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 |
|---|---|---|---|---|
1,995,663 | <p>My brother in law and I were discussing the four color theorem; neither of us are huge math geeks, but we both like a challenge, and tonight we were discussing the four color theorem and if there were a way to disprove it.</p>
<p>After some time scribbling on the back of an envelope and about an hour of trial-and-e... | dtldarek | 26,306 | <p>Starting at the top, going clockwise:</p>
<ul>
<li>center: 1, 2, 3</li>
<li>middle: 2,4,3,4,2,4</li>
<li>outside: 1</li>
</ul>
<p><a href="https://i.stack.imgur.com/0gjxt.png" rel="noreferrer"><img src="https://i.stack.imgur.com/0gjxt.png" alt="4colors"></a></p>
<p>I hope this helps $\ddot\smile$</p>
|
4,614,334 | <p>I'm trying to resolve this recurrence equation <span class="math-container">$T(n)=4T(\frac{n}{2})+cn$</span>.</p>
<p>The solution I discovered online is <span class="math-container">$T(n)=\theta(n^2)$</span>.</p>
<p>The steps I follow was these: <br/>
a) Create the tree of recurrence as this:</p>
<pre><code> ... | Abezhiko | 1,133,926 | <p>I'm not that familiar with the recursive tree method, but it seems that you only consider the longest path, when you should have calculated the total cost, which takes the form of a partial geometric series :<br />
<span class="math-container">$$
\sum_{k=0}^{\log_2(n)}2^kcn = cn\frac{2^{\log_2(n)+1}-1}{2-1} = cn(2n-... |
3,251,851 | <p>I need help solving these simultaneous equations:</p>
<p><span class="math-container">$$a^2 - b^2 = -16$$</span>
<span class="math-container">$$2ab = 30$$</span></p>
| John Hughes | 114,036 | <p>Let's concentrate on the reals, in fact on <span class="math-container">$\Bbb R^3$</span>. The vectors are column vectors containing three real numbers. Because <span class="math-container">$0$</span> and <span class="math-container">$1$</span> are special real numbers, it turns out to be really nice to work with ve... |
2,835,474 | <p>What is linear about a linear combination of things?. In linear algebra, the "things" we are dealing with are usually vectors and the linear combination gives the span of the vectors. Or it could be a linear combination of variables and functions. But why not just call it combination. Why is the term "linear" includ... | mweiss | 124,095 | <p>The word "linear" has two distinct senses, one <em>geometric</em> and one <em>algebraic</em>. Linear combinations are linear in <em>both</em> senses, which is why the phrase is so apt.</p>
<p><strong>Geometric Linearity</strong></p>
<p>Let's take the geometric sense first, because that is the one that has not yet... |
2,520,044 | <p>$$\lim_{x\to2}{\frac{\sqrt{3x-2}-\sqrt{5x-6}}{\sqrt{2x-1}-\sqrt{x+1}}}$$</p>
<p>Evaluate the limit.</p>
<p>Thanks for any help</p>
| Paolo Intuito | 395,372 | <p>That stinks a lot as homework. A general hint for solving such a problem is to "rationalize" both numerator and denominator i.e. multiply the whole thing by</p>
<p>$$\frac{\sqrt{3x-2}+\sqrt{5x-6}}{\sqrt{3x-2}+\sqrt{5x-6}}\cdot \frac{\sqrt{2x-1}+\sqrt{x+1}}{\sqrt{2x-1}+\sqrt{x+1}}$$</p>
|
2,595,418 | <p>Let there be a graph $G$ and it's complement $G'$ , if the degree of a vertex in $G$ is added with degree of the corresponding vertex of $G'$ , the sum will be $(n-1)$; where $n$ is the number of vertices. How to prove this ?</p>
| Rebecca J. Stones | 91,818 | <p>Given a vertex $u$, there are three types of vertices:</p>
<ul>
<li>the vertices which are adjacent to $u$,</li>
<li>the vertices which are <strong>not</strong> adjacent to $u$ (excluding $u$ itself), and</li>
<li>$u$.</li>
</ul>
<p>I depict this below:</p>
<p><a href="https://i.stack.imgur.com/Ta8N1.png" rel="no... |
3,047,670 | <blockquote>
<p>Prove or disprove that <span class="math-container">$$\lim_{n\to \infty} \left(\frac{x_{n+1}-l}{x_n-l}\right)=\lim_{n\to\infty}\left(\frac{x_{n+1}}{x_n}\right)$$</span> where <span class="math-container">$l=\lim_{n\to \infty} x_n$</span></p>
</blockquote>
<p>I think that the above result is true,but ... | Math-fun | 195,344 | <p>Let <span class="math-container">$x_n=\frac{2^{n+1}-1}{2^n}$</span>, for which <span class="math-container">$l=2$</span> and <span class="math-container">$$\lim\frac{x_{n+1}}{x_n}=1$$</span> and <span class="math-container">$$\lim\frac{x_{n+1}-2}{x_n-2}=\frac12.$$</span></p>
|
14,140 | <p>One of the most annoying "features" of <em>Mathematica</em> is that the <code>Plot</code> family does extrapolation on <code>InterpolatingFunction</code>s without any warning. I'm sure it was discussed to hell previously, but I cannot seem to find any reference. While I know how to simply overcome the problem by def... | Simon Woods | 862 | <p>As Rojo showed, the symbol <code>$Messages</code> appears to become unset somewhere inside the <code>Plot</code> internals. I wondered if it was possible to prevent this by setting the <code>Protected</code> and <code>Locked</code> attributes for <code>$Messages</code>, but it would seem that <code>Plot</code> has m... |
188,158 | <p>I am interested in a function such that <code>f[m, i] = n</code> where <code>m, n</code> are positive integers and <code>n</code> is the <code>i</code>-th number relatively prime with <code>m</code>.</p>
<p>Getting a sample of the possible outputs of <code>f</code> is straightforward. For example, let <code>m = 30<... | Αλέξανδρος Ζεγγ | 12,924 | <p>I give a naive implementation</p>
<pre><code>ithCoprime[m_, i_] := Module[{coprimes, j = 1},
coprimes = {1};
While[Length[coprimes] < i,
j++;
If[CoprimeQ[m, j], AppendTo[coprimes, j]]
... |
188,158 | <p>I am interested in a function such that <code>f[m, i] = n</code> where <code>m, n</code> are positive integers and <code>n</code> is the <code>i</code>-th number relatively prime with <code>m</code>.</p>
<p>Getting a sample of the possible outputs of <code>f</code> is straightforward. For example, let <code>m = 30<... | KennyColnago | 3,246 | <p>To find relative primes, I've found <code>Complement</code> to be generally faster than <code>GCD</code> or <code>CoprimeQ</code>.</p>
<pre><code>RelativePrimes[m_Integer] :=
Complement[
Range[m - 1],
Apply[Sequence, Map[Range[#, m - 1, #] &, FactorInteger[m][[All, 1]]]]]
</code></pre>
<p>Your f... |
2,534,999 | <p>I tried to solve $z^3=(iz+1)^3$. I noticed that $(iz+1)^3=i(z-1)^3$ so $(\frac{z-1}{z})^3=i$. How to finish it?</p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p>Factor as</p>
<p>$$z^3-(iz+1)^3=(z-(iz+1))(z^2+z(iz+1)+(iz+1)^2)$$ and you have a linear and a quadratic equation.</p>
<p>Alternatively, considering the three cubic roots of unity, solve</p>
<p>$$z=\sqrt[3]1(iz+1).$$</p>
|
865,598 | <p>How can I calculate this value?</p>
<p>$$\cot\left(\sin^{-1}\left(-\frac12\right)\right)$$</p>
| Thoth19 | 76,241 | <p>You should probably have memorized things like the sine of 30 degrees. We therefore know that $sin(30) = 0.5$ So $arcsin(-1/2)=-30$ degrees
Now we want to take the cotangent of that. Well Cotangent is cosine over sine.
$cos(-30) = cos(30) = \sqrt(3)/2$
$sin(-30)=-sin(30)=-1/2$
Thus, the final answer is$-\sqrt(3)$</p... |
865,598 | <p>How can I calculate this value?</p>
<p>$$\cot\left(\sin^{-1}\left(-\frac12\right)\right)$$</p>
| Steven Alexis Gregory | 75,410 | <p><a href="https://i.stack.imgur.com/JAVKR.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JAVKR.jpg" alt="enter image description here"></a></p>
<p>Arcsine is defined in the first and the fourth quadrants. If
<span class="math-container">$\theta = \arcsin\left(-\dfrac 12 \right)$</span>, then <sp... |
3,773,856 | <p>I'm having trouble with part of a question on Cardano's method for solving cubic polynomial equations. This is a multi-part question, and I have been able to answer most of it. But I am having trouble with the last part. I think I'll just post here the part of the question that I'm having trouble with.</p>
<p>We ha... | Paul Frost | 349,785 | <p>Let <span class="math-container">$w(\alpha) = \cos \alpha + i\sin \alpha$</span>. Then
<span class="math-container">$$w(\alpha) w(\beta) = (\cos\alpha + i \sin \alpha)(\cos \beta + i\sin \beta) \\ =\cos\alpha \cos \beta - \sin \alpha \sin \beta +i(\cos\alpha \sin \beta + \sin \alpha \cos \beta) = \cos(\alpha + \beta... |
2,972,355 | <p>How to convert sentence that contains “no more than 3” into predicate logic sentence?</p>
<p>For example: "No more than three <span class="math-container">$x$</span> satisfy <span class="math-container">$R(x)$</span>"
using predicate logic. </p>
<p>This is what I have for "exactly one <span class="math-container"... | Anguepa | 196,351 | <p><span class="math-container">$$
\forall x \forall y \forall z \forall u ((R(x)\wedge R(y) \wedge R(z) \wedge R(u)) \rightarrow (x=y \vee x=z \vee x=u \vee y=z \vee y=u \vee z=u))
$$</span></p>
|
756,236 | <p>The question is to write the general solution for this recurrence relation:</p>
<p>$y_{k+2} - 4y_{k+1} + 3y_{k} = -4k$.</p>
<p>I first solved the homogeneous equation $y_{k+2} - 4y_{k+1} + 3y_{k} = 0$, by writing the auxiliary equation $r^2 - 4r + 3 = (r-3)(r-1) = 0$. Thus $y_k^{h} = c_1(1)^k + c_2 (3)^k$. The gen... | Mark Bennet | 2,906 | <p>Your error is to put $b=2k$, when $b$ is a constant. Notice that one of the solutions you have to the homogeneous equation is $1^k$. This means that you have to increase the degree of your particular solution and try $a+bk+ck^2$.</p>
<p>Because $1$ is a root of the auxiliary equation, the term in $a$ will simply go... |
756,236 | <p>The question is to write the general solution for this recurrence relation:</p>
<p>$y_{k+2} - 4y_{k+1} + 3y_{k} = -4k$.</p>
<p>I first solved the homogeneous equation $y_{k+2} - 4y_{k+1} + 3y_{k} = 0$, by writing the auxiliary equation $r^2 - 4r + 3 = (r-3)(r-1) = 0$. Thus $y_k^{h} = c_1(1)^k + c_2 (3)^k$. The gen... | vonbrand | 43,946 | <p>As a side remark, it is best to use generating functions and solve the equation in one go. Define $g(z) = \sum_{k \ge 0} y_k z^k$, multiply the recurrence by $z^k$, sum over $k \ge 0$, and recognize a few sums:
\begin{align}
\sum_{k \ge 0} y_{k + r} z^k
&= \frac{g(z) - y_0 - y_1 z - \ldots - y_{r - 1} z^{r - ... |
4,252,431 | <p>Let the constant <span class="math-container">$\alpha > 0$</span> be the problem
<span class="math-container">$$\left\{\begin{array}{cll}
u_t + \alpha u_x & = & f(x,t); \ \ 0 < x < L; \ t > 0\\
u(0,t) & = & 0; \ \ t > 0;\\
u(x,0) & = & 0; \ \ 0 < x < L.
\end{array}\right.... | José Carlos Santos | 446,262 | <p>The approach is fine, but since you did not show us your computations, I cannot tell you whether or not the full solution is correct.</p>
<p>Here's how I would do it. Note that<span class="math-container">\begin{align}x+xy^2=y+yx^2&\iff x-y=yx^2-xy^2\\&\iff x-y=xy(x-y)\end{align}</span>and so if <span class=... |
4,252,431 | <p>Let the constant <span class="math-container">$\alpha > 0$</span> be the problem
<span class="math-container">$$\left\{\begin{array}{cll}
u_t + \alpha u_x & = & f(x,t); \ \ 0 < x < L; \ t > 0\\
u(0,t) & = & 0; \ \ t > 0;\\
u(x,0) & = & 0; \ \ 0 < x < L.
\end{array}\right.... | Donald Splutterwit | 404,247 | <p>Square the equation <span class="math-container">$x(1+y^2)=y(1+x^2)$</span> and we have (Note that <span class="math-container">$2x^2y^2$</span> will cancel)
<span class="math-container">\begin{eqnarray*}
x^2(1+y^4)=y^2(1+x^4).
\end{eqnarray*}</span>
Now divide by the first equation and we have <span class="math-con... |
1,767,682 | <p>I was thinking about sequences, and my mind came to one defined like this:</p>
<p>-1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, ...</p>
<p>Where the first term is -1, and after the nth occurrence of -1 in the sequence, the next n terms of the sequence are 1, followed by -1, and so on. Which led me to perhaps a str... | André Nicolas | 6,312 | <p>The <a href="https://en.wikipedia.org/wiki/Ces%C3%A0ro_mean" rel="nofollow">Cesaro mean</a> accomplishes something that is close to your intuition.</p>
<p>For the sequence $a_1,a_2,a_3,\cdots$, the Cesaro mean is the limit, if it exists, of the sequence $(b_n)$, where $b_n=\frac{a_1+\cdots+a_n}{n}$.</p>
<p>If the ... |
3,356,951 | <p>On SAT,scores range from 2000 to 2400, with two thirds of the scores falling in the range of 2200 to 2300. If we further assume that test scores are normally distributed in this range from 2000 to 2400, determine the mean and standard deviation.</p>
| BruceET | 221,800 | <p>I believe there is a mistake in the statement of your problem.
By symmetry, and for things to work out as intended '2200' has to be '2100'.</p>
<p>The Empirical Rule says that 68% of observations fall within one standard deviation of the mean. What you request can be done
only approximately, so we will use 68% inst... |
2,810,008 | <p>Can I investigate this limit and if yes, how? $${i^∞}$$</p>
<p>I am at a loss of ideas and maybe it is undefined?</p>
| Rhys Hughes | 487,658 | <p>The convergence of certain sequences is impossible to find if the range is limited to a cycle of a few values. In this case $i^n$ is equal to either $\pm1$ or $\pm i$ with no other values. $\infty$ being the <em>concept</em> that it is makes it impossible to determine which of the four solutions would be $\lim_{n\to... |
2,375,529 | <p>Let $H$ be a Hilbert space and let $T\in \mathcal{B}(H)$ such that $T$ is self-adjoint. I want to show that if $T$ is non-zero, then $T^n\neq 0$ for all $n\in \mathbb{N}$.</p>
<p>Suppose $n$ be the least positive integer such that $T^n=0$. Then for all $x,y\in H$, we have $\langle T^nx,y\rangle=0\implies \langle T^... | Michael L. | 153,693 | <p>This may be a bit heavy-handed, but we can consider the spectral theorem for bounded self-adjoint operators on Hilbert spaces:</p>
<p>For self-adjoint $T\in \mathcal{L}(H, H)$, there is an $L^2(X, \mathcal{M}, \mu)$, a unitary $U : L^2(X, \mathcal{M}, \mu)\to H$, and an essentially bounded measurable function $f : ... |
4,631,618 | <p>Consider this absolute value quadratic inequality</p>
<p><span class="math-container">$$ |x^2-4| < |x^2+2| $$</span></p>
<p>Right side is always positive for all real numbers,so the absolute value is not needed.</p>
<p>Now consider the cases for the left absolute value</p>
<ol>
<li><span class="math-container">$$... | Stas Volkov | 1,101,398 | <p>Let</p>
<p><span class="math-container">$$f(x)=\sum_{n=1}^\infty \frac{x^n}{(2n-2)!}=\frac{x(e^{\sqrt x}+ e^{-\sqrt x })}2$$</span></p>
<p>Then</p>
<p><span class="math-container">$$x f'(x)=x \sum_{n=1}^\infty \frac{n x^{n-1}}{(2n-2)!}=x \sum_{n=1}^\infty \frac{n x^n}{(2n-2)!}$$</span></p>
<p>and</p>
<p><span class=... |
2,938,424 | <p>Explain why the equation </p>
<p><span class="math-container">$$x^3 - 15x +1 = 0$$</span></p>
<p>has at least three solutions in the interval [-4,4].</p>
<p>My thoughts:</p>
<p><span class="math-container">$$f(x) = x^3 - 15x + 1$$</span>
<span class="math-container">$$f(-4) = -3 $$</span>
<span class="math-conta... | Will Jagy | 10,400 | <p>You may have taken a transpose where not appropriate. In both cases, four columns, the vector created by summing <span class="math-container">$w c_1 + x c_2 + y c_3 + z c_4$</span> is exactly the result of multiplying the matrix on the left times the column vector
<span class="math-container">$$
V =
\left(
\begin{ar... |
2,938,424 | <p>Explain why the equation </p>
<p><span class="math-container">$$x^3 - 15x +1 = 0$$</span></p>
<p>has at least three solutions in the interval [-4,4].</p>
<p>My thoughts:</p>
<p><span class="math-container">$$f(x) = x^3 - 15x + 1$$</span>
<span class="math-container">$$f(-4) = -3 $$</span>
<span class="math-conta... | egreg | 62,967 | <p>If you perform Gaussian elimination on the matrix with the given vectors as <em>columns</em>, then the columns with no leading <span class="math-container">$1$</span> can be removed.</p>
<p><span class="math-container">\begin{align}
\begin{bmatrix}
2 & 1 & 0 & 4 \\
-1 & 1 & 1 & -1 \\
2 &... |
4,451,894 | <p><strong>Problem</strong><br />
There is a knight on an infinite chessboard. After moving one step, there are <span class="math-container">$8$</span> possible positions, and after moving two steps, there are <span class="math-container">$33$</span> possible positions. The possible position after moving n steps is <sp... | Community | -1 | <p>Mordechai Katzman demonstrates in section <span class="math-container">$3$</span> of his paper <a href="https://arxiv.org/abs/math/0504113" rel="nofollow noreferrer">Counting monomials</a> (pages <span class="math-container">$5$</span> - <span class="math-container">$8$</span>) that</p>
<p><span class="math-containe... |
4,250,845 | <p>Considering a Quadrilateral <span class="math-container">$ABCD$</span> where <span class="math-container">$A(0,0,0), B(2,0,2), C(2,2\sqrt 2,2), D(0,2\sqrt2,0)$</span>. Basically I have to find the <strong>Area</strong> of <strong>projection</strong> of quadrilateral <span class="math-container">$ABCD$</span> on the ... | sirous | 346,566 | <p>You can also first find the projection of points and then calculate the area.</p>
<p>Hints for finding the projection of a point:</p>
<p><span class="math-container">$x+y-z=3$</span></p>
<p>So the normal vector is <span class="math-container">$N(1, 1, -1)$</span></p>
<p>the projection of any point on this plane is t... |
4,328,630 | <p>While preparing for a midterm, I came across this question</p>
<blockquote>
<p>Suppose a restaurant is visited by 10 clients per hour on average, and clients follow a homogeneous Poisson Process. Independantly of other client, each client has a 20% chance to eat here and 80% to take away. In average, how many client... | greg | 357,854 | <p><span class="math-container">$
\def\bbR#1{{\mathbb R}^{#1}}
\def\d{\delta}
\def\k{\sum_k}
\def\l{\sum_l}
\def\e{\varepsilon}
\def\n{\nabla}\def\o{{\tt1}}\def\p{\partial}
\def\E{{\cal E}}\def\F{{\cal F}}\def\G{{\cal G}}
\def\B{\Big}\def\L{\left}\def\R{\right}
\def\LR#1{\L(#1\R)}
\def\BR#1{\B(#1\B)}
\def\vecc#1{\opera... |
270,410 | <p>I have simplified the equations and decrease the variables to 5, and changed the parameters' value as I think the equations in <a href="https://mathematica.stackexchange.com/questions/270375/findrootjsing-encountered-a-singular-jacobian-at-the-point-when-solving-nonli">enter link description here</a> is because of t... | Ulrich Neumann | 53,677 | <p>Try <code>Reduce</code> to analyze the nonlinear equations:</p>
<pre><code>Reduce[equa /. parameters, vars]
(* False*)
</code></pre>
<p>The result confirms @Bill's helpful comment!</p>
|
2,629,408 | <p>How to evaluate this given expression?
$$\int\frac{du}{\sqrt{9e^{-2u}-1}}$$
I got so many tries but I'm not sure of my answer because somebody said that it was wrong, they told me that I used a wrong formula applied!
That's why I ask a support here I want correct explanation and answer of this given!</p>
<p>Thanks!... | D F | 501,035 | <p>try to substitute $\frac{1}{3}e^{u} = cost$ then $du = -tg(t)dt$. And the integral will be $\int \frac{-tgtdt}{\sqrt{\frac{1}{cos(t)^2} - 1}}$ then use that $(cos(t))^2 + (sin(t))^2 = 1$</p>
|
3,454,095 | <p>Minimize <span class="math-container">$\;\;\displaystyle \frac{(x^2+1)(y^2+1)(z^2+1)}{ (x+y+z)^2}$</span>, if <span class="math-container">$x,y,z>0$</span>.
By setting gradient to zero I found <span class="math-container">$x=y=z=\frac{1}{\displaystyle\sqrt{2}}$</span>, which could minimize the function.</p>
<bl... | dezdichado | 152,744 | <p>If you want some calculus/analysis argument:</p>
<p>After establishing there must exist a global minimum, let <span class="math-container">$p$</span> be the global minimum. Then we must have that
<span class="math-container">$$f(x) = x^2\left((y^2+1)(z^2+1) - p\right) - 2xp(y+z) + (y^2+1)(z^2+1) - p(y+z)^2\geq 0$$<... |
2,078,535 | <p>I'm kinda new with this and find hard to solve the problems related to LA although I can visually and conceptualize stuff easily. Please help me to find $\{(v_1, v_2, v_3) \in \Bbb R^3 \mid 5v_1 - 3v_2 + 2v_3 = 0\}$.</p>
| Noble Mushtak | 307,483 | <p>We want to solve the system represented by the following augmented matrix:
$$[5 \ -3 \ 2 \ | \ 0]$$
To get this into RREF form, divide the row by $5$:
$$[1 \ -\frac{3}{5} \ \frac{2}{5} \ | \ 0]$$
Now, the pivot variable is the first coordinate and the free variables are the second and third coordinate. From the seco... |
2,247,522 | <p>Suppose $X$ and $Y$ are discrete random variables. Show that $$E(X \mid Y)=E(X \mid Y^3).$$</p>
<p>The conditional expected value of a discrete random variable is expressed as
$$E(X \mid Y)=\sum xp_{X \mid Y}(x \mid y),$$
where$$p_{X \mid Y}(x \mid y)=\frac{p_{X,Y}(x,y)}{p_Y(y)}.$$ </p>
<p>Similarly, you can say t... | Amit | 378,131 | <p>Let $S$ denote the sample space. Notice that both $\mathbb{E}(X|Y):S\rightarrow \mathbb{R}$ and $\mathbb{E}(X|Y^3):S\rightarrow \mathbb{R}$ are random variables, and the following holds:</p>
<p>$\mathbb{E}(X|Y)(s) = \mathbb{E}(X|Y = Y(s)) = \mathbb{E}(X|Y^3 = (Y(s))^3) = \mathbb{E}(X|Y^3 = Y^3(s))= \mathbb{E}(X|Y^3... |
23,268 | <p>I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and m... | Nicolas Ford | 5,281 | <p>The way I think about limits and colimits is in terms of the most elementary examples of each: (co)products and (co)equalizers. Since any (co)limit can be built out of these, this is technically enough, and anyway I think it does give something of a feel for what the object should be. (I'm not being very picky about... |
67,460 | <p>Denote the system in $GF(2)$ as $Ax=b$, where:
$$
\begin{align}
A=&(A_{ij})_{m\times m}\\
A_{ij}=&
\begin{cases}
(1)_{n\times n}&\text{if }i=j\quad\text{(a matrix where entries are all 1's)}\\
I_n&\text{if }i\ne j\quad\text{(the identity matrix)}
\end{cases}
\end{align}
$$
that is, $A$ i... | user1551 | 1,551 | <p>Edit: I posted a wrong answer earlier. Hope I can get things fixed this time.</p>
<p>For convenience, write $x^T = (x_1^T, \ldots, x_m^T)$ where each $x_i$ is a vector of length $n$. Similarly, write $b^T = (b_1^T, \ldots, b_m^T)$. Let $J$ and $u$ be respectively the $n$-by-$n$ matrix and $n$-vector with all entrie... |
3,998,098 | <p>I was asked to determine the locus of the equation
<span class="math-container">$$b^2-2x^2=2xy+y^2$$</span></p>
<p>This is my work:</p>
<blockquote>
<p>Add <span class="math-container">$x^2$</span> to both sides:
<span class="math-container">$$\begin{align}
b^2-x^2 &=2xy+y^2+x^2\\
b^2-x^2 &=\left(x+y\right... | Quanto | 686,284 | <p>With variable changes
<span class="math-container">$$x= \sqrt{ \frac{5+\sqrt5}2}u + \sqrt{ \frac{5-\sqrt5}2}v,\>\>\>\>\>
y= \sqrt{ \frac{5-\sqrt5}2}u - \sqrt{ \frac{5+\sqrt5}2}v
$$</span>
the curve equation <span class="math-container">$b^2-2x^2=2xy+y^2$</span> can be recast as</p>
<p><span class="... |
257,821 | <p>The Kullback-Liebler divergence between two distributions with pdfs $f(x)$ and $g(x)$ is defined
by
$$\mathrm{KL}(F;G) = \int_{-\infty}^{\infty} \ln \left(\frac{f(x)}{g(x)}\right)f(x)\,dx$$</p>
<p>Compute the Kullback-Lieber divergence when $F$ is the standard normal distribution and $G$
is the normal distribution ... | Timmmm | 60,289 | <p>Ok, I've been searching for ages for the Kullback-Liebler divergence between two Normal distributions and didn't find it, but RS's answer enabled me to calculate it quite simply. Here's my derivation. I've also derived it for Beta distributions.</p>
<h1>Kullback-Liebler divergence of Normal Distributions</h1>
<p>S... |
204,842 | <p>A probability measure defined on a sample space $\Omega$ has the following properties:</p>
<ol>
<li>For each $E \subset \Omega$, $0 \le P(E) \le 1$</li>
<li>$P(\Omega) = 1$</li>
<li>If $E_1$ and $E_2$ are disjoint subsets $P(E_1 \cup E_2) = P(E_1) + P(E_2)$</li>
</ol>
<p>The above definition defines a measure that... | Michael | 155,065 | <p>This question was from years ago, but I was just about to ask a similar question (I found this page from the stackexchange list of similar questions). My own question is whether it is possible to have an <em>explicit</em> example. The answers above are all non-explicit. Here is another (non-explicit) answer in a d... |
419,625 | <p>Please help me, I have two functions:</p>
<blockquote>
<pre><code>a := x^3+x^2-x;
b := 20*sin(x^2)-5;
</code></pre>
</blockquote>
<p>and I would like to change a background color and fill the areas between two curves. I filled areas but I dont know how can I change the background, any idea?</p>
<blockquote>
<pre>... | Sujaan Kunalan | 77,862 | <p>Hmm, the closest thing I could find after a quick search on Maple Help was the ability to choose a colour between the curve and the x-axis. I don't know how helpful that would be to you though.</p>
<p>Alternatively, maybe you could try to make two plots- one your actual plot and the other a plot of solid colour for... |
2,363,733 | <p>I saw this notation $V= V_1\otimes V_2$ in a survey on universal algebra, where $V$ was a variety, but the survey in question didn't define this notation. Could anyone explain what it means ?</p>
| Eran | 70,857 | <p>I would have to read the survey in question to be sure, but I have most commonly seen this notation used when talking about decidable varieties.</p>
<p>You can find a definition here: <a href="https://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/DecidVar.pdf" rel="nofollow noreferrer">https://www.math.uwat... |
2,329,730 | <p>Let $G$, an algebraic group, act morphically on the affine variety $X$.</p>
<p>Then we can also have $G$ act on the affine algebra $K[X]$ as follows:
$$\tau_x(f(y))=f(x^{-1}\cdot y),\qquad (x\in G, y\in X)$$</p>
<p>Then $\tau:G\to GL(K[X]),\quad \tau:x\mapsto \tau_x$.</p>
<p>Humphreys says that the reason that th... | Björn Friedrich | 203,412 | <p>From $\color{blue}{27x} = \color{blue}{b}$ and $\color{blue}{b}x = 1024$ it follows that
$$
\color{blue}{(27x)}x = 1024 \;.
$$
If you now divide both sides by $27$, you will see that this expression is equivalent to
$$
x^2 = \dfrac{1024}{27} \;.
$$
Now you take the positive and the negative root, and you will obtain... |
4,201,477 | <blockquote>
<p>Integrate <span class="math-container">$$\int \frac{\cos 2x}{(\sin x+\cos x)^2}\mathrm dx$$</span></p>
</blockquote>
<p>I was integrating my own way.</p>
<p><span class="math-container">$$\int \frac{\cos 2x}{\sin^2x+2\sin x\cos x+cos^2}\mathrm dx$$</span>
<span class="math-container">$$\int \cot 2x \mat... | Henry Lee | 541,220 | <p>notice that:
<span class="math-container">$$\frac{\cos 2x}{\cos^2x+2\cos x\sin x+\sin^2x}=\frac{\cos 2x}{\color{red}{1}+2\cos x\sin x}=\frac{\cos 2x}{1+\sin2x}$$</span>
this is because:
<span class="math-container">$$\sin^2x+\cos^2x\equiv1$$</span>
and:
<span class="math-container">$$\sin2x=2\cos x\sin x$$</span></p... |
3,936,545 | <p>What happens if you have some error in your expression and you wish to take the limit? Does the big-oh go away or are you unable to take a limit at all?</p>
<p>This may seem very obvious as it is likely I am misunderstanding the definition of big-oh?</p>
| vonbrand | 43,946 | <p>Check your definitions. Big-Oh and limits should commute:</p>
<p><span class="math-container">$\begin{align*}
\lim_{n \to \infty} (f(n) + O(g(n)))
= \lim_{n \to \infty} f(n) + O(\lim_{n \to \infty} g(n))
\end{align*}$</span></p>
|
3,936,545 | <p>What happens if you have some error in your expression and you wish to take the limit? Does the big-oh go away or are you unable to take a limit at all?</p>
<p>This may seem very obvious as it is likely I am misunderstanding the definition of big-oh?</p>
| zkutch | 775,801 | <p>What you understand under <span class="math-container">$\lim_\limits{n \to \infty} O(g(n))$</span>? Set of limits of all members?</p>
<p>If exists <span class="math-container">$\lim_\limits{n \to \infty} g(n)$</span>, then <span class="math-container">$O(\lim_\limits{n \to \infty} g(n))=O(1)$</span>, so here are bo... |
2,903,359 | <p>I am trying to prove the following:</p>
<p>Given $1 \le d \le n$, a matrix $P \in R^n$ is a rank-$d$ orthogonal projection matrix. Prove that P is projection matrix iff there exists a $n$x$d$ matrix $U$ such that $P =UU^T$ and $U^TU = I$.</p>
<p>I know that this is an obvious fact about projection matrices but I a... | Dean Alderucci | 682,002 | <p>We can prove the equivalence by proving both ways separately.</p>
<hr>
<h2>1. <span class="math-container">$ P =UU^T,U^T U = I \Rightarrow $</span> P is a projection matrix</h2>
<p>Let's solve an objective similar to the one you stated:
<span class="math-container">$$
\arg \min_{b \in R^n} \lVert {v - Pb} \rVe... |
405,783 | <p>I saw the following in my lecture notes, and I am having difficulties
verifying the steps taken.</p>
<p>The question is:</p>
<blockquote>
<p>Assuming $0<\epsilon\ll1$ find all the roots of the polynomial
$$\epsilon^{2}x^{3}+x+1$$ which are $O(1)$ up to a precision of
$O(\epsilon^{2})$</p>
</blockquote>
<... | TMM | 11,176 | <p>This is not exactly as it is written above, but my approach would be as follows. </p>
<p>First, assume that $x(\epsilon) = O(1)$. Then we can say $x(\epsilon) = x_0 + f(\epsilon)$, where $f(\epsilon) = o(1)$ is some function in $\epsilon$, and $x_0$ is constant in $\epsilon$. Filling this in, and using that $x(\eps... |
2,825,522 | <p>I have this problem:</p>
<p><a href="https://i.stack.imgur.com/blD6N.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/blD6N.png" alt="enter image description here"></a></p>
<p>I have not managed to solve the exercise, but this is my breakthrough:</p>
<p><a href="https://i.stack.imgur.com/0dTdO.j... | dxiv | 291,201 | <p>Hint: draw the segments to the center of the circle, and recognize a rhombus there formed by two equilateral triangles with a common base.</p>
<p><a href="https://i.stack.imgur.com/IeOem.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IeOem.jpg" alt="enter image description here"></a></p>
|
2,825,522 | <p>I have this problem:</p>
<p><a href="https://i.stack.imgur.com/blD6N.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/blD6N.png" alt="enter image description here"></a></p>
<p>I have not managed to solve the exercise, but this is my breakthrough:</p>
<p><a href="https://i.stack.imgur.com/0dTdO.j... | random | 513,275 | <p>By using the symmetry of the figure.</p>
|
2,268,345 | <p>Find the value of $$S=\sum_{n=1}^{\infty}\left(\frac{2}{n}-\frac{4}{2n+1}\right)$$ </p>
<p>My Try:we have</p>
<p>$$S=2\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{2}{2n+1}\right)$$ </p>
<p>$$S=2\left(1-\frac{2}{3}+\frac{1}{2}-\frac{2}{5}+\frac{1}{3}-\frac{2}{7}+\cdots\right)$$ so</p>
<p>$$S=2\left(1+\frac{1}{2}-... | Simply Beautiful Art | 272,831 | <p>Note that this is not valid by the <a href="https://en.wikipedia.org/wiki/Riemann_series_theorem#Examples" rel="noreferrer">Riemann series theorem</a>, which shows you cannot group terms like that. In particular, the terms you are grouping tend to be farther and farther from each other, meaning you are "pulling" te... |
1,393,869 | <p>Given a cubic polynomial with real coefficients of the form $f(x) = Ax^3 + Bx^2 + Cx + D$ $(A \neq 0)$ I am trying to determine what the necessary conditions of the coefficients are so that $f(x)$ has exactly three distinct real roots. I am wondering if there is a way to change variables to simplify this problem and... | P Vanchinathan | 28,915 | <p>For simplicity assume $A>0$ (you can easily get analogous conditions for $A<0)$. Then the first turning point will be a maximum and the second one will be a minimum. Then the 3 distinct roots:</p>
<p>(i) the least root should be earlier than the point where maximum is attained</p>
<p>(ii) the middle root sho... |
3,497,679 | <p>I was given this function:
<span class="math-container">$$
f(x)=
\begin{cases}
x+x^2, & x\in\Bbb Q\\
x, & x\notin \Bbb Q
\end{cases}
$$</span>
I first proved that it is continuous at <span class="math-container">$x=0$</span>.</p>
<p>Now I need to prove that that for every <span class="math-container">$x_0 \... | Michael Hardy | 11,667 | <p>For <span class="math-container">$x\ne0,$</span> first note that <span class="math-container">$x\ne x^2 +x,$</span> and then let <span class="math-container">$\varepsilon$</span> be half the distance between <span class="math-container">$x$</span> and <span class="math-container">$x^2+x.$</span></p>
<p>Suppose ther... |
2,005,604 | <p>Showing $\sqrt a + $$\sqrt {\cos(\sin a)} = 2$</p>
<p>I've attempted various manipulations (multiplying by one, squaring, etc.) but cannot find a way to solve for a. Anyone have an idea how I can approach this problem? Thanks. </p>
| Robert Israel | 8,508 | <p>The solution is approximately $a = 1.5994958620742425268$, but is very unlikely to be expressible in closed form. Numerical methods (e.g. Newton's method) work well.</p>
|
605,277 | <p>I have an electronics project where I sample two sine waves. I would like to know what the amplitude (peak) and difference in phase is. Actually I just need to know the average product of the two waves.</p>
<p>A caveat I have is that the two sine waves have been rectified. (negatives cut off) Here is what I expect ... | Semjon Mössinger | 136,192 | <p>You could try to use a Least Squares Estimator (ls-estimator). It can fit a curve with unknown amplitude an phase to a signal. A least squares estimator always consists of an observation matrix. In this you can "design" your "cuttet sine wave" like this (if one period of your sine wave consists of 8 samples an your ... |
876,763 | <p>Let $R$ be a commutative ring with identity. Assume that for any two principal ideals $Ra$ and $Rb$ we have either $Ra\subseteq Rb$ or $Rb\subseteq Ra$. Show that for any two ideals $I$ and $J$ in $R$, we have either $I\subseteq J$ or $J\subseteq I$.</p>
<p>Initially i thought that if i could show that any ideal in... | Alex J Best | 31,917 | <p>I'm not sure trying to show every ideal is principal will work (though I can't verify a counterexample off the top of my head!) however I'll start you off a different way:</p>
<p>First assume $I \not\subseteq J$, then we can take some $x\in I\smallsetminus J$ and now consider the principal ideal $Rx$, what can we s... |
3,020,365 | <p>Let <span class="math-container">$A=\{t\sin(\frac{1}{t})\ |\ t\in (0,\frac{2}{\pi})\}$</span>.</p>
<p>Then </p>
<ol>
<li><p><span class="math-container">$\sup (A)<\frac{2}{\pi}+\frac{1}{n\pi}$</span> for all <span class="math-container">$n\ge 1$</span>.</p></li>
<li><p><span class="math-container">$\inf (A)>... | Shubham Johri | 551,962 | <p>Let <span class="math-container">$x=1/t\implies A=\{\frac{\sin x}x:x>\pi/2\}$</span></p>
<p>Let <span class="math-container">$f(x)=\frac{\sin x}x, x>\pi/2$</span></p>
<p><span class="math-container">$f(3\pi/2)=-\frac2{3\pi}<0, f'(3\pi/2)>0\implies f$</span> is strictly increasing at <span class="math-c... |
1,796,156 | <p>Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number<a href="http://mathworld.wolfram.com/FibonacciNumber.html" rel="noreferrer">$^{[1]}$</a><a href="http://en.wikipedia.org/wiki/Fibonacci_number" rel="noreferrer">$\!^{[2]}$</a><a href="http://oeis.org/A000045" rel="noreferrer">$\!^{[3]}$</a>. The Fibonacci number... | giobrach | 332,594 | <p>I think I've found a <strike>suitable</strike> [see <strong>EDIT</strong> below] way to generalize the fibonorial in the reals! (I've taken inspiration from an answer to <a href="https://math.stackexchange.com/questions/1842543/write-an-expression-in-terms-of-n-for-the-nth-term-in-the-following-sequence">this questi... |
425,460 | <p>While browsing through several pages of nlab(mainly on n-Categories), I encountered the notion "foo" several times. However, there seems to be article on nlab about this notion. Is this some kind of category theorist slang? Please explain to me what this term means.</p>
| Clive Newstead | 19,542 | <p>It's slang, which I've mostly seen used in the context of computing rather than category theory; <em>foo</em> is just a placeholder for something else, as is <em>bar</em>. A logician I know likes talking about <em>widgets</em> and <em>wombats</em> $-$ it all serves the same purpose.</p>
<p>For example, you might sa... |
1,490,219 | <blockquote>
<p>Given $S=\displaystyle \bigcap^{\infty}_{k=1}\left(1-\frac{1}{k}, 1+\frac{1}{k}\right)$, what is $\sup(S)$ and $\max(S)$?</p>
</blockquote>
<p>I reasoned that this is empty, since as $k$ goes to infinity, then $\frac{1}{k}$ goes to $0$. So ultimately, the intersection of all the intervals in $S$ is $... | Nitrogen | 189,200 | <p>Since $1-\frac{1}{k} < 1 < 1+\frac{1}{k}$ for all $k>0$, $1$ is indeed in the intersection. Now, suppose there is some $1\ne a \in S$ and suppose $a<1$ (the case $1<a$ is similar.) Then, there exists $k\in \Bbb{N}$ such that $\frac{1}{k}<1-a$. Thus $a<1-\frac{1}{k}$ and $a\notin S$.</p>
<p>Ther... |
3,877,652 | <p>Can anyone help me how to do this, like there are some examples in my book, but this exercise problem seems to be alittle difficult for me to approach:</p>
<p>Given a set <span class="math-container">$\{A_k|k\in\mathbb{N}\}$</span>:
<span class="math-container">$$A_k=\bigg\{x\in\mathbb{R}\bigg|\space\space 1-\frac{1... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$A_1=(0,2)$</span> and since <span class="math-container">$k\geqslant1\implies A_k\subset(0,2)$</span>, the union <span class="math-container">$\bigcup_{k\in\Bbb N}A_k$</span> is indeed <span class="math-container">$(0,2)$</span>.</p>
<p>And the intersection <span class="math-conta... |
3,452,249 | <p>I'd like to ask to check for a solution to a homework problem, below.</p>
<p>Define <span class="math-container">$\forall n \in \mathbb{N}, g_n(x):=\frac{x}{(1-x)^n}$</span>. Let <span class="math-container">$F_n$</span> be the associated Newton function, i.e. <span class="math-container">$F_n(x):= x - \frac{g_n(x)... | marty cohen | 13,079 | <p>Since</p>
<p><span class="math-container">$\begin{array}\\
F_n(x)
&=x-\dfrac{g_n'(x)}{g_n(x)}\\
&=x-\dfrac{\frac{ (n - 1) x + 1}{(1 - x)^{n + 1}}}{\frac{x}{(1-x)^n}}\\
&=x-\dfrac{(n - 1) x + 1}{x(1-x)}\\
&=x+\dfrac{(n - 1) x + 1}{x(x-1)}\\
&=x+\dfrac{n - 1}{x-1}+\dfrac{1}{x(x-1)}\\
&>x+\d... |
3,452,249 | <p>I'd like to ask to check for a solution to a homework problem, below.</p>
<p>Define <span class="math-container">$\forall n \in \mathbb{N}, g_n(x):=\frac{x}{(1-x)^n}$</span>. Let <span class="math-container">$F_n$</span> be the associated Newton function, i.e. <span class="math-container">$F_n(x):= x - \frac{g_n(x)... | Lee Mosher | 26,501 | <p><span class="math-container">$F'(\infty)$</span> does not make any sense, because <span class="math-container">$\infty$</span> is not a point in the domain of <span class="math-container">$F'$</span>. </p>
<p>For similar reasons, the behavior of <span class="math-container">$F'(x)$</span> as <span class="math-conta... |
3,802,269 | <p>Evaluate <span class="math-container">$\lim_{x\rightarrow \infty} x\int_{0}^{x}e^{t^2-x^2}dt$</span></p>
<p>My approach:</p>
<p><span class="math-container">$$
\lim_{x\rightarrow \infty} x\int_{0}^{x}e^{t^2-x^2}dt = \lim_{x\rightarrow \infty} \frac{\int_{0}^{x}e^{t^2}dt}{x^{-1}e^{x^2}}
$$</span></p>
<p>Both the nume... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
1,476,946 | <p>So, I'm just starting to peruse "Categories for the Working Mathematician", and there's one thing I'm uncertain on. Lets say I have three objects, $X,Y,Z$ and two arrows $f,g$ such that $X\overset {f} {\to}Y\overset {g} {\to}Z$. Does this necessitate the composition arrow exist so the diagram commutes, i.e mus... | Community | -1 | <p>The definition of category ensures that you can <em>always</em> "compose" two arrows if one's target is the other's source: that is, every diagram</p>
<p>$$ \begin{matrix} X & \xrightarrow{f} & Y
\\ & & \downarrow g\!\!\!\!\!
\\ & & Z
\end{matrix}$$</p>
<p>can be (uniquely!) completed to a ... |
1,707,929 | <p>How do I solve for the object distance to each receiver for three radar receivers on the ground, each the same distance from the other, and each receiving echoes, reflected from an object overhead, of a signal pulse from a single transmitter located on the ground at the exact center of the receivers? </p>
<p>Tran... | amd | 265,466 | <p>For each receiver, the possible locations of the target are on the surface of an ellipsoid of revolution that has the transmitter as one focus and the receiver as the other. The target will be at the intersection of these three surfaces.</p>
|
3,430,066 | <p><strong>Question:</strong></p>
<p>Calculate the integral </p>
<p><span class="math-container">$$\int_0^1 \frac{dx}{e^x-e^{-2x}+2}$$</span></p>
<p><strong>Attempted solution:</strong></p>
<p>I initially had two approaches. First was recognizing that the denominator looks like a quadratic equation. Perhaps we can ... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$-1$</span> is a root of <span class="math-container">$u^3+2u^2-1$</span>, you can write it as <span class="math-container">$u+1$</span> times a quadratic monic polynomial. It turns out that that polynomial is <span class="math-container">$u^2+u-1$</span>. Besides<span class="math-... |
251,466 | <p>Let $A$, $B$ and $C$ be three points in a disk,
does $f\left(A,B,C\right)=\mbox{Area}\left(\mbox{triangle}\,ABC\right)/\mbox{Perimeter}\left(\mbox{triangle}\,ABC\right)$
have maximum on
the boundary? </p>
| chloe_shi | 45,070 | <p>$\triangle ABC=\dfrac{r(a+b+c)}{2}=rs$ $~$ $\Longrightarrow$ $~$ $\dfrac{\triangle ABC}{s}=r$<br>
Euler theorem : $~$ $\underline{OI^{2}=R^{2}-2Rr}\geq 0$ $~$ $\Longrightarrow$ $~$ $r\leq\dfrac{R}{2}$ , which equality hold at $OI=0$ , $R=2r$<br>
That is, it's equilateral. </p>
<p>Now, for the ratio $R$ ($\trian... |
68,386 | <p>I'm looking for a theorem of the form </p>
<blockquote>
<p>If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.</p>
</blockquote>
<p>My attempts to do this purely algebraically are not working, so I started looking into methods fr... | Sándor Kovács | 10,076 | <p>$R[\frac 1v]$ corresponds to the open subset of $\mathrm{Spec}R$ where $v\neq 0$. So, all kind of things can happen. Here is an example: Let $R_1$ be a "nice" ring with a unity $v\in R_1$ of dimension $n$ and $R_2$ a "nice" ring with a unity $w\in R_2$ of dimension $m$. Let $R=R_1\oplus R_2$. Then $\dim R=\max \{n,m... |
2,083,347 | <p>Let's consider a linear operator
$$
Lu = -\frac{1}{w(x)}\Big(\frac{d}{dx}\Big[p(x)\frac{du}{dx}\Big] + q(x)u\Big)
$$
So the Sturm-Liouville equation can be written as
$$
Lu = \lambda u
$$
Why the proper setting for this problem is the weighted Hilbert space $L^2([a,b], w(x)dx)$?</p>
| Jimmy R. | 128,037 | <p>Despite the change of its form at $k=4$, the function $k(x)$ is monotone decreasing on $\mathbb R$ and continuous at $x=4$. Hence $$k^{-1}(x)=\begin{cases}-\dfrac{x-2}{4}, & x\le -14 \\ -\dfrac{x-6}{5}, & x>-14\end{cases}$$ where $-14=k(4)$.</p>
|
705,945 | <p>I have this expression:
$$\sum_{\{\vec{S}\}}\prod_{i=1}^{N}e^{\beta HS_{i}}=\prod_{i=1}^{N}\sum_{S_{i}\in\{-1,1\}}e^{\beta HS_{i}} \qquad (1)$$
Where $\sum_{\{\vec{S}\}}$ means a sum over all possible vectors $\vec{S}=(S_1,...,S_N)$ with the restriction that $S_i$ can only take the values $\{-1,+1\}$, i.e. the sum i... | Mx Glitter | 134,212 | <p>Try to go from the right formula to the left and use <a href="http://en.wikipedia.org/wiki/Distributivity" rel="nofollow">distributivity</a>. If you're not sure, try with N=2 to convince yourself.</p>
|
4,330,031 | <p>I want to prove that this degree sequence <span class="math-container">$(5,5,5,2,2,2,1)$</span> isn't valid to draw a graph from it, the graph needs to be simple. I am looking for a Theroem or a way to contradict the assumption that we can make a graph from it.</p>
<p>My solution was the following, for the given nod... | MachineLearner | 647,466 | <p>As <span class="math-container">$A$</span> is invertable we know that it is a square matrix and that its determinant is not equal to zero.</p>
<p>For square matrices we have</p>
<p><span class="math-container">$$\det(ABA)=\det A \det B \det A=0$$</span>
We know that <span class="math-container">$\det A \neq 0$</span... |
3,290,514 | <p>I need to make the navigation and guidance of a vehicle (a quadcopter) in a platform. This platform can be seen like this:</p>
<p><a href="https://i.stack.imgur.com/jeJ34.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jeJ34.png" alt="enter image description here"></a></p>
<p>where the blue dots... | Intelligenti pauca | 255,730 | <p>If I understand correctly, you have the coordinates of some points, all of them (or nearly so) lying on a circle, and you want to find the center of that circle.</p>
<p>A possible approach is the following: </p>
<ol>
<li><p>choose at random a triplet of points and find their circumcenter using <a href="https://en.... |
2,863,533 | <blockquote>
<p>Let $f(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients and all real roots, also $|f(i)|=1$ where $i=\sqrt{-1}$. Prove that all three roots of $f(x)=0$ are zero. Also prove that $a+b+c=0$.</p>
</blockquote>
<hr>
<p>As $f(i)=-i-a+ib+c=1$ and $f(i)=-i-a+ib+c=-1$<br><br>
I don't know how ... | vadim123 | 73,324 | <p>Hint: Let the three real roots of $f(x)$ be $r,s,t$ (not necessarily distinct). Then, we may write $$f(x)=(x-r)(x-s)(x-t)$$
Expand this, and set equal to $x^3+ax^2+bx+c$, and continue...</p>
|
2,863,533 | <blockquote>
<p>Let $f(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients and all real roots, also $|f(i)|=1$ where $i=\sqrt{-1}$. Prove that all three roots of $f(x)=0$ are zero. Also prove that $a+b+c=0$.</p>
</blockquote>
<hr>
<p>As $f(i)=-i-a+ib+c=1$ and $f(i)=-i-a+ib+c=-1$<br><br>
I don't know how ... | mechanodroid | 144,766 | <p>Let $x_1, x_2, x_3$ be the roots. We have $f(x) = (x - x_1)(x - x_2)(x-x_3)$.</p>
<p>Hence</p>
<p>\begin{align}
1 &= |f(i)|^2 \\
&= f(i)\overline{f(i)} \\
&= (i - x_1)(i - x_2)(i - x_3)(-i - x_1)(-i - x_2)(-i - x_3) \\
&= (x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1)
\end{align}</p>
<p>so $x_1 = x_2 = x_3 = 0... |
4,050,855 | <p>On <a href="https://math.stackexchange.com/a/4050373/878105">this answer</a>, the function <span class="math-container">$f_n(x)=x^n$</span> in the interval <span class="math-container">$[0,1]$</span> is given as a pathologic example with pointwise convergence.</p>
<p>Can I say that this Cauchy sequence does not (poi... | Elchanan Solomon | 647 | <p>The notion of a sequence being "Cauchy" is inherently a metric concept. When you talk about a Cauchy sequence, you need to be talking about convergence in a metric. Pointwise convergence does not correspond to any metric (this can be proven). If you use the <span class="math-container">$\|f\|_{\infty}$</sp... |
433,639 | <p>(What follows is motivated by an answer to <a href="https://mathoverflow.net/questions/433612/fourier-optimization-problem-related-to-the-prime-number-theorem?noredirect=1#comment1116702_433612">Fourier optimization problem related to the Prime Number Theorem</a>)</p>
<p>Let <span class="math-container">$f:\mathbb{R... | H A Helfgott | 398 | <p>Warning: the following answer is (a) maybe a bit careless in a nineteenth-century sort of way, (b) missing its final step (which may be obvious to others, and/or amount to looking things up in the right table)</p>
<p>We are looking for an even function <span class="math-container">$g:\mathbb{R}\to \mathbb{R}$</span>... |
877,646 | <p>Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. ,</p>
<p>Following are the given conditions</p>
<p>a) each $A_i$ is non invertible <strong>except $A_0$</strong> because their determinant is zero.</p>
<p>b) $\sum_{n=0}^\infty A_i$ is invertible and determinant is not zero</p>
<p>c) </p>
<... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\dsc}[1]{\displaystyle{\color{red}{#1}}}
\newcommand{\ex... |
241,871 | <p>A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying
$$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$</p>
<p>A Lévy process can be characterized by triples $(b, A, \nu)$ by
Lévy-Itô decomposition, then
$$X_{t} = bt + W_{A}(t) + \int_{B_{1}} x \tilde N(t, dx) + \int_{B_{1}^{c... | Joachim | 29,657 | <p>Using Jason Starr's comment I was able (I think) to figure out the case of $\pi_1(X)^0$. For anyone who stumbles across this with the same question in mind I add a sketch of the proof in an answer. For the experts, if you feel like leaving a comment whether my reasoning is correct, that would be great.</p>
<p>Let $... |
1,057,819 | <p>The number $128$ can be written as $2^n$ with integer $n$, and so can its every individual digit. Is this the only number with this property, apart from the one-digit numbers $1$, $2$, $4$ and $8$? </p>
<p>I have checked a lot, but I don't know how to prove or disprove it. </p>
| Robert Israel | 8,508 | <p>This seems to be an open question. See <a href="http://oeis.org/A130693">OEIS sequence A130693</a> and references there.</p>
|
3,931,831 | <p>For the scenario given below, I am confused about if the samples are dependent or independent since the scenario does not mention anything about the samples being paired/related or vice versa.</p>
<p>I am aware if terms such as paired, repeated measurements, within-subject effects, matched pairs, and pretest/posttes... | Théophile | 26,091 | <p>As others have said, you don't necessarily need to use Lagrange multipliers. But since you've set up the system, we can see what happens:</p>
<p><span class="math-container">$$
3x^2-3=\lambda\\
3y^2-3=2\lambda\\
x+2y-3=0
$$</span></p>
<p>From the first two equations, we have <span class="math-container">$3y^2-3=2(3x... |
2,362,477 | <p>Solve the equation $f(x) = 2$.
I reached the stage $\sin(x) = {2\over 3}$ but then (as I remember it was solved) using $x = \sin^{-1}(2/3)$ (sine inverse) I get the answer $x = 41.81$ but the correct answer is $x = 0.730$ or $2.41$. Why is this so? Sorry it might be a silly question but it had been long since I stud... | N. F. Taussig | 173,070 | <p>We are given the function $f(x) = 4 - 3\sin x$ defined on the interval $[0, 2\pi]$. Notice that $x$ is measured in radians. </p>
<p>Solving the equation $f(x) = 2$ yields
\begin{align*}
f(x) & = 2\\
4 - 3\sin x & = 2\\
-3\sin x & = -2\\
\sin x & = \frac{2}{3}
\end{align*}
Thus, we must find all ang... |
4,622,956 | <p>I think <span class="math-container">$\,9\!\cdot\!10^n+4\,$</span> can be a perfect square, since it is <span class="math-container">$0 \pmod 4$</span> (a quadratic residue modulo <span class="math-container">$4$</span>), and <span class="math-container">$1 \pmod 3$</span> (also a quadratic residue modulo <span cla... | B. Goddard | 362,009 | <p>If you reduce mod <span class="math-container">$11$</span> you get <span class="math-container">$(-2)(-1)^n+4 \equiv 2$</span> or <span class="math-container">$6 \pmod{11}$</span>. Neither <span class="math-container">$2$</span> nor <span class="math-container">$6$</span> is a quadratic residue mod <span class="mat... |
1,038,076 | <p>Solve the equation $7\times 13\times 19=a^2-ab+b^2$ for integers $a>b>0$. How many are there such solutions $(a,b)$?</p>
<p>I know that $a^2-ab+b^2$ is the norm of the Eisentein integer $z=a+b\omega$, but how can I make use of this? Thank you so much.</p>
| Dietrich Burde | 83,966 | <p>Note that $N(a+b\omega)=a^2-ab+b^2$ is the sum of squares, because
$$
a^2-ab+b^2=\frac{1}{4}((2a-b)^2+3b^2).
$$
Hence we have to solve the equation $(2a-b)^2+3b^2=4\cdot 7\cdot 13\cdot 19=6916$, which is
straightforward, since we only have to test a few integers $a,b \in \mathbb{N}$. In particular, $3b^2\le 6916$, s... |
1,038,076 | <p>Solve the equation $7\times 13\times 19=a^2-ab+b^2$ for integers $a>b>0$. How many are there such solutions $(a,b)$?</p>
<p>I know that $a^2-ab+b^2$ is the norm of the Eisentein integer $z=a+b\omega$, but how can I make use of this? Thank you so much.</p>
| achille hui | 59,379 | <p>It is known that the <a href="http://en.wikipedia.org/wiki/Eisenstein_integer" rel="nofollow">Eisenstein integers</a> $\mathbb{Z}[\omega]$ is an <a href="http://en.wikipedia.org/wiki/Unique_factorization_domain" rel="nofollow">unique factorization domain</a> and it has six <a href="http://en.wikipedia.org/wiki/Unit_... |
831,763 | <p>The following equation $$e^{i+z}e^{iz}=1$$ is to be solved for $z$. I have tried
$$
\begin{eqnarray}
e^{i+z+iz} = 1\\
i+z+iz=0\\
z= -{i \over 1+i} = -{i(1+i)\over 2} = \frac12-i\frac12
\end{eqnarray}
$$
However I am absolutely unsure, that's correct. Somehow I suspect trigonometry should creep in the answer.</p>
| Community | -1 | <p>Let $z=a+ib$ then the given equality becomes</p>
<p>$$e^{a+(b+1)i}e^{ai-b}=1\iff e^{a-b+(b+a+1)i}=1$$
hence we find
$$a-b=0\quad\text{and}\quad b+a+1\equiv 0\mod 2\pi$$
so
$$a=b\quad\text{and}\quad a\equiv-\frac12\mod \pi$$</p>
<p><strong>Added</strong> i.e.
$$a=b\quad\text{and}\quad a=-\frac12+k\pi,\quad k\in\Bb... |
3,545,548 | <p><span class="math-container">$\def\LIM{\operatorname{LIM}}$</span>
Let <span class="math-container">$(X,d)$</span> be a metric space and given any cauchy sequence <span class="math-container">$(x_n)_{n=1}^{\infty}$</span> in <span class="math-container">$X$</span> we introduce the formal limit <span class="math-cont... | Eric Towers | 123,905 | <p><span class="math-container">$x$</span>, <span class="math-container">$y$</span>, and <span class="math-container">$z$</span> are successive terms of an arithmetic progression. Consequently, there is an <span class="math-container">$s$</span> such that <span class="math-container">$y = x + s$</span> and <span class... |
3,989,878 | <p>I can't solve this problem. I tried to find <span class="math-container">$\tan x$</span> directly by solving cubic equations but I failed.</p>
<p>The problem is to find <span class="math-container">$\tan x\cot 2x$</span> given that
<span class="math-container">$$\tan x+ \tan 2x=\frac{2}{\sqrt{3}}, \>\>\>\&g... | J.G. | 56,861 | <p>If <span class="math-container">$t:=\tan x$</span> then <span class="math-container">$\frac{t(3-t^2)}{1-t^2}=\tfrac{2}{\sqrt{3}}$</span> so <span class="math-container">$t^3-\tfrac{2}{\sqrt{3}}t^2+3t+\tfrac{2}{\sqrt{3}}=0$</span>. This has <a href="https://www.wolframalpha.com/input/?i=solve+t%5E3-%282%2Fsqrt%283%29... |
2,904,603 | <p>I'm working on the following question:</p>
<blockquote>
<p>Show that $G$ is a group if and only if, for every $a, b \in G$,
the equations $xa = b$ and $ay = b$ have solutions $x, y \in G$.</p>
</blockquote>
<p>I'm having trouble getting started because I'm not understanding what it means for "the equations $xa... | stressed out | 436,477 | <p>What it means is as follows:
$$\forall a,b \in G, \exists x \in G: xa=b$$
$$\forall a,b \in G, \exists y \in G: ay=b$$</p>
<p>Assuming that $(G,\star)$ is a semi-group, i.e. $G \not= \emptyset$ is closed and associative under $\star$, you must prove the existence of the identity element and the existence of the inv... |
2,641,073 | <p>I want to define the scalar product by $\vec a \cdot \vec b = | \vec a | |\vec b | \cos(\varphi)$ and derive $\vec a \cdot \vec b = a_1 b_1 + a_2 b_2$. Algebraically, this is no problem (we can just expand the vectors in the standard basis).</p>
<p>How do I show this by using only elementary geometry? I've drawn th... | msm | 340,064 | <p>Yeah, I don't think the thing to do is to look at rectangles. First, prove that the dot product is invariant under rotations (this is a little circular, since some people would say rotations are defined by being linear maps preserving the dot product and orientation). </p>
<p>Then, line one of those vectors up with... |
2,641,073 | <p>I want to define the scalar product by $\vec a \cdot \vec b = | \vec a | |\vec b | \cos(\varphi)$ and derive $\vec a \cdot \vec b = a_1 b_1 + a_2 b_2$. Algebraically, this is no problem (we can just expand the vectors in the standard basis).</p>
<p>How do I show this by using only elementary geometry? I've drawn th... | pjs36 | 120,540 | <p>One way is to compute the (square of the) red length two ways, and equate the two.</p>
<p><a href="https://i.stack.imgur.com/vvIko.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/vvIko.png" alt="enter image description here"></a></p>
<p>On the one hand, Pythagoras tells us that the red length is... |
2,533,960 | <p>So, the given expression is
$$\binom{2n}{2} = 2\binom{n}{2}+n^2$$</p>
<p>The task is to give a combinatorical proof for it.</p>
<p>Left side of the identity is obviously equal to the number of options for choosing 2 elements out of the set with cardinality $2n$.</p>
<p>What issues me is that I can't think of any ... | Peter Szilas | 408,605 | <p>Consider 2 sets:</p>
<p>$A=${$a_1,a_2,....a_n$}, and </p>
<p>$B=${$b_1,b_2,..., b_n$}, all distinct elements.</p>
<p>LHS:</p>
<p>The number of ways to choose $ 2$ elements from $A\cup B$ :</p>
<p>$\binom{2n}{2}.$</p>
<p>RHS: </p>
<p>Choose $2$ elements from $A$, or from $B$ :</p>
<p>In $2\binom{n}{2}$ ways</... |
1,731,364 | <p>So the question asks:</p>
<blockquote>
<p>Let $X_1,X_2,X_3\sim \operatorname{Exp}(\lambda)$ be independent (exponential) random variables (with $\lambda> 0$).<br>
(a) Find the probability density function of the random variable $Z = \max \{X_1,X_2,X_3\}$.<br>
(b) Let $T = X_1+X_2/2+X_3/3$, use moment gener... | Em. | 290,196 | <p>(b) Just means use MGFs to show that $Z$ and $T$ have the same distribution.</p>
<blockquote>
<p>So is there a simple way to calculate the $E[Z]$ and $\text{Var} [Z]$ without literally solving the integration?</p>
</blockquote>
<p>Yes. Intuitively, if you have $3$ light bulbs with lifetimes that are iid exponent... |
66,068 | <p>I have a list like this. </p>
<pre><code>cdatalist = {{1., 0.898785, Failed, Failed, 50., 25., "serial"}, {1., 1.31175,1., Failed, 50., 25., "serial"}, {1., 18.8025, Failed, 0.490235, 50., 25., "serial"}, {1., 19.6628, 0.990079, Failed, 50., 25., "serial"}, {1., 39.547, Failed, Failed, 50., 25., "serial"}, {1., 39.... | Aisamu | 8,238 | <p>This matches your example, but I had to get only the first three elements of each line (you didn't mention it).</p>
<pre><code>Select[cdatalist, #[[3]] =!= Failed &][[All, ;; 3]]
</code></pre>
<p>Or, as per @belisarius suggestion (roughly twice as fast!)</p>
<pre><code>Cases[cdatalist, Except[{_, _, Failed, _... |
66,068 | <p>I have a list like this. </p>
<pre><code>cdatalist = {{1., 0.898785, Failed, Failed, 50., 25., "serial"}, {1., 1.31175,1., Failed, 50., 25., "serial"}, {1., 18.8025, Failed, 0.490235, 50., 25., "serial"}, {1., 19.6628, 0.990079, Failed, 50., 25., "serial"}, {1., 39.547, Failed, Failed, 50., 25., "serial"}, {1., 39.... | kglr | 125 | <pre><code>Pick[#, (#[[-1]] =!= Failed) & /@ #] &@cdatalist[[All, ;; 3]]
DeleteCases[cdatalist[[All, ;; 3]], {_, _, Failed}]
DeleteCases[cdatalist, {_, _, Failed, ___}][[All, ;; 3]]
</code></pre>
<p>all give</p>
<pre><code>(* {{1., 1.31175, 1.},
{1., 19.6628, 0.990079},
{1., 40.6208, 0.980588},
... |
2,252,090 | <p>Someone posed this question to me on a forum, and I have yet to figure it out. If $a,b,c,d$ are the zeroes of:</p>
<p>$$x^4-7x^3+2x^2+5x-1=0$$
Then what is the value of $$ \frac1a +\frac1b +\frac1c +\frac1d $$</p>
<p>I can figure out the zeroes, but they are wildly complex. I'm sure there must be an easier way. </... | Robert Israel | 8,508 | <p>$$(x-a)(x-b)(x-c)(x-d) = x^4 + \ldots - (abc+abd+acd+bcd)x + abcd$$
So $$abcd = -1$$ and
$$abc + abd + acd + bcd = abcd \left(\frac{1}{a}+\frac{1}{b}+\frac1c + \frac1d\right) = -5$$
making
$$\frac{1}{a}+\frac{1}{b}+\frac1c + \frac1d = 5$$</p>
|
3,290,095 | <p>Now first something that I already know;
<span class="math-container">\begin{eqnarray}
∞/ ∞ = undetermined ( ≠1 ) \\
∞- ∞ = undetermined (≠0)\\
\end{eqnarray}</span></p>
<p>So basically one reason for this is that the <span class="math-container">$∞$</span> I assume is not as same as the <span class="math-contain... | G Cab | 317,234 | <p>As the other answers explained <span class="math-container">$\infty$</span> is not a number. Already Galileo had to admit that.</p>
<p>However, it is true that <span class="math-container">$1=a/a$</span> is valid for all <span class="math-container">$0 \ne a$</span>, including the limit for <span class="math-contai... |
385,537 | <p>How would you go about proving the following?</p>
<p>$${1- \cos A \over \sin A } + { \sin A \over 1- \cos A} = 2 \operatorname{cosec} A $$</p>
<p>This is what I've done so far:</p>
<p>$$LHS = {1+\cos^2 A -2\cos A + 1 - \cos^2A \over \sin A(1-\cos A)}$$</p>
<p>....no idea how to proceed .... X_X</p>
| André Nicolas | 6,312 | <p>When we "cross-multiply," on top we get $(1-\cos A)^2+\sin^2 A$. Expand the square.</p>
<p>We get $1-2\cos A+\cos^2 A+\sin^2 A$. Replace $\cos^2 A+\sin^2 A$ by $1$. We get $2-2\cos A$ on top, and now it's over. </p>
|
2,276,907 | <p>If $\cos{x}=\frac{3}{5}$ and angle $x$ terminates in the fourth quadrants, find the exact value of each of the following:</p>
<p>A. $\sin{2x}$
B. $\cos{2x}$
C. $\tan {\frac{x}{2}}$</p>
<p>Okay, so I am going through my old exam reviews for the final exam I have this evening, and choosing problems I have trouble wi... | turkeyhundt | 115,823 | <p>I would draw it on the unit circle. Put a point at the point $(0.6,-0.8)$ From there you can get the values for $\sin{x}$ & $\tan{x}$ and use some basic trig identities to get the rest.</p>
<p><a href="https://i.stack.imgur.com/3f9Om.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3f9Om.jpg"... |
128,221 | <p>Let $v_1=[-3;-1]$ and $v_2= [-2;-1]$</p>
<p>Let $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation satisfying:</p>
<p>$T(v_1)=[15;-6]$ and $T(v_2)=[11;-3]$</p>
<p>Find the image of an arbitrary vector $[x;y]$</p>
| Community | -1 | <p>Note sure if <code>(homework)</code> yet. So <strong>hint:</strong></p>
<p>Let
$$
T =
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}
$$</p>
<p>We can re-interpret the given $T(v_1)$ and $T(v_2)$ as:</p>
<p>$$
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}
\begin{pmatrix}
-3 \\ -1
\end{pma... |
2,870,729 | <blockquote>
<p>Why does $|e^{ix}|^2 = 1$?</p>
</blockquote>
<p>The book said $e^{ix} = \cos x + i\sin x$, and square it, then $|e^{ix}|^2 = \cos^2x + \sin^2x = 1$.</p>
<p>But, when I calculated it, $ |e^{ix}|^2 = \left|\cos x + i\sin x\right|^2 = \cos^2x - \sin^2x + 2i\sin x\cos x$.</p>
<p>I can't make it to be e... | Nebo Alex | 218,007 | <p>you can first apply the modulus $$|e^{ix}|=\sqrt{\cos^2 x + \sin^2 x}$$ then square it whole you will get $$|e^{ix}|^2={\cos^2 x + \sin^2 x}=1$$</p>
|
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