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 |
|---|---|---|---|---|
34,487 | <p>A few years ago Lance Fortnow listed his favorite theorems in complexity theory:
<a href="http://blog.computationalcomplexity.org/2005/12/favorite-theorems-first-decade-recap.html" rel="nofollow">(1965-1974)</a>
<a href="http://blog.computationalcomplexity.org/2006/12/favorite-theorems-second-decade-recap.html" rel=... | Ryan Williams | 2,618 | <p>I think Lance's choices from the past are pretty comprehensive, although I might add a couple more from the lower bounds department which for some reason are not well-known:</p>
<blockquote>
<p>John E. Hopcroft, Wolfgang J. Paul, Leslie G. Valiant: On Time Versus Space. J. ACM 24(2): 332-337 (1977)</p>
<p>Wolfgang J... |
704,921 | <p>This is the question:
$$
\frac{(2^{3n+4})(8^{2n})(4^{n+1})}{(2^{n+5})(4^{8+n})} = 2
$$
I've tried several times but I can't get the answer by working out.I know $n =2$, can someone please give me some guidance? Usually I turn all the bases to 2, and then work with the powers, but I probaby make the same mistake ever... | William Chang | 133,204 | <p>$$
\frac{2^{11n}}{2*2^{2n}} = 32768
$$</p>
<p><em>And this is the furthest I get, what do I do now?</em></p>
<p>Multiply both sides by 2. Then you get
$$
\frac{2^{11n}}{2^{2n}} = 65536.
$$
$$\implies 2^{11n-2n}=2^{9n}=65536.$$
$$\therefore n=\frac{log_2(65536)}{9}=\frac{16}{9}.$$</p>
<p>EDIT: This is wrong, and ... |
1,114,502 | <p>I attempted the following solution to the birthday "paradox" problem. It is not correct, but I'd like to know where I went wrong.</p>
<p>Where $P(N)$ is the probability of any two people in a group of $N$ people having the same birthday, I consider the first few values.</p>
<p>For two people, the probability that ... | Thomas Andrews | 7,933 | <p>You are over-counting the cases where three people or more share a birthday.</p>
<p>You are also over-counting the case were $A$ and $B$ have the same birthday and $C$ and $D$ have the same birthday.</p>
<p>If $A, B, C$ have the same birthday, then you've counted that case $3$ times, when you only want to count it... |
312,847 | <p>Let <span class="math-container">$k$</span> be a global field, and let <span class="math-container">$G = \mathbf G(\mathbb A_k)$</span> for a connected, reductive group <span class="math-container">$\mathbf G$</span> over <span class="math-container">$k$</span>. In <a href="https://services.math.duke.edu/~hahn/Chap... | Uri Bader | 89,334 | <p>Considering a topological group <span class="math-container">$G$</span>, a Hilbert space <span class="math-container">$V$</span> and a corresponding unitary representation, that is a homomorphism <span class="math-container">$\pi:G\to U(V)$</span>, the following are equivalent:</p>
<ol>
<li><p><span class="math-con... |
110,078 | <p>Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then:
$ \left \| \int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha} } \right\|_{L^q(\mathbb{R}^n)}\leq$ $C\left\| f\right\| _{L^p(\mathbb{R^n})}$.</p>
| Bazin | 21,907 | <p>The function $\vert x\vert^{\alpha-n}$ is radial homogeneous of degree $\alpha-n$, so its Fourier transform is radial homogeneous of degree $-(\alpha-n)-n=-\alpha$ (both locally integrable since $\alpha >0$ and $-\alpha>-n$ so both are distributions which are easily seen as temperate: Fourier transforms make s... |
41,155 | <p>Lauren has 20 coins in her piggy bank, all dimes and quarters. The total amount of money is $3.05. How many of each coin does she have?</p>
| ubpdqn | 1,997 | <p>If you wanted to color code the relationships (using Murta's g):</p>
<p>Defining edge styles by weight:</p>
<pre><code>es = Join @@
MapThread[
Thread[#1 -> #2] &, {(#[[All, 1]] & /@
SortBy[GatherBy[{#, PropertyValue[{g, #}, EdgeWeight]} & /@
EdgeList[g], #[[2]] &], #[[... |
1,783,200 | <p>Prove or disprove the following statement:</p>
<p><strong>Statement.</strong> <em>Continuous for each variables, when other variables are fixed, implies continuous?</em> More clearly, prove or disprove the following problem:</p>
<p>Let $\displaystyle f:\left[ a,b \right]\times \left[ c,d \right]\to \mathbb{R}$ for... | MPW | 113,214 | <p>How about
$$f(x,y)=\begin{cases}\frac{xy}{x^2+y^2},&(x,y)\neq(0,0)\\
0,&(x,y)=(0,0)\\
\end{cases}$$</p>
|
69,476 | <p>Hello everybody !</p>
<p>I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer scientist each day) that if the result is exact this may not be the computationally fastest way possible to do it... | Daniel McLaury | 6,427 | <p><strong>EDIT:</strong> Looks like I overlooked that the OP stipulated he wants to minimize the total number of additions and multiplications. (Although he said he wanted to do that "to start," so arguably the below is still relevant.)</p>
<p>However, to address the question as stated, what you are essenti... |
69,476 | <p>Hello everybody !</p>
<p>I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer scientist each day) that if the result is exact this may not be the computationally fastest way possible to do it... | mathreadler | 89,237 | <p>If we are on an architecture which has <strong>multiple cores</strong>, <strong>CPU pipelines</strong> and <strong>multi-media extensions (MME)</strong> then Horner's method really doesn't have to be best.</p>
<p>If the polynomial is large, you can split into as many bins as you have processor cores. If you have tw... |
2,225,606 | <p>Solution: The eigenvalues for $\begin{bmatrix}1.25 & -.75 \\ -.75 & 1.25\end{bmatrix}$ are $2$ and $0.5$. </p>
<p>I'm confused on how it's not $1$ and $-1$. If we set up the characteristic matrix: $\begin{bmatrix}5/4 - \lambda & -3/4 \\ -3/4 & 5/4 - \lambda \end{bmatrix}$ </p>
<p>$ad-bc=0$</p>
<p>... | Dave | 334,366 | <p>Third last line should read $$\frac{25}{16}-\frac{5}{2}\lambda+\lambda^2-\frac{9}{16}=0$$
It looks like you just forgot to expand the bracket $\left(\frac{5}{4}-\lambda\right)^2$ properly using binomial expansion. </p>
|
1,336,937 | <p>I think: <em>A function $f$, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain $E$.</em></p>
<p>However Royden & Fitzpatrick’s book "Real Analysis" (4th ed) seems to say implicitly that “a function could be integrable without being Lebesgue measurable”. I... | Thang | 34,339 | <p>There is a subtle difference in defining Lebesgue integrals in Real analysis textbooks:</p>
<p><strong>I) The approach of Royden & Fitzpatrick (in “Real analysis” 4th ed), Stein & Shakarchi (in “Real Analysis: Measure Theory, Integration, And Hilbert Spaces”)</strong></p>
<p>Firstly, it defines Lebesgue... |
3,898,818 | <p>A (UK sixth form; final year of high school) student of mine raised the interesting question of how to prove that the total angle in the Spiral of Theodorus (formed by constructing successive right-angled triangles with hypotenuses of <span class="math-container">$\sqrt{n}$</span>), diverges.</p>
<p>He identified th... | Raffaele | 83,382 | <p>It is not "intuition". By MacLaurin expansion at <span class="math-container">$x=0$</span> we have</p>
<p><span class="math-container">$$\arctan\sqrt{x}= \sqrt{x}+O\left(x^{3/2}\right)$$</span>
Therefore
<span class="math-container">$$\arctan\sqrt{\frac{1}{n}}\sim \sqrt{\frac{1}{n}};\quad n\to\infty$$</spa... |
3,226,028 | <h2>Problem</h2>
<p>I want to know how to solve the differential equation
<span class="math-container">$$ \dot{x} + a\cdot x - b\cdot \sqrt{x} = 0 $$</span> for <span class="math-container">$a>0$</span> and both situations: for <span class="math-container">$b > 0$</span> and <span class="math-container">$b < ... | Community | -1 | <p>The next step is to integrate,</p>
<p><span class="math-container">$$t+c=\int\frac{dx}{b\sqrt x-ax}=\int\frac{d\sqrt x^2}{b\sqrt x-a\sqrt x^2}=\int\frac{2\,d\sqrt x}{b-a\sqrt x}=-\frac2a\log\left(\frac ba-\sqrt x\right).$$</span></p>
<p>From this you can draw <span class="math-container">$x$</span>,</p>
<p><span ... |
31,308 | <p>Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)</p>
<p>One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiabl... | Olumide | 7,486 | <p>Thanks everyone for answers given so far. Now for some really ignorant questions from me. I'm really trying to make sense of generalized functions, so here goes:</p>
<p>Its often said that the concept of generalized functions helps to assign integrals to otherwise integrable functions (pardon my phrasing). What con... |
2,077,958 | <p>Or more abstractly, let $T \in \mathcal{L}(U,V)$ be a linear map over finite dimensional vector spaces, I need to prove that $T^*$ and $T^* T$ have the same range.</p>
<p>The direction $v \in range(T^*T) \rightarrow v \in range(T^*)$ is obvious. I'm stuck on the other direction.
Suppose $u\in range(T^*)$, then the... | A.Γ. | 253,273 | <p>You know that $u=T^*v$. Let $Tw$ be the orthogonal projection of $v$ onto $\operatorname{range}T$, i.e.
$$
v-Tw\bot \operatorname{range}T\quad\Leftrightarrow\quad T^*(v-Tw)=0\quad\Leftrightarrow\quad T^*v=T^*Tw.
$$</p>
|
1,169,336 | <p>Using the formal definition of convergence, Prove that $\lim\limits_{n \to \infty} \frac{3n^2+5n}{4n^2 +2} = \frac{3}{4}$.</p>
<p>Workings:</p>
<p>If $n$ is large enough, $3n^2 + 5n$ behaves like $3n^2$</p>
<p>If $n$ is large enough $4n^2 + 2$ behaves like $4n^2$</p>
<p>More formally we can find $a,b$ such that... | kobe | 190,421 | <p>Note</p>
<p>$$\left|\frac{3n^2 + 5n}{4n^2 + 2} - \frac{3}{4}\right| = \left|\frac{20n - 6}{4(4n^2 + 2)}\right| = \frac{|10n - 3|}{4(2n^2 + 1)} < \frac{10n + 3}{8n^2} < \frac{20n}{8n^2} = \frac{5}{2n}.\tag{1}$$</p>
<p>Hence, given $\epsilon > 0$, setting $N > 2\epsilon/5$ will make the left hand side of... |
3,296,122 | <p>I was given a problem in which a matrix <span class="math-container">$A$</span> was specified along with its determinant value, now the determinant of another matrix <span class="math-container">$B$</span> was asked to be found out whose indices were scalar multiplies of <span class="math-container">$A$</span>. What... | amd | 265,466 | <p>One of the basic properties of determinants that you should have learned is that they are multilinear functions of the columns (or rows) of a matrix. Indeed, some treatments start with this and a few additional properties as the definition of a determinant and then derive from them the formulas that you’re likely fa... |
1,092,091 | <p>I wonder how I can calculate the distance between two coordinates in a $3D$ coordinate-system.
Like this. I've read about <em><a href="http://www.purplemath.com/modules/distform.htm" rel="nofollow">the distance formula</a></em>:</p>
<p>$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$</p>
<p>(How) Can I use that it $3D... | Praveen | 140,774 | <p>The distance between two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is given by</p>
<p>$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2+(z_2 - z_1)^2}$$</p>
|
52,299 | <p>Hello everybody.</p>
<p>I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.</p>
<p>Does anyone know such an example.</p>
<p>Best
CJ</p>
| Andrey Rekalo | 5,371 | <p>I am not sure if this would qualify as 'easy' but the first example of such a function was constructed by Lusin. It can be found in N. Lusin, J. Priwaloff, <a href="http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1925_3_42_/ASENS_1925_3_42__143_0/ASENS_1925_3_42__143_0.pdf" rel="noreferrer">Sur l'unicité et la multipl... |
150,180 | <p>I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:</p>
<p>Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered pairs $(E,E^{'})$ of elliptic curves together with cyclic isogeny $E\rightarrow E^{'}$ of degree $N$. Gross u... | Eric Wofsey | 75 | <p>Here's a negative answer: there can be no self-dual way to pass from the category of finitely generated projective modules to the category of all finitely generated modules (over, say, a Noetherian ring). To see this, note that the category of finitely generated projective modules is self-dual via the functor $Hom(... |
2,886,544 | <p>For some set $V \subset [a,b]^d$, define the convex hull of $V$ as the set</p>
<p>$$\{\lambda_1v_1 + ... + \lambda_kv_k: \ \lambda_i \ge 0, \ v_i \in V, \ \sum_{i=1}^k \lambda_i = 1, k = 1, 2, 3, ...\}.$$</p>
<p>I don't understand why exactly these vectors form the convex hull of $V$. Why wouldn't I be able to cho... | Bernard | 202,857 | <p>Any convex set $C$ which contains $v_1,\dots,v_k$, also contains the segments $[v_i,v_j]\;(1\le i,j\le k)$. These segments have a parametric representation $\;tv_i+(1-t)v_j\;(0\le t\le1)$, which we may rewrite, setting $\lambda_i=t$, $\lambda_j=1-t$:
$$\lambda_iv_i+\lambda_jv\in C,\quad\lambda_i,\lambda_j\ge 0,\e... |
638,012 | <p>I want to know if there exists a generalization of L'Hopital rule in $n$ dimensions? For example, let us consider this <a href="http://answers.yahoo.com/question/index?qid=20101028114107AAtmZ9l" rel="nofollow">problem</a>.</p>
<p>There it is just said that we should take separate path and see if they will end wit... | Lutz Lehmann | 115,115 | <p>l'Hopitals rule is, in its essence, an extension or corollary from the (extended) mean value theorem. There is no such simple mean value theorem in the vector case. </p>
<p>Your example</p>
<p>$$\frac{4x^2-y^2}{2x-y}=\frac{(2x-y)(2x+y)}{2x-y}$$ </p>
<p>works because you can cancel the common factor. But take any ... |
638,012 | <p>I want to know if there exists a generalization of L'Hopital rule in $n$ dimensions? For example, let us consider this <a href="http://answers.yahoo.com/question/index?qid=20101028114107AAtmZ9l" rel="nofollow">problem</a>.</p>
<p>There it is just said that we should take separate path and see if they will end wit... | pppqqq | 58,784 | <p>Some thoughts.</p>
<p>Let $f,g\colon \mathbb R ^2\to \mathbb R$ be $C^2$ functions over $\mathbb R ^2$ such that $f(0,0)=g(0,0)=0$. Suppose that $g$ is injective in a neighborood of $( 0,0 )$, so that $\frac{f(x,y )}{g(x,y)}$ is well defined.</p>
<p>Given a couple $(x,y)$ sufficiently near the origin, it is true t... |
4,084,624 | <p>A cool problem I was trying to solve today but I got stuck on:</p>
<p>Find the maximum possible value of <span class="math-container">$x + y + z$</span> in the following system of equations:</p>
<p><span class="math-container">$$\begin{align}
x^2 – (y– z)x – yz &= 0 \tag1 \\[4pt]
y^2 – \left(\frac8{z^2}– x\right... | spiral_being | 908,530 | <p>It will be edited, but let's begin with attempt..<br />
From (4): <span class="math-container">$y=\frac{x(x+z)}{x+z}=x$</span> (if <span class="math-container">$x \neq -z$</span>) because otherwise <span class="math-container">$x + z$</span> would be <span class="math-container">$0$</span>.<br />
From (7): <span cla... |
3,881,390 | <p>I tried multiplying both sided by 4a
which leads to <span class="math-container">$(6x+4)^2=40 \pmod{372}$</span>
now I'm stuck with how to find the square root of a modulo.</p>
| Robert Israel | 8,508 | <p>First multiply by <span class="math-container">$3^{-1} \equiv 21 \mod 31$</span> to get <span class="math-container">$x^2 + 22 x + 20 \equiv 0 \mod 31$</span>.
Then complete the square to get <span class="math-container">$(x+11)^2 \equiv 101 \equiv 8 \mod 31$</span>. Now the square roots of <span class="math-contai... |
1,828,729 | <p>I am trying to solve this summation problem .
$$\sum\limits_{k = 0}^\infty {\left( {\begin{array}{*{20}{l}}
{n + k}\\
{2k}
\end{array}} \right)} \left( {\begin{array}{*{20}{l}}
{2k}\\
k
\end{array}} \right)\frac{{{{( - 1)}^k}}}{{k + 1}}$$
It will be grateful if someone could help me !!</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>The equation of the tangent at $(t,t^4-2t^2-t)$ will be</p>
<p>$$\dfrac{y-(t^4-2t^2-t)}{x-t}=4t^3-4t-1$$</p>
<p>$$x(4t^3-4t-1)-y=3t^4-2t^2$$</p>
<p>So, the equation of the tangents at $(x_1,y_1),(x_2,y_2)$ will respectively be</p>
<p>$$x(4x_1^3-4x_1-1)-y=3x_1^4-2x_1^2\ \ \ \ (1)$$</p>
<p>$$x(4x_2^... |
152,620 | <p>The following question is from Golan's linear algebra book. I have posted a solution in the answers. </p>
<p><strong>Problem:</strong> Let $F$ ba field and let $V$ be a vector subspace of $F[x]$ consisting of all polynomials of degree at most 2. Let $\alpha:V\rightarrow F[x]$ be a linear transformation satisfying</... | Potato | 18,240 | <p>The idea here is to determine the action on each of the basis elements $1$, $x$, and $x^2$.. By linearity we see </p>
<p>$\alpha(x)=\alpha((x+1)-1)=\alpha(x+1)-\alpha(1)=x^5+x^3-x$</p>
<p>$\alpha(x^2)=\alpha((x^2+x+1)-(x+1))=\alpha(x^2+x+1)-\alpha(x+1)=-x^5+x^4-x^3-x^2+1$</p>
<p>Using linearity again we see</p>
... |
56,082 | <p>Suppose I have a nested list such as,</p>
<pre><code>{{{A, B}, {A, D}}, {{C, D}, {A, A}, {H, A}}, {{A, H}}}
</code></pre>
<p>Where the elements of interest are,</p>
<blockquote>
<pre><code>{{A, B}, {A, D}}
{{C, D}, {A, A}, {H, A}}
{{A, H}}
</code></pre>
</blockquote>
<p>How would I use select to pick up only eleme... | kglr | 125 | <pre><code>list = {{{a, b}, {a, d}}, {{c, d}, {a, a}, {h, a}}, {{a, h}}}
Pick[list, Count[#[[All, 1]], a] >= 2 & /@ list]
</code></pre>
<p>or</p>
<pre><code>Select[list, Count[#[[All, 1]], a] >= 2 &]
</code></pre>
<p>or</p>
<pre><code>Cases[list, _?(Count[#[[All, 1]], a] >= 2 &)]
</code></pre>... |
3,242,363 | <blockquote>
<p>Why does this function, <span class="math-container">$$\tan\left(x ^ {1/x}\right)$$</span>
have a maximum value at <span class="math-container">$x=e$</span>?</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/pqE0Q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pqE0Q.png" ... | Ovi | 64,460 | <p>This type of math is called "solving a system of linear equations". You should be able to find it in any sort of algebra or precalculus book. </p>
<p>You should take the bottom two equations</p>
<p><span class="math-container">$a=(x+b) \cdot 0.029+0.3$</span></p>
<p><span class="math-container">$b=(x+a) \cdot 0.0... |
446,197 | <blockquote>
<p>Dudley Do-Right is riding his horse at his top speed of $10m/s$ toward the bank, and is $100m$ away when the bank robber begins to accelerate away from the bank going in the same direction as Dudley Do-Right. The robber's distance, $d$, in metres away from the bank after $t$ seconds can be modelled by... | John Joy | 140,156 | <p>Lets, not make extra work for yourself. You already have a $\cot^2\theta$, so keep it.
$$\begin{array}{lll}
3(\cot^2\theta+1)-\csc^2\theta-1&=&(\cot^2\theta+1)+2(\cot^2\theta+1)-\csc^2\theta-1\\
&=&\cot^2\theta+2(\cot^2\theta+1)-\csc^2\theta\\
&=&\cot^2\theta+2\csc^2\theta-\csc^2\theta\\
&... |
719,681 | <p>There are 2 similar questions on <span class="math-container">$\log$</span> that I'm unable to solve. </p>
<ol>
<li><p>Given that <span class="math-container">$\log_a xy^2 = p$</span> and <span class="math-container">$\log_a x^2/y^3 = q $</span>. Express <span class="math-container">$\log_a 1/\sqrt{xy}$</span> or ... | Mike | 17,976 | <p>Hint: Try solving for $\log x$ and $\log y$ in terms of $p$ and $q$. Then use that result to get the values you're looking for.</p>
|
1,028,720 | <p>I was wondering the following:</p>
<blockquote>
<p><strong>Background Question:</strong> Does there exist a Banach space $X$ which contains a copy $X_0 \subset X$ of itself that is not complemented?</p>
</blockquote>
<p>By "$X_0$ is a copy of $X$", I mean $X_0 \cong X$ via an invertible, bounded, linear map. So... | Tomasz Kania | 17,929 | <p>By the <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_theorem" rel="nofollow noreferrer">Banach-Mazur theorem</a>, each separable Banach space embeds into $C[0,1]$–in particular $C[0,1]\oplus \ell_2$ does. However, no copy of this space in $C[0,1]$ can be complemented, because if it were, so would be $\... |
402,427 | <p><em>Sorry if I don't use the words properly, I haven't learnt these things in English, only some of the words.
Anyway, I'm practicing to one of my exams and sadly this task seemed more challanging for me than it should be. Some kind of explain would help a lot!</em></p>
<p>10 meters of clothes have 6 holes in it.</... | Ayman Hourieh | 4,583 | <p><strong>Hint</strong>: Define:</p>
<p>$$
\varphi(a + b \omega) = \begin{cases} 0 &: a \text{ even} \\
1 &: a \text{ odd}
\end{cases}
$$</p>
<p>Show that this is a homomorphism from $\Bbb Z[\omega]$ to $\Bbb Z/2\Bbb Z$ and find its kernel. Apply the first isomorphism theorem.</p>
|
402,427 | <p><em>Sorry if I don't use the words properly, I haven't learnt these things in English, only some of the words.
Anyway, I'm practicing to one of my exams and sadly this task seemed more challanging for me than it should be. Some kind of explain would help a lot!</em></p>
<p>10 meters of clothes have 6 holes in it.</... | DonAntonio | 31,254 | <p>Denoting $\,\Bbb Z_p:=$ the prime field of characteristic $\,p\,$ , try to follow and prove the following:</p>
<p>$$\Bbb Z[w]/(2,w)\cong\left(\Bbb Z[w]/(2)\right)/\left((2,w)/(2)\right)\cong\Bbb Z_2[w]/(w)\cong\Bbb Z_2$$</p>
|
1,936,043 | <p>I would like to prove that the sequence $n^{(-1)^{n}}$ is divergent. </p>
<p>My thoughts: I know $(-1)^n$ is divergent, so $n$ to the power of a divergent sequence is still divergent? I am not sure how to give a proper proof, pls help!</p>
| fleablood | 280,126 | <p>Or to be direct...</p>
<p>Let $r \in \mathbb R$. Let $\epsilon > 0$ </p>
<p>For any $M$ let $m > \max (r + \epsilon, M); m$ even. So $m - r > \epsilon > 0$.</p>
<p>Then $|m^{(-1)^m} - r| = |m -r|= m-r > \epsilon$. So the sequence doesn't converge to any real $r$.</p>
|
506,394 | <p>Let $A=\{g\in C([0,1]):\int_{0}^{1}|g(x)|dx<1\}$. If $p\in [0,\infty]$, is $A$ an open set of $(C([0,1]), \left\|{\cdot}\right\|_p)$?</p>
<p>Is it obvious that if $p=1$ then $A$ is open in $(C([0,1]), \left\|{\cdot}\right\|_1)$, because $A=B(0,1)$.</p>
<p>I think $A$ is not open if $p>1$. Any hint to show th... | njguliyev | 90,209 | <p>No, $A$ is open for $p>1$. Let $g \in A$ and $\varepsilon = \frac12\left(1-\int_0^1|g(x)|dx\right)$. Then for any $f \in C([0,1])$ with $\|f\|_p < \varepsilon$ we have $$\int_0^1|g(x)+f(x)|dx \le \int_0^1|g(x)|dx+\int_0^1|f(x)|dx \le \int_0^1|g(x)|dx+\|f\|_p < 1.$$</p>
|
1,441,905 | <blockquote>
<p>Find the range of values of $p$ for which the line $ y=-4-px$ does not intersect the curve $y=x^{2}+2x+2p$</p>
</blockquote>
<p>I think I probably have to find the discriminant of the curve but I don't get how that would help.</p>
| John_dydx | 82,134 | <p>Equate the two expressions for $y$ and then re-arrange to form a quadratic equation</p>
<p>For the equation to have no real root, the discrimant $b^2 - 4ac < 0$ must hold. From here, you can find the range of values for $p$ for which the line does not meet the curve. </p>
|
1,915,450 | <p>Can anyone help me to prove this? This is given as a fact, but I don't understand why it is true.</p>
<blockquote>
<p>For an integer $n$ greater than 1, let the prime factorization of $n$ be $$n=p_1^ap_2^bp_3^cp_4^d...p_k^m$$
Where a, b, c, d, ... and m are nonegative integers, $p_1, p_2, ..., p_k$ are prime nu... | SchrodingersCat | 278,967 | <p>Consider that all possible divisors of $n$ can be created by choosing from $p_1,p_2, \ldots, p_k$ in appropriate numbers.</p>
<p>So, for creating a particular divisor, we can choose $1$ $p_1$ or $2$ $p_1$'s or $3$ $p_1$'s and so on till a choice of all $a$ $p_1$'s. Then again we have the choice of not choosing any... |
4,327,729 | <p>In this question, I would like to investigate the location of the absolute value in the arcsecant integral.</p>
<p>Following <a href="https://math.stackexchange.com/questions/3735966/why-the-derivative-of-inverse-secant-has-an-absolute-value">this answer</a> and <a href="https://math.stackexchange.com/questions/1449... | ryang | 21,813 | <p><a href="https://i.stack.imgur.com/SK1n5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SK1n5.png" alt="enter image description here" /></a></p>
<p>For nonnegative <span class="math-container">$x,$</span> the required implication is self-evident; so, let's suppose that <span class="math-container... |
2,868,047 | <p>My question is in relation to a problem I am trying to solve <a href="https://math.stackexchange.com/questions/2867002/finding-mathbbpygx">here</a>. If $g(.)$ is a monotonically increasing function and $a <b$, is it always true that $a<g(a)<g(b)<b$? Why or why not?</p>
| Sentient | 193,446 | <p>If I understand your problem correctly, you're looking for a generalization of the binomial distribution to $n > 2$ where $n$ is the number of possible classes. The term for this is <a href="https://en.wikipedia.org/wiki/Multinomial_distribution" rel="nofollow noreferrer">a multinomial distribution</a>.</p>
<p>W... |
2,868,047 | <p>My question is in relation to a problem I am trying to solve <a href="https://math.stackexchange.com/questions/2867002/finding-mathbbpygx">here</a>. If $g(.)$ is a monotonically increasing function and $a <b$, is it always true that $a<g(a)<g(b)<b$? Why or why not?</p>
| awkward | 76,172 | <p>The OP is correct in saying that this is a generalization of the Birthday Problem and that it is harder because we are interested in cases where the number of people with the same birthday is greater than two. One approach is by way of exponential generating functions.</p>
<p>Let's take a concrete example. Suppos... |
2,011,181 | <blockquote>
<p><strong>Question:</strong> Find the area of the shaded region given $EB=2,CD=3,BC=10$ and $\angle EBC=\angle BCD=90^{\circ}$.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/BFf2h.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BFf2h.jpg" alt="Diagram"></a></p>
<p>I first ... | Community | -1 | <p><strong>HINT</strong></p>
<p>$\frac {BA}{BD} = \frac {BE}{BE + CD}$ because $\triangle AEB\sim\triangle ACD$</p>
|
4,219,303 | <p>I'm trying to solve <span class="math-container">$y''-3y^2 =0$</span>, i use the substitution <span class="math-container">$w=\frac{dy}{dx}$</span>.</p>
<p>Using the chain rule i have:
<span class="math-container">$$\frac{d^2y}{dx^2} = \frac{dw}{dx} = \frac{dw}{dy}\cdot \frac{dy}{dx} = w \cdot \frac{dw}{dy}$$</span>... | Yiorgos S. Smyrlis | 57,021 | <p><span class="math-container">$$
y''-3y^2=0\quad\Longrightarrow\quad y'y''-3y^2y'=0
\quad\Longrightarrow\quad \frac{1}{2}(y')^2-y^3=c
\quad\Longrightarrow\quad y'=\pm\sqrt{2y^3+c'}.
$$</span></p>
|
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Dominic van der Zypen | 8,628 | <p>In earlier times, it seems that great emphasis was put on technical calculation, such as checking for convergence, handling logarithms, and so on (this has been pointed out in an earlier answer).</p>
<p>As for today, the only thing I am convinced every mathematician should know is to formulate correct proofs and be... |
1,995,663 | <p>My brother in law and I were discussing the four color theorem; neither of us are huge math geeks, but we both like a challenge, and tonight we were discussing the four color theorem and if there were a way to disprove it.</p>
<p>After some time scribbling on the back of an envelope and about an hour of trial-and-e... | PellMel | 346,817 | <p>Although I wouldn't call it an <em>algorithm</em>, you can construct a solution equivalent to those of the other three answers via this rational approach:</p>
<ol>
<li><p>Assign a color <em>C<sub>1</sub></em> to the outer ring. It's a promising candidate because of the symmetry and topology of the figure.</p></li>... |
1,995,663 | <p>My brother in law and I were discussing the four color theorem; neither of us are huge math geeks, but we both like a challenge, and tonight we were discussing the four color theorem and if there were a way to disprove it.</p>
<p>After some time scribbling on the back of an envelope and about an hour of trial-and-e... | Tanner Swett | 13,524 | <p>A few people have commented that all of the answers given so far have been identical up to symmetry (either by exchanging colors, or by using a symmetry of the uncolored diagram). Here's a proof that the answer that everyone has given is the only possible answer, up to symmetry.</p>
<p>Let me number the regions, li... |
3,251,851 | <p>I need help solving these simultaneous equations:</p>
<p><span class="math-container">$$a^2 - b^2 = -16$$</span>
<span class="math-container">$$2ab = 30$$</span></p>
| José Carlos Santos | 446,262 | <p>They're <em>not</em> the same. The standard basis of <span class="math-container">$\mathbb R^n$</span> is<span class="math-container">$$\bigl\{(1,0,0,\ldots,0,0),(0,1,0,\ldots,0,0),\ldots,(0,0,0,\ldots,1,0),(0,0,0,\ldots,0,1)\bigr\}.$$</span>It turns out that it as <em>an</em> orthonormal basis, but there are others... |
1,210,018 | <p>$$ \begin{bmatrix}
1 & 1 \\
1 & 1 \\
\end{bmatrix}
\begin{Bmatrix}
v_1 \\
v_2 \\
\end{Bmatrix}=
\begin{Bmatrix}
0 \\
0 \\
\end{Bmatrix}$$</p>
<p>How can i solve this ?</p>
<p>I found it $$v_1+v_2=0$$ $$v_1+v_2=0$$ .</p>
<p>... | Ruby | 227,036 | <p>There are infinite solutions. For every $v_1$ there is a $v_2$ which is negative of it.</p>
|
2,965,821 | <p>Let <span class="math-container">$(x_k)_{k\in N}$</span> <span class="math-container">$\subset \mathbb{R^4} $</span>.</p>
<p>Then there's this series, which I have to check for convergence and its limit.</p>
<p><a href="https://i.stack.imgur.com/o6zpZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Recall that by chain rule</p>
<p><span class="math-container">$$f(x)=e^{g(x)}\implies f'(x)=g'(x)\cdot e^{g(x)}$$</span></p>
|
2,965,821 | <p>Let <span class="math-container">$(x_k)_{k\in N}$</span> <span class="math-container">$\subset \mathbb{R^4} $</span>.</p>
<p>Then there's this series, which I have to check for convergence and its limit.</p>
<p><a href="https://i.stack.imgur.com/o6zpZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.... | Mark Bennet | 2,906 | <p>If <span class="math-container">$y=e^x$</span> then <span class="math-container">$y'=e^x=y$</span> (we can also write <span class="math-container">$\frac {dy}{dx}$</span> instead of <span class="math-container">$y'$</span>, they are different notations for the same thing). This relationship can be used in the chain ... |
704,073 | <p>I encountered something interesting when trying to differentiate $F(x) = c$.</p>
<p>Consider: $\lim_{x→0}\frac0x$. </p>
<p>I understand that for any $x$, no matter how incredibly small, we will have $0$ as the quotient. But don't things change when one takes matters to infinitesimals?
I.e. why is the function $\fr... | Emo | 127,234 | <p>When we deal with limits we first see the result of the part after the sign $\lim$. In this case we have $\frac{0}{x}=0$. Than we have $\lim_{x\to0}{\frac{0}{x}}=\lim_{x\to0}0=0$.</p>
|
704,073 | <p>I encountered something interesting when trying to differentiate $F(x) = c$.</p>
<p>Consider: $\lim_{x→0}\frac0x$. </p>
<p>I understand that for any $x$, no matter how incredibly small, we will have $0$ as the quotient. But don't things change when one takes matters to infinitesimals?
I.e. why is the function $\fr... | JMCF125 | 65,964 | <p>The limits are not about a value at a point, but about the values <strong>approaching</strong> that point.</p>
<blockquote>
<p>I.e. why is the function $\frac0x = f(x)$, not undefined at $x=0$?</p>
</blockquote>
<p>It <strong>is</strong> undefined at that point. However, its "neighbourhood" is defined, and that'... |
704,073 | <p>I encountered something interesting when trying to differentiate $F(x) = c$.</p>
<p>Consider: $\lim_{x→0}\frac0x$. </p>
<p>I understand that for any $x$, no matter how incredibly small, we will have $0$ as the quotient. But don't things change when one takes matters to infinitesimals?
I.e. why is the function $\fr... | Kaladin | 133,789 | <p>The reason is if you look at the function $f(x,y)=y/x$ then when you approach $(0,0)$ this should be invariant to the path you take in your case you walk along the path where $y=0$ (constant $0$) but when approaching over the path $y=x$ then it is constant $1$. So we say it is not defined.</p>
|
865,598 | <p>How can I calculate this value?</p>
<p>$$\cot\left(\sin^{-1}\left(-\frac12\right)\right)$$</p>
| Jason | 164,082 | <p>Draw a right triangle (in the x>0,y<0 quadrant) with opposite edge -1 and hypotenuse 2. Then the adjacent side is $\sqrt{2^2-1^2}=\sqrt{3}$. cotangent is the ratio of adjacent side over opposite side.</p>
|
3,773,856 | <p>I'm having trouble with part of a question on Cardano's method for solving cubic polynomial equations. This is a multi-part question, and I have been able to answer most of it. But I am having trouble with the last part. I think I'll just post here the part of the question that I'm having trouble with.</p>
<p>We ha... | dan_fulea | 550,003 | <p>Here is my way to see the algebra beyond the solution of the cubic. It is based on the known algebraic identity:
<span class="math-container">$$
\tag{$*$}
t^3+x^3+y^3-3txy
=(t+x+y)(t+\omega x+\omega^2y)(t+\omega^2 x+\omega y)\ .
$$</span>
Then with the notations from the OP, taking <span class="math-container">$x,y... |
2,837,934 | <p>A cyclist gets left behind by $500$ meters every $minute$ by motorcyclist, because of that he takes $2$ $hour$ and $42$ $minute$ more than motorcyclist to cover $52$ $km$.
Find both of their speed.</p>
<p>My approach: $v_2-v_1=30km/h$ (converted 500 meter per minute to km/h)</p>
<p>$v_2=52/t$<br>
$v_1=52/(t+2.42)$... | Ross Millikan | 1,827 | <p>$2$ hours and $42$ minutes is not $2.42$ hours. It is $2.7$ hours. The denominator in your last equation should be $t+2.7.$ Otherwise you are doing fine.</p>
|
14,726 | <p>I am a tutor for a student and I work with him 7 days a week, for about 2-3 hours a day. The student severely struggles with math, although I am a tutor for every subject (he is in high school). His first chemistry exam he got a 77, and I brought it up to a 96. I studied with him for history and he received 100 on t... | Community | -1 | <p>A raw score on an exam doesn't mean anything in and of themselves, so we have no context for evaluating what a 73% means. If this is a class where the grading scale is 90%=A, 80%=B, 70%=C, then this student passed his math exam, which doesn't seem like a bad outcome for someone who really struggles in math. Grade in... |
14,726 | <p>I am a tutor for a student and I work with him 7 days a week, for about 2-3 hours a day. The student severely struggles with math, although I am a tutor for every subject (he is in high school). His first chemistry exam he got a 77, and I brought it up to a 96. I studied with him for history and he received 100 on t... | WeCanLearnAnything | 7,151 | <p>This is very hard to answer without details such as:</p>
<ul>
<li>Is this student generally a sense-making student or one who just seeks answers? How do you know?</li>
<li>Is the student lacking prior knowledge? Many students get high grades for years in math because they rote memorize procedures while understand ~... |
2,248,550 | <p>Will be the value in the form of $\frac{"0"}{"0"}$? Do I have to use the L'Hopital rule? Or can I say, that the limit doesn't exist?</p>
| mlc | 360,141 | <p>Rationalize the denominator
$$\lim_{\substack{x \rightarrow 0^{+} \\ y \rightarrow 1^{-}}} \frac{x+y-1}{\sqrt{x}-\sqrt{1-y}} \frac{\sqrt{x}+\sqrt{1-y}}{\sqrt{x}+\sqrt{1-y}} = \\
\lim_{\substack{x \rightarrow 0^{+} \\ y \rightarrow 1^{-}}} \frac{(x+y-1)(\sqrt{x}+\sqrt{1-y})}{x-(1-y)} = \\
\lim_{\substack{x \rightar... |
165,582 | <p>The three lines intersect in the point $(1, 1, 1)$: $(1 - t, 1 + 2t, 1 + t)$, $(u, 2u - 1, 3u - 2)$, and $(v - 1, 2v - 3, 3 - v)$. How can I find three planes which also intersect in the point $(1, 1, 1)$ such that each plane contains one and only one of the three lines?</p>
<p>Using the equation for a plane
$$a_i ... | Robert Israel | 8,508 | <p>$2k(k+5)/2 - k(k+3)=2k$ with $a=k(k+5)/2$, $x=k$, $y=k$, would mean $b=k+3$, not $k(k+3)$.
The conclusion is that $(k(k+5)/2,k+3) | 2k$, not $2$. For example, with $k=3$, $(24,6)=6$. </p>
|
165,582 | <p>The three lines intersect in the point $(1, 1, 1)$: $(1 - t, 1 + 2t, 1 + t)$, $(u, 2u - 1, 3u - 2)$, and $(v - 1, 2v - 3, 3 - v)$. How can I find three planes which also intersect in the point $(1, 1, 1)$ such that each plane contains one and only one of the three lines?</p>
<p>Using the equation for a plane
$$a_i ... | Bill Dubuque | 242 | <p>Let $\rm\:j = (k\!+\!3,\, k(k\!+\!5)/2).\,$ The solvability criterion is $\rm\,j\:|\:2k,\,$ not $\rm\:j\:|\:2.\,$ The two are equivalent only when $\rm\:(j,k) = 1\iff (k,3) = 1.$ Otherwise $\rm\:3\:|\:k\:|\:j\:$ thus $\rm \:j\nmid 2.$</p>
|
2,375,298 | <p>My question is as follows:</p>
<p>I have four different die and I'm trying to figure out how many possible combinations there are of (6,6,6,3)</p>
<p>My intuition tells me that there are 24 combinations. I'm imagining we have 4 spots:</p>
<hr>
<p>For the first spot there are 4 options (6,6,6,3)
For the second sp... | Robert Israel | 8,508 | <p>If $x_n$ is a weakly Cauchy sequence for each $n$, and $x_n \to x$ in $\ell_\infty(X)$, i.e. $x_n(j)$ converges uniformly to $x(j)$ as $n \to \infty$, I claim $x$ is weakly Cauchy. If not, there is $y \in X^*$ with (for convenience) $\|y\| \le 1$ such that
$y(x(j))$ is not a Cauchy sequence, i.e. there is $\epsilo... |
4,328,630 | <p>While preparing for a midterm, I came across this question</p>
<blockquote>
<p>Suppose a restaurant is visited by 10 clients per hour on average, and clients follow a homogeneous Poisson Process. Independantly of other client, each client has a 20% chance to eat here and 80% to take away. In average, how many client... | epi163sqrt | 132,007 | <p>We look somewhat more detailed at OPs first identity and calculate
<span class="math-container">\begin{align*}
\frac{\partial\left(\boldsymbol {EJE}^{T}\right)}{\partial\boldsymbol {E}}
\end{align*}</span></p>
<p>The matrix derivation used in OPs first cited <em><a href="https://lewisgroup.uta.edu/ee5329/lectures/Br... |
270,410 | <p>I have simplified the equations and decrease the variables to 5, and changed the parameters' value as I think the equations in <a href="https://mathematica.stackexchange.com/questions/270375/findrootjsing-encountered-a-singular-jacobian-at-the-point-when-solving-nonli">enter link description here</a> is because of t... | fhrl | 87,268 | <p>1, Surely even if <code>DmI[1]=0</code> and <code>Aki[1]=0</code>, there is still one
equation that cannot satisfy.</p>
<p>2, I have tried to change the first 2 equations,</p>
<pre><code> (AmI[1] - BmI[1]/R3^2) CmI[1] == k1 u0,
R3 AmI[1] DmI[1] - (BmI[1] DmI[1])/R3==0,
</code></pre>
<p>Even make <code... |
151,956 | <p>I'm looking for a general method to evaluate expressions of the form</p>
<p>$$\frac{\mathrm{d}(u^v)}{\mathrm{d}u}\text{ and }\frac{\mathrm{d}(u^v)}{\mathrm{d}v}\;.$$</p>
<p>I know that the answers to these are, respectively, $u^{v-1}v$ and $u^v\mathrm{ln}u$, but am unsure of how to obtain them, and how the chain r... | Gerry Myerson | 8,269 | <p>In the first one, if $v$ is a constant, then the chain rule is not involved. If $v$ is not a constant, but is a function of $u$, then your formula is wrong, and some form of the chain rule is needed. One way to go about it is $u^{v(u)}=e^{v(u)\log u}$; the derivative is $$e^{v(u)\log u}\times{d\over du}(v(u)\log u)=... |
23,268 | <p>I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and m... | Buschi Sergio | 6,262 | <p>consider a truncated cylinder as a succession of circles (say of radius 1): $(C_i)_{i\in [0,1]}$, then consider this as a functor on the set (discrete category) $[0,1]$ to $Set$ (category of sets), the a limit is the class of (no necessarily continuos) curves that are graphs of function $(f, g): [0, 1]\to R^2$ cont... |
3,909,005 | <p>I would like to ask what are the derivative values (first and second) of a function "log star": <span class="math-container">$f(n) = \log^*(n)$</span>?</p>
<p>I want to calculate some limit and use the De'l Hospital property, so that's why I need the derivative of "log star":
<span class="math-co... | Steven Stadnicki | 785 | <p>Hint: take <span class="math-container">$n=2^m$</span> in your limit (and make sure you understand why you can do this!). Then (using <span class="math-container">$\lg(x)$</span> for <span class="math-container">$\log_2(x)$</span>, which is a common convention) <span class="math-container">$\lg^*(n)=\lg^*(m)+1$</spa... |
2,921,927 | <p>We have the following function : <span class="math-container">$$f(z)=\frac{z^2}{1-\cos z}$$</span> where <span class="math-container">$z_0=0$</span> is a removable singularity since the limit as <span class="math-container">$z$</span> goes to <span class="math-container">$0$</span> is <span class="math-container">$2... | Qi Zhu | 470,938 | <p>Mark's answer is, of course, the best to approach this. Supposing it were not a removable singularity, then what I'll show will still sometimes work. In specific, you'll get the Laurent Series.</p>
<p>Recall Geometric Series.
$$ \frac{x^2}{1-\cos{x}} = x^2 (1+\cos(x)+\cos(x)^2+\dots) = x^2 \sum_{n=0}^\infty \left( ... |
4,386,087 | <p>There exists an elevator which starts off containing <span class="math-container">$p$</span> passengers.</p>
<p>There are <span class="math-container">$F$</span> floors.</p>
<p><em><span class="math-container">$\forall i: P_i = $</span>P(i. passenger exists on any of the floors)</em> = <span class="math-container">$... | leonbloy | 312 | <p>The problem is equivalent to the following (slightly neater) formulation: we place <span class="math-container">$p$</span> balls in <span class="math-container">$F$</span> urns, with uniform probability; which is the probability that all urns are occupied?</p>
<p>Your approach and your solution is fine.</p>
<p>It ca... |
204,842 | <p>A probability measure defined on a sample space $\Omega$ has the following properties:</p>
<ol>
<li>For each $E \subset \Omega$, $0 \le P(E) \le 1$</li>
<li>$P(\Omega) = 1$</li>
<li>If $E_1$ and $E_2$ are disjoint subsets $P(E_1 \cup E_2) = P(E_1) + P(E_2)$</li>
</ol>
<p>The above definition defines a measure that... | Anonymous | 57,088 | <p>In the following <a href="https://lirias.kuleuven.be/bitstream/123456789/267264/1/DPS1010.pdf">note</a> the author shows that a finitely additive diffused measure on $\mathcal{P}(\omega)$ can be used to define a non Ramsey family. Combining this with a result of Mathias, it follows that it is consistent with $ZFC$ t... |
4,280,424 | <p>The PDE:
<span class="math-container">$$\frac1D C_t-Q=\frac2rC_r+C_{rr}$$</span></p>
<p>on the domain <span class="math-container">$r \in [0,\bar{R}]$</span> and <span class="math-container">$t \in [0,+\infty]$</span> and where <span class="math-container">$D$</span> and <span class="math-container">$Q$</span> are R... | Atticus Stonestrom | 663,661 | <p>Yes, it is the case that <span class="math-container">$\operatorname{cl}(A)=\mathbb{R}$</span>. Here is an argument that works in more general contexts: by definition, a subset <span class="math-container">$X\subseteq\mathbb{R}$</span> is closed if and only if either <span class="math-container">$X$</span> is counta... |
4,280,424 | <p>The PDE:
<span class="math-container">$$\frac1D C_t-Q=\frac2rC_r+C_{rr}$$</span></p>
<p>on the domain <span class="math-container">$r \in [0,\bar{R}]$</span> and <span class="math-container">$t \in [0,+\infty]$</span> and where <span class="math-container">$D$</span> and <span class="math-container">$Q$</span> are R... | Henno Brandsma | 4,280 | <p>The closed sets are all sets that are at most countable, or <span class="math-container">$\Bbb R$</span>. So the <em>only</em> closed superset of an uncountable set <span class="math-container">$A$</span> is <span class="math-container">$\Bbb R$</span> so</p>
<p><span class="math-container">$$A \text{ uncountable } ... |
4,393,193 | <p>I am looking at the function
<span class="math-container">$$f(x) = \begin{cases}
\dfrac{x^2-1}{x^2-x} & x \ne 0,1\\
0 & x=0\\
2 &x=1
\end{cases}$$</span>
and am trying to show that <span class="math-container">$\lim_{x \to 0} f(x)$</span> DNE. This makes sense to me because <span class="math-container">... | Lubin | 17,760 | <p>Another general equation is
<span class="math-container">$$
Ax^2+Bxy+Cy^2+Dx+Ey+F=0\,,
$$</span>
in which the graph is an ellipse if <span class="math-container">$B^2-4AC<0$</span> and if the graph has at least two points (could be empty or a singleton, as you see in the cases <span class="math-container">$x^2+y^... |
2,363,733 | <p>I saw this notation $V= V_1\otimes V_2$ in a survey on universal algebra, where $V$ was a variety, but the survey in question didn't define this notation. Could anyone explain what it means ?</p>
| Keith Kearnes | 310,334 | <p>The tensor product notation, $V_1\otimes V_2$, for some kind of product of varieties is used in (at least) two different ways.</p>
<p><strong>Way 1.</strong></p>
<p>For the Kronecker product, or tensor product. ($V_1\otimes V_2$ is the variety of $V_1$ models in $V_2$, and conversely.) You can find it used this wa... |
3,965,834 | <p>Does this sum converge or diverge?</p>
<p><span class="math-container">$$ \sum_{n=0}^{\infty}\frac{\sin(n)\cdot(n^2+3)}{2^n} $$</span></p>
<p>To solve this I would use <span class="math-container">$$ \sin(z) = \sum \limits_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{(2n+1)!} $$</span></p>
<p>and make it to <span class="math... | José Carlos Santos | 446,262 | <p>By the same argument, since both series<span class="math-container">$$\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt n}\quad\text{and}\quad\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt n}$$</span>are convergent (yes, they're equal), then the series<span class="math-container">$$\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt n}\times\frac{(-1)^n... |
4,201,477 | <blockquote>
<p>Integrate <span class="math-container">$$\int \frac{\cos 2x}{(\sin x+\cos x)^2}\mathrm dx$$</span></p>
</blockquote>
<p>I was integrating my own way.</p>
<p><span class="math-container">$$\int \frac{\cos 2x}{\sin^2x+2\sin x\cos x+cos^2}\mathrm dx$$</span>
<span class="math-container">$$\int \cot 2x \mat... | xxxx036 | 850,363 | <p><span class="math-container">$\sin^2x+\cos^2x=1\neq0$</span>, there's a mistake in the first solution.</p>
|
31,158 | <p>To generate 3D mesh <a href="http://reference.wolfram.com/mathematica/TetGenLink/tutorial/UsingTetGenLink.html#167310445" rel="nofollow noreferrer">TetGen</a> can be easily used. Are there similar functions (or a way to use TetGen) to generate 2d mesh? I know that such functionality can be <a href="https://mathemati... | dwa | 136 | <p>Another approach is to use Imtek's package. These deal with both 2 and 3D with interfaces to Shewchuck's triangle and Tetgen respectively.</p>
<p>Imtek can be had from <a href="http://portal.uni-freiburg.de/imteksimulation/downloads/ims" rel="nofollow">the University of Freiburg</a>. Documentation is extensive.</... |
1,130,487 | <p>Jessica is playing a game where there are 4 blue markers and 6 red markers in a box. She is going to pick 3 markers without replacement.
If she picks all 3 red markers, she will win a total of 500 dollars. If the first marker she picks is red but not all 3 markers are red, she will win a total of 100 dollars. Under ... | vikas meena | 210,698 | <p>Just add a constant term and also notice that $\ln2$ is also a constant.</p>
|
2,825,522 | <p>I have this problem:</p>
<p><a href="https://i.stack.imgur.com/blD6N.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/blD6N.png" alt="enter image description here"></a></p>
<p>I have not managed to solve the exercise, but this is my breakthrough:</p>
<p><a href="https://i.stack.imgur.com/0dTdO.j... | Kenny Lau | 328,173 | <p>$$x = 360^\circ - 90^\circ - 120^\circ - 60^\circ = 90^\circ$$</p>
<p><img src="https://i.stack.imgur.com/xyTFx.png" alt=""></p>
|
2,268,345 | <p>Find the value of $$S=\sum_{n=1}^{\infty}\left(\frac{2}{n}-\frac{4}{2n+1}\right)$$ </p>
<p>My Try:we have</p>
<p>$$S=2\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{2}{2n+1}\right)$$ </p>
<p>$$S=2\left(1-\frac{2}{3}+\frac{1}{2}-\frac{2}{5}+\frac{1}{3}-\frac{2}{7}+\cdots\right)$$ so</p>
<p>$$S=2\left(1+\frac{1}{2}-... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,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{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
269,655 | <p>I am trying to find a nonlinear model from the data.</p>
<p><a href="https://i.stack.imgur.com/W6JEI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/W6JEI.png" alt="enter image description here" /></a></p>
<p>My code is below:</p>
<pre><code>data = {{0.0, 0.0}, {0.05, 0.87}, {0.1, 0.99}, {0.15, 0.... | Alex Trounev | 58,388 | <p>We can use <code>NMimimize</code> to solve this problem as follows</p>
<pre><code>data = {{0.0, 0.0}, {0.05, 0.87}, {0.1, 0.99}, {0.2,
0.98}, {0.2, 0.91}, {0.25, 0.81}, {0.3, 0.71}, {0.35, 0.62}, {0.4,
0.51}, {0.45, 0.31}, {0.5, 0.31}, {0.55, 0.23}, {0.6,
0.18}, {0.65, 0.14}, {0.7, 0.08}, {0.75, 0.05}... |
269,655 | <p>I am trying to find a nonlinear model from the data.</p>
<p><a href="https://i.stack.imgur.com/W6JEI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/W6JEI.png" alt="enter image description here" /></a></p>
<p>My code is below:</p>
<pre><code>data = {{0.0, 0.0}, {0.05, 0.87}, {0.1, 0.99}, {0.15, 0.... | xzczd | 1,871 | <pre><code>nlm = NonlinearModelFit[data, {model, 0 < a < 1}, a, x, Method -> NMinimize]
Plot[nlm[x], {x, 0, 1}]~Show~ListPlot@data
</code></pre>
<p><a href="https://i.stack.imgur.com/6GOzX.png" rel="noreferrer"><img src="https://i.stack.imgur.com/6GOzX.png" alt="enter image description here" /></a></p>
|
3,982,937 | <p>To avoid typos, please see my screen captures below, and the red underline. The question says <span class="math-container">$h \rightarrow 0$</span>, thus why <span class="math-container">$|h|$</span> in the solution? Mustn't that <span class="math-container">$|h|$</span> be <span class="math-container">$h$</span>?</... | Ethan Bolker | 72,858 | <p>Writing
<span class="math-container">$$
0 < |h| < \delta
$$</span>
is easier than writing
<span class="math-container">$$
-\delta < h < \delta \text{ and } h \ne 0 .
$$</span></p>
|
2,005,604 | <p>Showing $\sqrt a + $$\sqrt {\cos(\sin a)} = 2$</p>
<p>I've attempted various manipulations (multiplying by one, squaring, etc.) but cannot find a way to solve for a. Anyone have an idea how I can approach this problem? Thanks. </p>
| Simply Beautiful Art | 272,831 | <p>It isn't actually possible to solve for $a$, but we can do some simple fixed-point iteration:</p>
<p>$$\sqrt a=2-\sqrt{\cos(\sin a)}\implies a=\left(2-\sqrt{\cos(\sin a)}\right)^2$$</p>
<p>We rewrite this as</p>
<p>$$a_{n+1}=\left(2-\sqrt{\cos(\sin a_n)}\right)^2$$</p>
<p>And start off with a guess $a_0=1$.</p>
... |
605,277 | <p>I have an electronics project where I sample two sine waves. I would like to know what the amplitude (peak) and difference in phase is. Actually I just need to know the average product of the two waves.</p>
<p>A caveat I have is that the two sine waves have been rectified. (negatives cut off) Here is what I expect ... | cactus314 | 4,997 | <p>You don't even know the frequency, because of <a href="https://en.wikipedia.org/wiki/Aliasing" rel="nofollow noreferrer">aliasing</a>. </p>
<p>Here a sine wave gets undersampled two different ways. </p>
<p>The red curve could be made a perfect fit with <a href="https://en.wikipedia.org/wiki/Fast_Fourier_transfor... |
2,134,167 | <p>So we finished studying chapter 5 of Rudin on differentiation (Mean value theorem, Taylor's theorem etc) and this was given as a homework problem:</p>
<p>Let $ f(x) $ be continuously differentiable on $ [0, \infty) $ such that $ f $ satisfies $ f'(x) = \cos(x^2)f(x) $ for all $ x \geq 0 $, with $ f(0) = 1 $. Prove ... | Tsemo Aristide | 280,301 | <p>I assume $f\geq 0$ Hint: write $h(x)=e^{-x}f(x), h'(x)=e^{-x}f(x)(cos(x^2)-1)$</p>
<p>and $l(x)=e^{x}f(x)$, if $f\geq 0$, $h$ decreases and $l$ increases, so $h(x)\leq h(0)=1$ and $l(x)\geq l(0)=1$.</p>
|
2,697,729 | <p>Suppose that the probability that you will drop a penny on the ground is 1/5, and the probability that you will find a penny on the ground today is 1/4. If the two events are independent, what is the probability that at least one of the two events will occur?</p>
<p>First I tried simply $1/5+1/4$ which was incorrec... | Bensstats | 286,966 | <p><strong>Note:</strong>
$$P(\text{drop or find penny})=P(\text{drop a penny})+P(\text{find a penny})-P(\text{drop and find a penny})$$</p>
<p>$$\iff P(D\cup F)=P(D)+P(F)-P(D\cap F)$$
Thus,
$$=\frac{1}{4}+\frac{1}{5}-\frac{1}{4}*\frac{1}{5}=\frac{2}{5}$$</p>
<p>Hope this is helpful!</p>
|
4,400,261 | <p>If <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are <a href="https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Standard_normal_random_vector" rel="nofollow noreferrer">bivariate normal PDFs</a> having correlation coefficients <span class="math-container">$ρ_f$</spa... | Community | -1 | <p>Suppose <span class="math-container">$X \sim \mathcal{N}(\mu_1, \Sigma_1)$</span> has pdf <span class="math-container">$f$</span> and <span class="math-container">$Y \sim \mathcal{N}(\mu_2, \Sigma_2)$</span> has pdf <span class="math-container">$g$</span>. Then if <span class="math-container">$X$</span> and <span cl... |
16,749 | <p>I wanted to remove the <code>Ticks</code> in my coding but i can't. Here when i try to remove the <code>Ticks</code> the number also gone. I need numbers without <code>Ticks</code>, <code>Ticks</code> and <code>GridLines</code> should be automatic and don't use<code>PlotRange</code> .</p>
<pre><code>BarChart[{{1,... | Chris Degnen | 363 | <p>Due to <code>AbsoluteOptions</code> reporting <code>Ticks</code> and <code>GridLines</code> in an unuseable fashion, I've had to resort to <code>Rasterizing</code> to find out how many grid lines are automatically being produced.</p>
<pre><code>data = {{1, 2, 3}, {4, 5, 6}};
bc = BarChart[data, ImageSize -> 400... |
131,322 | <p>A knot in S^3 is small if its complement does not contain a closed incompressible surface. Is it a generic property for knots, meaning that among all knots with less than $n$ crossings, the proportion of small knots goes to 1 when $n$ goes to infinity?</p>
| Robert Bruner | 6,872 | <p>There are several issues to address here, but let me first point out that the formula for $B_n$ at the top of p. 71 of our Memoir has a typo: it should say
$B_n = (Z/p^{j+1})^s \oplus (Z/p^j)^{p^2-p-s} $ where $0 < s \leq p^2-p$ and
$n = 2j(p^2-p)+2s+2p-3$. The point is that there should be $(p^2-1) - (p-1) = ... |
1,490,219 | <blockquote>
<p>Given $S=\displaystyle \bigcap^{\infty}_{k=1}\left(1-\frac{1}{k}, 1+\frac{1}{k}\right)$, what is $\sup(S)$ and $\max(S)$?</p>
</blockquote>
<p>I reasoned that this is empty, since as $k$ goes to infinity, then $\frac{1}{k}$ goes to $0$. So ultimately, the intersection of all the intervals in $S$ is $... | Community | -1 | <p>It is easy to observe that $S=\{1\}.$ So $\inf S =\sup S =1.$</p>
|
2,957,315 | <p><span class="math-container">$j_{1,1}$</span> denotes the first zero of the first Bessel function of the first kind. (That's a lot of firsts!) It's approximately equal to <span class="math-container">$3.83$</span>. My question is, is there any closed form expression for its value? Even a infinite series or infin... | mathstackuser12 | 361,383 | <p>I like Claude's approach and approximation. However for variety (and possibly interest) here is another approach which i pinched from Watson who pinched from Rayleigh who pinched from Euler. And while Claude's approximation converges from above, here we'll approach from below. First we can write the Bessel functi... |
4,413,641 | <p>I'm trying to understand a proof that given a vector space <span class="math-container">$V$</span> over the field <span class="math-container">$F$</span> and <span class="math-container">$n$</span> vectors <span class="math-container">$v_1, \ldots, v_n$</span>, <span class="math-container">$\mathrm{span}(v_1, \ldots... | Mr.Gandalf Sauron | 683,801 | <p>The confusion you are having is that you are checking the equivalence of one definition with itself.</p>
<p>If you define <span class="math-container">$span\{v_{1},v_{2},...,v_{n}\}$</span> as the smallest subspace containing <span class="math-container">$v_{1},...v_{n}$</span> then by definition it is the intersect... |
3,877,652 | <p>Can anyone help me how to do this, like there are some examples in my book, but this exercise problem seems to be alittle difficult for me to approach:</p>
<p>Given a set <span class="math-container">$\{A_k|k\in\mathbb{N}\}$</span>:
<span class="math-container">$$A_k=\bigg\{x\in\mathbb{R}\bigg|\space\space 1-\frac{1... | Greg Martin | 16,078 | <p>If you want to show that <span class="math-container">$\bigcup_{k\in\Bbb N} A_k = (0,2)$</span>, then you have to prove two things:</p>
<ul>
<li>If <span class="math-container">$x\in \bigcup_{k\in\Bbb N} A_k$</span>, then <span class="math-container">$x\in (0,2)$</span>. In other words, if <span class="math-containe... |
2,882,696 | <p>$a,b,x$ are elements of a group .</p>
<p>$x$ is the inverse of $a$.</p>
<p>Here is my attempt to prove it :-</p>
<p>$a\cdot b = e$</p>
<p>$x\cdot (a\cdot b) = x\cdot e$</p>
<p>$(x\cdot a)\cdot b = x$</p>
<p>$e\cdot b = x$</p>
<p>$b = x$</p>
<p>Are my steps correct?
What I wanted to prove is that if $ab = e$,... | wjmolina | 25,134 | <p>Let $a,b\in G$ be such that $ab=e$, and let $x$ be the inverse of $a$. Then $ab=e=ax$, so $b=x$.</p>
<p><strong>Edit</strong>: If $ab=e$, then</p>
<p>$$\begin{align}ba&=ba\\&=bea\\&=b(ab)a\\&=(ba)(ba).\end{align}$$</p>
<p>Let $c$ be the inverse of $ba$. Then $ba=(ba)(ba)$ implies that $c(ba)=c(ba)... |
2,882,696 | <p>$a,b,x$ are elements of a group .</p>
<p>$x$ is the inverse of $a$.</p>
<p>Here is my attempt to prove it :-</p>
<p>$a\cdot b = e$</p>
<p>$x\cdot (a\cdot b) = x\cdot e$</p>
<p>$(x\cdot a)\cdot b = x$</p>
<p>$e\cdot b = x$</p>
<p>$b = x$</p>
<p>Are my steps correct?
What I wanted to prove is that if $ab = e$,... | Larry B. | 364,722 | <p>It implies that $x = b$. </p>
<p>Your reasoning is sound, and it is this exact reasoning that proves a group element's inverse is unique. You also proved this using only the identity, associativity, and inverse laws. Good job!</p>
|
2,882,696 | <p>$a,b,x$ are elements of a group .</p>
<p>$x$ is the inverse of $a$.</p>
<p>Here is my attempt to prove it :-</p>
<p>$a\cdot b = e$</p>
<p>$x\cdot (a\cdot b) = x\cdot e$</p>
<p>$(x\cdot a)\cdot b = x$</p>
<p>$e\cdot b = x$</p>
<p>$b = x$</p>
<p>Are my steps correct?
What I wanted to prove is that if $ab = e$,... | Community | -1 | <p>From $a\cdot b=e$ you draw $b=a^{-1}$, and the inverse is unique.</p>
|
1,707,929 | <p>How do I solve for the object distance to each receiver for three radar receivers on the ground, each the same distance from the other, and each receiving echoes, reflected from an object overhead, of a signal pulse from a single transmitter located on the ground at the exact center of the receivers? </p>
<p>Tran... | Nominal Animal | 318,422 | <p>Rewritten on 2016-04-19. The situation is as follows:</p>
<p><a href="https://i.stack.imgur.com/I3fXe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/I3fXe.png" alt="Three radars"></a><br>
<sub>(source: <a href="http://www.nominal-animal.net/answers/radars.png" rel="nofollow noreferrer">nominal-a... |
3,430,066 | <p><strong>Question:</strong></p>
<p>Calculate the integral </p>
<p><span class="math-container">$$\int_0^1 \frac{dx}{e^x-e^{-2x}+2}$$</span></p>
<p><strong>Attempted solution:</strong></p>
<p>I initially had two approaches. First was recognizing that the denominator looks like a quadratic equation. Perhaps we can ... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>If you put <span class="math-container">$u=e^x $</span>, the integral becomes</p>
<p><span class="math-container">$$\int_1^e\frac{u\,du}{u^3-1+2u^2}$$</span>
but</p>
<p><span class="math-container">$$u^3+2u^2-1=(u+1)(u^2+au+b)$$</span>
with
<span class="math-container">$$1+a=2$$</span... |
1,592,224 | <p>I need to understand how to find $a \times b = 72$ and $a + b = -17$. Or I am fine with any other example, even general form $a \times b = c$ and $a + b = d$, how to find $a$ and $b$.</p>
<p>Thanks!</p>
| seeker | 267,945 | <p>Since $a+b=-17\implies a=-17-b$. So put it back in the second equation</p>
<p>$(-17-b)b=72\implies b^2+17b+72=0$ which is a quadratic equation in $b$. Solving it you will get $b=-9$ or $b=-8$ thus giving $a=-8$ or $a=-9$.</p>
|
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