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 |
|---|---|---|---|---|
3,478,700 | <p>As you may know we can define the equation of a tangent line of a differentiable function at any point <span class="math-container">$a$</span> is given by:
<span class="math-container">$$y = f(a) + f'(a)(x-a)$$</span></p>
<blockquote>
<p>However how can I interpret this equation?
<span class="math-container">$$... | Lee Mosher | 26,501 | <p>The equation <span class="math-container">$y = f(a) + f'(a) (x-a)$</span> represents <strong>the only</strong> 1st degree polynomial function that has the same value and the same first derivative at <span class="math-container">$x=a$</span> as the function <span class="math-container">$f$</span>. Geometrically, this... |
3,276,572 | <p>Let be <span class="math-container">$\lVert \cdot \rVert$</span> a matrix norm (submultiplicative).</p>
<p>Do we have for all matrices of determinant 1, the following lower bound:</p>
<p><span class="math-container">$$\lVert M \rVert \geq 1$$</span></p>
<p>I'm very confused and could not find any counterexample a... | Community | -1 | <p>If <span class="math-container">$\| \cdot \|$</span> is a matrix norm then <span class="math-container">$\rho(A) \leq \| A\|$</span> and <span class="math-container">$\rho(A) = 1$</span></p>
|
112,021 | <p>Let $n$ be a positive integer.
The $n$ by $n$ Fourier matrix may be defined as follows:</p>
<p>$$
F^{*} = (1/\sqrt{n}) (w^{(i-1)(j-1)})
$$</p>
<p>where </p>
<p>$$
w = e^{2 i \pi /n}
$$</p>
<p>is the complex $n$-th root of unity with smaller positive argument
and $*$ means transpose -conjugate.</p>
<p>It is we... | Jeremy | 14,424 | <p>The vector $v = (1 - \sqrt{n}, 1, 1, 1, ...)$ is an eigenvector of $F(n)$ with an eigenvalue of -1 for all $n > 2$. To see this, note that the first row of $F(n)$ is $(\frac{1}{\sqrt{n}}, \frac{1}{\sqrt{n}}, \frac{1}{\sqrt{n}}, ...)$. From this is follows trivially that the first element of $F(n).v$ is $-1 + \sq... |
4,114,180 | <p>The Theorem is as follows:</p>
<p>For any numbers x and y, the following statements are true:</p>
<ol>
<li><span class="math-container">$|x|<y$</span> if and only if <span class="math-container">$-y<x<y$</span></li>
<li><span class="math-container">$|x|\leq{y}$</span> if and only if <span class="math-contai... | Eric Towers | 123,905 | <p>You already have one absolute value isolated on one side of the inequality.</p>
<p><span class="math-container">$$ |x-1| \leq 2|x| - 3 \text{.} $$</span>
So apply part 2 to obtain
<span class="math-container">$$ -(2|x| - 3) \leq x-1 \leq 2|x| - 3 \text{.} $$</span></p>
<p>Work on <span class="math-container">$... |
1,081,447 | <p>I'm talking about a Roulette wheel with $38$ equally probable outcomes. Someone mentioned that he guessed the correct number five times in a row, and said that this was surprising because the probability of this happening was $$\left(\frac{1}{38}\right)^5$$</p>
<p>This is true if you only play the game $5$ times. H... | Samrat Mukhopadhyay | 83,973 | <p>To answer your general question, if the events are independent then the probability of getting <em>only</em> $m$ successes at a row <em>once</em> is $$(n-m+1)p^m(1-p)^{n-m}$$ This is because one can have success $m$ times at a row out of $n$ plays in $n-m+1$ ways and in each of these events have a probability $p^m$ ... |
1,231,095 | <p>How does one find $\mathcal{L}^{-1}\{\ln[\frac{s^2+a^2}{s^2+b^2}]\}$?</p>
<p>I've tried splitting it up into $\mathcal{L}^{-1}\{\ln(s^2+a^2)\}-\mathcal{L}^{-1}\{\ln(s^2+b^2)\}$. However, I can't think of any way to actually take the inverse transform of $\mathcal{L}^{-1}\{\ln(s^2+a^2)\}$.</p>
| doraemonpaul | 30,938 | <p>$\mathcal{L}^{-1}\left\{\ln\dfrac{s^2+a^2}{s^2+b^2}\right\}$</p>
<p>$=\mathcal{L}^{-1}\left\{\int_s^\infty\left(\dfrac{2s}{s^2+a^2}-\dfrac{2s}{s^2+b^2}\right)ds\right\}$</p>
<p>$=\dfrac{1}{t}\mathcal{L}^{-1}\left\{\dfrac{2s}{s^2+a^2}-\dfrac{2s}{s^2+b^2}\right\}$</p>
<p>$=\dfrac{2\cos at-2\cos bt}{t}$</p>
|
2,751,819 | <p>I need some help solving this.
I have tried:</p>
<p>$$
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
=\frac{1}{\operatorname{det}A}\cdot \begin{bmatrix}
d & -b \\
-c & a \\
\end{bmatrix}$$
I ended up with $$a=\frac{d}{\operatorname{det}A},$$
and
$$d=\frac{a}{\operat... | Ben Grossmann | 81,360 | <p>This problem is easier if you think about it in terms of the eigenvalues of $A$. Note that if $x$ is an eigenvector of $A$, we have $A^{-1}x = Ax$. What does this tell you about that eigenvalue? </p>
|
2,151,937 | <p>Let $A$ be an $m\times n$ real matrix, $x$ an $n\times 1$ vector and $b$ an $m\times 1$ vector. I want to compute
\begin{equation}
\dfrac{\partial }{\partial x} \Vert Ax+b\Vert^{2}.
\end{equation}
First, I expanded
\begin{equation}
\Vert Ax+b\Vert^{2}=(Ax+b)^{T}(Ax+b)=x^{T}A^{T}Ax+2x^{T}A^{T}b+b^{T}b
\end{equation}
... | john316 | 418,508 | <p>Rather than expanding <em>first</em>, do the opposite. Define a new vector $$y=Ax+b$$ and write the function in terms of this new variable and the Frobenius product (which I'll denote by a colon). This approach reduces the visual "clutter". You can then expand the results <em>after</em> finding the derivative.</p>
... |
2,779,429 | <blockquote>
<p>Evaluate $$\int \frac {dx}{\sin \frac x2\sqrt {\cos^3 \frac x2}}$$</p>
</blockquote>
<p>My try </p>
<p>Write $t=\frac x2$ and hence $dx=2dt$</p>
<p>To change the integral to $$\int \frac {\csc t dt}{\cos^{\frac 32} t}$$</p>
<p>Multiplying both bottom and top by $\csc t$ and then using $\csc^2 t=1+... | Szeto | 512,032 | <p>Elaborating on answer of @JoseCarlosSantos:</p>
<p>By performing the substitution $t=u^2$,</p>
<p>$$\int\frac{dt}{(1-t^2)t\sqrt t}=\int\frac{2udu}{(1-u^4)(u^3)}=2\int\frac{du}{(1-u^4)u^2}$$</p>
<p>Performing partial fraction decomposition,</p>
<p>$$\frac1{(1-u^4)u^2}$$
$$=\frac1{u^2}+\frac{u^2}{1-u^4}$$
$$=\frac... |
3,152,021 | <p>I'm wondering if there are well known sorting techniques for the following problem.</p>
<p><strong>Problem</strong>:</p>
<p>Suppose you would like to sort a list of integer numbers <span class="math-container">$0, 1, 2, \ldots, d$</span>.
If one is only allowed to use swaps of adjacent positions the major part
of ... | MachineLearner | 647,466 | <p>The eigenvalue equation is given by</p>
<p><span class="math-container">$$Av_i=\lambda_i v_i \quad \text{ for all } i=1,\ldots,n.$$</span></p>
<p>If you combine all equations for all eigenvalues you can rewrite the previous equations in a single matrix equation as </p>
<p><span class="math-container">$$A[v_1,\ldo... |
1,463,419 | <p>A letter has come from exclusively LONDON or CLIFTON, but on the postmark only $2$ consecutive letters ''ON'' are found to be visible. What is the probability that the letter came from LONDON?</p>
<hr>
<p>This is a question of conditional probability. Let $A$ be the event that the letter has come from LONDON. Let... | joriki | 6,622 | <p>There is not enough information given to answer the question, since we don't know the prior probabilities of letters arriving from London or from Clifton. A reasonable assumption might be that letters are equally likely to come from any of the people living in those two places. London has a population of roughly $10... |
4,062,667 | <p>Let <span class="math-container">$A$</span> be a <span class="math-container">$n^{th}$</span> order square and skew-symmetric matrix, if <span class="math-container">$(E-A)$</span> is an invertible matrix show that
<span class="math-container">$(E+A)(E-A)^{-1}$</span> is an invertible matrix (where <span class="math... | user6725906 | 900,359 | <p>Hint: <span class="math-container">$\det(E+A)=0$</span> if and only if <span class="math-container">$A$</span> has an eigenvector with eigenvalue <span class="math-container">$-1$</span>. Assuming that for some <span class="math-container">$v\in \mathbb{R}^n\backslash\{0\}$</span> <span class="math-container">$$Av=-... |
11,073 | <p>I have three simple graphs in one Plot. Now I am trying to make a button for each graph so you can hide or show it in the plot. Until now I was just able to make a checkbox with the Manipulate function, but I don't now how to tell the checkbox that it should hide my graph when unchecked an display it when checked. <... | PlatoManiac | 240 | <p>As you mentioned that you wanted a GUI using <code>Button</code></p>
<pre><code>DynamicModule[{b = {True, True, True}, res,
funs = {0.5 x + 1, x, 2 x - 2}, pic},
fun[val_] := If[val == True, "Pressed", "DialogBox"];
res = Dynamic@
Row@{Button["f(x)", If[b[[1]... |
3,113,083 | <p>Why does every CNF formula for <span class="math-container">$(x_{1} \vee y_{1}) \wedge (x_{2} \vee y_{2})\wedge \ldots \wedge (x_{n} \vee y_{n})$</span> have at least <span class="math-container">$2^{n}$</span> terms?</p>
<p>This statement is on the Wikipedia page for DNF form here: <a href="https://en.wikipedia.or... | Bram28 | 256,001 | <p>If you think it is <span class="math-container">$(\frac{3}{6})^6$</span> because you assume these are all independent events ... then you are making a wrong assumption. For example, if the first can is put in the place of where the second can was, then the probability that the second can gets put in a place other th... |
2,860,360 | <p>It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book <em>Algebraic Number Theory</em>:</p>
<p>"<em>Example 7.20</em> For $K=\mathbb{Q}(\sqrt[3]{2} )$, the discriminant is 108, and $r_{2}=1$. So the Minkowski bound is $\approx 2... | Dhalsim | 579,015 | <p>The problem is that you cannot add probabilities when dealing with events that are not "disjoint". What you can do is multiply them, but <em>only</em> if the events are independent.</p>
<p>So, in your example, 6 people are given 300 attempts each to catch a pokemon each. First assume these are independent of each o... |
2,860,360 | <p>It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book <em>Algebraic Number Theory</em>:</p>
<p>"<em>Example 7.20</em> For $K=\mathbb{Q}(\sqrt[3]{2} )$, the discriminant is 108, and $r_{2}=1$. So the Minkowski bound is $\approx 2... | Alessandro Jeanteur | 579,032 | <p>I think that the easiest way for you to grasp where the 'missing' successes went is actually mostly in multiple successes.</p>
<p>If one event has 0.2 probability of success, twice the event has probabilities:</p>
<ul>
<li>0.64 -> no success at all (0.8*0.8)</li>
<li>0.04 -> two successes (0.2*0.2)</li>
<li>0.32 -... |
3,844,448 | <p>Find all values of <span class="math-container">$h$</span> such that rank(<span class="math-container">$A$</span>) = <span class="math-container">$2$</span>.</p>
<p><span class="math-container">$A$</span> = <span class="math-container">$\begin{bmatrix}
1 & h & -1\\
3 & -1 & 0\\
-4 & 1 & 3
\en... | Fred | 380,717 | <p>It is easy to see that the first and the second row of <span class="math-container">$A$</span> are independent, hence</p>
<p><span class="math-container">$$rank(A) \ge 2.$$</span></p>
<p>Furthermore, since <span class="math-container">$\det(A)=-2-9h,$</span></p>
<p><span class="math-container">$$rank(A) =2 \iff \det... |
3,844,448 | <p>Find all values of <span class="math-container">$h$</span> such that rank(<span class="math-container">$A$</span>) = <span class="math-container">$2$</span>.</p>
<p><span class="math-container">$A$</span> = <span class="math-container">$\begin{bmatrix}
1 & h & -1\\
3 & -1 & 0\\
-4 & 1 & 3
\en... | Aaron | 9,863 | <p>There are several ways to approach this problem. Here are two approaches that I will not use in this answer. (1) Rank is preserved by both row operations and column operations, and so we can do column operations to simpliffy the matrix further. (2) The matrix has rank 3 if and only if it is invertible, and we can... |
2,771,034 | <p>$\frac{a_n}{b_n} \rightarrow 1$ and $\sum_{n=1}^\infty b_n$ converges, can it be concluded that $\sum_{n=1}^\infty a_n$ converges?<br>
My attempt at an answer to this question: since $\sum_{n=1}^\infty b_n$ converges, $b_n \rightarrow 0$. Because of this, $a_n \rightarrow 0$ equally fast. However, I'm well aware tha... | Community | -1 | <p>I actually think the answer is no. Take $b_{n} = \frac{(-1)^{n}}{\sqrt{n}}$
and $a_{n} = \frac{(-1)^{n}}{\sqrt{n}} + \frac{1}{n}$. Then $\sum_{n}b_{n}$ converges and $\sum_{n}a_{n}$ diverges but $\frac{a_{n}}{b_{n}} \rightarrow 1$ as </p>
|
1,335,640 | <p>1) A disease has hit a city. The percentage of the population infected $t$ days after the disease arrives is approximated by $$p(t) = 12te^{\frac{-t}{7}} \qquad \mbox{for} \qquad0\leq t \leq 35.$$ </p>
<p>After how many days is the percentage of infected people a maximum? What is the maximum percent of the popu... | Graham Kemp | 135,106 | <p>The gradient (or "slope") at a local maximum, minimum, or point of inflection, will equal zero.</p>
<p>We have $p(t) = 12 t e^{-t/7}$ , so then by differentiation (using the chain rule): $\frac{\mathrm d p(t)}{\mathrm d t} = 12 e^{-t/7} - \dfrac{12 t}{7} e^{-t/7}$.</p>
<p>Solving for $t$ when $\frac{\mathrm d p(t)... |
1,812,675 | <p>Is there a recurrence solution to $a_n=\frac{n}{a_{n-1}}$? I'm wondering if it could be done in the form of an alternating series partial to $n$ or as a trigonometric function.</p>
| peter.petrov | 116,591 | <p>For $n=2k+1$ odd you have: </p>
<p>$$a_{2k+1} = \frac{c . (2k+1)!!}{ (2k)!!}$$ </p>
<p>For $n=2k$ even you have: </p>
<p>$$a_{2k} = \frac{(2k)!!}{c . (2k-1)!!}$$ </p>
<p>where $c = a_1$. </p>
<p>This pattern can be seen by writing down the first 5-6 terms.<br>
It should be easy to prove it by induction. ... |
1,259,853 | <p>Why the derivative of $n^{1/n} = \sqrt[n]{n}$ is $n^{1/n} \left( \frac{1}{n^2} - \frac{\log(n)}{n^2}\right)$ (according to Maxima and other tools online)?</p>
<p>I have tried to applied the chain rule, but it comes something completely different:</p>
<p>$$\frac{1}{n} n^{\frac{1}{n} - 1} \cdot 1 = \frac{1}{n} n^\fr... | E.H.E | 187,799 | <p>$$y=n^{1/n}$$
$$\log(y)=1/n\log(n)$$
$$\frac{y'}{y}=-\frac{1}{n^2}\log(n)+{1/n}(1/n)$$
multiply by $y$
$$y'=y(\frac{1}{n^2}-\frac{\log n}{n^2})$$</p>
|
894,476 | <p>I don't have a strong background in probability/statistics and I'm trying to understand the example at <a href="http://rationalwiki.org/wiki/Extraordinary_claims_require_extraordinary_evidence#Probability_theory" rel="nofollow">http://rationalwiki.org/wiki/Extraordinary_claims_require_extraordinary_evidence#Probabil... | Abhishek Murali | 432,569 | <p>In my Applied Numerical Methods course, we are taught using Simpson's method also. Both works equally well. However, Trapezoidal rule works for any interval length as compared to Simpson's rule which needs even number of intervals.
So if we have say 5 values of h, we can apply trapezoidal rule for all those 5 values... |
38,439 | <p>I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue integral, and then we switch framework to manifolds, and we have that trick of using partitions of unity to define inte... | G. Rodrigues | 2,562 | <p>My answer here is realy just a footnote to Paul Siegel's excellent answer, but it has become too long to fit in a comment box. Integrals are siamese brothers to measures; leaving them out seems rather perverse to me. Anyway, here is how I think of integrals. The objective here is to tackle the "categorical" part; th... |
2,658,691 | <p>I have one last question regarding permutation. I understand the problem and the rule of product but this problem seems to be in a different format compared with the two questions I asked before.</p>
<p>A committee of eight is to form a round table discussion group. In how many ways
may they be seated if the 2 memb... | CostaZach | 517,147 | <p>Let $T\in L(X,Y)$ (linear and bounded, $X,Y$ normed spaces). We can show that $$T^{**}\circ J_{X}=J_{Y}\circ T,$$
with $J_X$ the natural embedding from $X$ to $X^{**}$ via $(J_X x)(x^*)=x^*(x)$.</p>
<p>Back to your case: If $E$ is reflexive, then $S=J_{E}^{-1}\circ T^*\circ J_{E}$. With this you have that $T=S^*$ a... |
3,408,846 | <p>This is an example in Serge Lang "Introduction to Linear Algebra", page 48. I try to multiply these two <span class="math-container">$2$</span>x<span class="math-container">$3$</span> and <span class="math-container">$3$</span>x<span class="math-container">$2$</span> matrices but fail to obtain the result as mention... | callculus42 | 144,421 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$\left(\begin{array}{}\color{blue }2&\color{red }1&\color{green }5\end{array} \right) \cdot \left(\begin{array}{}\color{blue }4\\ \color{red }2 \\ \color{green }1\end{array} \right)=\color{blue }{2\cdot 4}+\color{red }{1\cdot 2}+\color{green }{5\c... |
200,279 | <p>I'm trying to show that given a set $\{\mathbf{a}, \mathbf{b}\}$ of orthonormal vectors in a 2-dimensional vector space, I can construct the identity matrix by computing $aa^\dagger + bb^\dagger$. This should be straightforward but it's not working out. I get that my conditions for orthonormality are $$|a_1|^2+|a_2|... | martini | 15,379 | <p>As $a$ and $b$ are orthonormal, they form a basis of your vector space $V$. We have
\[ (aa^* + bb^* )a = aa^* a + bb^* a = a|a|^2 + b0 = a
\]
and
\[(aa^* + bb^* )b = aa^* b + bb^* b = a0 + b|b|^2 = b
\]
So $(aa^* + bb^*)$ is the identity on a basis, hence on $V$, which gives $aa^* + bb^* = \mathrm{Id}$.</p>
<hr>... |
200,279 | <p>I'm trying to show that given a set $\{\mathbf{a}, \mathbf{b}\}$ of orthonormal vectors in a 2-dimensional vector space, I can construct the identity matrix by computing $aa^\dagger + bb^\dagger$. This should be straightforward but it's not working out. I get that my conditions for orthonormality are $$|a_1|^2+|a_2|... | Owen Biesel | 41,747 | <p>Denote the two vectors by $\{\mathbf{a},\mathbf{b}\}$ instead. Then orthonormality is the condition that $\mathbf{a}^\dagger\mathbf{b}=\mathbf{a}^\dagger\mathbf{b}=0$, while $\mathbf{a}^\dagger\mathbf{a}= \mathbf{b}^\dagger\mathbf{b}=1$. Now $\mathbf{a}\mathbf{a}^\dagger + \mathbf{b}\mathbf{b}^\dagger$ is the bloc... |
601,951 | <p><em><strong>2</strong> + <strong>5</strong> + <strong>8</strong> + . . . + <strong>(6n-1)</strong> = <strong>n(6n+1</strong>)</em></p>
<p>This is what I have so far. </p>
<p>The <strong>sum</strong> of <strong>(3j-1)</strong> from <strong>j=1</strong> to <em>something I`m not sure of</em>.</p>
| Grigory M | 152 | <p>LHS is the coefficient of $z^{2n}$ in
$$
(1+z)^m\frac{(1+\sqrt{1+z})^{2n+1}-(1-\sqrt{1+z})^{2n+1}}{2\sqrt{1+z}}.
$$
Such coefficient can be written as a residue,
$$
\operatorname{res}\left\{
(1+z)^m\frac{(1+\sqrt{1+z})^{2n+1}-(1-\sqrt{1+z})^{2n+1}}{2\sqrt{1+z}}\frac{dz}{z^{2n+1}}
\right\}.
$$
After the substitution ... |
3,347,264 | <p>I'm trying to find the distance between two points for a question in my textbook. The points are P(-2, 3) and Q(1, -3). Here's the working I have so far:</p>
<p>d = <span class="math-container">$\sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$</span></p>
<p>d = <span class="math-container">$\sqrt{(1 - (-2))^2 + (-3 - 3)^2}$<... | Andreas | 317,854 | <p>Consider the inequality in the form</p>
<p><span class="math-container">$$\Big(\frac{x^n+y^n+(\frac{x+y}{2})^n}{x^{n-1}+y^{n-1}+(\frac{x+y}{2})^{n-1}}\Big)^n+\Big(\frac{x+y}{2}\Big)^n\leq x^n+y^n
$$</span>
Due to homegeneity, let <span class="math-container">$\frac{x+y}{2} = 1$</span>. Denote <span class="math-conta... |
3,347,264 | <p>I'm trying to find the distance between two points for a question in my textbook. The points are P(-2, 3) and Q(1, -3). Here's the working I have so far:</p>
<p>d = <span class="math-container">$\sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$</span></p>
<p>d = <span class="math-container">$\sqrt{(1 - (-2))^2 + (-3 - 3)^2}$<... | Aforest | 337,161 | <p>Let's begin with your 2nd inequality with <span class="math-container">$x$</span> only.</p>
<p>Let <span class="math-container">$x = (1+p)/(1-p)$</span> and we have <span class="math-container">$x>0\iff p\in(-1,1)$</span>, then your inequality becomes</p>
<p><span class="math-container">$$\left(\frac{(1+p)^n+(1-p... |
1,435,196 | <p>In particular, I would like to know if the four velocity and the four acceleration are tensors. </p>
| Community | -1 | <p>"Tensors" are pretty general. Scalars are (rank 0) tensors. Vectors are (rank 1) tensors, including 4 vectors. So are matrices, etc.</p>
|
1,435,196 | <p>In particular, I would like to know if the four velocity and the four acceleration are tensors. </p>
| Mark Viola | 218,419 | <p>The four velocity and four acceleration are vectors in a space-time <a href="https://en.wikipedia.org/wiki/Minkowski_space" rel="nofollow">Minkowski space</a> and can be represented as a quadruplet $(u_0,u_1,u_2,u_3)$ and $(a_0,a_1,a_2,a_3)$ where the elements with subscripts $1-3$ represent the ordinary Euclidian v... |
619,890 | <p>I have a question.There is a group of 5 men and a group of 7 women.With how many ways can each of the 5 men get married with one of the 7 women?</p>
| yig | 463,312 | <p>I don't know of a computationally more efficient way to project points onto the affine subspace (flat) than what you describe ($p + P_{\textit{null}(W)}( x - p )$).</p>
<p>Given ${W}$ with orthonormal columns, $P_{\textit{null}(W)} = {W} {W}^\top$
and ${I} - P_{\textit{null}(W)}$ are projection matrices onto the nu... |
984,558 | <p>Is it possible to find a representation of the infinitesimal generators of the special unitary group SU(3) that contains 4 by 4 matrices, by say taking a Kronecker product of its irreducible representation(s) with itself?</p>
<p>I know this is possible for SU(2), where one can express the three 4 by 4 matrices span... | Jonathan Rayner | 90,675 | <p>As noted by the accepted answer and the comments, the answer is no.</p>
<p>Note, we can prove this without using Young Tableaux: we can check the possible dimensions as noted in Jyrki's comment above by directly using the <a href="https://en.wikipedia.org/wiki/Lie_algebra_representation#The_case_of_sl.283.2CC.29" r... |
357,520 | <p>How can I show that
$$\lim_{a\to{0}}\frac{{\pi}a\ \coth{\mathrm{{\pi}a}-1}}{2a^2}=\frac{\pi^2}{6}$$
I think the limit is in $\frac{0}{0}$ form, so I am using L'Hospital's rule, and then I cannot solve further, Please Help.</p>
<p>Thanks!</p>
| Community | -1 | <p>Start with Marvis' approach, and evaluate $$\lim_{t\to{0}}\frac{t\ \coth(t)-1}{2t^2}=\lim_{t\to{0}}\frac{t\ \cosh(t)-\sinh(t)}{2t^2 \sinh(t)}$$</p>
<p>It is usually easier to work with sines and cosines. Apply l'Hopital rule once :</p>
<p>$$\lim_{t\to{0}}\frac{t\ \cosh(t)-\sinh(t)}{2t^2 \sinh(t)}=\lim_{t\to{0}}\fr... |
14,508 | <p>Suppose that $f$ is a weight $k$ newform for $\Gamma_1(N)$ with attached $p$-adic Galois representation $\rho_f$. Denote by $\rho_{f,p}$ the restriction of $\rho_f$ to a decomposition group at $p$. When is $\rho_{f,p}$ semistable (as a representation of
$\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$?</p>
<... | Rob Harron | 1,021 | <p>Since <em>f</em> is potentially semi-stable, you can look at its attached filtered (φ, <em>N</em>, Gal(<em>L</em>/<strong>Q</strong><sub><em>p</em></sub>))-representation (where ρ<sub>f,p</sub> becomes semi-stable when restricted to <em>G</em><sub><em>L</em></sub>). If its <em>N</em> is zero, then it is pote... |
1,085,014 | <p>Let $G$ be a group. Prove that $g^2 = e$ for all $g \in G$, then $G$ is abelian. ($e$ is the identity element.)</p>
<p>My Solution: Let $a,b \in G$. Then $a(ab)b = a^2b^2 = e^2 =e$. Now I tried to reverse $ab$ in the brackets to get the same solution to show that the group is also commutative but I was not able to ... | ILoveMath | 42,344 | <p>Let $a,b \in G$ be arbitrary elements. Notice:</p>
<p>$$abab = (ab)^2 = e = a^2 b^2 = aabb$$</p>
<p>Cancellation gives $ba = ab$.</p>
|
1,085,014 | <p>Let $G$ be a group. Prove that $g^2 = e$ for all $g \in G$, then $G$ is abelian. ($e$ is the identity element.)</p>
<p>My Solution: Let $a,b \in G$. Then $a(ab)b = a^2b^2 = e^2 =e$. Now I tried to reverse $ab$ in the brackets to get the same solution to show that the group is also commutative but I was not able to ... | Leox | 97,339 | <p>we have that $ab ab= e$. Thus (multiply by $b$ two sides) $abab^2=b$ or $aba=b$. In the same way $aba^2=ba$ or $ab=ba.$</p>
|
1,575,764 | <p>$Q$ is a linear operator from $V \to V$ with $V$ being a finite dimensional complex inner-product-space. </p>
<p>Given: $Q^*=5Q$, $Q^*$ being the adjoint.</p>
<p>Show that $0$ is the only eigenvalue of $Q$. </p>
<p>I've been staring at this problem for quite a while now. I think the answer lies in the eigenvalues... | Community | -1 | <p>Try doing this by direct computation. Suppose we have an eigenvalue $\lambda$ with (unit) eigenvector $v$. Then since we have a question about adjoints, it's natural that inner products will be used somewhere. Try considering $v$:</p>
<p>$$\langle Qv, v\rangle = \langle \lambda v, v \rangle = \lambda$$</p>
<p>On t... |
131,283 | <p>I came across a question which required us to find $\displaystyle\sum_{n=3}^{\infty}\frac{1}{n^5-5n^3+4n}$. I simplified it to $\displaystyle\sum_{n=3}^{\infty}\frac{1}{(n-2)(n-1)n(n+1)(n+2)}$ which simplifies to $\displaystyle\sum_{n=3}^{\infty}\frac{(n-3)!}{(n+2)!}$. I thought it might have something to do with pa... | Prasad G | 25,314 | <p><code>(n−3)!/(n+2)! = 1/[(n+2)(n+1)n(n-1)(n-2)]</code> and you can easily solve by using fractional part.</p>
|
3,300,469 | <p>I have a problem counting all the possible ways of "pairing" two datasets of size n and m, including partial pairing. </p>
<p>Example:
Assume we have two sets <span class="math-container">$\{A,B\}$</span> and <span class="math-container">$\{1,2,3\}$</span>. My aim is to find all ways of pairing letters with numbers... | Boyku | 567,523 | <p>there is a solution involving the exponential generating function. </p>
<p>The description using combinatorial species is <span class="math-container">$E(X)\cdot E(XY) \cdot E(Y)$</span> where:</p>
<p>X is the sort of letters</p>
<p>Y is the sort of digits</p>
<p>and the formula means that there is a set of unpa... |
1,515,823 | <p>I am doing my research in Functional Analysis, especially in "Generalized inverse of Linear Maps".</p>
<p>I have come across Probability by studying only the methods or Distributions(like Binomial, poisson, normal,etc)</p>
<p>Now I wish to study the mathematical background and intuitive way of looking on i... | Jose Arnaldo Bebita Dris | 28,816 | <p>I would highly recommend <a href="http://rads.stackoverflow.com/amzn/click/032179477X" rel="nofollow">A First Course in Probability</a> by Sheldon Ross.</p>
<p>From the hyperlinked Amazon page:</p>
<blockquote>
<p><strong>A First Course in Probability</strong> features clear and intuitive explanations of the mat... |
409,689 | <p>I have $(x_1, y_1), (x_2, y_2)$.</p>
<p>How do I find the point that's $d$ distance away from $(x_1, y_1)$ on a straight line to $(x_2, y_2)$?</p>
<p>I know I can get the length of the line with Pythagoras. I know if I drew a circle I could use the radius as distance and the point would be where the line and the c... | John Douma | 69,810 | <p>If I can move one unit along a line I can move any distance along that line. We'll calculate how much we would have to add to each of $x_1$ and $y_1$ to move one unit along the line and then we'll multiply that by $d$ to get the answer.</p>
<p>Let $d^{\prime}$ be the distance between $(x_1,y_1)$ and $(x_2,y_2)$. It... |
950,485 | <p>I have been trying to solve the following limit but am completely stuck.</p>
<p>$$\lim_{\alpha \rightarrow \infty} 1-\left( \frac{y+\alpha}{\alpha-1} \right)^{-\alpha}$$</p>
<p>I have tried inverting the ratio and came up with the following expression:</p>
<p>$$ 1 - \lim_{\alpha \rightarrow \infty} \left( 1-\frac... | Community | -1 | <p>$$\lim_{\alpha \rightarrow \infty} \left( 1-\frac{y+1}{y+\alpha}\right)^\alpha=\lim_{\alpha \rightarrow \infty} \dfrac{\left( 1-\frac{y+1}{y+\alpha}\right)^{y+\alpha}}{\left( 1-\frac{y+1}{y+\alpha}\right)^{y}}=\exp(-(y+1))$$</p>
|
3,203,100 | <p>This equation just came to my mind, I tried solving it but can't find any solution to this problem. Can anyone please tell what is the process to approach this problem? </p>
| Claude Leibovici | 82,404 | <p>Welcome to the world of <a href="https://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow noreferrer">Lambert function</a> !</p>
<p>The solution of <span class="math-container">$x=n^x$</span> is given by
<span class="math-container">$$x=-\frac{W(-\log (n))}{\log (n)}$$</span> and, in the real domain, <span cl... |
3,653,979 | <p>Let A = <span class="math-container">$\begin{bmatrix}r_1 & r_2 & r_3 & r_4 & r_5\end{bmatrix}^T$</span> have rows <strong><span class="math-container">$r_1$</span></strong>, <strong><span class="math-container">$r_2$</span></strong>, <strong><span class="math-container">$r_3$</span></strong>, <strong... | Shiv Tavker | 687,825 | <p><strong>Hint 1:</strong> What happens to new matrix if you do <span class="math-container">$R_1 \to R_1 - R_2 - R_3 -R_4 -R_5$</span> where <span class="math-container">$R$</span> represents the rows of the matrix in part (i). </p>
<p><strong>Hint 2:</strong> The answer for the determinant is zero.</p>
|
2,545,516 | <p>So I have to assess the convergence of $$\displaystyle\sum_{n=1}^{\infty}\sin\left(\displaystyle\frac{1}{\sqrt{n}}\right).$$</p>
<p>I'm told that it diverges, but can't really see why.</p>
<p>The divergence test doesn't really help, because
$\lim\limits_{x\to\infty}\displaystyle\frac{1}{\sqrt{n}}=0$, so</p>
<p>... | user284331 | 284,331 | <p>Use the inequality that $\sin x\geq\dfrac{2}{\pi}x$ for $x\in[0,\pi/2]$.</p>
|
3,705,539 | <p>I am trying to learn more about probability and came across an interesting question that I am stuck on and can no longer find online. There are 20 numbered balls and 10 bins. Someone is trying to assign the balls to the bins, but does it with replacement on accident.</p>
<p>So they did the following: Place a ball i... | user | 293,846 | <p>After clarification of the question it can be answered as following:</p>
<ol>
<li><em>What is the probability that exactly 1 ball was assigned to exactly 4 bins?</em></li>
</ol>
<p>We have <span class="math-container">$\binom{20}{1}$</span> ways to choose the "4-fold" ball and <span class="math-container">$\binom{... |
2,209,438 | <p>I am trying to find this limit,</p>
<blockquote>
<p>$$\lim_{x \rightarrow 0} \frac{1}{x^4} \int_{\sin{x}}^{x} \arctan{t}dt$$</p>
</blockquote>
<p>Using the fundamental theorem of calculus, part 1,
$\arctan$ is a continuous function, so
$$F(x):=\int_0^x \arctan{t}dt$$
and I can change the limit to
$$\lim_{x \righ... | Stefano | 387,021 | <p>Since $F(0) = 0$ and everything is smooth, you can apply de l'Hopital and get</p>
<p>$$\lim_{x \to 0}\frac{F(x)-F(\sin x)}{x^4} = \lim_{x \to 0}\frac{\arctan x - \cos x\arctan \sin x}{4x^3}.$$</p>
<p>This last limit can be evaluated using Taylor series:</p>
<p>$$\arctan x = x-\frac{x^3}{3}+O(x^5) $$
and</p>
<p>$... |
376,796 | <p>This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures which have a reasonably simple mathematical formalization and even better if there is a related diagram that explains ... | Mohammad Ghomi | 68,969 | <p>The best way I know to illustrate the notion of curvature is via Toponogov's theorem. We can compare any (geodesic) triangle in a Riemannian manifold <span class="math-container">$M$</span> with one with the same edge lengths in Euclidean plane <span class="math-container">$R^2$</span>. The (sectional) curvature of... |
9,918 | <p>I recently flagged as "rude or offensive" the comment </p>
<blockquote>
<p>As the tone should suggest, he’s a crank. It’s a hysterical screed with a few nuggets of fact surrounded by a great deal of nonsense. E.g., he may find that set theory ‘doesn’t make sense’, but a great many of us have no trouble making sen... | Willie Wong | 1,543 | <p>I'm cleaning up <a href="https://math.stackexchange.com/questions/356264/infinite-sets-dont-exist">that comment thread</a>, since it veered a bit off topic. For the sake of transparency I'm preserving the comment thread below from screen caps. (To make sure that I don't miss any comments there are some over-laps in ... |
3,501,332 | <blockquote>
<p>A cryptoanalist, while trying to decipher a message, found that the
most frequent blocks were RH and NI, which must correspond to TH and
HE, which are the most common in the english language. Supposing the
text was codified using a 2x2 block cipher, what was the used matrix?</p>
</blockquote>
<... | Siong Thye Goh | 306,553 | <p>Note that we have<span class="math-container">\begin{align}\begin{bmatrix} 19 & 7 \\ 7 & 4\end{bmatrix}^{-1} &= (19 \cdot 4 - 7^2)^{-1}\begin{bmatrix} 4 & -7 \\ -7 & 19\end{bmatrix}\\&=(-7\cdot 4 - 7^2)^{-1}\begin{bmatrix} 4 & -7 \\ -7 & -7\end{bmatrix} \\
&= (-77)^{-1}\begin{bmat... |
275,539 | <p>Kind of leading on from my other question, how would I solve for $i$? Or how would I check that it is possible to have such an $i$?</p>
<p>First I had to check for all $2^i$ and clearly this doesn't happen as all $2^i$ are even and so I will just get even $x's$ such that $2^i \equiv x \mod 28$. So the next one I go... | DonAntonio | 31,254 | <p>$$3^1=3\pmod{28}$$</p>
<p>$$3^2=9\pmod{28}$$</p>
<p>$$3^3=27=-1\pmod{28}$$</p>
<p>$$3^4=3\cdot3^3=-3=25\pmod{28}$$</p>
<p>$$3^5=3^2\cdot3^3=-9=19\pmod{28}$$</p>
<p>$$3^6=3^3\cdot3^3=1\pmod{28}\,\,\ldots\text{etc}$$</p>
|
200,903 | <p>My teacher was explaining quadratics in my class and it was a little bit unclear to me. The problem was <br> <br>
Suppose $at^2 + 5t + 4 > 0$, show that $a > 25/16$ . <br> <br></p>
<p>My teacher said that there are no solutions for this function when it is greater than $0$ and used $b^2-4ac \lt 0$, and this ... | Ben | 27,132 | <p>The equation a$t^2$ + 5t + 4 = 0 may or may not have a solution. This depends on the value of <strong>a</strong> that you pick. Graphically, if this equation has a solution, the graph will touch and/or go through the x-axis. If there is not a solution, the graph will not have a solution. These x-intercepts are d... |
2,609,537 | <p>Is the following Proof Correct? In particular please comment on the correctness of the given formulas.</p>
<p><strong>Theorem.</strong> Given that $x$ is a real number, $x\neq 0$, and $x + \frac{1}{x}$ is an integer. For all $n\ge 1$, $x^n+\frac{1}{x^n}$ is an integer.</p>
<p><strong>Proof.</strong> We construct t... | Matthew Leingang | 2,785 | <p>I get the gist of the argument and it looks correct to me. To verify it, though, I had to stare at the summations in the middle of the two displayed equations for a while.</p>
<p>This would be annoying to a grader (graders are <a href="http://www.jimpryor.net/teaching/guidelines/writing.html" rel="nofollow norefer... |
4,495,950 | <blockquote>
<p>Why does <span class="math-container">$-\frac{1}{17-x}$</span> equal <span class="math-container">$\frac{1}{x-17}$</span>?</p>
</blockquote>
<p>Is there any simple computation to make this seem a little bit more intuitive? Right now, I cannot wrap my head around the fact that I can just switch signs of ... | JonathanZ supports MonicaC | 275,313 | <p>It comes from the following algebra facts:</p>
<p><span class="math-container">$$-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b}$$</span></p>
<p>and</p>
<p><span class="math-container">$$\begin{align}
-(c-d) &= (-c)-(-d)\\
&=-c+d\\
&=d-c
\end{align}$$</span></p>
<p>(For this second fact we first "distribute ... |
3,826,994 | <p>I would like to find <span class="math-container">$z$</span> which minimizes the below, when <span class="math-container">$x$</span> is held at a specific value.</p>
<p><span class="math-container">$f(x,z) =\sqrt{\sqrt{x^2 + z^2} - 0.25}$</span></p>
<p>For example; I would like to find the value of <span class="math... | Rezha Adrian Tanuharja | 751,970 | <p>Most of the time, polar form is better when dealing with complex numbers. Draw a circle with <span class="math-container">$2i$</span> and <span class="math-container">$6$</span> as the ends of its diameter. This diameter divide the circle into 2 parts, the lower half is the loci.</p>
<p>Hint: if <span class="math-co... |
1,572,045 | <p>This is maybe a stupid question, but I want to find the roots of:</p>
<blockquote>
<p>$$2(x+2)(x-1)^3-3(x-1)^2(x+2)^2=0$$</p>
</blockquote>
<p>What that I did:</p>
<p>$$\underbrace{2(x+2)(x-1)(x-1)(x-1)}_{A}-\underbrace{3(x-1)(x-1)(x+2)(x+2)}_{B}=0$$</p>
<p>So the roots are when $A$ and $B$ are both zeros when... | J.Gudal | 225,386 | <p>Well </p>
<p>$2(x+2)(x-1)^{3}-3(x-1)^{2}(x+2)^{2}=0 \Rightarrow (x-1)^{2}(x+2)[2(x-1)-3(x+2)]=0 \Rightarrow (x-1)^{2}(x+2)(-x-8)=0 $</p>
<p>From here it should be clear why $x=-8$ is also a root.</p>
<p>This is called <a href="https://www.mathsisfun.com/algebra/factoring.html" rel="nofollow">factoring</a>.</p>
|
1,307,085 | <p>How does one solve this equation?</p>
<blockquote>
<p>$$\cos {x}+\sin {x}-1=0$$</p>
</blockquote>
<p>I have no idea how to start it.</p>
<p>Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?</p>
<p>Thanks in advance!</p>
| Eric R. Anschuetz | 162,230 | <p>As $\sin\left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\left(\cos x+\sin x\right)$ by the angle addition formula we find that:
\begin{equation}
\begin{aligned}
\cos x+\sin x-1&=0\\
\implies\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)&=1\\
\implies\sin\left(x+\frac{\pi}{4}\right)&=\frac{\sqrt{2}}{2}\\
\implie... |
1,307,085 | <p>How does one solve this equation?</p>
<blockquote>
<p>$$\cos {x}+\sin {x}-1=0$$</p>
</blockquote>
<p>I have no idea how to start it.</p>
<p>Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?</p>
<p>Thanks in advance!</p>
| Brian Tung | 224,454 | <p>I'll throw my hat in the ring to get a picture in edgewise. :-)</p>
<p><a href="https://i.stack.imgur.com/0dRHT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0dRHT.png" alt="enter image description here"></a></p>
<p>I'll use $\theta$ for the angle, rather than $x$ (to avoid confusion with the ... |
846,108 | <p>How do you solve this equation: $2x+8=6x-12$ by using the guess and check method?</p>
<p>I divide $2x+8$ and I get $4$ then I divide $6x-12$ and I get $-2$ but I don't know what to do next or is it wrong?</p>
| paw88789 | 147,810 | <p>One thing that you can get by 'guess-and-check' is how linear functions grow. </p>
<p>$$\begin{array}{r|r|r} x &2x+8 &6x-12\\ \hline 0&8&-12\\1&10&-6\\2&12&0 \\ \vdots&\vdots&\vdots \end{array}$$ Note that every time $x$ increases by $1$, $2x+8$ increases by $2$ and $6x-12... |
177,144 | <p>If G is a finite group, I understand that the category of RO(G)-graded spectra, when rationalized, becomes Quillen equivalent to the category of Mackey functors valued in chain complexes of rational vector spaces.</p>
<p>How does RO(G) act on the category of Mackey functors? For instance, if F = F(G/H) is a Mackey... | Justin Noel | 8,818 | <p>Although I agree with Peter's comments, I believe I can add a few helpful comments of my own. First for $G$ finite, every rational Mackey functor is both injective and projective, so chain complexes are weakly equivalent to their homology. One can find this statement in appendix A of Greenlees-May 'Generalized Tate ... |
3,583,484 | <p>I am trying to work through a hw problem to show that the ring of gaussian integers <span class="math-container">$G=\{a+bi:a,b\in\mathbb{Z}\}$</span> is principal. To make it concrete I picked the ideal <span class="math-container">$I$</span> generated by <span class="math-container">$a=5$</span> and <span class="ma... | egreg | 62,967 | <p>You can note that
<span class="math-container">$$
(2+i)^2=4+4i-1=3+4i
$$</span>
and that <span class="math-container">$5=(2+i)(2-i)$</span>. So the greatest common divisor is <span class="math-container">$2+i$</span>, which is the generator of the ideal.</p>
|
3,583,484 | <p>I am trying to work through a hw problem to show that the ring of gaussian integers <span class="math-container">$G=\{a+bi:a,b\in\mathbb{Z}\}$</span> is principal. To make it concrete I picked the ideal <span class="math-container">$I$</span> generated by <span class="math-container">$a=5$</span> and <span class="ma... | Daniel Schepler | 337,888 | <p>Let us work through applying the Euclidean algorithm to find the gcd in <span class="math-container">$\mathbb{Z}[i]$</span> of <span class="math-container">$3+4i$</span> and <span class="math-container">$5$</span>. Now, <span class="math-container">$|3+4i|^2 = |5|^2 = 25$</span>, so neither element is really "small... |
2,592,007 | <blockquote>
<p>Let $V$ be a vector space and $W,U\subseteq V$ subspaces s.t $W\not \subseteq U$
$\dim(V)=5, \dim(W)=2, \dim(U)=4$ </p>
<p>Prove\Disprove: $\dim(U\cap W)=1$</p>
</blockquote>
<p>So I started with \begin{align} & \dim(W+U)=\dim(U)+\dim(W)-\dim(U\cap W) \\[10pt]
\iff & \dim(W+U)=4+2-\di... | Dan | 500,478 | <p>From $\dim(W+U)\leq \dim(V)=5$ and using that $1\leq \dim(U\cap W)\leq\dim(W)=2$ we get two cases. If $\dim(U\cap W)=2=\dim(W)$, then $W\cap U=W$, as $W\cap U \subset W$, so $W\subset U$, which contradicts our initial assumption. So $\dim(U\cap W)=1$.</p>
|
1,829,030 | <p>The limit isn't too bad using l'hospital's rule, but I was wondering if there was a way to do it without l'hospital's. </p>
<p>Looking around the section limits without lhopital's, it seems usually evaluating without requires some clever factoring, while here the $\arctan$ seems to muck things up. </p>
<p>Here is ... | zhw. | 228,045 | <p>Let $f(x) = 1+\arctan (x/2).$ Apply $\ln $ to the expression of interest to get</p>
<p>$$\frac{\ln (f(x))}{x/2} = 2\frac{\ln f(x) - \ln f(0)}{x-0}.$$</p>
<p>As $x\to 0,$ the last expression $\to 2(\ln f)'(0)$ by definition of the derivative (no L'Hopital used). That's easy enough to compute. Exponentiate back for ... |
872,889 | <p>What determinant is zero? What equation does this give for the plane?</p>
<p>I need some help here, am pretty stuck</p>
| Aritmo | 352,282 | <p>You can find plenty on information on those special continued fractions at
the following link (American Mathematical Monthly):</p>
<p><a href="https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/1055084257873891/?type=3&theater" rel="nofollow noreferrer">https://www.facebook.... |
2,024,997 | <blockquote>
<p>$$\lim_{x \rightarrow +\infty}\frac{\log_{1.1}x}{x}$$</p>
</blockquote>
<p>I can solve this easily by generating the graph with my calculator, but is there is a way to do this analytically?</p>
| hamam_Abdallah | 369,188 | <p>Let $g(x)=\ln(x)-\sqrt{x}$</p>
<p>we have $g(1)=0$ and</p>
<p>$$g'(x)=\frac{1}{2x}(2-\sqrt{x})$$</p>
<p>$g$ is decreasing at $[4,+\infty)$</p>
<p>thus</p>
<p>$$\forall x\geq 4 \;\;0<\frac{\log_{1.1}(x)}{x}\leq \frac{1}{\ln(1.1)\sqrt{x}}$$</p>
<p>and squeeze theorem.</p>
|
2,024,997 | <blockquote>
<p>$$\lim_{x \rightarrow +\infty}\frac{\log_{1.1}x}{x}$$</p>
</blockquote>
<p>I can solve this easily by generating the graph with my calculator, but is there is a way to do this analytically?</p>
| Mark Viola | 218,419 | <blockquote>
<p>I thought it might be instructive to present an approach that relies on elementary tools only. To that end, we begin with the following primer.</p>
<p><strong>PRIMER: BOUNDS FOR THE LOGARITHM FUNCTION</strong></p>
<p>In <a href="https://math.stackexchange.com/questions/1589429/how-to-prove-that-logxx-... |
2,849,643 | <p>Consider the following recurrence problem:
\begin{align}
d_{i-1} &= 2\varphi_{i+1}+4\varphi_i + 8d_i-7d_{i+1} - F \left( \delta_{i,N} + \delta_{i,N+1} \right) \, , \\
\varphi_{i-1} &= -7\varphi_{i+1}-16\varphi_{i} + 24 \left( d_{i+1}-d_{i} \right) + F \left( \delta_{i,N} + \delta_{i,N+1} \right) \, ,
\end{a... | Yuri Negometyanov | 297,350 | <p>$$\mathbf{\color{green}{The\ linear\ approach}}$$</p>
<p>As it follows from the comments, the issue task can be detalized in the form of
\begin{cases}
d_{i-1} = 2\varphi_{i+1}+4\varphi_i + 8d_i-7d_{i+1}\\[4pt]
\varphi_{i-1} = -7\varphi_{i+1}-16\varphi_{i} + 24 \left( d_{i+1}-d_{i} \right) \\[4pt]
i=2,3\dots N-1\\[4... |
635,077 | <p>$$\sin(a+b) = \sin(a) \cos(b) + \cos(a) \sin(b)$$</p>
<p>How can I prove this statement?</p>
| Edward ffitch | 26,243 | <p>$e^{i(a+b)}=\cos(a+b)+i\sin(a+b)$ by Euler's formula. But $e^{i(a+b)}=e^{ia}e^{ib}=(\cos(a)+i\sin(a))(\cos(b)+i\sin(b))= \cos(a)\cos(b)-\sin(a)\sin(b)+i(\sin(a)\cos(b) + \cos(a)\sin(b))$</p>
<p>So by comparing real and imaginary parts you obtain the trigonometric addition formulae for both $\sin$ and $\cos$.</p>
|
635,077 | <p>$$\sin(a+b) = \sin(a) \cos(b) + \cos(a) \sin(b)$$</p>
<p>How can I prove this statement?</p>
| dwarandae | 55,403 | <p>This is the way I learned it: a geometric proof like <a href="http://en.wikipedia.org/wiki/File%3aTrigSumFormula.svg" rel="nofollow">this</a> on Wikipedia.</p>
<p>The segment $OP$ has length $1$. We have then, $\sin(\alpha + \beta) = PB = PR + RB = \cos(\alpha) \sin(\beta) + \sin(\alpha) \cos(\beta)$.</p>
|
635,077 | <p>$$\sin(a+b) = \sin(a) \cos(b) + \cos(a) \sin(b)$$</p>
<p>How can I prove this statement?</p>
| Michael Hoppe | 93,935 | <p>This gem comes from E. Schmidt: Consider $f(x)=\sin(\alpha+\beta-x)\cos(x)+\cos(\alpha+\beta-x)\sin(x)$. Since $f'(x)=0$ we know that $f$ is constant, hence $f(0)=f(\beta)$.</p>
|
1,800,233 | <blockquote>
<p>If $n$ is composite and $\phi{(n)} | (n - 1)$ then prove that $n$ has at least four distinct prime factors.</p>
</blockquote>
<p><strong>Attempt:</strong></p>
<p>Since $n$ is not a prime, let's first take the case that $n$ is squarefree. Then $n = a_1 \cdot a_2 \cdots a_r$ where $a_i$ are the prime ... | user5713492 | 316,404 | <p>To eliminate $3$ prime factors is pretty simple because since $\phi(n)<n-1$ it implies that $\phi(n)\le\frac n2$. If $3$ isn't one of the factors the smallest $\frac{\phi(n)}n$ could be is $\frac45\frac67\frac{10}{11}=0.6234$. If $5$ isn't a factor, the smallest is $\frac23\frac67\frac{10}{11}=0.5165$. If both $3... |
3,310,038 | <p>If <span class="math-container">$U$</span> is an open set of <span class="math-container">$\mathbb{R}^{m}$</span>, do we have that <span class="math-container">$U\times \mathbb{R}^{n-m}$</span> is an open set of <span class="math-container">$\mathbb{R}^{n}$</span>? </p>
<p>Here <span class="math-container">$\mathbb... | ayeayemaung | 249,040 | <p>Yes. Note <span class="math-container">$U \times \Bbb R^{n - m} = \pi^{-1}(U)$</span> where <span class="math-container">$\pi$</span> is the projection map
<span class="math-container">$$
\pi : \Bbb R^n \ni (x_1, \ldots, x_m, \ldots, x_n) \mapsto (x_1, \ldots, x_m) \in \Bbb R^m
$$</span> and as projection maps are c... |
671,407 | <p>I have problem with equation: $4^x-3^x=1$. </p>
<p>So at once we can notice that $x=1$ is a solution to our equation. But is it the only solution to this problem? How to show that there aren't any other solutions? </p>
| IAmNoOne | 117,818 | <p>Prove it by contradiction. </p>
<p>Let $f(x) = 4^x - 3^x - 1$. This function is smooth and as you have shown, $x = 1$ is a root. By Rolle's theorem, suppose it had two other roots $f(a) = f(b) = 0$, then there exists $c$ in between $a$ and $b$ (where $a < 1 < b$) such that $f'(c) = 0$, but notice for $ x \geq... |
671,407 | <p>I have problem with equation: $4^x-3^x=1$. </p>
<p>So at once we can notice that $x=1$ is a solution to our equation. But is it the only solution to this problem? How to show that there aren't any other solutions? </p>
| N. S. | 9,176 | <p>It is easy to see that there is no negative solution.</p>
<p>Assume by contradiction there exists a solution $x=a$ with $0 < a \neq 1$. </p>
<p>Let $g(y)=y^a$. By the mean value Theorem, there exists some $c \in (3,4)$ such that</p>
<p>$$\frac{g(4)-g(3)}{4-3} =g'(c)$$</p>
<p>Therefore
$$1= g'(c)=ac^{a-1}$$</... |
708,604 | <blockquote>
<p>Let $f: \mathbb{R^4} \to \mathbb{R}$ be a linear transformation defined by $f(a,b,c,d)=a+b+c+d$. Find a basis for the $Im(f)$.</p>
</blockquote>
<p>So, $Im(f)=\{f(a,b,c,d) \in \mathbb{R}: (a,b,c,d) \in \mathbb{R^4} \}$.</p>
<p>Then $Im(f)=\{a+b+c+d \in \mathbb{R}: a,b,c,d \in \mathbb{R} \}=\mathbb{R... | mle | 66,744 | <p>if $\operatorname{im}(f)=\Bbb{R}^1$ then $\dim_\Bbb{R}(\operatorname{im}(f))=\dim_\Bbb{R}(\Bbb{R}^1)=1$, therefore a basis for the $\operatorname{im}(f)$ is a basis for $\Bbb{R}^1$</p>
<p><strong>Hint</strong>: let be $\{b_1,b_2,b_3,b_4\}$ a basis for $\Bbb{R}^4$, for example "<a href="http://de.wikipedia.org/wiki/... |
279,520 | <p>So I have to find the integral of $$ \int \frac{\sin^{-1}(x)}{\sqrt{1+x}} \; dx$$</p>
<p>I think I have to do this using the integration by parts..so I will take $f = \sin^{-1}(x)$ and $ \sqrt {1+x}=g' $...what about now? </p>
| gt6989b | 16,192 | <p>Integrate by parts, letting $dv = (x+1)^{-1/2}$ and $u = \arcsin(x)$, then $du = \frac{1}{\sqrt{1-x^2}}$ and $v = \int (x+1)^{-1/2} dx = 2 \sqrt{x+1}$.</p>
<p>Use $\int u dv = uv - \int v du$ to reduce this to</p>
<p>$$
\int v du = \int 2 \sqrt{ \frac{x+1}{1-x^2} } dx
= 2 \int \frac{dx}{\sqrt{1-x}}
= -4 \sqr... |
1,689,523 | <p>I need help with this Laplace question.
<span class="math-container">$$f(t) = e^{-t} \sin(t) $$</span></p>
<hr />
<p>Answer should be <span class="math-container">$\dfrac{1}{s^2 + 2s + 2}$</span></p>
<hr />
<p>What I'm currently doing is as follows:</p>
<p><span class="math-container">$u = \sin(t)\qquad$</span> ... | user873542 | 873,542 | <p>Here's the 'direct' proof:
<span class="math-container">$f(t)=e^{-t}\sin t.$</span>
<span class="math-container">\begin{eqnarray*}\mathcal{L}\{f(t)\}&=&\int_{0}^{+\infty}e^{-(s+1)t}\sin t\,dt~=~\lim_{\ell\to+\infty}\int_{0}^{\ell}e^{-(s+1)t}\sin t\,dt\\
&=&-\frac{1}{s+1}\lim_{\ell\to+\infty}\left[... |
510,080 | <p>This is not too obvious to me - what is the size of alternating group?</p>
<p>Following the hint in the comment, should it be $A_n = S_n/2$?</p>
<p>So I don't feel right up to here.....</p>
| Jared | 65,034 | <p>The map $\sigma:S_n\to\mathbb{Z}/2\mathbb{Z}$ defined by sending a permutation to $0$ if it has even parity, and $1$ if it has odd parity, is a group homomorphism. The kernel of this map is $A_n$, so by the first isomorphism theorem, we have $[S_n:A_n]=2$ for $n\ge 2$ (the map is not surjective for $n=1$). It foll... |
106,126 | <blockquote>
<p><strong>Problem</strong> Prove that $n! > \sqrt{n^n}, n \geq 3$. </p>
</blockquote>
<p>I'm currently have two ideas in mind, one is to use induction on $n$, two is to find $\displaystyle\lim_{n\to\infty}\dfrac{n!}{\sqrt{n^n}}$. However, both methods don't seem to get close to the answer. I wonder ... | Henry | 6,460 | <p>$(n!)^2 = (n \times 1) \times ((n-1)\times 2) \times \cdots \times (1 \times n) \gt n^n$</p>
<p>since $(n-1)\times 2 = 2n-2 \gt n$ iff $n \gt 2$. </p>
<p>Then take the square root. </p>
|
2,578,444 | <blockquote>
<p><span class="math-container">$\tan x> -\sqrt 3$</span></p>
</blockquote>
<p>How do I solve this inequality?</p>
<p>From the <a href="https://www.desmos.com/calculator/qb8bg1vbsf" rel="nofollow noreferrer">graph</a> it is evident that <span class="math-container">$\tan x>-\sqrt 3$</span> for <span ... | Martin Argerami | 22,857 | <p>You went wrong in reading the graph. The number $5\pi/3$ should have been $5\pi/2$. And the period is $\pi $ and not $2\pi $, which leads to the correct answer that you quoted: if you have $$\left( n\pi+\frac {5\pi}3,n\pi+\frac {5\pi}2\right), $$ Now,replacing $n $ with $n-2$ achieves the solution given.</p>
|
3,691,255 | <p>Pierre runs a game at a fair, where each player is guaranteed to win $10. </p>
<p>Players pay a certain amount each time they roll an unbiased die, and must keep rolling until a ‘6’ occurs. </p>
<p>When a ‘6’ occurs, Pierre gives the player $10 and the game concludes. </p>
<p>On average, Pierre wishes to make a p... | Math Comorbidity | 731,254 | <p>The following question was answered by u/Alkalannar on reddit. </p>
<p>Answer: </p>
<p>Assumption not stated: This is a 6-sided die.</p>
<p>Consider the general n-sided die, and you want to roll max (or 1).</p>
<p>Expected income for the game is [Sum from k = 1 to infinity of xk(1 - 1/n)k-1(1/n)] = xn, where x i... |
4,278,505 | <p>I would like to clear up a confusion which might be trivial. In a proof the author proved <span class="math-container">$T = T'$</span> as following:</p>
<p>The author showed if <span class="math-container">$x \in T$</span> then <span class="math-container">$x \in T'$</span>, the next line is -</p>
<blockquote>
<p>.... | Anonymous M | 934,527 | <p>Based on the comments it seems the issue is that you need to return to the definition of <span class="math-container">$T'$</span> and make sure you understand it thoroughly. The author proved <span class="math-container">$T \subseteq T'$</span> and then immediately concluded <span class="math-container">$T = T'$</sp... |
787,926 | <p>I need some help to solve this integral:</p>
<p>$$\int_0^1 dy\int_0^{1-y} \cos \left(\frac{x-y}{x+y} \right) \mathrm dx$$</p>
<p>Thank you.</p>
| DaveBlackston | 149,057 | <p>Unless I am missing something, Julian's algorithm will simply check all numbers from 1, 2, 3, ... until it finds the 1000th 5-smooth number. This means it requires $O(u_n)$ time.</p>
<p>A more efficient algorithm is as follows. Let $N$ be large, and define $S_2$, $S_3$ and $S_5$ s follows.</p>
<p>$S_2 = \cup_{i=... |
942,030 | <p>Given a Hilbert space $\mathcal{H}$.</p>
<p>Consider spectral measures:
$$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad E(\mathbb{C})=1$$</p>
<p>Define its support:
$$\operatorname{supp}(E):=\bigg(\bigcup_{U=\mathring{U}:E(U)=0}U\bigg)^\complement=\bigcap_{C=\overline{C}:E(C)=1}C$$</p>
<p>By second c... | Ri-Li | 152,715 | <p>You have $b(\theta) = \dot{\theta}^2$ .......(1)</p>
<p>Where $\dot{\theta}$ and $\ddot{\theta}$ are the first and second derivate respecting time.</p>
<p>In the above expression, you understand that $\dot{b}(\theta) = b'(\theta)\dot{\theta}$ because of the chain rule. Then $b'(\theta)\dot{\theta} = \dot{b}(\thet... |
942,030 | <p>Given a Hilbert space $\mathcal{H}$.</p>
<p>Consider spectral measures:
$$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad E(\mathbb{C})=1$$</p>
<p>Define its support:
$$\operatorname{supp}(E):=\bigg(\bigcup_{U=\mathring{U}:E(U)=0}U\bigg)^\complement=\bigcap_{C=\overline{C}:E(C)=1}C$$</p>
<p>By second c... | Chinny84 | 92,628 | <p>The chain rule is your friend
$$
\dfrac{d}{dt}b(\theta) = \dfrac{db}{d\theta}\cdot \dfrac{d\theta}{dt}
$$
but you also have $b=\dot{\theta}^2$</p>
|
3,045,899 | <p>In the lecture notes we have a fact:</p>
<blockquote>
<p>If <span class="math-container">$A$</span> has orthonormal columns then <span class="math-container">$||Ax||^2_2 = ||x||^2_2$</span></p>
</blockquote>
<p>Why is it the case? What properties of matrix-vector multiplication should I know to reason about this... | littleO | 40,119 | <p>The fact that <span class="math-container">$A$</span> has orthonormal columns is expressed concisely by the statement that <span class="math-container">$A^T A = I$</span>. It follows from this fact that
<span class="math-container">\begin{align}
\| Ax \|^2 &= (Ax)^T Ax \\
&= x^T A^T A x \\
&= x^T x \\
&a... |
4,616,559 | <p>For each day we store a snapshot of data in a database. We want to balance the storage costs with the densitiy of snapshots.
The older a time frame is the fewer snapshots from this time frame we need.
For example: if we store 10 snapshots from last year then we would like to store only 1 snpashot from the time ten y... | Alexander Tissen | 1,139,101 | <p>This is the idea how to compute the date when to delete a snapshot based on the snapshot's date only.
First we need the ability to assign a number ( or an index ) to every possible date. So, let's choose a start date (for example 2020-01-01) and assign it the index 0.
The choice of the start date is arbitary and doe... |
659,256 | <p>This might be a silly question to some, but I need some help in this topic. <br />
Iota, denoted as <em>'i'</em> is equal to the principal root of -1.
Therefore, </p>
<p>$\iota^2 = -1$</p>
<p>When studying Modulus, I was wondering..</p>
<p>$|\iota| = ?$</p>
<p>A Google search revealed that the value is <strong>... | heropup | 118,193 | <p>An axiomatic definition of $\mathbb C$ follows from its construction from $\mathbb R^2 = \{(x,y) : x \in \mathbb R, y \in \mathbb R\}$. An abbreviated treatment follows, and a rigorous treatment can be found in Walter Rudin's classic text <em>Principles of Mathematical Analysis</em>.</p>
<p>We <strong>define</stro... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| torbonde | 30,734 | <p>How about giving various proofs that $1+\cdots+n = n(n+1)/2$? There are at least a few different geometric, easily understandable ones (although I'm not sure about how easy, when it comes to 3rd-graders. You'll be the judge of that.)</p>
<p>I saw the one below recently, and it blew my mind. Using this, you will nee... |
962,242 | <p>$X \simeq Y$ reads as $X$ is <em>equivalent</em> to $Y$</p>
<p>If $X \simeq Y$, <strong>iff</strong> $X \leftrightarrow Y$ is a tautology.</p>
<p>Now given $X_1 \simeq X_2$, how do I prove,</p>
<ol>
<li>$\tilde X_1 \simeq \tilde X_2$</li>
<li>$X_1 \cap Y\simeq X_2 \cap Y$</li>
<li>$X_1 \cup Y\simeq X_2 \cup Y$</l... | DSinghvi | 148,018 | <p>TRY TO PROVE AB<em>CD=AC</em>CB </p>
<p>THEN square it use AB^2=AC^2+BC^2 </p>
<p>To prove AB * CD=AC * CB </p>
<p>Equate the areas of triangle in terms of AC and CB and the other one in AB and CD </p>
<p>1/2 AC * CB=1/2 CD * AB</p>
|
962,242 | <p>$X \simeq Y$ reads as $X$ is <em>equivalent</em> to $Y$</p>
<p>If $X \simeq Y$, <strong>iff</strong> $X \leftrightarrow Y$ is a tautology.</p>
<p>Now given $X_1 \simeq X_2$, how do I prove,</p>
<ol>
<li>$\tilde X_1 \simeq \tilde X_2$</li>
<li>$X_1 \cap Y\simeq X_2 \cap Y$</li>
<li>$X_1 \cup Y\simeq X_2 \cup Y$</l... | Irvan | 172,851 | <p>The area of the triangle can be expressed in two ways: $\displaystyle \frac{|AC| |BC|}{2}$ and $\displaystyle \frac{|AB| |CD|}{2}$. Thus, they must be equal:</p>
<p>$$\frac{|AC| |BC|}{2}=\frac{|AB| |CD|}{2}$$</p>
<p>Multiplying throughout by $2$ and squaring,</p>
<p>$$|AC|^2 |BC|^2 = |AB|^2 |CD|^2$$</p>
<p>$$|AC... |
4,206,286 | <p><span class="math-container">$$\int_0^1\int_0^\infty ye^{-xy}\sin x\,dx\,dy$$</span></p>
<p>How can I calculate out the value of this integral?</p>
<p>P.S. One easy way is to calculate this integral over <span class="math-container">$dy$</span> first, to get an integral form <span class="math-container">$\frac{1-e^{... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/7UmqH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7UmqH.png" alt="enter image description here" /></a></p>
<p>For the acute triangle <span class="math-container">$ABC$</span> with the circumcenter <span class="math-container">$O$</span></p>
<p><span class="m... |
2,196,936 | <p>How to prove that $7p + 3^p -4$ is not a perfect square? </p>
<p>I calculated: $\left(\frac{7p+3^p-4}{p}\right) = \left(\frac{-1}{p}\right)$. So if $p \equiv 3 \mod 4$, the result is $-1$. So in that case, $7p+3^p -4$ can't be a square. But what about the case $p \equiv 1 \mod 4$? Any hints? Thanks in advance.</p>
| Shaun | 104,041 | <p>The squares modulo <span class="math-container">$4$</span> are</p>
<p><span class="math-container">$$(4k)^2 = 4(4k^2) \equiv 0 \pmod{4},$$</span></p>
<p><span class="math-container">$$(4k+1)^2 = 4(4k^2 + 2k) + 1 \equiv 1 \pmod{4},$$</span></p>
<p><span class="math-container">$$(4k + 2)^2 = 4(4k^2 + 4k + 1) \equiv 0 ... |
1,722,995 | <blockquote>
<blockquote>
<p>Question: Given the circle $x^2+y^2=25$ is inscribed in triangle $\triangle ABC$, where vertex $B$ lies on the first quadrant. Slope of $AB$ is $\sqrt 3$ and has a positive y-coordinate, and $|AB|=|AC|$. Find the equations for $AC$ and $BC$</p>
</blockquote>
</blockquote>
<p>I foun... | chenbai | 59,487 | <p><a href="https://i.stack.imgur.com/t9eXK.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/t9eXK.jpg" alt="enter image description here"></a></p>
<p>I will suggest to take $\alpha$ as parameter to write the equation. and you need to take care of the range of $\alpha$ .</p>
|
1,722,995 | <blockquote>
<blockquote>
<p>Question: Given the circle $x^2+y^2=25$ is inscribed in triangle $\triangle ABC$, where vertex $B$ lies on the first quadrant. Slope of $AB$ is $\sqrt 3$ and has a positive y-coordinate, and $|AB|=|AC|$. Find the equations for $AC$ and $BC$</p>
</blockquote>
</blockquote>
<p>I foun... | Steven Alexis Gregory | 75,410 | <p><span class="math-container">$AB=AC$</span> suggests to me that line <span class="math-container">$BC$</span> is a vertical line and point <span class="math-container">$A$</span> is on the negative x-axis. See the picture.</p>
<blockquote>
<p><span class="math-container">$A$</span> has to be on the line perpendicula... |
278 | <p>If you take a look at our status in the <a href="http://area51.stackexchange.com/proposals/64216/mathematics-educators">area51</a>, all criteria seem to be satisfied (soon) but not the number of questions asked (which seems to be decreasing, actually). Do you think this is a problem for us? Is it something we should... | Robert Cartaino | 2 | <p>This pattern is not unusual at all. Most sites go through a <em>honeymoon period</em> where users pile in from the excitement of a new launch with their questions at the ready. Then you experience a sharp drop in activity… followed by a pattern of slow, steady growth as you continue to compile content and the... |
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