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 |
|---|---|---|---|---|
425,400 | <p>Consider the following integral expression:
<span class="math-container">$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$</span>
for <span class="math-container">$\epsilon>0$</span>, <span class="math-container">$f \in L^\infty(\mathbb R)$</span>, and <sp... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand{\ep}{\epsilon}\newcommand{\R}{\mathbb R}$</span>Added later by the OP:</p>
<blockquote>
<p>does the dependence on <span class="math-container">$\ep$</span> in the answer below improve if we further assume <span class="math-container">$f$</span> to be compactly supported?</p>... |
3,683,542 | <p>My question is whether following conjecture is true:</p>
<p>If <span class="math-container">$f: \>{\mathbb R}_{\geq0}\to{\mathbb R}$</span> is continuous, and <span class="math-container">$\lim_{x\to\infty}\bigl|f(x)\bigr|=\infty$</span>, then
<span class="math-container">$$\lim_{x\to\infty}{1\over x}\int_0^x\bi... | Barry Cipra | 86,747 | <p>The conjecture is not true. Let <span class="math-container">$f$</span> be a continuous approximation to the step function <span class="math-container">$\pi\lfloor x\rfloor$</span>, where the portions of the curve that connect consecutive horizontal steps get steeper and steeper as <span class="math-container">$x\to... |
3,683,542 | <p>My question is whether following conjecture is true:</p>
<p>If <span class="math-container">$f: \>{\mathbb R}_{\geq0}\to{\mathbb R}$</span> is continuous, and <span class="math-container">$\lim_{x\to\infty}\bigl|f(x)\bigr|=\infty$</span>, then
<span class="math-container">$$\lim_{x\to\infty}{1\over x}\int_0^x\bi... | Severin Schraven | 331,816 | <p>No, this conjecture is not true. Consider a continuous function such that <span class="math-container">$f(x) = 2\pi n$</span> for <span class="math-container">$x\in [n+1/n^2, n+1]$</span>. Then we get for <span class="math-container">$n\leq x \leq n+1$</span>
<span class="math-container">$$ \frac{1}{x} \int_0^x \ver... |
2,161,911 | <p>Find the limit :</p>
<p>$$\lim_{ n\to \infty }\sqrt[n]{\prod_{i=1}^n \frac{1}{\cos\frac{1}{i}}}=\,\,?$$</p>
<p>My try :</p>
<p>$$\lim_{ n\to \infty }\sqrt[n]{a} =1\,\, \text {for} \,\,a>0$$</p>
<p>and;</p>
<p>$$\prod_{i=1}^n \frac{1}{\cos\frac{1}{i}}>0$$</p>
<p>so :</p>
<p>$$\lim_{ n\to \infty }\sqrt[n... | Mark Viola | 218,419 | <p>Let $\epsilon>0$ be given. Take $N$ so large that $|\log(\cos(1/i))|<\epsilon/2$ whenever $i>N$. Then, we can write for $n>N+1$</p>
<p>$$\begin{align}
\left|\log\left(\sqrt[n]{\prod_{i=1}^n\sec(1/i)}\right)\right|&=\frac1n\left|\sum_{i=1}^n\log(\cos(1/i))\right|\\\\
&\le\frac1n\sum_{i=1}^N|\lo... |
238,052 | <p>If I have 4 different types of data such (that I get from an Excel file) as:</p>
<pre><code>https://pastebin.com/j3Bgfxqm
</code></pre>
<p>I am trying to implement a <code>Do</code> loop that extracts the data from the Excel file, superimposes the data in two different regions (as done here: <a href="https://mathema... | Daniel Huber | 46,318 | <p>In MMA using Loops is not advised. Instead, use Map and a function. Here is the procedure.</p>
<p>Assume that we want to overlay dataset2, dataset3,.. onto dataset1</p>
<p>We again need data:</p>
<pre><code>dat = Import["https://pastebin.com/j3Bgfxqm", "Data"][[1]];
dat = ToExpression[
Strin... |
1,666,295 | <p>I was wondering when we add partial pivoting to an $LU$ factorization to a matrix $A$ it supposedly changes the data structure but improves the overall algorithm since we get better numerical stability. I am curious to why this is? </p>
<p>Any feedback is appreciated, my apologies for not formally introducing the m... | SplitInfinity | 316,671 | <p>The algorithm for $LU$ factorization of a matrix $A$ (at least, the only one that I'm aware of) involves dividing several elements in $A$ by $a_{11}$. Without pivoting, this element might be equal to 0, and we obviously can't have that. In addition, this element might be very small, and dividing by very small number... |
4,314,443 | <p>I'm trying to proof the following statement coming from a book:</p>
<p>"Pushforward of the vector field <span class="math-container">$\dfrac{d}{dx}$</span> by exponential, i.e <span class="math-container">$exp_{*}\dfrac{d}{dx}$</span> = <span class="math-container">$x\dfrac{d}{dx}$</span> on <span class="math-c... | Masacroso | 173,262 | <p>The pushforward of a vector field is well-defined just by a diffeomorphism (or in a Lie group), that is, if <span class="math-container">$\varphi :M\to N$</span> is such diffeomorphism then <span class="math-container">$\varphi _*:\mathfrak{X}(M)\to \mathfrak{X}(N)$</span>, where <span class="math-container">$\mathf... |
4,314,443 | <p>I'm trying to proof the following statement coming from a book:</p>
<p>"Pushforward of the vector field <span class="math-container">$\dfrac{d}{dx}$</span> by exponential, i.e <span class="math-container">$exp_{*}\dfrac{d}{dx}$</span> = <span class="math-container">$x\dfrac{d}{dx}$</span> on <span class="math-c... | rych | 73,934 | <p><span class="math-container">$\def\R{\mathbb{R}}%$</span>@Masacroso's is the answer, but I think we could also use the Fréchet derivative of the map between two copies of <span class="math-container">$\R$</span>, <span class="math-container">$Dexp: \R_1\to\R_2$</span>, <span class="math-container">$Dexp_x=e^x=y$</sp... |
69,655 | <p>I'm facing a strange behavior of <code>HoldForm</code>.</p>
<p>I need to display <code>1/2*3/4</code> in LaTeX like this :
$$ \frac{1}{2} \times \frac{3}{4} $$</p>
<p>So I use Mathematica : <code>1/2* 3/4 // HoldForm // TeXForm</code>
BUT I get
$$ \frac{3}{2\ 4} $$</p>
<p>First the writing <code>2 space 4</code>... | bbgodfrey | 1,063 | <p>Use <code>HoldForm</code> applied to each fraction to keep the fractions from combining.</p>
<pre><code>HoldForm[1/2] HoldForm[3/4]
</code></pre>
<p>to produce $$ \frac{1}{2} \frac{3}{4} $$</p>
<p>or </p>
<pre><code>HoldForm[(1/2) (3/4)]
</code></pre>
<p>to produce $$ \frac{3}{2 \times 4} $$</p>
<p>Using <code... |
69,655 | <p>I'm facing a strange behavior of <code>HoldForm</code>.</p>
<p>I need to display <code>1/2*3/4</code> in LaTeX like this :
$$ \frac{1}{2} \times \frac{3}{4} $$</p>
<p>So I use Mathematica : <code>1/2* 3/4 // HoldForm // TeXForm</code>
BUT I get
$$ \frac{3}{2\ 4} $$</p>
<p>First the writing <code>2 space 4</code>... | Mr.Wizard | 121 | <p>The behavior you observe is due to the formatting rules associated with <code>Times</code>. Please start by reading my answer here: <a href="https://mathematica.stackexchange.com/q/7880/121">Returning an unevaluated expression with values substituted in</a>. We can apply a similar technique here though the result ... |
2,771,823 | <p>The question is if the modulus of a multiplication, i.e. $a*a*a$ modulus $n$, is the same when we take the modules at each step of the multiplication.
So if </p>
<p>$(((a \text{ mod } n)* a \text{ mod } n) * a \text{ mod } n) = a*a*a \text{ mod } n$?</p>
| AbstractNonsense | 429,931 | <p>If $a\equiv b(\mod n)$ and $a'\equiv b'(\mod n)$, then $aa'\equiv bb' (\mod n). $ So you can either multiply and then reduce $\mod n$ or vice versa. For an application, to see that you can reduce large computations to something with logarithmic runtime, e.g. have a look at <a href="https://en.wikipedia.org/wiki/Expo... |
1,220,800 | <blockquote>
<p>Calculation of x real root values from $ y(x)=\sqrt{x+1}-\sqrt{x-1}-\sqrt{4x-1} $</p>
</blockquote>
<p>$\bf{My\; Solution::}$ Here domain of equation is $\displaystyle x\geq 1$. So squaring both sides we get</p>
<p>$\displaystyle (x+1)+(x-1)-2\sqrt{x^2-1}=(4x-1)$.</p>
<p>$\displaystyle (1-2x)^2=4(... | Vincenzo Oliva | 170,489 | <p>Alternatively, isolating $ \sqrt{4x-1}$ and then multiplying both sides by $\sqrt{x+1}+\sqrt{x-1}$ makes it easier to conclude the LHS is smaller than the RHS: $$\require\cancel \cancel{x}+1-\cancel{x}+1=2<\sqrt{4x-1}\left(\sqrt{x+1}+\sqrt{x-1}\right).\tag{$\star$}$$</p>
<p>Once we check $(\star)$ holds for $x=1... |
1,220,800 | <blockquote>
<p>Calculation of x real root values from $ y(x)=\sqrt{x+1}-\sqrt{x-1}-\sqrt{4x-1} $</p>
</blockquote>
<p>$\bf{My\; Solution::}$ Here domain of equation is $\displaystyle x\geq 1$. So squaring both sides we get</p>
<p>$\displaystyle (x+1)+(x-1)-2\sqrt{x^2-1}=(4x-1)$.</p>
<p>$\displaystyle (1-2x)^2=4(... | Community | -1 | <p>For $x\ge1$,</p>
<p>$$l(x):=\sqrt{x+1}-\sqrt{x-1}=\dfrac2{\sqrt{x+1}+\sqrt{x-1}}\le\sqrt2$$ and</p>
<p>$$r(x):=\sqrt{4x-1}\ge\sqrt3.$$</p>
|
2,105,963 | <p>Suppose ${{A_1}}$=[1,3] and ${{A_2}}$=[2,4], then ${{A_1} \cup {A_2}}$=[1,4] now $\sup \left( {{A_1} \cup {A_2}} \right)$ is clearly 4. so, $\sup \left( {{A_1} \cup {A_2}} \right) = \max \left( {\sup {A_1},\sup {A_2}} \right)$ is true.</p>
<p>Confusion with definition: <br/>
<em>s</em> is least upper bound for a se... | Andrew D. Hwang | 86,418 | <p>$\newcommand{\eps}{\varepsilon}$<strong>Suggestion</strong>: To prove that a specific real number $c$ is the supremum of a non-empty set $A$ of real numbers (that is bounded above), it's often helpful to use the second condition of the definition in contrapositive form:</p>
<ol>
<li><p>$c$ is an upper bou... |
55,918 | <blockquote>
<p><strong>Zariski's Main Theorem</strong> (<a href="http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1966__28__5_0" rel="noreferrer">EGA IV</a>, Thm 8.12.6): Suppose $Y$ is a quasi-compact and quasi-separated scheme, and $f:X\to Y$ is quasi-finite, separated, and finitely presented. Then $f$ factors as ... | Qing Liu | 3,485 | <p>I think an initial object exists if you work with integral excellent schemes (maybe integral is not really necessarily, but then require that $X$ be schematically dense in $Z$). </p>
<p>So suppose $X, Y$ are integral and excellent. Consider all possible factorizations $X\to Z_{\alpha} \to Y$ with $Z_{\alpha}$ integ... |
9,990 | <p>Consider the following problem: </p>
<ul>
<li>Maria always buys ice-cream when she goes to the beach. She bought ice-cream today. So, she must have gone to the beach. </li>
</ul>
<p>Obviously this statement is wrong. Maria could have gone to other place and bought an ice-cream. You don't need any math tool to ar... | Daniel R. Collins | 5,563 | <p>You can say that this is "just reasoning", but the truth is that this is a specific application of basic logic, in particular the implication (if/then) relation. I have a colleague with a PhD in logic who says, "Implication is tricky!" when I bring this up. And I do think that it's a major proble... |
887,656 | <p>Is there a closed form (complex) solution $z(t)$ to the equation</p>
<p>\begin{align}
\frac{dz}{dt}=f(t)\bar{z},
\end{align}</p>
<p>(the bar means complex conjugate) for any given complex valued function $f$ of a real variable $t$? The usual approach to deal with separable equations gives
\begin{align}
\int\frac{1... | MrSlunk | 12,509 | <p>Here is an alternative approach.<br>
When you are given the equation
$$z' = f(t) \overline{z},$$
you are also implicitly given
$$\overline{z}' = \overline{f}(t) z$$
by conjugation.
So let $Z =(z,\overline{z})$, then we have
$$Z' = \left(\begin{matrix}0&f(t)\\ \overline{f}(t)&0
\end{matrix}\right)Z
$$
This i... |
1,653,416 | <p>We know that:
<a href="https://www.youtube.com/watch?v=w-I6XTVZXww" rel="nofollow">https://www.youtube.com/watch?v=w-I6XTVZXww</a>
$$S=1+2+3+4+\cdots = -\frac{1}{12}$$</p>
<p>So multiplying each terms in the left hand side by $2$ gives:
$$2S =2+4+6+8+\cdots = -\frac{1}{6}$$
This is the sum of the even numbers</p>
... | Count Iblis | 155,436 | <p>As pointed out in section 3, page 1191 of <a href="http://arxiv.org/abs/hep-ph/0510142">this article</a>, the rules for manipulating divergent series are more restrictive than those for convergent series. As pointed out in the article, to avoid problems you should work with the power series obtained by multiplying ... |
1,206,195 | <p>I am trying to find the maximum of $x^{1/x}$. I don't know how to find the derivative of this. I have plugged in some numbers and found that $e^{1/e}$ seems to be the maximum at around 1.44466786. I don't know if this is the maximum, and I would like an explanation of why it is/what the maximum is. essentially, how ... | lab bhattacharjee | 33,337 | <p>$$y=x^{1/x}\implies \ln(y)=\frac{\ln x}x$$</p>
<p>$$\frac{d(\ln y)}{dx}=\frac{d(\ln y)}{dy}\cdot\frac{dy}{dx}=\frac1y\cdot\frac{dy}{dx}$$</p>
<p>and $$\frac{d\left(\frac{\ln x}x\right)}{dx}=\frac{1-\ln x}{x^2}$$</p>
|
191,796 | <blockquote>
<p>I met with the following difficulty reading the paper <a href="http://www.cnki.com.cn/Article/CJFDTotal-ZZDZ198801008.htm" rel="nofollow">Li, Rong Xiu "The properties of a matrix order column" (1988)</a>:</p>
<p>Define the matrix $A=(a_{jk})_{n\times n}$, where
$$a_{jk}=\begin{cases}
j+k\cdot ... | Christian Remling | 48,839 | <p>OK, let me try again, maybe I'll get it right this time. I'll show that $P$ is positive definite. This will imply the claim because if $(P+iQ)(x+iy)=0$ with $x,y\in\mathbb R^n$, then $Px=Qy$, $Py=-Qx$, and by taking scalar products with $x$ and $y$, respectively, we see that $\langle x, Px \rangle = -\langle y, Py\r... |
2,820,696 | <p>We were just discussing with colleagues the number of combinations you could get with two "normal", $6$-sided dice. Almost all of my colleagues were saying $36$ ($6^2$), which I agree with as such, but you will get almost half of the possible combinations counted twice.
If I count, the number of different combinati... | Arthur | 15,500 | <p>If order doesn't count, then consider the following setup (I will do it for three 6-sided dice, but it can easily be generalized):</p>
<p>Put up five dividers ("bars"):
$$
|\quad|\quad|\quad|\quad|
$$
These five bars give us six regions (four gaps between the bars, as well as to the left and to the right). Each reg... |
2,762,391 | <p>Let $x,y>0$ s.t. $x^3+y^3\geq 2$.</p>
<p>Show that $x^2+y^2\geq x+y $.</p>
<p>I analyse the case when $x,y\geq 1$ but I don't know to solve the case when $x\geq 1\geq y $.</p>
| Cesareo | 397,348 | <p>By symmetry $x = y = z$ </p>
<p>then </p>
<p>$$
2z^3 \ge 2 \Rightarrow z^3 \ge 1 \Rightarrow z^2 \ge z
$$</p>
<p>Comparison with $x^3+y^3 \ge 2$ (light blue) and the circle $(x-\frac{1}{2})^2+(y-\frac{1}{2})^2 = 1/2$ (red)</p>
<p><a href="https://i.stack.imgur.com/uGGTb.jpg" rel="nofollow noreferrer"><img src="h... |
2,762,391 | <p>Let $x,y>0$ s.t. $x^3+y^3\geq 2$.</p>
<p>Show that $x^2+y^2\geq x+y $.</p>
<p>I analyse the case when $x,y\geq 1$ but I don't know to solve the case when $x\geq 1\geq y $.</p>
| A.Γ. | 253,273 | <p>Consider $f(x,y)=x^2+y^2-x-y$. The inequality is equivalent to the fact that the following minimum is non-negative
$$
\min f(x,y)\quad\text{subject to }x^3+y^3\ge 2,\ x\ge 0,\ y\ge 0.
$$</p>
<ol>
<li>As $f(x,y)=(x-\frac12)^2+(y-\frac12)^2-\frac12$ has compact sublevel sets, the minimum exists. Apply the necessary c... |
96,110 | <p><span class="math-container">$A = \begin{pmatrix}
0 & 1 &1 \\
1 & 0 &1 \\
1& 1 &0
\end{pmatrix} $</span></p>
<p>The matrix <span class="math-container">$(A+I)$</span> has rank <span class="math-container">$1$</span> , so <span class="math-container">$-1$</span> is an eigenvalue with an al... | Pierre-Yves Gaillard | 660 | <p>It is clear that $A+I$ is diagonalizable with $(0,0,3)$ on the diagonal. </p>
<p>Thus, $A$ is diagonalizable with $(-1,-1,2)$ on the diagonal. </p>
<p>Justification of the first claim:</p>
<p>A vector is in the kernel of $A+I$ if and only if the sum of its coordinates is zero. </p>
<p>The vector $(1,1,1)$ is an ... |
96,110 | <p><span class="math-container">$A = \begin{pmatrix}
0 & 1 &1 \\
1 & 0 &1 \\
1& 1 &0
\end{pmatrix} $</span></p>
<p>The matrix <span class="math-container">$(A+I)$</span> has rank <span class="math-container">$1$</span> , so <span class="math-container">$-1$</span> is an eigenvalue with an al... | Tapu | 17,142 | <p>Here is my trying (which may be less simple than the previous answers):</p>
<p>we note that $A+I=ee^t\text{ or }$ $$A=ee^t-I$$ post multiplication by $e$ readily shows that $e$ is an eigenvector of $A$ with eigenvalue $2$. Since we can find two other vectors orthogonal to $e$, post multiplications by those show tha... |
4,054,428 | <p>The question is</p>
<p>Let <span class="math-container">$X$</span> be a discrete random variable with probability mass function</p>
<p><a href="https://i.stack.imgur.com/DhJVI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DhJVI.png" alt="enter image description here" /></a></p>
<p>(a) Find <span... | Graham Kemp | 135,106 | <p>(a) Your calculation was off. It seems that you dropped the decimal.<span class="math-container">$$\mathsf E(X)=−2\cdot 0.3+−1\cdot \underbrace{0.6}+12\cdot0.1=0.0$$</span></p>
<p>(b) A negative variance is an error warning. You need to square <em>each</em> supported value when you multiply by the probability mas... |
62,526 | <p>I want to show the following:</p>
<p>$X$ $n$-connected $\iff $ any continuous map $f:K \rightarrow X$ where $K$ is a cell complex of dimension $\leq n$ is homotopic to a constant map</p>
<p>For this I think I can use the following:
$X$ $n$-connected $\iff $ every continuous map $f: S^n \rightarrow X$ is homotopic ... | SFeesh | 346,530 | <p>We show that <span class="math-container">$K^m \hookrightarrow K \rightarrow X$</span> is nullhomotopic for every <span class="math-container">$m \leqslant n$</span> by induction on <span class="math-container">$m$</span>. This is clear for <span class="math-container">$m = 0$</span> since <span class="math-containe... |
2,297,421 | <p>Probably a really simple question, but I am trynig to fit an air bed in a tent.</p>
<p>Circular tent with a diameter of $3$m and a central vertical pole in the middle.</p>
<p>The air bed measures $1.41$ m $\times$ $1.9$ m.</p>
<p>Will the air bed fit fully inside the tent without being obstructed by the central p... | hmakholm left over Monica | 14,366 | <p>Here are some helpful facts:</p>
<ol>
<li><p>It is sufficient to prove the property when $n$ is an odd prime.</p></li>
<li><p>A quadratic polynomial over a field is reducible exactly if it has a root.</p></li>
<li><p>The polynomials for different $k$ (modulo $n$) cannot have roots in common.</p></li>
<li><p>The pol... |
285,841 | <p>I would like to solve the following problem:</p>
<p>$$\begin{array}{ll} \text{minimize} & \mathbf{x}^T \mathbf{A} \mathbf{x}\\ \text{subject to} & \mathbf{x}^T\mathbf{B}\mathbf{x} = 0\\ & \mathbf{x}^T \mathbf{x} = 1\end{array}$$</p>
<p>where $\bf x$ is a vector, $\bf A, \bf B$ are square matrices, and ... | Mark L. Stone | 75,420 | <p>Because no one has offered a solution meeting your ideal of using a standard numerical linear algorithm, I will offer an approach using the global numerical nonlinear optimizer BARON.</p>
<p>Here is a solution using BARON as the solver under YALMIP under MATLAB. I will use the B provided by @Federico Poloni in his... |
52,480 | <p>The question comes from a statement in Concrete Mathematics by Graham, Knuth, and Patashnik on page 465.</p>
<p>$$\sum_{k \geq n} \frac{(\log k)^2}{k^2} = O \left(\frac{(\log n)^2}{n} \right).$$</p>
<p>How is this calculated?</p>
| Qiaochu Yuan | 232 | <p>Here's a pretty straightforward solution. The idea is to split the sum up into a main part and an error term, and generalizes to many sums where the integral test won't work because the corresponding integral is difficult. </p>
<p>First recall that we have $\sum_{k \ge n} \frac{1}{k^s} = O \left( \frac{1}{n^{s-1}} ... |
4,251,161 | <p><strong>Objective</strong><br />
I need to find roots of <span class="math-container">$$f(x)=c$$</span> in interval <span class="math-container">$[a,b]$</span>, where</p>
<ul>
<li><span class="math-container">$f(a)=0$</span> and <span class="math-container">$c<f(b)<1$</span></li>
<li><span class="math-containe... | Claude Leibovici | 82,404 | <p>I shall suppose that the computation of <span class="math-container">$f'(x)$</span> is more expensive that the computation of <span class="math-container">$f(x)$</span> itself.</p>
<p>For this kind of problems, I extensively used the so-called RTSAFE subroutine from <em>"Numerical Recipes"</em> (have a loo... |
1,276,264 | <p>So, I was wondering if it is possible to solve for $n$ in $2^n=8$ (or any other question where $n$ is a power) using $9^{th}$ grade math. Please excuse my naïveté if this is extremely stupid/simple. </p>
<p>Thanks so much in advance! –– come to think of it: Is it possible at all?</p>
| Chris | 235,548 | <p>Note that $8=2^3$. Compare it to $2^n$ and conclude that $n=3$.</p>
|
1,276,264 | <p>So, I was wondering if it is possible to solve for $n$ in $2^n=8$ (or any other question where $n$ is a power) using $9^{th}$ grade math. Please excuse my naïveté if this is extremely stupid/simple. </p>
<p>Thanks so much in advance! –– come to think of it: Is it possible at all?</p>
| passenger | 23,187 | <p>I don't know what is included in 9-th grade math, but here is a try to explain that $ 2^n, n \in \mathbb N $ tends to infinity as $n$ does so.</p>
<p>Obviously $n=3$ is a solution, since $2^3=8$. Now, if you try larger values of $n$ then $ 2^n$ becames larger and larger, since every time you multiply more times $2$... |
3,471,684 | <p>Is always correct statement that if natural numbers <span class="math-container">$a,b \in \Bbb N$</span> for which LCM<span class="math-container">$(a,b)=16\cdot(a,b)$</span>, then <span class="math-container">$a|b$</span> or <span class="math-container">$b|a$</span>?</p>
<p>I used formula that LCM<span class="math... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$ $</span> <em>Conceptually</em> it is obvious: a prime <span class="math-container">$\,p\,$</span> occurs in <span class="math-container">$\,c = {\rm lcm}(a,b)/\gcd(a,b)\!\iff\! p\, $</span> occurs to <em>different</em> powers in <span class="math-container">$\,a\,... |
220,996 | <p>I was wondering if for every NFA there exists an equivalent DFA? I think the answer is yes. How would one <em>prove</em> it? Since I'm just starting up learning about Automata I'm not confused about this and especially the proof of such a statement.</p>
| HabiSoft | 45,748 | <p>Indeed, every NFA can be converted to an equivalent DFA. In fact, DFAs, NFAs and regular expressions are all equivalent. One approach would be to observe the NFA and, if it is simple enough, determine the regular expression that it recognizes, then convert the regular expression to a DFA.</p>
<p>Yet, there are algo... |
135,012 | <p>How to prove (or to disprove) that all the roots of the polynomial of degree $n$ $$\sum_{k=0}^{k=n}(2k+1)x^k$$ belong to the disk $\{z:|z|<1\}?$ Numerical calculations confirm that, but I don't see any approach to a proof of so simply formulated statement. It would be useful in connection with an irreducibility p... | Gabriel Furstenheim | 5,556 | <p>The idea is taken from this other question <a href="https://mathoverflow.net/questions/18094/polynomial-with-the-primes-as-coefficients-irreducible">Polynomial with the primes as coefficients irreducible?</a></p>
<p>Show instead that $f(1/x)$ has all roots lying outside of the unit disk. For that, multiply by $(x-1... |
88,861 | <p>If $n$ is an integer, how do you know whether $n^n$ is a perfect square, without a calculator?</p>
<p>The actual question is: "<em>how many integers between $1$ and $100$ inclusive, raised to their own power, are perfect squares?</em>".</p>
| Gaurav Tiwari | 11,044 | <p>'$n^n$ is a perfect square' means that Square root of this number is a whole number. Let me solve this for both case of even and odd numbers as suggested by @pete:</p>
<p>If $n$ is an even number, then we may replace $n$ by $2m \ \forall m=1,2 \ldots$; and thus can write $n^n$ as ${(2m)}^{2m}$.Therefore, $\sqrt {{... |
4,267,862 | <p>If k balanced n-sided dice are rolled, what is the probability that each of the n different numbers will appear at least once?</p>
<p>A case of this was discussed <a href="https://math.stackexchange.com/questions/264408">here</a>, but I’m not sure how to extend this. Specifically, I’m not sure how the to calculate t... | G Cab | 317,234 | <p>(<em>Please allow me to use <span class="math-container">$n$</span> in place of your <span class="math-container">$k$</span>, and <span class="math-container">$m$</span> in place of your <span class="math-container">$n$</span></em>)</p>
<p>So we have <span class="math-container">$n$</span> fair <span class="math-con... |
652,446 | <p>I just ran into the next problem: The random variables $X$ and $Y$ are independent, where $X \sim Normal(1,1)$ and $Y \sim Gamma(\lambda,p)$ with $E(Y) = 1$ and $Var(Y) = 1/2$ How do we find $E(X+Y)^3$ ?? I've tried a convolution, which leads to a really ugly looking integral from which I then have to get the third ... | Tom-Tom | 116,182 | <p>Write $$\mathrm E\left[(X+Y)^3\right]=\mathrm E\left[X^3+3X^2Y+3XY^2+Y^3\right]$$
and use the fact that $X$ and $Y$ are independent, <em>i.e.</em>
$$\mathrm E\left[X^pY^q\right]=\mathrm E\left[X^p\right]\mathrm E\left[Y^q\right].$$</p>
|
1,731,382 | <p>Notice that the parabola, defined by certain properties, is also the trajectory of a cannon ball. Does the same sort of thing hold for the catenary? That is, is the catenary, defined by certain properties, also the trajectory of something?</p>
| pjs36 | 120,540 | <p>From the right perspective, maybe.</p>
<p><a href="https://i.stack.imgur.com/2ZPeI.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/2ZPeI.gif" alt="enter image description here"></a></p>
<p>(image from <a href="https://en.wikipedia.org/wiki/Square_wheel" rel="noreferrer">Wikipedia</a>)</p>
<p>I'm not exa... |
4,353,958 | <p><a href="https://i.stack.imgur.com/zxyBa.png" rel="nofollow noreferrer">In this example</a>, it says that the phase of the complex number <span class="math-container">$i$</span> is <span class="math-container">$\pi/2$</span> and <span class="math-container">$-1$</span> has phase <span class="math-container">$\pi$</s... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$z\in\Bbb C$</span> and <span class="math-container">$z=x+yi$</span>, with <span class="math-container">$x,y\in\Bbb R$</span>, then, if you want to write <span class="math-container">$z$</span> as <span class="math-container">$\rho\bigl(\cos(\varphi)+\sin(\varphi)i\bigr)$</span>, then... |
1,085,702 | <p>It's said that a computer program "prints" a set <span class="math-container">$A$</span> (<span class="math-container">$A \subseteq \mathbb N$</span>, positive integers.) if it prints every element of <span class="math-container">$A$</span> in ascending order (even if <span class="math-container">$A$</span... | WillO | 29,145 | <p>Jihad's answer proves that some such $A$ exists. For an explicit example, let $A$ be the set of Godel numbers of true statements about arithmetic (after fixing your favorite encoding).</p>
|
1,085,702 | <p>It's said that a computer program "prints" a set <span class="math-container">$A$</span> (<span class="math-container">$A \subseteq \mathbb N$</span>, positive integers.) if it prints every element of <span class="math-container">$A$</span> in ascending order (even if <span class="math-container">$A$</span... | KSmarts | 192,747 | <p>Any known model of computing, and thus any computer we can make, is equivalent to a Turing machine. Since the number of different possible inputs and states of a Turing machine is finite, there are, as Jihad said, countably many possible programs. Since there are uncountably many subsets of the natural numbers, ther... |
1,085,702 | <p>It's said that a computer program "prints" a set <span class="math-container">$A$</span> (<span class="math-container">$A \subseteq \mathbb N$</span>, positive integers.) if it prints every element of <span class="math-container">$A$</span> in ascending order (even if <span class="math-container">$A$</span... | Hanno | 81,567 | <p>The following is an elementary and informal approach which does not use set theory:</p>
<p>First, fix the syntax for your computer programs and enumerate them by natural numbers.</p>
<p><em>Definition:</em> Denote $A\subset{\mathbb N}$ the set of those numbers which correspond to a computer program that prints out... |
1,274,717 | <p>Say that $V$ is a finite dimensional vector space over a field and and $f : V \to V$ a linear map. There is an integer $i$ such that $\text{ker}(f^n) = \text{ker}(f^{n+1})$ for all $n \geq i$. You see that by noting that $\text{ker}(f^n) \subseteq \text{ker}(f^{n+1})$ for all $n$ and since $V$ is finite dimensional,... | Alex W | 230,729 | <p>Let $K_i=\ker(f^i)$. Assume, that $n\in\mathbb{Z}_{\geq 0}$ and $K_n=K_{n+1}$. We show that $K_{n+1}=K_{n+2}$. Clearly, this will prove your claim. Obviously, $K_{n+1}\subset K_{n+2}$. Conversely, let $a\in K_{n+2}$ and let's show that $a\in K_{n+1}$. Since $a\in K_{n+2}$, then
$0=f^{n+2}(a)=f^{n+1}(f(a))$, hence $... |
1,341,385 | <p>I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my own. My dad wants me to memorize about 30 trig identities (though some of them are very similar) since I'm missing tri... | Teoc | 190,244 | <p>To be honest, the only trig identities you really need are the definitions of the 6 trig functions, and Euler/De Moivre. You can prove almost all the trig identities from $e^{ix}$. Deriving an identity is much easier than just rote memorization, and with repeated deriviation, it eventually becomes memorized. </p>
|
171,565 | <p>Does anybody know whether there exists a proof by induction (or at least a proof that does not use Hilbert polynomials) that the degree of the Segre variety product of $n$ lines is $n!$ ?</p>
| Francesco Polizzi | 7,460 | <p>Let me give a proof by induction.</p>
<p>For $n=2$ we have $\mathbb{P}^1 \times \mathbb{P}^1$ embedded as a quadric in $\mathbb{P}^3$, so the claim is true in this case.</p>
<p>Now let $X_n$ be the Segre embedding of the product of $n$ lines $\mathbb{P}^1 \times \ldots \times \mathbb{P}^1$. An easy computation usi... |
171,565 | <p>Does anybody know whether there exists a proof by induction (or at least a proof that does not use Hilbert polynomials) that the degree of the Segre variety product of $n$ lines is $n!$ ?</p>
| abx | 40,297 | <p>The Segre embedding is defined by the line bundle $\ L:=\mathcal{O}_{\mathcal{P}^1}(1)\boxtimes\ldots \boxtimes \mathcal{O}_{\mathcal{P}^1}(1)\ $ on $(\mathbb{P}^1)^n$, therefore its degree is $L^n=(p_1^*h+\ldots +p_n^*h)^n$, where $h$ is the class of a point in $\mathrm{Pic}(\mathbb{P}^1)$ and $p_i$ the $i$-th proj... |
3,567,662 | <p>I have just learned how to convert a plane in R3 from Cartesian to parametric form, by setting 2 variables to 0 and solving for the 3rd one in order to obtain 3 points on the plane, and solve from there. However, this does not work when 1 or 2 of the variables are 0, as it is not possible to find 3 points on the pla... | Community | -1 | <p>Enumerate following the systematic sequence of <span class="math-container">$(x_i,y_j)$</span> indexes</p>
<p><span class="math-container">$$11,\ 21,12,\ 31,22,13,\ 41,32,23,14,\ 51,42,33,24,15,\ \cdots$$</span></p>
<p>or any other enumeration of <span class="math-container">$\mathbb N\times\mathbb N$</span>.</p>
|
232,276 | <p>I can prove with the triangle inequality that the unit sphere in $R^n$ is convex, but how to show that it is strictly convex?</p>
| Hagen von Eitzen | 39,174 | <p>For the euclidean metric, we see that the unit ball is <em>strictly</em> convex because for different vectors $a$ and $b$ we have that
$$f(t):=||ta+(1-t)b||^2\\ =\langle ta+(1-t)b, ta+(1-t)b\rangle\\ = t^2||a||^2+(1-t)^2||b||^2 + 2t(1-t)\langle a,b\rangle\\= (\ldots) t^2+(\ldots)t + (\ldots)$$
is a quadratic funct... |
78,368 | <p>Please, can anybody give a reference(s) to some good recent review papers about copulas and time series?</p>
| Fabrizio | 18,614 | <p>I would suggest to look at the following surveys (both available online):
H. Manner and O. Reznikova. "A survey on time-varying copulas: Specification, simulations and estimation. Econometric Reviews, forthcoming.
A. Patton. "Copula-Based Models for Financial Time Series", 2009, in T.G. Andersen, R.A. Davis, J.-P. K... |
3,274,766 | <p>I have been trying to do this problem:</p>
<p>Solve <span class="math-container">$$\sec(x)=\tan(x),\quad 0≤x<2π$$</span></p>
<p>I started by rewriting <span class="math-container">$\sec(x)$</span> as <span class="math-container">$\frac{1}{\cos(x)}$</span>.</p>
<p>I then rewrote <span class="math-container">$\t... | José Carlos Santos | 446,262 | <p>There is nothing wrong with what you did. The problem has no solution since, precisely, if <span class="math-container">$x\in\mathbb R$</span> is such that both <span class="math-container">$\tan(x)$</span> and <span class="math-container">$\sec(x)$</span> are defined, then <span class="math-container">$\sec(x)=\tan... |
4,544,450 | <p>im trying to solve this logical equation</p>
<p>p≡((p∧∼q)→q)→p
i know i have to solve for the right side and im pretty certain the final step must be the absorption law 10). But the ∼q is bugging me</p>
| Graham Kemp | 135,106 | <blockquote>
<p>im pretty certain the final step must be the absorption law 10). But the ∼q is bugging me</p>
</blockquote>
<p>Yes it is, but there is no reason to be bugged.</p>
<p>The relevant absorption law is that: for <em>any</em> predicates <span class="math-container">$A$</span> and <span class="math-container">... |
2,409,580 | <p>Recall that </p>
<blockquote>
<p><strong>Theorem (Bessel inequality).</strong> Let $(e_k)$ be an orthonormal sequence in an inner product space $X$. Then for every $x \in X $, $$\sum_{k=1}^{\infty} |\langle x,e_k \rangle|^2 \le \|x\|^2 .$$</p>
</blockquote>
<p>The proof results in $\le$ and not just $=$. Can $\l... | Angina Seng | 436,618 | <p>In general, equality holds (<a href="https://en.wikipedia.org/wiki/Parseval%27s_identity" rel="nofollow noreferrer">Parseval's theorem</a>) when the $e_k$
form a <strong>complete</strong> orthonormal sequence. For instance the $e^{inx}$
for $n\in\Bbb Z$ is a complete orthonormal sequence in $L^2[0,1]$.
If you take a... |
242,203 | <p>What's the derivative of the integral $$\int_1^x\sin(t) dt$$</p>
<p>Any ideas? I'm getting a little confused.</p>
| TonyK | 1,508 | <p>If $f$ is <em>any function at all</em> that can be integrated, then the derivative of the integral of $f(t)dt$ from $1$ to $x$ is $f(x)$. This wonderful fact is the Fundamental Theorem of Calculus.</p>
|
2,406,107 | <p><a href="https://i.stack.imgur.com/ccIr4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ccIr4.png" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/zCPUz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zCPUz.png" alt="enter image description he... | drhab | 75,923 | <p>In general if $R\subseteq A\times B$ and $S\subseteq B\times C$ then: $$S\circ R:=\{\langle a,c\rangle\mid\exists b\in B[\langle a,b\rangle\in R\wedge\langle b,c\rangle\in S\}\subseteq A\times C$$</p>
<p>So starting with relations $R\subseteq A\times B$, $S\subseteq B\times C$ and $T\subseteq C\times D$ we have $S\... |
3,135,440 | <p>This is throwing me off a bit I believe mainly because the way the question is worded? Would this simply be <span class="math-container">$4$</span> out of <span class="math-container">$36$</span>?</p>
| Michael Rybkin | 350,247 | <p>It means that you're doing something wrong.</p>
<p><span class="math-container">$$
\left(\frac{x+1}{x-1}\right)'=
\frac{(x+1)'(x-1)-(x+1)(x-1)'}{(x-1)^2}=\\
\frac{x-1-(x+1)}{(x-1)^2}=
\frac{x-1-x-1}{(x-1)^2}=-\frac{2}{(x-1)^2}
$$</span>
<hr>
<span class="math-container">$$
[(x+1)(x-1)^{-1}]'=\\
(x+1)'(x-1)^{-1}+(x+... |
3,909,191 | <p>I am not too sure as to what the relation is but I think <span class="math-container">$R = \{(1, 2), (2, 3), (3, 4), ..., (n - 1, n)\} $</span>.</p>
<p>Any guidance would be appreciated.</p>
| kevinkayaks | 449,944 | <p>In the next step, you want the cube root of the complex number <span class="math-container">$z = 1-i\sqrt{3}/2$</span>. First express this number in the form <span class="math-container">$z = re^{i\theta}$</span> using Euler's identity. For a number <span class="math-container">$z = a + i b$</span> we have <span cla... |
26,589 | <p>Whenever you open a post in Mathstack, you can not know how many answer that post has got. To know you need to scroll down to the end of the post. In general when I see a post I how long is the scroll bar, to predict if the question has got an answer or not. <strong>But sometimes</strong> OP post very long question,... | Asaf Karagila | 622 | <p>Between the comments on the questions and the answers itself, this thing literally exists.</p>
<p>The number of answers, and you can choose how to order them. Moreover, if there is an accepted answer, it will appear first (with an exception for when the answer was posted by the author of the question, in which case... |
26,589 | <p>Whenever you open a post in Mathstack, you can not know how many answer that post has got. To know you need to scroll down to the end of the post. In general when I see a post I how long is the scroll bar, to predict if the question has got an answer or not. <strong>But sometimes</strong> OP post very long question,... | amWhy | 9,003 | <p>Yes: "Whenever you open a post" you will have first clicked on the post in order to open it. </p>
<p>But every question listed on the main page (from which you open a post) <em>already shows</em> the net votes the question has earned thus far, the number of answers received, and when the number of answers received... |
488,983 | <p>I'm trying to prove the group isomorphism $(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$.</p>
<p>Obviously I tried to establish a ring isomorphism from $\Bbb Z[x]/(x^{n+1})$ to some ring $R$, a direct product of easier rings, and prove that $R^\times$ equals the RHS of the original i... | Ryan Reich | 3,547 | <p>This problem is stated deceptively in the sense that it gives a highly polished <em>description</em> of what $\def\Z{\mathbb{Z}}(\Z[x]/(x^{n + 1}))^\times$ is, but doesn't actually say what it is as a subset of $\Z[x]/(x^{n + 1})$ itself. You will have an easier time proving it if you sit down and compute what the ... |
1,732,526 | <p>I start by expanding the denominator and separating the real and imaginary but get stuck when deciding what my $u$ and $v$ should be.</p>
<p>Thanks.</p>
| Robert Israel | 8,508 | <p>Avoid looking at real and imaginary parts.
Sum and product of analytic functions are analytic. Quotient is analytic where the denominator is nonzero.</p>
|
440,844 | <p>Suppose we have a linear map $A \colon V \to V$ on a finite- dimensional vector space, and $W \leq V$ it's invariant subspace. Then we have obviously $\operatorname{Ker} A + W \subseteq A^{-1}(W)$.</p>
<p>Is it then necessary $\operatorname{Ker} A + W = A^{-1}(W)$ ?</p>
<p>I can prove it in case $A$ is a projector... | Asaf Karagila | 622 | <p>Suppose that $f\colon A\to B$ is surjective, then for every $b\in B$ the set $F_b=\{a\in A\mid f(a)=b\}$ is non-empty. Therefore, using the axiom of choice, there is some $g$ which selects an element from $F_b$, that is $g(F_b)\in F_b$.</p>
<p>Now show that $g$ is actually a function from $B$ into $A$, and that $g$... |
4,193,578 | <p>This is maybe a very easy one, but I can't find a solution...</p>
<p>I'm looking for a sequence <span class="math-container">$a_1,...,a_n$</span> such that <span class="math-container">$0\leq a_1<\cdots<a_n<1$</span> and <span class="math-container">$\sum_{k=1}^na_k=1$</span>. Of course, this should work fo... | Arthur | 15,500 | <p>Here is a very general, pretty easy way to get what you want.</p>
<p>Take any strictly increasing sequence of <span class="math-container">$n$</span> positive numbers. Find their sum. Divide each term by that sum. You now have a strictly increasing sequence of positive numbers wich sums to <span class="math-containe... |
4,006,571 | <p>Prove this statement using a proof by contradiction: <br />
Let <span class="math-container">$n$</span> be a natural number. If <span class="math-container">$x_1,\ldots,x_n \in \mathbb{N} \cup \{0\}$</span> and <span class="math-container">$\sum_{i=1}^{n}{x_i} = n+1$</span> then there is an <span class="math-contain... | J. Dunivin | 203,407 | <p>The mistake is that you are focusing only on <span class="math-container">$\sum_{i=1}^1 x_i = x_1$</span> to get your contradiction, but we don't know anything about this sum. The sum that we do know a lot about is <span class="math-container">$\sum_{i=1}^n x_i = x_1 + x_2 + \cdots + x_n = n+1$</span>, so we should ... |
362,801 | <blockquote>
<p>$f:[0,1]\to\mathbb{R}^2$ is continuous, $f(0) \in B_{1}(0,0)$ and $f(1) \in B_{1}(10,10)$. Prove there exists $t \in [0,1]$ such that $f(t) \in \{(x,y): x+y=5\}$. </p>
</blockquote>
<p>I am thinking we need to use extreme value theorem or intermediate value theorem. Which one and how? </p>
<p>Just ... | Hagen von Eitzen | 39,174 | <p><strong>Hint:</strong> $g\colon\mathbb R^2\to\mathbb R$, $(x,y)\mapsto x+y-5$ is continuous and so is $g\circ f$.</p>
<p><strong>Second hint:</strong> If $(x,y)\in B_r(a,b)$, then $a-r<x<a+r$ and $b-r<y<b+r$.</p>
|
350,910 | <p>Is there any condition based on the coefficients of terms that guarantees all real solutions to a general cubic polynomial? e.g. $$ax^3+bx^2+cx+d=0\, ?$$</p>
<p>If not, are there methods rather than explicit formula to determine it?</p>
<p>Thank You.</p>
| lsp | 64,509 | <p>Discriminant, $D = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2$</p>
<p>If $D < 0$, then the polynomial has two complex roots, </p>
<p>If $D = 0$, then there are three real roots, and two of them are definitely equal,</p>
<p>If $D > 0$, then there are three distinct real roots.</p>
<p>Refer <a href="http://e... |
2,300,049 | <p>these are my toughs:</p>
<p>$$z^2 = 1 + 2i \Longrightarrow (x+yi)(x+yi) = 1 + 2i$$</p>
<p>so: $x^2-y^2 = 1$ and $2xy = 2$</p>
<p>then i got that $x = 1/y$ but i cant continue to find the real- and imaginary part of z anymore. Appriciated any help</p>
| sharding4 | 254,075 | <p>An alternative approach would be to use DeMoivre's Theorem.
$1+2i=\sqrt{5}e^{i\arctan{2}}$ so $z=\sqrt[4]{5}e^{i\arctan 2/2}$ or $z=\sqrt[4]{5} e^{(i\arctan 2 +2\pi i)/2}$ If $\tan \theta =2, \tan \frac{\theta}{2} = \frac{2}{1+\sqrt{5}}$ giving $$z=\sqrt[4]{5}\frac{1+\sqrt{5}}{\sqrt{10+2\sqrt{5}}}+i\sqrt[4]{5}\fr... |
2,543,215 | <p>I’ve read a post asking whether a subring of a PID is always a PID. The answer is no, but the post itself gave me more questions.</p>
<ol>
<li><p>Is that possible for a PID that is a subring of a non-PID?</p></li>
<li><p>Is that possible for a subring of a PID that is not a UFD? </p></li>
</ol>
<p>Some hints or ex... | Hagen von Eitzen | 39,174 | <ol>
<li>$\Bbb Z\times \Bbb Z$ has $\Bbb Z\times 0\cong \Bbb Z$ as a subring.</li>
<li>Every PID is a UFD</li>
</ol>
|
3,214,331 | <p>If i take the complex number <span class="math-container">$e^{i(3+2i)}$</span>, it's conjugate is <span class="math-container">$e^{i(-3+2i)}$</span>.</p>
<p>However, the conjugate of the function f, defined as <span class="math-container">$f(x+iy)=e^{i(x+iy)}$</span>, is, according to my book: <span class="math-con... | MSDG | 447,520 | <p>Well write it out:
<span class="math-container">\begin{align}
\overline{ e^{i(x+iy)} } &= \overline{e^{ix-y}} = e^{-y}\overline{e^{ix}} = e^{-y}\overline{(\cos x+ i \sin x)} = e^{-y}(\cos x-i\sin x)\\
&= e^{-y}(\cos(-x) + i \sin (-x)) = e^{-y}e^{-ix} = e^{i(-x+iy)},
\end{align}</span>
so the book likely has... |
3,082,635 | <p>Prove that for a given prime <span class="math-container">$p$</span> and each <span class="math-container">$0 < r < p-1$</span>, there exists a <span class="math-container">$q$</span> such that </p>
<p><span class="math-container">$$rq \equiv 1 \bmod p$$</span></p>
<p>I've only taken one intro number theory ... | Will Jagy | 10,400 | <p>Continued fraction for <span class="math-container">$\frac{137}{73}$</span></p>
<p><span class="math-container">$$ \frac{ 137 }{ 73 } = 1 + \frac{ 64 }{ 73 } $$</span>
<span class="math-container">$$ \frac{ 73 }{ 64 } = 1 + \frac{ 9 }{ 64 } $$</span>
<span class="math-container">$$ \frac{ 64 }{ 9 } = 7 + \fr... |
2,572,032 | <p>I'm looking for help with <strong>(b)</strong> and <strong>(c)</strong> specifically. I'm posting <strong>(a)</strong> for completeness.</p>
<p><strong>(a)</strong> Show convergence for $a_n=\sqrt{n+1}-\sqrt{n}$ towards $0$ and test $\sqrt{n}a_n$ for convergence.</p>
<p><strong>(b)</strong> Show $b_n=\sqrt[k]{n+1}... | Nosrati | 108,128 | <p><strong>Hint:</strong> It might be useful applying mean-value theorem for continuous function $f(x)=\sqrt[k]{x}$ on $[1,\infty)$ then there exists $\xi\in[n,n+1]$ such that
$$\sqrt[k]{n+1}-\sqrt[k]{n}=\dfrac{1}{k\sqrt[k]{\xi}}\leq\dfrac{1}{k\sqrt[k]{n}}$$</p>
|
24,704 | <p>It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least <span class="math-container">$n_p(p^n-1)$</span> elements, where <span class="math-container">$n_p$</span> is the number of Sylow <span class="math-container">$p$</spa... | Myself | 5,189 | <p>It simply isn't true, Sylow p-subgroups can very well intersect non-trivially, Plop gave an example thereof.</p>
<p>Well, it seems like you actually cannot say the following, see comments. I'm just leaving it here as a mistake one shouldn't make, so I won't mind if a moderator deletes it since it's not an actual an... |
624,002 | <p>Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not.</p>
<p>pf) Suppose that these are isomorphic. Note that $\mathbb{Z}\times \mathbb{Z}$ is a subgroup of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times\left \{ 0 \righ... | Edward ffitch | 26,243 | <p>Suppose $\mathbb{Z}\times \mathbb{Z}$ is isomorphic to $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$, via a map $\phi$. Then, as $(1,0)$, $(0,1)$ generate $\mathbb{Z}\times \mathbb{Z}$, $\phi(1,0)$ and $\phi(0,1)$ generate $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$. But $\mathbb{Z}\times \mathbb{Z}\times \ma... |
458,922 | <p>Recently I've stumbled across this claim:</p>
<blockquote>
<p>Peano axioms can be deduced in ZFC</p>
</blockquote>
<p>I found a lot of info regarding this claim (e.g. what would (one version of) the natural numbers look like within the universe of sets: $0 = \emptyset$, $n + 1 = n \cup \{n\}$), but not what the ... | Carl Mummert | 630 | <p>The most direct, model-theoretic method to prove the existence of a model of PA within ZFC is as follows:</p>
<ol>
<li><p>First, we formalize the syntax of PA within ZFC. The method is similar to the one used for formalize syntax for the second incompleteness theorem. The main result is a formula $S(n,m)$ which say... |
3,045,491 | <p>The set <span class="math-container">$\{(x,y,z) \in R^3: x^8+y^4+z^8-16=0\}$</span> is a bounded set?
I guess it isn't a bounded set because from <span class="math-container">$x,y,z \geq 0$</span> i suppose it's only inferiorly bounded.
Is it correct? Please tell me the correct answer.</p>
| Clive Newstead | 19,542 | <p><strong>Hint:</strong> Note that <span class="math-container">$x^8$</span>, <span class="math-container">$y^4$</span> and <span class="math-container">$z^8$</span> are all nonnegative. What can you say about the quantity <span class="math-container">$x^8+y^4+z^8$</span> if <span class="math-container">$|x|>\sqrt{... |
2,741,229 | <p>I have searched a lot, but i haven't found any proof about that statement. I have checked the proof of</p>
<blockquote>
<p>If <span class="math-container">$f$</span> is differentiable, then <span class="math-container">$f$</span> is continuous</p>
</blockquote>
<p>but it's not the same argument I think. Also, I ... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$f$</span> is differentiable, then <span class="math-container">$f$</span> is continuous. The continuity of <span class="math-container">$f'$</span> is irrelevant here.</p>
<p>In particular, <em>even if <span class="math-container">$f'$</span> is discontinuous, <span class="math-cont... |
2,741,229 | <p>I have searched a lot, but i haven't found any proof about that statement. I have checked the proof of</p>
<blockquote>
<p>If <span class="math-container">$f$</span> is differentiable, then <span class="math-container">$f$</span> is continuous</p>
</blockquote>
<p>but it's not the same argument I think. Also, I ... | arp | 370,974 | <p>Answering only part of the question: "If derivative of f is not continuous, then f is not continuous": As perhaps the simplest counter-example, the absolute value function is continuous but not continuously differentiable. </p>
<p>This does not disprove the opposite statement, of course.</p>
|
129,132 | <p>Both the ratio test and the root test define a number (via a limit).</p>
<p>If both limits exist (and shows that the series is convergent), what (if any) is the relation between the 2 numbers ? are they equal ?
What is the relation (if any) between them and the original series (other than the fact that they say th... | David Mitra | 18,986 | <p>For your first question:</p>
<p>If both limits exist, they must be equal to each other. In fact, for a sequence of positive terms <span class="math-container">$(a_n)$</span>, if <span class="math-container">$\lim\limits_{n\rightarrow\infty} {a_{n+1}\over a_n}$</span> exists, then so does
<span class="math-container"... |
2,794,962 | <p>Give is: $C$ which is a closed curve which forms the surface $\Sigma$., $\vec{v} $ which is a constant vector. </p>
<p>I should prove the following expression without using Stokes' Theorem: </p>
<p>$$\oint_C \vec{v} \cdot d\vec{l} = 0$$</p>
<p>How do I go about doing it for an arbitrarily closed (even overlappin... | Alecto Irene Perez | 242,788 | <p><strong>Problem statement:</strong></p>
<ul>
<li>You have a biased coin. </li>
<li>If you flip the coin, then with probability $p$ the coin will come up
heads, and otherwise it'll come up tails.</li>
<li>You're allowed to continue flipping the coin until it comes up tails, or you've flipped it $N$ times (whichever ... |
3,728,963 | <p>(Exercise 21 Chapter 2, Baby Rudin) I am trying to prove</p>
<blockquote>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be separated subsets of some <span class="math-container">$\mathbb{R}^k$</span>, suppose <span class="math-container">$\textbf{a} \in A$</span>, <span c... | Justin Young | 17,892 | <p>Ok, here we go: this is a general proof of the following:</p>
<p>If <span class="math-container">$p:X\to Y$</span> is a continuous function and <span class="math-container">$S\subseteq Y$</span> is a subset, then <span class="math-container">$\overline{p^{-1}(S)} \subseteq p^{-1}\left (\overline S \right )$</span>.<... |
3,728,963 | <p>(Exercise 21 Chapter 2, Baby Rudin) I am trying to prove</p>
<blockquote>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be separated subsets of some <span class="math-container">$\mathbb{R}^k$</span>, suppose <span class="math-container">$\textbf{a} \in A$</span>, <span c... | DanielWainfleet | 254,665 | <p>This is a response to a query by the proposer in a comment to the A from @WilliamElliot.</p>
<p>Sets <span class="math-container">$A,B$</span> are separated iff <span class="math-container">$A\cap \bar B=B\cap \bar A=\phi.$</span> Sets <span class="math-container">$A,B$</span> are completely separated iff there exi... |
2,508,499 | <p>How is this hold that $\mathbb R \subseteq B(0,2)$ where
$\big<\mathbb R,d\big>$ and d is a discrete metric?</p>
<p>By doing so we showed that $\mathbb R $ is bounded</p>
| K.Power | 306,685 | <p>let $x\in \mathbb R$. Then $d(x,0)=1$ if $x\neq 0$, and $d(x,0)=0$ otherwise. Hence $d(x,0)\leq 1 <2$ for all $x\in \mathbb R$.</p>
|
2,144,140 | <p>We know that $(a,b)$ are open by definition. How do you prove that some arbitrary union of $(a,b)$ cannot give you $[c,d]$ ?</p>
| User8976 | 98,414 | <p>In $\Bbb R$ both open and closed sets are $\Bbb R$ and $\phi$. Now see the complement of $[c,d]$ is $(-\infty ,c) \cup (d, + \infty)$ which is open. Hence $[c,d]$ is closed.</p>
|
4,187,498 | <p>I am studying the proof of the Prime Number Theorem and I want to show that the function <span class="math-container">$\frac{\zeta'(s)}{\zeta(s)}$</span> has a simple pole at <span class="math-container">$s=1$</span>.</p>
<p>I think that if I can find the Laurent series expansion of <span class="math-container">$\ze... | TravorLZH | 748,964 | <p>Finding Laurent expansion for <span class="math-container">$\zeta(s)$</span> is equivalent to finding a power series representation for</p>
<p><span class="math-container">$$
F(s)=\zeta(s)-{1\over s-1}
$$</span></p>
<p>at <span class="math-container">$s=1$</span>. This means that we need to develop strategies allowi... |
1,298,730 | <p>Find functions $f$ and $\alpha$ such that the improper Riemann-Stieltjes integral $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist?</p>
<p>I'm really not sure how to start this problem, and I haven't been able to find another post on here that has considered this.</p>
<p>EDIT: I ... | Community | -1 | <p>How about the function:</p>
<p>$$f(x)=\begin{cases}\frac{1}{\sqrt{n}} \text{ for } x \in [n,n+1) \text{ and odd } n\\
-\frac{1}{\sqrt{n}} \text{ for } x \in [n,n+1) \text{ and even } n \end{cases}$$</p>
<p>and let $\alpha(x)=xf(x)$. </p>
<p>Then for odd $n$, </p>
<p>$$\int_{n}^{n+1} f(x) d\alpha(x)=\int_{n}^{n+1... |
1,711,087 | <p>Find the number of elements in the set if the average of these numbers is seven less than the number of elements in the set.</p>
| Sam | 286,799 | <p>Let the number of elements in the set be $x$.</p>
<p>Then we have $$\frac{30}{x}=x-7$$
Implying, $$30 = x^2 - 7x$$
$$x^2-7x-30=0$$
$$(x-10)(x+3)=0$$
$$x=10,-3$$</p>
<p>We cannot have a negative number of elements in the set so there are 10 elements.</p>
|
3,528,370 | <p>Maybe this is too obvious, but I what to be sure... Let <span class="math-container">$Y$</span> be a <span class="math-container">$p\times p$</span> symmetric random matrix (i.e. you can think about <span class="math-container">$Y$</span> as a matrix with random entries). Define <span class="math-container">$E[Y]$</... | Ian | 83,396 | <p>Consider <span class="math-container">$p=1$</span> and <span class="math-container">$Y$</span> equal to the 1x1 matrix 1 with probability 1/2 or the 1x1 matrix -1 with probability 1/2.</p>
|
1,438,999 | <p>If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$?</p>
<p>At the end when you get $4ax=0$, can I divide by $4x$ to cancel out $4$ and $x$?</p>
| Ennar | 122,131 | <p>If $(x-a)^2=(x+a)^2$ for all $x$, then graphs of functions $x\mapsto (x-a)^2$ and $x\mapsto (x+a)^2$ coincide but these are just parabolas with roots at $a$ and $-a$, respectively. Since they must coincide, $a = -a$ which implies $a = 0$.</p>
|
1,514,628 | <p>I've been looking over some old assignments in my analysis course to get ready for my upcoming exam - I've just run into something that I have no idea how to solve, though, mainly because it looks nothing like anything I've done before. The assignment is as follows:</p>
<p>"Let $H$ be a Hilbert space, and let $(e_n... | Mark Fischler | 150,362 | <p>Your idea of using Graham-Schmidt for determining the orthogonal basis is absolutely right, but the answer you present has gone wrong. All you know about the $e_i$ is that for all $i$, $e_i \cdot e_i = 1$ and for all $i \neq j$, $e_i \cdot e_j = 0$. But that is quite a lot to know, and enough to do G.S.</p>
<p>Sta... |
1,705,453 | <p>I have a list of prime numbers which can be expressed in the form of $3x+1$. One such prime of form $3x+1$ satisfies the expression: $a^2+b^2-ab$.</p>
<p>Now I am having list of prime numbers of form $3x+1$ (i.e., $7,19 \ldots$). But I am unable to find the $a$ and $b$ which satisfy the above expression.</p>
<p>Th... | mathreadler | 213,607 | <p><strong>Hint</strong> You get cancellation effects. A difference which is close enough to zero will lose almost all precision the terms could have. An example: say you have 16 bits precision and subtract two numbers which are of same magnitude and equal in the first 12 bits. That leaves only 4 possible bits of preci... |
3,403,272 | <p><p> I'm currently taking abstract algebra and I'm very lost.</p>
<blockquote>
<p>Let <span class="math-container">$G = (\Bbb Z/18\Bbb Z, +)$</span> be a cyclic group of order <span class="math-container">$18$</span>.</p>
<p>(1) Find a subgroup <span class="math-container">$H$</span> of <span class="math-cont... | Locally unskillful | 494,915 | <p><span class="math-container">$G/H$</span> has 6 elements since <span class="math-container">$|G/H| =|G|/|H|=\frac{18}{3}=6$</span>. We we are looking for a group with 6 elements. We say that <span class="math-container">$x,y \in G$</span> are in the same equivalence class if <span class="math-container">$x-y \in H$... |
2,722,609 | <p>In a past thread it was mentioned that $x \in A$ is a predicate. I know $\exists x$ and $\forall x$ are quantifiers but are they also predicates themselves? What about when combined with "in" itself (or whatever this operator is called)? e.g. $\exists x \in A$ or $\forall x \in A$</p>
| Bram28 | 256,001 | <p>No, the quantifiers are not predicates. Rather, combined with predicates, quantifiers can form claims. E.g. $\exists x \ x \in A$ would be the claim that there is some object $x$ that is an element of $A$</p>
<p>This is not the same as $\exists x \in A$ though, which is a restricted quantifier. You'd need to combi... |
3,691,147 | <p>Consider the wave equation in one dimension <span class="math-container">$u_{tt}-u_{xx}=0$</span> together with a Fourier Transform along <span class="math-container">$t$</span>, ie <span class="math-container">$$\text{FT}[u](x,\omega)=\int_{-\infty}^{+\infty}u(x,t)\exp(-i\omega t)\mathrm{d}t.\tag{1}$$</span> The ab... | pluton | 30,598 | <p>A partial answer to the above question is available in the book "Fourier Analysis, by TW Körner, Cambridge University Press, 1988, page 268, Theorem 53.5" (where <span class="math-container">$x$</span> and <span class="math-container">$t$</span> should be interchanged to comply with the question):</p>
<p>Let <span ... |
1,103,239 | <p>For example, if I multiply the value of a base squared by four, I also get twice the base if it's squared. Look:$$6^2\cdot4=12^2$$ because $$36\cdot4=144$$and $36$ is the square of $6$ and $144$ is the square of $12$. Why does this always happen?</p>
| Neal | 20,569 | <p>Because $4 = 2\cdot 2$ and multiplication works like this:
$$ 6^2 \cdot 4 = 6\cdot 6 \cdot 4 = 6\cdot 6 \cdot 2\cdot 2 = 6\cdot 2\cdot 6\cdot 2 = (6\cdot 2)^2$$</p>
|
200,658 | <p>What is the value of :</p>
<p>$$\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$$</p>
| Beni Bogosel | 7,327 | <p>You have for $|x|<1$ </p>
<p>$$ \sum_{k=0}^\infty x^k=\frac{1}{1-x}$$</p>
<p>$$ \sum_{k=0}^\infty kx^k=x\cdot \left(\frac{1}{1-x}\right )'$$</p>
<p>$$ \sum_{k=0}^\infty k^2x^k=x \cdot \left( x \cdot \left( \frac{1}{1-x}\right )'\right )'$$</p>
<p>Replace $x$ with $1/3$ and you will get the result.</p>
|
2,661,443 | <p>For the equation $2^x = 7$</p>
<p>The textbook says to use log base ten to solve it like this $\log 2^x = \log 7$. </p>
<p>I then re-arrange it so that it reads $x \log 2 = \log 7$ then divide the RHS by $\log 2$ to isolate the $x$. I understand this part.</p>
<p>I can alternatively solve it in an easier way by s... | James K | 92,207 | <p>Another common type of question in logarithms is</p>
<p>$$2^{x+2} = 3^{2x}$$</p>
<p>Now suppose you have learnt always to use $\log_2$ if the base is 2, and $\log_3$ if the base is 3. You are now stuck! Which base should you use?</p>
<p>On the other hand suppose you have learned always to use $\log_{10}$, then yo... |
77,379 | <p>It is to show for an $a\in \mathbb{C}^{\ast}$ that $aB_{1}(1)= B_{|a|}(a)$ </p>
<p>where B denotes a disc </p>
<p>Okay, maybe this is correct: </p>
<p>$aB_{1}(1) = a(e^{i\phi}) = ae^{i\phi} = |a|e^{i\phi} = B_{|a|}(a)$</p>
<p>But this seems very wrong! </p>
<p>V</p>
| VVV | 18,298 | <p>Following the second attempt: </p>
<p>So we look at this system of inequalities: </p>
<p>$aB_{1}(1)$: $|az-a|$ and $|z-1|<1 $</p>
<p>what we want to show is that this equals $|z-a|<|a|$</p>
<p>Then stuck. </p>
<p>Following the first attempt: </p>
<p>for $a \in \mathbb{C}^{*} = \mathbb{C}\backslash \{0\}... |
218,933 | <p>The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as
$$
L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}),
$$
where
$$
L^2_k (\mathbb{R}^2; \mathbb{C}) = \bigl\{ f \in L^2 (\mathbb{R}^2;\mathbb{C}) : \text{for almost every \(z \in \mathbb{R}^2 \simeq \mathb... | Abdelmalek Abdesselam | 7,410 | <p>Maybe: isotypical decomposition of the representation $L^2(\mathbb{R}^2,\mathbb{C})$ of the group $U(1)$ might do.</p>
|
1,712,457 | <blockquote>
<p>Assume $f$ is differentiable over an open interval $I$. Suppose $a<b$ are two numbers in $I$ with $f'(a) < f'(b)$. Show that if $f'(a) < 0 <f'(b)$, then neither $f(a)$ nor $f(b)$ can be the minimum value of $f$ over $[a,b]$.</p>
</blockquote>
<p>Intuitively this makes sense: $f$ must ch... | bgins | 20,321 | <p>First, by considering $f(-x)$ or $f(a+b-x)$, it's enough to show that $f(a)$ cannot be the minimum on $[a,b]$. Now the derivative of $f$ at $a$ is equal to the right derivative there,
$$0>f'(a)=f'(a^+)=\lim_{h\to0^+}\frac{f(a+h)-f(a)}h.$$
But this means that for all $h>0$ sufficiently small,
the numerator must... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.