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 |
|---|---|---|---|---|
2,617,244 | <p>I'm trying to get the <em>least x</em> from a system of congruences by applying the Chinese Remainder Theorem. Keep running into issues.</p>
<p>System of congruences:
$$
x \equiv 0 (_{mod} 7) \\
x \equiv 5 (_{mod} 6) \\
x \equiv 4 (_{mod} 5) \\
x \equiv 3 (_{mod} 4) \\
x \equiv 2 (_{mod} 3) \\
x \equiv 1 (_{mod} 2... | Steven Alexis Gregory | 75,410 | <p>This is very similar to aid78's solution.
<span class="math-container">\begin{align}
x &\equiv 0 \pmod 7 \\
x &\equiv 5 \pmod 6 \\
x &\equiv 4 \pmod 5 \\
x &\equiv 3 \pmod 4 \\
x &\equiv 2 \pmod 3 \\
x &\equiv 1 \pmod 2
\end{align}</span></p>
<p>can be rewritten as</p>
<p><span class="math-co... |
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Petya | 2,823 | <p>"Development of mathematics in the 19th century" by Felix Klein is a great book. Is it a good example here?</p>
|
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Peter Arndt | 733 | <p>Peter Johnstone's <a href="http://books.google.com/books?id=TLHfQPHNs0QC&dq=johnstone+sketches+of+an+elephant&source=bl&ots=yqamFhZrMo&sig=A_1ZVyvl4kcFv2A_SLJMQvSpO6E&hl=de&ei=s2KRTLz3O5DEOPfEjMUM&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBsQ6AEwAQ" rel="nofollow">"Sket... |
2,936,899 | <p>Let <span class="math-container">$X=\{(x,y)\in \mathbb{R}^2| x^2+y^2=1\}$</span> the unit circle. For each vector <span class="math-container">$v\in X$</span> define <span class="math-container">$H_v=\{w\in \mathbb{R}^2|w\cdot v\geq 0\}$</span> where <span class="math-container">$\cdot$</span> is the dot product. H... | William Elliot | 426,203 | <p><span class="math-container">$H_v$</span> is the set of vectors covering the half plane bounded by a line through (1,0) with the slope of the tangent to the unit circle at v.</p>
|
3,618,029 | <p>In <em>Cormen's Introduction to Algorithm's book</em>, I'm attempting to work the following problem:</p>
<blockquote>
<p>Show that the solution to the recurrence relation <span class="math-container">$T(n) = T(n-1) + n$</span> is <span class="math-container">$O(n_2)$</span> using substitution
<span class="math-... | J.G. | 56,861 | <p>@CHAMSI obtained <span class="math-container">$T(n)$</span> exactly, but in general big-<span class="math-container">$O$</span> results aren't achieved this way. It's often sufficient to argue from calculus approximations: in this case, <span class="math-container">$T^\prime(n)\approx T(n)-T(n-1)=n\implies T(n)\sim\... |
720,368 | <p>I want to express
$$\prod_{k=1}^n \left( k - \frac{1}{2} \right)$$
using the gamma function. I think this is equivalent to $\left(k-\frac{1}{2}\right)!$ so I set $a=k-1$
and then used the identity $$\Gamma \left(n+\frac{1}{2}\right) = {(2n)! \over 4^n n!} \sqrt{\pi}$$
to get
$$\prod_{k=1}^n \left( k - \frac{1}{2} \r... | Lutz Lehmann | 115,115 | <p>The first type of identity needs no gamma function at all to arrive at a correct result.
$$
\prod_{k=1}^n\left(k-\tfrac12\right)=\frac1{2^n}\prod_{k=1}^n(2k-1)
=\frac1{4^nn!}\prod_{k=1}^n(2k)(2k-1)=\frac{(2n)!}{4^nn!}
$$</p>
|
18,831 | <p>I provided an answer (whether it really is one or not is a matter of dispute)
to the following question:
<a href="https://math.stackexchange.com/questions/1074861/why-isnt-a-noncommutative-ring-with-only-trivial-ideals-a-division-ring/1074867#1074867">Where does the proof for commutative rings break down in the non-... | dustin | 78,317 | <p>I think answers that are considered low quality and not an answer needs to be thoroughly addressed. There is a meta post floating around about hints being an answer but they are constantly flagged as low quality any ways. I have noticed that posts that are as long as hints but don't contain the word <strong>Hint</st... |
2,948,984 | <p>We have five different pairs of gloves. Three people choose at random one left and one right glove. What is probability, that each person doesn't have a pair.
My attempt:
<span class="math-container">$|\Omega| = {{10}\choose{6}}$</span>
First person can choose the left glove in five ways, right glove-four ways.
Seco... | Phil H | 554,494 | <p>Yet another way of looking at it:</p>
<p>It doesn't really matter which <span class="math-container">$3$</span> left hand gloves are chosen, the math can be limited to the selection of the <span class="math-container">$3$</span> right hand gloves.</p>
<p>When the <span class="math-container">$3$</span> right hand ... |
4,009,431 | <p>Let <span class="math-container">$V$</span> and <span class="math-container">$W$</span> be finite dimensional complex vector spaces and let <span class="math-container">$f: V\xrightarrow[]{}W$</span> and <span class="math-container">$g: W\xrightarrow[]{}V$</span> be linear maps. Suppose that <span class="math-contai... | José Carlos Santos | 446,262 | <p>Let <span class="math-container">$V_1=\ker f$</span> and let <span class="math-container">$V_0=g(W)$</span>.</p>
<ol>
<li>If <span class="math-container">$v\in V_1\cap V_0$</span>, then <span class="math-container">$v=g(w)$</span> for some <span class="math-container">$w\in W$</span> and <span class="math-container"... |
13,370 | <p>I've read the formula written on the title on a mathematics' book and it doesn't seem correct for me:</p>
<p><strong>For the first part of the formula [A ∨ (B∧C)] I have the following possible values:</strong></p>
<p><em>A; B and C; A and B and C (the Or is not exclusive)</em></p>
<p><strong>For the second part of t... | Fredrik Meyer | 4,284 | <p>Draw a Venn diagram and think of $A,B,C$ as sets. (consisting of True,False). Then you'll see that the formula is correct.</p>
<p>Or you could draw a truth table.
<a href="http://en.wikipedia.org/wiki/Truth_table" rel="nofollow">http://en.wikipedia.org/wiki/Truth_table</a> </p>
|
13,370 | <p>I've read the formula written on the title on a mathematics' book and it doesn't seem correct for me:</p>
<p><strong>For the first part of the formula [A ∨ (B∧C)] I have the following possible values:</strong></p>
<p><em>A; B and C; A and B and C (the Or is not exclusive)</em></p>
<p><strong>For the second part of t... | Alex Basson | 506 | <p>Suppose you have $A \vee (B \wedge C)$. Then you either have $A$ or $B \wedge C$. If you have $A$, then you have $A \vee B$ and $A \vee C$. If, on the other hand, you have $B \wedge C$, then you have both $B$ and $C$, in which case you have both $A \vee B$ and $A \vee C$. So $A \vee (B \wedge C) \Rightarrow (A \... |
4,057,243 | <p>My textbook lists two theorems and I'm not sure how I'm supposed to interpret them. I don't need a proof; I'm only trying to figure out what information I'm being told by each theorem.</p>
<blockquote>
<p>Let <span class="math-container">$p$</span> be a prime and let <span class="math-container">$a$</span> be an int... | fleablood | 280,126 | <p>Okay, what does it mean.</p>
<p>A complete residue system <span class="math-container">$\mod p$</span>. What's that?</p>
<p>If you compare any integer <span class="math-container">$n \pmod p$</span> that are only <span class="math-container">$p$</span> options. Either <span class="math-container">$n\equiv 0\pmod p... |
3,735,973 | <p>This is taken as a side question from Rudin's book on Real and Complex Analysis.
I need to prove that</p>
<p><span class="math-container">$$f_n(x)=\frac{\sin{(x)}\sin{(nx)}}{x^2}$$</span></p>
<p>has an <span class="math-container">$L^1$</span> norm that tends to infinity as <span class="math-container">$n\to\infty$<... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$\int |f_n(x)|dx=n\int \frac{|sin y| |\sin (\frac y n)| } {y^{2}}dy$</span> by the substitution <span class="math-container">$y=nx$</span>. Hence <span class="math-container">$\int |f_n(x)|dx \geq n\int_{n-1}^{n+1} \frac{|\sin y| |sin (\frac y n)| } {y^{2}}dy$</span>. Since <span c... |
652,960 | <p>I am lost on this one. I'm still new to ring theory, as we're only a couple weeks into the course, but it's already well over my head. I know that $R$ is an integral domain, so the additive and multiplicative identities are not equal, and if both $a$ and $b$ are nonzero, then their product will be nonzero. So $u$ ca... | Henry Swanson | 55,540 | <p>Because $(a) = (b)$, $b \in (a)$. By the definition of $(a)$, there exists $u \in R$ such that $b = ua$. Similarly, there is a $v \in R$ such that $a = vb$. Combining these gives $a = vua$, and since this is an integral domain, we can cancel*, getting $vu = 1$. Thus, $u$ is a unit.</p>
<p>*We have to watch out for ... |
2,599,982 | <p>Can anybody help me finding a good way to (approximately) figure out the first, lets say $200$, <strong>positive</strong> roots of $$\tan(x) + 2 \ell x - \ell ^2 x^2 \tan(x) = 0,$$
where $\ell$ is just a constant?</p>
<p>I believe there will be no analytic expression, so is there a better idea than just running <e... | RideTheWavelet | 394,393 | <p>Since the function for which you are trying to find roots has many poles, I have serious doubts about how well Newton's method will succeed on its own. I would recommend something like a bisection method until you get $|f(x)|<10^{-3}$ and then one or two Newton iterations (if you need better precision than this) ... |
1,774,181 | <p>I searched leap years online and found that 1900 is not, contrary to what <em>I</em> thought, a leap year. But, why is it not if 1900 is divisible by 4:<br><br>
$\frac{1900}{4} = 475$<br>
<br> My brother was working on his math (and he obviously got it wrong and asked me for help, so.. here I am), and the question w... | Arthur | 15,500 | <p>If you google "leap year" you will find that a year $n$ is a leap year if $n$ is divisible by $4$, <em>except</em> if $n$ is divisible by $100$. If it's divisible by $400$, though, it's still a leap year.</p>
<p>$1900$ is divisible by $100$, but not $400$, so it is not a leap year. $2000$, however, <em>was</em> a l... |
270,600 | <p>I'm a software engineer with math classes through differential equations about 15 years in my past, and I've gotten stuck trying to invert an equation.</p>
<p>The equation:
<span class="math-container">$y = x + (0.022 - x)^{1.414}$</span>.</p>
<p>In Mathematica form:</p>
<pre><code>sapcClamp[y] := y + ((22 / 100) - ... | Bob Hanlon | 9,362 | <pre><code>$Version
(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)
Clear["Global`*"]
</code></pre>
<p>Note that <code>0.022</code> is <code>22/1000</code> and <code>1.414</code> is <code>1414/1000</code></p>
<pre><code>0.022 == 22/1000 && 1.414 == 1414/1000
(* True *)
sapcClamp... |
569,328 | <p>I want to ask if I'm understand few subgroups correct:</p>
<ol>
<li>$\langle 2\rangle$ of $\mathbb{R}$ is $\left\{2\cdot n\big|n\in \mathbb{Z} \right\}$</li>
<li>$\langle 2\rangle$ of $\mathbb{R}^{*}$ is $\left\{2^{n}\big|n\in \mathbb{Z} \right\}$</li>
<li>$\langle i\rangle$ of $\mathbb{C}$ is $\left\{i\cdot n\big|... | amWhy | 9,003 | <p>Assuming you mean $\mathbb R$ and $\mathbb C$ to be the additive group of reals and the additive group of complex numbers, and by $\mathbb R^*$ and $\mathbb C^*$ the non-zero reals and complex numbers under multiplication, then yes, you're correct in your understanding of the subgroups generated by $2$ in the first ... |
569,328 | <p>I want to ask if I'm understand few subgroups correct:</p>
<ol>
<li>$\langle 2\rangle$ of $\mathbb{R}$ is $\left\{2\cdot n\big|n\in \mathbb{Z} \right\}$</li>
<li>$\langle 2\rangle$ of $\mathbb{R}^{*}$ is $\left\{2^{n}\big|n\in \mathbb{Z} \right\}$</li>
<li>$\langle i\rangle$ of $\mathbb{C}$ is $\left\{i\cdot n\big|... | Mikasa | 8,581 | <p>General hint:</p>
<p>If $G$ is a group and $\emptyset\neq M\subseteq G$, then $$\langle M\rangle=\{x_1^{\epsilon_1}x_2^{\epsilon_2}\cdots x_n^{\epsilon_n}\mid x_i\in M,~\epsilon_i\in\mathbb Z, n\in\mathbb N\cup\{0\}\}$$ for example in $(\mathbb Z,+)$, $\langle 2\rangle=\{2k\mid k\in\mathbb Z\}$. or in $(\mathbb Q,+... |
1,393,694 | <p>I am having a rather tough time wrapping my head around any possible logical fallacy in my solution to the following question as my answer is wrong:</p>
<p>1 percent of children have autism. A test for autism is developed such that 90% of autistic children are correctly identified as having autism but 3% of non-aut... | Robert Israel | 8,508 | <p>Actually, when dealing with complex numbers your proposal is essentially what is used: for a complex number $z$ there is no single number that is called $\sqrt{z}$, rather we say $z$ has two square roots, and $\sqrt{z}$ is a "multivalued function". Sometimes we designate one of the two as the "principal branch", bu... |
26,083 | <p>For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in understanding arithmetic questions?</p>
| Charles Matthews | 6,153 | <p>If we think about Diophantine equations in general, the situation is "hopeless". That's a theorem. Nevertheless in number theory we want to study such equations, in special cases at least, so some ideas are required to sort out "Diophantine equation space" (DES) into parts that we might come to understand, and parts... |
16,342 | <p>Why do you need logarithms?
In what situations do you use them?</p>
| Community | -1 | <p>Logarithms are primarily used for two thing:</p>
<p>i) Representation of large numbers. For example pH(the number of hydrogen atoms present) is too large (up to 10 digits). To allow easier representation of these numbers, logarithms are used.</p>
<p>For example let's say the pH of the substance is $10000000000$.
... |
2,521,031 | <p>I'm supposed to find a counterexample to counter this claim</p>
<p>Suppose $f: X \mapsto Y$ is a continuous function. Suppose X is bounded then $f(x)$ is bounded. My counterexample is this:</p>
<p>Let $X = (-\pi/2, \pi/2)$. A is bounded by $B_{\pi/2}(0)$. Let $f(x) = \tan(x)$ and let $Y = \mathbb{R}$ and $\mathbb{... | Peter Szilas | 408,605 | <p>Here is more:</p>
<p>$f(x) = 1/x +1/(x-1)$, $x \in (0,1).$</p>
<p>The trick is to choose an interval open, or half open.</p>
<p>Can you find a closed interval $[a,b]$, bounded, where $f$, continuos, is not bounded? :))</p>
|
245,083 | <p>I'm new to integral calculus, I started literally 15 minutes ago, and I need help with this question:</p>
<p>$$\int \dfrac{\ln(x)^2}{x} dx $$</p>
<p>My first step was:</p>
<p>$$\int \dfrac{1}{x}\ln(x)^2 dx $$</p>
<p>However, what to do next, how to solve this using the reverse chain rule? </p>
| Michael Hardy | 11,667 | <p>Here's a hint:
$$
\int (\ln x)^2 \Big( \frac1x\,dx\Big).
$$
To understand how to use the "reverse chain rule", also called integration by substitution, is to understand this kind of hint.</p>
<p>The next step is to go from the hint above to this:
$$
\int u^2 \, du.
$$</p>
|
368,763 | <p>A part two, you could say, of <a href="https://math.stackexchange.com/questions/368343/j2-1-but-j-neq-1-what-is-j">my previous question</a>.</p>
<p>I was watching a Numberphile video about the favorite number of some mathematicians, and at one point, the creator of ViHart said the following -</p>
<blockquote>
<p... | Ross Millikan | 1,827 | <p>The version of $*$ I know is described in Winning Ways for your Mathematical Plays by Elwyn R. Berlekamp, John H. Conway and Richard K. Guy. It is a terrific book. They start by describing games as sets that look like $\{|\}$. The things to the left of the bar are options of one player, called Left. The things... |
4,187,128 | <p>On page 411 in <a href="https://people.smp.uq.edu.au/DirkKroese/DSML/" rel="nofollow noreferrer">this book</a>, the authors give the following BFGS formula <span class="math-container">$$ \boxed{\boldsymbol C_{\textrm{BFGS}} = \boldsymbol C + \underbrace{\frac{\boldsymbol g^\top\boldsymbol\delta+\boldsymbol g^\top\b... | P. Camilleri | 167,258 | <p>What we want to minimize is the Stein loss, aka the KL divergence between two Gaussians of same mean, aka the Bregman divergence induced by the negative logdet.
I'll write in slightly different notation because those are the ones I have already typed in my class.</p>
<p><span class="math-container">$$\min_H - \mathr... |
462,590 | <p>Obtain a grammar for the language</p>
<ul>
<li>$L = \{a^m b^n \mid m ≠ n , m > 0 , n > 0 \}$.</li>
</ul>
<p>Please help me with the solution.</p>
| dtldarek | 26,306 | <p><strong>Hint:</strong></p>
<ul>
<li>Let $L_0 = \{a^nb^n \mid n > 0\}$.</li>
<li>Let $L_1 = \{a^k \mid k > 0\} \circ L_0$ and
$L_2 = L_0 \circ \{b^k \mid k > 0\}$.</li>
<li>Then $L = L_1 \cup L_2$.</li>
</ul>
<p>I hope this helps $\ddot\smile$</p>
|
211,379 | <p>The numbers 7, 8, 9, apart from being part of a really lame math joke, also have a unique property. Consecutively, they are a prime number, followed by the cube of a prime, followed by the square of a prime. Firstly, does this occurence happen with any other triplet of consecutive numbers? More importantly, is there... | Erick Wong | 30,402 | <p>Of course this exact pattern cannot recur as Patrick Li easily observed, but there are plenty of neat coincidences to be found in any group of small numbers. My favorite: of 243, 343, 443, the first is the fifth power of a prime, the second is the cube of a prime, and the third is a prime.</p>
|
211,379 | <p>The numbers 7, 8, 9, apart from being part of a really lame math joke, also have a unique property. Consecutively, they are a prime number, followed by the cube of a prime, followed by the square of a prime. Firstly, does this occurence happen with any other triplet of consecutive numbers? More importantly, is there... | Gerry Myerson | 8,269 | <p>If you'll settle for a prime, cube of a prime, square of a prime in arithmetic progression (instead of consecutive), you've got $$5,27=3^3,49=7^2\qquad \rm{(common\ difference\ 22)}$$ and $$157,\ 343=7^3,\ 529=23^2 \qquad \rm{(common\ difference\ 186)}$$ and, no doubt, many more where those came from. A bit more... |
3,724,383 | <p>I encountered this SAT type Math question and do not know how to progress.</p>
<p>Before a plum is dried to become a prune, it is 92% water. A prune is just 20% water. If only water is evaporated in the drying process, how many pounds of prunes can be made with 100 pounds of plums?</p>
<p>My attempt:</p>
<p>Since a ... | Integrand | 207,050 | <p>If <span class="math-container">$|z|<1$</span>, multiply by <span class="math-container">$1-z$</span> and FOIL it out.</p>
|
3,724,383 | <p>I encountered this SAT type Math question and do not know how to progress.</p>
<p>Before a plum is dried to become a prune, it is 92% water. A prune is just 20% water. If only water is evaporated in the drying process, how many pounds of prunes can be made with 100 pounds of plums?</p>
<p>My attempt:</p>
<p>Since a ... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
325,582 | <p>I am looking for examples of constructions for transfinite towers <span class="math-container">$(X_{\alpha})_{\alpha}$</span> generated by structures <span class="math-container">$X$</span> where the problem of determining whether the tower <span class="math-container">$(X_{\alpha})_{\alpha}$</span> stops growing is... | Jeremy Brazas | 5,801 | <p>The following is an example where is it difficult to identify where stabilization occurs. It meets all of your technical criteria but I suppose the word "growing" is not exactly accurate.</p>
<p>A quasitopological group <span class="math-container">$G$</span> is a group with a topology <span class="math-container">... |
2,630,628 | <pre><code>2/(x+y) + 2/(y+z) + 2/(z+x) >= 9/(x+y+z)
</code></pre>
<p>I am using the following vectors:</p>
<p>(x+y y+z z+x) and (1 1 1)</p>
<p>I obtain something resembling the aforementioned inequality, but maybe my choice of vectors is totally wrong. Let me know if I should post a picture of my work!</p>
<p>Th... | Keith McClary | 252,672 | <p>For small $x$ the RHS $\approx x$ so $\epsilon^{-1} x^3 \approx x$ so the roots are $\approx \pm \epsilon^{1/2}$ and $0$.</p>
|
846,020 | <blockquote>
<p>Let $(\mathfrak a_i)$ be an infinite family of ideals in commutative ring $R$. Is $\bigcap\limits_{i=1}^\infty \mathfrak a_i$ not defined?</p>
</blockquote>
<p>I am trying to understand Zariski topology. Here, $V(\bigcap_i \mathfrak a_i)= \bigcup\limits_{i} V(\mathfrak a_i)$. </p>
<p>If $\bigcap\lim... | Martin Brandenburg | 1,650 | <p>$V(\cap_i \mathfrak{a}_i)$ is the closure of $\cup_i V(\mathfrak{a}_i)$. In general, $\cup_i V(\mathfrak{a_i})$ might be not closed. For example, take $\mathbb{Z}$ so that $\mathrm{Spec}(\mathbb{Z})$ carries the cofinite topology except for the generic point $\eta$ which is contained in every non-empty open subet. W... |
49,633 | <p>I know the construction of the Hodge star operator in the context of (pseudo-)euclidean real vector spaces. Apart from the scalar product it involves a orientation of the vector space, which one has to choose (at least if one is not willing to deal with forms of odd parity). </p>
<p>While I was wondering how to ext... | Tom Goodwillie | 6,666 | <p>An orientation of the $n$-dimensional real vector space $V$ is an equivalence class of generators of the $1$-dimensional vector space $det(V)=\Lambda^n(V)$ under the relation $\omega\sim c\omega$, $c>0$.</p>
<p>A basis-free description of the usual Hodge star for a real vector space with positive inner product i... |
4,335,831 | <blockquote>
<p>Let <span class="math-container">$f:\mathbb R\rightarrow \mathbb R$</span> be a differentiable function, and suppose <span class="math-container">$f=f'$</span> and <span class="math-container">$f(0)=1$</span>. Then prove <span class="math-container">$f(x)\neq 0$</span> for all <span class="math-containe... | Severin Schraven | 331,816 | <p>There is a shorter proof if you know the Picard-Lindelöf theorem. Assume that there exists <span class="math-container">$x_0\in \mathbb{R}$</span> such that <span class="math-container">$f(x_0)=0$</span>. Then <span class="math-container">$f$</span> solves the ODE
<span class="math-container">$$ \begin{cases} f'&... |
4,335,831 | <blockquote>
<p>Let <span class="math-container">$f:\mathbb R\rightarrow \mathbb R$</span> be a differentiable function, and suppose <span class="math-container">$f=f'$</span> and <span class="math-container">$f(0)=1$</span>. Then prove <span class="math-container">$f(x)\neq 0$</span> for all <span class="math-containe... | Pilcrow | 760,823 | <p>Certainly the shortest solution involves <span class="math-container">$e^x$</span>. However, depending on one's definitions, such a solution may very well involve a circular argument: <span class="math-container">$e^x$</span> is often <em>defined directly</em> as the unique function <span class="math-container">$f$<... |
3,722,748 | <p>I would appreciate it if someone could help me with the following problem. I can not understand how a delta function <span class="math-container">$\delta(x)$</span> is integrated from zero to infinity. Because the integration interval should contain zero.</p>
| user151522 | 553,318 | <p>For the integration <span class="math-container">$\int_0^\infty$</span>, you do not use the usual definition of the delta function, which Barton, Reference 1, pg 12 , refers to as the ‘weak definition’ of <span class="math-container">$\delta(x)$</span>, and which he gives in (1.1.1) on pg 10 as ( I quote )</p>
<p><s... |
4,004,612 | <p>I need to examine convergence of that sum:
<span class="math-container">$$\sum_{n=1}^{\infty} \left(\frac{1+ 3\cos(n^2)}{7+2\sin^2n} \right)^{3n - \ln^2n}$$</span></p>
<p>I can rewrite it as:
<span class="math-container">$$\sum_{n=1}^{\infty} \frac{\left(\frac{1+ 3\cos(n^2)}{7+2\sin^2n}\right)^{3n}}{\left(\frac{1+ 3... | Raffaele | 83,382 | <p>Apply the <a href="https://en.wikipedia.org/wiki/Root_test" rel="nofollow noreferrer">root test</a></p>
<p>We have
<span class="math-container">$$\underset{n\to \infty }{\text{lim}}\frac{3 n-\log ^2 n}{n}=3$$</span></p>
<p>and
<span class="math-container">$$\left| \frac{3 \cos \left(n^2\right)+1}{2 \sin ^2(n)+7}\rig... |
1,971,031 | <p>Stewart - Calculus</p>
<hr>
<p><a href="https://i.stack.imgur.com/IDZAA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IDZAA.png" alt="enter image description here"></a></p>
<hr>
<p>It does not appear that work being change of kinetic energy depends on $C$ or that $F$ being conservative is as... | jnyan | 365,230 | <p>It depends against which force you are doing the work. If you have a curl free force field then and only then it is path independent. If the concept of curl is still not introduced then you can just remember that workis path independent in only certain cases like gravity, electrostatics. Simple counter example is im... |
547,151 | <p>I am attempting to show that if $R$ is commutative, local notherian ring with $J$ nilpotent then $R$ is Artinian.</p>
<p>Clearly as $\exists k$ such that $J^k=0$ then if there is no other ideal other than $R$ and $J^i$ then we are done but how do I show this?</p>
<p>That is how do I show that if $I$ is an ideal of... | Community | -1 | <p>$\tan(\theta)= 1$ when $\theta = \frac{(1+4n)\pi}{4}$ where $n$ is an integer starting at $0$, so the zeros for your problem are $2x = \frac{(1+4n)\pi}{4} = \frac{(1+4n)\pi}{8}$. For the range $x$ you gave, that would be the first two values $\frac{\pi}{8},\frac{5\pi}{8}$</p>
|
2,652,562 | <p><a href="https://i.stack.imgur.com/Okrbw.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Okrbw.jpg" alt="enter image description here"></a></p>
<p>When is $\ln x = x^{\frac13}$?</p>
<p>Is there any way to find this solution that isn't just an estimate? </p>
<p>--</p>
<p>Wolfram says it has to ... | Claude Leibovici | 82,404 | <p>If you do not want to use Lambert function and still look for the solution(s) of equation $$\log(x) = x^{\frac13}$$ consider that you look for the zero(s) of function
$$f(x)=\log(x) - x^{\frac13}$$ $$f'(x)=\frac{1}{x}-\frac{1}{3 x^{2/3}}$$
$$f''(x)=\frac{2}{9 x^{5/3}}-\frac{1}{x^2} $$The first derivative cancels wh... |
2,064,812 | <p>I would like to know what properties were used in making this answer?</p>
<p>$$\sin(x)+\cos(x)=\sqrt 2 \left( \frac{1}{\sqrt{2}}\sin(x)+\frac{1}{\sqrt{2}}\cos(x)\right)=\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)$$</p>
<p>Because I am a bit clueless on the $\sqrt{2}\sin(x+\frac{\pi}4)$ was reached.</p>
| Spirine | 244,050 | <p>You know that for any $a, b \in \mathbb{R}$, $\sin (a+b) = \sin (a) \cos(b) + \sin(b) \cos(a)$. Also, $\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$. You have then</p>
<p>$$ \frac{1}{\sqrt{2}} \sin(x) + \frac{1}{\sqrt{2}}\cos(x) = \cos(\frac{\pi}{4})\sin(x) + \sin(\frac{\pi}{4})\cos(x) = \sin(x + \frac{\pi}{4})$$</p>
|
11,178 | <p>I have the trigonometric equation
\begin{equation*}
\sin^8 x + 2\cos^8 x -\dfrac{1}{2}\cos^2 2x + 4\sin^2 x= 0.
\end{equation*}
By putting $t = \cos 2x$, I have
\begin{equation*}
\dfrac{3}{16} t^4+ \dfrac{1}{4}t^3 + \dfrac{5}{8}t^2 -\dfrac{7}{4}t + \dfrac{35}{16} = 0.
\end{equation*}
How do I tell Mathematica to ... | halirutan | 187 | <p>You can use <code>TrigExpand</code> to expand all trigonometric functions to fundamental forms and then <code>Eliminate</code> solves the rest</p>
<pre><code>eq1 = Sin[x]^8 + 2 Cos[x]^8 - 1/2 Cos[2 x]^2 + 4 Sin[x]^2 == 0;
eq2 = t == Cos[2 x]
Eliminate[TrigExpand[{eq1, eq2}], x]
</code></pre>
|
568,870 | <p>I've been asked to prove that</p>
<p>$$ \int^\infty_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0, \space \forall x > n \geq 2.$$</p>
<p>My approach so far has been to use a theorem proved in class that, for a random variable $X$ with characteristic function $\phi(t)$ and $a,b\in\mathbb{R}$,</p>
<p>$... | Random Variable | 16,033 | <p>A complex analysis approach is to integrate the function $e^{ixz} \left(\frac{\sin z}{z} \right)^{n} $ around a contour that consists of the real axis and the large semicircle above it.</p>
<p>To show that the integral along the semicircle vanishes as the radius of the semicircle goes to $\infty$, we only need to s... |
284,322 | <p>Let <span class="math-container">$f$</span> be a continuous and integrable function over <span class="math-container">$[a,b]$</span>. Prove or disprove that</p>
<p><span class="math-container">$$\int_a^b |f(x)|\ \mathrm{d}x\geq \left | \int_a^b f(x)\ \mathrm{d}x\right|
$$</span></p>
| Hanul Jeon | 53,976 | <p>If <span class="math-container">$f$</span> is a real Riemann-integrable function, this inequality is true. (and if <span class="math-container">$f$</span> is complex Riemann-integrable, then this inequality holds.)</p>
<p>By properties of the modulus function, we have
<span class="math-container">$$-|f(x)|\le f(x)\l... |
2,478,173 | <p>I have a sample mean given by:</p>
<p>$$S_n=\frac{1}{n}\sum_{i=1}^nX_i$$
Where $X_i$ are i.i.d. Gaussian random variable, i.e., each of them has pdf:</p>
<p>$$p(X_i=x_i)=\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x_i-\mu)^2/2\sigma^2}$$</p>
<p>The parameters $\mu, \sigma^2$ are, of course, mean and variance.</p>
<p>The ... | Sunyam | 463,614 | <p>Note : Sum of Gaussian pdfs of $n$ random variables is not a Gaussian pdf, only the sum of those random variables is distributed according to Gaussain pdf.
I will give a quick sketch to arrive at the result, you can fill up the gaps yourself.</p>
<p>Moment generating function of $S_{n}^{}$ i can be defined as :
$$... |
1,815,344 | <p>We want to prove there exists a finite field of $p^n$ elements ($p$ is prime and $n>0$). Take $q=p^n$ and $g(x)=x^q-x\in\mathbb{Z}_p[x]$, and let $E$ be a field that contains $\mathbb{Z}_p$ and all roots of $g(x)$. Let $F=\{a\in E:a^q=a\}$.</p>
<p>I understand that $F$ is closed under addition (since $E$ has cha... | Ashwin Ganesan | 157,927 | <p>Let $g(x)=x^{p^n}-x$. You want to prove that if $g(\beta)=0$, then $g(-\beta)=0$. So suppose $g(\beta)=\beta^{p^n}-\beta=0$. Then $g(-\beta) = (-\beta)^{p^n}-(-\beta)=(-\beta)^{p^n}+\beta$. </p>
<p>If $p=2$, then $g(-\beta)=(-\beta)^{p^n}+\beta = (\beta)^{p^n}+\beta$ (because we are raising to an even power). Th... |
828,057 | <p>Let $X\geq 0$ be a real random variable and $h:\mathbb{R} \rightarrow \mathbb{R}$ a monotonously growing, continuously differentiable function with $h(0)=0$.</p>
<p>Show:</p>
<p>$E[h(X)] = \int_0^{\infty} h'(t)P[X>t]dt$</p>
<p>With the given hint I got </p>
<p>$E[h(X)] = \int_0^{\infty} \int_{0}^x h'(t)dt dF(... | Did | 6,179 | <p>The only explicit question seems to ask to explain where the following computation is faulty:</p>
<blockquote>
<p>$E[X^2] = \int_0^{\infty} x^2 F(x) dx = \int_0^{\infty}\int_0^{x}2x F(x) dx dt = 2\int_0^{\infty}\int_t^{\infty} x F(x) dx dt$</p>
</blockquote>
<p>Already the first equal sign is wrong. If $X$ has a... |
3,657,887 | <p>I have been doing some practice questions for an upcoming Maths Challenge. There's one question I can't seem to grasp. I'm not sure entirely sure where to start. I don't know how to approach this one. Any help would be appreciated</p>
<p><span class="math-container">$n$</span> in the form <span class="math-containe... | joriki | 6,622 | <p>Any sequence <span class="math-container">$c_n$</span> that converges to <span class="math-container">$\pi$</span> can be used to define such a cut, with</p>
<p><span class="math-container">$$
A=\{a\in\mathbb Q\mid a\lt c_n\text{ infinitely often}\}\hphantom{\;.}
$$</span></p>
<p>and</p>
<p><span class="math-cont... |
253,306 | <p>Let $\beta: H^n(X, \mathbb{Z}_2)\to H^{n+1}(X, \mathbb{Z})$ be the Bockstein homomorphism. Is it possible to define a cohomology operation $f: H^{n+1}(X, \mathbb{Z})\to H^{n+k+1}(X, \mathbb{Z})$ such that
\begin{eqnarray*}\beta \text{Sq}^k=f\beta\end{eqnarray*}
where $\text{Sq}^k: H^n(X, \mathbb{Z}_2)\to H^{n+k}(X,... | Tyler Lawson | 360 | <p>Such an $f$ only exists in two cases: when $k=1$ (when $\beta Sq^1 = 0$) and when $k=0$ (when $\beta Sq^0 = \beta$). Here is a proof, which will take a little work with the Steenrod algebra.</p>
<p>Let's suppose that $X$ and $Y$ are either $H\Bbb Z$ or $H\Bbb Z/2$ (not necessarily the same). Then the map
$$
[X,Y]_t... |
1,848,573 | <p>Let $a \in \mathbb{C}$ and $r > 0$. Prove then that $B = \left\{x \in \mathbb{C} \mid | x- a | \leq r \right\}$ is closed.</p>
<p><strong>Attempt:</strong> In class we defined a closed set as one whose complement is open. And a set $A \subset \mathbb{C}$ is open if $$ \forall x \in A, \exists \delta > 0, \for... | florence | 343,842 | <p>For part (a), it can be easily deduced that $f$ does not have degree 2. Further, $f$ cannot have degree 3, as an odd degree polynomial has no global minimum. So assume
$$f(x) = sx^4 + bx^3 + cx^2 + dx + e$$
with $f(a) = -\sqrt{2}, f'(a) = 0$. As one last piece of guesswork, let $a = k\sqrt{2}$. Then we have
$$4sk^... |
24,608 | <p>Hi. Are there nice/simple examples of cyclic extensions $L/K$ (that is, Galois extensions with cyclic Galois group) for which $L$ cannot be written as $K(a)$ with $a^n\in K$?</p>
<p>Thanks.</p>
| P Vanchinathan | 22,878 | <p>Take $\zeta = e^{2\pi i/p}$ for a prime number $p\equiv1$ (mod 3), e.g. $p=7$.
Then $Q(\zeta+\bar\zeta)$ is a totally real cyclic Galois extension of $\mathbf{Q}$ of degree a multiple of 3, hence contains a cubic extension $L$ that is Galois with cyclic Galois group.
Being totally real it cannot be the splitting fi... |
1,128,316 | <p>I'm having a bit of difficulty understanding the proof that in a Hausdorff space, a compact subset is closed. The proof I'm looking at uses the fact that a finite intersection of open sets is open (e.g. <a href="https://math.stackexchange.com/questions/83355/how-to-prove-that-a-compact-set-in-a-hausdorff-topological... | Brian M. Scott | 12,042 | <p>You’ve misunderstood the basic idea of the argument. I’ll refer to the answer at your first link for notation. We show that $K$ is closed by showing that if $x$ is any point of $\Bbb X$ not in $K$, then $x$ has an open nbhd disjoint from $K$; this is one of the most basic ways of showing that a set is closed. If you... |
644,714 | <p>given is
$f(x,y) = ( \frac{y}{x^2+1}, \frac{x^2}{y^2-1} ) $. I have to study the continuity of the function for$ (x,y) \to (0,1)$.</p>
<p>First function $f_1$ is continuous, since $lim f_1 = 1/1 = 1$ so the limit exists. And the function is defined on whole $\mathbb{R}^2$.</p>
<p>Secon function $f_2$.. Ok, here I ... | monroej | 121,862 | <p>\begin{align*}
h'(x)&=\frac{d}{dx}\left(\int_5^{1/x}10\arctan(t)dt\right)\\
&=10\arctan\left(\frac{1}{x}\right)\frac{d}{dx}\left(\frac{1}{x}\right)\\
&=10\arctan\left(\frac{1}{x}\right)\left(\frac{-1}{x^2}\right)
\end{align*}</p>
|
3,959,781 | <p>I did</p>
<p><span class="math-container">\begin{align}
& \left( \frac{t^2}{t^2+2}i+\frac{2}{t^2+2}j+\frac{2t}{t^2+2}k \right) \times \frac{2ti-2tj-(t^2+2)k}{t^2+2} \\[8pt]
= {} & \frac{2t^2i-4tj-2t(t^2+2)k}{(t^2+2)^2} \\[8pt]
= {} & \frac{2t^2i-4tj-2t^3k+4tk}{(t^2+2)^2} \\[8pt]
= {} & \frac{-4tj+4tk... | Doug M | 317,162 | <p><span class="math-container">$(\frac{t^2}{t^2+2}i+\frac{2}{t^2+2}j+\frac{2t}{t^2+2}k) \times \frac{2ti-2tj-(t^2+2)k}{t^2+2}$</span></p>
<p><span class="math-container">$\frac {1}{(t^2 + 2)^2} ((-2(t^2+2)+4t^2) \mathbf i + (4t^2+t^2(t^2+2))\mathbf j + (-2t^3-4t) \mathbf k)$</span></p>
<p><span class="math-container">... |
3,959,781 | <p>I did</p>
<p><span class="math-container">\begin{align}
& \left( \frac{t^2}{t^2+2}i+\frac{2}{t^2+2}j+\frac{2t}{t^2+2}k \right) \times \frac{2ti-2tj-(t^2+2)k}{t^2+2} \\[8pt]
= {} & \frac{2t^2i-4tj-2t(t^2+2)k}{(t^2+2)^2} \\[8pt]
= {} & \frac{2t^2i-4tj-2t^3k+4tk}{(t^2+2)^2} \\[8pt]
= {} & \frac{-4tj+4tk... | Michael Hardy | 11,667 | <p><span class="math-container">\begin{align}
i\times j & = k = -j\times i \\
j\times k & = i = -k\times j \\
k\times i & = j = - i\times k \\
i \times i & = j \times j = k\times k = 0
\end{align}</span></p>
|
1,036,176 | <p>I need to solve Problem 3.5 - 11 p. 164 of the book <em>Partial Differential Equations</em> by Lawrence C. Evans (2nd ed., AMS, 2010):</p>
<blockquote>
<ol start="11">
<li>Show that
<span class="math-container">$$
u(x,t) = \begin{cases}
-\dfrac{2}{3}\left(t+\sqrt{3x+t^2}\right); & \text{if } 4x ... | Ѕᴀᴀᴅ | 302,797 | <p><span class="math-container">$\DeclareMathOperator{\supp}{supp}\def\d{\mathrm{d}}\def\peq{\mathrm{\phantom{=}}{}}$</span>Note that <span class="math-container">$u_t + u u_x = u_t + \left( \dfrac{u^2}{2} \right)_x = 0$</span> holds for <span class="math-container">$x > -\dfrac{t^2}{4}$</span>, and <span class="mat... |
3,824,456 | <p>Let <span class="math-container">$a, b$</span> be two real numbers. If the function <span class="math-container">$x(t)$</span> is a solution of
<span class="math-container">$$\frac{d^2 x}{dt^2}+ 2a\frac{dx}{dt}+ bx = 0$$</span>
with <span class="math-container">$x(0) = x(1) = 0$</span>, then show that <span class="m... | DonAntonio | 31,254 | <p>For the inductive step> observe that</p>
<p><span class="math-container">$$(n+1)^p=\sum_{k=0}^p\binom pk n^k=1+n^p+\sum_{k=1}^{p-1}\binom pk n^k$$</span></p>
<p>But every term in the last sum is sivisible by <span class="math-container">$\;p\;$</span> (check this: this is the gist and this is <em>not always</em> ... |
3,824,456 | <p>Let <span class="math-container">$a, b$</span> be two real numbers. If the function <span class="math-container">$x(t)$</span> is a solution of
<span class="math-container">$$\frac{d^2 x}{dt^2}+ 2a\frac{dx}{dt}+ bx = 0$$</span>
with <span class="math-container">$x(0) = x(1) = 0$</span>, then show that <span class="m... | Community | -1 | <p>Not only primes but also Carmichael numbers. The necessary and sufficient condition that a number "n" is Carmichael number is that it should be a squarefree composite number and if p is a prime dividing n, then p-1 must divide n-1.</p>
|
1,650,131 | <p>Suppose we are given a finitely generated free module $M$ over a ring $R$. Assume $N \subseteq M$ is a free submodule of $M$.</p>
<p>If $B$ is a basis for $M$, does it follow that there exists a subset $A \subseteq B$ such that $A$ is a basis for $N$?</p>
<p>As stated, I'm pretty sure the question is incorrect: Ju... | Qiaochu Yuan | 232 | <p>This is not even true for vector spaces, which is just about the nicest possible case. Take $k$ a field, $M = k^2$ with basis $e_1, e_2$, and $N = \text{span}(e_1 + e_2)$. </p>
|
1,277,567 | <p>Q: Let R be a commutative ring with unity. Prove that if A is an ideal of R and A contains a unit, then A=R.</p>
<p>This is my attempt at an answer:
It suffices to show that all the elements in R are in A.</p>
<p>Let a be and element of A, and r be an element of R.
Since A is an ideal of R then all elements ar and... | Giwrgos K | 877,935 | <p>It's: <span class="math-container">$I=R \Leftrightarrow \exists x\in I: \; x\in U(R)$</span> <br />
The way "=>" is common
The other way "<=" is alike yours: if <span class="math-container">$x\in I$</span> and <span class="math-container">$x\in U(R),\;$</span> because <span class="math-cont... |
119,321 | <p>The idea is to make axes 3d, with a more subdued in black mesh, and shafts with their respective numbers with a maximum of 10 in each (black to print after the size of half a sheet craft, vine also print two per page )</p>
<p>z in the second drawing up</p>
<p>(mathematica ver 10.0.1.0 64 bits)</p>
<p><a href="htt... | Feyre | 7,312 | <p>Well, maybe you can make something with this?</p>
<pre><code>a1 := SliceContourPlot3D[z,
x == 0, {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, Background -> Black,
ContourShading -> White, Contours -> 9,
TicksStyle -> {Red, Green, Blue}]
a2 := SliceContourPlot3D[z,
y == 0, {x, -5, 5}, {y, -5, 5}, {z, ... |
119,321 | <p>The idea is to make axes 3d, with a more subdued in black mesh, and shafts with their respective numbers with a maximum of 10 in each (black to print after the size of half a sheet craft, vine also print two per page )</p>
<p>z in the second drawing up</p>
<p>(mathematica ver 10.0.1.0 64 bits)</p>
<p><a href="htt... | yode | 21,532 | <p>Tricks to my mind,Suppose your version is 10.2 or later,although I don't sure you will like</p>
<pre><code>Show[SliceContourPlot3D[#,
"CenterPlanes", {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
ContourShading -> White] & /@ {x, y, z}, Axes -> True,
Boxed -> False, AxesOrigin -> {0, 0, 0}]
</code... |
3,429,338 | <p>Please help. I've missed some lectures, and now I'm stuck (my fault!). The lectures notes don't explain elaborately, and I can't find good tutorials online. I've somehow managed to arrive at <span class="math-container">$(Q \lor P) \land P$</span>. If this is correct, can this be simplified further? Thanks heaps.</p... | Randy Marsh | 391,136 | <p>It is equal to one because it contains <span class="math-container">$K_5$</span> and we can exhibit an embedding in genus 1. In general the problem of determining the genus of a graph is NP-hard.</p>
<p>Here is an approach for finding an embedding in orientable genus 1:</p>
<p>Cut out two open disks from the spher... |
3,517,714 | <p>Suppose <span class="math-container">$X = \lbrace 1,2,3,4,5 \rbrace$</span> and <span class="math-container">$\tau = \lbrace X, \emptyset , \lbrace 1 \rbrace , \lbrace 1,2 \rbrace , \lbrace 1,3,4 \rbrace , \lbrace 1,2,3,4 \rbrace , \lbrace 1,2,5 \rbrace \rbrace$</span>. </p>
<p>For <span class="math-container">$\ta... | Donald Splutterwit | 404,247 | <p>You have <span class="math-container">$\tau_{M} = \lbrace \lbrace 1,3,5 \rbrace , \emptyset , \lbrace 1 \rbrace \rbrace$</span>. You missed <span class="math-container">$\{1,3\}$</span> and <span class="math-container">$\{1,5\}$</span>.</p>
<p><span class="math-container">$(f \mid_{M})^{-1}(0) = \{1,3\}$</span> an... |
1,468,395 | <p>I have $\Omega={1,2,3,...}$ and the possibility of each number is $P(A) = 2^{-n}$, $n=1,2,3...$
I have to prove $P(\Omega) = 1$ . I can understand that from the graph, but how do I actually prove it?</p>
| Daniel Valenzuela | 156,302 | <p>So I am not sure which bundle interpretation you are referring to, so I will try to exploit this terminology in hope to find what you were looking for.</p>
<p>First try: As soon as you have a non-trivial map $G\to H$ which is like a bundle, i.e. all fibers are non-trivial, you have a normal subgroup, namely the fib... |
224,559 | <p>Given two probability distributions $p,q$ on a finite set $X$, the quantity variously known as <strong>relative information</strong>, <strong>relative entropy</strong>, <strong>information gain</strong> or <strong>Kullback–Leibler divergence</strong> is defined to be </p>
<p>$$ D_{KL}(p\|q) = \sum_{i... | usul | 29,697 | <p>I just want to make a point (and apologies if this seems already clear or obvious) that many such characterizations wil have a simple feature in common: a form of additivity axiom.</p>
<p>This goes right back to Shannon's 1948 characterization of entropy (and is common to any number of subsequent variations). You s... |
3,037,161 | <p>Let <span class="math-container">$$f:\overline{B(0,1)}\rightarrow\mathbb{C}$$</span> be continuous and holomorphic on <span class="math-container">$B(0,1)$</span>. Consider the function <span class="math-container">$$z\mapsto F(z):=f(z)\overline{f(\overline{z})}.$$</span> Show </p>
<p>(i) <span class="math-containe... | farruhota | 425,072 | <p>Note that the equation is a circle with center <span class="math-container">$O(5,0)$</span> and radius <span class="math-container">$3$</span>:
<span class="math-container">$$x^2 -10x+y^2 +16=0 \iff (x-5)^2+y^2=9$$</span>
The objective function is <span class="math-container">$\frac yx=k \iff y=kx$</span>, whose con... |
1,868,495 | <p>Consider $\lim_{ x \to -\infty} \sqrt{x^2-x+1}+x$ </p>
<p>Rationalising, one will get, $\lim_{x \to -\infty} \frac{1-x}{\sqrt{x^2-x+1}-x}$, which after taking x common and cancelling out gives $-\infty$.</p>
<p>Now, replace $x$ by $-x$, so the limit becomes, $\lim_{x \to \infty} \frac{1+x}{\sqrt{x^2+x+1}+x}$, whic... | N. F. Taussig | 173,070 | <p>\begin{align*}
\lim_{x \to -\infty} \left(\sqrt{x^2 - x + 1} + x\right) & = \lim_{x \to -\infty} \left(\sqrt{x^2 - x + 1} + x\right) \cdot \frac{\sqrt{x^2 - x + 1} - x}{\sqrt{x^2 - x + 1} - x}\\
& = \lim_{x \to -\infty} \frac{x^2 - x + 1 - x^2}{\sqrt{x^2 - x + 1} - x}\\
& = \lim_{x \to -\infty} \frac{-x ... |
5,249 | <p>Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangements</a> of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( 1-\frac{1}{(1)_q!}+\frac1{(2)_q!}-\frac1{(3)_q!}+\cdots+(-1)^{n}\fr... | Greg Marks | 6,521 | <p>In homage to Serge Lang, I might suggest that your friend pick up any book on Morita theory and solve all the exercises.  Less facetiously, I might point out that many interesting ring-theoretic properties can be characterized, sometimes unexpectedly, by properties of the (right) module category over the ring.... |
3,829,161 | <p>Using the Axiom of choice, one can show that (see <a href="https://math.stackexchange.com/questions/288075/on-every-infinite-dimensional-banach-space-there-exists-a-discontinuous-linear-f?rq=1">here</a>) every infinite dimensional normed vector space has discontinuous functionals. My question is: Is this also true f... | Tsemo Aristide | 280,301 | <p><span class="math-container">$V^x\times W^y$</span> subset <span class="math-container">$U$</span> does not implies that <span class="math-container">$V^x\times W^{y'}\subset U$</span> if <span class="math-container">$(x,y)\neq (x',y')$</span>.</p>
|
1,377,192 | <p>I have stumbled upon the following reasoning, but I'm not sure if it's correct. Here it goes:
Domain X</p>
<ol>
<li>$\forall x :\phi(x)⟹\gamma(x)$</li>
<li>Let $E\subseteq X⟹[\forall x\in E :\phi(x)⟹\gamma(x)]$</li>
<li>Suppose I know, by some property of $E$, that $\forall x\in E :\phi(x)$, that is for every $x\in... | Sepideh Abadpour | 93,266 | <p>for the second limit:<br>
from mac-lauren series we have: $x\to 0\Rightarrow \ln(1+x)\approx x+\frac{x^2}{2}$ so:<br>
$$\lim\limits_{n→\infty} \left(1+ \frac {1}{n}\right)^n=\lim_{n\to +\infty}e^{n\ln(1+\frac{1}{n})}=\lim_{n\to +\infty}e^{n(\frac{1}{n}+\frac{1}{2n^2})}=\lim_{n\to\infty}e^{1+\frac{1}{2n}}=e^1=e$$</p>... |
585,114 | <p>I have read a bit about Gauss, who was well known for being careful in only publishing work he had perfected (or in his own words "few, but ripe"). What is interesting to me about Gauss though is that all accounts of his students and contemporaries essentially make him appear flawless. Dedekind's recollections of ho... | Grady Player | 87,555 | <p>how about errors of omission? Gauss was famous for not publishing monumental discoveries until after they were rediscovered.</p>
|
1,264,613 | <p>While reading in a Discrete maths text book, there was this question:</p>
<blockquote>
<p>How many bit strings of length n are palindromes?</p>
</blockquote>
<p>The answer is:</p>
<blockquote>
<p><span class="math-container">$2^\frac{n+1}{2}$</span> for odd and <span class="math-container">$2^\frac{n}{2}$</span> for... | bashfuloctopus | 42,974 | <p>If a string is length $n$, then we can write it as either being length $2k+1$ if $n$ is odd, or $2*k$ if $n$ is even, where $k \in \mathbb Z$. </p>
<p>In either case, the first half of the digits determine the values of the second half of the digits. Indeed, let $$(a_m)_{m=1}^{n}$$ be a string that is a palindrome.... |
490,556 | <p>I'm solving some practice problems to prepare for a competitive exam. Here is one which I'm trying to do for some time but still haven't found a solution to:</p>
<p>"In the given figure, <span class="math-container">$∠ABC = 2∠ACB$</span> and <span class="math-container">$AB = DC$</span>. Also, <span class="math-con... | Sanjay Jain | 135,621 | <p>let angles *=y and o=x, AE=DE so base angles are equal. and by CPCT angle EDC will be 2y. So from triangle ADC y+y+2y+x=180 and from triangle ABC x+x+y+y+x=180, solving these x=36 and hence 2x=72 degrees </p>
|
19,766 | <p>I know that for a series of non-negative, continuous functions $u_{n}(x)$, a sufficient condition for uniform convergence of $\sum u_{n}(x)$ to $u(x)$ is for $u(x)$ to be continuous in $I\subset \mathbb{R}$. </p>
<p>But I can't think of an example where $\sum f_{n}(x)\to f(x)$ uniformly in $I\subset \mathbb{R}$ but... | NebulousReveal | 2,548 | <p>This follows from the following:</p>
<p><strong>Theorem.</strong> If the limit function $f(x)$ of a pointwise convergent sequence of continuous functions $\{f_{n}(x) \}$ is discontinuous, then the convergence of the sequence $\{f_{n}(x) \}$ is nonuniform.</p>
<p>So the contrapositive is: $\{f_{n}(x) \} \to f \ \te... |
283,310 | <p>For $n\in\mathbb{N}^{+}$, let $c_{n}$ denote the number of simple non-isomorphic cycle matroids of graphs on $n$ vertices. That is, let</p>
<p>$$A(n)=\{M(G)\;;\;G\text{ is a graph on }n\text{ vertices}\},$$</p>
<p>and let $B(n)$ be a largest subset of $A(n)$ such that no two elements of $B(n)$ are isomorphic (as m... | Tony Huynh | 2,233 | <p>There are $2^{\binom{n}{2}}$ labelled graphs on $n$ vertices. Since isomorphic graphs have isomorphic graphic matroids, $c_n$ is at most the number of non-isomorphic graphs on $n$ vertices (see <a href="https://oeis.org/A000088" rel="noreferrer">OEIS A000088</a>). In particular, $c_n$ is at most $2^{\binom{n}{2}}$... |
1,780,856 | <p>The least positive integer that is divisible by $2, 3 ,4,$ and $5,$ and is also a perfect square, perfect cube, $4^{th}$ power, and $5^{th}$ power, can be written in the form $a^b$ for positive integers $a$ and $b$. What is the least possible value of $a+b$?</p>
<p>The answer is $90$.</p>
<p>Even completely cheat... | Josh Hunt | 282,747 | <p>I'm not sure if this is the source of your confusion, but a "5th power" is something of the form $n^5$, not $5^n$ (a "power of 5").</p>
|
66,525 | <p>Let $f(x)=a+bx^2$. Define $f_n(x)$ to be the $n$-fold composition of $f$. That is
$$f_1(x)=f(x)$$
$$f_2(x)=f \circ f(x)$$
$$f_n(x)=f \circ f_{n-1}(x), n \ge 2$$</p>
<p>Is there a way to find a formula for $f_n$?</p>
<p>I tried to write down $f_2$, $f_3,\ldots$, but I don't see any pattern.</p>
| Shaun Ault | 13,074 | <p>I don't believe there is a "nice" formula for $f_n$ or even a pattern. Here's my reasoning:</p>
<p>If the graph of $y = a + bx^2$ intersects the line $y=x$, then there can be chaotic behavior in the values $f_n(x)$ for general $x$. See the neat animation on the Wiki article for "cobweb plot":</p>
<p><a href="htt... |
4,379,228 | <p>I came across an example of an integral by change of variable that reads as follows</p>
<p>Find <span class="math-container">$$\int_a^{b} \frac{1}{\sqrt[2]{(x-a)(b-x)}}dx$$</span></p>
<p>What they do is to assume that <span class="math-container">$a<b$</span>. They claim that the convenient change of variable con... | lab bhattacharjee | 33,337 | <p>Like mio,</p>
<p>as we need <span class="math-container">$$a< x<b$$</span></p>
<p><span class="math-container">$$\iff -\dfrac{b-a}2=a-\dfrac{a+b}2<x-\dfrac{a+b}2<b-\dfrac{a+b}2=\dfrac{b-a}2$$</span></p>
<p>WLOG <span class="math-container">$x-\dfrac{a+b}2=\dfrac{b-a}2\cdot\cos2t$</span> where <span class... |
191,738 | <p>I have the following limit:</p>
<p>$$\lim_{n\rightarrow\infty}e^{-\alpha\sqrt{n}}\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k}\sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}$$</p>
<p>where $\alpha>0$.</p>
<p>Evaluating this in Mathematica suggests that this converges, but I don't know how to evaluate it. Any he... | Did | 6,179 | <p>Let $X_n$ denote a Poisson random variable with parameter $\alpha\sqrt{n}$ and $Y_n$ a <a href="http://en.wikipedia.org/wiki/Negative_binomial_distribution" rel="nofollow">negative binomial random variable</a> with parameters $(n,\frac12)$. Recall that this means that, for every $k\geqslant0$,
$$
P[X_n=k]=\mathrm e^... |
4,245,883 | <p>Here's a problem from my probability textbook:</p>
<blockquote>
<p>Of three independent events the chance that the first <em>only</em> should happen is <em>a</em>; the chance of the second <em>only</em> is <span class="math-container">$b$</span>; the chance of the third <em>only</em> is <span class="math-container">... | Ian | 83,396 | <p>The accepted answer gets the correct result largely by coincidence. There is not a priori any relationship between the time when the expected number of heads and tails are equal vs. the expected time when the number of heads and tails are equal, and the latter is what is asked for.</p>
<p>You first convert to a rand... |
11,149 | <p>I could sample a set of m elements from the uniform distribution over a universe $U$ of n >> m elements. Alternately, I could select a random probability distribution $\mathcal{D}$, and sample $m$ elements from $\mathcal{D}$. </p>
<p>EDIT (per Michael Lugo): When I say "select a random probability distribution", I ... | Chris Godsil | 1,266 | <p>My favorite undecidable problem in graph theory is the following: given a finite set of graphs $L$, is there a graph $G$ such that the neighborhood of each vertex in $G$ is isomorphic to a graph in $L$.</p>
<p>For a proof see: Peter M. Winkler. Existence of graphs with a given set of r-neighborhoods
Journal of Comb... |
262,684 | <p>If $S$ is any ring, and $P$ is a projective $S$ module, and $Q$ is any $S$ module, then will $Q \otimes_{S} P$ be a projective $S$-module?</p>
| rschwieb | 29,335 | <p>No. Let $S$ be a ring with nonprojective module $Q$. Then $Q\otimes_S S\cong Q$ is not projective.</p>
|
262,684 | <p>If $S$ is any ring, and $P$ is a projective $S$ module, and $Q$ is any $S$ module, then will $Q \otimes_{S} P$ be a projective $S$-module?</p>
| Brandon Carter | 1,016 | <p>No, let $S = P = \mathbf Z$ and $Q = \mathbf{Z}/ 2 \mathbf{Z}$. $P$ is free, hence projective, but $P \otimes Q \simeq \mathbf{Z}/2\mathbf{Z}$ is not.</p>
|
323,435 | <p>Please help me find answer for the following task:</p>
<p>Prove, that $A\in M_n(\mathbb{Q})$ satisfy $A^5=I$, and 1 isn't an eigenvalue. Show, that $4 \mid n$</p>
| P.. | 39,722 | <p>If $P(x)=x^5-1=(x-1)(x^4+x^3+x^2+x+1)$ then $P(A)=0$.
<br>
Prove that:</p>
<ul>
<li>$A-I$ is invertible,</li>
<li>If $Q(x)=x^4+x^3+x^2+x+1$ then $Q(A)=0$,</li>
<li>$Q(x)$ is irreducible over $\mathbb Q$,</li>
<li>If $\chi_A(x)$ is the characteristic polynomial of $A$ then $\chi_A(x)=Q^k(x)$ for some $k\in\mathbb N... |
1,450,737 | <p>I think that this expression very easy, but i don't know how resolve it. Please, help me, guys. So, there is:
$$
\dfrac{\cos 3\alpha - \sin 3\alpha}{\cos \alpha + \sin \alpha}, \;\;\; \mbox{if} \;\;\; \sin \left(\dfrac{\pi}{4} - \alpha\right) = 0,1.
$$
I make following transformation:
\begin{gather}
\dfrac{\cos 3\al... | Harish Chandra Rajpoot | 210,295 | <p>Notice, </p>
<ol>
<li>If $$\sin\left(\frac{\pi}{4}-\alpha\right)=0$$$$ \sin\alpha\cos\frac{\pi}{4}-\cos\alpha\sin\frac{\pi}{4}=0\implies \tan\alpha=1$$
Hence, $$\color{red}{1-2\sin 2\alpha}=1-2\frac{2\tan \alpha}{1+\tan^2\alpha}=1-2\frac{2(1)}{1+(1)^2}=\color{red}{-1}$$</li>
<li>If $$\sin\left(\frac{\pi}{4}-\alpha\... |
1,227,180 | <p>I would appreciate if somebody could help me with the following problem:</p>
<blockquote>
<p>Let $f(a)$ area of S, $A(a,a^2)$, $B(b,b^2)$ and $\overline{AB}=1$, given that:</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/nyWcH.png" alt="enter image description here"></p>
<p>Find that $$\lim_{a\to \inft... | Greg Martin | 16,078 | <p>It can be shown that the area above an upward-facing parabola, defined by $y=q(x)$, that is cut off by the secant line between $x=a$ and $x=b$ is equal to
$$
(b-a) \bigg( \frac13 q(a) - \frac23 q\bigg(\frac{a+b}2\bigg) + \frac13 q(b) \bigg).
$$
(Indeed, this calculation is what goes into Simpson's Rule!)</p>
<p>In ... |
1,227,180 | <p>I would appreciate if somebody could help me with the following problem:</p>
<blockquote>
<p>Let $f(a)$ area of S, $A(a,a^2)$, $B(b,b^2)$ and $\overline{AB}=1$, given that:</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/nyWcH.png" alt="enter image description here"></p>
<p>Find that $$\lim_{a\to \inft... | Ron Gordon | 53,268 | <p>The area of S is $\frac12 (b-a) (b^2+a^2) - \frac13 (b^3-a^3) = \frac16 (b-a)^3 $.</p>
<p>As $a \to \infty$, the difference betwen $b$ and $a$ vanishes. In this case,
$$\left |\overline{AB}\right |^2 = 1 = (a-b)^2 \left [ 1+(a+b)^2 \right ] \approx (b-a)^2 (1+4 a^2) $$ </p>
<p>Thus, </p>
<p>$$\lim_{a \to \inft... |
2,580,214 | <p><strong><em>Edit</em></strong> : I'm ''derivation'' and ''integration'' beginner. So I don't have any techniques to solve equations like that.This why I think there have to be a clever trick to solve this. a), b), c) was okay. You can see my solution below. I want to emphasize that I don't want to see the solution. ... | John Doe | 399,334 | <p>Hints:</p>
<p>a) This can be rearranged as $$\frac{df}{dx}-f=e^x$$
You can then use the method of <a href="http://weber.itn.liu.se/~krzma/DS2017/Integrating%20factor%20method.pdf" rel="nofollow noreferrer">integrating factors</a> to solve it.</p>
<p>b) This can be rearranged as $$\frac1f\frac{df}{dx}=e^x$$Then int... |
3,274,974 | <p>I want to read Carlitz paper on arithmetic functions and he defines a 'complement' of a function <span class="math-container">$f$</span> by <span class="math-container">$f'$</span>. He defines it this way: </p>
<p>If <span class="math-container">$f$</span> is a function on the set <span class="math-container">$N$<... | Robert Israel | 8,508 | <p>In fact, all you need is that <span class="math-container">$f$</span> is continuous. Consider two paths from <span class="math-container">$(-1,0)$</span> to <span class="math-container">$(1,0)$</span>, say the top and bottom halves of the unit circle. By the intermediate value theorem, for any <span class="math-co... |
195,547 | <p>What numerical methods can be used to study the initial value problem for the continuity equation where <span class="math-container">$ u = u(t, x) $</span></p>
<p><span class="math-container">$$
u_t + \nabla\cdot(\boldsymbol b u) = 0, \qquad t \in [0,T], \quad x=(x_1,x_2) \in \mathbb{R}^2
$$</span>
where <span clas... | Alex Trounev | 58,388 | <p>To illustrate the problem, I will give an example that differs from the one proposed by Riku. But in this case, numerical instability is better seen. The result is similar to erosion. Perhaps geologists will like this.</p>
<pre><code>b = {1, HeavisideTheta[x - y]}; L = 4; reg =
DiscretizeRegion[Rectangle[{-L, -L}... |
1,133,385 | <h3>Question</h3>
<p>Given a torsion-free $\mathbb{Z}$-module (aka. abelian group) $G$, let $i: \mathbb{Q} \hookrightarrow \mathbb{R}$ be the inclusion. I want to show that
$$
i \otimes \mathrm{id}: \mathbb{Q} \mathbin{\otimes_\mathbb{Z}} G \hookrightarrow \mathbb{R} \mathbin{\otimes_\mathbb{Z}} G
$$
is injective.</p>... | MooS | 211,913 | <p>Torsion-free modules over Dedekind domains are flat, in particular torsion-free abelian groups are flat.</p>
|
754,522 | <p>I'm trying to prove that that for any $\epsilon>0$, there exists rational $\frac{j}{k}$ such that $0<\frac{j}{k}<\epsilon$. Obviously $j,k\in \mathbb{N}$. This is not for homework, it's a lemma I needed for another proof I was working on. I was trying to avoid limits.</p>
<p>Proof:</p>
<p>Assume we can fi... | Christoph | 86,801 | <p>You are right, but in my opinion "$k < \frac{1}{\varepsilon}$ not for all $k\in\mathbb N$" is on the same level of "being clear" as the existence of a $k\in\mathbb N$ such that $\frac{1}{k} < \varepsilon$, so you could have just said the statement is clearly true to begin with.</p>
<p>A strict proof would use... |
1,118,458 | <p>A fairly pretty technique of showing that<br>
$$\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt{\pi}$$ is to square the integral, writing that square as the product of two integrals with integration variables $x$, and $y$, treating that as an integral over the whole plane $\Bbb{R}^2$, and then changing to polar coordina... | KCd | 619 | <p>In the 1720s, de Moivre had discovered the normal approximation to the binomial distribution. In the course of this work de Moivre showed $\binom{2n}{n}/2^{2n} \sim C/\sqrt{n}$ for some $C$ that he could estimate numerically using infinite series. Stirling identified $C$ as $1/\sqrt{\pi}$ (which is equivalent to Sti... |
3,820,722 | <p>I found a problem in calculus lecture notes where it is asked to find the supremum for the follownig set <span class="math-container">$X = [0, \sqrt{2}] ∩ Q$</span></p>
<p>I assumed that the <span class="math-container">$\sup X = \sqrt{2}$</span></p>
<p>Then I wrote (according to definition of supremum)</p>
<ol>
<li... | ilovebulbasaur | 586,948 | <p>Here's a more direct argument.</p>
<ol>
<li><p>Clearly if <span class="math-container">$x\in [0,\sqrt{2}]\cap \mathbb{Q}$</span>, we have <span class="math-container">$x\in [0,\sqrt{2}]$</span>, so <span class="math-container">$x\leq \sqrt{2}$</span>.</p>
</li>
<li><p>Now for all <span class="math-container">$\epsil... |
3,595,782 | <p>From Probability through problems by Marek Capinski,Jerzy Zastawniak,:</p>
<p>Find <span class="math-container">$\limsup_{n \to \infty}A_n$</span> and <span class="math-container">$\liminf_{n \to \infty}A_n$</span>,where </p>
<p><span class="math-container">\begin{eqnarray*}
A_n &=&\left(\frac 13-\frac1{n+... | Davide Giraudo | 9,849 | <p>We can start by the <span class="math-container">$\liminf$</span>. Let <span class="math-container">$x\in\mathbb R$</span>. Saying that <span class="math-container">$x\in\liminf A_n$</span> means that there exists <span class="math-container">$N$</span> such that <span class="math-container">$x\in A_n$</span> for al... |
3,106,501 | <p>In <span class="math-container">$\mathbb R \setminus \mathbb Q$</span> and <span class="math-container">$\mathbb R /\mathbb Q$</span>, what do these ("<span class="math-container">$\setminus$</span>","<span class="math-container">$/$</span>") symbols between the sets of real and rational numbers mean?</p>
| Henno Brandsma | 4,280 | <p>Depends on the context: </p>
<p><span class="math-container">$\mathbb{R} \setminus \mathbb{Q}$</span> is the set difference
between the reals and the rationals, so it equals the set of irrationals.</p>
<p><span class="math-container">$\mathbb{R}/\mathbb{Q}$</span> can mean the quotient of the group of reals by it... |
3,892,622 | <p>The following function
<span class="math-container">$$f(x)=\left\{\begin{array}{ll} 50, &0\leq x\leq10 \\ 10+6x-0.2x^2, & 10<x\leq30\end{array}\right.$$</span>
gives me the number of products that are sold in the <span class="math-container">$x$</span>-th day of a month.</p>
<p>How am I supposed to find t... | TheSilverDoe | 594,484 | <p>Yes.</p>
<p><span class="math-container">$$\sum_{k=0}^{30} f(k)= \sum_{k=0}^{10} f(k) + \sum_{k=11}^{30} f(k) = \sum_{k=0}^{10} 50 + \sum_{k=11}^{30}(10 + 6k -0.2k^2)$$</span>
<span class="math-container">$$=50 \times 11 + 10 \times 20 + 6 \left( \frac{30\times 31}{2} -\frac{10\times 11}{2} \right) - 0.2 \left( \fr... |
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