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 |
|---|---|---|---|---|
1,392,661 | <p>For a National Board Exam Review: </p>
<blockquote>
<p>Find the equation of the perpendicular bisector of the line joining
(4,0) and (-6, -3)</p>
</blockquote>
<p>Answer is 20x + 6y + 29 = 0</p>
<p>I dont know where I went wrong. This is supposed to be very easy:</p>
<p>Find slope between two points:</p>
<p... | msinghal | 194,655 | <p>To find the midpoint, you don't need to negate the coordinates. So the midpoint is $\left(\frac{-6+4}{2}, \frac{0+(-3)}{2}\right)=\left(-1, \frac{-3}{2}\right)$</p>
|
1,392,661 | <p>For a National Board Exam Review: </p>
<blockquote>
<p>Find the equation of the perpendicular bisector of the line joining
(4,0) and (-6, -3)</p>
</blockquote>
<p>Answer is 20x + 6y + 29 = 0</p>
<p>I dont know where I went wrong. This is supposed to be very easy:</p>
<p>Find slope between two points:</p>
<p... | user21820 | 21,820 | <p>First you computed midpoint of $X$ wrongly.</p>
<p>Later you evaluated $b$ wrongly.</p>
|
2,961,686 | <p>Consider a matrix <span class="math-container">$A$</span> which we subject to a small perturbation <span class="math-container">$\partial A$</span>. If <span class="math-container">$\partial A$</span> is small, then we have <span class="math-container">$(A + \partial A)^{-1} \approx A^{-1} - A^{-1} \partial A A^{... | weijun yin | 605,934 | <p>In fact, you can understand the problem by the solution, given by mjqxxxx, of the related problem you give. The primary problem is to find a way to estimate the term in the right hand. Why not consider the determinant of the term. The determinant of the product equals the product of the corresponding determinant. T... |
2,979,103 | <blockquote>
<p>Let <span class="math-container">$S$</span> be the region <span class="math-container">$\{z:0<|z|<\sqrt{2}, \ 0 < \text{arg}(z) <
\pi/4\}$</span>. Determine the image of <span class="math-container">$S$</span> under the transformation</p>
<p><span class="math-container">$$f(z)=\frac{... | Martin R | 42,969 | <p>It helps to consider not only those segments or arcs, but the whole line or
circle that they are part of.</p>
<p><span class="math-container">$M$</span> maps <span class="math-container">$(0, 2, \infty)$</span> on the real line to <span class="math-container">$(2, \frac 43, 1)$</span>, therefore the segment <span ... |
2,817,507 | <p>I came across the following argument in my discrete maths textbook:</p>
<p>Since $n=O(n), 2n=O(n)$ etc., we have:
$$
S(n)=\sum_{k=1}^nkn=\sum_{k=1}^nO(n)=O(n^2)
$$</p>
<p>The accompanying question in the book is: <strong>What is wrong with the above argument?</strong></p>
<p><strong>Attempt:</strong> Performing t... | Régis | 254,227 | <p>It is a problem of quantifiers $f(n)=O(n)$ means that there exists a constant $M$ that is <em>independant of n</em> such that for all $n$ large enough $|f(n)| \le Mn$. In your case, the implied constant for each term is $k$, which <em>does</em> depend on $n$ since it ranges between $1$ and $n$. So you cannot get th... |
2,817,507 | <p>I came across the following argument in my discrete maths textbook:</p>
<p>Since $n=O(n), 2n=O(n)$ etc., we have:
$$
S(n)=\sum_{k=1}^nkn=\sum_{k=1}^nO(n)=O(n^2)
$$</p>
<p>The accompanying question in the book is: <strong>What is wrong with the above argument?</strong></p>
<p><strong>Attempt:</strong> Performing t... | Chinny84 | 92,628 | <p>Assuming that we have
$$
O(kn)
$$
using your logic, then we can say if $n<5$ then choosing $k = 10$ we are saying that
$$
kn > n^2 \;\;\forall\; n
$$</p>
<p>alternatively if I have $k=2$ then
$$
kn < n^2 \;\;\forall \;n > 2
$$
so by choosing a fixed parameter we changing when $n^2$ is less than $kn$ ... |
3,413,261 | <p>I know this was answered before but I'm having one particular problem on the proof that I'm not getting.</p>
<p>My Understanding of the distribution law on the absorption law is making me nuts, by the answers of the proof it should be like this.</p>
<p>A∨(A∧B)=(A∧T)∨(A∧B)=A∧(T∨B)=A∧T=A</p>
<p>This should prove th... | Henno Brandsma | 4,280 | <p>Lindelöf and second countable are saying that a space is "small" in some sense; so one way to find non-examples is to take products of lots of spaces, such products (or powers) are "big".</p>
<p><span class="math-container">$\Bbb R^I$</span> is not Lindelöf for <span class="math-container">$I$</s... |
1,356,900 | <p>For section 1 on Fields, there is a question 2c:</p>
<p>2.</p>
<p>a) Is the set of all positive integers a field?</p>
<p>b) What about the set of all integers?</p>
<p>c) Can the answers to both these question be changed by re-defining addition or multiplication (or both)?</p>
<p>My answer initially to 2c was th... | coldnumber | 251,386 | <p>You point out correctly the field axioms that $\Bbb{N}$ and $\Bbb{Z}$ do not satisfy. Just a (maybe nitpicky) note on wording: it's better to say that they are not fields because they do not satisfy the field axioms than to say they would be fields if the definition of fields were different.</p>
<p>Now onto the lin... |
38,586 | <p>The $n$-th Mersenne number $M_n$ is defined as
$$M_n=2^n-1$$
A great deal of research focuses on Mersenne primes. What is known in the opposite direction about Mersenne numbers with only small factors (i.e. smooth numbers)? In particular, if we let $P_n$ denote the largest prime factor of $M_n$, are any results kn... | Gjergji Zaimi | 2,384 | <p>I guess lower bounds on the largest prime factor of Mersenne numbers are not only interesting in number theory but also in coding theory (see this article of K. Kedlaya and S. Yekhanin <a href="http://arxiv.org/abs/0704.1694" rel="nofollow">here</a>). They say the current strongest lower bound is
$$P_n>\epsilon(n... |
652,660 | <p>Show $\lnot(p\land q) \equiv \lnot p \lor \lnot q$</p>
<p>this is my solution . Check it please </p>
<p><img src="https://i.stack.imgur.com/1y7DB.jpg" alt="enter image description here"></p>
| Jon | 124,068 | <p>Your solution is right and covers all cases that arise in classical logic. Classically, a formula is false (in a model) if and only if it is not true (in the model). So, the falsity case mentioned in a comment reduces to showing that: If $\neg(p \wedge q)$ is not true, then $\neg p \vee \neg q$ is not true. But this... |
3,884,098 | <p>Suppose <span class="math-container">$A$</span> is a <span class="math-container">$n \times n$</span> symmetric real matrix with eigenvalues <span class="math-container">$\lambda_1, \lambda_2, \ldots, \lambda_n$</span>, what are the eigenvalues of <span class="math-container">$(I - A)^{3}$</span>?</p>
<p>Are they <s... | Jake Mirra | 278,017 | <p>This is a perfect problem for proving the <em>contrapositive</em>. Suppose <span class="math-container">$ f $</span> is non-negative but <em>not</em> identically zero, and show that the integral is positive. To accomplish this, let <span class="math-container">$ x_0 $</span> be any point where <span class="math-c... |
794,736 | <blockquote>
<p>Let $60$ students and $10$ teachers. How many arrangements are there, such that, between two teachers must be exactly $6$ students? </p>
</blockquote>
<p>I know that there are $10!$ permutations for the teachers, and there are $54$ places between them for the students. Nothing said about the edges. ... | Alexander Gruber | 12,952 | <p>My phrasing would be as follows:</p>
<blockquote>
<p>Since $g(x)=g(x^2)$ for any $x$, given $x_0\ne 0$, we have that $$g(x_0)=g(x_0^2)=g((-x_0)^2)=g(-x_0),$$ so it suffices to consider $x_0>0$. In this case we have $$g(x_0)=g(x_0^{1/2})=g(x_0^{1/2})=g(x_0^{1/4})=\cdots =g(x_0^{1/2^n})=\cdots$$
Given an $\ep... |
1,115,222 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is a continuous, strictly increasing function defined on a closed interval <span class="math-container">$[a,b]$</span> such that <span class="math-container">$f^{-1}$</span> is the inverse function of <span class="math-container">$f$</span>. Prove that,
... | Vim | 191,404 | <p>I think it is very natural from a geometrical point of view. It's just about the addition of two areas, which make up a big rectangle substracting a small one. See the graph below:<img src="https://i.stack.imgur.com/nDdaL.png" alt="enter image description here"></p>
<p><br/>
Now, obviously, in the case shown in my ... |
1,115,222 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is a continuous, strictly increasing function defined on a closed interval <span class="math-container">$[a,b]$</span> such that <span class="math-container">$f^{-1}$</span> is the inverse function of <span class="math-container">$f$</span>. Prove that,
... | Micah | 30,836 | <p>I think you want to start by proving three things:</p>
<ol>
<li>The theorem holds for linear functions.</li>
<li>The theorem holds for piecewise-defined functions, if it holds for each individual piece and everything remains monotone and continuous.</li>
<li>The trapezoidal rule can be used to approximate the integ... |
1,115,222 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is a continuous, strictly increasing function defined on a closed interval <span class="math-container">$[a,b]$</span> such that <span class="math-container">$f^{-1}$</span> is the inverse function of <span class="math-container">$f$</span>. Prove that,
... | adbforlife | 859,736 | <p>Let <span class="math-container">$P' = \{f(x_0=a), f(x_1), \cdots, f(x_n=b)\}$</span> be a partition of <span class="math-container">$f^{-1}$</span> on <span class="math-container">$[f(a), f(b)]$</span>. Then we know <span class="math-container">$P = \{x_0, \cdots, x_n\}$</span> is a partition of <span class="math-c... |
1,903,235 | <p>According to Wikipedia, </p>
<blockquote>
<p>Hilbert space [...] extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions</p>
</blockquote>
<p>However, the article on Euclidean space states a... | Pand | 1,143,192 | <ol>
<li>A Hilbert space does not have to be infinite dimensional (it could be).</li>
<li>The Euclidean space is an example of a finite dimensional (n- dimensional) Hilbert space where the scalar field is the set of real numbers, i.e., <span class="math-container">$\mathbb{R}^n$</span>.</li>
<li>It is best to leave out... |
2,324,850 | <p>How to find the shortest distance from line to parabola?</p>
<p>parabola: $$2x^2-4xy+2y^2-x-y=0$$and the line is: $$9x-7y+16=0$$
Already tried use this formula for distance:
$$\frac{|ax_{0}+by_{0}+c|}{\sqrt{a^2+b^2}}$$</p>
| farruhota | 425,072 | <p>Look at the graphs:</p>
<p><a href="https://i.stack.imgur.com/N2IGw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/N2IGw.png" alt="enter image description here" /></a></p>
<p>The tangent line to the parabola at the point <span class="math-container">$(x_0,y_0)$</span> is: <span class="math-contai... |
270,495 | <p>From <a href="http://en.wikipedia.org/wiki/Ultrafilter#Generalization_to_partial_orders">Wikipedia</a></p>
<blockquote>
<p>an ultrafilter $U$ on a set $X$ is a collection of subsets of $X$ that is a filter, that cannot be enlarged (as a filter). <strong>An ultrafilter may be considered as a finitely additive meas... | Brian M. Scott | 12,042 | <p>It does not say that every finitely additive measure induces an ultrafilter. However, it is true that every non-trivial <span class="math-container">$\{0,1\}$</span>-valued finitely additive measure on <span class="math-container">$\wp(X)$</span> induces an ultrafilter on <span class="math-container">$X$</span>.</p>... |
270,495 | <p>From <a href="http://en.wikipedia.org/wiki/Ultrafilter#Generalization_to_partial_orders">Wikipedia</a></p>
<blockquote>
<p>an ultrafilter $U$ on a set $X$ is a collection of subsets of $X$ that is a filter, that cannot be enlarged (as a filter). <strong>An ultrafilter may be considered as a finitely additive meas... | André Nicolas | 6,312 | <p>Yes, it is enough (if we insist that the whole set has measure $1$). We need to check that any superset of a set of measure $1$ has measure $1$ (easy), and that the intersection of two sets of measure $1$ has measure $1$. </p>
<p>So let $A$ and $B$ have measure $1$. Then $A\cup B$ has measure $1$, and is the disjoi... |
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Makoto Kato | 28,422 | <p>Let $p$ be an odd prime number.
$p = x^2 + y^2$ has integer solutions if and only if $p \equiv 1$ (mod $4$).
This can be elegantly proved by using the properties of the ring $\mathbb{Z}[i]$.</p>
<p><a href="http://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares" rel="nofollow">http://en.wi... |
3,407,489 | <p><span class="math-container">$\neg\left (\neg{\left (A\setminus A \right )}\setminus A \right )$</span></p>
<p><span class="math-container">$A\setminus A $</span> is simply empty set and <span class="math-container">$\neg$</span> of that is again empty set. Empty set <span class="math-container">$\setminus$</span... | Bram28 | 256,001 | <p>If by 'negation' you mean complement, then the 'negation' of the empty set is not the empty set, but the universal set ... or at least: the set of all things you are talking about: the Universe of Discourse. If we call that <span class="math-container">$U$</span>, we get:</p>
<p><span class="math-container">$((A\se... |
567,683 | <p>Let $F:\mathbb R^2\to \mathbb R^2$ be the force field with </p>
<p>$$F(x,y) = -\frac{(x,y)}{\sqrt{x^2 + y^2}}$$</p>
<p>the unit vector in the direction from $(x,y)$ to the origin. Calculate the work done against the force field in moving a particle from $(2a,0)$ to the origin along the top half of the circle $(x−a... | user108946 | 108,946 | <p>Your vector field is conservative: $\nabla \times F = 0$. Thus the integral is path independent. This should simply your calculation considerably—choose the easy straight line path from $(2a,0)$ to $(0,0)$ and integrate.</p>
|
241,612 | <blockquote>
<p>Find all eigenvalues and eigenvectors:</p>
<p>a.) $\pmatrix{i&1\\0&-1+i}$</p>
<p>b.) $\pmatrix{\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta}$</p>
</blockquote>
<p>For a I got:
$$\operatorname{det} \pmatrix{i-\lambda&1\\0&-1+i-\lambda}= \lambda^{2} - 2\lambda ... | Smajl | 43,803 | <p>You can find eigen values by putting $\det(A-\lambda E)=0$.</p>
<p>If you want to find corresponding eigenvectors, too, try solving this equation:</p>
<p>$Av=\lambda v$, where v is an eigenvector for $\lambda$ in this equation. In other termss:
$Av_i=\lambda_i v_i$</p>
|
4,076,006 | <p>I would like to know the number of valuation rings of <span class="math-container">$\Bbb Q_p((T))$</span>.
I know <span class="math-container">$\Bbb Q_p$</span> has <span class="math-container">$2$</span> valuation rings, that is,<span class="math-container">$\Bbb Q_p$</span> and <span class="math-container">$\Bbb ... | Torsten Schoeneberg | 96,384 | <p>A <a href="https://en.wikipedia.org/wiki/Valuation_ring" rel="nofollow noreferrer">valuation ring</a> as you define it in a comment can also be described via a valuation, i.e. a surjective homomorphism <span class="math-container">$v: K^\times \twoheadrightarrow \Gamma$</span> onto a totally ordered abelian group <s... |
2,267,165 | <p>Find the distance between the two parallel planes?</p>
<p>$$a: x-2y+3z-2=0$$
$$b: 2x-4y+6z-1=0$$</p>
<p>The given answer is: $\dfrac{3}{\sqrt{56}}$</p>
| Dr. Sonnhard Graubner | 175,066 | <p>converting the equation of one plane into the Hessian normal form we get
$$\frac{x-2y+3x-2}{\pm \sqrt{1+4+9}}=0$$ and taking a Point from the other plane, for.e.g $$P(\frac{1}{2};0;0)$$ plugging this in the Hessian form we get
$$\frac{\frac{3}{2}}{\sqrt{14}}$$</p>
|
3,933,296 | <p>What I already have,</p>
<ol>
<li>Palindrome in form XYZYX, where X can’t be 0.</li>
<li>Divisibility rule of 9: sum of digits is divisible by 9. So, we have 2(X+Y)+Z = 9M.</li>
<li>The first part is divisible by 9 if and only if X+Y is divisible by 9. So, we have 10 pairs out of 90. And each such pair the total sum... | Explorer | 630,833 | <p>As you have rightly mentioned, we need to figure out all tuples of <span class="math-container">$(X,Y,Z)$</span> such that <span class="math-container">$$2(X+Y)+Z\bmod 9 = 0 \implies 2(X+Y)\bmod 9 = (9-Z)\bmod 9.\tag{1}$$</span>
Note that for any value of <span class="math-container">$2(X+Y)\bmod 9=0,1,\ldots,8$</sp... |
3,933,296 | <p>What I already have,</p>
<ol>
<li>Palindrome in form XYZYX, where X can’t be 0.</li>
<li>Divisibility rule of 9: sum of digits is divisible by 9. So, we have 2(X+Y)+Z = 9M.</li>
<li>The first part is divisible by 9 if and only if X+Y is divisible by 9. So, we have 10 pairs out of 90. And each such pair the total sum... | paw88789 | 147,810 | <p>Choose digits <span class="math-container">$Y$</span> and <span class="math-container">$Z$</span> freely (<span class="math-container">$100$</span> ways to do this). Then there is a unique nonzero digit <span class="math-container">$X$</span> so that <span class="math-container">$2X+2Y+Z\equiv 0 \pmod{9}$</span>.</p... |
885,450 | <blockquote>
<p>After covering a distance of 30Km with a uniform speed, there got some
defect in train engine and therefore its speed is reduced to 4/5 of
its original speed. Consequently, the train reaches its destination 45
minutes late. If it had happened after covering 18Km of distance, the
train would h... | cactus314 | 4,997 | <p>$$\log \left(1+\tfrac{1}{2^{2^n}}\right)\left(1+\tfrac{1}{2^{2^n}+2}\right)\cdots\left(1+\tfrac{1}{2^{
2^n+1}}\right)
= \log \left(1+\tfrac{1}{2^{2^n}}\right)+
\log \left(1+\tfrac{1}{2^{2^n}+2}\right)+\cdots + \log\left(1+\tfrac{1}{2^{
2^n+1}}\right)$$</p>
<p>Expanding on Alex' idea let $t = 2^{2^n}$ which is getti... |
1,024,068 | <p>I need to solve these two equations . </p>
<p>$ 2x + 4y + 3x^{2} + 4xy =0$</p>
<p>$ 4x + 8y + 2x^{2} + 4y^{3}$ = $0 $</p>
<p>I have added them , subtracted them . Nothing is helping here . Can anyone give hints ? Thanks</p>
| Empy2 | 81,790 | <p>Rearrange the first equation to get $y$ as a function of $x$. Plug that into the second equation.<br>
I think that, after rearrangement, you should get a polynomial in $x$ of degree 6.</p>
|
1,189,216 | <p>Wikipedia and other sources claim that </p>
<p>$PA +\neg G_{PA}$</p>
<p>can be consistent, where $\neg G_{PA}$ is the Gödel statement for PA.</p>
<p>So what is the error in my reasoning?</p>
<p>$G_{PA}$ = "$G_{PA}$ is unprovable in PA"</p>
<p>$\neg G_{PA} $</p>
<p>$\implies$ $\neg$ "$G_{PA}$ is unprovable in P... | Mauro ALLEGRANZA | 108,274 | <p><em>Long comment</em>, regarding the "deductive flaw" in your argument.</p>
<p>We have that <a href="http://plato.stanford.edu/entries/goedel-incompleteness/" rel="nofollow">Gödel's First Incompleteness Theorem</a> needs the Gödel's sentence $G_{\mathsf {PA}}$ such that :</p>
<blockquote>
<p>$\mathsf {PA} \vdas... |
1,248,331 | <p>That's the question :</p>
<p>Let $a$ be a cardinality such that this following statment is true :</p>
<p>For every $A, C$, if $ A \subseteq C$, $|A| = a$ and $|C| > a$, then $|C \setminus A| > |A|$.</p>
<p>Without using cardinality arithmethics, prove that $a + a = a$.</p>
<p>This is how the question is wr... | martini | 15,379 | <p>You gave <strong>one</strong> $A$, $C$, <em>but</em>: The statement says <em>all</em> $A$, $C$. For $a=2$, $C = \{1,2,3\}$, $A = \{1,2\}$, is a counterexample, as $|C\setminus A| = 1$.</p>
|
1,248,331 | <p>That's the question :</p>
<p>Let $a$ be a cardinality such that this following statment is true :</p>
<p>For every $A, C$, if $ A \subseteq C$, $|A| = a$ and $|C| > a$, then $|C \setminus A| > |A|$.</p>
<p>Without using cardinality arithmethics, prove that $a + a = a$.</p>
<p>This is how the question is wr... | Asaf Karagila | 622 | <p><strong>HINT:</strong> Take $C=A\times\{0,1\}$. If $|C|>|A|$, then it cannot be that $C$ can be split into two parts both of size $A$. </p>
|
3,846,717 | <p>Denote <span class="math-container">$\mathbb{F}=\mathbb{C}$</span> or <span class="math-container">$\mathbb{R}$</span>.</p>
<p><strong>Theorem (Cauchy - Schwarz Inequality).</strong> <em>If <span class="math-container">$\langle\cdot,\cdot\rangle$</span> is a semi-inner product on a vector space <span class="math-con... | Bernard | 202,857 | <p>There's a high-school theorem on the sign of a quadratic polynomial:</p>
<blockquote>
<p>A quadratic polynomial <span class="math-container">$p(x)=ax^2+bx+c\quad(a\ne 0)$</span> has the sign of its leading coefficient, except between its (real) roots, if any.</p>
</blockquote>
|
1,855,748 | <blockquote>
<blockquote>
<p>Find a solution of the differential equation: $$\frac{d\left(x^2\frac{dy}{dx}\right)}{dx}=x\frac{dy}{dx}-y+5$$</p>
</blockquote>
</blockquote>
<hr>
<p>What I have attempted:</p>
<p>Consider: $$\frac{d\left(x^2\frac{dy}{dx}\right)}{dx}=x\frac{dy}{dx}-y+5$$</p>
<p>$$ \frac{d}{dx}... | Behrouz Maleki | 343,616 | <p><strong>Hint:</strong></p>
<p>set $x=e^{t}$ we have
$$\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=\frac{1}{x}\frac{dy}{dt}$$
$$\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{1}{x}\frac{dy}{dt}\right)=-\frac{1}{x^2}\frac{dy}{dt}+\frac{1}{x^2}\frac{d^2y}{dt^2}$$
we have
$$\frac{d^2y}{dt^2}+y=5$$</p>
|
1,855,748 | <blockquote>
<blockquote>
<p>Find a solution of the differential equation: $$\frac{d\left(x^2\frac{dy}{dx}\right)}{dx}=x\frac{dy}{dx}-y+5$$</p>
</blockquote>
</blockquote>
<hr>
<p>What I have attempted:</p>
<p>Consider: $$\frac{d\left(x^2\frac{dy}{dx}\right)}{dx}=x\frac{dy}{dx}-y+5$$</p>
<p>$$ \frac{d}{dx}... | Jan Eerland | 226,665 | <p>$$\frac{\text{d}}{\text{d}x}\left(x^2\cdot\frac{\text{d}}{\text{d}x}\left(y(x)\right)\right)=x\cdot\frac{\text{d}}{\text{d}x}\left(y(x)\right)-y(x)+5\Longleftrightarrow$$
$$x(xy''(x)+2y'(x))=xy'(x)-y(x)+5\Longleftrightarrow$$</p>
<hr>
<p>The general solution will be the sum of the complementary solution
and partic... |
741,436 | <p>I get stuck at the following question:</p>
<p>Consider the matrix<br>
$$A=\begin{bmatrix}
0 & 2 & 0 \\
1 & 1 & -1 \\
-1 & 1 & 1\\
\end{bmatrix}$$</p>
<p>Find $A^{1000}$ by using the Cayley-Hamilton theorem.</p>
<p>I find the characteristic polynomial by $P(A) = -A^{3} + 2A^2 = 0$ (by Cayle... | Martín-Blas Pérez Pinilla | 98,199 | <p>$$A^{1000}= A(A^3)^{333}=A (-2A^2)^{333}=(-2)^{333}A^{667}=\cdots$$</p>
|
566,993 | <p>Suppose $f(z)=1/(1+z^2)$ and we want to find the power series in $a=1$. I think we have to write $1/(1+z^2)=1/(1+(z-1)+1)^2=1/(1+(1+(z-1)^2+2(z-1)))$, but I'm stuck here.</p>
| medicu | 65,848 | <p>Elementary solution to this problem. </p>
<p>The problem is equivalent to that the $x^2 \in Z(G)$ for any $x\in G$, as stated in a previous response. </p>
<p>Under these conditions we show that $(xyx^{-1}y^{-1})^4=e.$
$$(xyx^{-1}y^{-1})^4=(xyx^{-1}y^{-1})^2(xyx^{-1}y^{-1})(xyx^{-1}y^{-1})=$$
$$=(xyx^{-1})(xyx^{-1}... |
11,244 | <p>In order to evaluate new educational material the contentment of students with this material is often measured. However, just because a student is contented doesn't mean that he/she has actually learned something. Is there any research investigating the correlation between students contentment and the educational qu... | Daniel R. Collins | 5,563 | <p>From Clark, Richard, Paul A. Kirschner, and John Sweller. "Putting students on the path to learning: The case for fully guided instruction." (2012):</p>
<blockquote>
<p>Even more disturbing is evidence that when learners are asked to
select between a more-guided or less-guided version of the same
course, less... |
3,592,747 | <p>How to solve this equation for <span class="math-container">$x$</span> in reals, without using theory of complex number?
<span class="math-container">$$\frac{a}{\left(x+\frac{1}{x}\right)^{2}}+\frac{b}{\left(x-\frac{1}{x}\right)^{2}}=1$$</span>
Where <span class="math-container">$a$</span> and <span class="math-cont... | Jean Marie | 305,862 | <p>Here is an approach yielding solutions by reducing the issue <strong>to solve successive quadratic equations</strong> :</p>
<p>Let us write the expression under the form :</p>
<p><span class="math-container">$$\frac{a}{X^{2}}+\frac{b}{X^{2}-4}=1\tag{1}$$</span></p>
<p>where we have set : </p>
<p><span class="ma... |
1,227,609 | <p>Let $X,Y,Z$ be finite sets, and consider probability distributions $p$ over $X\times Y\times Z$. If we know the marginals of $p$ over all the pairs $X\times Y$, $X\times Z$ and $Y\times Z$, is that enough to pin down $p$ uniquely?</p>
| celtschk | 34,930 | <p>No. Consider $X=Y=Z=\{0,1\}$ and the following two probability distributions:</p>
<ul>
<li>$p_1(x,y,z) = \frac18$</li>
<li>$p_2(x,y,z) = \frac18\left(1 + (-1)^{x+y+z}\right)$</li>
</ul>
<p>It is easily checked that in both cases, all marginal distributions are equally distributed.</p>
|
204,106 | <p>I would like to know what the definition of a short proof is.</p>
<p>In Lance Fortnow’s article “<a href="http://cacm.acm.org/magazines/2009/9/38904-the-status-of-the-p-versus-np-problem/fulltext" rel="nofollow">The Status of the P Versus NP Problem</a>”, Communications of the ACM, Vol. 52 No. 9, he says,</p>
<blo... | Andreas Blass | 6,794 | <p>The statement you quoted is somewhat sloppy, since there is no precise notion of a short proof for a single formula. There is, however, a notion of short proofs for a class $C$ of formulas, when the class contains formulas of arbitrarily high length. One says that $C$ admits short proofs if there is a polynomial $... |
667,371 | <p>I try to solve this equation:
$$\sqrt{x+2}+\sqrt{x-3}=\sqrt{3x+4}$$</p>
<p>So what i did was:</p>
<p>$$x+2+2*\sqrt{x+2}*\sqrt{x-3}+x-3=3x+4$$</p>
<p>$$2*\sqrt{x+2}*\sqrt{x-3}=x+5$$</p>
<p>$$4*{(x+2)}*(x-3)=x^2+25+10x$$</p>
<p>$$4x^2-4x-24=x^2+25+10x$$</p>
<p>$$3x^2-14x-49$$</p>
<p>But this seems to be wrong! ... | Magdiragdag | 35,584 | <p>Note: the original question read $= \sqrt{3x + 5}$ instead of $= \sqrt{3x + 4}$. There is no problem with the fixed question. Maybe just that it's unfinished. The final line should read $3x^2 - 14x + 49 = 0$ rather than just $x^2 - 14x + 49$. After that, solve the quadratic; only one of the solutions ($7$) to the qu... |
215,858 | <p>If, $$\mathcal L \left\{ \frac{\cos(2\sqrt{3t})}{\sqrt{\pi t}} \right\}= \frac{\exp\big(\frac{-3}{s}\big)}{\sqrt{s}}$$, $$\mathcal L^{-1} \left\{ \frac{\exp\big(\frac{-1}{s}\big)}{\sqrt{s}}\right\}=?$$</p>
<p>could help</p>
<p><a href="https://math.stackexchange.com/questions/215831/laplace-transform-proof-that-l-... | Fly by Night | 38,495 | <p>A slight modification of the method you used to compute the first transform will give you:</p>
<p>$$\mathcal{L}\left(\frac{\cos\left(k\sqrt{t}\right)}{\sqrt{\pi t}}\right) = \frac{e^{-k^2/4s}}{\sqrt{s}} \, . $$</p>
<p>For your inverse transform you need $k=2$.</p>
|
215,858 | <p>If, $$\mathcal L \left\{ \frac{\cos(2\sqrt{3t})}{\sqrt{\pi t}} \right\}= \frac{\exp\big(\frac{-3}{s}\big)}{\sqrt{s}}$$, $$\mathcal L^{-1} \left\{ \frac{\exp\big(\frac{-1}{s}\big)}{\sqrt{s}}\right\}=?$$</p>
<p>could help</p>
<p><a href="https://math.stackexchange.com/questions/215831/laplace-transform-proof-that-l-... | Mikasa | 8,581 | <p>You noted that: $$\mathcal L \{ \frac{\cos(2\sqrt{3t})}{\sqrt{\pi t}} \}= \frac{\exp(\frac{-3}{s})}{\sqrt{s}}$$ and $$\mathcal L \{ \frac{1}{k}f\bigg(\frac{t}{k}\bigg) \}= F(ks)$$ and $F(s)=\frac{\exp(\frac{-3}{s})}{\sqrt{s}}$. Set $s$ to $3s$ in $F(s)$, so you have $$\sqrt{3}F(3s)=\frac{\exp(\frac{-1}{s})}{\sqrt{s... |
228,889 | <p><em>[Attention! This question requires some reading and it's answer probably is in form of a "soft-answer", i.e. it can't be translated into a hard mathematical proposition. (I hope I haven't scared away all readers with this.)]</em></p>
<p>Consider the following three examples:</p>
<p>1) <em>[If this example seem... | GEdgar | 442 | <p>A Bernstein set, $A\subseteq[0,1]$ has the property that both $A$ and $[0,1]\setminus A$ have outer measure $1$.</p>
|
2,129,362 | <p>I have this question below ,I don't know if this thing right or not but i've just wonders about it
If we have
$$
A=\int \sqrt(f_x)
$$
Can we do this move
$$
A^2=\int f_x
$$
Thanks </p>
| Shinja | 327,661 | <p>No</p>
<p>Counterexample:</p>
<p>$\int xdx = \frac{1}{2}x^2$<br>
$\int x^2dx = \frac{1}{3}x^3$</p>
<p>$(\frac{1}{2}x^2)^2=\frac{1}{4}x^4\neq\frac{1}{3}x^3$</p>
|
2,129,362 | <p>I have this question below ,I don't know if this thing right or not but i've just wonders about it
If we have
$$
A=\int \sqrt(f_x)
$$
Can we do this move
$$
A^2=\int f_x
$$
Thanks </p>
| Starz | 408,702 | <p>No. As an analogy, we can consider A as a simple summation.
$$A = x_1 + x_2 + ... + x_n$$
$$A^2 = (x_1 + x_2 + ... + x_n)^2\neq x_1^2 + x_2^2 + ... + x_n^2$$</p>
|
2,361,920 | <p>The question is as follows:
<a href="https://i.stack.imgur.com/RHPwG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RHPwG.png" alt="enter image description here"></a> </p>
<p>I can not solve this question so I am asking what exactly in the probability theory that I must revise so I could solve i... | Fimpellizzeri | 173,410 | <p>This is not so much a question on probability as it is on combinatorics.
One usually solves this kind of question as follows:</p>
<p>$\quad(1)$: Calculate the <strong>number of possible outcomes</strong>. </p>
<p>In our case, we're drawing $2$ socks out of a drawer with $8$ socks, so the number is $\binom{8}{2}=2... |
22 | <p>By matrix-defined, I mean</p>
<p>$$\left<a,b,c\right>\times\left<d,e,f\right> = \left|
\begin{array}{ccc}
i & j & k\\
a & b & c\\
d & e & f
\end{array}
\right|$$</p>
<p>...instead of the definition of the product of the magnitudes multiplied by the sign of their angle... | Zach Conn | 251 | <p>Here's an explanation in terms of the Hodge dual and the exterior (wedge) product.</p>
<p>Let ${e_1, e_2, e_3}$ be the standard orthonormal basis for $\mathbb{R}^3$. Consider the two vectors $a = a_1 e_1 + a_2 e_2 + a_3 e_3$ and $b = b_1 e_1 + b_2 e_2 + b_3 e_3$. From the matrix computation we obtain the familiar f... |
2,076 | <p>I'm attempting for the first time to create a map within <em>Mathematica</em>. In particular, I would like to take an output of points and plot them according to their lat/long values over a geographic map. I have a series of latitude/longitude values like so:</p>
<pre><code> {{32.6123, -117.041}, {40.6973, -111.9}... | Mark McClure | 36 | <h3>Here's a start.</h3>
<pre><code>latLngs={{32.6123,-117.041},{40.6973,-111.9},{34.0276,-118.046},
{40.8231,-111.986},{34.0446,-117.94},{33.7389,-118.024},
{34.122,-118.088},{37.3881,-122.252},{44.9325,-122.966},
{32.6029,-117.154},{44.7165,-123.062},{37.8475,-122.47},
{32.6833,-117.098},{44.4881,-122.797},{3... |
2,076 | <p>I'm attempting for the first time to create a map within <em>Mathematica</em>. In particular, I would like to take an output of points and plot them according to their lat/long values over a geographic map. I have a series of latitude/longitude values like so:</p>
<pre><code> {{32.6123, -117.041}, {40.6973, -111.9}... | Vitaliy Kaurov | 13 | <p>There is nice way to to put your data on rotatable 3D globe. Your data:</p>
<pre><code>centers = {{32.6123, -117.041}, {40.6973, -111.9}, {34.0276, \
-118.046}, {40.8231, -111.986}, {34.0446, -117.94}, {33.7389, \
-118.024}, {34.122, -118.088}, {37.3881, -122.252}, {44.9325, \
-122.966}, {32.6029, -117.154}, {44.71... |
552,155 | <p>I've been struggling to figure out this integral.
$$\int \frac {1}{x\sqrt{5-x^2}}$$</p>
<p>I'm almost certain it has something to do with this fact:</p>
<p>$$\int \frac 1{\sqrt{1-x^2}} = \sin^{-1}(x) + C$$</p>
<p>But I can't figure out how to use that to my advantage. No obvious substitution is jumping out at me... | André Nicolas | 6,312 | <p>A natural substitution is $u=\sqrt{5-x^2}$, though I prefer the version $u^2=5-x^2$. Then $2u\,du=-2x\,dx$.</p>
<p>Rewrite our integral as
$$\int \frac{x\,dx}{x^2\sqrt{5-x^2}}.$$
After we make the substitution, we end up with
$$\int \frac{du}{u^2-5}.$$
Factor $u^2-5$ as $(u-\sqrt{5})(u+\sqrt{5})$ and use partial f... |
552,155 | <p>I've been struggling to figure out this integral.
$$\int \frac {1}{x\sqrt{5-x^2}}$$</p>
<p>I'm almost certain it has something to do with this fact:</p>
<p>$$\int \frac 1{\sqrt{1-x^2}} = \sin^{-1}(x) + C$$</p>
<p>But I can't figure out how to use that to my advantage. No obvious substitution is jumping out at me... | Manny265 | 96,690 | <p>So substitute $ \sqrt{5-x^2} $ as u therefore ${u^2} = 5-x^2 $ right. From this keep in mind that x can be made subject of the formula and that will make $ x= \sqrt{5-u^2} $ .
<br>$ -2x dx = 2u du$ therefore $ x dx = - u du$ . You can multiply the equation by 1,which in fundamentals of mathematics there is no change... |
1,212,262 | <p>The statement goes as follow: </p>
<p>$ B ∩ C ⊆ A ⇒ (C − A) ∩ (B − A) = ∅. $</p>
<p>First, the sign "=>" represents a tautology, no? ( apparently I get it confuse with the 3 bar sign, if you know what I mean).</p>
<p>Second, the fact that it equals to no solution, how do I prove that? Seems to contradict itself, ... | Clément Guérin | 224,918 | <p>You must enumerate all the case (as you did I find the same as yours).</p>
<p>Case 1 : (2,3) 3 boxes with 1 color and 2 boxes with another color.</p>
<p>Case 2 : (1,1,3) 3 boxes with 1 color, 1 box with another color and another 1 box with another color.</p>
<p>Case 3 : (1,2,2)...</p>
<p>Case 4 : (1,1,1,2)...</p... |
1,557,733 | <p>I have a function $f(\mathbf{u}, \Sigma)$ where $\mathbf{u}$ is a $p \times 1$ vector and $\Sigma$ is a $p \times p$ real symmetric matrix (positive semi-definite).</p>
<p>I somehow successfully computed the partial derivatives $\frac{\partial f}{\partial \mathbf{u}}$ and $\frac{\partial f}{\partial \Sigma}$.</p>
... | cr001 | 254,175 | <p>As the second part is proven in the other answer, I will add the proof for the first part here. I will use the same point system as described in the other answer.</p>
<p>WLOG we consider the sum $m_a+m_b$.</p>
<p>Let the midpoint of $CF$ be $R$. Connect $BR.AR$.</p>
<p>First we show $BR+AR>BG+AG$. This is true... |
200,278 | <p>Say I have two TimeSeries:</p>
<pre><code>x = TimeSeries[{2, 4, 1, 10}, {{1, 2, 4, 5}}]
y = TimeSeries[{6, 2, 6, 3, 9}, {{1, 2, 3, 4, 5}}]
</code></pre>
<p>x has a value at times: 1,2,4,5</p>
<p>y has a value at times: 1,2,3,4,5</p>
<p>I would like to build a list of pairs {<span class="math-container">$x_i$</sp... | Carl Lange | 57,593 | <p>One way you can do this is by combining the two <code>TimeSeries</code> into <code>TemporalData</code> with no resampling:</p>
<pre><code>x = TimeSeries[{2, 4, 1, 10}, {{1, 2, 4, 5}}]
y = TimeSeries[{6, 2, 6, 3, 9}, {{1, 2, 3, 4, 5}}]
td = TemporalData[{x, y}, ResamplingMethod -> None]
</code></pre>
<p>Then, w... |
200,278 | <p>Say I have two TimeSeries:</p>
<pre><code>x = TimeSeries[{2, 4, 1, 10}, {{1, 2, 4, 5}}]
y = TimeSeries[{6, 2, 6, 3, 9}, {{1, 2, 3, 4, 5}}]
</code></pre>
<p>x has a value at times: 1,2,4,5</p>
<p>y has a value at times: 1,2,3,4,5</p>
<p>I would like to build a list of pairs {<span class="math-container">$x_i$</sp... | Anjan Kumar | 19,742 | <p>You can simply normalize the data (<code>x</code> and <code>y</code>), convert it to an <a href="https://reference.wolfram.com/language/ref/Association.html" rel="nofollow noreferrer"><code>Association</code></a> and later <a href="https://reference.wolfram.com/language/ref/Merge.html" rel="nofollow noreferrer"><cod... |
3,122,732 | <p>Let <span class="math-container">$C[a,b]$</span> denote the set of all continuous, real-valued maps on the interval <span class="math-container">$[a,b]$</span>. Let <span class="math-container">$P_n$</span> denote the set of all real polynomials on <span class="math-container">$[a,b]$</span> which have a <em>maximum... | John Hughes | 114,036 | <p><strong>A Broad Hint, but not a complete answer</strong></p>
<p>Suggestions: </p>
<ol>
<li><p>Simplify to <span class="math-container">$a = 0, b = 1$</span>, or <span class="math-container">$a = -1, b = 1$</span> to make the notation nicer. </p></li>
<li><p>You might want to look at something like <span class="mat... |
3,122,732 | <p>Let <span class="math-container">$C[a,b]$</span> denote the set of all continuous, real-valued maps on the interval <span class="math-container">$[a,b]$</span>. Let <span class="math-container">$P_n$</span> denote the set of all real polynomials on <span class="math-container">$[a,b]$</span> which have a <em>maximum... | Eric Wofsey | 86,856 | <p>First, a comment: your thinking is kind of backwards. Since <span class="math-container">$P_n$</span> is closed in <span class="math-container">$C[a,b]$</span>, <em>any</em> continuous function <span class="math-container">$f$</span> that is not a polynomial of degree <span class="math-container">$\leq n$</span> is... |
2,077,694 | <p>How to find $A = M^{A}_{B}$ in linear transformation $F = \mathbb{P_{2}} \rightarrow \mathbb{R^{2}} $, where $ F(p(t)) = \begin{pmatrix} p(0) \\ P(1) \end{pmatrix},$
$ A = \{1,t,t^{2}\},$
$B=\left \{ \begin{pmatrix} 1\\ 0\end{pmatrix},\begin{pmatrix} 0\\ 1\end{pmatrix} \right \}$?</p>
| Asinomás | 33,907 | <p>If $n=p_1^{a_1}p_2^{a_2}\dots p_r^{a_r}$ then $\frac{\sigma(n)}{n}=\prod\limits_{j=1}^r \frac{p^{a_j+1}-1}{p^j(p-1)}$.</p>
<p>Clearly each factor is an increasing function with respect to $a_j$ and every factor is greater than or equal to $1$. So increasing the number of factors or increasing the exponents only mak... |
138,658 | <p>Suppose $X$ is a topological space, and $\mu$ is a Borel measure on $X$. Also suppose we have an $n$-dimensional vector bundle $E \to X$, with an inner product $\langle \cdot,\cdot \rangle_x$ on the fibre $E_x$ for all $x \in X$, in such a way that each $E_x$ is complete and such that there exists a vector bundle tr... | Peter Michor | 26,935 | <p>Maybe the following helps:
Theorem 3.12 (page 20) in the following source has such a related result, albeit for higher Sobolev spaces.
There are quite subtle requirements for the trivialising atlas and the partition of unity which are used in the proof.</p>
<ul>
<li><a href="http://www.ams.org/mathscinet-getitem?mr=... |
1,784,679 | <p>if $p,q,r$ are three positive integers prove that</p>
<p>$$LCM(p,q,r)=\frac{pqr \times HCF(p,q,r)}{HCF(p,q) \times HCF(q,r) \times HCF(r,p)}$$</p>
<p>I tried in this way;</p>
<p>Let $HCF(p,q)=x$ hence $p=xm$ and $q=xn$ where $m$ and $n$ are relatively prime.</p>
<p>similarly let $HCF(q,r)=y$ hence $q=ym_1$ and $... | Ryder Rude | 445,404 | <p>Let <span class="math-container">$P, Q$</span> and <span class="math-container">$R$</span> be the sets of prime factors of the numbers <span class="math-container">$p,q$</span> and <span class="math-container">$r$</span> respectively. Let the universal set be <span class="math-container">$P\cup Q\cup R$</span>. Let ... |
66,370 | <p>Let $(X,\mathcal{E},\mu)$ be a measure space. Let $u,v$ be $\mu$-measurable functions. If $0 \leq u \leq v$ and $\int_X v d\mu$ exists we know that $\int_X u d\mu \leq \int_X v d\mu$.</p>
<p>I wanted to know if $0 \leq u < v$ and $\int_X v d\mu$ exists then is it true that
$\int_X u d\mu < \int_X v d\mu$? Th... | Did | 6,179 | <p>By contraposition, you might want to prove that if $w\ge0$ and $\displaystyle\int\limits_Xw\mathrm d\mu=0$ then $w=0$ $\mu$-almost everywhere. To see this, consider $A_n=\{x\mid w(x)\ge1/n\}$ and note that $w\ge n^{-1}\mathbf 1_{A_n}$ hence $\displaystyle\int\limits_Xw\mathrm d\mu\ge n^{-1}\mu(A_n)$ hence $\mu(A_n)=... |
2,512,556 | <p>What would be the solution of $ y''+y=\cos (ax) \ $ if $ \ a \to 1 \ $. </p>
<p><strong>Answer:</strong></p>
<p>I have found the complementary function $ y_c \ $ </p>
<p>$ y_c(x)=A \cos x+B \sin x \ $</p>
<p>But How can I find the particular integral if $ a \to 1 \ $ </p>
| Dr. Sonnhard Graubner | 175,066 | <p>a possible particular solution is given by $$y_P=\frac{\cos ^2(x) (-\cos (a x))-\sin ^2(x) \cos (a x)}{a^2-1}$$ now you can consider the case if $a$ tends to $1$</p>
|
3,500,799 | <p>what is the dimension of the vector space spanned by the set of vectors <span class="math-container">$(a,b,c) $</span>where <span class="math-container">$a^2+b^2=c$</span>?</p>
| KRL | 585,628 | <p>One approach is to come up with vectors <span class="math-container">$(a,b,c)$</span> that satisfy the equation and figure out if they are linearly independent. </p>
<p>Note that in your question the dimension is at most three. Here, vectors (1,0,1), (0,1,1), (1,-1,2) are linearly independent vectors that satisfy t... |
250,687 | <p>I'm doing a sanity check of the following equation:
<span class="math-container">$$\sum_{j=2}^\infty \frac{(-x)^j}{j!}\zeta(j) \approx x(\log x + 2 \gamma -1)$$</span></p>
<p>Naive comparison of the two shows a bad match but I suspect one of the graphs is incorrect.</p>
<ol>
<li>Why isn't there a warning?</li>
<li>H... | J. M.'s persistent exhaustion | 50 | <p>Using the <span class="math-container">$d$</span>-type <a href="https://doi.org/10.1016/0167-7977(89)90011-7" rel="nofollow noreferrer">Weniger transformation</a>, as implemented in <a href="https://resources.wolframcloud.com/FunctionRepository/resources/WenigerSum/" rel="nofollow noreferrer"><code>ResourceFunction[... |
3,251,337 | <p>Be <span class="math-container">$E,F,K , L,$</span> points in the sides <span class="math-container">$AB,BC,CD,DA$</span> of a square <span class="math-container">$ABCD$</span>, respectively. Show that if <span class="math-container">$EK$</span> <span class="math-container">$\perp$</span> <span class="math-containe... | DanielWainfleet | 254,665 | <p>In any topological space <span class="math-container">$X$</span> and any <span class="math-container">$E\subset X,$</span> the 3 sets <span class="math-container">$int(E),\, int(X\setminus E),\, \partial E)$</span> are pair-wise disjoint and their union is <span class="math-container">$X.$</span></p>
<p>So if <sp... |
68,803 | <p>I am trying to understand how all the players in the title relate, but with all the grading shifts,and difficult isomorphisms involved in the subject I am having a hard time being sure that I have the picture right. I am going to write what I think is true, and if someone would confirm or deny it, that would be real... | Eigenbunny | 16,006 | <p>Let me rephrase: we consider the map from the symplectic cohomology of D*N, which
(Viterbo 97, et al) is the homology of the free loop space of N, to the Hochschild
cohomology of the Fukaya category consisting of compact Lagrangian submanifolds.
If we allowed only the zero-section N, that Hochschild cohomology would... |
121,541 | <p>I'd like to pick <em>k</em> points from a set of points in <em>n</em>-dimensions that are approximately "maximally apart" (sum of pairwise distances is almost maxed). What is an efficient way to do this in MMA? Using the solution from C Woods, for example:</p>
<pre><code>KFN[list_, k_Integer?Positive] := Module[{kT... | mikado | 36,788 | <p>It is possible to write the total distance between the <code>k</code> points as a quadratic form based on the <code>DistanceMatrix</code> <code>M</code>. Finding the <code>k</code> points that maximise the total distance between them is then a matter of finding a vector <code>V</code> of zeros and ones that maximis... |
1,945,116 | <p>I need a simple definition of Disjoint cycles in Symmetric Groups.I already understand what cycles and Transpositions are. I need a simple definition and if possible,give a clear example.Thanks in advance Mathematician</p>
| paw88789 | 147,810 | <p>Disjoint cycles have no cycle elements in common. For example $(1, 2, 3)$ and $(4,5,6,7)$ are disjoint cycles. </p>
<p>By contrast, $(1,2,3)$ and $(3,4,5,6)$ are not disjoint because they have the $3$ in common.</p>
|
2,963,587 | <p>I'm working on a relatively low-level math project, and for one part of it I need to find to a function that returns how many many configurations are reachable within n moves. from the solved state.</p>
<p>Because there are 18 moves ( using the double moves metric ), one form of the function could be <span class="... | hmakholm left over Monica | 14,366 | <p><a href="http://cube20.org/" rel="noreferrer">http://cube20.org/</a> shows exact counts for <span class="math-container">$n$</span> up to 15, but only has approximate counts above that.</p>
<p>This probably means no nice formula that would make those higher values easy to compute is known. </p>
<p>(If there were a... |
366,096 | <p>Let's consider $J\subset \mathbb R^2$ such that J is convex and such that it's boundary it's a curve $\gamma$. Let's suppose that $\gamma$ is anti-clockwise oriented, let's consider it signed curvature $k_s$. I want to prove the intuitive following fact:</p>
<p>$$
\int\limits_\alpha {k_s } \left( s \right)ds \geqs... | Brian Rushton | 51,970 | <p>If the curvature is negative, there must be a point with negative curvature. As you zoom up to that point, it looks more and more like the complement of a circle, which means that there are two points which are not connected by a straight line in the set.</p>
|
2,294,548 | <p><strong>Problem:</strong> Solve $y'=\sqrt{xy}$ with the initial condition $y(0)=1$.</p>
<p><strong>Attempt:</strong> Using $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$, I get that the DE is separable by dividing both sides by $\sqrt{y}:$ $$y'=\sqrt{x}\cdot\sqrt{y}\Leftrightarrow\frac{y'}{\sqrt{y}}=\sqrt{x}$$</p>
<p>which can... | Robert Z | 299,698 | <p>A solution of this Cauchy problem can not be defined in an interval $(-r,0)$ with $r>0$ because $y(0)=1$ and, by continuity, $y(x)$ is positive in a neighborhood of $0$ whereas $x<0$ so $xy<0$ and the square root on the RHS is not defined.</p>
<p>Moreover, for $x>0$, $y'(x)=\sqrt{xy(x)}\geq 0$ implies t... |
2,663,303 | <blockquote>
<p>Let $G$ be finite. Suppose that $\left\vert \{x\in G\mid x^n =1\}\right\vert \le n$ for all $n\in \mathbb{N}$. Then $G$ is cyclic.</p>
</blockquote>
<p>What I have attempted was the fact that every element is contained in a maximal subgroup following that <a href="https://groupprops.subwiki.org/wiki/... | user120527 | 530,843 | <p>Here are some hints of a possible way to prove this:</p>
<p>1) Show that for all $n$ dividing $|G|$, there is at most one cyclic subgroup of cardinal $n$ in $G$. Call it $H_n$ (when it exists).</p>
<p>2) Look at the map $\Psi: G\to \{\text{cyclic subgps} \}$, $x\mapsto \langle x \rangle$. Compute $|\Psi^{-1}(H_n)|... |
694,668 | <p>Let (X,Y) be uniformly distributed in a circle of radius 1. Show that if R is the distance from the center of the circle to (X,Y) then $R^2$ is uniform on (0,1). </p>
<p>This is question from the Simulation text of Prof. Sheldon Ross. Any hints? </p>
| Unwisdom | 124,220 | <p>There are lots of approaches one could take, but the simplest one I can think of is to consider the shape of the CDF of $R^2$. </p>
<p>What is the probability that $R^{2}<t$? Well, this is clearly $0$ for $t<0$ and $1$ for $t>1$. For $t\in [0,1]$, the probability is the area of the circle of radius $R$ (di... |
445 | <p>Under what circumstances should a question be made community wiki?</p>
<p>Probably any question asking for a list of something (e.g. <a href="https://math.stackexchange.com/questions/81/list-of-interesting-math-blogs">1</a>) must be CW. What else? What about questions asking for a list of applications of something ... | Grigory M | 152 | <p>I think we still need a “CW-policy”. Suggestion:</p>
<ol>
<li><code>big-list</code> questions, asking for sorted list of resources/books should be CW
<ul>
<li>and, probably, questions asking for the best example/intuition/etc in some field (including questions of most interesting propeties of some objec... |
3,978,606 | <p>Question says</p>
<blockquote>
<p>For <span class="math-container">$(C[0,1], \Vert\cdot\Vert_{\infty})$</span>, let <span class="math-container">$B=\{f\in C[0,1] :
\Vert f\Vert_{\infty} \leq 1\}$</span>. Find all <span class="math-container">$f\in B$</span> such that
there exist <span class="math-container">$g,h\in... | Kavi Rama Murthy | 142,385 | <p>Hint for a rigorous proof: Suppose <span class="math-container">$|f(c)| <1$</span> for some <span class="math-container">$c$</span>. Let <span class="math-container">$r=1-|f(c)|$</span>. Let <span class="math-container">$g_n$</span> be a piece-wise linear function such that <span class="math-container">$g_n(c)=\f... |
2,502,617 | <p>My teacher said this is the case because $A_\infty$ is generated by elements of order $3$ (the 3-cycles), and $S_\infty$ is not. I understand that the 3-cycles do not generate $S_n$, and that $\phi(x) = y \implies \text{order}(x) = \text{order}(y)$, but why can't there be an isomorphism between another set of genera... | Derek Holt | 2,820 | <p>Suppose that $\phi:A_\infty \to S_\infty$ is an isomorphism. We know that $A_\infty = \langle X \rangle$, where $X$ is the set of $3$-cycles. It follows that $S_\infty = \phi(\langle X \rangle) = \langle \{\phi(x) : x \in X \} \rangle$.</p>
<p>Since $\phi(x)$ has order $3$ for all $x \in X$, it follows that $S_\inf... |
2,502,617 | <p>My teacher said this is the case because $A_\infty$ is generated by elements of order $3$ (the 3-cycles), and $S_\infty$ is not. I understand that the 3-cycles do not generate $S_n$, and that $\phi(x) = y \implies \text{order}(x) = \text{order}(y)$, but why can't there be an isomorphism between another set of genera... | orangeskid | 168,051 | <p>Every element in $A_{\infty}$ is a product of an even number of transpositions. If $\tau_1$, $\tau_2$, $\tau$ are transpositions, we have
$\tau_1 \tau_2 = (\tau_1 \tau)( \tau \tau_2)$. Hence every element in $A_{\infty}$ can be written as a product of an even number of elements of the form $\eta_1 \eta_2$, where $... |
94,525 | <p>I am trying to solve the equation
$$z^n = 1.$$</p>
<p>Taking $\log$ on both sides I get $n\log(z) = \log(1) = 0$.</p>
<p>$\implies$ $n = 0$ or $\log(z) = 0$</p>
<p>$\implies$ $n = 0$ or $z = 1$.</p>
<p>But I clearly missed out $(-1)^{\text{even numbers}}$ which is equal to $1$.</p>
<p>How do I solve this equati... | nb1 | 15,767 | <p>First of all, note that, apart from $z=1$, all other answers will be complex. For the equation</p>
<p>$z^n=1$, we use the theorems $e^{ix}=cosx + i sin x$ and that $(cosx + i sin x)^{n}=cos(nx)+ sin(nx)$.</p>
<p>Then, $z^n=1$ implies $r^n(cos(nx)+ sin(nx))=1$, where $z=re^{ix}$. Hence, $r=1$ and</p>
<p>$cos(nx)+ ... |
3,744,560 | <p>Suppose I have a function <span class="math-container">$\Lambda(t)$</span> for any <span class="math-container">$t>0$</span>. This function has the following three properties:</p>
<ol>
<li><span class="math-container">$\Lambda(t)$</span> is differentiable.</li>
<li><span class="math-container">$\Lambda(t)$</span>... | Cardinal | 254,200 | <p><span class="math-container">$\frac{d}{dt}\Lambda(t) > 0 \rightarrow \Lambda(t) = C t + u$</span></p>
<p><span class="math-container">$\Lambda(S+T) = \Lambda(S) + \Lambda(T) \rightarrow u=0$</span></p>
<p>Hence</p>
<p><span class="math-container">$\Lambda(t) = Ct$</span></p>
|
3,744,560 | <p>Suppose I have a function <span class="math-container">$\Lambda(t)$</span> for any <span class="math-container">$t>0$</span>. This function has the following three properties:</p>
<ol>
<li><span class="math-container">$\Lambda(t)$</span> is differentiable.</li>
<li><span class="math-container">$\Lambda(t)$</span>... | ir7 | 26,651 | <p><strong>Hint:</strong> Try proving these properties first, by giving <span class="math-container">$S$</span> and <span class="math-container">$T$</span> various values:
<span class="math-container">$$\Lambda(0)=0$$</span>
<span class="math-container">$$\Lambda(2t) = 2\Lambda(t)$$</span>
<span class="math-container">... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Akhil Mathew | 344 | <p>A groupoid is a category where every morphism is invertible. If such a category has one object, then it is a group. Unfortunately I don't know why people are so interested in them, so perhaps this is not helpful.</p>
<p>An example is the fundamental groupoid of path classes in a topological space; the objects ar... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Harrison Brown | 382 | <p>I also unfortunately don't really understand why people care so much about them, although I should probably go back and read old TWFs. </p>
<p>Sort of a combinatorial example of the fundamental groupoid is the category assigned to a graph where the objects are vertices and the morphisms are directed paths, and v->w... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Yemon Choi | 763 | <p>In addition to the answers already given: Alan Weinstein wrote a nice <a href="http://www.ams.org/notices/199607/weinstein.pdf">article for the Notices of the AMS</a> which tries to give some motivating examples:</p>
<p>It seems that in certain situations where taking the quotient by a group action "destroys too mu... |
2,409,918 | <p>I need your help in evaluating the following integral in <strong>closed form</strong>. <span class="math-container">$$\displaystyle\int\limits_{0.5}^{1}
\frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$$</span></p>
<p>Since the function is singular at <span class="math-container">$x = 0.5$... | James Arathoon | 448,397 | <p>This provisional answer is an horribly inelegant conjecture with no proof attached</p>
<p>$$\int\limits_{0.5}^{1}\frac{Li_2(x) \ln(2x-1)}{x} dx
=-\frac{1}{2}\sum_{n=1}^{\infty}\frac{\sum_{k=0}^{n-1} \frac{\binom{n-1}{k}}{(k+1)^2}}{ \sum_{k=0}^n (k (2 k-1)) \binom{n}{k}}\tag1$$</p>
<p>I have checked this solution u... |
2,409,918 | <p>I need your help in evaluating the following integral in <strong>closed form</strong>. <span class="math-container">$$\displaystyle\int\limits_{0.5}^{1}
\frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$$</span></p>
<p>Since the function is singular at <span class="math-container">$x = 0.5$... | Knas | 634,505 | <p>I will be using next integral
<span class="math-container">$$
\operatorname{Lv}_{n}(x,\alpha) =\int\limits_{0}^{x} \dfrac{\ln^n t}{1-\alpha t}\,\mathrm{d}t = \dfrac{n!}{\alpha}\sum\limits_{k\,=\,0}^{n}\dfrac{(-1)^k}{\left(n-k\right)!}\ln^{n-k}x\operatorname{Li}_{k+1}(\alpha x)
$$</span>
with <span class="math-contai... |
2,409,918 | <p>I need your help in evaluating the following integral in <strong>closed form</strong>. <span class="math-container">$$\displaystyle\int\limits_{0.5}^{1}
\frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$$</span></p>
<p>Since the function is singular at <span class="math-container">$x = 0.5$... | Ali Shadhar | 432,085 | <p>Start with subbing <span class="math-container">$ 2x-1\to x$</span> then integrate by parts we have </p>
<p><span class="math-container">$$I=-\frac54\zeta(4)+\int_0^1\frac{\text{Li}_2(-x)}{1+x}\ln\left(\frac{1-x}{2}\right)\ dx+\int_0^1\frac{\ln(x)\ln(1+x)}{1+x}\ln\left(\frac{1-x}{2}\right)\ dx$$</span></p>
<p><spa... |
929,532 | <p>Okay so I want some hints (not solutions) on figuring out whether these sets are open, closed or neither.</p>
<p>$A = \{ (x,y,z) \in \mathbb{R}^3\ \ | \ \ |x^2+y^2+z^2|\lt2 \ and \ |z| \lt 1 \} \\ B = \{(x,y) \in \mathbb{R}^2 \ | \ y=2x^2\}$</p>
<p>Okay so since this question is the last part of the question where... | Mike J | 271,776 | <p>We set $\Omega=N$, $F$=$2^{N}$, $\mu (A)=0$, if $|A|<\infty$; $\mu (A)=\infty$, if $|A|=\infty$.</p>
<p>We can see that $\mu$ is finite additive. Given a finite collection of disjoint $A_k$,1$\leq$ k $\leq$ n. If every $|A_k|<\infty$, then $|\bigcup_{k=1}^n A_k|<\infty$, so\ $\mu(\bigcup_{k=1}^n A_k)=0=\su... |
744,952 | <p>Is it true that a map between ${\bf T1}$ topological spaces $f:X \to Y$ is surjective iff the induced geometric morphism $f:Sh(Y) \to Sh(X)$ is a surjection (i.e. its inverse image part $f^*$ is faithful)?</p>
<p>In "Sheaves in Geometry and Logic" a proof is given, but the the "if" part leaves me a bit unsatisfied ... | Lutz Lehmann | 115,115 | <p>I'm not really sure that there is a simple solution for that. In some sense, this limit is equivalent to the theorem about the inverse Fourier transform, so there will be no simple solutions. But use Parseval/Plancherel (I never remember which one is about the Fourier series).</p>
<p>Or proceeding more elementary: ... |
744,952 | <p>Is it true that a map between ${\bf T1}$ topological spaces $f:X \to Y$ is surjective iff the induced geometric morphism $f:Sh(Y) \to Sh(X)$ is a surjection (i.e. its inverse image part $f^*$ is faithful)?</p>
<p>In "Sheaves in Geometry and Logic" a proof is given, but the the "if" part leaves me a bit unsatisfied ... | Thorben | 135,025 | <p>Im really not sure if this works but i thought I try,</p>
<p>As you mentioned the claim, $$lim_{a\rightarrow\infty}\int_0^{\infty}f(x)\frac{sin(ax)}{x}dx=\pi/2f(0)$$
follows after your substitution as $f(x)$ is continuous at $0$ when we can move the limit inside.</p>
<p>So I started to construct $g(x)\in L^1(\math... |
94,501 | <p>The well-known Vandermonde convolution gives us the closed form <span class="math-container">$$\sum_{k=0}^n {r\choose k}{s\choose n-k} = {r+s \choose n}$$</span>
For the case <span class="math-container">$r=s$</span>, it is also known that <span class="math-container">$$\sum_{k=0}^n (-1)^k {r \choose k} {r \choose n... | anon | 11,763 | <p>$$(1-x)^r(1+x)^s=\left(\sum_{g=0}^r (-x)^g{r\choose g}\right)\left(\sum_{h=0}^sx^h{s\choose h}\right)$$</p>
<p>$$\implies \sum_{k=0}^n(-1)^k{r\choose k}{s\choose n-k}=[x^n](1-x)^r(1+x)^s.$$</p>
<p>How closed would you consider this? I'm not sure if it gets simpler, but obviously it tells us</p>
<p>$$\sum_{k=0}^n(... |
879,886 | <p>If one number is thrice the other and their sum is $16$, find the numbers.</p>
<p>I tried,
Let the first number be $x$ and the second number be $y$
Acc. to question </p>
<p>$$
\begin{align}
x&=3y &\iff x-3y=0 &&(1)\\
x&=16-3y&&&(2)
\end{align}
$$</p>
| Mikasa | 8,581 | <p>I assume you have $$x=\color{red}{3y}, ~~x+y=16$$ Then $3(x+\color{red}{y})=3\times 16=48$ and so $3x+\color{red}{3y}=48$ and so $3x+x=48$ and so $4x=48$...</p>
|
879,886 | <p>If one number is thrice the other and their sum is $16$, find the numbers.</p>
<p>I tried,
Let the first number be $x$ and the second number be $y$
Acc. to question </p>
<p>$$
\begin{align}
x&=3y &\iff x-3y=0 &&(1)\\
x&=16-3y&&&(2)
\end{align}
$$</p>
| Community | -1 | <p>The problem statement says
$$x+3x=16,$$
hence
$$x=4,\\3x=12.$$</p>
|
879,886 | <p>If one number is thrice the other and their sum is $16$, find the numbers.</p>
<p>I tried,
Let the first number be $x$ and the second number be $y$
Acc. to question </p>
<p>$$
\begin{align}
x&=3y &\iff x-3y=0 &&(1)\\
x&=16-3y&&&(2)
\end{align}
$$</p>
| Rachit Gupta | 558,628 | <p>Let the first number be <span class="math-container">$x$</span>.<p>
Let the second number be <span class="math-container">$y$</span>.<p>
According to question</p>
<p><span class="math-container">$$ \tag{1}
x+y=16
$$</span>
<span class="math-container">$$
\tag{2}
x=3y
$$</span>
So, <span class="math-container"... |
879,886 | <p>If one number is thrice the other and their sum is $16$, find the numbers.</p>
<p>I tried,
Let the first number be $x$ and the second number be $y$
Acc. to question </p>
<p>$$
\begin{align}
x&=3y &\iff x-3y=0 &&(1)\\
x&=16-3y&&&(2)
\end{align}
$$</p>
| user637720 | 637,720 | <p>Let the 1st number be <span class="math-container">$x$</span> and the 2nd number be <span class="math-container">$3x$</span>. Since <span class="math-container">$x + 3x = 16$</span>, <span class="math-container">$4x = 16$</span>, so <span class="math-container">$x=4$</span>. Therefore, 1st number is <span class="mat... |
40,920 | <p>We have functions $f_n\in L^1$ such that $\int f_ng$ has a limit for every $g\in L^\infty$. Does there exist a function $f\in L^1$ such that the limit equals $\int fg$? I think this is not true in general (really? - why?), then can this be true if we also know that $f_n$ belong to a certain subspace of $L^1$?</p>
| Nate Eldredge | 822 | <p>Perhaps surprisingly, the answer is yes.</p>
<p>More generally, given any Banach space $X$, a sequence $\{x_n\} \subset X$ is said to be <strong>weakly Cauchy</strong> if, for every $\ell \in X^*$, the sequence $\{\ell(f_n)\} \subset \mathbb{R}$ (or $\mathbb{C}$) is Cauchy. If every weakly Cauchy sequence is weakl... |
1,885,068 | <p>Prove $$\int_0^1 \frac{x-1}{(x+1)\log{x}} \text{d}x = \log{\frac{\pi}{2}}$$</p>
<p>Tried contouring but couldn't get anywhere with a keyhole contour.</p>
<p>Geometric Series Expansion does not look very promising either.</p>
| Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
3,494,758 | <p>If the eigen value of <span class="math-container">$n\times n$</span> matrix are not all distinct then does that imply eigen vectors are linearly dependent and hence not diagonalizable?</p>
| pre-kidney | 34,662 | <p>The graph of a <span class="math-container">$C^k$</span> function <span class="math-container">$f\colon \mathbb R\to\mathbb R$</span> is an example of a <span class="math-container">$C^k$</span> manifold. For instance, consider the graphs of the functions <span class="math-container">$|x|\cdot x^k$</span>.</p>
|
3,494,758 | <p>If the eigen value of <span class="math-container">$n\times n$</span> matrix are not all distinct then does that imply eigen vectors are linearly dependent and hence not diagonalizable?</p>
| Eric Wofsey | 86,856 | <p>Well, you can take any smooth manifold, and just consider it as a <span class="math-container">$C^k$</span> manifold for any <span class="math-container">$k<\infty$</span>. (If you define a manifold in terms of a maximal atlas, this means you enlarge the atlas to contain all charts that are <span class="math-con... |
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