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 |
|---|---|---|---|---|
173,387 | <p>How can I indent properly long code in <em>Mathematica</em>?
Are there some best practices?</p>
| b3m2a1 | 38,205 | <p>Here's a quick plug for some stuff in that question I linked. I have it all built into my GitHub so you can load a thing to let your cells be indentable by:</p>
<pre><code>loadIndenter[] :=
(
BeginPackage["Indenter`"];
Indenter`MakeIndentable::usage = "Makes indentable";
BeginPackage["`Package`"];
Indenter... |
4,574,692 | <p>The theorem goes: Let <span class="math-container">$A_{1}, A_{2} ... \in \mathcal{A}$</span> with <span class="math-container">$A_{N}$</span> increasing to <span class="math-container">$\Omega$</span> and <span class="math-container">$\mu (A_{N}) < \infty$</span> for all <span class="math-container">$N \in \mathb... | Tom | 986,425 | <p>Thank @geetha290krm for your hint!</p>
<p>Following this hint, fix <span class="math-container">$m \in \mathbb{N}$</span>, we have that</p>
<p><span class="math-container">$\tilde{d}(f, f_{n}) \leq \frac{1}{2^{m}} + \sum_{N = 1}^{m}\frac{2^{-N}}{1 + \mu(A_{N})} \tilde{d}_{N}(f, f_{n})$</span>.</p>
<p>Then, if all <s... |
1,894,867 | <p>Let $n=3^{1000}+1$. Is n prime?</p>
<p>My working so far:</p>
<p>$n=3^{1000}+1 \cong 1 \mod 3$</p>
<p>I notice that n is of form; $n=3^n+1$</p>
<p>Seeking advice tips, and methods on progressing this.</p>
| Ennar | 122,131 | <p>Since we have</p>
<p>$$3\equiv 1 \pmod 2 \implies 3^{1000} \equiv 1 \pmod 2 \implies 3^{1000}+1\equiv 0\pmod 2$$</p>
<p>$3^{1000}+1$ is not a prime.</p>
|
1,102,928 | <p>Let $\mathcal{H}$ be a Hilbert space. I am trying to show that every self-adjoint idempotent continuous linear transformation is the orthogonal projection onto some closed subspace of $\mathcal{H}$. If $P$ is such an operator, the obvious thing is to consider $S=\{Px:x\in\mathcal{H}\}$. However, I'm having trouble s... | tomasz | 30,222 | <p><strong>Hint</strong>: use continuity of $P$. Then show what the kernel of $P$ is orthogonal to $S$.</p>
|
1,842,340 | <p>A polynomial with integer coefficients is called
primitive if its coefficients are relatively prime. For
example, $$3{x^2} + 7x + 9$$ is primitive while $$10{x^2} + 5x + 15$$
is not.</p>
<p>(a) Prove that the product of two primitive polynomials is primitive.</p>
<p>(b) Use this to prove Gauss's Lemma: If a polyno... | Benjamin Lindqvist | 96,816 | <p>This will not be possible because the new code would be MDS. It is known that such codes do not exist for $n>k+1$.</p>
|
2,964,359 | <blockquote>
<p>Let <span class="math-container">$(X, d)$</span> be a metric space with no isolated points, and let <span class="math-container">$A$</span>
be a relatively discrete subset of <span class="math-container">$X$</span>. Prove that <span class="math-container">$A$</span> is nowhere
dense in <span class... | José Carlos Santos | 446,262 | <p>Let <span class="math-container">$x\in\mathring{\overline A}$</span>. Then there is a <span class="math-container">$r>0$</span> such that <span class="math-container">$B_r(x)\subset\overline A$</span>. Since <span class="math-container">$B_r(x)$</span> is an open set which is contained in <span class="math-contai... |
2,946,384 | <p>How to prove that any integer n which is not divisible by 2 or 3 is not divisible by 6?</p>
<p>The point was to prove separately inverse, converse and contrapositive statements of the given statement: "for all integers n, if n is divisible by 6, then n is divisible by 3 and n is divisible by 2".
I have the proof f... | Bernard | 202,857 | <p>If you make the substitution <span class="math-container">$\;t=\mathrm e^x\iff x=\ln t$</span>, so that <span class="math-container">$\;\mathrm dx=\dfrac{\mathrm d t}t$</span>, we obtain
<span class="math-container">$$\int_{1}^{\infty}\frac{\mathrm e^{x}+\mathrm e^{3x}}{\mathrm e^{x}-\mathrm e^{5x}}\,\mathrm dx=\int... |
2,946,384 | <p>How to prove that any integer n which is not divisible by 2 or 3 is not divisible by 6?</p>
<p>The point was to prove separately inverse, converse and contrapositive statements of the given statement: "for all integers n, if n is divisible by 6, then n is divisible by 3 and n is divisible by 2".
I have the proof f... | Henry Lee | 541,220 | <p>I think it should be:
<span class="math-container">$$I=\int_1^\infty\frac{e^x+e^{3x}}{e^x-e^{5x}}dx$$</span>
<span class="math-container">$u=e^x$</span> so <span class="math-container">$dx=\frac{du}{u}$</span> so:
<span class="math-container">$$I=\int_e^\infty\frac{u+u^3}{u-u^5}\frac{1}{u}du=\int_1^\infty\frac{1+u^2... |
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | Billy | 39,970 | <p>$\mathbb{R}^2$ and $\mathbb{C}$ have the same cardinality, so there are (lots of) bijective maps from one to the other. In fact, there is one (or perhaps a few) that you might call "obvious" or "natural" bijections, e.g. $(a,b) \mapsto a+bi$. This is more than just a bijection:</p>
<ul>
<li>$\mathbb{R}^2$ and $\mat... |
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | James S. Cook | 36,530 | <p>My thought is this: $\mathbb{C}$ is not $\mathbb{R}^2$. However, $\mathbb{R}^2$ paired with the operation $(a,b) \star (c,d) = (ac-bd, ac+bd)$ provides a <strong>model</strong> of the complex numbers. However, there are others. For example, a colleague of mine insists that complex numbers are $2 \times 2$ matrices ... |
106,219 | <blockquote>
<p>Define a sequence of functions $f_n: (0,1)\rightarrow\mathbb{R}$ by<br>
$\
f(x) =
\begin{cases}
1/q^n
& \text{if } x =p/q \space(\space\mathrm{nonzero})\\
0 & \text{otherwise}
\end{cases}
$<br>
Find the pointwise limit $f$ of $\{f_n\}$ and show $\{f_n\}$ converges ... | VSJ | 21,330 | <p>For every irrational $x \in (0,1)$, $$\lim_{n \to \infty} f_n(x) = 0$$ is obvious. For a rational $x = \frac{p}{q}$, its again easy to see that $$\lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{p}{q^n} = 0$$<br>
Thus the pointwise limit is $0$ at all $x \in (0,1)$.<br>
To show uniform convergence, note that $... |
2,038,520 | <p>I know that the series b. converges as $\sum \frac{1}{n^p}$ converges for $p>1$, So a. also converges. I want to know the sum.</p>
<blockquote>
<blockquote>
<p>a.$1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+.....$</p>
<p>$b.1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+.....$</p>
</blockquote>
</blockquote... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>$$1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+\cdots=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{49}+\cdots-(\frac{1}{4}+\frac{1}{16}+\frac{1}{36}+\cdots)$$ Now $$\frac{1}{4}+\frac{1}{16}+\frac{1}{36}+\cdots=\frac{1}{4}(1+\frac{1}{4}+\frac{1}{9}+\cdots)$$</p>
|
189,069 | <p>The Survival Probability for a walker starting at the origin is defined as the probability that the walker stays positive through n steps. Thanks to the Sparre-Andersen Theorem I know this PDF is given by</p>
<pre><code>Plot[Binomial[2 n, n]*2^(-2 n), {n, 0, 100}]
</code></pre>
<p>However, I want to validate this ... | m_goldberg | 3,066 | <p>It seems to me that this is a problem to which <code>Catch</code> and <code>Throw</code> can be usefully applied.</p>
<pre><code>SeedRandom[1];
Module[{result = {0}, s},
Catch[
Fold[
If[#2 < 0, Throw[Null], result = {result, s = #1 + #2}; s] &,
0,
Accumulate[RandomVariate[NormalDistri... |
1,121,845 | <p>let $G$ be a multiplicative group of non-zero complex analysis.consider the group homomorphism $\phi:G\rightarrow G$ defined by $\phi(z)=z^4$.</p>
<p>1.Identify kernel of $\phi=H$.</p>
<p>2.Identify $G/H$</p>
<p>My try:</p>
<p>let $z\in \ker \phi$ then $\phi(z)=1\implies z^4=1$
let $z=re^{i\theta}\implies r^4... | Andreas Caranti | 58,401 | <p>You are right about the first point, just finish off by noting the the kernel has four elements, and try and list them (this will turn out to be easy).</p>
<p>As to the second point, remember the so-called fundamental theorem of algebra: in particular, for each complex number $a$, there is $z$ such that $z^{4} = a$... |
1,488,388 | <p><strong>The Statement of the Problem:</strong></p>
<p>Let $G$ be a finite abelian group. Let $w$ be the product of all the elements in $G$. Prove that $w^2 = 1$.</p>
<p><strong>Where I Am:</strong></p>
<p>Well, I know that the commutator subgroup of $G$, call it $G'$, is simply the identity element, i.e. $1$. But... | Jacob Maibach | 159,592 | <p>Consider the following "proof" that $w = 1$. See if you can patch it up to reach to conclusion that $w^{2} = 1$ instead.</p>
<p>We partition the non-identity elements of $G$ into two sets, which we call $S = \{g_{1}, g_{2}, \dots\}$ and $S' = \{g_{1}^{-1}, g_{2}^{-1}, \dots\}$.
We do this by iteratively building up... |
51,509 | <p>Here is a problem due to Feynman. If you take 1 divided by 243 you get 0.004115226337 .... It goes a little cockeyed after 559 when you're carrying out the decimal expansion, but it soon straightens itself out and repreats itself nicely. Now I want to see how many times it repeats itself. Does it do this indefinitel... | Bob Hanlon | 9,362 | <pre><code>NumberForm[N[1/243,135],DigitBlock->27]
</code></pre>
<blockquote>
<p>0.004115226337448559670781893 004115226337448559670781893 004115226337448559670781893 004115226337448559670781893 004115226337448559670781893 00</p>
</blockquote>
<p>let x = 0.004115226337448559670781893... then for it to repeat for... |
1,537,648 | <p>For example let's say we have a password combination of (a,b,c,d), if the password length was 1 then we'll have 4 possible passwords (a,b,c,d), now if the length was 2 then we'll have 20 possible passwords (a,b,...,dc,dd), I calculated this manually, I want the rule of calculating probability?</p>
| yoki | 28,262 | <p>Every addition of a character multiplies the number of possibilities by $4$. For instance, if you have $N$ possibilities of passwords, and I now add another letter, then for any seuqence $[xyz...w]$ you can now generate four times:
$$[xyz...wa], [xyz...wb], [xyz...wc], [xyz...wd]$$.</p>
<p>So, for a password of len... |
3,042,149 | <p>We can't exactly draw a line of length square root of 2 but in an isosceles right angle triangle of sides 1 unit each, the length of hypotenuse will be the square root of 2. Now does it mean we can get the line of exact such length?</p>
<p>How is it possible? How can we get a line of exact length square root of 2 w... | Keith Backman | 29,783 | <p>The apparent paradox results from the difference between the ideal triangle you can construct in your mind, with a hypotenuse of <span class="math-container">$\sqrt{2}$</span>, and an actual figure that you can draw, where making a two legs of <em>exactly</em> unit length, meeting at a <em>perfectly</em> right angle... |
1,115,645 | <p>I understand that a primitive polynomial is a polynomial that generates all elements of an extension field from a base field. However I am not sure how to apply this definition to answer my question. Can someone explain to me how I need to start please?</p>
| user208259 | 208,259 | <p>The polyomial is irreducible for the reasons given in Will Brooks' answer. $F_{49}^{*}$ is cyclic of order 48. You want to check that a root $\alpha$ of the polynomial must have order 48. Since every element of $F_{49}^{*}$ has order dividing $48$, it's enough to check that $\alpha^{16} \ne 1$ and $\alpha^{24} \ne 1... |
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | Dr. Sonnhard Graubner | 175,066 | <p>it is $$44<\sqrt{2017}<45$$ since $$44^2=1936$$ and $$45^2=2025$$</p>
|
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | Simon Fraser | 717,270 | <p><span class="math-container">$\sqrt{1600} = 40$</span> and <span class="math-container">$\sqrt{2500} = 50$</span>. <span class="math-container">$(40+4)^2 = 40^2 + 8\cdot 40 + 16 = 1936$</span> and <span class="math-container">$(40+5)^2 = 40^2 + 10\cdot 40 +25 = 2025$</span>. Hence the desired answer is <span class="... |
2,788,498 | <p>Suppose $T([a,-b])=[−x,y]$ and $T([a,b])=[x,y]$. Find a matrix $A$ such that $T(x)=Ax$ for all $x\in\mathbb{R}^2$.</p>
| MR ASSASSINS117 | 546,265 | <p>$$y''+y=4xe^x$$</p>
<p>the characteristic equation is $m^2+1=0$ with solutions $m_{12}=\pm\ i$</p>
<p>$$\mathbf{y_{h}(x)=C_1 \cos(x)+C_2\sin(x)}$$</p>
<p>from here you can use <em>Undetermined Coefficients</em> or <em>Variation of parameters</em>
\begin{align}
y_p(x)&=(Ax+B)e^x
\\~
\\
y'_p(x)&=Ae^x+Axe^x... |
3,575,334 | <p>I am trying to show that <span class="math-container">$\int_{-b}^{b} \frac{f(N+\frac{1}{2} + it)}{e^{2\pi i(N+\frac{1}{2} + it)}-1} dt \to 0$</span> as <span class="math-container">$N \to \infty$</span> where <span class="math-container">$|f(N+1/2+it)| \le A/(1+(N+1/2)^2)$</span> for some constant <span class="math-... | fleablood | 280,126 | <p><span class="math-container">$\color{magenta}{(A\setminus C)}\cap \color{green}{(C\setminus B)}$</span></p>
<p><a href="https://i.stack.imgur.com/tzQ3j.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tzQ3j.jpg" alt="enter image description here"></a></p>
<p>If <em>roundsquare</em> wants to steal... |
2,674,102 | <p>Is the following Proof Correct?</p>
<blockquote>
<p>Given that $T\in\mathcal{L}(\mathbf{R}^2)$ defined by $T(x,y) =
(-3y,x)$. $T$ has no eigenvalues.</p>
</blockquote>
<p><em>Proof.</em> Let $\sigma_T$ denote the set of all eigenvalues of $T$ and assume that $\sigma_T\neq\varnothing$ then for some $\lambda\in\s... | Jesse Meng | 536,610 | <p>Proof looks correct to me. Though you could have also just added the case for $\lambda$ is zero since otherwise to multiply both sides by $0$ would reduce the solution set for $x$ and $y$.</p>
<p>$Edit:$ The case is not necessary as you are actually using substitution instead of multiplying both sides by $\lambda$.... |
1,278,719 | <p>This is a problem from Artin's book "Algebra". In the fifth miscellaneous problem of the chapter "Vector spaces", he has asked to prove that:</p>
<p>If $\alpha$ is a cube root of $2$, then the real numbers $a+b\alpha +c\alpha ^2$ with $a,b,c \in \mathbb{Q}$ form a field.</p>
<p>I am stuck at proving this. For exam... | Sam Christopher | 239,327 | <p>Try Leibniz theorem for alternating series</p>
<p>$\sum_{n=0}^\infty$ $(-1)^{n}u_{n}$ converges when</p>
<p>i) $lim_{n \to \infty}{u_n} =0$ and</p>
<p>ii){$u_n$} is monotonically decreasing sequence. </p>
|
4,572,505 | <p>There are many approximations of <span class="math-container">$\pi$</span> using trigonometric and rational numbers. But I created this one: <span class="math-container">$$\pi \approx \sqrt[11]{294204}$$</span> Which is correct to almost <span class="math-container">$8$</span> decimal places. Are there any other app... | Claude Leibovici | 82,404 | <p>Playing your game, let me define
<span class="math-container">$$A_n=\text{Round}\left[\pi ^{p_n}\right]$$</span> and compute
<span class="math-container">$$\Delta_n=\log_{10}\Bigg[\left| A^{\frac{1}{p_n}}-\pi \right| \Bigg]$$</span> where <span class="math-container">$p_n$</span> is the <span class="math-container... |
4,572,505 | <p>There are many approximations of <span class="math-container">$\pi$</span> using trigonometric and rational numbers. But I created this one: <span class="math-container">$$\pi \approx \sqrt[11]{294204}$$</span> Which is correct to almost <span class="math-container">$8$</span> decimal places. Are there any other app... | Piquito | 219,998 | <p>COMMENT: By way of simple pertinent information.</p>
<p>There are many remarkable approximations of <span class="math-container">$\pi$</span>. For example
<span class="math-container">$$\pi\approx\dfrac{22}{17}+\dfrac{37}{47}+\dfrac{88}{43}\\\pi\approx\sqrt[4]{\frac{2143}{22}}\\\pi\approx\sqrt[9]{\frac{3404135027487... |
2,602,799 | <p>This is exactly what is written in Walter Rudin chapter 2, Theorem 2.41:</p>
<p>If $E$ is not closed, then there is a point $\mathbf{x}_o \in \mathbb{R}^k$ which is a limit point of $E$ but not a point of $E$. For $n = 1,2,3, \dots $ there are points $\mathbf{x}_n \in E$ such that $|\mathbf{x}_n-\mathbf{x}_o| < ... | Sahiba Arora | 266,110 | <p>Suppose $S$ is finite, say $S=\{y_1,\cdots,y_k\}\subseteq\{x_n:n \in \mathbb N\}\subseteq E.$ Then the possible values each $|x_n-x_0|$ can take are $$|y_1-x_0|,\cdots,|y_k-x_0|.$$</p>
<p>So there will exists $i \in \{1,\cdots,k\}$ such that $|x_n-x_0|=|y_i-x_0|$ for infinitely many $n.$ So we have $$|y_i-x_0|\leq ... |
1,679,920 | <p>I'm working for a firm, who can only use straight lines and (parts of) circles.</p>
<p>Now I would like to do the following: imagine a square of size $5\times5$. I would like to expand it with $2$ in the $x$-direction and $1$ in the $y$-direction. The expected result is a rectangle of size $7\times9$. Until here, e... | Jean Marie | 305,862 | <p>Being given any prescribed ellipse curve, it is possible to find a parametric family of circles having this ellipse as its <em>envelope</em> (see figure 2 below). The more circles you take, the more precise you are.</p>
<p>How are these circles obtained? As an oblique projection of level sets of an ellipsoid (Figure... |
1,679,920 | <p>I'm working for a firm, who can only use straight lines and (parts of) circles.</p>
<p>Now I would like to do the following: imagine a square of size $5\times5$. I would like to expand it with $2$ in the $x$-direction and $1$ in the $y$-direction. The expected result is a rectangle of size $7\times9$. Until here, e... | bubba | 31,744 | <p>If you want to control the error in the approximation, then biarc interpolation/approximation is what you need, as indicated in the answer from @Paul H.</p>
<p>A biarc curve is usually constructed from two points and two tangent vectors. This actually leaves one degree of freedom unfixed, and there are several diff... |
242,636 | <p>I am interested in the proof of the following result:
Suppose that $A > 1$, $\lambda \in \mathbb{R}$, and for $0 < Z \leq 1$, let $U(Z)$ be the number of integer solutions $v$ of
\begin{eqnarray}
|v| < ZA \ \ \ \text{ and } \ \ \ \| \lambda v \| < Z A^{-1}.
\end{eqnarray}
Then, if $0 < Z_1 < Z... | Anton | 22,733 | <p>If you don't mind, I'll reformulate your problem slightly. Let $X = ZA$, $B = A^2$. Then $XB^{-1} = ZA^{-1}$. We would like to know the number of integer solutions $U'(X)$ to the system of inequalities</p>
<p>$$
\begin{cases}|v| < X;\\\|\lambda v\|<XB^{-1}.\end{cases}
$$</p>
<p>Let $\lfloor x \rfloor$ denote... |
1,713,778 | <p>Let $P=\{p_1,p_2,\ldots ,p_n\}$ the set of the first $n$ prime numbers and let $S\subseteq P$. Let
$$A=\prod_{p\in S}p$$ and $$B=\prod_{p\in P-S}p.$$ Show that if
$A+B<p_{n+1}^2$, then the number $A+B$ is prime.
Also, if
$$1<|A-B|<p_{n+1}^2,$$ then the number $|A-B|$ is prime.</p>
| S.C.B. | 310,930 | <p><strong>HINT</strong></p>
<p>For any geeral $n$, $n<(p_{k+1})^2$ where $p_{k+1}$ is the $k+1$th prime than we just need to check if $n$ is divisble by $p_i$ where $1 \le i \le n$.</p>
<p>But note that for any $p_i$ $$p_i \in S, P-S$$ is a contradiction. Thus, since $p_i$ is a prime it can only divide one of $A$... |
3,183,274 | <p>This is a reinterpretation of my old question <a href="https://math.stackexchange.com/questions/3177594/fit-data-to-function-gt-frac1001-alpha-e-beta-t-by-using-least-s">Fit data to function $g(t) = \frac{100}{1+\alpha e^{-\beta t}}$ by using least squares method (projection/orthogonal families of polynomials)</a>. ... | Yuri Negometyanov | 297,350 | <p><span class="math-container">$\color{brown}{\textbf{Via linear model}}$</span></p>
<p>Let
<span class="math-container">$$h(t) = \ln\left(\dfrac{100}{g(t)}-1\right),\tag1$$</span>
then the data table is
<span class="math-container">\begin{vmatrix}
i & 1 & 2 & 3 & 4 & 5 & 6 & 7\\
t... |
194,134 | <p>For some FittedModel, the "BestFitParameters" are given in terms of the symbols used to define the model. </p>
<pre><code>fit = NonlinearModelFit[{10,11,12},a*x+c,{a,c},x];
fit["BestFitParameters"]
</code></pre>
<p>returns <code>{a->1.,c->9.}</code></p>
<p>This can be problematic if I define <code>a</code> ... | N.J.Evans | 11,777 | <p>One straightfoward way to handle this is to accept the unique keys generated inside the module and write a function that replaces these with de-unique-ified strings when the best fit parameters are needed:</p>
<pre><code>getBestFit[fit_FittedModel] := Module[
{a, c, bf, newKeys, x, oldkeys},
bf = fit["BestFit... |
1,855,824 | <blockquote>
<p>Given $a_1=1$ and $a_n=a_{n-1}+4$ where $n\geq2$ calculate,
$$\lim_{n\to \infty }\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\cdots+\frac{1}{a_na_{n-1}}$$</p>
</blockquote>
<p>First I calculated few terms $a_1=1$, $a_2=5$, $a_3=9,a_4=13$ etc. So
$$\lim_{n\to \infty }\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\cdots+... | Behrouz Maleki | 343,616 | <p>As lab bhattacharjee mentioned for every $n\in\mathbb{N}$, we have
$$a_n-a_{n-1}=4$$
$$\begin{align}
& {{I}_{n}}=\sum\limits_{i=1}^{n-1}{\frac{1}{{{a}_{i}}{{a}_{i+1}}}}=\sum\limits_{i=1}^{n-1}{\frac{1}{{{a}_{i+1}}-{{a}_{i}}}\left( \frac{1}{{{a}_{i}}}-\frac{1}{{{a}_{i+1}}} \right)}=\frac{1}{4}\sum\limits_{i=1}... |
156,179 | <p>Let $A$ be a closed subset of $\mathbb{R}^{n}$. Can the quotient space $\mathbb{R}^{n}/A$ be embedded in some Euclidean space $\mathbb R^{m}$? In particular, assume that $A$ is an algebraic variety of degree $k$, can we control $m$ in term of $n$ and $k$? </p>
| Daniele Zuddas | 23,193 | <p>For $A$ compact the answer is that given by Joseph Van Name. If $A$ is not compact then the answer is negative. For example take the standard $\Bbb R \subset \Bbb R^2$. Then the quotient is not II-countable (the same holds for any unbounded closed subset of $\Bbb R^n$, with unbounded complement).</p>
<p>In the comp... |
3,789,494 | <p>I'm stuck in solving this strange and beautiful formula : <span class="math-container">$3= 3^{z}$</span> since it says 'Solve' and not 'Prove'</p>
<p>Also i really don't understand what does it means by saying <span class="math-container">$3^{z}$</span>? Will <span class="math-container">$3^z$</span> form a set ?</p... | Muhammad | 804,057 | <p>Since <span class="math-container">$z$</span> is a complex number then <span class="math-container">$z=x+iy$</span> where <span class="math-container">$x,y \in \mathbb{R}$</span></p>
<p>Also Since <span class="math-container">$0,1 \in \mathbb{R}$</span> then <span class="math-container">$z=x+iy=(1)+i(0)=1$</span></p... |
106,031 | <p>I need to "monochromize" a large amount of plots (mostly coming from <code>ListPlot</code>) and export them to PDF. The problem is that I no longer have the data used to generate the plots, I only have notebooks that contain the plots. I attempted copy-pasting one plot and then something along the lines of <code>Sho... | Aisamu | 8,238 | <p>In the same spirit of <a href="https://mathematica.stackexchange.com/users/57/sjoerd-c-de-vries">Sjoerd's</a> answer, you can "steal" the theme-generated directives and replace them in the target plot.</p>
<p>With a dummy plot created using the desired theme,</p>
<pre><code>dummyPlot = Plot[{1, 2, 3}, {x, 0, 1}, P... |
19,598 | <p>I have two independent ODE systems. </p>
<pre><code>A = NDsolve[..., {x, y}, {t, 0, 10}];
B = NDsolve[..., {a, b}, {t, 0, 10}];
</code></pre>
<p>I can draw a <code>ParametricPlot</code> from one ODE. That is, </p>
<pre><code>ParametricPlot[Evaluate[{x[t], y[t]} /. A], {t, 0, 10}]
</code></pre>
<p>I wonder if I c... | 2island | 6,590 | <p>Simply you can use the following command:</p>
<pre><code>sol = NDSolve[{x'[t] == Sin[t], a'[t] == Cos[t], a[0] == 1,
x[0] == 1}, {x, a}, {t, 0, 10}];
ParametricPlot[{x[t], a[t]} /. sol, {t, 0, 10},
AxesLabel -> {x[t], a[t]}]
</code></pre>
<p>Firstly, you may consider <em>one</em> system so as to use the... |
322,140 | <p>$$\int \left ( r\sqrt{R^2-r^2} \right )dr$$</p>
<p>It looks simple. I know that the derivative of </p>
<p>$$\left (R^2-r^2 \right )^\frac{3}{2}$$</p>
<p>Is the stuff in the integral.</p>
<p>However, what about if I don't know?</p>
<p>How in general do we solve integral of</p>
<p>$$G(r)^n$$</p>
| Bombyx mori | 32,240 | <p>This kind of integral is usually dealt with standard trignometry substitutions. You can use $r=R\cos[\theta]$, $\sqrt{R^{2}-r^{2}}=R\sin[\theta]$, $dr=-R\sin[\theta]d\theta$, for example. I am sure you can find some other ingenius ways to do the job as well. </p>
<p>The main difficulty for this approach is to integ... |
1,399,935 | <p>I'm reading Kleene's introduction to logic and in the beginning he mentions something that I have thought about for a while. The question is how can we treat logic mathematically without using logic in the treatment? He mentions that in order to deal with this what we do is that we separate the logic we are studying... | Andreas Blass | 48,510 | <p>We use logic to <em>study</em> logic, not to <em>create</em> logic. Our study is usually not intended to justify some logic but rather to understand how it works. For example, we might try to prove that, whenever a conclusion $c$ follows from an infinite set $H$ of hypotheses then $c$ already follows from a finite ... |
2,728,248 | <blockquote>
<p>Let $K=\mathbb{Q}(\sqrt{-2})$. Show that $\mathcal{O}(K)$ is a principal ideal domain. Deduce that every prime $p\equiv 1, 3$ (mod 8) can be written as $p = x^2 + 2y^2$ with $x, y \in \mathbb{Z}$.</p>
</blockquote>
<p>As $−2$ is squarefree $6\equiv 1$ (mod 4) we have $\mathcal{O}(K) = \mathbb{Z}[
\s... | Jack D'Aurizio | 44,121 | <p>An elementary proof starting from scratch. If $p\equiv 1\pmod{8}$, in $\mathbb{F}_p^*$ there is an element with order $8$, which we may call $\alpha$. Since $\alpha^4+1=0$ we have $(\alpha+\alpha^{-1})^2=0 $ and $-2$ is a quadratic residue $\pmod{p}$. If $p\equiv 3\pmod{8}$ we may consider the degree of the splittin... |
2,065,254 | <p>Let $f: \mathbb{R} \to \mathbb{R}$ be a function that is twice differentiable.</p>
<p>We know that:
$$\lim_{x\to-\infty}\ f(x) = 1$$</p>
<p>$$\lim_{x\to\infty}\ f(x) = 0$$</p>
<p>$$f(0) = \pi$$</p>
<p>We have to prove that there exist at least two points of the function in which $f''(x) = 0$.</p>
<p>How could w... | Yiorgos S. Smyrlis | 57,021 | <p>Since the limits of $f$, as $x$ tends to $\infty$ and $-\infty$ both exist, and
$$
\lim_{x\to\infty}f(x),\,\,\lim_{x\to-\infty}f(x)<f(0)=\pi,
$$
then $f$ attains a total maximum, say at $x_0$, with $f(x_0)\ge\pi$, and thus $f'(x_0)=0$. </p>
<p>The fact that $\,\lim_{x\to-\infty}f(x)<f(0)$, implies the existe... |
3,043,296 | <p>Prop: For sets A and B, say A ~ B iff there exists a bijection from A to B. Then ~ is an equivalence relation on sets.</p>
<p>I understand that an equivalence relation holds the properties of reflexive, symmetric, and transitive. I am also aware of their definitions, however, I am struggling to write a proof for th... | Sambo | 454,855 | <p>Showing that these properties hold is a straightforward application of the definitions, with some elementary properties of bijections. So, I suspect what you are having trouble with is formulating a proper proof. I will show reflexivity as an example.</p>
<p>We want to show that for any set <span class="math-contai... |
4,498,199 | <p>Exercise 1.2.1(vii) from Page 5 of Keith Devlin's "The Joy of Sets":</p>
<blockquote>
<p>Prove the following assertion directly from the definitions. The
drawing of "Venn diagrams" is forbidden; this is an exercise in the
manipulation of logical formalisms.
<span class="math-container">$$(x\subse... | ryang | 21,813 | <blockquote>
<p>Exercise 1.2.1(vii) from <strong>page 5</strong> of Keith Devlin's "The Joy of Sets":</p>
<blockquote>
<p>Prove the following assertion <strong>directly from the definitions</strong>. this is an exercise in the <strong>manipulation of logical formalisms</strong>.
<span class="math-container">... |
1,837,220 | <p>In this post:
<a href="https://math.stackexchange.com/questions/1056058/computing-int-sqrt14x2-dx">Computing $\int \sqrt{1+4x^2} \, dx$</a>
someone mentioned Euler substitution to compute the following integral:</p>
<p>$$\int \sqrt{1+4x^2} \, dx$$</p>
<p>I tried to follow this advice and got very nice result, name... | egreg | 62,967 | <p>If you set $\sqrt{1+4x^2}=t-2x$, you have
$$
1+4x^2=t^2-4tx+4x^2
$$
so $4tx=t^2-1$ and therefore
$$
x=\frac{t^2-1}{4t}=\frac{t}{4}-\frac{1}{4t}
$$
Thus
$$
dx=\left(\frac{1}{4}+\frac{1}{4t^2}\right)\,dt=\frac{t^2+1}{4t^2}\,dt
$$
and
$$
\sqrt{1+4x^2}=t-\frac{t}{2}+\frac{1}{2t}=\frac{t^2+1}{2t}
$$
so the integral becom... |
1,808,441 | <p>I want to know for two circles why the ratio of arc length is equal to the ratio of the two central angles in geometry. It must have something to do with the concept of similarity in geometry. I have scoured the Internet looking answers but only found the ratio of circumferences to the ratio of their diameter.</p>
... | Xaver | 302,955 | <p>The arc length of a circle with radius $r$ is $2r\pi$. The full circle has a central angel of $\varphi=2\pi$. Therefore, given a central angel $\varphi$, you can calculate the arc length by $r\varphi$. So the following statements hold:</p>
<ul>
<li>For a fixed central angel $\varphi$, the arc length $r\varphi$ is p... |
1,528,235 | <p>Recall that <a href="http://en.wikipedia.org/wiki/Tetration" rel="noreferrer">tetration</a> ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$. </p>
<p>Its inverse function with respect to $x$ is called <a href="http://en.wikipedia.org/wiki/Tetration#Super-root" rel="noreferrer">s... | mick | 39,261 | <p>I got a message from Tommy1729.</p>
<p>He considered nonzero $T$ such that :</p>
<p>$$T = \lim \frac{A^n}{(f(n,2) - \sqrt 2 - L \ln 2 ^n)}$$.</p>
<p>Where $f(n,2)$ is the nth superroot of $2$ , $L$ is the constant from the Op and $A$ is a constant.</p>
<p>If the limit $T$ does not exist at least the best fitting... |
4,587,657 | <p>The call function is defined as</p>
<p><span class="math-container">$$
\text{call}: \begin{cases}
(\mathbb{R}^{I}\times I) \to \mathbb{R} \\
(f,x) \mapsto f(x)
\end{cases}
$$</span></p>
<p>is "<span class="math-container">$\text{call}$</span>" a measurable function? In other words: for a random fie... | Felix B. | 445,105 | <p>EDIT: previously this presented a dead end, but with @FShrike's answer this does not make any sense anymore. Instead I want to elaborate on @FShrike's answer. Partly to explain it to myself.</p>
<h1>The Continuity Approach</h1>
<p>The main idea is: A common <strong>sufficient</strong> criterion for measurability wi... |
1,795,836 | <p>Let's say that $A \subset X$ is a deformation retract. It follows that $A$ is both a retract and a space homotopically equivalent to $X$. Is the converse true? Probably not, but I couldn't find any example yet.</p>
<p>More specifically the converse would be:</p>
<p>If $A \subset X$ is a retract which is homotopic ... | Faraad Armwood | 317,914 | <p>If your question is whether every space $Y$ which is homotopically equivalent to a space $X$ must be a retract of $X$ then this is certainly not true. One main requirement for a deformation retract is that $A \subset X$. One can easily produce homeomorphic spaces which aren't subspaces of each other. Consider two di... |
1,838,002 | <p>There is famous <a href="https://en.wikipedia.org/wiki/Quillen-Suslin_theorem" rel="nofollow">Quillen-Suslin theorem</a> which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free.</p>
<p>I have never carefully read a proof of this theorem... | Joel92 | 349,727 | <p>It should be true for any PID. See the book by Lam, Serre's Conjecture</p>
|
2,259,109 | <p>If the value of $f(z_0)$ or $f^\prime(z_0)$ is complex number then is $f(z)$ analytic at $z_0$?</p>
| Lutz Lehmann | 115,115 | <p>No. But if $f(z)$ is differentiable as a 2D real function and its derivative $f'(z)$ is expressible as a complex number, and that is true for all points in a neighborhood of $z_0$, then $f$ is holomorphic or analytical in that open set and thus in $z_0$.</p>
|
1,705,081 | <p>It's a matrix solved with least squares equations (probaly). I used some calculator but can't get his outcome. If you have a way how to get to this please explain how.</p>
<p>[The math on that image is:
$$A = \left[\matrix{4&3&1&0&1\cr 5&2&1&0&1\cr 4&2&1&1&1\cr 3&... | Alexander | 316,927 | <p>Your answer is correct because
$101^{16}$ is fixed in terms of growing.
$n^{100}$ is slower than $1.5^{n}$.
And $(n!)^{2}$ is significantly large.</p>
|
2,328,505 | <p>Let $X$ be an exponential random variable with $\lambda =5$ and $Y$ a uniformly distributed random variable on $(-3,X)$. Find $\mathbb E(Y)$.</p>
<p>My attempt:</p>
<p>$$\mathbb E(Y)= \mathbb E(\mathbb E(Y|X))$$ </p>
<p>$$\mathbb E(Y|X) = \int^{x}_{-3} y \frac{1}{x+3} dy = \frac{x^2+9}{2(x+3)}$$</p>
<p>$$ \mathb... | Yining Wang | 158,147 | <p>You made a mistake in your calculation of $\mathbb E[Y|X]$. The correct calculation is
$$
E[Y|X=x] = \frac{1}{x+3}\int_{-3}^x{y dy} = \frac{1}{x+1} \left(\frac{1}{2}y^2\right)\bigg|_{-3}^x = \frac{x^2-9}{2(x+3)} = \frac{x-3}{2}.
$$</p>
<p>We then have that
$$
\mathbb E[Y] = \int_0^{\infty}\frac{x-3}{2}\cdot 5e^{-5x... |
226,323 | <p>Let $X$ and $Y$ be complex Banach spaces and $B(X,Y)$ be the Banach space
of all bounded operators. An operator $T\in B(X,Y)$ is weakly compact if
$T(\{ x\in X;\; \| x\| \leq 1\})$ is relatively compact in the weak topology
of $Y$. If $X$ or $Y$ is reflexive, then every operator in $B(X,Y)$ is weakly
compact. I gues... | M.González | 39,421 | <p>The fact that each $T\in B(X,Y)$ is weakly compact does not imply $X$ or $Y$ reflexive. For example, every non-weakly compact operator $T:\ell_\infty\to Y$ is an isomorphism on a subspace isomorphic to $\ell_\infty$ (See Prop. 2.f.4 in Classical Banach spaces I, by Lindenstrauss ans Tzafriri). </p>
<p>Thus if $Y$ i... |
1,439,429 | <p>Is it possible to calculate the Sagitta, knowing the Segment Area and Radius?
Alternatively, is there a way to calculate the Chord Length, knowing the Segment Area and Radius?</p>
| Rajat | 177,357 | <ol>
<li><p>Non-Convex</p></li>
<li><p>Gradient descent is an unconstrained optimization method, but it's success depends on various conditions over the functions, like differentiability of the function, Lipschitz condition etc. If a function is not convex, then you can not guarantee about the global optima.</p></li>
<... |
1,651,991 | <p>Let $p(x)$ be an odd degree polynomial and let $q(x)=(p(x))^2+ 2p(x)-2$ </p>
<p>a) The equation $q(x)=p(x)$ admits atleast two distinct real solutions.</p>
<p>b) The equation $q(x)=0$ admits atleast two distinct real solutions.</p>
<p>c) The equation $p(x)q(x)=4$ admits atleast two distinct real solutions.</p>
<... | Claude Leibovici | 82,404 | <p>The equation of a straight line is $$y=a + b x$$ So, just apply to each point</p>
<p>$$-0.5=a-2b$$ $$0.25=a+0.5b$$ Solve for $a$ and $b$ and apply $$?=a - b$$</p>
<p>I am sure that you can take from here.</p>
|
3,328,737 | <p>For any rational number, <span class="math-container">$\frac{p}{q}$</span> , <span class="math-container">$p$</span> and <span class="math-container">$q$</span> should be integers, <span class="math-container">$q\neq0$</span> and <span class="math-container">$p,q$</span> should not have any common factors.
Now, if w... | Mark Bennet | 2,906 | <p>The fact is that if you have <span class="math-container">$p$</span> and <span class="math-container">$q\neq 0$</span> integers then <span class="math-container">$|p|$</span> and <span class="math-container">$|q|$</span> are positive integers, or <span class="math-container">$p=0$</span> when <span class="math-conta... |
439,941 | <p>I ran into this question and I am finding it very difficult to solve:</p>
<blockquote>
<p>How many different expressions can you get by inserting parentheses into:
$$x_{1}-x_{2}-\cdots-x_{n}\quad ?$$</p>
</blockquote>
<p>For example:</p>
<p>$$\begin{align*}
x_{1}-(x_{2}-x_{3}) &= x_{1}-x_{2}+x_{3}\\
(x_{1... | Ross Millikan | 1,827 | <p>The answer is $2^{n-2}$. $x_1$ must always be positive and $x_2$ must always be negative. Then you can pick the signs on all the rest any way you want, starting with $x_3$. For a string of length $n$, start with a string of length $n-1$ that has the signs the way you want up to there. If you want the sign before... |
2,281,894 | <blockquote>
<p>The Hardy space <span class="math-container">$H^2(\mathbb{D})$</span> is defined to be the space of all functions <span class="math-container">$f$</span> >holomorphic on the unit disk <span class="math-container">$\mathbb{D}$</span> with the norm <span class="math-container">$\lVert \cdot \rVert_H$</... | Teebro Prokash | 481,770 | <p>Try using this fact (called the <em>polarization identity</em>): </p>
<blockquote>
<p>A <em>Banach space</em> $\mathcal{B}$ with <em>norm</em> $\parallel \ .\parallel$ is a <em>Hilbert space</em> iff $$\forall f,g \in \mathcal{B}, \ \ \ \ \parallel f+g\ \parallel +\parallel f-g \ \parallel = 2\left(\parallel f \... |
2,325,436 | <p>I was reading <em>Introduction to quantum mechanics</em> by David J. Griffiths and came across following paragraph:</p>
<blockquote>
<p><span class="math-container">$3$</span>. The eigenvectors of a hermitian transformation span the space.</p>
<p>As we have seen, this is equivalent to the statement that any hermitia... | Robert Israel | 8,508 | <p>Physicists persist in writing things that do not make sense mathematically, but there is a mathematically rigorous version of it: the Spectral Theorem for densely defined self-adjoint operators on Hilbert space. Most functional analysis texts will cover it.</p>
|
4,479,972 | <p>The suspension <span class="math-container">$SX$</span> of a topological space <span class="math-container">$X$</span> is defined as follows:
<span class="math-container">$${\displaystyle S(X)=(X\times I)/\{(x_{1},0)\sim (x_{2},0){\mbox{ and }}(x_{1},1)\sim (x_{2},1){\mbox{ for all }}x_{1},x_{2}\in X\}}.$$</span></p... | GSofer | 509,052 | <p>The dimension of the suspension is equal to the dimensions of <span class="math-container">$\mathbb{R}P^2$</span> plus <span class="math-container">$1$</span>, so <span class="math-container">$3$</span>. This is usually the case with 'nice' spaces - taking the suspension increases the dimension by <span class="math-... |
4,479,972 | <p>The suspension <span class="math-container">$SX$</span> of a topological space <span class="math-container">$X$</span> is defined as follows:
<span class="math-container">$${\displaystyle S(X)=(X\times I)/\{(x_{1},0)\sim (x_{2},0){\mbox{ and }}(x_{1},1)\sim (x_{2},1){\mbox{ for all }}x_{1},x_{2}\in X\}}.$$</span></p... | Mihail | 201,204 | <p>It is not correct to talk about the dimension of a suspension because it is not a manifold in our example. See <a href="https://math.stackexchange.com/questions/784962/easier-proof-about-suspension-of-a-manifold">this</a></p>
|
2,038,323 | <p>I am a first year college student studying linear algebra. </p>
<p>I understand that all linear transformations can be represented by a matrix mapping, and more specifically, the matrix mapping can be constructed by taking the column vectors of the images of the standard basis vectors. However, if the transformatio... | q.Then | 222,237 | <p>By definition, </p>
<blockquote>
<p>Any linear transformation can be represented as a matrix...</p>
</blockquote>
<p>Any rotation can be represented as an orthogonal matrix in the general form of
$$\begin{bmatrix} cos\theta & -sin\theta \\ sin\theta &cos\theta \end{bmatrix}$$
And can be generalized to $... |
3,643 | <p>Is there a quick method to transpose uneven lists without conditionals?</p>
<p>With:</p>
<pre><code>Drop[Table[q, {10}], #] & /@ Range[10]
</code></pre>
<p>Thus the first list would have the first element of all the lists, the 2nd list would have all the 2nd elements of all the lists, etc. If there are no ele... | Sjoerd C. de Vries | 57 | <p>Just to show that there are always a zillion ways to do things in Mathematica, here is my version. Actually, I myself would have used <code>Flatten</code> and its mind-shattering second argument after having learned of its existence a couple of months ago. </p>
<p>Contrary to the <code>Flatten</code> method this on... |
3,987,718 | <p>Let <span class="math-container">$L \in \mathbb{R}$</span> and let <span class="math-container">$f$</span> be a function that is differentiable on a deleted neighborhood of <span class="math-container">$x_{0} \in \mathbb{R}$</span> such that <span class="math-container">$\lim_{x \to x_{0}}f'(x)=L$</span>.</p>
<p>Fin... | leoli1 | 649,658 | <p>Take <span class="math-container">$f:\Bbb R\to\Bbb R$</span>, with <span class="math-container">$$f(x)=\begin{cases}Lx~~~\text{ for }x<x_0\\Lx+1~\text{for }x\geq x_0\end{cases}$$</span></p>
|
23,502 | <p><em>Edit: I wrote the following question and then immediately realized an answer to it, and moonface gave the same answer in the comments. Namely, $\mathbb C(t)$, the field of rational functions of $\mathbb C$, gives a nice counterexample. Note that it is of dimension $2^{\mathbb N}$.</em></p>
<p>The following is... | Kevin McGerty | 1,878 | <p>This is standard stuff: for example, if $A$ is an associative algebra over a field $k$ and $M$ is a simple module over $A$ whose dimension as a $k$-vector space is smaller than the cardinality of $k$, then any element of $\text{End}_k(M)$ is algebraic over $k$ (one just needs to consider the $k$-dimension of $k(\alp... |
2,285,299 | <p>For $ c> b>a>0 $
Is this inequality true?
$$ c^2+ab> ac+bc $$</p>
<p>If yes can anybody please provide hint so I can solve it? </p>
| Dr. Sonnhard Graubner | 175,066 | <p>we have $$c^2-ac+ab-bc>0$$ and this is equivalent to
$$c(c-a)+b(a-c)>0$$ and this is equivalent to
$$(c-a)(c-b)>0$$
this is true, since we have $$c>b>a>0$$</p>
|
3,299,469 | <p>Consider the set <span class="math-container">$A=\{n\ a \}$</span> where <span class="math-container">$a>0$</span> is a constant and <span class="math-container">$n \in \mathbb{N}$</span></p>
<p><strong>How shall we write this set <span class="math-container">$A$</span> in set theory?</strong></p>
<p>If we writ... | Chinnapparaj R | 378,881 | <ul>
<li>If both <span class="math-container">$n$</span> and <span class="math-container">$a$</span> are fixed, then the set is , the singleton <span class="math-container">$\{na\}$</span></li>
<li>If <span class="math-container">$a$</span> is fixed and <span class="math-container">$n \in \Bbb N$</span> is a varying qu... |
4,049,293 | <p>I am learning about the cross entropy, defined by Wikipedia as
<span class="math-container">$$H(P,Q)=-\text{E}_P[\log Q]$$</span>
for distributions <span class="math-container">$P,Q$</span>.</p>
<p>I'm not happy with that notation, because it implies symmetry, <span class="math-container">$H(X,Y)$</span> is often us... | Somos | 438,089 | <p>The question asks how to prove a polynomial
<span class="math-container">$\,f(n)\,$</span> takes only integer values for any
integer <span class="math-container">$\,n\,.$</span> If the polynomial is of
degree <span class="math-container">$0$</span>, then it is a constant and if
that constant is an integer we are don... |
1,138,212 | <p>I am given $f(x) = 1 + x - \frac{sin(x)}{(x e^x)} $ and am asked to solve this for when x ≃ 0.</p>
<p>I'm doing the following steps but am getting stuck halfway through:</p>
<p>$$f(x) = 1 + x - \frac {x - \frac{x^3}{6} + \frac{x^5}{120}}{xe^x} $$</p>
<p>$$= 1 + x - \frac{e^{-x} (x - \frac{x^3}{6} + \frac{x^5}{120... | Emilio Novati | 187,568 | <p>Since
$$
\lim_{x\rightarrow 0}\dfrac{\sin x}{x}=1
$$
You have
$$
\lim_{x\rightarrow 0}\dfrac{\sin x}{xe^x}=1
$$
and
$$
\lim_{x\rightarrow 0}\left(1+x-\dfrac{\sin x}{xe^x}\right)=1+0-1=0
$$
so, for $x \simeq 0 $ you have $f(x)<\epsilon \qquad\forall \epsilon>0$ </p>
|
1,290,176 | <p>Can anybody help me with this limit?
I think the answer should be $0$ as $0$ to the power $1$ should be $0$ but it doesn't match with the book's answer.</p>
<p>$$ \lim_{x\to 0} |x|^{\lfloor\cos{x}\rfloor}$$</p>
| Wolfgang Brehm | 223,307 | <p>$$
\lim_{x\rightarrow0}|x|^{\lfloor\cos{x}\rfloor}\\
\lim_{x\rightarrow0}\lfloor\cos{x}\rfloor=0\\
y^0 = 1
$$
The floor function means this is a special case... for values of x>0 it will have the limit 1 but for x=1 it will have the value 0. This means that the limit is 1 but the value of the function at x=0 is 0 as... |
1,563,518 | <p>Give an example of a natural number $n > 1$ and a polynomial $f(x) ∈ \Bbb Z_n[x]$ of degree $> 0$ that is a unit in $\Bbb Z_n[x]$.</p>
<p>I am trying to understand how units work in polynomial rings. My book doesn't really define it and I need a bit of help with this.</p>
| Alekos Robotis | 252,284 | <p>Your answer is completely correct. More generally, the solution is $45+77k: k\in \mathbb{Z}$ as stated above, because
$$ 45+77k\equiv 45\mod 77.$$
If you're concerned about getting the negative answer first, it is simple to just add $77$ as you did to find the first positive value.</p>
|
1,563,518 | <p>Give an example of a natural number $n > 1$ and a polynomial $f(x) ∈ \Bbb Z_n[x]$ of degree $> 0$ that is a unit in $\Bbb Z_n[x]$.</p>
<p>I am trying to understand how units work in polynomial rings. My book doesn't really define it and I need a bit of help with this.</p>
| ale | 270,082 | <p>You must to find the inverse of 12 in $Z$/77$Z$ and for that you can use Euclid'S algorithm .That is a general way to find the solutions.</p>
|
194,671 | <p>I'm searching for two symbols - considering they exist - (1) unknown value; (2) unknown probability.</p>
<p><strong>Note</strong>: I thought that $x$ was used in a temporary context, whenever I see it, it remains unknown until an evaluation is made. I was thinking in a "unknown and impossible to be known" context. ... | kjetil b halvorsen | 32,967 | <p>This arises often in statistics, for example, the statistical programming language R has two special values: NaN (mentioned in another answer) and NA. This can be taken as
NaN Not a number, can be a result of 0/0 and other illegal operations
NA Not available, used to represent "Do logically have a value, but we d... |
2,818,427 | <p>Let $f \in \mathrm{End} (\mathbb{C^2})$ be defined by its image on the standard basis $(e_1,e_2)$: </p>
<p>$f(e_1)=e_1+e_2$</p>
<p>$f(e_2)=e_2-e_1$</p>
<p>I want to determine all eigenvalues of f and the bases of the associated eigenspaces.</p>
<p>First of all how does the transformation matrix of $f$ look like?... | Dylan | 135,643 | <p>The method is very simple. Start with the general form of a homogeneous, second-order ODE</p>
<p>$$ y'' + a(x)y' + b(x)y = 0 $$</p>
<p>You know the two solutions, so you can plug them into the equation to get</p>
<p>\begin{align} 2 + 2xa(x) + x^2b(x) &= 0 \\ e^{-x} - e^{-x}a(x) + e^{-x}b(x) &= 0 \end{alig... |
178,302 | <p>Assume that $H$ is a separable Hilbert space.
Is there a polynomial $p(z)\in \mathbb{C}[x]$ with $deg(p)>1$ with the following property?:</p>
<p>Every densely defined operator $A:D(A)\to D(A),\;D(A)\subset H$ with $p(A)=0$ is necessarily a bounded operator on $H$.</p>
<p>That is the polynomial-operator e... | user52733 | 56,229 | <p>I do not think so. </p>
<p><strong>Observation:</strong> Without loss of generality, $p(x)$ can be taken to be monic (constant multiples won't affect either $p(A) = 0$ or boundedness). </p>
<p><strong>Case 1:</strong> $p$ is degree $2$.</p>
<p>By the above reduction, $p(x) = (x - \lambda)(x - \mu)$ for some $\l... |
232,424 | <p>Are there any claims and counterclaims to mathematics being in some certain cases a result of common sense thinking? Or can some mathematical results be figured out using just pure common sense i.e. no mathematical methods? </p>
<p>I'd also appreciate any mentions relating to sciences, social sciences or ordinary l... | glebovg | 36,367 | <p>Many basic theorems can be proven using common sense, not to mention that almost all axioms in mathematics, except for axioms of set theory are based on common sense. According to <a href="http://mathworld.wolfram.com/Axiom.html" rel="nofollow">MathWorld</a>, an axiom is a proposition regarded as self-evidently true... |
38,193 | <p>For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:</p>
<p>ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not prove Con(ZFC), where Con(ZFC) is a formula that expresses that
ZFC is consistent.</p>
<p>(Here ZFC can be replac... | Kaveh | 7,507 | <p>The answer is the following observation due to Hilbert: </p>
<blockquote>
<p>If we can prove the consistency of $ZFC$ using <em>elementary</em> methods, then any <em>elementary theorem</em> of $ZFC$ has an <em>elementary proof</em>, i.e. we don't need <em>ideal/abstract objects</em> like sets or real number for d... |
38,193 | <p>For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:</p>
<p>ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not prove Con(ZFC), where Con(ZFC) is a formula that expresses that
ZFC is consistent.</p>
<p>(Here ZFC can be replac... | Carl Mummert | 5,442 | <p>The fact that the second incompleteness theorem refers to consistency is important for several applications, both philosophical and mathematical. </p>
<p>Philosophically, the second incompleteness theorem is what lets us know that we cannot, in general, prove the existence of a (set) model of ZFC within ZFC itself.... |
38,193 | <p>For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:</p>
<p>ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not prove Con(ZFC), where Con(ZFC) is a formula that expresses that
ZFC is consistent.</p>
<p>(Here ZFC can be replac... | user8248 | 8,248 | <p>John H Conway proves and discusses the incompleteness theorem is his badass wolf prize lectures: <a href="http://www.math.princeton.edu/facultypapers/Conway/" rel="nofollow">http://www.math.princeton.edu/facultypapers/Conway/</a>
Anyone who hasn't seen these talks is missing out. </p>
|
38,193 | <p>For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:</p>
<p>ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not prove Con(ZFC), where Con(ZFC) is a formula that expresses that
ZFC is consistent.</p>
<p>(Here ZFC can be replac... | nickname | 5,462 | <p>There is another nice consequence of the Goedel first incompleteness theorem. Indeed by proving that there exists an undecidable sentence, the theorem is offering a formal proof of the consistency of ZFC (if it were not consistent then it would prove whatever). The only problem is that it is doing so <em>inside</em... |
4,253,640 | <p>For example:
suppose we need to find <strong>x</strong> given that <strong>x mod 7 = 5</strong> and <strong>x mod 13 = 8</strong>.</p>
<p><strong>x = 47</strong> is a solution but needs hit and trial.</p>
<p>Is there any shortcut to calculate such number?</p>
| Roddy MacPhee | 903,195 | <p>There exist a lot of ways to do this :</p>
<ul>
<li>Sequentially try values of <span class="math-container">$x$</span> (basic, 47 guesses maximum)</li>
<li>Use the lowest congruence and work up in sequence( meh,13 guesses maximum )</li>
<li>Use the highest congruence and work down sequence (okay 7 guesses maximum)</... |
1,216,983 | <p>Let $f(x)$ be a polynomial with complex coefficients such that $\exists n_0 \in \mathbb Z^+$ such that $f(n) \in \mathbb Z , \forall n \ge n_0$, then is it true that $f(n) \in \mathbb Z , \forall n \in \mathbb Z$ ?</p>
| Clément Guérin | 224,918 | <p>Intersting question, but I think, only constant sequences verify this. take $(a_n)$ a sequence verifying a recurrence equation :</p>
<p>$$a_n=\sum_{k=2}^r\lambda_ka_{\lfloor \frac{n}{k} \rfloor}$$</p>
<p>Then, evaluating it for $n=0$ we get :</p>
<p>$$a_0=\sum_{k=2}^r\lambda_ka_0 $$</p>
<p>That is either $a_0=0$... |
3,325,658 | <blockquote>
<p>Count the number of 5 cards such that there's exactly 2 suits</p>
</blockquote>
<p>Suppose we draw five cards from a standard deck of 52 cards. I want to count the number of ways I can draw five cards such that the hand contains exactly 2 suits.</p>
<p>Here's my intuition:<br/>
There are two cases, ... | RobPratt | 683,666 | <p>Yes, this is correct without multiplying by 2. You can check by computing a different way. Choose two suits, choose all five cards from these two suits, and subtract the ways that yield only one suit:
<span class="math-container">$$\binom{4}{2}\left(\binom{26}{5}-\binom{2}{1}\binom{13}{5}\right)=379236$$</span></p... |
4,196,583 | <p>More precisely:</p>
<blockquote>
<p><strong>Definition.</strong><br />
A subset <span class="math-container">$S \subset \Bbb R$</span> is called <em>good</em> if the following hold:</p>
<ol>
<li>if <span class="math-container">$x, y \in S$</span>, then <span class="math-container">$x + y \in S,$</span> and</li>
<li>... | Sangchul Lee | 9,340 | <p>Write <span class="math-container">$S^{\times} = S\setminus\{0\}$</span>. Then we will prove the following claim:</p>
<blockquote>
<p><strong>Claim.</strong> A subset <span class="math-container">$S$</span> of <span class="math-container">$\mathbb{R}$</span> is good if and only if <span class="math-container">$S$</s... |
210,735 | <p>The Cantor set is closed, so its complement is open. So the complement can be written as a countable union of disjoint open intervals. Why can we not just enumerate all endpoints of the countably many intervals, and conclude the Cantor set is countable?</p>
| user642796 | 8,348 | <p>Because the Cantor set includes numbers which are not the endpoints of any intervals removed. For example, the number $\frac{1}{4}$ (0.02020202020... in ternary) belongs to the Cantor set, but is not an endpoint of any interval removed.</p>
|
59,954 | <p>I can rather easily imagine that some mathematician/logician had the idea to symbolize "it <strong>E</strong> xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the idea to symbolize "for <strong>A</strong> ll" by $\forall$ - a reversed A. Or vice versa. (Maybe it w... | Bill Dubuque | 242 | <p>See <a href="http://jeff560.tripod.com/set.html">Earliest Uses of Symbols of Set Theory and Logic</a> for this and much more.</p>
|
59,954 | <p>I can rather easily imagine that some mathematician/logician had the idea to symbolize "it <strong>E</strong> xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the idea to symbolize "for <strong>A</strong> ll" by $\forall$ - a reversed A. Or vice versa. (Maybe it w... | Doug Spoonwood | 11,300 | <p>I've misplaced my copy of it, but I recall S. C. Kleene in his Mathematical Logic noting that "v" came as an abbreviation of "vel". In Latin "vel" is one of the words which commonly gets translated to the English word "or", and at least people believed that the Latin word "vel" comes closer to alternation (or equi... |
59,954 | <p>I can rather easily imagine that some mathematician/logician had the idea to symbolize "it <strong>E</strong> xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the idea to symbolize "for <strong>A</strong> ll" by $\forall$ - a reversed A. Or vice versa. (Maybe it w... | robjohn | 13,854 | <p>My understanding of the quantifier symbols $\bigvee$ ("there exists") and $\bigwedge$ ("for all") was that they were supposed to be large versions of $\vee$ ("or") and $\wedge$ ("and"). Then $\bigvee_{x\in X}Fx$ would mean $Fx_1\vee Fx_2\vee Fx_3\vee\dots$ whereas $\bigwedge_{x\in X}Fx$ would mean $Fx_1\wedge Fx_2\w... |
59,954 | <p>I can rather easily imagine that some mathematician/logician had the idea to symbolize "it <strong>E</strong> xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the idea to symbolize "for <strong>A</strong> ll" by $\forall$ - a reversed A. Or vice versa. (Maybe it w... | Ricardo | 145,198 | <p>That's a nice question, but you misunderstood the creation of the universal quantifier "all" and "there exists". It appears to be derived from the letter A, and I guess it is, but it didn't emerge that way. The first one to introduce quantifiers the way we know today was Gottlob Frege, who was a german mathematician... |
2,089,502 | <blockquote>
<p>How many numbers are there from $1$ to $1400$ which maintain these conditions:
when divided by $5$ the remainder is $3$ and when divided by $7$ the remainder is $2$?</p>
</blockquote>
<p>How can I start? I am newbie in modular arithmetics. I can just figure out that the number $= 5k_1+3 = 7k_2+2$. ... | Max | 130,322 | <p>$\left|f(x)-f(y)\right|\leq \left(\sup\limits_{z\in [x,y]}\left|f'(z)\right|\right)\cdot \left|x-y\right|$</p>
|
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Bill Dubuque | 6,716 | <p>Some of the prettiest examples of Dedekind's structuralism arise from revisiting proofs in elementary number theory from a highbrow viewpoint, e.g. by reformulating them after noticing hidden structure (ideals, modules, etc). A striking example of such is the generalization and unification of elementary irrationalit... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Andreas Blass | 6,794 | <p>When I was a student, I once watched a professor (a famous and brilliant mathematician) spend a whole class period proving that the functor $M\otimes-$ is right exact. (This was in the context of modules over a commutative ring.) He was working from the generators-and-relations definition of the tensor product. W... |
1,818,976 | <p>Let there be many numbers $a_1,a_2,a_3,\dots,a_n$.</p>
<p>I want to find the first digit of their product, i.e. of $A=a_1\times a_2\times a_3\times a_4\times \dots\times a_n$.</p>
<p>These numbers are huge and multiplying all of them exceeds the time limit.</p>
<p>Is there any shortcut to find the most significan... | hmakholm left over Monica | 14,366 | <p><strong>Hint:</strong> The unit circle (or even the closed unit disc) is compact. And if $r$ is irrational, then the $e^{i2\pi rn}$s are all different ...</p>
|
78,569 | <p><img src="https://i.stack.imgur.com/FhX2B.png" alt="Limit of both sides of function"></p>
<p>I need to solve for <code>c</code> such that the function is continuous at <code>x=2</code>. How do I do this automatically?</p>
<p>I have expressions for the limit of both sides of the function as x->2, but how would i u... | Nasser | 70 | <pre><code>myfunc[x_] := Piecewise[{{c x^2 + 2 x, x <= 2}, {x^3 - c x, True}}];
lim1 = Limit[myfunc[x], x -> 2, Direction -> -1]
lim2 = Limit[myfunc[x], x -> 2, Direction -> 1]
sol = c /. First@Solve[{lim1 == lim2}, c]
(*2/3*)
Plot[myfunc[x] /. c -> sol, {x, 1, 3},
Epilog -> {Red, PointSize[.... |
3,257,799 | <blockquote>
<p>Find all values of <span class="math-container">$a$</span> for which the equation
<span class="math-container">$$ (a-1)4^x + (2a-3)6^x = (3a-4)9^x $$</span>
has only one solution.</p>
</blockquote>
<p><br>
I have two cases, one when <span class="math-container">$a = 1$</span> and other when Discr... | nonuser | 463,553 | <p>Here is another way to solve this. </p>
<p>Let <span class="math-container">$t=2^x/3^x>0$</span>, then we get <span class="math-container">$$a(t^2+2t-3) = t^2+3t-4$$</span> so <span class="math-container">$$a(t+3)(t-1)=(t+4)(t-1)$$</span></p>
<p>Then for each <span class="math-container">$a$</span> number <span... |
69,448 | <p>What <code>Method</code> options are allowed for <code>DensityPlot</code> and <code>ContourPlot</code>? I am unable to find this information either in MMA documentation or in SE. Thanks.</p>
| Community | -1 | <p>One should not be confused with method or option.</p>
<p>A method in the sense of Mathematica (See: <a href="http://reference.wolfram.com/language/ref/Method.html" rel="nofollow noreferrer">Method</a>)</p>
<p><img src="https://i.stack.imgur.com/9JTnC.png" alt="enter image description here"></p>
<p>Options in the ... |
4,549,340 | <p>I have heard people say that the flight time from Fort Lauderdale to Seattle is the longest possible flight time within the continental United States. However, upon further consideration, I realized that the curvature of the Earth may cause the visible distance on a map to decrease when traveling north (the circumfe... | Damian Pavlyshyn | 154,826 | <p>When doing induction arguments, it is helpful to write down exactly what your induction hypothesis <span class="math-container">$P(p)$</span> is and how it depends on the variable <span class="math-container">$p$</span>.
The way that you've written it is not very clear, and this is what's causing you to (correctly) ... |
191,984 | <p>In this context composition series means the same thing as defined <a href="http://en.wikipedia.org/wiki/Composition_series#For_groups" rel="noreferrer">here.</a></p>
<p>As the title says given a finite group <span class="math-container">$G$</span> and <span class="math-container">$H \unlhd G$</span> I would like to... | Marc van Leeuwen | 18,880 | <p>Yes your proof is essentially correct. You can make it look less messy as follows. Noticing that the proof that $G$ has a composition series in the first place has the same structure as your proof, you could do a two-in-one-blow induction on the order of $G$ by showing: "$G$ has a composition series, and if $H$ is a... |
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