that this move captures the conjugate root as well? So we ha... |
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... | Terry Tao | 766 | <p><a href="https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem" rel="noreferrer">Szemerédi's theorem</a> asserts that every set <span class="math-container">$A$</span> of integers of positive upper density (thus <span class="math-container">$\limsup_{N \to \infty} \frac{|A \cap [-N,N]|}{|[-N,N]|} > 0$</span>) ... |
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... | Stanley Yao Xiao | 10,898 | <p>In general, the probabilistic method of Erdos follows exactly this philosophy: prove that an object with a certain property of number theoretic interest exists by showing that the probability a random set satisfies the desired property with positive probability (usually the probability is one!)</p>
<p>Example: a sub... |
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>... | W_D | 85,348 | <p>No, no, dear MethodManX, while computing probabilities, addition refers to "or", multiplication - to "and". Here you have "and": the first card is a King AND the second is a King AND the fird is a King, so it's rather $\frac{4}{52}\cdot\frac{3}{52}\cdot\frac{2}{52}$.</p>
|
918,788 | <p>How to do this integral</p>
<p>$$\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$$</p>
<p>for any $k > 0$ ?.</p>
<p>I tried to use gamma function, but sometimes the series doesn't converge.</p>
| Jack D'Aurizio | 44,121 | <p>The integral equals $\sqrt{\pi}e^{-k^2/4}$. To show this, just consider that:
$$ I =\Re\int_{-\infty}^{+\infty}e^{ikx-x^2}\,dx = e^{-k^2/4}\cdot \Re\int_{-\infty}^{+\infty}e^{-(x-ik/2)^2}\,dx $$
and prove that the complex shift does not affect the value of the integral:
$$\int_{-\infty}^{+\infty}e^{-(x-ik/2)^2}\,dx ... |
240,700 | <p>How can I prove that every maximal ideal of $B= \mathbb{Z} [(1+\sqrt{5})/2] $ is a principal?</p>
<p>I know if I show that B has division with remainder, that means it is a Euclidean domain. It follows that B is PID, and then every maximal ideal is principal ideal in PID. </p>
<p>However, I haven't been able to sh... | Davide Giraudo | 9,849 | <p>Consider $e^n$ the sequence whose unique non-zero entry is the $n$-th (which is $1$), and the sequence $x_n=\sum_{j=1}^n2^{-j}e^j\in\Bbb R^{\infty}$. This sequence is Cauchy for the $\ell^p$ norm for all $p$, but doesn't converge to an element of $\Bbb R^{\infty}$ for the $\ell^p$ norm. Indeed, this converge implies... |
2,882,985 | <p>Let $f_n(x)=\frac{1}{n}\boldsymbol 1_{[0,n]}(x)$. This sequence is bounded in $L^1(\mathbb R)$ since $\|f_n\|_{L^1}=1$. But why is there no subsequence that convergent weakly ? I know that if such subsequence exist (still denote $f_n$), then $\|f_n\|_{L^1}=1$ Let denote $f$ it's limit. Then, since $f_n\to 0$ pointwi... | Dzoooks | 403,583 | <p>Let $\{n_k\}_{k \geq 1}$ be any subsequence. Then we readily compute $$\lim_{k \to \infty} \int_{\mathbb{R}} 1 \cdot \left(f_{n_k}(x) - 0\right)dx = \lim_{k \to \infty} 1 = 1 \neq 0.$$</p>
|
1,196,424 | <p>So I'm reviewing my notes and I just realized that I can't think of how to show that a particular integer mod group is abelian. I know how to do it with symmetric but not with integers themselves.</p>
<p>For example, lets say I was asked to show $\mathbb{Z_5}$ is abelian.</p>
<p>I know for symmetric groups, lets s... | David Wheeler | 23,285 | <p>What you really want to show, for any cyclic group $G = \langle a\rangle$, is that:</p>
<p>$a^ka^m = a^ma^k$ for any integers $k,m$.</p>
<p>This takes care of $\Bbb Z$ <em>and</em> $\Bbb Z_n$ (for any $n$) "all at once".</p>
<p>Can you show how to leverage the "rules of exponents" and commutativity of addition in... |
3,207,767 | <p>What is the general solution of differential equation <span class="math-container">$y\frac{d^{2}y}{dx^2} - (\frac{dy}{dx})^2 = y^2 log(y)$</span>.</p>
<p>The answer to this DE is <span class="math-container">$log(y) = c_1 e^x + c_2 e^{-x}$</span></p>
<p>I don't know the method to solve differential equation with d... | Michael Rozenberg | 190,319 | <p>Another way for the proof of the inequality <span class="math-container">$(x^2+3)(y^2+3)(z^2+3)(t^2+3)\geq16(x+y+z+t)^2.$</span></p>
<p>We obtain: <span class="math-container">$$(x^2+3)(y^2+3)=x^2y^2+3(x^2+y^2)+9=$$</span>
<span class="math-container">$$=(xy-1)^2+(x-y)^2+2(x+y)^2+8\geq2((x+y)^2+4).$$</span>
By the s... |
1,134,145 | <p>A set S is bounded if every point in S lies inside some circle |z| = R other it is unbound. Without appealing to any limit laws, theorems, or tools from calculus, prove or disprove that the set {$\frac{z}{z^2 + 1}$; z in R} is bounded.</p>
<p>I imagine that it's simple, but I have no clue where to start due to the ... | user2566092 | 87,313 | <p>Hint: First assume $|z| < 1$ and prove that the set is contained in some bounded interval, say $[-1,1]$. Then assume $|z| \geq 1$ and prove the set is contained in the interval $[-1,1]$.</p>
|
1,600,307 | <p>Let $n$ be an integer greater than 1, $\alpha$ be a real number, and consider the quadratic form $Q_{\alpha}$ given by: </p>
<p>for every $(x_1, ... , x_n) \in R^n$, </p>
<p>$$Q_{\alpha}(x_1,...,x_n)= \sum_{i=1}^n x_i^2 - \alpha(\sum_{i=1}^n x_i)^2$$</p>
<p>Find all the eigenvalues of $Q_{\alpha}$ in terms of $\a... | K. Miller | 264,375 | <p>Here is a hint. Let $x = (x_1,\ldots,x_n)^T$ and let $e$ denote the $n$-vector of all ones. Then</p>
<p>$$
Q_\alpha(x) = x^Tx - \alpha (x^Te)(e^Tx) = x^T(I - \alpha ee^T)x
$$</p>
<p>So you need to find the eigenvalues of the matrix $A_\alpha = I - \alpha ee^T$.</p>
|
1,220,923 | <p>Find the value of the integral
$$\int_0^\infty \frac{x^{\frac25}}{1+x^2}dx.$$
I tried the substitution $x=t^5$ to obtain
$$\int_0^\infty \frac{5t^6}{1+t^{10}}dt.$$
Now we can factor the denominator to polynomials of degree two (because we can easily find all roots of polynomial occured in the denominator of the form... | Chappers | 221,811 | <p>Your integral is
$$ \int_0^{\infty} \frac{x^{7/5}}{1+x^2} \frac{dx}{x}. $$
Substituting $u=x^2$, $du/u=2dx/x$, the integral becomes
$$ \frac{1}{2}\int_0^{\infty} \frac{u^{7/10}}{1+u} \, du $$
Now,
$$ \frac{1}{1+u}= \int_0^{\infty} e^{-(1+u)\alpha} \, d\alpha, $$
and interchanging the order of integration gives
$$ \f... |
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... | Hagen von Eitzen | 39,174 | <p>Finding the limits is easy: If $a$ is the limit of such a sequence, then certainly $f(a)=a$. This leads to $a=a^2+\frac13a$, i.e., $a=0$ or $a=\frac23$.</p>
<p>If $x>\frac23$, then $f(x)=x^2+\frac13x>\frac23x+\frac13x=x$, i.e., $a_0>\frac23$ produces a strictly increasing sequence. This could only converge... |
42,787 | <p>I am using <code>ListPlot</code> to display from 5 to 12 lines of busy data. The individual time series in my data are not easy to distinguish visually, as may be evident below, because the colors are not sufficiently different.</p>
<p><img src="https://i.stack.imgur.com/PiMMh.png" alt="enter image description here... | Mike Honeychurch | 77 | <p>I just use <code>"ColorList"</code></p>
<pre><code>ListPlot[Table[i*Range[0, 10], {i, 1, 5, 0.5}], Frame -> True,
Joined -> True, PlotRange -> All,
PlotStyle -> ColorData[3, "ColorList"]]
</code></pre>
<p><img src="https://i.stack.imgur.com/NuzKE.png" alt="enter image description here"></p>
<p>You... |
3,213,464 | <p>Does 22.449 approximate to 22 or 23?
If we see it one way
<span class="math-container">$22.449≈22$</span>
But on the other hand
<span class="math-container">$22.449≈22.45≈22.5≈23$</span>
Which one is correct?</p>
| Wrzlprmft | 65,502 | <p>Your problem seems to be that you implicitly expected that rounding behaves like equality, “less than”, and similar.
Speaking in notation, you seem to assume that the <span class="math-container">$≈$</span> relation (meaning something like “the left-hand side rounded yields the right-hand side”) behaves like the <sp... |
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... | Martín Vacas Vignolo | 297,060 | <p>Total possible results: $6\times6=36$</p>
<p>Favorable results: $1-3,2-4,3-5,4-6$ and opposites, $8$.</p>
<p>Then the probability is $8/36=2/9$.</p>
|
2,426,263 | <p>Nowadays, the most widely-taught model of computation (at least in the English-speaking world) is that of Turing Machines, however, it wasn't the first Turing-Complete model out there: μ-recursive functions came a few years earlier, and λ-calculus came a year earlier. Why is it that Turing machines are so popular to... | Bram28 | 256,001 | <p>If you read Turing's original 1936 paper, you will see how he tried to get at the fundamental ingredients of a human following some systematic process of symbol manipulation. The word 'effective' is sometimes used in this context: a computation or algorithm is 'effective' when a human can perform it. E.g Long divisi... |
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>
| Pedro | 23,350 | <p>I think some things can be written in a clearer manner. First, I would change </p>
<blockquote>
<p>This implies that there is a sequence $\left\{x_{n}\right\}_{n\in\mathbb{N}}$ where $x_{n}\in A$.</p>
</blockquote>
<p>For</p>
<blockquote>
<p>This implies we can write $A=\{x_n:n\geqslant 1\}$ </p>
</blockquote... |
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>
| CopyPasteIt | 432,081 | <blockquote>
<p>Let <span class="math-container">$A$</span> be a countably infinite set and <span class="math-container">$E\subset A$</span> be an infinite subset. Then <span class="math-container">$E$</span> is countably infinite.</p>
</blockquote>
<p>The OP is using the 'can be bijectively enumerated' definition f... |
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... | user284331 | 284,331 | <p>\begin{align*}
\int_{0}^{2\pi}\sin(nx)\sin(mx)dx&=-\dfrac{1}{4}\int_{0}^{2\pi}(e^{inx}-e^{-inx})(e^{imx}-e^{-imx})dx\\
&=-\dfrac{1}{4}\int_{0}^{2\pi}(e^{i(m+n)x}-e^{-i(m+n)x}-e^{-i(n-m)x}-e^{-i(m-n)x})dx\\
&=-\dfrac{1}{4}\int_{0}^{2\pi}(-e^{-i(n-m)x}-e^{-i(m-n)x})dx\\
&=-\dfrac{1}{4}\delta_{m,n}(-2\p... |
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... | Martín-Blas Pérez Pinilla | 98,199 | <p>The trick for trigonometric integrals:
$$
z = e^{ix}\implies\sin(nx) =
\frac12(e^{inx} − e^{−inx}) = \frac12(z^n - z^{-n})
\qquad dz = iz\,dx
$$
$$
\int_{0}^{2\pi}\sin(nx)\sin(mx)\,dx =
\frac14\int_{|z|=1}(z^n - z^{-n})(z^m - z^{-m})\frac1{iz}\,dz =
$$
$$
\frac1{4i}\int_{|z|=1}(z^{m-n-1} + z^{n-m-1} - z^{m+n-1} - z... |
122,471 | <p>Can anyone explain how I can prove that either $\phi(t) = \left|\cos (t)\right|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.</p>
| Sasha | 11,069 | <p>Since $\phi(t) = | \cos(t) |$ is periodic with period $\pi$ and even and if it is valid, it should correspond to a symmetric discrete random variable. </p>
<p>It is not hard to establish that:
$$
| \cos(t) | = \frac{2}{\pi} + \frac{4}{\pi} \sum_{m=1}^\infty \frac{(-1)^{m-1}}{4 m^2-1} \cos(2 m t)
$$
<img src="htt... |
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... | Robert Z | 299,698 | <p>Let <span class="math-container">$A_n=\{x/n:x\in\mathbb{Z}\}$</span> with <span class="math-container">$n$</span> any integer greater than <span class="math-container">$1$</span> then it is easy to see that <span class="math-container">$\mathbb{Z}\subset A_n\subset \mathbb{Q}$</span>. We claim that
<span class="mat... |
1,395,619 | <p>One of my friend asked this doubt.Even in lower class we use both as synonyms,he says that these two concepts have difference.Empty set $\{ \}$ is a set which does not contain any elements,while null set ,$\emptyset$ says about a set which does not contain any elements.</p>
<p>I could not make out that...is his arg... | Christoph | 86,801 | <p>In measure theory, a <a href="https://en.wikipedia.org/wiki/Null_set" rel="nofollow noreferrer">null set</a> refers to a set of measure zero. For example, in the reals, <span class="math-container">$\mathbb R$</span> with its standard measure (Lebesgue measure), the set of rationals <span class="math-container">$\ma... |
320,452 | <p>For any positive integer <span class="math-container">$n\in\mathbb{N}$</span> let <span class="math-container">$S_n$</span> denote the set of all bijective maps <span class="math-container">$\pi:\{1,\ldots,n\}\to\{1,\ldots,n\}$</span>. For <span class="math-container">$n>1$</span> and <span class="math-container"... | Wojowu | 30,186 | <p><span class="math-container">$\newcommand{\e}{\varepsilon}$</span>
For any fixed <span class="math-container">$\e>0$</span>, permutations with <span class="math-container">$N_n(\pi)>\e n$</span> asymptotically have density zero. Indeed, consider values <span class="math-container">$\pi(1),\dots,\pi(k)$</span>.... |
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... | dustin | 78,317 | <p>Since this post was tagged with complex analysis, I can provide a contour integration solution as well. Again, I will exploit the identity given to you by @idm.
$$
\int_0^{\pi/2}\ln(\sin(\theta))d\theta = \frac{1}{2}\int_0^{\pi}\ln(\sin(\theta))d\theta
$$
Consider $1 - e^{2iz} = -2ie^{iz}\sin(z)$. We can write $1 - ... |
3,834,894 | <p>I understand a continuous function may not be differentiable. But does every continuous function have directional derivative at every point? Thanks!</p>
| Marius S.L. | 760,240 | <p>There is no need to distinguish between one and more directions. If a continuous function like <span class="math-container">$x\longmapsto |x|$</span> isn't differentiable at <span class="math-container">$x=0,$</span> then there is no directional derivative in <span class="math-container">$x$</span>-direction. <br />... |
2,153,743 | <p>I have the relation $R = \{(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\}$ on the set $\{1,2,3,4\}$. I have to find [1] and [4] but I don't really understand what that means. </p>
<p>I get that $[a]$ is the set of all elements of $A$ related (by $R$) to $a$ so $[a]=\{x\in A : x$ $R$ $a\}$ right? But I don't get the signific... | ConMan | 82,793 | <p>$[1]$ is the set of all things that are equivalent to $1$, given the relation. So, since $1R1$ and $1R2$ are the only relations given that involve $1$, $[1] = \{1, 2\}$ (and in particular note that $[1] = [2]$). You can do the same thing for $[4]$.</p>
|
706,980 | <p>If I know that $f(z)$ is differentiable at $z_0$, $z_0 = x_0 + iy_0$.
How do I prove that $g(z) = \overline{f(\overline{z})}$ is differentiable at $\overline z_0$?</p>
| Christian Blatter | 1,303 | <p>The function $f$ is defined in a neighborhood of $z_0$. Therefore the function $g$ is defined in a neighborhood of $\bar z_0$.</p>
<p>By assumption there is a number $a\in{\mathbb C}$ such that
$$f(z_0+\zeta)=f(z_0)+ a\zeta +o(\zeta)\qquad(\zeta\to0)\ .$$</p>
<p>Using the definition of $g$ we therefore have
$$g(\b... |
3,288,815 | <p>I'm reading theorem 3.11 in Rudin's RCA what says <span class="math-container">$L^p$</span> space is a complete metric space.
At the end of the proof, Rudin says that "Then <span class="math-container">$\mu(E)=0$</span>, and on the complement of <span class="math-container">$E$</span> the sequence <span class="math... | mechanodroid | 144,766 | <p>By definition of <span class="math-container">$\|\cdot\|_\infty$</span>, the sets
<span class="math-container">$$A_k=\{x \in X : |f_k(x)| > \|f_k\|_\infty\},\quad B_{m,n}=\{x \in X : |f_n(x)-f_m(x)| > \|f_n-f_m\|_\infty\}$$</span>
are <span class="math-container">$\mu$</span>-null sets so <span class="math-con... |
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>
| Mark Bennet | 2,906 | <p>The equation $y=3$ defines a line in $\mathbb R^2$, but a plane in $\mathbb R^3$. Likewise $z=5$ defines a plane in $\mathbb R^3$. Each equation can be seen as constraining a point in one dimension, leaving it free to be located in a two-dimensional space (in this case a plane).</p>
<p>The two equations $y=3, z=5$ ... |
1,182,953 | <p>Does anyone know the provenance of or the answer to
the following integral</p>
<p>$$\int_0^\infty\ \frac{\ln|\cos(x)|}{x^2} dx $$</p>
<p>Thanks.</p>
| Lucian | 93,448 | <p><strong>Hint:</strong> Let $I(n)=\displaystyle\int_0^\infty\frac{1-\cos^{2n}x}{x^2}~dx.~$ Prove first that $I(n)=n\pi~\dfrac{\displaystyle{2n\choose n}}{4^n}~,~$ then evaluate $I'(0)$.</p>
|
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... | Mark | 470,733 | <p>I guess you defined <span class="math-container">$f([a]_{24})=([a]_4, [a]_6)$</span>? This is clearly not a bijection. For example, <span class="math-container">$[0]_{24}$</span> and <span class="math-container">$[12]_{24}$</span> are mapped to the same element.</p>
<p>The rings are not isomorphic because the additi... |
311,677 | <p>The problem from the book. </p>
<blockquote>
<p>$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$ </p>
</blockquote>
<p>I understand the solution till this part. </p>
<p>$\ln \vert 6 - y \vert = x + C$ </p>
<p>The solution in the book is $6 - Ce^{-x}$ </p>
<p>My issue this that this solution, from the book, doesn't s... | David Mitra | 18,986 | <p>You should have, as your general solution,
$$
-\ln|6-y|=x+C\ \quad\iff\quad |6-y|=e^C e^{-x} .
$$</p>
<p>If $y-6>0$, you have the solution
$$y-6= e^Ce^{-x}\ \quad\iff\quad y=6+ e^Ce^{-x} .
$$</p>
<p>If $y-6<0$, you have the solution
$$6-y= e^Ce^{-x}\ \quad\iff\quad y=6- e^Ce^{-x} .
$$</p>
<p>In either... |
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... | Mary Star | 80,708 | <p><strong>EDIT:</strong></p>
<p>$x=t(x')^2+2tx'$</p>
<p>$p=x'$</p>
<p>$x=tp^2+2tp$</p>
<p>We differentiate in respect to $t$:</p>
<p>$p=p^2+t2pp'+2p+2tp' \Rightarrow p'(2tp+2t)=(p-p^2-2p) \Rightarrow p'(p+1)2t=-(p^2+p) \Rightarrow p'(p+1)2t=-p(p+1) \Rightarrow p'(p+1)2t+p(p+1)=0 \Rightarrow (p+1)(2tp'+p)=0 \\ \Ri... |
3,819,202 | <p>Can anyone explain to solve the identity posted by my friend <span class="math-container">$$2\cos12°= \sqrt{2+{\sqrt{2+\sqrt{2-\sqrt{2-...}}} }}$$</span> which is an infinite nested square roots of 2. <strong>(Pattern <span class="math-container">$++--$</span> repeating infinitely)</strong></p>
<p>Converging to fini... | saulspatz | 235,128 | <p>If the value of the radical is <span class="math-container">$x$</span>, then we have <span class="math-container">$$x=\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2-x}}}}\tag1$$</span> Repeated squaring gives
<span class="math-container">$$\left(\left(\left(x^2-2\right)^2-2\right)^2-2\right)^2=2-x\tag2$$</span></p>
<p>Now, <span ... |
1,436,867 | <p>I don´t know an example wich $ \rho (Ax,Ay)< \rho (x,y) $ $ \forall x\neq y $ is not sufficient for the existence of a fixed point .
can anybody help me? please</p>
| Community | -1 | <p>I was about to answer this question <a href="https://math.stackexchange.com/questions/1444036/dtx-ty-dx-y-does-not-guarantee-existence-of-fixed-point">here</a>, but it got marked as a duplicate of this one. Thus, I will answer it on this thread.</p>
<blockquote>
<p>Let $T$ be a mapping of a complete metric space ... |
2,038,189 | <p>(Note: I didn't learn how to solve equations the conventional way; instead I was just taught to "move numbers from side to side", inverting the sign or the operation accordingly. I am learning the conventional way though because I think it makes the process of solving equations clearer. That being said, I apologize ... | qwr | 122,489 | <p>Multiply by $x$: $$5x=2$$</p>
<p>Divide by $5$: $$\frac{2}{5} = x$$</p>
<p>More generally, $$\frac{x}{b} = a \iff x=ab \iff \frac{x}{a} = b$$</p>
|
2,038,189 | <p>(Note: I didn't learn how to solve equations the conventional way; instead I was just taught to "move numbers from side to side", inverting the sign or the operation accordingly. I am learning the conventional way though because I think it makes the process of solving equations clearer. That being said, I apologize ... | Frank | 332,250 | <p>From $5=\frac 2x$ to get $x=\frac 25$, you're multiplying both sides by $x$ to get $$5x=2$$
And dividing by $5$, we get $x=\frac 25$.</p>
|
2,334,215 | <p><a href="https://math.stackexchange.com/questions/20223/getting-an-x-for-chinese-remainder-theorem-crt">Getting an X for Chinese Remainder Theorem (CRT)</a></p>
<p>In the "Easy CRT" part of the answer to this problem, the author demonstrates that (-3/77) mod 65 is equal to 16. I don't understand - how is this accur... | Bernard | 202,857 | <p>In the ring of integers mod. $n$, there is no ‘real’ fraction. Some elements have reciprocals, others don't, depending whether they're coprime with the modulus or not.</p>
<p>For instance, modulo $14$, $5$ is a unit, since $3\cdot 5\equiv 1\pmod{14}$, and $5^{-1}=3$, so one can be tempted to write, say, $\dfrac 4... |
2,334,215 | <p><a href="https://math.stackexchange.com/questions/20223/getting-an-x-for-chinese-remainder-theorem-crt">Getting an X for Chinese Remainder Theorem (CRT)</a></p>
<p>In the "Easy CRT" part of the answer to this problem, the author demonstrates that (-3/77) mod 65 is equal to 16. I don't understand - how is this accur... | Trevor Gunn | 437,127 | <p>The definition of $\frac{1}{x}$ is that $\frac{1}{x}$ is the quantity such that $x \cdot \frac1x = 1$ (which may or may not exist). Therefore
$$ \frac{-3}{77} \equiv 16 \pmod {65} \text{ if and only if } -3 \equiv 77 \cdot 16 \pmod {65} $$
This happens if and only if
$$77 \cdot 16 + 3\equiv 0 \pmod {65}$$
which by d... |
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... | Shubham Johri | 551,962 | <p><span class="math-container">$P(k):k$</span> is irrational</p>
<p><span class="math-container">$P(k): \{k\}>0$</span>, where <span class="math-container">$\{k\}$</span> denotes the fractional part of <span class="math-container">$k$</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... | fleablood | 280,126 | <p>It's easier than you think:</p>
<p><span class="math-container">$n \le n-2$</span></p>
<p>Then <span class="math-container">$n+1 \le (n-2) + 1 = n-1 = (n+1) -2$</span>.</p>
<p><span class="math-container">$n$</span> is not an integer.</p>
<p><span class="math-container">$n + 1$</span> is not an integer.</p>
<p>... |
3,630,421 | <p>If <span class="math-container">$x+y = 5$</span>, <span class="math-container">$xy = 1$</span> and <span class="math-container">$x > y$</span>, then <span class="math-container">$\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}= ?$</span> The answer key gives for the asnwer <span class="math-container">$\frac{\sqrt{... | Soyeb Jim | 774,258 | <p>If you want to find the volume using integration you can integrate in <span class="math-container">$z$</span> axis by summing the trapezoidal cross sections. [Assuming the base of the pyramid is a square]</p>
<p>Suppose highest point of the pyramid is <span class="math-container">$(0,0,0)$</span>. Assume the pyrami... |
22,753 | <p>I've learned the process of orthogonal diagonalisation in an algebra course I'm taking...but I just realised I have no idea what the point of it is.</p>
<p>The definition is basically this: "A matrix <span class="math-container">$A$</span> is orthogonally diagonalisable if there exists a matrix <span class="math-co... | Yuval Filmus | 1,277 | <p><b>If a matrix $A$ is unitarily diagonalizable, then one can define a "Fourier transform" for which $A$ is a "convolution" matrix.</b></p>
<p>Here is an example. We have a family $F$ of subsets of some finite set $S$, i.e. $F \subset 2^S$, such that any two sets in $F$ agree in some coordinate, i.e. for any two $A,... |
1,452,943 | <p>I'm working on problem where I want to use the continuity of $f'$ to assert that $f'(x)$ cannot be zero ("bounded away from zero"?) near $x = 0$. We know that $(f'(0))^2 >3$.</p>
<p>So, I think that what I really want to ask is this: if $f'$ is cts, must $f'$-squared also be continuous? </p>
<p>Can I use t... | Cameron Buie | 28,900 | <p>Well, $x\mapsto f'(x)$ is continuous, and $y\mapsto y^2$ is also continuous, so what can you say about $x\mapsto\left( f'(x)\right)^2$?</p>
|
1,452,943 | <p>I'm working on problem where I want to use the continuity of $f'$ to assert that $f'(x)$ cannot be zero ("bounded away from zero"?) near $x = 0$. We know that $(f'(0))^2 >3$.</p>
<p>So, I think that what I really want to ask is this: if $f'$ is cts, must $f'$-squared also be continuous? </p>
<p>Can I use t... | Hasan Saad | 62,977 | <p>$f'(0)^2>3\implies |f'(0)|>\sqrt{3}$</p>
<p>However, the function is continuous at $x=0$, so for some $\delta$, we have $|f'(x)-f'(0)|<\sqrt{3}$ whenever $|x|<\delta$.</p>
<p>However, this asserts that, $|f'(0)-f'(x)|<\sqrt{3}\implies |f'(0)|-|f'(x)|<\sqrt{3}$ by using the triangle inequality.</p... |
19,495 | <p>I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called "Feynman path integral", which, as far as I understand, means "integrating" a functional (action... | user1504 | 35,508 | <p>It's not accurate to say that no theory of integration on infinite-dimensional spaces exists. The Euclidean-signature Feynman measure has been constructed -- as a measure on a space of distributions -- in a number of non-trivial cases, mainly by the Constructive QFT school in the 70s. </p>
<p>The mathematical co... |
115,483 | <p>Edited:</p>
<p>I guess </p>
<p>$$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$</p>
<p>We know that if $\operatorname{Supp} H^i_I(M)\subseteq V(I)\cap \operatorname{Supp}(M)$, then
$$\operatorname{Supp} H^2_{(x,y)}\frac{\Bbb Z[x,y]}{(5x+4y)})\subseteq V((x,y))\cap V((5x+4y))=V((x,y))=\lbrace(x,y) \rbr... | user26857 | 23,950 | <p>$P=(x,y,p)$ implies $P\cap\mathbb Z=p\mathbb Z$ (here $p$ is a prime or $0$). As localization commutes with local cohomology
$$H^2_{(x,y)}\left(\frac{\mathbb{Z}[x,y]}{(5x+4y)}\right)_P\simeq H^2_{(x,y)}\left(\frac{\mathbb{Z}[x,y]_P}{(5x+4y)}\right).$$
But $\mathbb Z[x,y]_P\simeq\mathbb Z_{(p)}[x,y]_{\overline{P}}$... |
1,874,634 | <blockquote>
<p>Corollary (of Schur's Lemma): Every irreducible complex representation of a finite abelian group G is one-dimensional.</p>
</blockquote>
<p>My question is now, why has the group to be abelian? As far as I know, we want the representation $\rho(g)$ to be a $Hom_G(V,V)$, where $V$ is the representation... | Maik Pickl | 317,129 | <p>The trick here is, that for a abelian group every element is a intertwining operator. This means let $h \in G$, then $\rho(h)\rho(g)\rho(h^{-1})=\rho(g)$ for all $g$ and therefore by Schur's lemma $\rho(h)=\lambda id$. Since your representation was assumed to be irreducible it follows that it is one dimensional. Not... |
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>
| Alex Wertheim | 73,817 | <p>Since $[G:H]$, $H$ has two distinct right cosets of $H$ in $G$: one is $H$, and the other can be written as $Hk$ for any $k \notin H, k \in K$. The right cosets of $H$ in $G$ partition $G$, so what can you say about $HK$, which obviously contains $H \cup 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... | CA Trevillian | 63,039 | <p>You can use <a href="https://reference.wolfram.com/language/ref/Save.html" rel="noreferrer"><code>Save</code></a> in this way:</p>
<pre><code>func[i_] := func[i] = i;
Do[func[i], {i, 1, 3}];
FullDefinition@func
(*
func[1] = 1
func[2] = 2
func[3] = 3
func[i_] := func[i] = i
*)
Save["out.m",func];
ClearAll[func];... |
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... | Chris Culter | 87,023 | <p>Starting from a prime $p_n$, the first candidate for a twin prime that the pattern generates is $3p_n-4$. If this formula does indeed produce prime numbers, then by iterating it, we have a simple way to generate arbitrarily large primes. That would be a big deal all by itself, as no such generator is known today. So... |
2,821,323 | <blockquote>
<p>How to show that a rational polynomial is irreducible in $\mathbb{Q}[a,b,c]$? For example, I try to show this polynomial $$p(a,b,c)=a(a+c)(a+b)+b(b+c)(b+a)+c(c+a)(c+b)-4(a+b)(a+c)(b+c)(*)$$ is irreducible, where $a,b,c\in \mathbb{Q}$.</p>
</blockquote>
<p>The related problem is <a href="https://math.... | Will Jagy | 10,400 | <p>To ask about complete (three linear factor) reducibility over the complexes, we take the Hessian matrix of second partials. The entries are linear in the named variables. Next, let $\Delta$ be the determinant of the Hessian. This $\Delta$ is once again a cubic form. The original cubic (homogeneous) ternary form fact... |
1,942,364 | <p>How many squares exist in an $n \times n$ grid? There are obviously $n^2$ small squares, and $4$ squares of size $(n-1) \times (n-1)$.</p>
<p>How can I go about counting the number of squares of each size?</p>
| marty cohen | 13,079 | <p>Another way of counting:</p>
<p>For each
$(i, j, k)$
with
$1 \le i, j \le n$
and
$0 \le k \le \min(n-i, n-j)$
there is a square
with upper left corner
at $(i, j)$
and lower right corner at
$(i+k, j+k)$.</p>
<p>Therefore the total is</p>
<p>$\begin{array}\\
s(n)
&=\sum_{i=1}^n \sum_{j=1}^n (1+\min(n-i, n-j))\\... |
2,626,506 | <p><strong>Proof: There is no other prime triple then $3,5,7$</strong></p>
<p>There are already lots of questions about this proof, but I can't find the answer to my question.</p>
<p>The complete the proof, we consider mod $3$ so $p=3k; p=3k+1; p=3k+2$ </p>
<p>But why do we look at divisibility by $3$?</p>
<p>Do we... | robjohn | 13,854 | <p>$$
\begin{align}
p(p+2)(p+4)
&=p^3+6p^2+8p\\
&=3\left[5\binom{p}{1}+6\binom{p}{2}+2\binom{p}{3}\right]
\end{align}
$$
is divisible by $3$, so unless one of the factors <em>is</em> $3$, one of the factors is not prime.</p>
|
2,626,506 | <p><strong>Proof: There is no other prime triple then $3,5,7$</strong></p>
<p>There are already lots of questions about this proof, but I can't find the answer to my question.</p>
<p>The complete the proof, we consider mod $3$ so $p=3k; p=3k+1; p=3k+2$ </p>
<p>But why do we look at divisibility by $3$?</p>
<p>Do we... | Adam Bailey | 22,062 | <p>Suppose $p,p+2,p+4$ are all prime. </p>
<p>Consider $p+2,p+3,p+4$. Since these are three consecutive integers, one of them must be divisible by $3$ (that's why we consider divisibility by $3$ and not by some other number). It can't be $p+2$ or $p+4$ because they are primes and not equal to $3$ (otherwise $p$ wou... |
210,655 | <p>The lower density of $A\subseteq\mathbb{N}$ is defined to be $\lambda(A)=\lim\text{inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$. We set $${\cal C} = \{A\subseteq \mathbb{N}: \lambda(\mathbb{N}\setminus A) = 1 - \lambda(A)\}.$$</p>
<p>Do both ${\cal C}$ and ${\cal P}(\mathbb{N})\setminus {\cal C}$ have cardinal... | Salvo Tringali | 16,537 | <p>Let $\alpha \in {]0,1]}$. On the one hand, the set $\{\lfloor \alpha^{-1}n \rfloor: n \in \mathbf N\}$ has (natural) density equal to $\alpha$, so it belongs to $\mathcal C$. On the other hand, the set $\bigcup_{n \ge 1} [\![(2n-1)!\,\alpha + (2n)!\,(1-\alpha) + 1, (2n)!+1]\!]$ has lower (natural) density equal to $... |
3,624,524 | <p>I want to figure out the process for showing why the function <span class="math-container">$\cos(1-\frac{1}{z})$</span> has an essential singularity at <span class="math-container">$z=0$</span> without using knowledge of the Laurent expansion. I know the process should be to rule out the possibility of removable sin... | Mnifldz | 210,719 | <p>Yes, it is true for all tangent bundles on all manifolds. This stems from the fact that tangent bundles are specific examples of vector bundles. Vector bundles are defined by having locally trivial neighborhoods.</p>
|
3,055,208 | <p>I am trying to compute the below limit through Taylor series:
<span class="math-container">$\lim \limits_{x\to \infty} ((2x^3-2x^2+x)e^{1/x}-\sqrt{x^6+3})$</span></p>
<p>What I have already tried is first of all change the variable x to
<span class="math-container">$x=1/t$</span> and the limit to t limits to 0, so ... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>The square root becomes</p>
<p><span class="math-container">$$\frac{\sqrt{1+3t^6}}{|t^3|}=$$</span></p>
<p><span class="math-container">$$\frac{1}{|t^3|}\Bigl(1+\frac{3t^6}{2}-\frac 98t^{12}+t^{12}\epsilon(t)\Bigr)$$</span></p>
|
3,055,208 | <p>I am trying to compute the below limit through Taylor series:
<span class="math-container">$\lim \limits_{x\to \infty} ((2x^3-2x^2+x)e^{1/x}-\sqrt{x^6+3})$</span></p>
<p>What I have already tried is first of all change the variable x to
<span class="math-container">$x=1/t$</span> and the limit to t limits to 0, so ... | Doug M | 317,162 | <p><span class="math-container">$y = \frac 1x$</span></p>
<p>then we have</p>
<p><span class="math-container">$\lim_\limits{y\to 0^+} \frac {(2 -2y+ y^2)e^y - \sqrt {1+3y^6}}{y^3}$</span></p>
<p>Now if you want to do a Taylor expansion...</p>
<p><span class="math-container">$\lim_\limits{y\to 0^+} \frac {(2 -2y+ y^... |
3,413,261 | <p>I know this was answered before but I'm having one particular problem on the proof that I'm not getting.</p>
<p>My Understanding of the distribution law on the absorption law is making me nuts, by the answers of the proof it should be like this.</p>
<p>A∨(A∧B)=(A∧T)∨(A∧B)=A∧(T∨B)=A∧T=A</p>
<p>This should prove th... | Mirko | 188,367 | <p><a href="https://en.wikipedia.org/wiki/Lower_limit_topology" rel="noreferrer">The Sorgenfrey line</a>, also called the lower-limit topology on the real line. It has a basis of intervals <span class="math-container">$[a,b)$</span> (or some authors prefer <span class="math-container">$(a,b]$</span>, upper-limit topolo... |
2,467,327 | <p>How to prove that $441 \mid a^2 + b^2$ if it is known that $21 \mid a^2 + b^2$.<br>
I've tried to present $441$ as $21 \cdot 21$, but it is not sufficient.</p>
| Michael Rozenberg | 190,319 | <p>If $a^2+b^2$ is divisible by $3$ then $a$ and $b$ are divisible by $3$ because
$$x^2\equiv0,1\pmod3.$$
Let $A=\{0,1\}$.</p>
<p>Thus, $3\not\in A+A$ and $0\in A+A$ for $0=0+0$ only.</p>
<p>Similarly, </p>
<p>if $a^2+b^2$ is divisible by $7$ then $a$ and $b$ are divisible by $7$ because
$$x^2\equiv0,1,2,4\pmod7.$$<... |
2,132,936 | <p>How do you simplify this problem?
$$ \frac {\mathrm d}{\mathrm dx}\left[(3x+1)^3\sqrt{x}\right] $$
$$= \frac {(3x+1)^3}{2\sqrt {x}} + 9\sqrt{x} (3x+1)^2 $$
$$\frac{(3x+1)^2(21x+1)}{2\sqrt x} $$</p>
| spaceisdarkgreen | 397,125 | <p>Multiplying top and bottom of the term $9\sqrt x(3x+1)^2$ by $2\sqrt x$ gives $$ \frac{18x(3x+1)^2}{2\sqrt x}.$$</p>
<p>Now can combine with the first term to get $$\frac{(3x+1)^3 + 18x(3x+1)^2}{2\sqrt x}.$$</p>
<p>Then we can factor a $(3x+1)^2$ out of the numerator, giving $$\frac{(3x+1)^2((3x+1)+18x)}{2\sqrt x}... |
3,960,404 | <p>Let <span class="math-container">$T$</span> be a tree.</p>
<p>Suppose that <span class="math-container">$T$</span> doesn’t have a perfect matching and let <span class="math-container">$M$</span> be a matching of <span class="math-container">$T$</span>, <span class="math-container">$|M| = k$</span>. Prove
that there ... | Misha Lavrov | 383,078 | <p>First, here are some general ideas. In any graph, not just a tree, given a matching <span class="math-container">$M$</span> that is not perfect, we can try to find an augmenting path to improve the matching.</p>
<p>This is a path <span class="math-container">$v_0 v_1 v_2 \dots v_{2k+1}$</span> with the property that... |
3,043,780 | <p><a href="https://i.stack.imgur.com/h1M7D.png" rel="nofollow noreferrer">the image shows right-angled triangles in semi-circle</a></p>
<p>In Definite Integration, we know that area can be found by adding up the total area of each small divided parts.</p>
<p>So, base on the Definite Integration, we may say the area ... | Oldboy | 401,277 | <p>Your method is totally inefficient and involves <span class="math-container">$\arctan$</span> function which is expensive from the computation point of view. </p>
<p>It's better to check angle orientations:</p>
<p><a href="https://i.stack.imgur.com/XNznF.jpg" rel="nofollow noreferrer"><img src="https://i.stack.img... |
1,115,222 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is a continuous, strictly increasing function defined on a closed interval <span class="math-container">$[a,b]$</span> such that <span class="math-container">$f^{-1}$</span> is the inverse function of <span class="math-container">$f$</span>. Prove that,
... | kobe | 190,421 | <p>Let $C$ be the graph of $y = f(x)$ over the interval $[a,b]$. Then $\int_a^b f(x)\, dx$ is the line integral $\int_C y\, dx$, and $\int_{f(a)}^{f(b)} f^{-1}(y)\, dy$ is the line integral $\int_C x\, dy$. Thus $$\int_a^b f(x)\, dx + \int_{f(a)}^{f(b)} f^{-1}(y)\, dy = \int_C x\, dy + y\, dx = \int_C d(xy) = bf(b) -... |
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... | C. E. | 731 | <pre><code>vertexShape[n_] := Graphics[{
EdgeForm[Black],
FaceForm[Lighter@Gray],
Disk[{0, 0}],
White,
Disk[0.5 #, 0.2] & /@ CirclePoints[n]
}]
shapes = Thread[Range[0, 11] -> Table[
vertexShape@RandomInteger[4],
12]];
SeedRandom[110]
g = TreeGraph[
RandomInteger[#] \[UndirectedEd... |
3,979,674 | <p>Let
<span class="math-container">$$I=\int\frac{dx}{\sqrt{ax^2+bx+c}}$$</span>
I know this can be either
<span class="math-container">$$\displaystyle I=\frac{1}{\sqrt{a}}\ln\left({2\sqrt{a}\sqrt{ax^2+bx+c}+2ax+b}\right)+C$$</span>
<span class="math-container">$$\displaystyle I=-\frac{1}{\sqrt{-a}}\arcsin{\left(\frac{... | GEdgar | 442 | <p>Suppose <span class="math-container">$a>0$</span>. Complete the square. You get one of these cases:
<span class="math-container">$$
\frac{1}{\sqrt{a}}\int\frac{dx}{\sqrt{(x-\beta)^2}},\qquad \beta\in \mathbb R,\\
\frac{1}{\sqrt{a}}\int\frac{dx}{\sqrt{(x-\beta)^2+\gamma^2}},\qquad \beta\in \mathbb R, \gamma >... |
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... | KCd | 619 | <p>To rationalize denominators that are not quadratic is a nice application of linear algebra over fields. For example, if asked to rewrite
$$
\frac{1}{1-5\sqrt[3]{2}}
$$
with a rational denominator, the best way to approach the problem is to work in the field ${\mathbf Q}(\sqrt[3]{2})$ with ${\mathbf Q}$-basis $1,\sq... |
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... | kjetil b halvorsen | 32,967 | <p>Group theory is useful in statistics, in experimental design: Look at the book by Box, Hunter & Hunter: "Statistics for Experimenters" (maybe the best book you will ever find about statistics, so you should have a look!) The first chapters is about $2^p$ designs and $2^{n-p}$ fractional designs. Everything they ... |
453,295 | <p>I wanna show that the non-zero elements of $\mathbb Z_p$ ($p$ prime) form a group of order $p-1$ under multiplication, i.e., the elements of this group are $\{\overline1,\ldots,\overline{p-1}\}$. I'm trying to prove that every element is invertible in the following manner:</p>
<blockquote>
<p><strong>Proof (a)</s... | davidlowryduda | 9,754 | <p>With respect to (a) first:</p>
<p>Why isn't $\bar0$ defined? It seems you're working in $\mathbb{Z}/p\mathbb{Z}$ and not only the group of units; thus $\bar{0}$ is defined just fine. Secondly, it seems you are deliberately forgetting that $\mathbb{Z}/p\mathbb{Z}$ is a ring and not just a group, i.e. that you have b... |
1,511,078 | <p><strong>Show that the product of two upper (lower) triangular matrices is again upper (lower) triangular.</strong></p>
<p>I have problems in formulating proofs - although I am not 100% sure if this text requires one, as it uses the verb "show" instead of "prove". However, I have found on the internet the proof belo... | Bernard | 202,857 | <p>I would say this:</p>
<p>In $\displaystyle\sum_{k=1}^nu_{ik}v_{kj}\enspace(i>j)$, then</p>
<ul>
<li>either $i>k$ and $u_{ik}=0$, hence $u_{ik}v_{kj}=0$,</li>
<li>or $k\ge i>j$ and $v_{kj}=0$, hence $u_{ik}v_{kj}=0$.</li>
</ul>
<p>In plain English, it says that in each term of the sum at least one of $u_... |
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>
| lab bhattacharjee | 33,337 | <p>Setting $x=\dfrac1y,$</p>
<p>$$I=\int_0^\infty\frac1{1+x^6}dx=\cdots=\int_0^\infty\frac{x^4}{1+x^6}dx$$</p>
<p>$$2I=\int_0^\infty\frac{1+x^4}{1+x^6}dx=\int_0^\infty\frac{(1+x^2)^2-2x^2}{1+x^6}dx$$</p>
<p>$$=\int_0^\infty\frac{1+x^2}{1-x^2+x^4}dx-\frac23\int_0^\infty\frac{3x^2}{1+x^6}dx$$</p>
<p>$$I_1=\int_0^\inf... |
3,935,811 | <p>While solving a bigger problem, I stumbeled upon a system of parametric equations
<span class="math-container">$$
\left\{
\begin{array}{ll}
\dfrac{x-a}{\sqrt{\left(x-a\right)^2+\left(y-b\right)^2}} + \dfrac{x-c}{\sqrt{\left(x-c\right)^2+\left(y-d\right)^2}} = 0\\
\dfrac{y-b}{\sqrt{\left(x-a\right)^2+\left(y-b\right)... | Quanto | 686,284 | <p>Note that <span class="math-container">$d_1=\sqrt{\left(x-a\right)^2+\left(y-b\right)^2}$</span> and <span class="math-container">$d_2=\sqrt{\left(x-c\right)^2+\left(y-d\right)^2}$</span> are the distances from
the point <span class="math-container">$(x,y)$</span> to the points <span class="math-container">$(a,b)$</... |
762,651 | <p>I have to prove that "any straight line $\alpha$ contained on a surface $S$ is an asymptotic curve and geodesic (modulo parametrization) of that surface $S$". Can I have hints at tackling this problem? It seems so general that I am not sure even how to formulate it well, let alone prove it. Intuitively, I imagine ... | Karl | 437,524 | <p>"Likewise when proving: If $0\leq a<b$, and $a^2<b^2$, then $a<b$. Why isn't taking the square root of both sides done?"</p>
<p>There is an intermediate step to this.</p>
<p>$$a^2<b^2$$</p>
<p>$$\sqrt{a^2}<\sqrt{b^2}$$
... notice that the square is inside the square root. Recall that this is equal... |
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... | lonza leggiera | 632,373 | <p>Just use the laws of operator precedence to evaluate the result of each binary or unary operation in the proper order:
<span class="math-container">\begin{align}
A\setminus A&=\emptyset\\
\therefore \neg(A\setminus A)&=\Omega\ \ \ \text{(universal set)}\\
\therefore \neg(A\setminus A)\setminus A&=\Omega\... |
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