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 |
|---|---|---|---|---|
3,382,464 | <p>Let <span class="math-container">$g$</span> be a <strong>smooth</strong> Riemannian metric on the closed <span class="math-container">$n$</span>-dimensional unit disk <span class="math-container">$\mathbb{D}^n$</span>. Let <span class="math-container">$f$</span> be a harmonic function w.r.t <span class="math-contain... | Community | -1 | <p>Given that <span class="math-container">$a^5 - a^3 + a=2$</span> we write <span class="math-container">$$a^3\left(a^2-1+\frac{1}{a}\right)=2$$</span>
This implies <span class="math-container">$$a^3\left(\left(a+\frac{1}{a}\right)^2-3\right)=2$$</span>
Now by A.M-G.M inequality <span class="math-container">$a+\frac{... |
1,939,382 | <p>I've read about integration, and i believe i understood concept correctly. But, unfortunately, the simplest exercise already got my stumbled. I need to find an integral of $x{\sqrt {x+x^2}}$. So i proceed as follows,</p>
<p>By the fundamental theorem of calculus:</p>
<p>$f(x)=\int[f'(x)]=\int[x\sqrt{x+x^2}]$,</p>
... | Enrico M. | 266,764 | <p>Hint</p>
<p>$$x\sqrt{x + x^2} = x\sqrt{\left(x + \frac{1}{2}\right)^2 - \frac{1}{4}}$$</p>
<p>Then you may think about setting</p>
<p>$$x + \frac{1}{2} = y$$</p>
<p>Et cetera.</p>
|
3,996,218 | <p>Well I wanted to know whether or not <span class="math-container">$y = x^2 + x + 7$</span> is a quadratic equation since the general form is <span class="math-container">$ax^2 + bx + c = 0$</span> here the equation
<span class="math-container">$y=x^2+x+7$</span>. Isn't equal to zero so I'm a bit confused</p>
| Toby Mak | 285,313 | <p>The general form is not <span class="math-container">$ax^2+bx+c=0$</span>, but <span class="math-container">$y = ax^2+bx+c$</span>. To have a function, you must be able to put one number in (<span class="math-container">$x$</span> in this case), and output one number out (<span class="math-container">$y$</span>).</p... |
3,996,218 | <p>Well I wanted to know whether or not <span class="math-container">$y = x^2 + x + 7$</span> is a quadratic equation since the general form is <span class="math-container">$ax^2 + bx + c = 0$</span> here the equation
<span class="math-container">$y=x^2+x+7$</span>. Isn't equal to zero so I'm a bit confused</p>
| Deepak | 151,732 | <p>When you write <span class="math-container">$y=x^2 + x+7$</span>, that is <em>not</em> generally considered a "quadratic equation" in the commonly used sense. Most of the time, that is taken to mean a functional relationship between two variables, namely <span class="math-container">$y$</span> and <span cl... |
1,431,464 | <p>Does anyone know a good reference where it is shown that the Schwartz class $\mathcal{S}(\mathbb R)$ is a dense subset of $L^2(\mathbb R)$?</p>
<p>Many thanks</p>
| Silvia Ghinassi | 258,310 | <p>Dan gave a good bunch of references. Another proof can be found in Lieb and Loss' "Analysis", Lemma 2.19. The following is a quick sketch of how the proof goes.</p>
<hr>
<p>In Rudin's "Real and Complex Analysis", Theorem 3.14, it is proved that $C_c(\mathbb R)$ is dense in $L^p (\mathbb R)$ (you can find this also... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| 7-adic | 2,666 | <p>I would say Fourier analysis, by Javier Duoandikoetxea, AMS.</p>
|
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| John Stillwell | 1,587 | <p>For a really gentle introduction I would recommend
Kolmogorov and Fomin's <em>Introductory Real Analysis</em>,
available as a Dover paperback. They have a nice
introduction to distributions as "generalized functions"
in Section 21.</p>
|
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Michael Hardy | 6,316 | <p>There's the book by Ian Richard and Heekyung Youn. It describes itself as a "non-technical introduction", which apparently means you don't need to know measure theory, topology, or functional analysis. Nonetheless you do need to think more like a mathematician than a physicist or the like in order to appreciate th... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| user45664 | 94,943 | <p>"Mathematics for the Physical Sciences", Laurent Schwartz, Dover 2008 is a simplified English language book that covers some (maybe even much) of Schwartz's theory of distributions. Very readable, helpful and interesting (also $19.95). The title sounds more general than it actually is--really is focused on distribut... |
842,266 | <p>I have a tiny little doubt related to one proof given in Ahlfors' textbook. I'll copy the statement and the first part of the proof, which is the part where my doubt lies on.</p>
<p><strong>Statement</strong>
The stereographic projection transforms every straight line in the $z$-plane into a circle on $S$ which pas... | Emily | 31,475 | <p>$0 \le a_0< 1$ is satisfied because the plane must intersect the Riemann sphere, whereon the maximum component if any point is bounded by 1.</p>
|
4,411,096 | <p>I know closure of connected set in a topological space must be connected as well. However, I can't understand why this counterexample fails.
Take <span class="math-container">$X=[0,2)\cup\{3\}, B_2(1)=(0,2)$</span> which is connected. Now take the closed ball <span class="math-container">$C_2(1)=[0,2)\cup \{3\}$</sp... | José Carlos Santos | 446,262 | <p>It turns out that the closure of <span class="math-container">$(0,2)$</span> in <span class="math-container">$[0,2)\cup\{3\}$</span> is <span class="math-container">$[0,2)$</span>, not <span class="math-container">$[0,2)\cup\{3\}$</span>. The closure of an open ball is not always the corresponding closed ball.</p>
|
3,369,069 | <p>Let <span class="math-container">$l_1$</span> and <span class="math-container">$l_2$</span> be two distributions in disjoint variables <span class="math-container">$x_1, ..., x_n$</span> and <span class="math-container">$y_1, ..., y_m$</span>. Then it is said to be possible to define a product distribution.</p>
<p>... | Robert Furber | 184,596 | <p>As mathcounterexamples.net says, this is related to the tensor product of distributions. Here is an outline of how to do this. The full proof is rather involved, depending on your background knowledge, so I'll give a reference.</p>
<p>Let's write <span class="math-container">$\newcommand{\D}{\mathcal{D}}\newcommand... |
214,486 | <p><a href="https://i.stack.imgur.com/rZXpG.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/rZXpG.gif" alt="enter image description here"></a></p>
<p>I made it by another software, and met some problems to change it into MMA code.</p>
<pre><code>f[x_] := Graphics[
Line[AnglePath[{90 °, -90 °}[[
1 + Nest[... | chyanog | 2,090 | <p>I think OP may want animation with transition effects. Compare these two effects:<br>
<a href="https://i.stack.imgur.com/c81s4.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/c81s4.gif" alt="enter image description here"></a><br>
Then translation<br>
<a href="https://i.stack.imgur.com/kbTXW.gif" rel="noref... |
214,486 | <p><a href="https://i.stack.imgur.com/rZXpG.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/rZXpG.gif" alt="enter image description here"></a></p>
<p>I made it by another software, and met some problems to change it into MMA code.</p>
<pre><code>f[x_] := Graphics[
Line[AnglePath[{90 °, -90 °}[[
1 + Nest[... | A little mouse on the pampas | 42,417 | <p>I have a similar code made by <a href="https://github.com/ChenMinQi/chenminqi.github.io/tree/master/Koch%E6%9B%B2%E7%BA%BF%E7%9A%84%E5%8A%A8%E6%80%81%E5%8C%96" rel="nofollow noreferrer">Apple</a>, just for reference.</p>
<pre><code>Clear["Global`*"]
rotate[p4_, p2_] := Evaluate[Simplify@RotationTransform[1. Pi/3, p... |
3,761,689 | <p>I was watching a YouTube video where it showed how length of daylight changes depending on the time of year, and I was curious and wanted to try calculating the value of how long the daylight is in the Tropic of Cancer (23.5 degrees latitude) during the winter solstice, apparently 10 hours and 33 minutes or so accor... | Hagen von Eitzen | 39,174 | <ul>
<li>The purple line at latitude <span class="math-container">$\alpha$</span> is <span class="math-container">$r\sin\alpha$</span></li>
<li>Then the orange line is <span class="math-container">$r\sin\alpha\tan\alpha$</span></li>
<li>The radius of the latitude circle is <span class="math-container">$r\cos\alpha$</sp... |
2,035,186 | <p>This is a probability question where I am asked to integrate a region that represents the probability of a scenario. X, Y, and U are random variables, where U = X-Y. I need to find the probability </p>
<p>$P(U \leq u) = P(X-Y \leq u)$, where the density function I'm integrating over is defined by f(x,y) = 1, for 0 ... | rogerl | 27,542 | <p>The area of the circle is $400\pi\ \text{cm}^2$.</p>
<p>Since the area of the triangle is $60$, the other leg is $8$, so that the hypotenuse is $17$. Now, call the two non-right-angle vertices of the triangle $A$ and $B$, and let the three points of tangency be $P$, $Q$, and $R$, all right-to-left. Then $AP = AQ$ a... |
161,024 | <p>I was recently having a discussion with someone, and we found that we could not agree on what an exponential function is, and thus we could not agree on what exponential growth is. </p>
<p>Wikipedia claims it is $e^x$, whereas I thought it was $k^x$, where k could be any unchanging number. For example, when I'm doi... | hmakholm left over Monica | 14,366 | <p>$e^x$ is <strong>the</strong> exponential function, but $c\cdot k^x$ is <strong>an</strong> exponential function for any $k$ ($> 0, \ne1$) and $c$ ($\ne 0$).</p>
<p>The terminology is a bit confusing, but is so well settled that one just has to get used to it.</p>
|
935,454 | <p>Suppose we have the integral operator $T$ defined by</p>
<p>$$Tf(y) = \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}}f(xy)\,dx,$$</p>
<p>where $f$ is assumed to be continuous and of polynomial growth at most (just to guarantee the integral is well-defined). If we are to inspect that the kernel of the operator, we would... | capea | 86,132 | <p>easy there are infinity solutions like $$y^2 \sin (x)+x \cos (x)$$ $$e^{-x} y^2+e^{-x} x$$
done easy</p>
|
947,254 | <p>The problem is part (b):</p>
<p><b>1.4.7.</b> A pair of dice is cast until either the sum of seven or eigh appears.</p>
<p> <b>(a)</b> Show that the probability of a seven before an eight is 6/11.</p>
<p> <b>(b)</b> Next, this pair of dice is cast until a seven appears twice or until each of a six and e... | Alijah Ahmed | 124,032 | <p>Drawing a state diagram in terms of Markov chains will help in calculating the probabilities to some extent, and you are right in that we need to sum all the branches. </p>
<p>The scenario of an indefinite number of rolls can be dealt with by realising that we will end up with a sum to infinity of geometric progres... |
42,040 | <p>Suppose the polynomial $t^k - a$ has a root (hence splits) in $\mathbb{Q}(\zeta_k)$. For which $k$ does it follow that one of the roots of $t^k - a$ is rational? In particular, are there infinitely many such $k$? </p>
<p>A counting argument shows this is true whenever $k$ has the property that $\varphi(k)$ is a pow... | Gerry Myerson | 8,269 | <p>I take it $a$ is to be a (rational) integer, otherwise you could take any old $\beta$ in ${\bf Q}(\zeta_k)$ and let $a=\beta^k$ and then $t^k-a$ would have a root in ${\bf Q}(\zeta_k)$ but, in general, not in $\bf Q$. </p>
<p>Now $t^k-a$ is irreducible over the rationals unless it is of the form $t^{pr}-b^p$ for so... |
2,969,203 | <p>Let <span class="math-container">$f$</span> be a <span class="math-container">$C''$</span> function on <span class="math-container">$(a, b)$</span> and suppose there is a point <span class="math-container">$c$</span> in (a, b) with <span class="math-container">$$f(c)= f'(c)=f''(c) = 0$$</span> Show that there is a c... | seamp | 606,999 | <p>Define <span class="math-container">$h(x) = \frac{f(x)}{(x-c)^2}$</span> for all <span class="math-container">$x \in (a,b)$</span> different from <span class="math-container">$c$</span>. Then try to show that <span class="math-container">$h$</span> can be extended by continuity at <span class="math-container">$x = c... |
1,977,588 | <p>In books like Calculus (Larson), in the theorems'definitions like Rolle's theorem, when they talk about continuity, they use closed intervals [a,b]. But when they talk about differentiability they use open brackets (a,b). </p>
<p>Why are closed intervals used for continuity and open intervals for differentiability... | H. H. Rugh | 355,946 | <p>In the particular case of Rolle's theorem you need continuity on $[a,b]$, but you only need differentiability in $(a,b)$. This being said, in ${\Bbb R}$ there is no problem in defining differentiability on $[a,b]$ (differentiability from the right/left). </p>
<p>In higher dimensions this gets more complicated. It i... |
878,939 | <p>I have found the eigen vaues, I also know that you can find the eigenvectors through a Gausian Jordan.
-- x1, gauss jordan gives me rows(1 -1/3 ,, 0 0 ), so [a, b] = [1,3]
For vector x2, GJ gives (1 -2/5 ,, 0 0 ), I would assume [a,b] = [2,5], but why did they choose to go with [-2,-5]. I don't get it?</p>
<p... | Community | -1 | <p>Because $n = 5q + r$ is actually a digression from what you're trying to prove.</p>
<p>The thing is, with $n = 4q + r$, you can theoretically set $q$ and $r$ to any integer values you want. But, if $|r| \geq 4$, you can choose a different value of $q$ so that then $0 \leq r < 4$ and then that way you reduce the ... |
22,207 | <p>How to make a defined symbol stay in symbol form?</p>
<pre><code>w = 3; g = 4;
{w, g}[[2]]
</code></pre>
<blockquote>
<p><code>3</code></p>
</blockquote>
<p>I want the output to be <strong><code>g</code></strong> and not <code>3</code>. For example, if I want to save different definitions by <code>DumpSave</co... | Jens | 245 | <p>You have to sacrifice <em>something</em>, but it depends on your preferences <em>what</em> you want to keep and what you want to give up. Let's assume you don't want to sacrifice being able to use assignments such as <code>w=3</code>, then you may have to give up using <code>{...}</code> as a wrapper grouping the <e... |
2,951,825 | <p>I want to show formally that </p>
<p><span class="math-container">$$M =\{(t, \vert t \vert) \text{ }\vert t \in \mathbb{R} \} $$</span> </p>
<p>is not a smooth <span class="math-container">$C^{\infty}$</span>-submanifold of <span class="math-container">$\mathbb{R}^2$</span>. </p>
<p>My attempts: Intuitively it's ... | Nate Eldredge | 822 | <p>Reduce it to calculus: <span class="math-container">$\gamma$</span> is a smooth map from an open subset of <span class="math-container">$\mathbb{R}^1$</span> into <span class="math-container">$\mathbb{R}^2$</span>; i.e. it's a smooth curve. Write <span class="math-container">$\gamma(s) = (x(s), y(s))$</span> and su... |
372,198 | <blockquote>
<p>If $G$ is a group, $H$ and $K$ both subgroups of $G$, $K \subseteq H$, $\left[G:H\right]$ and $\left[H:K\right]$ both finite then $\left[G:K\right]=\left[G:H\right]\cdot\left[ H:K \right].$</p>
</blockquote>
<p>I am not sure if this is standard notation but $\left[ G : K \right]$ denotes the number o... | Elchanan Solomon | 647 | <p>Suppose that $f(x)$ has a root $y$ in $F$. Then $y^{p} =a$. If $p$ is odd,</p>
<p>$$f(x) = x^p - a = x^p - y^p = x^{p} + (-y)^{p} = (x-y)^p$$</p>
<p>So $f(x)$ splits. If $p$ is even, then it is $2$, and if you have one root of a quadratic, you have the other.</p>
|
46,837 | <p>I am looking for jokes which involve some serious mathematics. Sometimes, a totally absurd argument is surprisingly convincing and this makes you laugh. I am looking for jokes which make you laugh and think at the same time. </p>
<p>I know that a similar <a href="https://mathoverflow.net/questions/1083/do-good-math... | Mikael Vejdemo-Johansson | 102 | <p>The first time I ran into the <em>carry</em> operation from grade school addition presented as a non-trivial group cocycle generating part of the group cohomology of <span class="math-container">$\mathbb Z/10$</span>, it was introduced as a joke embedded completely within mathematics.</p>
<p>Specifically, for those ... |
46,837 | <p>I am looking for jokes which involve some serious mathematics. Sometimes, a totally absurd argument is surprisingly convincing and this makes you laugh. I am looking for jokes which make you laugh and think at the same time. </p>
<p>I know that a similar <a href="https://mathoverflow.net/questions/1083/do-good-math... | Quadrescence | 556 | <p><em>This is from <a href="http://symbo1ics.com/blog/?p=389%20%22my%20blog%22">my blog</a>, which I interestingly just posted today (at the time of this posting).</em></p>
<p>Several mathematicians are asked, "how do you put an elephant in a refrigerator?"</p>
<p><strong>Real Analyst</strong>: Let $\epsilon\gt0$. T... |
3,552,555 | <p>Let <span class="math-container">$S$</span> be the set of all column matrices
<span class="math-container">$
\begin{bmatrix}
b_1 \\
b_2 \\
b_3
\end{bmatrix}
$</span>
such that <span class="math-container">$b_1,b_2,b_3 \in \mathbb{R}$</span> and the system of equations (in real variables)
<span class... | Peter Rasmussen | 750,159 | <p>You start by making it a single fraction:</p>
<p><span class="math-container">$$\frac pq = \frac{a}{b+\sqrt c} + \frac d{\sqrt c} = \frac{(d+a)\sqrt c + bd}{b\sqrt c + c}$$</span></p>
<p>with <span class="math-container">$p, q\in \mathbb Z$</span> and <span class="math-container">$q\neq 0$</span>. Then rewrite as
... |
4,058,884 | <p>I have an orthonormal basis <span class="math-container">${\bf{b}}_1$</span> and <span class="math-container">${\bf{b}}_2$</span> in <span class="math-container">$\mathbb{R}^2$</span>. I want to find out the angle of rotation. I added a little picture here. I essentially want to find <span class="math-container">$\t... | Somos | 438,089 | <p>You may want to consider using the complex plane in this situation. Each
point in the <span class="math-container">$\,x\,y\,$</span> plane is associated with a complex number. Thus,
<span class="math-container">$\,\mathbf{e}_1 \equiv 1\,$</span> and <span class="math-container">$\,\mathbf{e}_2 \equiv i.\,$</span> Si... |
2,646,363 | <p>Let $A_1, A_2, \ldots , A_{63}$ be the 63 nonempty subsets of $\{ 1,2,3,4,5,6 \}$. For each of these sets $A_i$, let $\pi(A_i)$ denote the product of all the elements in $A_i$. Then what is the value of $\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})$?</p>
<p>Here is the solution </p>
<p>For size 1: sum of the elements, whi... | quasi | 400,434 | <p>It's just one less than the product
$$(1+1)(1+2)(1+3)(1+4)(1+5)(1+6)$$
Equivalently, it's $f(1)-1$ where
$$f(x) = (x+1)(x+2)(x+3)(x+4)(x+5)(x+6)$$
By Vieta's formulas, the coefficients of all powers of $x$, other than $x^6,\;$in the expanded form of $f(x)$ are the sums of products that you want.
<p>
More precisely, ... |
2,646,363 | <p>Let $A_1, A_2, \ldots , A_{63}$ be the 63 nonempty subsets of $\{ 1,2,3,4,5,6 \}$. For each of these sets $A_i$, let $\pi(A_i)$ denote the product of all the elements in $A_i$. Then what is the value of $\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})$?</p>
<p>Here is the solution </p>
<p>For size 1: sum of the elements, whi... | Donald Splutterwit | 404,247 | <p>Include the empty set in the sum (we can subtract at the end) </p>
<p>The contribution from each of the subsets will correspond to a term in the following product
\begin{eqnarray*}
(1+1)(1+2)(1+3)(1+4)(1+5)(1+6)
\end{eqnarray*}
So the answer is $\color{red}{5039}$.</p>
<p>In your question the values should be $21,... |
1,282,489 | <p>I have a simple problem that I need to solve. Given a height (in blue), and an angle (eg: 60-degrees), I need to determine the length of the line in red, based on where the green line ends. The green line comes from the top of the blue line and is always 90-degrees.</p>
<p>The height of the blue line is variable.... | Sufyan Naeem | 199,112 | <p>Use Law of sines, $$\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}$$</p>
<p>Let, </p>
<p>$\angle{A}=60^o$</p>
<p>$\angle{B}=90^o$</p>
<p>$\angle{C}=30^o$</p>
<p>$a=?$</p>
<p>$b=?$</p>
<p>$c=10cm$</p>
<p>From Law of sines we have,</p>
<p>$$\frac{a}{\sin{A}}=\frac{c}{\sin{C}}$$</p>
<p>Put the values a... |
3,085,842 | <p>What can be said about the uniform Convergence of <span class="math-container">$\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1][nx+1]}$</span> in the interval <span class="math-container">$[0,1]$</span>?</p>
<p>The sequence inside the summation bracket doesn't seem to yield to root or ratio tests. The pointwise convergence ... | José Carlos Santos | 446,262 | <p><strong>Hint:</strong> <span class="math-container">$\displaystyle\frac1{\bigl((n-1)x+1\bigr)(nx+1)}=\frac1{(n-1)x+1}-\frac1{nx+1}$</span>.</p>
|
143,324 | <p>I want to know how to simplify the following expression by using the fact that $\sum_{i=0}^\infty \frac{X^i}{i!}=e^X$. The expression to be simplified is as follows:</p>
<p>$$\sum_{i=0}^{\infty} \sum_{j=0}^i \frac{X^{i-j}}{(i-j)!} \cdot \frac{Y^j}{j!}\;,$$ where $X$ and $Y$ are square matrices (not commutative). (... | Qiaochu Yuan | 232 | <p>Even if $X$ and $Y$ don't commute, it's still true that this expression is equal to $e^X e^Y$; it's just not true that this is equal to $e^{X+Y}$. </p>
|
4,247,888 | <p>I'm having a lot of trouble about an apparently simple task. I have the following trigonometric equation:</p>
<p><span class="math-container">$A\cos(\omega_1t+\phi_1)=B\cos(\omega_2t+\phi_2)$</span></p>
<p>which holds for every <span class="math-container">$t \in [0,+\infty)$</span>, where <span class="math-containe... | The_Sympathizer | 11,172 | <p>The easiest way to prove this is <em>not</em> with a direct solution, but with a <em>contrapositive</em> proof. That is, instead of trying to prove that if</p>
<p><span class="math-container">$$\forall t \in [0, \infty)\ [A \cos(\omega_1 t + \phi_1) = B \cos(\omega_2 + \phi_2)]$$</span></p>
<p>implies</p>
<p><span c... |
1,427,816 | <p>This is kinda of a philosophical question I guess. But are the elmements of the topological closure inside the linear space $X$ all the time? Or do they become apperent when we introduce the topology? And hence introduce the topolgy to control these elements of the space which are there but out of control when we on... | Eric Auld | 76,333 | <p>Here's how I would put it. We start out with a large linear topological space $\Omega$. Within $\Omega$ we isolate a linear subspace $X$ with nice properties (perhaps $X$ is the linear span of countably many elements, for example), and it may or may not be the case that each point of $\Omega$ is a topological limit ... |
3,225,784 | <p>Solve for x:</p>
<blockquote>
<p><span class="math-container">$$2\sin(x) + 3\sin(2x) = 0 $$</span></p>
<p><span class="math-container">$$2\sin(x)(1 + 3\cos(x)) = 0$$</span></p>
</blockquote>
<p>Stuck here. The solution mentions some arccos function, but I need a detailed explanation on this one.</p>
| Robert Israel | 8,508 | <p>Hint: if the product of two numbers is <span class="math-container">$0$</span>, at least one of them is <span class="math-container">$0$</span>.</p>
|
573,964 | <blockquote>
<p>Let set $S$ be the set of all functions $f:\mathbb{Z_+} \rightarrow \mathbb{Z_+}$. Define a realtion $R$ on $S$ by $(f,g)\in R$ iff there is a constant $M$ such that $\forall n (\frac{1}{M} < \frac{f(n)}{g(n)}<M). $ Prove that $R$ is an equivalence relation and that there are infinitely mane equ... | BaronVT | 39,526 | <p>Subspaces have to be closed under more general linear combinations than just $x + y$. That is, you have to have </p>
<p>$$
c_1 x + c_2 y \in S
$$
whenever $x,y \in S$ for any $c_1,c_2 \in \mathbb R$ (since the original vector space is a real-vector space, the scalars for linear combinations are, in general, real nu... |
783,502 | <p>Here in my exercise I have to study the function and draw its graph. Can you please tell me what's the best method to do this, because I don't think that's reasonable to use the input output method, it's quite imprecise.
$$f(x)={|x+1|\over x}$$</p>
<p>Thank you!!!</p>
| Caleb Stanford | 68,107 | <ul>
<li><p>There is a zero at $x = -1$.</p></li>
<li><p>The function does not exist at $0$ and grows infinitely large near $0$.</p></li>
<li><p>The function is positive when $x > 0$ and negative when $x < 0$.</p></li>
<li><p>When the function hits the $x$-axis at $x = -1$ it is a sharp corner rather than a smoo... |
1,920,994 | <p>My calculus teacher gave us this interesting problem: Calculate</p>
<p>$$ \int_{0}^{1}F(x)\,dx,\ $$ where $$F(x) = \int_{1}^{x}e^{-t^2}\,dt $$</p>
<p>The only thing I can think of is using the Taylor series for $e^{-t^2}$ and go from there, but since we've never talked about uniform convergence and term by term in... | mickep | 97,236 | <p>No fancy stuff is needed. I think you could just integrate by parts.</p>
<p>$$\int_0^1 F(x)\,dx=[xF(x)]_0^1-\int_0^1 xF'(x)\,dx$$</p>
<p>The outintegrated part cancel, and using the fundamental theorem of calculus, $F'(x)=e^{-x^2}$. Thus</p>
<p>$$\int_0^1 F(x)\,dx=-\int_0^1 xe^{-x^2}\,dx
$$
from where I think you... |
1,162,147 | <p>The definition of open set is different in metric space and topological space, though metric space is a special case of topological space. The definition in metric space seems to convey the idea that all the points isolated from outside from outside, while the definition in topological space is intended to separate ... | Mnifldz | 210,719 | <p>The definition of a <em>topology</em> $\mathcal{T}$ on a set $X$ is given as follows: Let $\mathcal{T}$ be a collection of subsets of $X$ satisfying the following:</p>
<ol>
<li>Both the empty set and $X$ itself belong to $\mathcal{T}$.</li>
<li>The union of any collection of subsets $\{U_\alpha \; | \; \alpha \in I... |
1,415,505 | <p>I was trying to find out how to prove </p>
<p>$$ \sin(A-\arcsin(0.3 \ \sin \ A)) \ \cdot \ \sin(A+\arcsin(0.3 \ \sin \ A)) \ = \ 0.91 \ \sin^2 \ A \ \ . $$
When I put this equation into my calculator both sides appear to be exactly the same, but I have no idea how to prove it.</p>
| haqnatural | 247,767 | <p>$$\sin { \left( A-\arcsin { \left( 0.3\sin { \left( A \right) } \right) } \right) } \cdot \sin { \left( A+\arcsin { \left( 0.3\sin { \left( A \right) } \right) } \right) =0.91\sin ^{ 2 }{ \left( A \right) } }$$</p>
<p><strong>Solution</strong>
:
$$\left( \sin { A\cos { \left( \arcsin { \left( 0.3\sin { ... |
122,274 | <p>I have a question, I think it concerns with field theory.</p>
<blockquote>
<p>Why the polynomial $$x^{p^n}-x+1$$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$?</p>
</blockquote>
<p>Thanks in advance. It bothers me for several days. </p>
| Hoang Nguyen | 934,337 | <p>I have another solution that might be easier to follow.</p>
<p>Let <span class="math-container">$\alpha$</span> be a root of <span class="math-container">$q(x)=x^{p^n}-x+1$</span>. Note that <span class="math-container">$\alpha + a$</span> is also a root of <span class="math-container">$q(x)$</span> for all <span cl... |
3,396,882 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a non-negative random variable, and suppose that <span class="math-container">$P(X \geq
n) \geq 1/n$</span> for each <span class="math-container">$n \in \mathbb{N}$</span>. Prove that <span class="math-container">$E(X) = \infty$</span>.</p>
</blockquote>... | GReyes | 633,848 | <p>I am giving a proof for the continuous case. Integrating by parts in the definition of expected value and observing that <span class="math-container">$P(X<-t)=0$</span> for any <span class="math-container">$t>0$</span> you have
<span class="math-container">$$
\mathbb{E}[X]=\int\limits_0^{\infty}P(X>t)\,dt\g... |
154,722 | <p>Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter.</p>
<p>Now consider the family of representations $r_{\alpha}$ of the free group on two generators $F_2 = \langle a,b\rangle$ in $\mathrm{SL}(2, \mathbb{C})$ setting ... | Venkataramana | 23,291 | <p>When $\alpha$ is very large one can see that the two elements play ping pong:</p>
<p>It is clear that if $\alpha ,n$ are large, $A,B^n$ play ping pong; so they are free. The proof of freeness also shows that the group generated by $A,B^n$ acts properly discontinuously on a piece of ${\mathbb P}^1({\mathbb C})$ and ... |
1,118,259 | <p>Consider a sphere of radius $a$ with 2 cylindrical holes of radius $b<a$ drilled such that both pass through the center of the sphere and are orthogonal to one another. What is the volume of the remaining solid?</p>
<p>Can someone help me at least setting up the integral? I know that there is a similar problem... | achille hui | 59,379 | <p>WOLOG, choose the coordinate system such that the sphere of radius $a$ is centered at origin $O$ and the axes of the two holes are aligned along the $x$ and $y$ axis. Let</p>
<p>$$c = \sqrt{a^2 - b^2},\quad
d = \begin{cases}\sqrt{b^2 - c^2},& b > \frac{a}{\sqrt{2}}\\0,& \text{otherwise}\end{cases}
\quad\... |
4,545,364 | <blockquote>
<p>Solve the quartic polynomial :
<span class="math-container">$$x^4+x^3-2x+1=0$$</span>
where <span class="math-container">$x\in\Bbb C$</span>.</p>
<p>Algebraic, trigonometric and all possible methods are allowed.</p>
</blockquote>
<hr />
<p>I am aware that, there exist a general quartic formula. (Ferrari... | Bob Dobbs | 221,315 | <p><span class="math-container">$(x^2+e^{\theta i}x+e^{\phi i})(x^2+e^{-\theta i}x+e^{-\phi i})=x^4+2\cos\theta x^3+(1+2\cos\phi)x^2+2\cos(\theta-\phi)x+1$</span>.</p>
<p>Then <span class="math-container">$\cos\theta=\frac{1}{2}$</span>, <span class="math-container">$\cos\phi=-\frac{1}{2}$</span>, <span class="math-con... |
2,179,253 | <p>$$n{n-1 \choose 2}={n \choose 2}{(n-2)}$$
Give a conceptual
explanation of why this formula is true.</p>
| Dando18 | 274,085 | <p>Here's a derivation:
$$ n \binom{n-1}{2} = \frac{n (n-1)!}{2!(n-3)!} = \frac{n!}{2!(n-3)!} = \frac{1}{2}(n-2)(n-1)n$$</p>
<p>$$ \binom{n}{2}(n-2) = \frac{n!(n-2)}{2!(n-2)!} = \frac{n!}{2!(n-3)!} = \frac{1}{2}(n-2)(n-1)n $$</p>
<p>So it follows that:</p>
<p>$$ n \binom{n-1}{2} = \binom{n}{2}(n-2) $$</p>
|
68,145 | <p>All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on;</p>
<p>It is well-known that every regular ring is Gorenstein and every Gorenstein ring is Cohen-Macaulay. There are some examples to demonstrate that the converse of the above statements do not hold. For... | Sándor Kovács | 10,076 | <p>All of these conditions are very important in algebraic geometry. I don't know much about the algebraic combinatorics aspect of these notions, but my feeling is that came from geometry and not vice versa.</p>
<p>The reason we care about these notions is that even though it would be nice to always work with non-sing... |
2,068,986 | <p>Consider the function</p>
<p>$$K(u) = \frac 1 {\sqrt {2\pi}} \left( \Bbb e ^{-\frac 1 2 \left( \frac {u-5.3} h \right)^2 } + \Bbb e ^{-\frac 1 2 \left( \frac {u-1.6} h \right)^2 } + \Bbb e ^{-\frac 1 2 \left( \frac {u-2.1} h \right)^2 } + \Bbb e ^{-\frac 1 2 \left( \frac {u-1.7} h \right)^2 } + \Bbb e ^{-\frac 1 2 ... | Barry Cipra | 86,747 | <p>Try </p>
<p>$$a_n=\lfloor2^n\ln2\rfloor$$</p>
<p>and use the inequalities $2^n\ln2-1\lt a_n\le2^n\ln2$ in the Squeeze Theorem: Since $1-{1\over2^n}\lt1$, we have</p>
<p>$$\left(1-{1\over2^n}\right)^{2^n\ln2}\le\left(1-{1\over2^n}\right)^{a_n}\lt\left(1-{1\over2^n}\right)^{2^n\ln2-1}$$</p>
<p>The left- and right... |
2,741,686 | <p>If I have the following vector space $ V, \text{{$e_0, e_1, e_2$}} \text{ where } e_0(x) = 1, e_1(x) = x \text{ and } e_2(x) = x^2$.I want to know the linear dependency of it how can I proceed? I thought of following the definition of linearly independent $$c_0e_0 + c_1e_1 + c_2e_2 = c_0+ c_1x + c_2x^2=0\iff c_0 = c... | user | 505,767 | <p>Note that for the <a href="http://mathworld.wolfram.com/ZeroPolynomial.html" rel="nofollow noreferrer">zero polynomial</a> property</p>
<p>$$c_0e_0 + c_1e_1 + c_2e_2 = c_0+ c_1x + c_2x^2=0 \quad \forall x\iff c_0 = c_1 = c_2 = 0$$</p>
|
510,130 | <p>Let $(r_i)_{i=1}^m$ be a sequence
of positive reals such that
$\sum_i r_i < 1$
and let $t$ be a positive real.
Consider the sequence $T(n)$
defined by $T(0) = t$,
$T(n) = \sum_i T(\lfloor r_i n \rfloor) $
for $n \ge 1$.</p>
<p>Show that
$T(n) = o(n)$,
that is,
$\lim_{n \to \infty} \dfrac{T(n)}{n}
= 0
$.</p>
<p>... | Marko Riedel | 44,883 | <p>I have some very exciting news for Mr. M. Cohen and other potential readers of this thread. I hope it gets viewed often because the result is really very pretty. We show that we can solve a more general case than in the first post using Dirichlet series and Mellin transforms, getting exact formulas for $T(n)$ and fo... |
510,130 | <p>Let $(r_i)_{i=1}^m$ be a sequence
of positive reals such that
$\sum_i r_i < 1$
and let $t$ be a positive real.
Consider the sequence $T(n)$
defined by $T(0) = t$,
$T(n) = \sum_i T(\lfloor r_i n \rfloor) $
for $n \ge 1$.</p>
<p>Show that
$T(n) = o(n)$,
that is,
$\lim_{n \to \infty} \dfrac{T(n)}{n}
= 0
$.</p>
<p>... | Marko Riedel | 44,883 | <p>I am presenting an important addendum. The code for my first answer does not work properly when some of the $p_k$ are repeated even though the math is right, and it is not all that efficient. I have remedied this defect and I am presenting code that works for duplicate $p_k$ and is amazingly fast even for large argu... |
1,448,585 | <blockquote>
<p>Let $\alpha \in \mathbb{R}^n$, $n \geq 2$, be a non-zero vector. Define a reflection in the hyperplane perpendicular to $\alpha$ by:
$$\sigma_{\alpha}(v) = v - \dfrac{2(v, \alpha)}{(\alpha, \alpha)} \cdot \alpha$$
($(x, y)$ is the usual inner product on $\mathbb{R}^n$).</p>
<p>1) Show $\sigma... | Lee Mosher | 26,501 | <p>For your first subquestion of 1), the hyperplane is described in the question: it is the hyperplane orthogonal to $\alpha$. You know from linear algebra that this hyperplane is the solution of the equation $\alpha \cdot v = 0$. So your goal is to take any $v$ in that hyperplane, i.e. take any $\nu$ such that $\alpha... |
1,448,585 | <blockquote>
<p>Let $\alpha \in \mathbb{R}^n$, $n \geq 2$, be a non-zero vector. Define a reflection in the hyperplane perpendicular to $\alpha$ by:
$$\sigma_{\alpha}(v) = v - \dfrac{2(v, \alpha)}{(\alpha, \alpha)} \cdot \alpha$$
($(x, y)$ is the usual inner product on $\mathbb{R}^n$).</p>
<p>1) Show $\sigma... | whacka | 169,605 | <p>Get some intuition from three dimensions first. Say the intersection of the two planes is the axis spanned by $\gamma$. Then $\{\alpha,\beta,\gamma\}$ is a basis, and the reflections only act on the $\alpha$ and $\beta$ components of any vector. This generalizes: prove that ${\rm span}\{\alpha,\beta\}$ is the orthog... |
2,193,171 | <p>Question: Let $\{a_n\}$ and $\{b_n\}$ be convergent sequences with $a_n \Rightarrow L$ and $b_n \Rightarrow M$ as $n \Rightarrow \infty$. </p>
<p>Prove that $a_nb_n \Rightarrow LM$</p>
<p>Solution: (My Attempt). Instead of redoing it could someone just tell me what I'm doing wrong. Thx</p>
<p>WTS: </p>
<p>(1) $\... | Mark Viola | 218,419 | <p>Note that the exponential satisfies the inequality </p>
<p>$$\begin{align}
e^x&\ge 1+x+\cdot +\frac{x^{k+1}}{(k+1)!}\\\\
&>\frac{x^{k+1}}{(k+1)!}\tag 1
\end{align}$$</p>
<p>Using $(1)$, it is easy to see that</p>
<p>$$\begin{align}
\frac{e^{-1/t}}{t^k}&=\frac{1}{t^ke^{1/t}}\\\\
&\le \frac{1}{t^... |
2,636,712 | <p>I have question about my proof. I could not tell whether it is sufficient enough since my professor approached it differently. </p>
<p><strong>The problem:</strong></p>
<blockquote>
<p>Let $z \in \mathbb{C}^{*}$. If $|z| \neq 1$, prove that the order of $z$ is infinite. </p>
</blockquote>
<p><strong>My proof:</... | egreg | 62,967 | <p>The proof is good, but has several redundant steps.</p>
<blockquote>
<p>Suppose $z$ has finite order. Then there exists an integer $m>0$ such that $z^m=1$. Hence $|z|^m=1$ which implies $|z|=1$.</p>
</blockquote>
<p>No contradiction, but “contrapositive”: if $z$ has finite order, then $|z|=1$.</p>
|
7,237 | <p>this came up in class yesterday and I feel like my explanation could have been more clear/rigorous. The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and one of the students did $$\frac{6x^2}{6x}=\frac{12x}{6x}$$ $$x = 2$$ which is a valid solution but this method elimin... | Aeryk | 401 | <p>Just before dividing, you can reason "Either $x=0$ or I can divide by $x$." This creates two separate cases to be analyzed.</p>
<p>This works for dividing by anything. You want to divide by $\sin(x)$? You need to make two cases: $\sin(x) \neq 0$ and $\sin(x)=0$. And then analyze each independently.</p>
|
2,638,679 | <p><a href="https://i.stack.imgur.com/S4p0Y.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/S4p0Y.jpg" alt="enter image description here"></a></p>
<p>Due apologies for this rustic image. But while drawing this lattice arrangement about the "square numbers" , I discovered a pattern here wherein if I ... | N. S. | 9,176 | <p>The characteristic Polynomial for $A^{-1}$ for any invertible $n \times n$ matrix is
$$P_{A^{-1}}(X)=\det(xI-A^{-1})=\det(A^{-1}) \det(xA-I)=x^n\det(A^{-1})\det(A-\frac{1}{x}I)\\=(-1)^nx^n \det(A^{-1}) P_{A}(\frac{1}{x})$$</p>
<p>Now use the fact that for a $2\times 2$ matrix the characteristic polynomial is
$$P_B... |
121,450 | <p>I am trying to prove that the series <span class="math-container">$\sum \dfrac {1} {\left( m_{1}^{2}+m_{2}^{2}+\cdots +m_{r }^{2}\right)^{\mu} } $</span> in which the summation extends over all positive and negative integral values and zero values of <span class="math-container">$m_1, m_2,\dots, m_r$</span>, except ... | Community | -1 | <p>What about comparing with an integral on $\mathbb{R}^r$? And then an appropriate change of variable?</p>
|
3,541,524 | <blockquote>
<p>Decide whether the following ie true or false
<span class="math-container">$$\lvert\arcsin z \rvert \le \left\lvert \frac {\pi z} {2} \right\rvert $$</span>
whenever <span class="math-container">$z\in\Bbb C$</span> . </p>
</blockquote>
<p><span class="math-container">$\arcsin z =-i \text{Log } (... | River Li | 584,414 | <p><strong>Proof</strong>: We split into four cases:</p>
<p>1) <span class="math-container">$z = x \in (1, +\infty)$</span>: From 4.23.20 in [1], we have
<span class="math-container">$$|\arcsin x| = \sqrt{\frac{1}{4}\pi^2 + \ln^2 (\sqrt{x^2-1} + x)}.\tag{1}$$</span>
It suffices to prove that
<span class="math-containe... |
1,747,696 | <p>First of all: beginner here, sorry if this is trivial.</p>
<p>We know that $ 1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2 $ .</p>
<p>My question is: what if instead of moving by 1, we moved by an arbitrary number, say 3 or 11? $ 11+22+33+44+\ldots+11n = $ ?
The way I've understood the usual formula is that the first n... | Gottfried Helms | 1,714 | <p>The key here is to look at the constant(!) differences. In $1+2+3+4+...n$ it is $d=1$, and the last number is, calling the first number $a$, $a+(n-1)*d=1+n-1=n$. In your next example the difference is $d=11$, and with $a=11$ you have the last number $a+(n-1)*d=11+(n-1)*11 = 11n $ and the sum can then analogically b... |
4,045,074 | <p><strong>Let <span class="math-container">$X$</span> be the random variable whose cumulative distribution function is
<span class="math-container">$$
F_X (x) = \begin{cases}
0, & \text{for} \space x\lt 0 \\
\frac{1}{2}, & \text{for} \space 0\le x\le 1 \\
1, & \text{for} \space x\gt 1 \\
\end{c... | reuns | 276,986 | <p><span class="math-container">$$(1+O(\frac1{\log n}))n\log n = \log n! = \sum_{p^k \le n} \lfloor n/p^k \rfloor\log p $$</span> <span class="math-container">$$= (1+O(\frac1{\log n}))\sum_{p \le n} n \frac{\log p}{p}\tag{1}$$</span>
Followed by a partial summation
<span class="math-container">$$\sum_{p\le n} \frac1p =... |
1,932,961 | <p>Prove by mathematical induction that
$$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$
holds $\forall n\in\mathbb{N}$.</p>
<hr>
<p>(1) Assume that $n=1$. Then left side is $1^2 =1$ and right side is $6/6 = 1$, so both sided are equal and expression holds for $n = 1$.</p>
<p>(2) Let $k \in \mathbb{N}$ is given. As... | Tintarn | 197,823 | <p>Substitute $a=\frac{x}{x-1}, b=\frac{y}{y-1}, c=\frac{z}{z-1}$.</p>
<p>Then we have $x=\frac{a}{a-1}$ and the similar identities so that the condition implies $abc=(a-1)(b-1)(c-1)$ and hence $ab+ac+bc=a+b+c-1$.</p>
<p>We want to prove $a^2+b^2+c^2 \ge 1$ which is equivalent to $(a+b+c)^2 - 2(ab+ac+bc)-1 \ge 0$ or,... |
104,375 | <p>How I am supposed to transform the following function in order to apply the laplace transform.</p>
<p>$f(t) = t[u(t)-u(t-1)]+2t[u(t-1) - u(t-2)]$</p>
<p>I know that it has to be like this</p>
<p>$L\{f(t-t_0)u(t-t_0)\} = e^{-st_0}F(s), F(s) = L\{f(t)\}$</p>
| Community | -1 | <p>I'll try to put it this way:</p>
<blockquote>
<p>Define a relation <span class="math-container">$\sim$</span> on <span class="math-container">$\mathbb Z$</span>, such that <span class="math-container">$a \sim b \iff \exists k \in \mathbb Z ~~ \text{such that}~~~~a-b=3k$</span></p>
<p>What does this say?</p>
<p>Integ... |
3,142,417 | <p>If <span class="math-container">$a , b , c$</span> and <span class="math-container">$d$</span> are positive integers,
and <span class="math-container">$ab$</span> is greater than <span class="math-container">$cd$</span>,
then, is <span class="math-container">$a+b$</span> greater than or equal to <span class="ma... | Yanko | 426,577 | <p>Not at all.</p>
<p>Consider <span class="math-container">$a=b=3$</span> and <span class="math-container">$c=1,d=8$</span>. Then <span class="math-container">$ab=9$</span> while <span class="math-container">$cd=8$</span> however <span class="math-container">$a+b=6$</span> while <span class="math-container">$c+d=9$</... |
1,768,700 | <p>According to my knowledge, to prove that $24^{31}$ is congruent to $23^{32}$ mod 19, we must show that both numbers are divisible by 19 i.e. their remainders must be equal with mod 19. Please correct me if I'm wrong.</p>
<p>So, I was able to reduce $23^{32}$ and find its mod 19, which is 17 but I am having a bit of... | lhf | 589 | <p>$
24 \equiv 5 \bmod 19
$</p>
<p>$
23 \equiv 4 \bmod 19
$</p>
<p>$
5 \cdot 4 \equiv 1 \bmod 19
$</p>
<p>$
5^{-31} 4^{32} \equiv 4^{31} 4^{32} \equiv 4^{63} \equiv 4^9 = 2^{18} \equiv 1 \bmod 19
$</p>
|
965,851 | <p>I have a computer problem that I was able to reduce to an equation in quadratic form, and thus I can solve the problem, but it's a little messy. I was just wondering if anybody sees any tricks to simplify it?</p>
<p>$$\sin^2\beta ⋅ d^4 + c^2\left(\cos^2\beta⋅\cos^2\alpha-\frac{\cos^2\beta}{2}-\frac12\right)d^2 + ... | Alexandru Ionescu | 596,292 | <pre><code>We see by observation that x = 2 and x = 4 are clearly solutions.
We will prove by induction that 2^n > n^2 for all n >= 5.
For the base case, n = 5 gives 32 > 25 which is true.
For the inductive case, we know that 2^n > n^2 and we want to show that 2^(n+1) > (n+1)^2.
We will have that 2^(n... |
4,450,470 | <p>Let <span class="math-container">$X$</span> a vectorial space y let <span class="math-container">$\Gamma \subset X^{\ast}$</span>. We will say that <span class="math-container">$\Gamma$</span> is <strong>total</strong> in <span class="math-container">$X$</span> if <span class="math-container">$f(x)=0$</span>, <span ... | angryavian | 43,949 | <p>Let <span class="math-container">$x_0 := (x_1+x_2)/2$</span> and let <span class="math-container">$h = (x_2-x_1)/2$</span>.
The desired inequality is <span class="math-container">$f(x_0) \le \frac{1}{2}(f(x_0-h) + f(x_0+h))$</span>,
which can be rearranged as
<span class="math-container">$$f(x_0) - f(x_0-h) \le f(x_... |
4,450,470 | <p>Let <span class="math-container">$X$</span> a vectorial space y let <span class="math-container">$\Gamma \subset X^{\ast}$</span>. We will say that <span class="math-container">$\Gamma$</span> is <strong>total</strong> in <span class="math-container">$X$</span> if <span class="math-container">$f(x)=0$</span>, <span ... | B. S. Thomson | 281,004 | <blockquote>
<p><strong>Definition</strong>. a function <span class="math-container">$f$</span> on an interval <span class="math-container">$(a,b)$</span> is said to be
<strong>midpoint convex</strong> if <span class="math-container">$$f\left(\frac{x_1 + x_2} {2}\right) \le \frac{1}{2}[f(x_1) + f(x_2)] \tag{1}$$</span... |
3,449,274 | <p>I have equation <span class="math-container">$2b^2 - 72b - 406=0$</span>. I divided it with 2 and I got <span class="math-container">$b^2 - 36b - 203=0$</span>. My teacher then wrote <span class="math-container">$(b-29)(b-7)=0$</span> but I don’t understand how he got that. When I try to solve that equation I get <s... | Quanto | 686,284 | <p>Note that the function does not cross the <span class="math-container">$y$</span>-axis due to the singularity at <span class="math-container">$x=0$</span>, which makes <span class="math-container">$f(x)$</span> discontinuous. For <span class="math-container">$x>0$</span>, <span class="math-container">$f(x)$</span... |
825,703 | <p>I have been working with vector spaces for a while and I now take for granted what the vector space does. I feel like I dont really understand why multiplication and addition must be defined on a vector space. For example, it feels like adding two vectors and having their sum contained within the space is just a nam... | mathematician | 98,943 | <p>There are lots of "natural" spaces that happen to satisfy the properties of a vector space. For example $\mathbb{R}^n$ and $C(\mathbb{R})$. Spaces with this kind of addition and scalar multiplication come up so much that we came up with the abstract definition of a vector space. That way we can just do one proof ... |
1,737,674 | <p>I am trying to understand how to find all congruence classes in $\mathbb{F}_2[x]$ modulo $x^2$. How can I compute them ? Can someone get me started with this? I am having trouble understanding $\mathbb{F}_2[x] $ is it the set $\{ f(x) = a_nx^n + ...+ a_1 x + a_0 : a_i = 0,1 \} $?</p>
| Ángel Mario Gallegos | 67,622 | <p>If you only want the measure of the angle (and after deduce with a geometric/trigonometric procedure) Geogebra gives the following:</p>
<p><a href="https://i.stack.imgur.com/pZQlP.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pZQlP.png" alt="enter image description here"></a></p>
|
3,027,286 | <p>I am a little confused as to proving that <span class="math-container">$(C^*)^{-1} = (C^{-1})^*$</span> where <span class="math-container">$C$</span> is an invertible matrix which is complex. </p>
<p>Initially, I thought that it would have something to do with the identity matrix where <span class="math-container">... | Sri-Amirthan Theivendran | 302,692 | <p>The statement is false. Put
<span class="math-container">$$
p_j=\frac{6}{\pi^2}\frac{1}{j^2}\quad (j\geq 1)
$$</span>
where the constant is for normalization. Let <span class="math-container">$N$</span> be distributed according to this pmf. Then
<span class="math-container">$$
\sum_{j=1}^\infty jp_j=EN=\frac{6}{\pi... |
4,271,909 | <p>I'm struggling with improper integrals (Calc I). I've calculated the following:</p>
<p>If <span class="math-container">$a = 0$</span>:
<span class="math-container">$$\int_{0}^{\infty}\cos(x)dx =\lim\limits_{R\to\infty} \int_{0}^{R}\cos(x)dx =\lim\limits_{R\to\infty} \sin(R) $$</span>
Which diverges?</p>
<p>If <span ... | Jean Marie | 305,862 | <p>Let us write this integral under the more general form:</p>
<p><span class="math-container">$$\int_{0}^{\infty}e^{-sx}\cos(Ax)\,dx \ \text{with} \ A=1$$</span></p>
<p>This integral is classical : it is the Laplace Transform (see formula 8 <a href="https://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf" rel="nofollow ... |
4,215,724 | <p><span class="math-container">$f\colon \mathbb{R}^2\to \mathbb{R}$</span> such that <span class="math-container">$f_x(2,-1)=1$</span> and <span class="math-container">$f_y(2,-1)=1$</span> and <span class="math-container">$g(x,y)=\langle x^2y,x-y\rangle$</span> and <span class="math-container">$h = f\circ g$</span> th... | Dhanvi Sreenivasan | 332,720 | <p>Since <span class="math-container">$f$</span> is a multivariable function, and <span class="math-container">$g$</span> is a vector function, we can write it as</p>
<p><span class="math-container">$$\frac{\partial h}{\partial y} = \vec{\nabla f} . \vec{\frac{\partial g}{\partial y}}$$</span></p>
|
2,264,614 | <p>Is there a way to evaluate, </p>
<p>$$
\large \cos x \cdot \cos \frac{x}{2} \cdot \cos \frac{x}{4} ... \cdot \cos \frac{x}{2^{n-1}} \tag*{(1)}
$$</p>
<p>I asked this to one of my teachers and what he told is something like this, </p>
<p>Multiply and divide the last term of $(1)$ with $\boxed{\sin \frac{x}{2^{n-... | Doug M | 317,162 | <p>How does step 2 happen:</p>
<p>multiplying the last factor by <span class="math-container">$\frac {\sin \frac{x}{2^{n-1}}}{\sin \frac{x}{2^{n-1}}}$</span></p>
<p>gives us</p>
<p><span class="math-container">$\cos x \cdot \cos \frac x2\cdots \cos \frac x{2^{n-2}}\cdot \frac{\cos \frac{x}{2^{n-1}} \cdot \sin \frac{x... |
401,002 | <p>$\forall x \neg A \implies \neg \exists xA$<br>
I won't ask you to solve this for me, but can you please give some guiding lines on how to approach a proof in NDFOL?<br>
There are many tricks that the TA shows in class, that I could not dream of...</p>
<p>P.S. I managed to proof $\neg \exists xA \implies \forall x ... | Lord_Farin | 43,351 | <p>To prove an implication, the general guideline for constructing formal proofs is: Assume the premise and the negation of the consequence, and derive a contradiction.</p>
<p>In your present case:</p>
<hr>
<p>Assume $\exists x A(x)$. By Existential Instantiation, we have $A(t)$ for some (unspecified but fixed) $t$.... |
3,098,838 | <blockquote>
<p>The displacement of a particle varies according to <span class="math-container">$x=3(\cos t +\sin t)$</span>.
Then find the amplitude of the oscillation of the particle.</p>
</blockquote>
<p>Can someone kindly explain the concept of amplitude and oscillation and how to solve it?</p>
<p>Any hints ... | David Holden | 79,543 | <p><span class="math-container">$$
ax^2 + 2bxy + dy^2 = a(x + \frac{by}a)^2 + \frac1a(ac-b^2)y^2
$$</span></p>
|
254,695 | <p>The concept of dimension seems to be:</p>
<blockquote>
<p>In physics and mathematics, the dimension of a space or object is
informally defined as the minimum number of coordinates needed to
specify any point within it.</p>
</blockquote>
<p>According to <a href="http://en.wikipedia.org/wiki/Dimension_%28mathe... | Wolphram jonny | 43,048 | <p>It is easy to be confused because the different meanings often given to the "fourth dimension". In the simplest an more natural case, the fourth dimension is just another spatial dimension, and you can have as many dimensions as you want. An introductory book on linear algebra should make it easy to understand (look... |
254,695 | <p>The concept of dimension seems to be:</p>
<blockquote>
<p>In physics and mathematics, the dimension of a space or object is
informally defined as the minimum number of coordinates needed to
specify any point within it.</p>
</blockquote>
<p>According to <a href="http://en.wikipedia.org/wiki/Dimension_%28mathe... | Ross Millikan | 1,827 | <p>There are two different concepts of the fourth dimension in what you talk about. Four dimensional Euclidean space, $\mathbb R^4$ has four equivalent coordinates. To find the distance between two points, you sum the squares of the differences of coordinates and take the square root. Many things are similar to your... |
648,607 | <p>I would like to determine whether the following series is absolut convergent or not. I´m not sure how to begin generally. I would say no, because when taking the absolut value of the fraction and add all of them together the series doesnt converge...could someone give me a general road plan how to manage this.</p>
... | Lost1 | 44,877 | <p>Do a comparison: sum of absolute value of your each term against $\sum \frac{1}{3n}$. Note the latter diverges, because?</p>
|
648,607 | <p>I would like to determine whether the following series is absolut convergent or not. I´m not sure how to begin generally. I would say no, because when taking the absolut value of the fraction and add all of them together the series doesnt converge...could someone give me a general road plan how to manage this.</p>
... | user76568 | 74,917 | <p>The meaning of a series $\sum_{n=1}^{\infty}c_n$ being absolutely convergent is equivalent (defined as) to $\sum_{n=1}^{\infty}|c_n|$ converging. </p>
<p>So,we compare $|a_n|=|\frac{(-1)^n}{2n+1}|=\frac{1}{2n+1}$ with $b_n=\frac{1}{n}$:
$$\lim_{n \to \infty}\frac{b_n}{|a_n|}=\lim_{n \to \infty}\frac{2n+1}{n}=\lim... |
3,466,680 | <p>I'm solving a problem in ODE:</p>
<blockquote>
<p>Solve in <span class="math-container">$\left (-\dfrac{\pi}{2},\dfrac{\pi}{2} \right )$</span> the ODE <span class="math-container">$y''(t) \cos t + y (t) \cos t=1$</span></p>
</blockquote>
<p>In my lecture, we are given three theorems:</p>
<blockquote>
<p><span class... | nmasanta | 623,924 | <p><span class="math-container">\begin{equation} y''(t) \cos t + y (t) \cos t=1\\
\implies y''(t) + y (t)=\sec t\\
\implies (D^2+1)y=\sec t\tag1
\end{equation}</span>
where <span class="math-container">$~D\equiv \dfrac{d}{dt}~$</span></p>
<p>Let <span class="math-container">$~y=e^{mt}~$</span> be solution of <span cla... |
462,983 | <h2>The Question:</h2>
<p>This is a very fundamental and commonly used result in linear algebra, but I haven't been able to find a proof or prove it myself. The statement is as follows:</p>
<blockquote>
<p>let $A$ be an $n\times n$ square matrix, and suppose that $B=\operatorname{LeftInv}(A)$ is a matrix such that... | Hagen von Eitzen | 39,174 | <p>Assume $A,B$ are $n\times n$ matrices with $BA=I$.
Let $\alpha\colon V\to V$ with $V=K^n$ be the endomorphism described by $A$ and similarly with $\beta$ for $B$. Then we are given that $\beta\circ\alpha=\operatorname{id}_V$, hence $\alpha$ is injective. The image of the standard basis of $V$ is therefore a linearly... |
3,237,337 | <p>I have the function <span class="math-container">$f(x)= \frac{\sqrt{x^2-1}}{x+\log x}$</span> in the set <span class="math-container">$E=[1,+ \infty)$</span>and I have to discuss the uniform continuity of f in E.</p>
<p>I've calculated the derivative <span class="math-container">$y'$</span> and it tends to <span c... | Peter | 82,961 | <p>This site</p>
<p><a href="https://en.wikipedia.org/wiki/Linnik%27s_theorem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Linnik%27s_theorem</a></p>
<p>shows what has been found out for the least prime in an arithmetic progression. The conjecture <span class="math-container">$$p(a,d)<d^2$$</span> woul... |
3,237,337 | <p>I have the function <span class="math-container">$f(x)= \frac{\sqrt{x^2-1}}{x+\log x}$</span> in the set <span class="math-container">$E=[1,+ \infty)$</span>and I have to discuss the uniform continuity of f in E.</p>
<p>I've calculated the derivative <span class="math-container">$y'$</span> and it tends to <span c... | Community | -1 | <p><span class="math-container">$$n\equiv-(x^{-1})\bmod p$$</span> are knocked out as lead coefficients. They create a number divisible by p any time p is not a factor of x. That means just 57 survive for x=100, p<10 and only 23 survive that for x=101 . Accounting for overlap is really the only tricky part for me. I... |
838,690 | <p>True or false question</p>
<p>If B is a subset of A then {B} is an element of power set A. </p>
<p>I think this is true.</p>
<p>Because B is {1,2} say A {1,2,3} then power set of includes </p>
<p>$\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{3,2\},\{1,2,3\},\emptyset\}$</p>
<p>Unless {B} means $\{\{1,2\}\}$</p>
| Muhammad Kashif | 269,168 | <p>I think it is true... Because the in asked question the {B} is a power set of A. It meabs all the elements of B are exist in A as Set B us the subset of set A. So, the power set of A will also contain all the sub sets of Set B.</p>
|
547,971 | <p>I have to show that for $f,g$ analytic on some domain and $a$ a double zero of $g$, we have:</p>
<p>$$\operatorname{Res} \left(\frac{f(z)}{g(z)}, z=a\right) = \frac{6f'(a)g''(a)-2f(a)g'''(a)}{3[g''(a)]^2}.$$</p>
<p>The problem is that direct calculation using the formula (for pole of order $2$):</p>
<p>$$\operato... | Daniel Fischer | 83,702 | <p>A little Taylor expansion takes you a long way. Say</p>
<p>$$g(z) = (z-a)^2\left(a_2 + a_3(z-a) + (z-a)^2\cdot \tilde{g}(z)\right)$$</p>
<p>with $a_2 \neq 0$, and</p>
<p>$$f(z) = b_0 + b_1(z-a) + (z-a)^2\cdot \tilde{f}(z).$$</p>
<p>Then</p>
<p>$$\begin{align}
\frac{f(z)}{g(z)} &= \frac{b_0 + b_1(z-a) +(z-a)... |
316,965 | <p>How do I interpret following types of matrices as special types of transformations?
I mean what are the transformative properties of following types of matrices, from $\mathbb{R}^n $ to $ \mathbb{R}^n$, or $\mathbb{C^n}$ to $\mathbb{C^n}$?</p>
<p><strong>Normal and Anti Hermitian Matrices</strong>?</p>
<p><strong>... | Christian Blatter | 1,303 | <p><a href="http://en.wikipedia.org/wiki/Normal_matrix" rel="nofollow">Wikipedia</a> lists about 10 properties of a linear transformation that are equivalent with normality. Here is a property with a geometrical touch that is not mentioned there (it's an exercice in Halmos' <em>Finite-dimensional vector spaces</em>):</... |
62,000 | <p>Let $I,J,K$ be three non-void sets, and let $\gamma$:$I\times J\times K\rightarrow\mathbb{N}$.
Is there some nonempty set $X$, together with some functions {$\{ f_{i}:X\rightarrow X;i\in I\} $},
some subsets {$\{ \Omega_{j}\subset X;j\in J\} $}, and some
points {$\{p_{k}\in X;k\in K} $} s.t. $\mid f_{i}^{-1}\left(p_... | Mark Meilstrup | 14,479 | <p>This is basically a detailed description of a solution, based on Gerhard's answer.</p>
<p>Let $X=\mathbb{N}_0 \times I_0 \times K $, where $\mathbb{N}_0$ includes $0$ and similarly $I_0=I\cup 0$ with $0\not \in I$.</p>
<p>Let $p_k =(0,0,k)$, and let $\displaystyle \Omega_j=\bigcup_i \bigcup_k \bigcup_{n=1}^{\gamma... |
4,513,678 | <p>Suppose <span class="math-container">$f(x) = ax^3 + bx^2 + cx + d$</span> is a cubic equation with roots <span class="math-container">$\alpha, \beta, \gamma.$</span> Then we have:</p>
<p><span class="math-container">$\alpha + \beta + \gamma= -\frac{b}{a}\quad (1)$</span></p>
<p><span class="math-container">$\alpha\b... | Jyrki Lahtonen | 11,619 | <p>You know the elementary symmetric polynomials evaluated at the roots (by the Vieta relations):
<span class="math-container">$$
\begin{aligned}s_1&=\alpha+\beta+\gamma=-b/a,\\
s_2&=\alpha\beta+\beta\gamma+\gamma\alpha=c/a,\\
s_3&=\alpha\beta\gamma=-d/a.
\end{aligned}
$$</span>
The fundamental theorem of s... |
2,583,047 | <p>I have a $m$ dimensional vector space $V$. And we define $\wedge^rV^*$ as the collection $r$ antisymmetric tensors. Why is $\wedge^rV^* = 0$ if $r>m$?</p>
<p>I have no idea.</p>
| C. Falcon | 285,416 | <p>Let $(e_1,\ldots,e_m)$ be a basis of $V$, then for any integer $r$, $\Lambda^rV^*$ is spanned by elements of the form:
$$\mathrm{d}e_{i_1}\wedge\ldots\wedge\mathrm{d}e_{i_r},$$
where $i_1,\ldots,i_r$ are elements of $\{1,\ldots,m\}$. Now, if $r>m$, for each choice of $i_1,\ldots,i_r$, at least twice the same inde... |
2,583,047 | <p>I have a $m$ dimensional vector space $V$. And we define $\wedge^rV^*$ as the collection $r$ antisymmetric tensors. Why is $\wedge^rV^* = 0$ if $r>m$?</p>
<p>I have no idea.</p>
| Community | -1 | <p>Because:</p>
<ul>
<li>$V^*$ is also an $m$-dimensional vector space.</li>
<li>$\Lambda^r U \cong 0$ for <em>any</em> vector space $U$ of dimension less than $r$</li>
</ul>
<p>An easy way to see the latter fact is to use (multi-)linearity decompose any wedge product into a linear combination of wedges of basis vect... |
1,979,226 | <p>Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places.
Y1, Y2, Y3 form a partition of S.</p>
<p>P(X | Y1) = .8, P(X | Y2) = .1, P(X | Y3) = .9, P(Y1) = .1, P(Y2) = .4. </p>
<p>Find P(Y1 | X).</p>
<p>P(Y1 | X) =</p>
<p>For this one I thought that all... | Nicolas FRANCOIS | 288,125 | <p>Another way of seeing this is to note that $(u_n)$ also verifies the recurrence formula
$$u_{n+1}=\sqrt{u_n+u_n^2}$$
and study the function $f:x\mapsto \sqrt{x+x^2}$. This function stabilizes the interval $[0,+\infty[$, is strictly increasing and it is easy to prove that $x>0$ implies $f(x)>x$.</p>
<p>So the ... |
2,098,810 | <p>In a triangle,what is the ratio of the distance between a vertex and the orthocenter and the distance of the circumcenter from the side opposite vertex.</p>
| Simply Beautiful Art | 272,831 | <p>Notice that</p>
<p>$$\int_{-1}^1\sqrt{1+x^2}\ dx<\int_{-1}^1\sqrt{1+1^2}\ dx=2\sqrt2$$</p>
<p>Likewise,</p>
<p>$$\int_{-1}^1\sqrt{1+x^2}\ dx>\int_{-1}^1\sqrt{1+0^2}\ dx=2$$</p>
<p>where we used</p>
<p>$$\int_a^b\min_{t\in(a,b)}f(t)\ dx\le\int_a^bf(x)\ dx\le\int_a^b\max_{t\in(a,b)}f(t)\ dt$$</p>
|
2,740,954 | <p>Determine price elasticity of demand and marginal revenue if $q = 30-4p-p^2$, where q is quantity demanded and p is price and p=3.</p>
<p>I solved it for first part-</p>
<p>Price elasticity of demand = $-\frac{p}{q} \frac{dq}{dp}$</p>
<p>on solving above i got answer as $\frac{10}{3}$</p>
<p>But on solving for M... | user636814 | 636,814 | <p>marginal revenue <span class="math-container">$= p(1+1/elasticity) = 3(1-3/10)= 21/10$</span>.</p>
|
1,760,687 | <p>Can anyone explain me why this equality is true?</p>
<p>$x^k(1-x)^{-k} = \sum_{n = k}^{\infty}{{n-1}\choose{k-1}}x^n$</p>
<p>I really don't see how any manipulation could give me this result. </p>
<p>Thanks!</p>
| Arthur | 15,500 | <p>That depends on your definition. Some would say you need to have two terms in order to have a well-defined difference. Some would say that $1$ is an arithmetic progression (of length $1$ and any difference you like). </p>
<p>Personally, I consider the empty sequence $\{\}$ an arithmetic progression as well, simply... |
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | M. Imaninezhad | 61,045 | <p><a href="https://i.stack.imgur.com/FKAvf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FKAvf.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$$\lim_{n\to\infty}{\alpha_n}=\frac{\pi}{2}$$</span>
So we cannot rotate the graph of <span class="math-container">$y=x^... |
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