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 |
|---|---|---|---|---|
3,752,402 | <p>I want to find out the existence of the solutions in diophantine equations of the style:</p>
<p><span class="math-container">$$-259y ^2+ 2400yx + 1817y + 2122x =
$$</span>
<span class="math-container">$$1057364602723981500371957207036553770637547302056514367123547565680640946707606178926389130616$$</span></p>
<p>Th... | Dmitry Ezhov | 602,207 | <p><span class="math-container">$$-259y ^2+ 2400yx + 1817y + 2122x =
$$</span>
<span class="math-container">$$1057364602723981500371957207036553770637547302056514367123547565680640946707606178926389130616$$</span></p>
<p><span class="math-container">$$\implies \Bigl(1440000 x + 1364999\Bigr)^2-\Bigl(600(2400 x - 518 y... |
4,520,388 | <p>I'm stuck on this multivariable equation:</p>
<p><span class="math-container">$$
\frac{d}{dx}\left(\int^x_af(g(b,t),t)dt\right)
$$</span></p>
<p>where a and b are just constants.</p>
<p>If this involved a single variable, it looks like one would just apply the fundamental theorem of calculus. Is there an equivalent ... | Steven | 606,584 | <p>The function under the integral can be seen as a function of one variable <span class="math-container">$t$</span>, and then the fundamental theorem applies directly. That is, if we define <span class="math-container">$h(t) = f(g(b,t),t)$</span>, then <span class="math-container">$$\frac{d}{dx}\int_a^x h(t) \, dt = h... |
4,473,543 | <p>For example, given <span class="math-container">$\color{green}{l_1:5x-2y-8=0}$</span> and <span class="math-container">$\color{blue}{l_2:3x+8y-8=0}$</span>,</p>
<p><a href="https://i.stack.imgur.com/sxZJ1m.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sxZJ1m.jpg" alt="img1" /></a></p>
<p>We can ... | aerile | 282,796 | <p>One explanation is to show that for any point <span class="math-container">$(p,q)$</span> outside the line <span class="math-container">$L_2(x,y)=0$</span>, we can set <span class="math-container">$\lambda$</span> so that the the point is on the line <span class="math-container">$L_1(x,y)+\lambda L_2(x,y)=0$</span>.... |
23,911 | <p>I am teaching a course on Riemann Surfaces next term, and would <strong>like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties</strong> (over, say, $\mathbb{C}$). I am looking for examples that would be meaningful to 2nd year US g... | roy smith | 9,449 | <p>Some of these properties are local, and distinguish analytic and algebraic functions, from smooth functions.</p>
<p>Others are global and distinguish compact manifolds from projective manifolds by their difference in containment of many subvarieties.</p>
<p>some are as simple as contrasting the dimension, and homo... |
683,513 | <p>There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. My impression has been that the mathematical community often holds an upbeat opinion on the success of student recepti... | Ellya | 135,305 | <p>I'm really just putting forth my opinions on $\epsilon$-$\delta$, and how I think it should be introduced to people who have not seen it before.</p>
<p>I believe the entire difficulty that the $\epsilon-\delta$ approach puts forth is the idea of a constant that depends on something, since before undergrad level (i.... |
3,943,199 | <p>How to put <span class="math-container">$(-\sqrt{3}-i)^{\frac{5}{7}}$</span> into polar form and find all roots.</p>
<p>What I tried:</p>
<p><span class="math-container">$$w = -\sqrt{3}-i$$</span>
<span class="math-container">$$\arg(w)=\arctan(\frac{1}{\sqrt 3})-\pi = \frac{\pi}{6} - \pi = \frac{-5\pi}{6}$$</span>
<... | Christian Blatter | 1,303 | <p>It seems you have a complicated definition <span class="math-container">$F(\phi)$</span> of <span class="math-container">$\sin\phi$</span> in terms of simpler things defined before, and in the same way you have a complicated definition <span class="math-container">$G(\phi)$</span> of <span class="math-container">$e^... |
2,355,852 | <p>Given are $m$ bins with equal probability of choosing one of them. Unknown number of balls $n$ is placed into the bins, and, at the end of placement, we observe number of empty bins $m_e$ and non-empty bins $m_{n}$.</p>
<p>Given $m$, $m_e$, $m_n$, what is the most likely number of balls $n$, which have been placed ... | Thanassis | 8,153 | <p>Let's say we have $m$ bins out of which $m_e$ are empty (and the non-empty bins are $m_n=m-m_e$). Let's also assume that $m_e > 0$, otherwise there is very little information we can get, as you discussed in the comments. </p>
<p>If we could compute the probability of having $i$ balls, given this observation, the... |
2,853,989 | <p>I'm trying to demonstrate that $\left( 1+\frac1 n \right)^n$ is bigger than $2$. I have tried to prove that $\left( 1+\frac1 n \right)^n$ is smaller than $\left( 1+\frac1{n+1} \right)^{n+1}$ by expanding
$\left( 1+\frac1n \right)^n = \sum\limits_{i=0}^n \left( \frac{n}{k} \right) \frac{1}{n^k}$ and $\left( 1+\frac1... | Donald Splutterwit | 404,247 | <p>You just need to consider the first $2$ terms in the binomial expansion
\begin{eqnarray*}
\left( 1+ \frac{1}{n} \right)^{n} = \sum_{i=0}^{n}\binom{n}{i}\frac{1}{n^i}=1+n \frac{1}{n} +\cdots \geq 2.
\end{eqnarray*}</p>
|
1,365,384 | <p>How can the sum of factorials $(n+1)!+n!$ be simplified?</p>
| izœc | 83,639 | <p><strong>HINT:</strong> $(n+1)! = (n+1)\cdot n!$. So...
$$
(n+1)! + n! = (n+1)\cdot n! + n! = ...
$$</p>
|
1,365,384 | <p>How can the sum of factorials $(n+1)!+n!$ be simplified?</p>
| alkabary | 96,332 | <p>$$(n+1)! = (n+1) \times n!$$</p>
<p>and so $$(n+1)! + n! = (n+1) \times n! + n!$$</p>
<p>and $$(n+1) \times n! + n! = n! ((n+1) +1) = n!(n+2)$$</p>
<p>and hence $$\color{blue}{(n+1)! +n! = n!(n+2)}$$</p>
|
3,246,244 | <p>Consider the action of <span class="math-container">$G$</span> on <span class="math-container">$X$</span>.</p>
<p>Let it be a property of <span class="math-container">$G,X$</span> that <span class="math-container">$\forall x,y,\exists g:g\cdot x=g\cdot y$</span>. This is not quite a transitive action - it describe... | Wojowu | 127,263 | <p>No group action (on a set with more than one element) satisfies this. If <span class="math-container">$g\cdot x=g\cdot y$</span>, then <span class="math-container">$x=g^{-1}\cdot(g\cdot x)=g^{-1}\cdot(g\cdot y)=y$</span>, so your condition implies <span class="math-container">$x=y$</span> for all <span class="math-c... |
4,178,548 | <p>I'm starting a Linear Algebra course and I'm a bit confused.</p>
<p>Say we have a vector
<span class="math-container">$x = \begin{pmatrix}
x_1\\
x_2\\
\end{pmatrix}$</span>, and another vector <span class="math-container">$y = \begin{pmatrix}
y_1\\
y_2\\
\end{pmatrix}$</span></p>
<p>When we have a matrix composed of... | Eric | 909,833 | <p>This just depends on context and definitions. Both are feasible in the right context:
<a href="https://en.m.wikipedia.org/wiki/Row_and_column_spaces" rel="nofollow noreferrer">https://en.m.wikipedia.org/wiki/Row_and_column_spaces</a></p>
|
1,804,042 | <p><strong>Edit:</strong> Here is the original problem; it is possible that my recurrence for the stationary distribution $\pi$ is incorrect.</p>
<blockquote>
<p>Consider a single server queue where customers arrive according to a Poisson process with intensity $\lambda$ and request i.i.d. $\mathsf{Exp}(\mu)$ servic... | Semiclassical | 137,524 | <p>Note: The present form of the answer reflects an earlier version of the problem. I plan to modify it to reflect the changes, hopefully sooner rather than later...</p>
<p>Let $P(x)=\sum_{n=0}^\infty \pi_n x^n$. The first few lines of this recurrence are
\begin{align}
\pi_1+\frac{1}{\mu}\left(\frac{\alpha\theta}{\al... |
818,169 | <p>The variational distance is defined by,
$$
V(P,Q)=\sum _{i}|p_{i} -q_{i} |
$$
where $P=(p_{1} ,...,p_{n})$ and $Q=(q_{1} ,...,q_{n} )$ are discrete distributions.</p>
<p>It is fairly easy to see that $V$ is a metric, and in particular that it satisfies the triangle inequality. I now introduce class prior, $0<\... | Johannes Schindelin | 166,166 | <p>But $V(\alpha, P, Q) != V(\alpha, Q, P)$ in general, therefore it would not be a metric, right? If that is true, what is the point of proving or disproving the triangle inequality?</p>
|
462,921 | <p>I simply don't get the following question answered:</p>
<p>How can i proof the equality $\lim_{a\to 0}\sup_{z\in\mathbb{Z}}2-2\cos(2\pi a z)=0$?</p>
<p>Or is it even false?</p>
<p>Thanks in advance!</p>
| Supriyo | 70,120 | <p>Think little bit differently.</p>
<p>$2 - 2\cos(2\pi a z) = 2(1-\cos(2 \pi a z)) = 4\sin^2(\pi a z)$$</p>
<p>Now in complex plain $\sin(z)$ is an unbounded function having a zero at origin. So the value of the expression should be $0$.</p>
|
1,977,306 | <p>This is from a math competition so it must not be something really long
If a parabola touches the lines $y=x$ and $y=-x$ at $A(3,3) $ and $b(1,-1)$ respectively, then </p>
<p>(A) equation of axis of parabola is $2x+y=0$ </p>
<p>(B)slope of tangent at vertex is $1/2$</p>
<p>(C) Focus is $(6/5,-3/5)$</p>
<p>(D) ... | bubba | 31,744 | <p>It's well known that any rational quadratic Bézier curve is a conic section (ellipse, hyperbola, or parabola), and any (polynomial) quadratic Bézier curve is a parabola. There are fairly simple formulae for obtaining the geometric characteristics (directrix, focus, vertex) directly from the control points of the Béz... |
730,198 | <blockquote>
<p>Show $f(x)=\sqrt{x^4+1} - \sqrt{x^4+x^2} \rightarrow -1/2$ for $x \rightarrow \infty$, $x \in \mathbb R$.</p>
</blockquote>
<p>I've tried $$\frac {(\sqrt{x^4+1} - \sqrt{x^4+x^2})(\sqrt{x^4+1} + \sqrt{x^4+x^2})}{\sqrt{x^4+1} + \sqrt{x^4+x^2} } = \frac {1-x^2} {\sqrt{x^4+1} + \sqrt{x^4+x^2}}$$</p>
<p>... | Amir Parvardi | 6,715 | <p>In this kind of problems which want you to calculate the limit when $x \to \infty$, constants are not very important and you have to pay attention to powers of $x$ (or anything related to $x$). That makes sense because, e.g., when $x=10$, you have $x^4 = 10000$, which is way greater than $1$.</p>
<p>Try to convert... |
878,961 | <p>I'm asked to prove a theorem (if that is the right word) about double derivatives. I'm still struggling with understanding Leibniz notation and I could use a push in the right direction. It's easy enough for me to differentiate the function when I write it down as $f(g(x))$ but not so much with Leibniz notation.</p>... | Paul Sundheim | 88,038 | <p>The answer to your last question is that $\frac{dy}{dx}=\frac{d}{dx}y$ so the second derivative is $\frac{d}{dx}\frac{d}{dx}y=\frac{d^2}{dx^2}y$. This should answer the first question as well.</p>
|
1,853,464 | <p>I am using the Lorentz Force Equation and the electric-cross-magnetic field velocity equation] to solve for the $E$ and $B$ fields given the known path of a particle moving in 3D. </p>
<p>So with that I have the following equations where a and v are known:
<a href="https://i.stack.imgur.com/oL7fg.gif" rel="nofollow... | Ng Chung Tak | 299,599 | <p>One interesting case when a charged particles is at rest initially in the crossed electric and magnetic fields, it moves in a cycloid.</p>
<p>See the video <a href="https://www.youtube.com/watch?v=1eOn5K9qZu4" rel="nofollow">here</a>.</p>
<p>See also a detailed discussion in Chapter 2 of <strong>Fundamentals of Pl... |
1,369,641 | <p>I do not know how to set this problem up. Any insight as to how to get the equation would be great. </p>
<p>It is John's birthday and his parents want to make him a cake in the shape of a rectangular box. The height of the cake will be $15$ centimeters, and $2$ times the width plus $2$ times the length will be $180... | DeepSea | 101,504 | <p>$L + W = 90 \to V =f(L)= L\times W\times H = 15L(90-L)\to f'(L) = 15(90-2L)=0 \iff L = 45 \to W = 45$. Thus $V_{max} = 15\times 45\times 45 = ....$</p>
|
1,369,641 | <p>I do not know how to set this problem up. Any insight as to how to get the equation would be great. </p>
<p>It is John's birthday and his parents want to make him a cake in the shape of a rectangular box. The height of the cake will be $15$ centimeters, and $2$ times the width plus $2$ times the length will be $180... | JMP | 210,189 | <p>We have $W+L=90$. Consider $W=L=45$ and then consider $W=45-k, L=45+k$. It is well known that $(45-k)(45+k)$ is maximum when $k=0$. In fact, this justifies the initial assumption.</p>
|
456,892 | <p>Find all solutions of $4\cos^2(x)-4\sin(x)-5=0$ in the interval $(6\pi, 8\pi)$.</p>
<p>I tried to work it out and got: $4y^2-4y -9 = 0$, but I can't figure out what $\cos x = $from there to finish the problem.</p>
| amWhy | 9,003 | <p>$$\begin{align} 4\cos^2 x - 4\sin x - 5 & = 0 \\ \\\iff 4(1 - \sin^2 x) & = 4\sin x + 5 \\ \\ \iff 4\sin^2 x + 4\sin x + 1 &= 0\end{align}$$</p>
<hr>
<p>$$\begin{align} y = \sin x \implies 4y^2 + 4y + 1 & = 0 \\ \\
(2y+1)^2 & = 0 \\ \\
\implies y & = -1/2\end{align}$$</p>
<hr>
<p>Therefo... |
796,262 | <p>So, I am computing something seemingly simple involving complex gaussians and constants, but I am getting a big contradiction in my calculations. </p>
<p><strong>The setup:</strong></p>
<ul>
<li>Let $C$ be a complex constant, that is, $C = c_r + jc_i$. </li>
<li>Let $G$ be a complex gaussian variable, $G = g_r + j... | N. S. | 9,176 | <p>Any spanning tree has $6$ edges. Thus, the sum of degrees of the $3$ respectively $4$ edges is $6$.</p>
<p>Thus, the degrees of the $4$ vertices can be $(3,1,1,1)$ or $(2,2,1,1)$. Count them from here.</p>
|
204,365 | <p>Consider a positive matrix <code>M</code> and a positive vector <code>b</code>, e.g.</p>
<pre><code>nn = 1000;
M = Table[RandomReal[{0, 100}], {i, 1, nn}, {j, 1, nn}];
b = Table[RandomReal[{0, 100}], {i, 1, nn}];
</code></pre>
<p>I would like to find a positive vector <code>X</code></p>
<pre><code>X = Array[x,... | KennyColnago | 3,246 | <p>Perhaps you could use the non-negative least-squares algorithm (NNLS) of Lawson and Hanson, discussed <a href="https://mathematica.stackexchange.com/questions/26336/how-to-perform-a-multi-peak-fitting/26338#26338">here</a>. See the links in the comments.</p>
<p>NNLS was ported to Mathematica by Michael Woodhams, an... |
2,304,379 | <p>My textbook give the following definition.</p>
<blockquote>
<p>Let $G$ be any topological group. A representation of $G$ on a nonzero complex Hilbert space $V$ is a group homomorphism $\phi$ of $G$ into the group of bounded linear operators on $V$ with bounded inverses, such that the resulting map $ G\times V\to ... | bof | 111,012 | <p>The empty set is closed and does not contain any nonempty open set. Probably you meant to ask for a <strong>nonempty</strong> closed set which does not contain any nonempty open set. The following theorem answers this question in a general topological space X.</p>
<p><strong>Theorem.</strong> If in the space $X$ th... |
3,065,818 | <blockquote>
<p>If <span class="math-container">$$z=\dfrac{\sqrt{3}-i}{2}$$</span> then <span class="math-container">$$(z^{95}+i^{67})^{94}=z^n$$</span> then, <span class="math-container">$\text{find the smallest positive integral value of}$</span> <span class="math-container">$n$</span> <span class="math-container">... | N. S. | 9,176 | <p><strong>Hint</strong> You can find some <span class="math-container">$k,l \in \mathbb Z$</span> such that
<span class="math-container">$$ka+lb=1$$</span></p>
<p>Then
<span class="math-container">$$n=n^{ka+lb}=(n^a)^k\cdot(n^b)^l$$</span></p>
<p>Now, if you want <span class="math-container">$x = n^b, y=n^a$</span>... |
3,362,115 | <blockquote>
<p>Find the maximum value of <span class="math-container">$y/x$</span> if it satisfies <span class="math-container">$(x-5)^2+(y-4)^2=6$</span>.</p>
</blockquote>
<p>Geometrically, this is finding the slope of the tangent from the origin to the circle. Other than solving this equation with <span class="m... | albert chan | 696,342 | <p>Let a line be <span class="math-container">$y = mx$</span></p>
<p><span class="math-container">$$(x-5)^2 + (mx-4)^2 - 6 = 0$$</span>
<span class="math-container">$$(m^2+1)x^2 + (-8m-10)x + 35 = 0$$</span></p>
<p>If the line does not touch the circle, above have no solution.<br>
In other words, discriminant is nega... |
3,554,188 | <blockquote>
<p>I was given <span class="math-container">$$|ax - 11| = 4x - 10$$</span> has a positive integral solution and <span class="math-container">$a$</span> is a positive integer.</p>
<p>I was asked what was <span class="math-container">$x, a$</span></p>
</blockquote>
<p><span class="math-container">$$ax > 1... | VIVID | 752,069 | <p>I think you made a little error because you had to have <span class="math-container">$x=\frac{21}{a+4}$</span> when <span class="math-container">$ax<11$</span>.
Hence <span class="math-container">$(x,a)=(3,3)$</span>.</p>
|
4,326,073 | <p><a href="https://i.stack.imgur.com/RTyOy.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RTyOy.jpg" alt="enter image description here" /></a>
I came across questions in the free module section of my abstract algebra text. In the text, the notation <span class="math-container">$End_{R}(V)$</span> d... | user284331 | 284,331 | <p>A few comments to the answers provided by @QC_QAOA and @Zieac.</p>
<p>Personally I think one should go through by <span class="math-container">$\limsup$</span> and <span class="math-container">$\liminf$</span> argument.</p>
<p>Note that
<span class="math-container">\begin{align*}
\liminf(u_{n}+v_{n})=\liminf u_{n}+\... |
1,938,253 | <p>Consider the following limit:$$\lim_{x \to\infty}\frac{2+2x+\sin(2x)}{(2x+\sin(2x))e^{\sin(x)}}$$</p>
<p>If we apply L'hospital's rule then we get:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/slrYm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/slrYm.png" alt="Blockquote"></a></p>
... | zxcvber | 329,909 | <p>Since</p>
<p>$$-1 \leq -\sin x \leq 1, \ \ \ 1/e\leq e^{-\sin x}\leq e$$</p>
<p>Thus,</p>
<p>$$\frac{4\cos{x}}{2x+4\cos{x}+\sin{2x}}\cdot\frac{1}{e}\leq \frac{4\cos{x}}{2x+4\cos{x}+\sin{2x}}e^{-\sin{x}}\leq\frac{4\cos{x}}{2x+4\cos{x}+\sin{2x}}e$$</p>
<p>We know that
$$\lim_{x\to\infty}\frac{4\cos{x}}{2x+4\cos{x}... |
1,938,253 | <p>Consider the following limit:$$\lim_{x \to\infty}\frac{2+2x+\sin(2x)}{(2x+\sin(2x))e^{\sin(x)}}$$</p>
<p>If we apply L'hospital's rule then we get:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/slrYm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/slrYm.png" alt="Blockquote"></a></p>
... | Paramanand Singh | 72,031 | <p>First of all +1 for coming up with an example which seems to violate the L'Hospital's Rule. However note that if $f(x)/g(x)$ is the original expression then $$\frac{f'(x)}{g'(x)} = \frac{4\cos^{2}x}{4\cos^{2}x + (2x + \sin 2x)\cos x}\cdot e^{-\sin x}$$ The problem now is that the limit of $f'(x)/g'(x)$ does not exis... |
2,653,726 | <p>Prove that $1, r, r^2,\ldots, r^{n-1}, t, tr, tr^2,\ldots, tr^{n-1}$ are $2n$ distinct symmetries of the regular n-gon.</p>
<p>It is clear that there are always $2n$ symmetries for any regular $n$-gon, with half of them being rotations and the other half being their reflections. Therefore for an $n$-gon with an eve... | MCT | 92,774 | <p>You have showed that there are at least $2n$ symmetries, but there could be more. Note that these symmetries are distance preserving, in that if the distance between vertices $a$ and $b$ is $D$, then after applying the symmetry $s$ the vertices $s(a)$ and $s(b)$ will still have distance $D$. </p>
<p>Ok so now pick ... |
2,653,726 | <p>Prove that $1, r, r^2,\ldots, r^{n-1}, t, tr, tr^2,\ldots, tr^{n-1}$ are $2n$ distinct symmetries of the regular n-gon.</p>
<p>It is clear that there are always $2n$ symmetries for any regular $n$-gon, with half of them being rotations and the other half being their reflections. Therefore for an $n$-gon with an eve... | Siong Thye Goh | 306,553 | <p>Prove that $1,r,r^2,\ldots,r^{n−1},t,tr,tr^2, \ldots,tr^{n−1}$, are $2n$ <strong>distinct</strong> symmetries of the regular n-gon.</p>
<p>I would focus more on proving that they are distinct. </p>
<p>If $r^i=r^j, j> i, \{i,j\} \subseteq \{ 0,n-1\}$, then $r^{j-i}=1$ but the order of $r$ is $n$.</p>
<p>We als... |
27,271 | <p>A few days ago I recall finding a visual calendar of my sign-in history on Math Stack Exchange. It looked like an ordinary monthly calendar, where the days were white if I hadn't signed in, and green if I had. (I found it when I wondering how close I've been in the past to getting the "fanatic" badge.)</p>
<p>Howev... | Martin Sleziak | 8,297 | <p>A brief answer is that you should go to <a href="https://math.stackexchange.com/users/current?tab=profile">your profile</a> and click on the number of consecutive days. (Remember that you'll find this in the profile tab, not in the activity tab.)</p>
<p>You can find more details here:</p>
<ul>
<li><a href="https:/... |
2,186,076 | <p><strong>Question.</strong></p>
<blockquote>
<p>If $p$ is a polynomial of degree $n$ with $p(\alpha)=0$, what do we know of the polynomial $q$ (with degree $n-1$) such that the numbers $(q^k(\alpha))_{k=1}^n$ contain all of the zeroes of $p$?</p>
</blockquote>
<p>Here I denote $q(q(\cdots q(\alpha)))=q^k(\alpha)$... | mercio | 17,445 | <p>For any sequence of $n$ distinct (complex) numbers $\alpha_1 \ldots \alpha_n$, there is a unique interpolating polynomial $Q$ of degree $n-1$ with $P(\alpha_i) = \alpha_{i+1}$.</p>
<p>In the generic case, the coefficients of $Q$ are in $\Bbb Q[\alpha_i, \delta^{-1}]$, where $\delta = \prod_{i<j} (\alpha_i - \alp... |
1,425,907 | <p>Show that $$1+e^{-j\theta} =2e^{-j\theta/2}*\cos{\frac{\theta}2}$$</p>
<p>I know of the Euler equation: $e^{j\theta}=\cos(\theta)+j\sin(\theta)$ but am unsure how to simply show that the above are equal.</p>
| ajotatxe | 132,456 | <p>Substitute the expresion for $a_n$ in the formula of $S_n$:</p>
<p>$$363=\frac n2\big(-12+[-12+(n-1)9]\big)$$</p>
|
3,662,286 | <p>Can you raise the imaginary number i to a power that is an irrational number?</p>
| Community | -1 | <p>Short answer:</p>
<p><span class="math-container">$$(e^{i(\pi/2+2k\pi)})^x=e^{ix(\pi/2+2k\pi)}.$$</span></p>
|
1,161,631 | <p>Let $z \in \mathbb{C}$ be a root of real polynomial $p(x)=\sum_{k=0}^{n} a_k x^k$ ,$a_k\in \mathbb{R} \forall k$.</p>
<p>How to proove, that $\overline{z}$ is also a root of given polynomial? Is that true for complex polynomials, too?</p>
| quid | 85,306 | <p>Recall that complex conjugation can be interchanged with products and sums. Plug in $\overline{z}$ an use this repeatedly. </p>
<p>Note that $a \overline{z}= \overline{az}$ when $a$ is real yet not in general. </p>
|
1,161,631 | <p>Let $z \in \mathbb{C}$ be a root of real polynomial $p(x)=\sum_{k=0}^{n} a_k x^k$ ,$a_k\in \mathbb{R} \forall k$.</p>
<p>How to proove, that $\overline{z}$ is also a root of given polynomial? Is that true for complex polynomials, too?</p>
| kryomaxim | 212,743 | <p>May be $z_0=x+iy$ a root of the polynomial $p$. Then, if $\bar{z_0} = x-iy$ is also a root then the polynomial can be divided by $(z-z_0)(z-\bar{z_0}) = (z-x)^2+y^2 \in R$ because the numbers $x,y$ are real numbers.</p>
<p>For complex polynomials, this relation does not hold. </p>
|
1,161,631 | <p>Let $z \in \mathbb{C}$ be a root of real polynomial $p(x)=\sum_{k=0}^{n} a_k x^k$ ,$a_k\in \mathbb{R} \forall k$.</p>
<p>How to proove, that $\overline{z}$ is also a root of given polynomial? Is that true for complex polynomials, too?</p>
| Bumblebee | 156,886 | <p>Suppose $z=re^{i\theta}.$ </p>
<blockquote>
<p><strong>If you set $b_k=ra_k$ then the polynomial
$$q(x)=\sum_{k=0}^{n} b_k x^k$$ has the zero $e^{i\theta}$ if and only if $$p(x)=\sum_{k=0}^{n} a_k x^k$$ has the zero $z=re^{i\theta}.$</strong> </p>
</blockquote>
<p>If $b_k\in\mathbb{R},$ then all $a_k\in\math... |
3,735,798 | <blockquote>
<p><strong>QUESTION:</strong> Given a square <span class="math-container">$ABCD$</span> with two consecutive vertices, say <span class="math-container">$A$</span> and <span class="math-container">$B$</span> on the positive <span class="math-container">$x$</span>-axis and positive <span class="math-containe... | atul ganju | 788,520 | <p>You can calculate it like this:</p>
<p><span class="math-container">$$(u-v)^2+u^2$$</span></p>
<p>To understand why, lets define two variables: <span class="math-container">$x, y$</span>. Given the fact that the square is "leaning" on the y-axis, <span class="math-container">$x$</span> is the distance from... |
433,816 | <p>While I was studying the measurements of pressure at earth's atmosphere,I found the barometric formula which is more complex equation ($P'=Pe^{-mgh/kT}$) than what I used so far ($p=h\rho g$).</p>
<p>So I want to know how this complex formula build up? I could reach at the point of
$${dP \over dh}=-{mgP \over kT}$$... | Robert Lewis | 67,071 | <p>If </p>
<p>$\frac{dP}{dh} = (-\frac{mgP}{kT})$,</p>
<p>then</p>
<p>$\frac{1}{P} \frac{dP}{dh} = (-\frac{mg}{kT})$,</p>
<p>or</p>
<p>$\frac{d(ln P)}{dh} = (-\frac{mg}{kT})$.</p>
<p>Integrating with respect to $h$ over the interval $[h_0, h]$ yields</p>
<p>$ln(P(h)) - ln(P(h_0)) = (-\frac{mg}{kT})(h - h_0)$,</p... |
3,817,340 | <p>I'm trying to draw the bio-hazard symbol for <a href="https://codegolf.stackexchange.com/questions/191294/draw-the-biohazard-symbol">a codegolf challenge</a> in Java, for which I've been given the following picture (later referred to as unit diagram):</p>
<p><a href="https://i.stack.imgur.com/fIsNl.png" rel="nofollo... | Kevin Cruijssen | 368,863 | <p>I've been able to figure out all calculations. As I mentioned earlier, I've used 10 times the units of the picture in the challenge description, so those sizes are: <span class="math-container">$A=10, B=35, C=40, D=60, E=110, F=150, G=210, H=300$</span>. I've also assumed the very center is at coordinate <span class... |
1,675 | <p>This is a follow-up to <a href="https://mathoverflow.net/questions/1039/explicit-direct-summands-in-the-decomposition-theorem">this post</a> on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general.</p>
<p>My question is how does one use the De... | Geordie Williamson | 919 | <p>To supplement Ben's answer, basically every aspect of the decomposition theorem is hard.</p>
<p>To give you a simple example of something which is implied by the decomposition theorem but is far from trivial is the following statement: given a proper smooth map of smooth varieties f : X -> Y the direct image of the... |
2,793,384 | <p>I am taking a course in Algebraic Topology next semester, so I thought of starting to read about it on my own.</p>
<p><strong>My concern:</strong> I have read <em>Topology</em> by Munkres, and read the first chapter on algebraic topology from that book, so I have an idea about the fundamental group and covering sp... | Osama Ghani | 341,911 | <p>A cursory glance through the book suggests it is has a strong differential flavour. If you’re familiar with multivariable calculus especially as formulated through differential forms and exterior derivatives, this isn’t necessarily a bad thing, and does help in understand what it means for two cycles to be homologou... |
1,560,050 | <p>I want to solve the homogenous part of a stretched string problem where $y=y(x)$.</p>
<p>$$y'' + y = 0$$</p>
<p>with the boundary conditions such that: $y(0)=y(\pi/2)=0$</p>
<p>The differential equation gives rise to a solution on the form:
$$y = a \cos(x) + b \sin(x)$$</p>
<p>But when applying the boundary con... | Martin Argerami | 22,857 | <p>Yes, the only solution to that boundary value problem is $y(x)=0$. </p>
|
1,585,408 | <p>I have the equation: (1-x<sup>2</sup>)u<sup>''</sup> -xu<sup>'</sup>+ku=0,
where ' represents differentiation with respect to x and k is a constant.</p>
<p>I am asked to show that cos(k<sup>1/2</sup>cos<sup>-1</sup>x) is a solution to this equation.</p>
<p>I assumed to show this you need to set u=cos(k<sup>1/2</su... | David Holden | 79,543 | <p>if you make the substitution $x=\cos t$ then
$$
u_t=-u_x\sin t
$$
and
$$
u_{tt} = -u_x\cos t +u_{xx}\sin^2 t = u_{xx}(1-x^2) - x u_x
$$
the equation becomes:
$$
u_{tt} +ku =0
$$</p>
|
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | ShreevatsaR | 205 | <p>Another example: <a href="http://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture" rel="noreferrer">Euler's sum of powers conjecture</a>, a generalization of Fermat's Last Theorem. It states:<br>
If the equation <span class="math-container">$\sum_{i=1}^kx_i^n=z^n$</span> has a solution in positive integers, ... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Bill Dubuque | 242 | <p>Another class of examples arise from diophantine equations with huge minimal solutions. Thus the conjecture that such an equation is unsolvable in integers has only huge counterexamples. Well-known examples arise from Pell equations, e.g. the smallest solution to the classic <a href="http://en.wikipedia.org/wiki/Arc... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Oscar Lanzi | 248,217 | <p>A case where you can "dial in" a large counterexample involves this theorem:</p>
<p>"A natural number is square if and only if it is a <span class="math-container">$p$</span>-adic square for all primes <span class="math-container">$p$</span>."</p>
<p>A <span class="math-container">$p$</span>-adic... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | BlueRaja - Danny Pflughoeft | 136 | <p>Here's a recent one I didn't see on either page. The following are true statements:</p>
<p><span class="math-container">$\displaystyle{ \int_0^\infty \frac{\sin t}{t} \, dt = \frac{\pi}{2} }$</span><br>
<span class="math-container">$\displaystyle{ \int_0^\infty \frac{\sin t}{t} \, \frac{\sin \left(\frac{t}{101}\ri... |
1,246,522 | <p>What are the prime ideals of $\mathbb F_p[x]/(x^2)$? I have been told that the only one is $(x)$, but I would like a proof of this. I want to say that a prime ideal of $\mathbb F_p[x]/(x^2)$ corresponds to a <strong>prime</strong> ideal $P$ of $\mathbb F_p[x]$ containing $(x^2)$. And then $P$ contains $(x)$ since i... | Spencer Leslie | 129,418 | <p>Prime ideals do correspond under the correspondence theorem, so your argument suffices.</p>
<p>To see this,</p>
<p>Let $I\subset P\subset R$ be any prime in $R$ containing an ideal $I$, then
$R/P \cong \frac{R/I}{P/I}$ by the 3rd isomorphism theorem. Since $P$ is prime, $R/P$ is an integral domain, hence so is $\... |
40,116 | <p>I'm building a program that calculates the cost of an item based on it's size (let's say a bamboo pole). As the customer requests a longer pole, it gets hard to find a bamboo, plus requires more resources to grow, therefore, I would want to charge more per inch for the piece of bamboo based on it's length approachin... | zmaril | 11,124 | <p>Try looking into statistics distributions. By messing with the degrees of freedom you can get something that could do what you want. If you take the inverse of the probability of something happening, then you can get your arbitrarily large number. Bonus: you can probably set 75" as the average parameter for the dist... |
3,506,659 | <p>Let <span class="math-container">$\sigma_i$</span> denote the <a href="https://en.wikipedia.org/wiki/Pauli_matrices" rel="nofollow noreferrer">Pauli matrices</a>:
<span class="math-container">$$
\sigma_1\equiv \begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad
\sigma_2\equiv \begin{pmatrix}0&-i\\i&0\end{pma... | Ben Grossmann | 81,360 | <p>We define an inner product over <span class="math-container">$\Bbb C^{n \times n}$</span> by <span class="math-container">$\frac 1n \langle A,B \rangle = \operatorname{tr}(A^\dagger B)$</span>; this is (a normalized version of what is) known as the <a href="https://en.wikipedia.org/wiki/Frobenius_inner_product" rel=... |
3,753,474 | <p><strong>Question:</strong></p>
<blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma$</span> are the roots of the equation, <span class="math-container">$x^3+x+1=0$</span>, then find the equation whose roots are: <span class="math-container">$({\alpha}-{\beta})^2,({\beta}-{\gamma})^2,({\gamma}-{\alpha... | Z Ahmed | 671,540 | <p>Let <span class="math-container">$a,b,c$</span> be the roots of <span class="math-container">$x^3+x+1=0$</span> so we have <span class="math-container">$a+b+c=0, ab+bc+ca=1,abc=-1$</span>, so <span class="math-container">$a^2+b^2+c^2=-2$</span> and <span class="math-container">$c^3=-c-1$</span></p>
<p>We would explo... |
3,653,212 | <p>Due to Covid -19 , in our university quizzes are held online and it's hard to ask questions. </p>
<p>3 Days back in my Combinatorics quiz this question was asked on which I am struck. I couldn't solve it in the time alloted and struggled to find a proper strategy. </p>
<p>Question is ->Determine the number of non ... | Robert Z | 299,698 | <p>The symmetric group is <span class="math-container">$S_4$</span> with <span class="math-container">$4!=24$</span> elements:</p>
<ul>
<li>6 with cycle structure <span class="math-container">$(abcd)$</span> then <span class="math-container">$C(f)=k$</span>;</li>
<li>8 with cycle structure <span class="math-container"... |
897,043 | <p>I'm having issues getting my head around cartesian products and their cardinalities.</p>
<p>$A = \{0, 1, \{2, 3, 4\}\}$<br>
$B = \{1,5\}$<br>
$D = B \times N$ (where $N$ is the set of natural numbers)</p>
<p><strong>The first problem:</strong> What is the cardinality of:</p>
<p>(a) $A \times B$ (cartesian produ... | AlexR | 86,940 | <p>$\mathbb N$ is <strong>not</strong> a subset of $D$. Subsets of $D$ alike to $\mathbb N$ are, for example
$$\{1\} \times \mathbb N, \{5\} \times \mathbb N$$
The difference is that elements from $D$ look like $(1, a)$ or $(5, a)$ with $a\in\mathbb N$ whereas elements from $\mathbb N$ are just natural numbers (no tupl... |
125,610 | <p>I have question about sets. I need to prove that: $$X \cap (Y - Z) = (X \cap Y) - (X \cap Z)$$</p>
<p>Now, I tried to prove that from both sides of the equation but had no luck.</p>
<p>For example, I tried to do something like this: $$X \cap (Y - Z) = X \cap (Y \cap Z')$$ but now I don't know how to continue.</p>
... | MarnixKlooster ReinstateMonica | 11,994 | <p>Here is a third approach, which first translates to the element level, and then uses logic manipulation.</p>
<p>Starting at the most complex side, we have for any $\;a\;$
\begin{align}
& a \in (X \cap Y) - (X \cap Z) \\
\equiv & \;\;\;\;\;\text{"expand definitions of $\;-\;$ and of $\;\cap\;$ (twice)"} \\
&... |
1,661,986 | <p>I was out all last week sick with the flu and am trying to get caught up in my Discrete Mathematics course. One set of questions in my book goes as follows:</p>
<p>Find the least integer $n$ such that $f(x) \in O(x^n)$ for each of the functions:</p>
<p>(a). $f(x) = 8x+4$<br>
(b). $f(x) = x\sqrt(x)$<br>
(c). $f(x) ... | user137481 | 137,481 | <p>grand_chat has already explained the concept of Big-O to you.
However, for your exercise, it might be easier to understand if you use the following definition:</p>
<p>$f(x) \in O(g(x))$ if there exists positive constants $C$ and $x_0$ such that: </p>
<p>$\quad\quad 0 \le f(x) \le Cg(x)$ for all $x \gt x_0$</p>
... |
1,264,195 | <p>So there is a way of extending the set $N$ of natural numbers with 0, equipped with ordinary multiplication, to its Grothendieck group, the group of integers with respect to addition.</p>
<p>This group, $Z$, is the <em>best</em> extension of $N$ in the sense that ...? (question)</p>
<p>So, here are my questions:</... | Michael Albanese | 39,599 | <p>Not necessarily. Consider the <a href="http://en.wikipedia.org/wiki/Peano_curve" rel="nofollow">Peano curve</a>, a continuous surjective map $f : [0, 1] \to [0, 1]^2$ (the codomain can be identified with a subset of $\mathbb{C}$ is you prefer). Not every element of $[0, 1]^2$ is a boundary point.</p>
|
3,755,288 | <p>I'm trying to solve this:</p>
<blockquote>
<p>Which of the following is the closest to the value of this integral?</p>
<p><span class="math-container">$$\int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \ dx$$</span></p>
<p>(A) 1</p>
<p>(B) 1.2</p>
<p>(C) 1.6</p>
<p>(D) 2</p>
<p>(E) The integral doesn't converge.</p>
</blockquot... | Community | -1 | <p><span class="math-container">$$\int_0^1\sqrt{1+\dfrac1{3x}}dx=2\int_0^1\sqrt{t^2+\dfrac13}dt$$</span> proves convergence.</p>
<p>Then</p>
<p><span class="math-container">$$\frac1{\sqrt 3}\le\sqrt{t^2+\frac13}\le t+\frac1{\sqrt3}$$</span>
implies</p>
<p><span class="math-container">$$\frac2{\sqrt 3}\approx 1.155\le I... |
3,755,288 | <p>I'm trying to solve this:</p>
<blockquote>
<p>Which of the following is the closest to the value of this integral?</p>
<p><span class="math-container">$$\int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \ dx$$</span></p>
<p>(A) 1</p>
<p>(B) 1.2</p>
<p>(C) 1.6</p>
<p>(D) 2</p>
<p>(E) The integral doesn't converge.</p>
</blockquot... | Anonymous | 807,989 | <p>We know that,</p>
<p><span class="math-container">${\displaystyle\int}\sqrt{\dfrac{1}{3x}+1}\,\mathrm{d}x$</span>=<span class="math-container">$=\class{steps-node}{\cssId{steps-node-1}{\dfrac{1}{\sqrt{3}}}}{\displaystyle\int}\sqrt{\dfrac{1}{x}+3}\,\mathrm{d}x$</span></p>
<p>Substitute <span class="math-container">$u... |
4,095,831 | <p>If <span class="math-container">$\frac{dy}{dx}=y\sec^2x$</span> and <span class="math-container">$y=5$</span> when <span class="math-container">$x=0$</span>, then <span class="math-container">$y=$</span>?
A) <span class="math-container">$e^{\tan x} + 4$</span></p>
<p>B) <span class="math-container">$e^{\tan x} + 4$<... | GReyes | 633,848 | <p>When you take the exponential function, you actually get
<span class="math-container">$$
|y|=e^{\tan x}e^C
$$</span>
or
<span class="math-container">$$
y=De^{\tan x}
$$</span>
where <span class="math-container">$D=\pm e^C\ne 0$</span>.Since <span class="math-container">$y=0$</span> is also a solution, you can includ... |
61,509 | <p>I'm trying to read Elias Stein's "Singular Integrals" book, and in the beginning of the second chapter, he states two results classifying bounded linear operators that commute (on $L^1$ and $L^2$ respectively).</p>
<p>The first one reads:</p>
<p>Let $T: L^1(\mathbb{R}^n) \to L^1(\mathbb{R}^n)$ be a bounded linear ... | Willie Wong | 1,543 | <p>Let me just give a literature reference, in case that's what you are interested in. The classical paper on translation invariant operators on Lebesgue spaces is Hormander's 1960 <em>Acta</em> paper "Estimates for translation invariant operators in $L^p$ spaces". If you have access to MathSciNet, <a href="http://www.... |
2,361,602 | <p>The "Heine–Cantor theorem" states: If $f : M → N$ is a continuous function between two metric spaces, and $M$ is compact, then $f$ is uniformly continuous.</p>
<p>I do not doubt its validity, of course, just trying to understand <strong>why</strong> it is valid.</p>
<p>If we, say, take the function: $y = x^4$.
It ... | AbstractNonsense | 429,931 | <p>you can also consider the following idea(take for simplicty $N = \mathbb{R})$: since $f$ is continouus and $M$ is compact, the function $f$ obtains its <strong>maximal value</strong>, which is some real number, inside $M$ (see also <a href="https://en.wikipedia.org/wiki/Extreme_value_theorem" rel="nofollow noreferre... |
20,942 | <p>If I had a recursive function (<code>f(n) = f(n-1) + 2*f(n-2)</code> for example), how would I derive a formula to solve this? For example, with the Fibonacci sequence, Binet's Formula can be used to find the nth term.</p>
| Community | -1 | <p>If the recurrence relation is linear, homogeneous and has constant coefficients, here is the way to solve it. First obtain the characteristic equation. To do this, assume $f(n) = m^n$. Plug it in to get a quadratic in $m$. Solve for $m$. Get the two roots say $m_1$ and $m_2$. Now the general solution is given by the... |
1,719,840 | <p>If there is a group $G$ with order $a$, having a subgroup $H_1$ with order $b$, and $H_2$ with order $c$, and $bc=a$, $H_1 \cap H_2 = e $. Is $H_1 H_2 =G$? </p>
| Math1000 | 38,584 | <p>Since $H_1$ and $H_2$ are subgroups of $G$, we have $$|H_1H_2|=\frac{|H_1||H_2|}{|H_1\cap H_2|}=|H_1||H_2|=|G|,\tag1 $$ so $H_1H_2=G$.</p>
<p>(See @Bungo's answer for a proof of the identity in (1))</p>
|
1,303,485 | <p>Evaluate the integral $$\int_{C}\frac{z^2}{z^2+9}dz$$
where C is the circle $|z|=4$</p>
<p>I know that if f is analytic in simply connected domain $D$, $C$ a simple closed positively oriented contour that lies in D and $z_o$ lies interior to $C$, then
$$\int_{C}\frac{f(z)}{z-z_o}dz=2\pi i f(z_o)$$</p>
<p>But for t... | Kaj Hansen | 138,538 | <p><strong>Hint</strong>:</p>
<p>Note that $z^2+9 = (z+3i)(z-3i)$. So with partial fraction decomposition, we get:</p>
<p>$$\frac{z^2}{(z^2+9)} = \frac{iz^2}{6(z+3i)} - \frac{iz^2}{6(z-3i)}$$</p>
<p>Therefore:</p>
<p>$$\int_C \frac{z^2}{z^2 + 9} \ dz = \int_C \frac{iz^2}{6(z+3i)} \ dz - \int_C \frac{iz^2}{6(z-3i)}... |
188,492 | <p>$A$ is an $n\times n$ matrix (not symmetric). If $\rho(A)$, spectral radius of $A$, is less than or equal to 1, can we say that $x^TAx\leq x^Tx$? </p>
<p>In another word,</p>
<p>if $\rho(A)\leq 1$, then $\frac{1}{2}\rho(A+A^T)\leq 1$?</p>
| copper.hat | 27,978 | <p>No. Choose $A$ to be a matrix of all zeroes except $[A]_{1n} = n+1$. Let $x = (1,...,1)^T$. Then $x^T A x = n+1$, but $x^T x = n$. The spectral radius is $\rho (A) = 0$.</p>
|
2,809,090 | <p>The stochastic vector $(X,Y)$ has a continuous distribution with pdf:
$$f(x,y) =
\begin{cases}
xe^{-x(y+1)} & \text{if $x,y>0$} \\[2ex]
0 & \text{otherwise}
\end{cases}$$<br>
Define $Z:=XY$.<br>
I would like to know what exactly $XY$ is. It seems to me that the function $f(x,y)$ has but one output, so wh... | Mythomorphic | 152,277 | <p><strong>Hint:</strong>
$$\cos A\cos B+\sin A\sin B=\cos(A-B)$$
$$\cos^2 A-\sin^2 A=\cos2A$$</p>
|
2,878,553 | <p>I was trying to think about what $\pi$ actually is. There are a lot of ways to get $\pi$ for example $4(1-\frac{1}{3}+\frac{1}{5}-\cdots)$.</p>
<p>But there is no one way to define it. </p>
<p>On the other hand a fraction like $\frac{1}{2}$ also has multiple definitions e.g. $\frac{2}{4}$ or $\frac{3}{6}$. Althoug... | YeatsL | 563,841 | <p>Just as rationals are defined by the equivalence class of ordered pairs of integers under the equivalence relation $\langle a, b\rangle \sim \langle c, d\rangle$ iff $a \cdot d = b \cdot c$, reals are defined as the equivalence class of Cauchy sequences of rational numbers under the equivalence relation $f : \mathbb... |
2,878,553 | <p>I was trying to think about what $\pi$ actually is. There are a lot of ways to get $\pi$ for example $4(1-\frac{1}{3}+\frac{1}{5}-\cdots)$.</p>
<p>But there is no one way to define it. </p>
<p>On the other hand a fraction like $\frac{1}{2}$ also has multiple definitions e.g. $\frac{2}{4}$ or $\frac{3}{6}$. Althoug... | Michael Hardy | 11,667 | <p>The answer is that with $\pi$ as with many mathematical concepts, there is no objective reason to single out one characterization as being "the definition". One establishes a definition within a context; for example one may write "For the purposes of this paper [or this textbook, or this web page, etc.] we take $\pi... |
306,788 | <p>I understand determinants but I cannot understand the following question, can someone explain it to me ?</p>
<p>Suppose that a $4 x 4$ matrix with rows $v_1,v_2,v_3$ and $v_4$ has determinant det A = -1. Find the following determinants: </p>
<p>$$det \begin{bmatrix}v_1\\6v_2\\v_3\\v_4 \end{bmatrix}=$$
$$det \beg... | Julien | 38,053 | <p>Hints:</p>
<ul>
<li>the determinant is multilinear on the rows.</li>
<li>the sign changes when you swap two rows.</li>
<li>the determinant is zero when two rows are the same</li>
</ul>
<p>With this, you can compute all three determinants in your question</p>
<p>Note: multilinear means $\det(ar_1+br_1',r_2,r_3,r_4... |
2,883,370 | <p>If I want to determine whether a sequence, ${a_n}$, is bounded above $\forall n \in \Bbb{N} $, is it enough to find a sequence that is larger than $a_n$, and show that it converges and is therefore bounded? For example:</p>
<p>$\forall n \in \Bbb{N}, let,$</p>
<p>$$
a_n = \frac{1}{n+1} + \frac{1}{n+2} + ...+\frac{... | miracle173 | 11,206 | <p>I think you mess up some ideas.</p>
<p>You say "and since $\lim\limits_{n\to\infty}1 = 1$", but you never showed that $\lim\limits_{n\to\infty}1 = 1$. And if you check the <a href="https://math.stackexchange.com/users/6460/henry">comment of Henry</a> this seems to be wrong. But you don't need the limes.</p>
<p>You... |
4,435,088 | <p>In an <span class="math-container">$8×8$</span> table one of the square is colored black and all the others are white . Prove that one cannot make all the boxes white by recoloring the rows and columns . "Recoloring" is the operation of changing the color of all boxes in a row or in a column .</p>
<p>This ... | user2661923 | 464,411 | <p>Your analysis for the <span class="math-container">$8 \times 8$</span> square amounts to saying that each operation leaves the parity of the total number of black squares unchanged. This assertion is accurate, and therefore, the analysis is valid. The assertion critically depends on the idea that <span class="math... |
2,418,916 | <blockquote>
<p>Find how many terms there are in this geometric sequence:</p>
<p><span class="math-container">$-1, 2, -4, 8, ..., -16777216$</span></p>
</blockquote>
<p>My attempt:</p>
<p><span class="math-container">$a_k=a.r^{k-1}$</span></p>
<p>And in this sequence:</p>
<p><span class="math-container">$a=-1$</span>, ... | farruhota | 425,072 | <p>It is:
$$-1,2,-4,8,...,(-1)\cdot 2^{24}$$</p>
|
3,896,817 | <blockquote>
<p>Solve <span class="math-container">$x^2 \equiv 12 \pmod {13}$</span></p>
</blockquote>
<p>By guessing I can say that the solutions are <span class="math-container">$5$</span> and <span class="math-container">$8$</span>, but is there another way to find the solution besides guessing?</p>
| Nathanael Skrepek | 423,961 | <p>As cosmo5 mentioned in his comment. You can do the following which is a little bit better than guessing since you probably can instantly recognise square numbers.
For every natural number <span class="math-container">$n$</span> we have
<span class="math-container">$$
x^2 \equiv 12 \mod 13
\quad\Leftrightarrow\quad
... |
2,732,562 | <p>I'm trying to figure out how to find the number of ternary strings of length $n$ that have 3 or more consecutive 2's.
So far I've been able to establish that there is $n(2^{n-1})$ with a single 2.
And I think (but am not certain) that this can be extrapolated to the number of strings with a single group of 2's of le... | epi163sqrt | 132,007 | <blockquote>
<p>We consider the alphabet $V=\{0,1,2\}$ and we are looking for the number $c_n$ of strings of length $n$ having runs of $2$ less than three. The wanted number is $$a_n=3^n-c_n$$</p>
<p>We derive a generating function $C(z)=\sum_{n=0}^\infty c_nz^n$ from which the number $a_n$ can be obtained ea... |
264,745 | <p>When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a distribution. I realized that while I know the definition of a function as a relation and I have an intuitive notion ... | Michael Hardy | 11,667 | <p>"$y$ is a function of $x$" means the value of $y$ is determined by that of $x$. For example, to say that the area of a circle is a function of the radius implies that all circles with the same radius have the same area.</p>
|
264,745 | <p>When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a distribution. I realized that while I know the definition of a function as a relation and I have an intuitive notion ... | Michael Bächtold | 1,984 | <p>Very much the <a href="https://mathoverflow.net/questions/307947">same question</a> was asked several years later on mathoverflow and received a few interesting answers, including one from a fields medalist.</p>
|
2,203,008 | <p>Consider the quadratic program</p>
<blockquote>
<p><span class="math-container">$ \min$</span> <span class="math-container">$ f(x) $</span></p>
<p><span class="math-container">$ \text{s.t.} \space Ax=c$</span></p>
</blockquote>
<p>Prove that <span class="math-container">$ x^* $</span> is a local minimum if and only ... | copper.hat | 27,978 | <p>If there is a feasible point $x_0$ then the feasible set is given by
$\{x_0\} + \ker A$ which can be written as $\{x_0 + Ly \}_y$ for some $L$ with ${\cal R} L = \ker A$.</p>
<p>Hence we can define $\phi(y) = f(x_0+Ly)$, note that $\phi$ is also
quadratic and the problem is to show that $y_0$ is a local $\min$
<str... |
389,460 | <blockquote>
<p>For $n>0$ let $A(n) = \underbrace{111 \ldots 11}_{n}$. Prove that if $A(n)$ is divisible by a prime number $p>3$, then $\gcd(n, p-1) > 1$.</p>
</blockquote>
<p>It is no huge discovery that if $n$ is even, then $2$ is a common divisor of $n$ and $p-1$, thus the implication holds. I don't know... | lsr314 | 66,964 | <p>If $10^n\equiv 1 \pmod p,(n,p-1)=1,(p,10)=1$,then there exist integer $x,y$, satisfies $xn+y(p-1)=1,$ hence
$$10^1=10^{xn+y(p-1)}=(10^n)^x\cdot(10^{p-1})^y\equiv1^x\cdot 1^y=1 \pmod p$$
We get $p=3.$</p>
|
3,191,402 | <p>I have tried to answer by taking change the variable <span class="math-container">$\theta$</span> to <span class="math-container">$\theta/2$</span>, so the integration is now over unit circle, then I have taken <span class="math-container">$z=e^{i\theta}$</span>. Now I tried to use residue formula for integration, b... | eyeballfrog | 395,748 | <p>Winther notes in the comments that integration by parts gives
<span class="math-container">$$
(n+1)I_{n+1} = (2n+1)a I_n - n(a^2-1)I_{n-1}
$$</span>
This is suspiciously close to the recursion relation for the Legendre Polynomials,
<span class="math-container">$$
(n+1)P_{n+1}(x) = (2n+1)x P_n(x) - nP_{n-1}(x).
$$</s... |
374,209 | <p>Need to show that if $f$ is a glide reflection then there is only one line $L$
such that $f(L) = L$</p>
<p>What I know is that a glide reflection is an isometry </p>
<p>$$f(z)=a\bar{z}+b,$$ such that $|a|=1$ and $a\bar{b}+b\neq0$.</p>
<p>Now assume that two lines $L_1$ and $L_2$ such that are axes for this glide ... | MvG | 35,416 | <p>What the proof idea in your question really shows is that $f$ is injective.</p>
<p>One problem here is the fact that it isn't easy to describe a “line” in the complex number plane. At least not as easy as say the real Euclidean plane. Most formalisms don't provide a nice way to compare lines for equality.</p>
<p>O... |
4,350,015 | <p>I'm trying to learn calculus through self study and I happened upon the following exercise:</p>
<p><span class="math-container">$$ \int_{0}^{2} \sqrt{4 - x^2}\cdot\operatorname{sgn}(x-1) \,dx $$</span></p>
<p>Seeing the sgn I thought: well this is easy and concluded that since <span class="math-container">$ \int_{0}... | José Carlos Santos | 446,262 | <p>That integral is equal to<span class="math-container">$$-\int_0^1\sqrt{4-x^2}\,\mathrm dx+\int_1^2\sqrt{4-x^2}\,\mathrm dx.$$</span>So, all you have to do is to compute those two integrals. In order to do that, it is usful to know that<span class="math-container">$$\int\sqrt{4-x^2}\,\mathrm dx=\frac12x\sqrt{4-x^2}+2... |
1,685,523 | <p>What is the domain of $f(x)=x^x$ ? </p>
<p>I used Wolfram alpha where it is said that the domain is all positive real numbers. Isn't $(-1)^{(-1)} = -1$ ? Why does the domain not include negative real numbers as well?</p>
<p>I also checked graph and its visible for only $x>0$ . Can someone help me clarify this?<... | John Bentin | 875 | <p>The expression $x^y$ can be assigned a reasonable meaning for all real $x$ and all rational numbers of the form $y=m/n$, where $m$ is even and $n$ is odd and positive. Thus $x^y=(x^m)^{1/n}$, interpreted as the unique real $n$th root of $x^m$ (define $0^0$ to be $1$). Since every real number can be arbitrarily well ... |
4,198,805 | <p>A <a href="https://en.wikipedia.org/wiki/Sober_space" rel="nofollow noreferrer">sober space</a> is a topological space such that every irreducible closed subset is the closure of exactly one point. Looking for examples I convinced myself that the following is true.</p>
<blockquote>
<p>Every finite <span class="math... | Eric Wofsey | 86,856 | <p>Suppose <span class="math-container">$X$</span> is a finite <span class="math-container">$T_0$</span> space and <span class="math-container">$A\subseteq X$</span> is an irreducible closed subset. Let <span class="math-container">$B\subset A$</span> be a maximal closed proper subset (which exists by finiteness), and... |
1,627,713 | <p>This is maybe math $101$ question:</p>
<p>Let $z_1=1+i$.</p>
<p>I know that $r=\sqrt 2$ and $\theta=\arctan(1/1)=\pi/4$ so $$z_1=\color{blue}{\sqrt 2e^{i\pi/4}} .$$</p>
<p>But now if I take a look at</p>
<p>$z_2=-1-i$,</p>
<p>I know that $r=\sqrt 2$ and $\theta=\arctan(-1/-1)=\pi/4$ so $$z_1=\color{blue}{\sqrt ... | Balarka Sen | 117,002 | <p>Path-connectedness, by definition, means for every pair of points $a, b$ in $\Bbb R^2 \setminus E$, there is a continuous map $f : [0, 1] \to \Bbb R^2 \setminus E$ with $f(0) = a$, $f(1) = b$. So what you said is not wrong.</p>
<p>To prove that such paths exist, note that there are uncountably many lines passing th... |
163,873 | <p>With the exception of a few miscellaneous cases, the axioms (and/or schemeta) of ZFC can roughly be divided into two kinds:</p>
<ol>
<li><p>Those that guarantee the existence of more complicated sets, given that simpler sets are already around (e.g. separation, replacement schema). Also, uniqueness of these entitie... | Monroe Eskew | 11,145 | <p>Axiom: $0^\sharp$ exists.</p>
<p>$0^\sharp$ is a pivotal principle in the large cardinal hierarchy, but it is actually a set of natural numbers. If it exists, it is unique.</p>
|
2,793,983 | <p>For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,</p>
<p>Is this the quickest way? </p>
<p>$x\in \left[ 1,50\right] \cap \mathbb{N} $</p>
<p>Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$,
$\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.</p>
| Tesla | 338,187 | <p>It depends on your own preference on how to write things down, there are countless variations, for example</p>
<p>$x \in \{ n \in \mathbb N : 1 ≤ n ≤ 50\}$</p>
<p>$x \in \{1,2,...,50\}$</p>
<p>$x \in \mathbb N_1^{50}$ </p>
|
1,627,619 | <p>Could anyone please check my solution to the following problem?</p>
<blockquote>
<p><strong>Problem:</strong> Let $f(x, y) = (x^2 + y^2)e^{-(x^2 + y^2)}$. Find global extrema of $f$ on $M = {\mathbf R}^2$.</p>
</blockquote>
<p><strong>Proposed solution:</strong> Taking partial derivatives of $f$, we conclude tha... | levap | 32,262 | <p>Looks good. Note that you can also start by translating the problem into a one-dimensional problem by letting $g(r) := re^{-r}$ and finding the global maximum and minimum of $g$ on $[0, \infty)$ and then note that $f(x,y) = g(x^2 + y^2)$ and conclude.</p>
|
60,152 | <p>What is the motivation behind topology?</p>
<p>For instance, in real analysis, we are interested in rigorously studying about limits so that we can use them appropriately. Similarly, in number theory, we are interested in patterns and structure possessed by algebraic integers and algebraic prime numbers.</p>
<p>So... | Matt E | 221 | <p><em>Topology</em> can mean different things in mathematics, depending on the context. </p>
<p>If a mathematician describes themselves as a <em>topologist</em>, this likely means that they study various kinds of shapes (technically, <em>manifolds</em>, or related kinds of spaces), with an eye perhaps to classifying ... |
2,567,607 | <p>$$\arctan 2x +\arctan 3x = \left(\frac{\pi}{4}\right)$$
$$\arctan \left(\frac{2x+3x}{1-2x*3x}\right)=\frac {\pi}{4}$$
$$\frac {5x}{1-6x^2}=\tan \frac{\pi}{4}=1$$
$$6x^2 + 5x -1 = 0$$
$$(6x-1)(x+1)=0$$
$$x=-1, \frac{1}{6}$$</p>
<p>The answer however rejects the solution $x=-1$ saying that it makes the L.H.S of the e... | Vivek | 546,817 | <p>Concep: Let $f$ be a function, which means $$x_1=x_2\Rightarrow f(x_1)=f(x_2)$$
But note that if $f(x_1)=f(x_2)$, then $x_1=x_2$ is one of the possibility, among perhaps many possibilities if the function is not injective</p>
<p>Now Given $\tan^{-1}(2x)+\tan^{-1}(3x)=\frac{\pi}{4}$, taking $\tan$ of both sides(sin... |
1,307,269 | <p>In my textbook, before introducing the epsilon delta definition, they gave a working definition of what a limit is. The definition sounded something like this "$\lim \limits_{x \to a}f(x) = L$, if when $x$ gets closer to $a$, $f(x)$ gets closer to $L$" </p>
<hr>
<p>But is that always the case with limits? What if... | Community | -1 | <p>Probably a better intuitive definition is $f(x)$ can be made arbitrarily close to $L$ by making $x$ close enough to $a$.</p>
<p>You avoid the awkwardness about the constant case. Additionally this emphasizes that it is for ALL $\epsilon$. It's not just that $f(x)$ gets "closer", it's that it can be made as close as... |
1,740,458 | <p>I saw somewhere on Math Stack that there was a way of finding integrals in the form $$\int \frac{dx}{a+b \cos x}$$ <strong>without</strong> using Weierstrass substitution, which is the usual technique. </p>
<p>When $a,b=1$ we can just multiply the numerator and denominator by $1-\cos x$ and that solves the problem ... | C. Dubussy | 310,801 | <p>One usual trick is the substitution $x=2y$. Note that $$\frac{1}{a+b\cos(2y)}=\frac{1}{a+b(2\cos^2(y)-1)}=\frac{\sec^2(y)}{2b+(a-b)\sec^2(y)}=\frac{\sec^2(y)}{(a+b)+(a-b)\tan^2(y)}.$$ Hence $$\int \frac{dx}{a+b\cos(x)}=\int \frac{\sec^2(y)}{(a+b)+(a-b)\tan^2(y)} \, dy.$$ Now conclude with the substitution $t=\tan(y)... |
19,325 | <p>I'm looking for a simple way to define mathematics to primary/elementary school teachers and explain some of the confusion children have.</p>
<p>I'm hoping some Algebraist could help me properly state the following:</p>
<blockquote>
<p>A number in and of itself has no true meaning. True in the sense that it relates ... | user52817 | 1,680 | <p>I highly recommend the book <em>The Number Sense: How the Mind Creates Mathematics</em>, by Stanislas Dehaene and published by Oxford University Press. Another book that comes to mind is <em>The Language Instinct: How The Mind Creates Language</em>, by Steven Pinker. Both books have had broad impact on scholarship ... |
56,847 | <p>What are the angles formed at the center of a tetrahedron if you draw lines to the vertices?</p>
<p>I'm trying to make these:</p>
<p><img src="https://i.stack.imgur.com/FRUi8.jpg" alt="caltrop"> </p>
<p>I need to know what angles to bend the metal.</p>
| Nick Alger | 3,060 | <p>The angle is
<span class="math-container">$$\theta = \arccos \left(-\frac{1}{3}\right) \approx 109.47.$$</span>
You can find this angle by writing the volume of a regular tetrahedron in two different ways.</p>
<p>(this is an adaptation of a blog post I wrote for a website that has since been deleted)</p>
<hr>
<p... |
1,822,336 | <p>My friend asked me what the roots of $y=x^3+x^2-2x-1$ was.</p>
<p>I didn't really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots were $2\cos\left(\frac {2\pi}{7}\right), 2\cos\left(\frac {4\pi}{7}\right)$ and $2\cos\left(\frac {8\pi}{7}\ri... | David Quinn | 187,299 | <p>Consider the equation $$\cos4\theta=\cos3\theta$$ whose roots are $$\theta=n\cdot\frac{2\pi}{7}$$</p>
<p>Representing this as a polynomial in $c=\cos\theta$, we have $$8c^4-4c^3-8c^2+3c+1=0$$
$$\Rightarrow (c-1)(8c^3+4c^2-4c-1)=0$$</p>
<p>Now write $x=2c$ and we see that the polynomial equation $$x^3+x^2-2x-1=0$$ ... |
3,001,335 | <p>I know little formal math terminology and don't understand much of anything about complex analysis. Also, if this isn't a good starting point for complex integration feel free to say (I'm learning about it partly for Cauchy's residue theorem). </p>
<p>My first and intuitive idea of residue has to do with remainder,... | Nodt Greenish | 614,664 | <p>The residue (latin residuere - remain) is named that way because
<span class="math-container">$\frac{1}{2\pi i}\int_{|w-z_0|=r}f(w)dw=\sum_{n=-\infty}^{\infty} a_n\int_{|w-z_0|=r}(w-z_0)^n\,dw= a_{-1}$</span> is what remains after integration.</p>
|
254,030 | <p>I am following a course in basic algebra, and we have covered rings & groups in class, but I am having trouble visualising them. Are there applications of group &/or ring theory that can be more easily visualized than the abstract object? For instance, are there objects, or properties of objects, that behave... | Bill Dubuque | 242 | <p>Group theory may be viewed roughly as a general study of symmetry. In chemistry this applies to crystals via the study of crystallographic groups, and in art via <a href="http://en.wikipedia.org/wiki/Wallpaper_group" rel="nofollow">wallpaper groups.</a> For an example in physics, the Lie symmetry groups of partial ... |
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