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 |
|---|---|---|---|---|
203,614 | <p>I have two matrices </p>
<p><span class="math-container">$$
A=\begin{pmatrix}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & c
\end{pmatrix}
\quad
\text{ and }
\quad
B=\begin{pmatrix}
d & e & f \\
d & e & f \\
d & e & f
\end{pmatrix}
$$</span></p>
<p>In reality mine are... | Michael E2 | 4,999 | <p>It doesn't happen here:</p>
<pre><code>SeedRandom[0];
aa = RandomReal[{-10, 10}, {1000, 1000}];
bb = ConstantArray[RandomReal[{-10, 10}, {1000}], {1000}];
eva = Eigenvalues@aa;
evc = Eigenvalues[aa + bb];
ListPlot[{ReIm@eva, ReIm@evc}, ImageSize -> Large, MaxPlotPoints -> 1000]
</code></pre>
<p><a href="ht... |
3,156,359 | <p>I am currently attempting to solve a system of quadratic (and linear) systems that I have run into while trying to triangulate sound.</p>
<p>My hypothetical setup includes 3 sensors on a perfectly equilateral triangle, with one sensor located at <span class="math-container">$(0,0)$</span> and the other two located ... | Rohit Pandey | 155,881 | <p>Solving system of quadratic and indeed, general polynomial equations is possible with techniques like Buchberger's algorithm. See the first two chapters of <a href="http://people.dm.unipi.it/caboara/Misc/Cox,%20Little,%20O%27Shea%20-%20Ideals,%20varieties%20and%20algorithms.pdf" rel="nofollow noreferrer">the book</a... |
76,505 | <p>In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra. </p>
<p>He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie de mon énergie a été consacrée à un travail de réflexion sur les <em>fondements de l'algèbre (co)homologique no... | David Roberts | 4,177 | <p>The geometric realisation functor (read: homotopy colimit for nice situations) from simplicial spaces to $Top$ preserves pullbacks only when you take the $k$-ification of the product in $Top$, or work with compactly generated spaces (Edit: or a convenient category of spaces). This is false in the category of all spa... |
692,582 | <p>When I was reading a paper, I found an strange derivation like
$$\int^{+\infty}_{-\infty}\mathrm{ln}(1+e^w)f(w)dw\\=\int^0_{-\infty}\ln(1+e^w)f(w)+\int^\infty_0[\ln(1+e^{-w})+w]f(w)dw$$
when $w$ is the normal random variable and $f(w)$ is the normal density.</p>
<p>Why is that natural log integration broken up into... | bubba | 31,744 | <p>Use a <em>parametric</em> spline, in which $x$ and $y$ are spline functions (or even just polynomial functions) of some independent parameter $t$. </p>
<p>Here is a parametric cubic spline with 4 segments created in Powerpoint. Or, looking at it another way, this is just a string of four cubic Bezier curves that jo... |
1,877,567 | <p>I need help calculating two integrals</p>
<p>1)
$$\int_1^2 \sqrt{4+ \frac{1}{x}}\mathrm{d}x$$
2)
$$\int_0^{\frac{\pi}{2}}x^n sin(x)\mathrm{d}x$$</p>
<p>So I think on the 1st one I will have to use substitution, but I don't know what to do to get something similar to something that's in the known basic integrals.
2... | David Quinn | 187,299 | <p>HINT...for the first one, try substituting $$\frac 1x=4\tan^2\theta$$</p>
<p>For the second, try integrating by parts twice to obtain a reduction formula</p>
|
1,877,567 | <p>I need help calculating two integrals</p>
<p>1)
$$\int_1^2 \sqrt{4+ \frac{1}{x}}\mathrm{d}x$$
2)
$$\int_0^{\frac{\pi}{2}}x^n sin(x)\mathrm{d}x$$</p>
<p>So I think on the 1st one I will have to use substitution, but I don't know what to do to get something similar to something that's in the known basic integrals.
2... | Zau | 307,565 | <p>Hint:</p>
<p><strong>For the first problem:</strong>
$$\int \sqrt{4+ \frac{1}{x}}\mathrm{d}x = \int \frac{\sqrt{4x+ 1 }}{\sqrt x}\mathrm{d}x$$</p>
<p>Let $u = \sqrt x$ , $x = u^2$, $2u du =dx $</p>
<p>$$ \int \frac{\sqrt{4u^2+ 1 }}u 2u \mathrm{d}u$$</p>
<p>Try some trigonometric substitution.</p>
<p><strong>For... |
3,017,602 | <p>I am trying to solve the following differential equation:
<span class="math-container">$$\frac{dy}{dx}=\frac{y^{1/2}}{2}, \quad y>0$$</span></p>
<p>Here is what I tried:
<span class="math-container">$$
\begin{split}
\frac{dy}{dx} &= \frac{y^{1/2}}{2} \\
2y^{1/2}dy &= dx\\
6y^{3/2} &= x+c\\
y &= \... | hamam_Abdallah | 369,188 | <p>You made a mistake. compare with</p>
<p><span class="math-container">$$\frac{1}{\sqrt{y}}dy=\frac 12 dx$$</span></p>
<p><span class="math-container">$$\frac{dy}{2\sqrt{y}}=\frac 14 dx$$</span></p>
<p><span class="math-container">$$d(\sqrt{y})=d(\frac x4+C)$$</span></p>
<p><span class="math-container">$$\sqrt{y}=... |
3,330,938 | <p>On Wikipedia page about Weierstrass factorization theorem one can find a sentence which mentions a generalized version so that it should work for meromorphic functions. I mean:</p>
<blockquote>
<p>We have sets of zeros and poles of function <span class="math-container">$f$</span>. How could we use that sets to f... | David C. Ullrich | 248,223 | <p>First, a meromorphic function in the plane <em>is</em> the quotient of two entire functions: Say <span class="math-container">$f$</span> is entire except for poles at <span class="math-container">$p_j$</span>. Say the pole at <span class="math-container">$p_j$</span> has order <span class="math-container">$n_j$</s... |
4,249,281 | <p>In a game, 6 balls are chosen from a set of 40 balls numbered from 1 to 40. Find the probability that the number 30 is drawn and it is the highest number drawn in at least one of the next five games.</p>
<p>I have <span class="math-container">$X\sim \operatorname{Bin}(5,6/29)$</span> and <span class="math-container"... | user71207 | 814,679 | <blockquote>
<p>For a particular game, you want the probability that 30 is drawn and the other five numbers drawn are smaller.
one way is to look at the probability all six are 30 or less, and the probability all six are 29 or less</p>
</blockquote>
<p>That means <span class="math-container">$P(X≥1 \text{and others are... |
2,236,717 | <p>Let $S$ be a regular domain of characteristic $p>0$ with fraction field $K$. Assume that $K$ is $F$-finite, meaning that $K$ is a finite module over $K^p$. Does it follow that $S$ is also $F$-finite?</p>
<p>Diego</p>
| Yoël | 401,292 | <p>I believe the result must be true though I don't have the answer right now, but maybe this can help (or not):</p>
<p>Denote $\phi: A\rightarrow A$, $x\mapsto x^p$ the Frobenius morphism, and still denote $\phi: K\rightarrow K$ its extension to $K$.</p>
<p>Let $n\ge 1$ be an integer and assume $\mathcal{B}=\{f_1, .... |
2,632,273 | <p>so basically I want to know why when we have something like:</p>
<p>$$v(x) = x - y + 1$$
If we take the derivative with respect to x, it yields:</p>
<p>$$v'(x) = 1 - \frac{dy}{dx}$$</p>
<p>Now I still don't understand why when it comes to implicit differentiation, we need to tag a $y'$ or $\frac{dy}{dx}$ after ev... | ElfHog | 527,282 | <p>Basically when we are "taking differentiation with respect to $x$", we mean that (intuitively) "when $x$ has a small change, how will the function change". </p>
<p>Now $y$ can be a function of $x$, e.g. $y=x^2$ or $y=e^{e^x}$. So a small change in $x$ will cause a change (called $\frac{dy}{dx}$) in $y$. </p>
<p... |
373,068 | <p>For a real number $a$ and a positive integer $k$, denote by $(a)^{(k)}$ the number $a(a+1)\cdots (a+k-1)$ and $(a)_k$ the number
$a(a-1)\cdots (a-k+1)$. Let $m$ be a positive integer $\ge k$. Can anyone show me, or point me to a reference, why the number
$$ \frac{(m)^{(k)}(m)_k}{(1/2)^{(k)} k!}= \frac{2^{2k}(m)^{(k... | Shaswata | 68,110 | <p>We use the idea- product of n consecutive integers is divisible by $n!$.</p>
<p>The numerator = $$2^{2k}(m-k+1)(m-k+2)\cdots (m-1)(m)(m)(m+1)\cdots (m+k-2)(m+k-1)$$</p>
<p>= 2k-1 consecutive integers $\times m$</p>
<p>This must be divisible by $2^{2k}(2k-1)!\cdot m$</p>
|
91,700 | <p>Suppose that $A,C$ are $C^*$-algebras and $\phi:A \to C$ is a completely positive, orthogonality-preserving linear map.
(Orthogonality preserving means: if $a,b \in A$ satisfy $ab=0$ then $\phi(a)\phi(b) = 0$.)
Then:</p>
<p>(i) For any $a,b,c \in A$,
$$ \phi\left(ab\right)\phi\left(c\right) = \phi\left(a\right)\phi... | jorge | 100,313 | <p>The general form of a (bounded) orthogonality preserving linear mapping between C*-algebras is obtained here:
<a href="http://www.sciencedirect.com/science/article/pii/S0022247X08007245" rel="nofollow">http://www.sciencedirect.com/science/article/pii/S0022247X08007245</a></p>
|
2,036,301 | <p>Can someone please help me prove that this series is convergent? <br></p>
<p>The problem is I don't know what to do with sin.<br> </p>
<p>$$\sum_{n=1}^{\infty} 2^n \sin{\frac{\pi}{3^n}} $$</p>
| Community | -1 | <p>As $\sin x<x$ for $x>0$ we can estimate
$$
\sum_{n=1}^\infty2^n\sin(\pi/3^n)\leq\sum_{n=1}^\infty2^n\frac{\pi}{3^n}.
$$
The RHS is finite by the geometric series so our (positive) series is convergent.</p>
|
2,036,301 | <p>Can someone please help me prove that this series is convergent? <br></p>
<p>The problem is I don't know what to do with sin.<br> </p>
<p>$$\sum_{n=1}^{\infty} 2^n \sin{\frac{\pi}{3^n}} $$</p>
| hamam_Abdallah | 369,188 | <p>We have $$\sin(X)\sim X \;(X\to 0)$$</p>
<p>$$\implies 2^n\sin(\frac{\pi}{3^n})\sim (\frac{2}{3})^n\;(n\to+\infty)$$</p>
<p>$\implies \sum 2^n\sin(\frac{\pi}{3^n})$ is convergent since the geometric series $\sum (\frac{2}{3})^n$ is positive and convergent.</p>
|
67,513 | <p>When processing a larger Dataset I came up do a point where I want to form a dataset with culumn heads from an intermediate structure. Here is an example of this structure:</p>
<pre><code>test = {<|"name" -> "alpha",
"group" -> "one"|> -> {<|"value" -> 459|>}, <|"name" -> "beta",... | WReach | 142 | <p>If we assume that we are starting from the exhibited dataset:</p>
<pre><code>dataset = Dataset@assoc;
</code></pre>
<p>... then we can reshape it like this:</p>
<pre><code>dataset[All, Apply@Association]
</code></pre>
<p>Or, equivalently:</p>
<pre><code>dataset[Map[Apply@Association]]
</code></pre>
<p><img src... |
187,395 | <p>I can't find my dumb mistake.</p>
<p>I'm figuring the definite integral from first principles of $2x+3$ with limits $x=1$ to $x=4$. No big deal! But for some reason I can't find where my arithmetic went screwy. (Maybe because it's 2:46am @_@).</p>
<p>so </p>
<p>$\delta x=\frac{3}{n}$ and $x_i^*=\frac{3i}{n}$</... | Mikasa | 8,581 | <p><strong>Hint:</strong> $$\int_a^bf(x)dx=\lim_{n\to\infty}\sum_1^nf(a+\frac{b-a}{n}i)\frac{b-a}{n}$$</p>
|
4,046,532 | <p><strong>QUESTION 1:</strong> Let <span class="math-container">$f, g: S\rightarrow \mathbb{R}^m$</span> be differentiable vector-valued functions and let <span class="math-container">$\lambda\in \mathbb{R}$</span>. Prove that the function <span class="math-container">$(f+g):S\rightarrow \mathbb{R}^m$</span> is also d... | Jakobian | 476,484 | <ol>
<li><p>This is correct. Distributivity comes pretty much directly from definition, try calculating <span class="math-container">$(f+g)\circ X(p)$</span>. Recall that <span class="math-container">$(f+g)(q) := f(q)+g(q)$</span>.</p>
</li>
<li><p>This is wrong. Recall that differentiability is a local property. That ... |
24,195 | <p>I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce.
Does someone know some good references (article, book)? It would be very helpful for me.
To be more precise, consider a countable group $G$ and a 2-cocycle $\phi :G^2\rightarrow S^1$ where $S^1$ is the group of complex number of... | Roland Bacher | 4,556 | <p>This is not at all a complete answer but a remark which can be improved.
It is a completely rewritten and replaces bullshit (explaining the first comments.)</p>
<p>Suppose the answer to Segerman's question is no. There exists thus a counterexample given by
a decreasing sequence $r_1\geq \dots \geq r_n$ of radii su... |
24,195 | <p>I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce.
Does someone know some good references (article, book)? It would be very helpful for me.
To be more precise, consider a countable group $G$ and a 2-cocycle $\phi :G^2\rightarrow S^1$ where $S^1$ is the group of complex number of... | David Eppstein | 440 | <p>It's still not a yes-or-no answer to your question, but it seems to be true that a collection of circles with area 1/9 can always fit into a circle of area 1. Or, more strongly, if the circles are placed largest-first, then no matter how the larger circles are placed (by a malicious adversary trying to prevent the p... |
24,195 | <p>I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce.
Does someone know some good references (article, book)? It would be very helpful for me.
To be more precise, consider a countable group $G$ and a 2-cocycle $\phi :G^2\rightarrow S^1$ where $S^1$ is the group of complex number of... | Gerhard Paseman | 3,402 | <p>After weeks of sweating through computations without success, I think I have an auxillary result which I can use to tackle the problem posted above. However, the path to the conclusion is so surprising that I invite verification, in particular to point out any show-stopper mistake that may or may not be lurking.</p... |
2,834,864 | <p>Is it safe to assume that if $a\equiv b \pmod {35 =5\times7}$</p>
<p>then $a\equiv b\pmod 5$ is also true?</p>
| Vera | 169,789 | <p>If $hH=kK$ and $e$ denotes the identity then $e\in H=h^{-1}kK$.</p>
<p>This implies that $h^{-1}kK=K$ because a coset of a subgroup that contains the indentity must be the subgroup itself.</p>
<p>So we end up with $H=K$.</p>
|
2,668,839 | <blockquote>
<p>Finding range of $$f(x)=\frac{\sin^2 x+4\sin x+5}{2\sin^2 x+8\sin x+8}$$</p>
</blockquote>
<p>Try: put $\sin x=t$ and $-1\leq t\leq 1$</p>
<p>So $$y=\frac{t^2+4t+5}{2t^2+8t+8}$$</p>
<p>$$2yt^2+8yt+8y=t^2+4t+5$$</p>
<p>$$(2y-1)t^2+4(2y-1)t+(8y-5)=0$$</p>
<p>For real roots $D\geq 0$</p>
<p>So $$16... | MrYouMath | 262,304 | <p>Hint: Use
$$\sin^2 x +4\sin x+5 = (\sin x +2)^2 +1$$
and
$$2\sin^2 x +8 \sin x +8 = 2\left(\sin x + 2 \right)^2.$$</p>
<p>Also, break the fraction into two pieces</p>
<p>$$\dfrac{(\sin x +2)^2 +1}{2\left(\sin x + 2 \right)^2}=\dfrac{1}{2}+\dfrac{1}{2}\dfrac{1}{\left(\sin x + 2 \right)^2}$$</p>
|
557,426 | <p>I have 5 ring oscillators whose frequencies are f1, f2, ..., f5. Each ring oscillator (RO) has 5 inverters. For each RO, I just randomly pick 3 inverters out of 5 inverters. For example, in RO1, I pick inverter 1,3,5 (Notation: RO1(1,3,5)). So I have the following:</p>
<p>RO1(1,3,5) (I call this is the configuratio... | John Dvorak | 49,851 | <p>To encode a permutation into a number, you can use the <a href="http://en.wikipedia.org/wiki/Factoradic" rel="nofollow noreferrer">factoradic base</a>. A factoradic representation of a positive integer $x$ is the unique sequence of digits $d_n..d_0$ such that each digit $d_n$ is at most $n$ and the sum $\sum_{i=0}^{... |
2,541,997 | <p>For what values of n can {1, 2, . . . , n} be partitioned into three subsets
with equal sums?</p>
<p>I noticed that somehow the sum from 1 to n hast to be a multiple of 3 and the common sum among these 3 subset is this sum divided by 3, but it's still not a convincing argument. How do you prove there exists 3 subse... | Asinomás | 33,907 | <p>Clearly we need $n\equiv 0,2\bmod 3$ for the sum to be a multiple of $3$.</p>
<p>We need two disjoint sets with sum $s=\frac{n(n+1)}{6}$.</p>
<p>For the first set $A$ take $1,3,5,\dots$ until the next number would exceed $s$.</p>
<p>For the second set $B$ take $2,4,\dots $ until the next number would exceed $s$.<... |
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | Ilmari Karonen | 9,602 | <p><a href="http://www.proofwiki.org/" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GH4r0.png" alt="ProofWiki, the online compendium of mathematical proofs"></a></p>
|
2,221,807 | <p>I know that this question has been answered before, however I have not seen a response that satisfies me on whether my proof will work.</p>
<p><strong>Proof</strong></p>
<p>Suppose $A \cup B$ is a separation of $X$. Then WLOG $X-A=B$ and is finite, but this implies that $X-B$ is infinite thus $B$ is not a open set... | DMcMor | 155,622 | <p>This is fine, but you don't need to say 'WLOG'. If $A\cup B$ is a separation of $X$ then it must be the case that $B$ is finite because $X\setminus A=B$ and $A$ is open, and this doesn't cause any possible generality issues.</p>
|
2,221,807 | <p>I know that this question has been answered before, however I have not seen a response that satisfies me on whether my proof will work.</p>
<p><strong>Proof</strong></p>
<p>Suppose $A \cup B$ is a separation of $X$. Then WLOG $X-A=B$ and is finite, but this implies that $X-B$ is infinite thus $B$ is not a open set... | A. Thomas Yerger | 112,357 | <p>A slick way of saying what you've said is that a set is disconnected if it can be partitioned into two non-empty clopen sets. But in this topology no set can be clopen, as a set cannot be both finite and infinite.</p>
|
145,429 | <p>I have expression like this:</p>
<pre><code>expr = xuyz;
</code></pre>
<p>then</p>
<pre><code>Head[expr] = xuyz
</code></pre>
<p>But I wanted the product of four factors, so it should have been written as
<code>x*u*y*z</code> or <code>x u y z</code>, because Mathematica understands multiplication of four single... | Bob Hanlon | 9,362 | <pre><code>ClearAll[testVars]
testVars::usage =
"testVars[expr, allowedVars] returns variables appearing in expr that \
are not in the list allowedVars.";
testVars[expr_, allowedVars_?VectorQ] :=
Complement[Variables[Level[expr, {-1}]], allowedVars]
vars = {x, y, z};
testVars[xyx, vars]
(* {xyx} *)
</code></... |
368,789 | <p>Suppose I have a family of elliptic curves $E_{n}/\mathbb{Q}$. I would like to determine the torsion subgroup of $E_{n}(\mathbb{Q})$ denoted by $E_{n}(\mathbb{Q})_{\textrm{tors}}$. Two ways to do this are using Nagell-Lutz and computing the number of points over $\mathbb{F}_{\ell}$ for various $\ell$. Are there othe... | Álvaro Lozano-Robledo | 14,699 | <p>The two ways that you mention, plus looking your curve up in <a href="http://www.lmfdb.org/EllipticCurve/Q" rel="noreferrer">a database</a> (as Matt E. suggests), are the most practical and efficient ways I can think of. Here are two other ways to do this, one practical (but not as efficient) and one which is by no ... |
3,768,086 | <p>Show that <span class="math-container">$(X_n)_n$</span> converges in probability to <span class="math-container">$X$</span> if and only if for every continuous function <span class="math-container">$f$</span> with compact support, <span class="math-container">$f(X_n)$</span> converges in probability to <span class="... | lulu | 252,071 | <p>The problem is not clear as stated.</p>
<p>Interpretation <span class="math-container">$\#1$</span>: If you interpret it as "find the probability that the game end in an evenly numbered round" you can reason recursively.</p>
<p>Let <span class="math-container">$P$</span> denote the answer. The probability... |
1,428,377 | <p>So I was watching the show Numb3rs, and the math genius was teaching, and something he did just stumped me.</p>
<p>He was asking his class (more specifically a student) on which of the three cards is the car. The other two cards have an animal on them. Now, the student picked the middle card to begin with. So the c... | Graham Kemp | 135,106 | <p>The situation your first intuition tells you happens is that if the host picks a card <em>entirely at random</em> then there's an equal chance of either other cards revealing a car.</p>
<p>However, the <em>actual</em> situation is that the host picks a card <em>after</em> you made the first selection.</p>
<p>So, y... |
498,694 | <p>So, I'm learning limits right now in calculus class.</p>
<p>When $x$ approaches infinity, what does this expression approach?</p>
<p>$$\frac{(x^x)}{(x!)}$$</p>
<p>Why? Since, the bottom is $x!$, doesn't it mean that the bottom goes to zero faster, therefore the whole thing approaches 0?</p>
| Ron Gordon | 53,268 | <p>Here's $10^{10}$:</p>
<p>$$10 \cdot 10 \cdot 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10$$</p>
<p>Here's $10!$:</p>
<p>$$10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$$</p>
<p>Which one is bigger? Carry that thought out for larger and larger numbers, and you'll see th... |
1,672,080 | <p>I have troubles understanding the concepts of quotient topology and product topology (in the infinite case). </p>
<p>I know that we want to give a topology to new spaces built from the old ones, but the thing is that I can't figure out why is the definition for quotient topology natural since we only require that t... | Alex Provost | 59,556 | <p>A guiding principle in modern mathematics is that studying morphisms between objects is even more important than studying the objects themselves. In the category of topological spaces, this means that one should be more interested in continuous maps between spaces than in the spaces themselves. From this point of vi... |
3,612,351 | <p>It is given that a function f(x) satisfy:
<span class="math-container">$$f(x)=3f(x+1)-3f(x+2)\quad \text{ and } \quad f(3)=3^{1000}$$</span> then find value of <span class="math-container">$f(2019)$</span>.</p>
<p>I further wanted to ask that is there some general method to solve such equation. The method that I kn... | Gareth Ma | 623,901 | <p><span class="math-container">$$ \begin{align} f(x)&=f(x-1)-\frac{f(x-2)}{3}, f(3)=3^{1000}\\
f(x)&=f(x-1)-\frac{1}{3}f(x-2)\\
f(x+1)&=f(x)-\frac{1}{3}f(x-1)=(f(x-1)-\frac{1}{3}f(x-2))-\frac{1}{3}f(x-1)\\
f(x+1) &= \frac{2}{3}f(x-1)-\frac{1}{3}f(x-2)\\
f(x+2)&=f(x+1)-\frac{1}{3}f(x)=(\frac{2}{3}f(... |
3,612,351 | <p>It is given that a function f(x) satisfy:
<span class="math-container">$$f(x)=3f(x+1)-3f(x+2)\quad \text{ and } \quad f(3)=3^{1000}$$</span> then find value of <span class="math-container">$f(2019)$</span>.</p>
<p>I further wanted to ask that is there some general method to solve such equation. The method that I kn... | Bumblebee | 156,886 | <p>Note that, we can rewrite the given recurrence such that <span class="math-container">$$af(n+1)-f(n)=b(af(n)-f(n-1))$$</span> with complex numbers <span class="math-container">$a=\sqrt{3}e^{i\pi/6}, b=\dfrac{\sqrt{3}}{3}e^{i\pi/6}.$</span> Inductively we have that <span class="math-container">$$af(n+1)-f(n)=b^{k+1}(... |
2,236,008 | <p>Suppose $Z$ is a Gaussian distribution $N(0,\sigma^2)$. Is there a formula of upper bound for $P(Z\in [a,b])$, or do we know this probability is integral with respect to $\sigma\in \mathbf{R}$?</p>
| spaceisdarkgreen | 397,125 | <p>The probability is given by $$ P(Z\in[a,b]) =\int_a^b \frac{1}{\sqrt{2\pi\sigma^2}}e^{-x^2/(2\sigma^2)}dx$$ i.e. you integrate the density of the $N(0,\sigma^2)$ over the interval. You can also substitute $u = x/\sigma$ into the formula and get $$ P(Z\in[a,b]) = \int_{a/\sigma}^{b/\sigma} \frac{1}{\sqrt{2\pi}}e^{-u^... |
654,617 | <p>$v$ being a vector.
I never understood what they mean and haven't found online resources. Just a quick question.</p>
<p>Thought it was absolute and magnitude respectively when regarding vectors. need confirmation</p>
| AlexR | 86,940 | <p>I have found $|\cdot|$ to almost always represent the euclidean ($2$)-norm of a vector in $\mathbb K^n$, and $\Vert\cdot\Vert$ is a general sign for a norm. In different context, both may be used for different norms, though. I have even seen $|||\cdot |||$ for a "special" norm</p>
|
277,250 | <p>Let $\mathbb{N}$ be the set of natural numbers and $\beta \mathbb N$ denotes the Stone-Cech compactification of $\mathbb N$. </p>
<p>Is it then true that $\beta \mathbb N\cong \beta \mathbb N \times \beta \mathbb N $ ? </p>
| მამუკა ჯიბლაძე | 41,291 | <p>(Just noticed - already done by Todd Trimble in a comment:)</p>
<p>A proof by Stone duality: the dual question is whether the Boolean algebras $\mathscr P\mathbb N$ and $\mathscr P\mathbb N\otimes\mathscr P\mathbb N$ are isomorphic. (Here "$\otimes$" is the coproduct in Boolean algebras.)</p>
<p>The answer is no s... |
207,040 | <p>Is there some way I can solve the following equation with <span class="math-container">$d-by-d$</span> matrices in Mathematica in reasonable time?</p>
<p><span class="math-container">$$AX+X'B=C$$</span></p>
<p>My solution below calls linsolve on <span class="math-container">$d^2,d^2$</span> matrix, which is too ex... | Alex Trounev | 58,388 | <p>Your equation is not in the correct form for FEM. Homogeneous Neumann conditions are applied automatically.</p>
<pre><code>Needs["NDSolve`FEM`"]
Bi = 0.5; xf = 5; reg = Rectangle[{0, 0}, {xf, 1}];
mesh = ToElementMesh[reg, MaxCellMeasure -> 0.0001];
PDE = D[M[t, x, y], t] - 1/(x^2 + y^2)*D[(x^2 + 1)*D[M[t, x, y]... |
4,573,600 | <p>I need help to start solving a differential equation</p>
<p><span class="math-container">$$x^2y'+xy=\sqrt{x^2y^2+1}.$$</span></p>
<p>I would divide the equation with <span class="math-container">$x^2.$</span> Then the equation looks like a homogeneous equation, but I get under the square root <span class="math-conta... | MDCCXXIX | 1,118,767 | <p>Set some variable, lets call him q, as <span class="math-container">$$q = xy$$</span>
then, <span class="math-container">$$\frac{dy}{dx} = \frac{xq'-q}{x^{2}}$$</span>
<span class="math-container">$$xq' = \sqrt{q^{2}+1}$$</span>
<span class="math-container">$$\displaystyle \int \frac{1}{\sqrt{q^2+1}} \,dq = \display... |
3,375,181 | <p>How do I graph f(x)=1/(1+e^(1/x)) except for replacing variable x with numbers?
Besides, I get the picture of the answer online <a href="https://i.stack.imgur.com/gZb7X.png" rel="nofollow noreferrer">enter image description here</a> and do not understand why x = 0 exists on this graph.</p>
| Claude Leibovici | 82,404 | <p>Starting from Certainly not a dog's answer
<span class="math-container">$$y’-y\tan{(x+c)}=0\implies \frac{y'}y=\tan{(x+c)}$$</span> Integrate both sides to get
<span class="math-container">$$\log(y)=-\log (\cos (x+c))+d\implies y=d \sec(x+c)$$</span></p>
|
459,428 | <p>How does one evaluate a function in the form of
$$\int \ln^nx\space dx$$
My trusty friend Wolfram Alpha is blabbering about $\Gamma$ functions and I am having trouble following. Is there a method for indefinitely integrating such and expression? Or if there isn't a method how would you tackle the problem?</p>
| Community | -1 | <p>Let
$$F_n=\int \log^n(x) dx$$
so by integration by parts (we derivate $\log^n(x)$) we have
$$F_n=x\log^n(x)-n\int\log^{n-1}(x)dx=x\log^n(x)-nF_{n-1}$$
so we find $F_n$ by induction by the relation:</p>
<p>$$\left\{\begin{array}\\
F_0=x+C\\
F_{n}=x\log^n(x)-nF_{n-1},\quad n\geq 1
\end{array}\right.$$</p>
<p><stron... |
459,428 | <p>How does one evaluate a function in the form of
$$\int \ln^nx\space dx$$
My trusty friend Wolfram Alpha is blabbering about $\Gamma$ functions and I am having trouble following. Is there a method for indefinitely integrating such and expression? Or if there isn't a method how would you tackle the problem?</p>
| The_Sympathizer | 11,172 | <p>I notice these answers do not also explain why Wolfram Alpha gives results involving the gamma function. I'll provide that here.</p>
<p>The reason is that Wolfram Alpha interprets $n$ as an arbitrary <i>real or complex number</i>, not just a nonnegative integer. In that case, there is no elementary solution and we ... |
822,711 | <p>In case of Riemannian geometry the connection $\Gamma^i_{jk}$ as is derived from the derivatives of the metric tensor $g_{ij}$ is ought to be symmetric wrt to its lower two indices. But in the case of Non-Riemannian Geometry that need not be the case, so the question is how do you actually construct such connections... | Phillip Andreae | 141,493 | <p>Here's one way to construct a connection on the tangent bundle (a similar construction works on more general vector bundles). Let $\{\rho_\alpha \}$ be a partition of unity subordinate to a locally finite coordinate cover $\{U_\alpha \}$. On each $U_\alpha$, choose coordinates $x_1^\alpha, \dots, x_n^\alpha$, giving... |
4,298,951 | <p>Let us define a sequence <span class="math-container">$(a_n)$</span> as follows:</p>
<p><span class="math-container">$$a_1 = 1, a_2 = 2 \text{ and } a_{n} = \frac14 a_{n-2} + \frac34 a_{n-1}$$</span></p>
<p>Prove that the sequence <span class="math-container">$(a_n)$</span> is Cauchy and find the limit.</p>
<hr />
<... | Surjeet Singh | 809,487 | <p>Given that <span class="math-container">$a_1=1$</span> and <span class="math-container">$a_2=2$</span> such that <span class="math-container">$\displaystyle a_n=\frac{1}{4}a_{n-2}+\frac{3}{4}a_{n-1}$</span>
for <span class="math-container">$n\geq3$</span></p>
<p>Now <span class="math-container">$\displaystyle a_{n}-... |
355,888 | <p>Consider
$x''-2x'+x= te^t$</p>
<p>Determine the solution with initial values $x(1) = e,$ $x'(1) = 0.$</p>
<p>I know this looks like and probably is a very easy question, but i'm not getting the right answer when i try and solve putting into quadratic form. Could someone please demonstrate or show me a different m... | obataku | 54,050 | <p>First we solve the complementary homogeneous equation $x'' - 2x'+x=0$ by presuming a solution of the form $x=e^{rt}$ to yield:</p>
<p>$$e^{rt}\left(r^2-2r+1\right)=0\\(r-1)^2=0$$</p>
<p>So we have repeated roots of our characteristic polynomial yielding a complementary solution $x=c_1e^{t}+c_2te^{t}$.</p>
<p>Reco... |
2,316,448 | <p>I was working on the infinite sum
$$\sum_{x=1}^\infty \frac{1}{x(2x+1)}$$
and I used partial fractions to split up the fraction
$$\frac{1}{x(2x+1)}=\frac{1}{x}-\frac{2}{2x+1}$$
and then I wrote out the sum in expanded form:
$$1-\frac{2}{3}+\frac{1}{2}-\frac{2}{5}+\frac{1}{3}-\frac{2}{7}+...$$
and then rearranged it... | Jack D'Aurizio | 44,121 | <p>By the <a href="https://en.wikipedia.org/wiki/Riemann_series_theorem" rel="nofollow noreferrer">Riemann rearrangement theorem</a>, you have to be very careful when permuting terms of a conditionally but not absolutely convergent series. In your case it is probably simpler to notice that</p>
<p>$$ S=\sum_{n\geq 1}\f... |
2,316,448 | <p>I was working on the infinite sum
$$\sum_{x=1}^\infty \frac{1}{x(2x+1)}$$
and I used partial fractions to split up the fraction
$$\frac{1}{x(2x+1)}=\frac{1}{x}-\frac{2}{2x+1}$$
and then I wrote out the sum in expanded form:
$$1-\frac{2}{3}+\frac{1}{2}-\frac{2}{5}+\frac{1}{3}-\frac{2}{7}+...$$
and then rearranged it... | grand_chat | 215,011 | <p>When you split the series in question via partial fractions you create an alternating series that happens to be conditionally convergent (it's not absolutely convergent), so rearranging 'a bit' is not allowed. What went wrong is exactly what you surmised: you've lost the correspondence between terms in the two 'halv... |
346,432 | <p>I will think of <span class="math-container">$ \mathbb{R}^{n+m}$</span> as <span class="math-container">$\mathbb{R}^n \times \mathbb{R}^m$</span>.</p>
<p>Let <span class="math-container">$ V \subset \mathbb{R}^{n+m}$</span> be open and <span class="math-container">$g:V \to U \subset \mathbb{R}^{n+m} $</span> be a... | Ben McKay | 13,268 | <p>You can do this nicely with differential forms: see the chapter on Fubini's theorem in my lecture notes on <a href="https://euclid.ucc.ie/mckay/analysis/analysis.pdf" rel="nofollow noreferrer">Stokes's theorem</a>.</p>
|
1,080,858 | <p>Why do we have</p>
<ul>
<li>$u_n=\dfrac{1}{\sqrt{n^2-1}}-\dfrac{1}{\sqrt{n^2+1}}=O\left(\dfrac{1}{n^3}\right)$</li>
<li>$u_n=e-\left(1+\frac{1}{n}\right)^n\sim \dfrac{e}{2n}$</li>
</ul>
<p>any help would be appreciated</p>
| Alex Ravsky | 71,850 | <p>$$u_n=\dfrac{1}{\sqrt{n^2-1}}-\dfrac{1}{\sqrt{n^2+1}}=$$
$$\dfrac{\sqrt{n^2+1}-\sqrt{n^2-1}}{\sqrt{n^2-1}\sqrt{n^2+1}}=$$
$$\dfrac{(\sqrt{n^2+1}-\sqrt{n^2-1})(\sqrt{n^2+1}+\sqrt{n^2-1})}{\sqrt{n^2-1}\sqrt{n^2+1}(\sqrt{n^2+1}+\sqrt{n^2-1})}=$$
$$\dfrac{n^2+1-n^2+1}{\sqrt{n^2-1}\sqrt{n^2+1}(\sqrt{n^2+1}+\sqrt{n^2-1})... |
907,879 | <p>Calculate the limit $\lim\limits_{x\to\infty} (a^x+b^x-c^x)^{\frac{1}{x}}$ where $a>b>c>0$.</p>
<p>First,
$$\exp\left( \lim\limits_{x\to\infty} \frac{\ln(a^x+b^x-c^x)}{x} \right)$$</p>
<p>Next,
$$\lim\limits_{x\to\infty} a^x + b^x - c^x = \lim\limits_{x\to\infty} a^x \left[1 + (b/a)^x - (c/a)^x \right] = ... | Travis Willse | 155,629 | <p>You're using the correct trick but in the wrong place: Since $a > b > c$, the term $a^x$ will dominate the other two in the parenthetical expression as $x \to \infty$.</p>
<p>Factoring that term out gives</p>
<p>$\lim_{x \to \infty} \left[(a^x)^{\frac{1}{x}} \left(1 + \left(\frac{b}{a}\right)^x - \left(\frac... |
25,137 | <p>I want to find an intuitive analogy to explain how binary addition (more precise: an adder circuit in a computer) works. The point here is to explain the abstract process of <em>adding</em> something by comparing it to something that isn't abstract itself.</p>
<p>In principle: An everyday object or an action that is... | guest troll | 19,769 | <p>Weighing/massing is something that has this aspect. Really any extrinsic property. <a href="https://en.wikipedia.org/wiki/Intrinsic_and_extrinsic_properties" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Intrinsic_and_extrinsic_properties</a> But I think weighing is easy to understand because you see th... |
25,137 | <p>I want to find an intuitive analogy to explain how binary addition (more precise: an adder circuit in a computer) works. The point here is to explain the abstract process of <em>adding</em> something by comparing it to something that isn't abstract itself.</p>
<p>In principle: An everyday object or an action that is... | The_Sympathizer | 7,650 | <p>Why would you need an analogy? Binary is just another way of encoding numbers - so the question is, "why do you need something that adds numbers?"</p>
<p>I think what people get hung up on with this issue is they keep wanting to associate "numbers" with <em>decimals</em>.</p>
<p>And to that end, ... |
1,562,503 | <p>Can anyone help me here?</p>
<p>Question: "X is a normed space and A is a subset dense in the dual of X.
x belongs to X and the sequence (x_n) of X is bounded of E such that f(x_n) converges to f(x) for all f in A. Show that x_n converges to x weakly"</p>
<p>My try: I think that if I show that A=cl(A) so I prove ... | Tien Truong | 281,875 | <p>We want to prove </p>
<p>$g(x_n) \to g(x)$, for every $g \in X'$,</p>
<p>where $X'$ is the dual space of $X$. Equivalently, we can prove</p>
<p>$|g(x_n) - g(x)| \to 0$, as $n \to \infty$. </p>
<p>Let $\{f_k\}$ be a sequence in $A$ such that </p>
<p>$\|f_k - g\|_{X'} = \sup_{\|x\| \leq 1} |f_k(x) - g(x)| \to 0$,... |
3,088,766 | <p>I need to prove that the premise <span class="math-container">$A \to (B \vee C)$</span> leads to the conclusion <span class="math-container">$(A \to B) \vee (A \to C)$</span>. Here's what I have so far.</p>
<p><a href="https://i.stack.imgur.com/1AgTZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com... | Graham Kemp | 135,106 | <blockquote>
<p>I need to prove that the premise <span class="math-container">$A \to (B \vee C)$</span> leads to the conclusion <span class="math-container">$(A \to B) \vee (A \to C)$</span>. Here's what I have so far. ...</p>
</blockquote>
<p>A disjunction is usually proven by reduction to absurdity. Assume ... |
501,660 | <p>In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?</p>
<p>Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.</p>... | Alex Peter | 579,318 | <p><strong>Tailored Taylor</strong></p>
<p>You can use Taylor but first you need to pack your angle into the region $x_1=0,2\pi$. simply by $x \mod 2\pi$</p>
<p>Once you are there if $x_1>\pi$ take the result as $\sin(x_1)=-\sin(x_1 - \pi)$ reducing it to $x_2=0,\pi.$</p>
<p>Now if $x_2>\frac{\pi}{2}$ calculat... |
617,747 | <p>In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural numbers can be understood as putting together two apples and two oranges makes four fruits. How can we apply this thinking... | Dylan Yott | 62,865 | <p>There are lots of ways to develop intuition with complex numbers and they've been mentioned above, so I'll try to say something different. I don't think it matters whether or not complex numbers "exist", it simply matters that they are very useful and therefore worth studying. In fact, its hard to say whether any nu... |
2,791,204 | <p>I am trying to understand whether or not the product of two positive semidefinite matrices is also positive semidefinite. This topic has already been discussed in the past <a href="https://math.stackexchange.com/q/113859">here</a>. For me $A$ is positive definite" means $x^T A x > 0$ for all nonzero real vectors ... | mechanodroid | 144,766 | <p>A real matrix $A$ is positive-semidefinite if $A$ is symmetric and $x^TAx \ge 0$ for all $x \in \mathbb{R}^n$.</p>
<blockquote>
<p>Product of two positive-semidefinite matrices $A,B$ is again a positive-semidefinite matrix if and only if $AB = BA$.</p>
</blockquote>
<p>Proof.</p>
<p>Assume $AB = BA$. </p>
<p>T... |
1,873,370 | <p>I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the following way: consider an automorphism of raising each entry to the $q$-th power, (taking $n = 2$ as an example)</p>
<p>... | paul garrett | 12,291 | <p>A standard heuristic to get a feeling for such a question in my world is to replace $\mathbb F_{q^2}$ by $\mathbb F_q \oplus \mathbb F_q$, so that the question is about $GL(n,\mathbb F_q)\times GL(n,\mathbb F_q)$ modulo a diagonal copy of $GL(n,\mathbb F_q)$, and about the corresponding double cosets. The first ques... |
2,729,617 | <blockquote>
<p>Find the 5th-order Maclaurin polynomial $P_5(x)$ for $f(x) = e^x$.</p>
</blockquote>
<p>I got
$$P_5(x) = 1 + x +\frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + O(x^6) $$</p>
<p>From this answer, I'm supposed to approximate $f(-1)$, correct to the fifth decimal place. Is it right ... | Mark Fischler | 150,362 | <p>You almost have it right. The condition is better stated without referring to derivatives. A function $f(x)$ is strictly increasing if for all $(x,y)$ such that $y>x$,</p>
<p>$$
f(y) > f(x)
$$</p>
<p>and is monotonic increasing if for all $(x,y)$ such that $y>x$,
$$
f(y) \geq f(x)
$$</p>
<p>Your defini... |
1,079,356 | <p>My question can be summarized as:</p>
<blockquote>
<p>I want to prove that closed immersions are stable under base change.</p>
</blockquote>
<p>This is exercise II.3.11.a in Hartshorne's Algebraic Geometry. I researched this for about half a day. I consulted a number of books and online notes, but I found the pr... | Babai | 36,789 | <p><strong>Fact</strong>: <span class="math-container">$X, Y$</span> are schemes over <span class="math-container">$S$</span>, if <span class="math-container">$U\subset X$</span> is an open subset and if the product <span class="math-container">$X\times _S Y$</span> exists then <span class="math-container">$p_1 ^{-1}(U... |
1,079,356 | <p>My question can be summarized as:</p>
<blockquote>
<p>I want to prove that closed immersions are stable under base change.</p>
</blockquote>
<p>This is exercise II.3.11.a in Hartshorne's Algebraic Geometry. I researched this for about half a day. I consulted a number of books and online notes, but I found the pr... | Takumi Murayama | 116,766 | <p>This is not an optimal solution, but if you didn't know that closed immersions can be checked affine locally (like I didn't), then this would be something you can do: check each condition for a closed immersion separately.</p>
<p>Let $X = \operatorname{Spec} R$, $X' = \operatorname{Spec} A$, and $Y = \operatorname{... |
1,079,356 | <p>My question can be summarized as:</p>
<blockquote>
<p>I want to prove that closed immersions are stable under base change.</p>
</blockquote>
<p>This is exercise II.3.11.a in Hartshorne's Algebraic Geometry. I researched this for about half a day. I consulted a number of books and online notes, but I found the pr... | Wang Samuel | 422,510 | <p>In the below, we consider a criterion for stability of base change that works in this exercise.</p>
<p>Proof of this criterion using abstract non-sense is given at the end.</p>
<hr />
<p><strong>Criterion for Stability under Base Change</strong></p>
<p>Let <span class="math-container">$\mathbf{P}$</span> be a proper... |
3,183,617 | <p>I have an equation that looks like <span class="math-container">$$X' = a \sin(X) + b \cos(X) + c$$</span> where <span class="math-container">$a,b$</span> and <span class="math-container">$c$</span> are constants. For given values of <span class="math-container">$a, b$</span> and <span class="math-container">$c$</spa... | Claude Leibovici | 82,404 | <p>If you rewrite the equation as
<span class="math-container">$$\frac 1 {t'}=a \sin(x)+b \cos(x)+c$$</span> you should get
<span class="math-container">$$t+k=-\frac{2 \tanh ^{-1}\left(\frac{a+(c-b) \tan
\left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\sqrt{a^2+b^2-c^2}}$$</span> where <span class="math-contai... |
377,393 | <p>Two players play a game. Player 1 goes first, and chooses a number between 1 and 30 (inclusive). Player 2 chooses second; he can't choose Player 1's number. A fair 30-sided die is rolled. The player that chose the number closest to the value of the roll takes that value (say, in dollars) from the other player. Would... | A. R. Caputo III | 612,152 | <p>Here is some Python code that simulates this game using Monte Carlo and then uses CommonerG's method to solve for the optimal strategy for any n-sided die. Note that the Monte Carlo strategy sometimes converges to a local maximum, but with enough iterations it is usually correct.</p>
<p><a href="https://github.com/... |
2,880,384 | <p>Look at the following definition.</p>
<p><strong>Definition.</strong> Let $\kappa$ be an infinite cardinal. A theory $T$ is called $\kappa$-stable if for all model $M\models T$ and all $A\subset M$ with $|A|\leq \kappa$ we have $|S_n^M(A)|\leq \kappa$.
A theory $T$ is called stable if it is $\kappa$-stable for some... | Noah Schweber | 28,111 | <p><em>I recommend <a href="http://www.math.ucla.edu/~chernikov/teaching/StabilityTheory285D/StabilityNotes.pdf" rel="noreferrer">this survey of Chernikov</a> as a source.</em></p>
<p>Stability, in my opinion, should be thought of in the context of the overall <strong>classification program</strong>$^1$ - in particula... |
2,418,547 | <p>The following is a proof of
$$\frac{\partial(u,v)}{\partial(x,y)}.\frac{\partial(x,y)}{\partial(u,v)} = 1$$
<a href="https://i.stack.imgur.com/fLfRX.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fLfRX.jpg" alt="Jacobian proof"></a></p>
<p>In the above proof, I cannot understand why $\frac{\par... | Community | -1 | <p>The expressions $\frac{\mathrm{d}v}{\mathrm{d}u}$ and $\frac{\mathrm{d}u}{\mathrm{d}v}$ are indeed meaningless because the differentials $\mathrm{d}u$ and ${\mathrm{d}v}$ are not ratios of one another.</p>
<p>But those are not the expressions appearing in the formulas.</p>
<p>When given a system of coordinates suc... |
3,144,757 | <p>I got these definite integrals from the moments.</p>
<p>I am required to calculate the definite integrals <span class="math-container">$$\int_{0}^{\infty}x^kf(x)dx$$</span>, where <span class="math-container">$f(x)=e^{-x^{\frac{1}{4}}}sin(x^{\frac{1}{4}})$</span> and <span class="math-container">$k\in\mathbb N$</sp... | Travis Willse | 155,629 | <p><strong>Hint</strong> The appearance of the quantity <span class="math-container">$\require{cancel}x^{1 / 4}$</span> inside the arguments of <span class="math-container">$\exp$</span> and <span class="math-container">$\sin$</span> suggest the substitution <span class="math-container">$$x = u^4, \qquad dx = 4 u^3 du,... |
3,144,757 | <p>I got these definite integrals from the moments.</p>
<p>I am required to calculate the definite integrals <span class="math-container">$$\int_{0}^{\infty}x^kf(x)dx$$</span>, where <span class="math-container">$f(x)=e^{-x^{\frac{1}{4}}}sin(x^{\frac{1}{4}})$</span> and <span class="math-container">$k\in\mathbb N$</sp... | Community | -1 | <p>An alternative approach. Here your integral is:
<span class="math-container">\begin{equation}
I = 4\int_0^\infty x^{4k + 3} e^{-x} \sin(x)\:dx\nonumber
\end{equation}</span>
Here we will employ Feynman's Trick by introducing the function
<span class="math-container">\begin{equation}
J(t) = 4\int_0^\infty x^{4k + ... |
481,421 | <p>Find the limit of:
$$\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$$</p>
| ILikeMath | 86,744 | <p>As an alternative to NightRa's answer</p>
<p>$$\mathop {\lim}\limits_{h \to 0} \frac{-\sin h}{-2 \sin {2h}}=\mathop{\lim}\limits_{h\to 0}\frac{-\sin h}{-4 \sin h \cos h}=\mathop{\lim}\limits_{h\to 0}\frac{1}{4\cos h}=\frac{1}{4}.$$</p>
|
481,421 | <p>Find the limit of:
$$\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$$</p>
| André Nicolas | 6,312 | <p>Let $h=\frac{1}{x}$. We want to find
$$\lim_{h\to 0^+} \frac{\cos h-1}{\cos 2h-1}.$$
From the identity $\cos 2h=2\cos^2h-1$, we see that we want
$$\lim_{h\to 0^+} \frac{\cos h-1}{2\cos^2 h-2}.$$
But $2(\cos^2 h-1)=2(\cos h-1)(\cos h+1)$, so we want
$$\lim_{h\to 0^+} \frac{1}{2(\cos h+1)}.$$
This limit is $\dfrac{1... |
4,362,741 | <p>Let <span class="math-container">$f:(0,\infty)\rightarrow \mathbb{R}$</span> be a real-valued function, such that for some <span class="math-container">$t,C>0$</span>,
<span class="math-container">\begin{equation}
\limsup_{x\rightarrow\infty} f(x+t)\leq C
\end{equation}</span>
Is it also true that <span class="ma... | Mason | 752,243 | <p>This is easy to prove straight from the definition of <span class="math-container">$\limsup$</span>. By definition,
<span class="math-container">$$\limsup_{x \to \infty}f(x + t) = \lim_{x \to \infty}\sup_{y > x}f(y + t).$$</span>
Thus
<span class="math-container">$$\limsup_{x \to \infty}f(x + t) = \lim_{x \to \in... |
1,382,087 | <p>Problem:</p>
<p>A bag contains $4$ red and $5$ white balls. Balls are drawn from the bag without replacement.</p>
<p>Let $A$ be the event that first ball drawn is white and let $B$ denote the event that the second ball drawn is red. Find </p>
<p>(i) $P(B\mid A)$</p>
<p>(ii) $P(A\mid B)$</p>
<p>My confusion is t... | drhab | 75,923 | <p>In general:</p>
<p>$$P(B|A)P(A)=P(A\cap B)=P(A|B)P(B)$$</p>
<p>If you can find $P(A),P(B)$ and $P(A\cap B)$ then this enables you to find $P(A|B)$ and $P(B|A)$.</p>
<p>Note that $P(A|B)=P(A)$ leads to $P(A\cap B)=P(A)P(B)$ i.e. independence of $A$ and $B$. </p>
<p>In your question $A$ and $B$ are not independent... |
1,382,087 | <p>Problem:</p>
<p>A bag contains $4$ red and $5$ white balls. Balls are drawn from the bag without replacement.</p>
<p>Let $A$ be the event that first ball drawn is white and let $B$ denote the event that the second ball drawn is red. Find </p>
<p>(i) $P(B\mid A)$</p>
<p>(ii) $P(A\mid B)$</p>
<p>My confusion is t... | wythagoras | 236,048 | <p>$P(A \mid B) \neq P(A)$. </p>
<p>$$P(A \mid B)= \frac{P(A \cap B)}{P(B)} = \frac{\frac5{18}}{\frac59\times\frac48+\frac49\times\frac38} = \frac{5}{8}$$</p>
<p>We have $P(A)=\frac{5}{9}$. The intuition behind this is that $B$ makes it more likely that a white ball has been drawn the first time, because then $B$ is ... |
3,531,809 | <p>In how many ways can we place 7 identical red balls and 7 identical blue balls into 5 distinct urns if each urn has at least 1 ball?</p>
<p>This is how I approached the problem:</p>
<p>1) Compute the number of total combinations if there were no constraints:</p>
<p>Placing just the red balls, allowing for empty u... | joriki | 6,622 | <p>Your error lies in multiplying the number of ways to distribute <span class="math-container">$7$</span> red balls over <span class="math-container">$k$</span> non-empty urns by the number of ways to distribute <span class="math-container">$7$</span> blue balls over <span class="math-container">$k$</span> non-empty u... |
1,572,126 | <p>I did solve, I got four solutions, but the book says there are only 3.</p>
<p>I considered the cases $| x - 3 | = 1$ or $3x^2 -10x + 3 = 0$.</p>
<p>I got for $x\leq 0$: $~2 , 3 , \frac13$</p>
<p>I got for $x > 0$: $~4$ </p>
<p>Am I wrong? Is $0^0 = 1$ or NOT?</p>
<p>Considering the fact that : $ 2^2 = 2 \cd... | Jan Eerland | 226,665 | <p>First of all notice that $0^0\ne 1$</p>
<p>$$|x-3|^{3x^2-10x+3}=1\Longleftrightarrow$$
$$\ln\left(|x-3|^{3x^2-10x+3}\right)=\ln(1)\Longleftrightarrow$$
$$\ln\left(|x-3|\right)\left(3x^2-10x+3\right)=0\Longleftrightarrow$$
$$\ln\left(|x-3|\right)\left(x-3\right)\left(3x-1\right)=0$$</p>
<hr>
<p>Split $\ln\left(|x-... |
2,573,458 | <p>Given $n$ prime numbers, $p_1, p_2, p_3,\ldots,p_n$, then $p_1p_2p_3\cdots p_n+1$ is not divisible by any of the primes $p_i, i=1,2,3,\ldots,n.$ I dont understand why. Can somebody give me a hint or an Explanation ? Thanks.</p>
| openspace | 243,510 | <p>Suppose it divide bt $p_{i}$. Then $n \equiv 0 \mod p_{i}$. </p>
<p>But $n = p_{1} \dots p_{n} + 1 \equiv 1 \mod p_{i}$.</p>
|
1,396,322 | <p>For example I have eight kids,</p>
<pre><code>A,B,C,D,E,F,G,H
</code></pre>
<p>If I ask them to go into groups of two, their choices are</p>
<pre><code>A->B
B->C
C->B
D->B
E->A
F->A
G->H
H->C
</code></pre>
<p>How to make sure they get their choices as much as possible?</p>
<p>Or similarl... | Community | -1 | <p>We have using integration by parts</p>
<p>$$\sin(x)=\int_0^x\cos(t)dt=x-\int_0^x(x-t)\sin(t)dt$$
and by integrating twice by parts we get
$$\sin x=x-\frac{x^3}{6}+\int_0^x\frac{(x-t)^3}{6}\sin(t)dt$$
Finally since
$$\left|\int_0^x\frac{(x-t)^3}{6}\sin(t)dt\right|\le \int_0^x\frac{(x-t)^3}{6}dt=\frac{x^4}{24}$$
we g... |
1,396,322 | <p>For example I have eight kids,</p>
<pre><code>A,B,C,D,E,F,G,H
</code></pre>
<p>If I ask them to go into groups of two, their choices are</p>
<pre><code>A->B
B->C
C->B
D->B
E->A
F->A
G->H
H->C
</code></pre>
<p>How to make sure they get their choices as much as possible?</p>
<p>Or similarl... | Marconius | 232,988 | <p>I offer an alternative route based on a trigonometric identity.</p>
<p><em>You will need to prove that the limit $L$ exists.</em></p>
<p>Put $x=3u$ in $$L = \lim_{x\to0}{\frac{x-\sin x}{x^3}}$$</p>
<p>to get</p>
<p>$$L = \lim_{u\to 0}{\frac{3u-\sin(3u)}{(3u)^3}}$$</p>
<p>and since $$\sin(3u)=-4\sin^3 u + 3\sin ... |
240,741 | <p>I'm trying to include the legends inside the frame of the plot like this</p>
<p><a href="https://i.stack.imgur.com/7K5aa.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/7K5aa.jpg" alt="hehe" /></a></p>
<p>Here is my Attempt:</p>
<pre><code>ListPlot[{{2, 5, 2, 8, 6, 8, 3}, {1, 2, 5, 2, 3, 4, 3}},
PlotMark... | Sumit | 8,070 | <p>You did not put any location for <code>Placed</code></p>
<pre><code>ListPlot[{{2, 5, 2, 8, 6, 8, 3}, {1, 2, 5, 2, 3, 4, 3}},
PlotMarkers -> {"\[SixPointedStar]", 15}, Joined -> True,
PlotStyle -> {Orange, Green},
PlotLegends -> Placed[LineLegend[{"line1", "line2"},
L... |
2,877,080 | <p>Let A denote a commutative ring and let e denote an element of A such that $e^2 = e$.
How to prove that $eA \times (1 - e)A \simeq A$?
I thought that $\phi: A \mapsto eA \times (1 - e)A, \ \phi(a) = (ea, (1-e)a)$ is an isomorphism but I don't know how to prove that $\phi$ is a bijection.</p>
| David C. Ullrich | 248,223 | <p>You need MVT to prove just about anything about derivatives. For example, $f'=0$ implies $f$ is constant. (Of course explaining what the need for MVT is depends on the audience. In a calculus class the students probably think it's obvious that $f'=0$ implies that $f$ is constant. In a context like that where they do... |
169,531 | <p>Let me preface this by saying that I have essentially no background in logic, an I apologize in advance if this question is unintelligent. Perhaps the correct answer to my question is "go look it up in a textbook"; the reasons I haven't done so are that I wouldn't know which textbook to look in and I wouldn't know ... | André Nicolas | 6,312 | <p>Since the sentence $G$ that is added is a sentence of the language of $T$, the same diagonalization procedure that got you $G$ can be used to produce a sentence $G'$ of the kind you described. The language has not been extended, so $G'$ is definitely "a sentence of $T$." More properly put, it is a sentence of the la... |
284,809 | <p>$F$ is a field and $F[X^2, X^3]$ is a subring of $F[X]$, the polynomial ring. I need to show that nonzero prime ideals of $F[X^2, X^3]$ are maximal.</p>
<p>A classmate suggested taking a nonzero prime ideal $\mathfrak{p}$ of $F[X^2, X^3]$ and embedding $F[X^2,X^3]/\mathfrak{p} \hookrightarrow F[X]/(\mathfrak{p})$ a... | user1551 | 1,551 | <p>Let $Q^{1/2}$ be a Hermitian square root of $Q$. Then by completing square, we get
\begin{align*}
x^HQx-2~\Re{(x^Hb)}+1
&=x^HQx-x^Hb-b^Hx+1\\
&=\|Q^{1/2}x - Q^{-1/2}b\|^2 + (1 - b^HQ^{-1}b).
\end{align*}
Hence the minimum occurs at $x = Q^{-1}b$ and the minimum value is $1 - b^HQ^{-1}b$.</p>
|
2,134,928 | <p>Let <span class="math-container">$ \ C[0,1] \ $</span> stands for the real vector space of continuous functions <span class="math-container">$ \ [0,1] \to [0,1] \ $</span> on the unit interval with the usual subspace topology from <span class="math-container">$\mathbb{R}$</span>. Let <span class="math-container">$$\... | Kanwaljit Singh | 401,635 | <p>Let $x$ be correct answers then $10-x$ are incorrect answers.</p>
<p>Then marks for correct answers = $3 × x = 3x$</p>
<p>And marks deducted for incorrect answers = $1 × (10-x) = 10 - x$</p>
<p>Now after deducting negative marks she got 18 marks.</p>
<p>$3x - (10-x) = 18$</p>
<p>$3x - 10 + x = 18$</p>
<p>$4x =... |
1,849,577 | <p>I recently asked for <a href="https://math.stackexchange.com/questions/1848739/a-topology-on-the-set-of-lines">natural topologies on the set of lines</a> in $\mathbb R^2$. Now I'm aiming for a similar question on the set $S_p$ of conic sections in $\mathbb R^2$ sharing the same focus $p$ (but not necessary having th... | Community | -1 | <p>Consider as "conics" the zero set of equation $ax^2 + by^2 + cxy + dx + ey + f = 0$. </p>
<p>Since $(a,b,c,d,e,f)$ and $(\lambda a, \lambda b, \lambda c, \lambda d, \lambda e, \lambda f)$ define the same equation for $\lambda \neq 0$, a conic define naturally a point in $\mathbb P^5$, the real projective space of d... |
211,803 | <p>I ended up with a differential equation that looks like this:
$$\frac{d^2y}{dx^2} + \frac 1 x \frac{dy}{dx} - \frac{ay}{x^2} + \left(b -\frac c x - e x \right )y = 0.$$
I tried with Mathematica. But could not get the sensible answer. May you help me out how to solve it or give me some references that I can go over... | Pedro | 23,350 | <p>Let $x=e^u$. I changed $e$ to $f$ in the equation to avoid confusions. Then, multiplying by $x^2$ gives $${x^2}\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}} - ay + \left( {b{x^2} - cx - f{x^3}} \right)y = 0$$</p>
<p>Now, if $x=e^u$, then $$\eqalign{
& x\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cr
& {x^2... |
63,052 | <p>Suppose I have a square matrix $M$, which you can think of as the weighted adjacency matrix of a graph $G$. I want to order the vertices of $G$ in such a way that the entries of the matrix $M$ are clustered. By this I mean that the weights that are close in value should appear close in $M$.</p>
<p>I know Mathematic... | Karl Va4 | 72,907 | <p>"By this I mean that the weights that are close in value should appear close in M." Does this make sense if M represents a weight adjacency matrix? The clustered matrix would not be one anymore. You can just cluster the flattened list. Is the point of clustering a weighted adjacency matrix not rather that strongly c... |
3,733,757 | <p>I'm proving that given a nonempty set <span class="math-container">$I$</span>, and given a filter <span class="math-container">$F$</span>, there exists an ultrafilter <span class="math-container">$D$</span> on <span class="math-container">$I$</span> such that <span class="math-container">$F \subseteq D$</span>. I us... | Anonymous | 559,302 | <p>I like FiMePr's answer, but here is an alternative route which avoids invoking the finite meet property.</p>
<p>Either <span class="math-container">$A\in D'$</span> or <span class="math-container">$A\notin D'$</span>. If <span class="math-container">$A\in D'$</span> then we are done so suppose <span class="math-cont... |
3,595,451 | <p><strong>Question:</strong></p>
<p>Let <span class="math-container">$P_{3}(\mathbb{R})$</span> have the standard inner product and <span class="math-container">$U$</span> be the subset spanned by the two vectors (which are polynomials) <span class="math-container">$u_{1}=1+2x-3x^2$</span> and <span class="math-conta... | angryavian | 43,949 | <p>It is easier to find the CDF first (and then it is easy to find the PDF from there). Try to compute <span class="math-container">$P(R \le r)$</span> for any real number <span class="math-container">$r$</span>. (This will be a ratio of areas.)</p>
|
3,920,469 | <p>The topic of <a href="https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers" rel="nofollow noreferrer">odd perfect numbers</a> likely needs no introduction.</p>
<p>The question is as is in the title:</p>
<blockquote>
<p>If <span class="math-container">$p^k m^2$</span> is an odd perfect number with special... | Jose Arnaldo Bebita Dris | 28,816 | <p>Let <span class="math-container">$p^k m^2$</span> be an odd perfect number with special prime <span class="math-container">$p$</span>.</p>
<p>Suppose to the contrary that
<span class="math-container">$$\frac{\sigma(m^2)}{p^k} > \frac{m^2 - p^k}{C}$$</span>
and that <span class="math-container">$C=2$</span>.</p>
<... |
3,920,469 | <p>The topic of <a href="https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers" rel="nofollow noreferrer">odd perfect numbers</a> likely needs no introduction.</p>
<p>The question is as is in the title:</p>
<blockquote>
<p>If <span class="math-container">$p^k m^2$</span> is an odd perfect number with special... | Jose Arnaldo Bebita Dris | 28,816 | <p>(<em>Last updated on March 2, 2021 - 5:04 PM Manila time</em>)</p>
<p>Let <span class="math-container">$p^k m^2$</span> be an odd perfect number with special prime <span class="math-container">$p$</span>.</p>
<p>Suppose to the contrary that
<span class="math-container">$$\frac{\sigma(m^2)}{p^k} > \frac{m^2 - p^k}... |
15,063 | <p>Let $F:R \to S$ be an étale morphism of rings. It follows with some work that $f$ is flat. </p>
<p>However, faithful flatness is another story. It's not hard to show that faithful + flat is weaker than being faithfully flat. An equivalent condition to being faithfully flat is being surjective on spectra. </p>
... | Dustin Clausen | 3,931 | <p>so yeah, look, i was trying to be funny & also trying to highlight the absurdly haughty nature of the caveats in the question. to be serious i would say that if F is etale then it is faithfully flat iff it is surjective on separably-closed field valued points, but also remark that the same is true with "etale" ... |
1,216,619 | <p>Why are the rings $\mathbb{R}$ and $\mathbb{R}[ x ]$ not isomorphic to eachother ?</p>
<p>Think it might have to do with multiplicative inverses but I'm not sure.</p>
| Alexandre Halm | 177,651 | <p>Well $\Bbb R$ is a field while $\Bbb R[X]$ is not.</p>
<p>Question: if a ring is ring-isomorphic to a field, is it necessary a field? </p>
|
2,978,988 | <p>I'm stuck at a question. </p>
<p>The question states that <span class="math-container">$K$</span> is a field like <span class="math-container">$\mathbb Q, \mathbb R, \mathbb C$</span> or <span class="math-container">$\mathbb Z/p\mathbb Z$</span> with <span class="math-container">$p$</span> a prime. <span class="mat... | drhab | 75,923 | <p><span class="math-container">$$\sum_{k=0}^n(n^2+k)=(n+1)n^2+\sum_{k=0}^nk=n(n^2+n)+\sum_{k=1}^nk=\sum_{k=1}^n(n^2+n+k)$$</span></p>
|
84,204 | <p>Say I have some object or quantity and an instance or special case of it, how to formally write this down? </p>
<p>I don't (just) mean that $X$ is a set and $x$ an element, i.e. $x\in X$ is not it. I'm dealing with things as general like "<em>the specific group $g$ is a group/is a case of a group</em>". Or "<em>the... | Asaf Karagila | 622 | <p>If you can formalize the special case as "having property $p(x)$" the you can say $$\forall x\bigg( p(x)\lor \ldots\bigg)$$
Where the ellipses handle the general case. </p>
<p>If however you cannot express the specific case with such property or the case you want to handle is predefined (if the set is empty, the nu... |
370,212 | <p>Let <span class="math-container">$\mathbb{N}$</span> denote the set of positive integers. For <span class="math-container">$\alpha\in \; ]0,1[\;$</span>, let <span class="math-container">$$\mu(n,\alpha) = \min\big\{|\alpha-\frac{b}{n}|: b\in\mathbb{N}\cup\{0\}\big\}.$$</span> (Note that we could have written <span c... | Emil Jeřábek | 12,705 | <p>There is no such <span class="math-container">$\alpha$</span>.</p>
<p>If <span class="math-container">$\alpha\in\mathbb Q$</span>, there is <span class="math-container">$n$</span> such that <span class="math-container">$\mu(n,\alpha)=0$</span>, thus <span class="math-container">$\mu(n+1,\alpha)<\mu(n,\alpha)$</sp... |
186,638 | <p>$f(x)=\max(2x+1,3-4x)$, where $x \in \mathbb{R}$. what is the minimum possible value of $f(x)$.</p>
<p>when, $2x+1=3-4x$, we have $x=\frac{1}{3}$</p>
| Jeff Yontz | 34,893 | <p>Since $2x+1$ is strictly increasing and $3-4x$ is strictly decreasing, they must intersect at a some point, $z$. For any $\epsilon > 0, x =z+\epsilon$ implies that $2x+1 > 3-4x$ and similarly $x = z-\epsilon$ implies that $2x+1 < 3-4x$. Thus, the minimum of $f(x)$ must be at $z$. In your case, $z = \frac{1... |
64,780 | <p>I need to sum values that belongs to same week. For example, I have the list x with one column and n rows. Format: </p>
<pre><code>{{2007,1,3},0.2},{2007,1,4},0.1},{2007,1,5},0.14},{2007,1,8},0.}, ... {2014,10,17},-0.2},{2014,10,18},0.2},{2014,10,19},0.2}}.
</code></pre>
<p>Dates in list are sorted in the form fro... | Nasser | 70 | <p>Using the function by Heike from <a href="https://mathematica.stackexchange.com/questions/4551/determining-the-week-of-a-year-from-a-given-date">Determining the week of a year from a given date</a> which gives the week number from a date:</p>
<pre><code>data = {{{2007, 1, 3}, 0.2}, {{2007, 1, 4}, 0.1}, {{2007, 1, ... |
4,428,142 | <p>Applying integration by parts splits the integral into 3 integrals,
<span class="math-container">$\displaystyle \begin{aligned}I&=\int_{0}^{1} \frac{\sin ^{-1} x \ln (1+x)}{x^{2}} d x\\&=-\int_{0}^{1} \sin ^{-1} x \ln (1+x) d\left(\frac{1}{x}\right) \\&=-\left[\frac{\sin ^{-1} x \ln (1+x)}{x}\right]_{0}^... | WAH | 449,818 | <p>Perhaps not 100% satisfactory, as I've performed each step via Mathematica rather than a step-by-step derivation, but the following could be considered a more simple solution. Note that <span class="math-container">$$\ln(1+x) = \sum_{n=1}^\infty(-1)^{n+1}\frac{x^n}{n}$$</span> and <span class="math-container">$$\in... |
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