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 |
|---|---|---|---|---|
1,534,246 | <p>I'm trying to simplify this boolean expression:</p>
<p>$$(AB)+(A'C)+(BC)$$</p>
<p>I'm told by every calculator online that this would be logically equivalent:</p>
<p>$(AB)+(A'C)$</p>
<p>But so far, following the rules of boolean algebra, the best that I could get to was this: </p>
<p>$(B+A')(B+C)(A+C)$</p>
<p>... | Dhruv Joshi | 522,283 | <p>Though this has already some good answer, I know an interesting way to look at boolean expression with less than 4 elements(for greater than 3 it becomes <a href="https://math.stackexchange.com/questions/1475/why-can-a-venn-diagram-for-4-sets-not-be-constructed-using-circles">messy</a> ),it might be helpful for any ... |
2,306,319 | <p>I have point $P_2(x_2, y_2, z_2)$, which is situated in euclidean coordinate system $xyz$. I want to transform this point to similar coordinate system $x'y'z'$ arbitrarily rotated (three directions are some known vectors) and centered at point $P_1(x_1, y_1, z_1)$ - center relative to system $xyz$. So new coordinate... | JMJ | 295,405 | <p>If $x,y,z$ are the vectors in the first coordinate system and $x',y',z'$ the vectors in the second the rotation matrix is given by
$$
R =
\begin{bmatrix}
x'\cdot x & x'\cdot y & x'\cdot z\\
y'\cdot x & y'\cdot y & y'\cdot z\\
z'\cdot x & z'\cdot y & z'\cdot z\\
\end{bmatrix}
$$</p>
<p... |
2,306,319 | <p>I have point $P_2(x_2, y_2, z_2)$, which is situated in euclidean coordinate system $xyz$. I want to transform this point to similar coordinate system $x'y'z'$ arbitrarily rotated (three directions are some known vectors) and centered at point $P_1(x_1, y_1, z_1)$ - center relative to system $xyz$. So new coordinate... | amd | 265,466 | <p>This coordinate transformation consists of a translation followed by a rotation: $(x',y',z')=R(P_2-P_1)$. To construct the rotation matrix $R$, recall that the columns of a transformation matrix are images of the basis vectors. Let $\mathbf u_{x'}$, $\mathbf u_{y'}$ and $\mathbf u_{z'}$ be unit vectors in the origin... |
3,298,282 | <p>Find the number of terms in the expansion <span class="math-container">$(1+a^3+a^{-3})^{100}$</span></p>
<p>I used the concept <span class="math-container">$a^3+a^{-3}=T$</span>, while using this I have 101 terms, from <span class="math-container">$T^2$</span> to <span class="math-container">$T^{100}$</span> how do... | lab bhattacharjee | 33,337 | <p>Write <span class="math-container">$a^3=b$</span></p>
<p>Multinomial theorem says</p>
<p>The general term of <span class="math-container">$$(1+b+b^{-1})^{100}$$</span> is <span class="math-container">$$\dfrac{100!}{p!q!r!}b^{p-q}$$</span> with</p>
<p><span class="math-container">$p+q+r=100$</span> and <span class... |
1,930,933 | <blockquote>
<p>Does there exist an $n \in \mathbb{N}$ greater than $1$ such that $\sqrt[n]{n!}$ is an integer?</p>
</blockquote>
<p>The expression seems to be increasing, so I was wondering if it is ever an integer. How could we prove that or what is the smallest value where it is an integer?</p>
| ctst | 363,720 | <p>This is (for $n>2$) never an integer, since you always have a single prime number (meaning with exponent 1) and hence its $n$th root, namely the biggest prime smaller than $n$. Since the product of roots of different prime numbers are never integer, this number will never be an integer (for $n>2$).</p>
|
72,651 | <p>$G$ is a group and $H$ is a subgroup of $G$ such that $\forall a, b$ in $G, ab\in H\implies ba\in H$. Show that $H$ is normal in $G$</p>
| Bill Cook | 16,423 | <p>Your goal is to show $xH=Hx$ for all $x\in G$ so you should show set containment both ways.</p>
<p>Suppose $y \in xH$. This means that there exists some $h \in H$ such that $y=xh$. Notice that $x^{-1}y=h \in H$. Now invoke your hypothesis. I'll let you finish from there.</p>
<p>Do be careful though. If $ab=h_1 \in... |
3,831,702 | <p>One can prove that for <span class="math-container">$x\in \mathbb{R}$</span>, the sequence
<span class="math-container">$$
u_0=x\text{ and } \forall n\in \mathbb{N},\qquad u_{n+1}=\frac{e^{u_n}}{n+1}
$$</span>
converges to <span class="math-container">$0$</span> if <span class="math-container">$x \in ]-\infty,\delta... | Simply Beautiful Art | 272,831 | <p>Numerical results:</p>
<p>It seems that <span class="math-container">$\delta\approx0.3132776395465557$</span>. This was computed using root-finding techniques to find when <span class="math-container">$u_n=y$</span> for <span class="math-container">$y=0.1,1,10$</span> and <span class="math-container">$n=10,100,1000$... |
84,312 | <p>In the topological sense, I understand that the unit circle $S^1$ is a retract of $\mathbb{R}^2 \backslash \{\mathbb{0}\}$ where $\mathbb{0}$ is the origin. This is because a continuous map defined by $r(x)= x/|x|$ is a retraction of the punctured plane $\mathbb{R}^2 \backslash \{\mathbb{0}\}$ onto the unit circle $... | Matthias Klupsch | 19,700 | <p>Indeed there is no retraction $r: \mathbb{R^2} \rightarrow S^1$
because if $\iota :S^1 \rightarrow \mathbb{R^2}$ is the inclusion
you would have a monomorphism $\iota^* : \pi(S^1) \rightarrow \pi(\mathbb{R^2})$ between the fundamental groups, i.e. a monomorphism $\mathbb{Z} \rightarrow \{0\}$ which would be absurd.... |
84,312 | <p>In the topological sense, I understand that the unit circle $S^1$ is a retract of $\mathbb{R}^2 \backslash \{\mathbb{0}\}$ where $\mathbb{0}$ is the origin. This is because a continuous map defined by $r(x)= x/|x|$ is a retraction of the punctured plane $\mathbb{R}^2 \backslash \{\mathbb{0}\}$ onto the unit circle $... | Thomas Andrews | 7,933 | <p>You can use the <a href="http://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem" rel="nofollow noreferrer">Brouwer Fixed Point Theorem</a> to show that $S^1$ is not a retraction of the unit disk, and hence not a retraction of the entire plane (since, if $X\subset Y\subset Z$, and $j:Z\rightarrow X$ is a retraction... |
1,946,637 | <p>I have come across this integral and have been unable to solve it so far.</p>
<p>$$I=\int_{-\infty}^{0.29881}e^{-0.5x^2}\,\mathrm dx$$</p>
| kobe | 190,421 | <p>If we express your series as $\sum\limits_{j = 0}^\infty a_j z^j$, then $$a_j = \begin{cases}\dfrac{2^{j/2}}{3^{j/2} + 4^{j/2}}& \text{if $j$ is even}\\0&\text{otherwise}\end{cases}.$$</p>
<p>Since $\lim\limits_{j\to \infty} \lvert a_{2j-1}\rvert^{1/(2j-1)} = 0$ and </p>
<p>$$\lim_{j\to \infty} \lvert a_{2... |
866,885 | <p>I try to write math notes as clearly as possible. In practice, this means using letters and notation similar to what the reader is already familiar with.</p>
<p>A real function is often $f(x)$, an angle is often $\theta$, a matrix has size $m\times n$, and $i$ is often an index. The full theoretical list is long an... | Fred Daniel Kline | 28,555 | <p>I just got: <a href="http://rads.stackoverflow.com/amzn/click/1466230525" rel="nofollow">Mathematical Notation</a> which is surprisingly comprehensive for such a thin book. It covers several mathmatical topics. It even has $\LaTeX$ examples. Edit: it even has conventions for notes and blackboards.</p>
|
2,584,405 | <p>During an excercise session in a basic course of probability it was shown that the secretary problem can be reduced to solving the following task: For a given natural $n$ optimize $2\leq k\leq n-1$ so that
$\frac{k-1}{n}\sum^n_{i=k}\frac 1{i-1} = \frac{k-1}{n}(\operatorname{H}_{n-1}-\operatorname{H}_{k-2})$ is the ... | Dr. Wolfgang Hintze | 198,592 | <p>EDIT: graph and approximation improved</p>
<p>The following graph compares the exact optimal value of k with the approximations $k_{appr1} = n/e$ and $k_{appr2}=\left\lfloor (n+1)/e\right\rfloor$</p>
<p><a href="https://i.stack.imgur.com/5AYFe.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5AYF... |
1,051,464 | <p>I was asked to find the truth value of the statement: </p>
<blockquote>
<p>$$
\pi + e \; \text{ is rational or } \pi - e\; \text{ is rational }
$$</p>
</blockquote>
<p>I am only allowed to use the fact that $\pi, e $ are irrational numbers and cannot use the theory of transcendental numbers.</p>
<p>Cannot proce... | Martin Brandenburg | 1,650 | <p>This seems to be an open problem. It is a conjecture that the statement is false, i.e. that $\pi + e$ and $\pi - e$ are irrational. According to <a href="http://en.wikipedia.org/wiki/Transcendental_number">Wikipedia</a> this remains unproven. (Just imagine the impact of the discovery of an equation such as $\pi=e+\f... |
1,895,721 | <p>How to find $3\times3$ matrices that satisfy the matrix equation $A^2=I_3$? Can anyone please show me steps to do this question?</p>
| Asinomás | 33,907 | <p>All of the matrices you want can be obtained by taking $A$ of the following form:</p>
<p>$\begin{pmatrix}
a_1 & 0 & 0 \\
0 & a_2 & 0 \\
0 & 0 & a_3\\
\end{pmatrix}$</p>
<p>where $a_1,a_2$ and $a_3$ are $1$ or $-1$.</p>
<p>And then taking $B$ an invertible matrix (possibly with complex entr... |
434,614 | <p>Is there a name for a type of grid you might find in Battleship? Where coordinates don't relate to points on a grid but rather the squares themselves?</p>
| amWhy | 9,003 | <p>"Grid" is as good a name as any: See <a href="http://en.wikipedia.org/wiki/Regular_grid" rel="nofollow">Regular Grid</a> in Wikipedia: In particular, see the "related" grid: the Cartesian Grid: </p>
<blockquote>
<p>"A Cartesian grid is a special case where the <strong>elements are unit squares or unit cubes</stro... |
4,297,177 | <p>I have a polynomial:</p>
<p><span class="math-container">$f(n)=\frac{1}{4}n^2-24n-16$</span></p>
<p>I am supposed to show that it is <span class="math-container">$\Omega(n^2)$</span> and <span class="math-container">$O(n^2)$</span>. Once I prove those, it should be easy to show that it is also <span class="math-cont... | Keeley Hoek | 88,896 | <p>For checking that <span class="math-container">$f(n)$</span> is <span class="math-container">$O(n^2)$</span> I think you have just made a simple mistake and it seems like you understand what to do: if <span class="math-container">$C = \frac{1}{4}$</span> then <span class="math-container">$f(n) \leq C g(n)$</span> be... |
156,013 | <p>I would like to <code>FoldList</code> a simple function, with desired output:</p>
<pre><code>f[a,b,1]
f[b,c,1]
f[c,a,1]
f[f[a,b,1],f[b,c,1],2]
f[f[b,c,1],f[c,a,1],2]
f[f[c,a,1],f[a,b,1],2]
f[f[f[a,b,1],f[b,c,1],2],f[f[b,c,1],f[c,a,1],2],3]
f[f[f[b,c,1],f[c,a,1],2],f[f[c,a,1],f[a,b,1],2],3]
f[f[f[c,a,1],f[a,b,1],2... | Carl Woll | 45,431 | <p>Here is one possibility. Create a helper function:</p>
<pre><code>g[l_List,i_] := f[##, i]& @@@ Partition[l, 2, 1, 1]
</code></pre>
<p>Then, fold this helper function:</p>
<pre><code>FoldList[g, {a, b, c}, {1, 2, 3}]
</code></pre>
<blockquote>
<p>{{a, b, c}, {f[a, b, 1], f[b, c, 1],
f[c, a, 1]}, {f[f[... |
166,666 | <p>For which values of the coefficient $c$ does the quantity
$$
\cos\alpha\cos\beta- c\sin\alpha\sin\beta
$$
depend on $\alpha$ and $\beta$ only through their sum?</p>
<p>(I'll post a quick answer below. This will be the first time I've posted a question with intent to immediately post an answer.)</p>
| Michael Hardy | 11,667 | <p>Let $f(\alpha,\beta) = \cos\alpha\cos\beta- c\sin\alpha\sin\beta$. Since this depends on $\alpha$ and $\beta$ only through their sum we have</p>
<p>$$f(\alpha,\beta)=f(\alpha+\beta,0).$$</p>
<p>Then</p>
<p>$$
f(\alpha+\beta,0) = \cos(\alpha+\beta)\cos 0 - c\sin(\alpha+\beta)\sin 0 = \cos(\alpha+\beta).
$$</p>
<... |
166,666 | <p>For which values of the coefficient $c$ does the quantity
$$
\cos\alpha\cos\beta- c\sin\alpha\sin\beta
$$
depend on $\alpha$ and $\beta$ only through their sum?</p>
<p>(I'll post a quick answer below. This will be the first time I've posted a question with intent to immediately post an answer.)</p>
| Community | -1 | <p>We want $f(\alpha, \beta) = g(\alpha + \beta)$. This means that $f(\alpha,-\alpha) = g(0) = f(0,0)$, $\forall \alpha$.</p>
<p>$f(\alpha,-\alpha) = f(0,0)$ for all $\alpha$. Hence, we get that $$\cos^2(\alpha) + c \, \sin^2(\alpha) = 1, \, \, \forall \alpha$$
$$c \, \sin^2(\alpha) = \sin^2(\alpha), \, \, \forall \al... |
913,998 | <p>What can we do with this function, so the function will be continuous in $(0,0)$?</p>
<p>$f:\mathbb{R}^2\rightarrow\mathbb{R}:(x,y) \mapsto \frac{x^2+y^2-x^3y^3}{x^2+y^2}$</p>
<p>What I think we should do, is:</p>
<p>approximate $(0,0)$ via the line $y=x$, so substitute $y=x$ and take the limit of that function,... | lab bhattacharjee | 33,337 | <p>$$x^4+2x^2+2=(x^2+1)^2+1$$</p>
<p>Now for real $x,x^2\ge0$</p>
|
710,045 | <p><a href="https://i.stack.imgur.com/7rJAB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7rJAB.png" alt="question" /></a></p>
<p>For (a) I have said</p>
<p>By MVT:</p>
<p><span class="math-container">$|f(x)-f(y)|\le K|x-y|$</span></p>
<p>Choose <span class="math-container">$\delta=\epsilon/K$</sp... | Joshua Pepper | 43,511 | <p>Every finite-dimensional normed space of dimension $n$ over $\mathbb{F}$ is isomorphic to $\mathbb{F}^n$, and hence complete (assuming $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$).</p>
<p>Your student's argument works! The space that this norm defines (in $n$ dimensions) is somtimes called $l_1^n$ (which is complete). ... |
2,767,392 | <p>I have the following curve:</p>
<p>$x^4=a^2(x^2-y^2)$</p>
<p>Prove that the area of its loop is $\frac{2a^2}{3}$.</p>
<p><strong>My approach</strong></p>
<p>This curve has four loops. So the required area should be:</p>
<p>$4\int_{0}^{a}\frac{x}{a}\sqrt{a^2-x^2} dx$</p>
<p>But, After solving, the area turned o... | user | 293,846 | <p>For the proof it suffices to show that
$$
\sum_{i=0}^k (-1)^i\binom{n}{i}=(-1)^k\binom{n-1}{k}.
$$</p>
<p>The simplest way to demonstrate this is by induction. The statement is obviously true for $k=0$. Assumption that it is true for $k$ implies that it is true for $k+1$ as well:
$$
\sum_{i=0}^{k+1} (-1)^i\binom{n}... |
6,340 | <p>Just to clarify, I want to show that:</p>
<p>If $f$ is entire and $\int_{\mathbb{C}} |f|^p dxdy <\infty$, then $f=0$.</p>
<p>I think I can show that this is the case for $p=2$, but I'm not sure about other values of $p$...</p>
| Braindead | 2,499 | <p>Right! Thank you, I didn't think about using Hölder's.</p>
<p>In short, </p>
<p>\begin{equation}
|f(0)| = \left|\int_{|z|=R} \frac{f(z)dz}{z} \right| < \left(\int_{|z|=R} |f(z)|^p |dz|\right)^{1/p} \left(\int_{|z|=R} |z|^{-q} |dz|\right)^{1/q}
\end{equation}</p>
<p>And $\left(\int_{|z|=R} |z|^{-q} |dz|\right)... |
6,340 | <p>Just to clarify, I want to show that:</p>
<p>If $f$ is entire and $\int_{\mathbb{C}} |f|^p dxdy <\infty$, then $f=0$.</p>
<p>I think I can show that this is the case for $p=2$, but I'm not sure about other values of $p$...</p>
| kahen | 1,269 | <p>$\newcommand\RR{\mathbb{R}}$
$\newcommand\NN{\mathbb{N}}$
$\newcommand\e{\varepsilon}$
$\newcommand\lbg{\lambda}$</p>
<p>If $f: \RR \to \RR$ is an analytic unbounded Lebesgue integrable function, then it must in particular satisfy this property (because it is continuous):
\[
\forall \e,M > 0 \; \exists x_0 : |x|... |
3,065,331 | <p>In <a href="https://rads.stackoverflow.com/amzn/click/0199208255" rel="nofollow noreferrer">Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers</a> one of the very first stated equations are, as in the title of the question,</p>
<p><span class="math-container">$$
\ddot{x} = \fr... | Robert Lewis | 67,071 | <p><span class="math-container">$\dfrac{d}{dx} \left (\dfrac{1}{2} \dot x^2 \right ) = \dfrac{d \dot x}{dx}\dot x = \dfrac{d\dot x}{dx} \dfrac{dx}{dt} = \dfrac{d \dot x(x(t))}{dt} = \dfrac{d \dot x}{dt} = \dfrac{d^2 x}{dt^2} = \ddot x \tag 1$</span></p>
<p>by the chain rule.</p>
<p>The key here is the observation t... |
1,423,728 | <p>The definition of a limit in <a href="http://rads.stackoverflow.com/amzn/click/0321888545" rel="nofollow noreferrer">this book</a> stated like this </p>
<p><a href="https://i.stack.imgur.com/tKjaa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tKjaa.png" alt="enter image description here"></a><... | Steven Alexis Gregory | 75,410 | <p>(1) Because they want to approach $x_0$ from both sides.</p>
<p>(2) If you define $f:\mathbb Q \to \mathbb R$ by $f(x) = 1$ for all $x \in \mathbb Q$, then your definition of limit will not allow for $\displaystyle \lim_{x \to x_0}f(x)$ to exist. In topology, for example, the domain of the function is taken into ac... |
1,423,728 | <p>The definition of a limit in <a href="http://rads.stackoverflow.com/amzn/click/0321888545" rel="nofollow noreferrer">this book</a> stated like this </p>
<p><a href="https://i.stack.imgur.com/tKjaa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tKjaa.png" alt="enter image description here"></a><... | Paramanand Singh | 72,031 | <p>The idea of a <em>limit of a function $f$ at a point $x_{0}$</em> is invented to give answer to the following question:</p>
<p><em>How does function $f$ behave near the point $x_{0}$?</em></p>
<p>Note that the question does not ask as to what happens at $x_{0}$ but rather what happens near $x_{0}$. Hence it does n... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Dustin G. Mixon | 29,873 | <p>I don't have the time to regularly edit Wikipedia. I've made time for MO lately because asking and answering these questions positively affects my research. Based on my experience writing lecture notes, I find that organizing and presenting math that I already understand really well takes a lot of time compared to... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | ex0du5 | 24,133 | <p>There are also mathematical models of why the number of editors is decreasing that have nothing to do with a decreasing opinion of Wikipedia or a migration elsewhere. In the beginning, Wikipedia was tabula rasa, and editors could add sections on anything. Addition, multiplication, trigonometry, basic set theory, e... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Salix alba | 34,268 | <p>Replying to Dick Palais: Any edit to Wikipedia, no matter how small, is good even if it's just correcting a typo. One thing I've done is to make a small edit to every page I visit. There is always something to be fixed. Once you've done a number of small edits, you can do something more substantial. The mathematics ... |
361,060 | <blockquote>
<p>Consider the ring of Gaussian integers $D=\lbrace a+bi\mid a,b \in \mathbb{Z \rbrace}$, where $i \in \mathbb{C}$ such that $i^2=-1$. Consider the map $f$ from $D$ to $\mathbb{Z}[x]/(x^2+1)$ sending $i$ to the class of $x$ modulo $x^2+1$. Show that $f$ is a ring isomorphism.</p>
</blockquote>
<p>I got... | baharampuri | 50,080 | <p>The map in general format will be $f(a+ib)=a+bx + \langle x^2 +1 \rangle$
it is a ring homomorphism since $f(a+ib +c +id )= (a+bx +c+dx) + \langle x^2 +1 \rangle = a+bx +\langle x^2 +1 \rangle c+dx +\langle x^2 +1 \rangle =f(a+ib)+f(c+id)$</p>
|
57,642 | <p>I'm looking for the shortest and the clearest proof for this following theorem:</p>
<p>For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , I need to show $V= \ker T^n \oplus $ Im $T^n$.</p>
<p>Any hints? I don't know where to start from.</p>
<p>Thank you.</p>
| Robert Israel | 8,508 | <p>The main point is that ${\rm Ker}\ T^n \cap {\rm Im}\ T^n = \{0\}$. Show that if $T^k x \ne 0$ and $T^{k+1} x = 0$, then $x, Tx, \ldots T^k x$ are linearly independent, and therefore $k \le n-1$. No need here for Jordan Canonical Form - in particular this works over any field.</p>
|
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | mau | 89 | <p>unknown source:</p>
<p>Could the plane be colored with two different colors (say, red and blue) so that there is no equilateral triangle whose vertices are all of the same color?</p>
|
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | ThudanBlunder | 1,770 | <p>Quite simply, a monkey's mother is twice as old as the monkey will be when the monkey's father is twice as old as the monkey will be when the monkey's mother is less by the difference in ages between the monkey's mother and the monkey's father than three times as old as the monkey will be when the monkey's father is... |
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | Steven Stadnicki | 785 | <p>A personal favorite (albeit not ideally-worded): Jamie has a windowbox where he plants a row of iris flowers. He plants in just two colors - blue and yellow - but never plants two yellow irises next to each other (the result is just too garish). Assuming that he wants to keep the same number of flowers in his box ... |
844,887 | <p>I am trying to prove by induction that every non-zero natural number has at least one predecessor. However, I don't know what to use as a base case, since 0 is not non-zero and I haven't yet established that 1 is the number following zero.</p>
<p>My axioms are: </p>
<ol>
<li>$0$ is a natural number. </li>
<li>if $... | martini | 15,379 | <p><strong>Hint</strong>: Let $P(n) ={}$ "all numbers $<n$ have a predecessor". Then $P(0)$ is true (as there are no numbers $< 0$. I'm sure you can do the induction step.</p>
|
4,561,863 | <p>This limit is proposed to be solved without using the L'Hopital's rule or Taylor series:
<span class="math-container">$$
\lim\limits_{x\to 0} \left( \frac{a^{\sin x}+b^{\sin x} }{2} \right)^{\frac1{x}},
$$</span>
where <span class="math-container">$a>0$</span>, <span class="math-container">$b>0$</span> are som... | Feng | 624,428 | <p>Taking logarithm, then using <span class="math-container">$\sin x\sim x$</span> and <span class="math-container">$\ln(1+x)\sim x$</span> when <span class="math-container">$x\to0$</span>, we have
<span class="math-container">\begin{align*}
\frac1x\cdot\ln\left( \frac{a^{\sin x}+b^{\sin x} }{2} \right)&\sim\frac1{... |
506,767 | <p>What is the largest number on multiplying with itself gives the same number as last digits of the product?</p>
<p>i.e., $(376 \times 376) = 141376$</p>
<p>i.e., $(25\times 25) = 625$</p>
<p>If the largest number cant be found out can you prove that there is always a number greater than any given number? (only in ... | mercio | 17,445 | <p>For any integer $n$,there are $2$ solutions to $x^2 = x \pmod {2^n}$ and $x^2 = x \pmod {5^n}$, which are $0$ and $1$.</p>
<p>Hence by the chinese remainder theorem, there are $4$ solutions to $x^2 = x \pmod {10^n}$. Two of those are the obvious $0$ and $1$, there is one solution which is $0$ mod $5^n$ and $1$ mod ... |
2,084,671 | <p>I've tried simplifying the term by <strong>substituting</strong> $y := x^2$ and also to use a certain <strong>algorithm</strong> that outputs the f(a), but it gets too big for the input r, that is a variable and not a number to begin with (added little example):</p>
<p><a href="https://i.stack.imgur.com/8C9sV.png" ... | Maverick | 171,392 | <p>Let $y=x^2$ which reduces the equation to $y^2-6qy+q^2=0$</p>
<p>$y=\frac{6q\pm\sqrt{36q^2-4q^2}}{2}$</p>
<p>$x^2=\left(3 \pm 2\sqrt{2}\right)q$</p>
<p>$x^2=\left(\sqrt{2}\pm1\right)^2 q$</p>
<p>$x=\pm\left(\sqrt{2}\pm1\right)\sqrt{q}$</p>
<p>$x=\pm\left(\sqrt{2q}\pm\sqrt{q}\right)$</p>
<p>Q.E.D</p>
|
2,703,323 | <p>How can one show that the limit of the following is $1$?</p>
<p>$$\lim_{x\to 0}\frac{\frac{1}{1-x}-1}{x}=1$$</p>
| user | 505,767 | <p>By definition of derivative with </p>
<p>$$f(x)=\frac{1}{1-x}\implies f'(x)=\frac{1}{(1-x)^2}$$</p>
<p>we have</p>
<p>$$\lim_{x\to 0}\frac{\frac{1}{1-x}-1}{x-0}=f'(0)=1$$</p>
|
4,196,125 | <p>Can someone tell me where this calculation goes wrong?
I get (2 3 4)(1 2 3 4 5 6)^-1 = (1 6 5 2).
My book and Mathematica get (1 6 5 4).
I have read several explanations of how to multiply permutations in cycle notation and have
worked dozens of examples successfully, but I always get this one wrong.</p>
<p>(2 3 4)(... | Math Lover | 801,574 | <p>Please note that you have a circle</p>
<p><span class="math-container">$x^2 + (y-2)^2 \leq 1$</span></p>
<p>As the circle forms for <span class="math-container">$-1 \leq x \leq 1$</span>, your bounds for <span class="math-container">$x$</span> is incorrect.</p>
<p>Also for limits of <span class="math-container">$y$<... |
114,898 | <p>A) Let $f:F\rightarrow S$ be a flat proper morphism of schemes with geometrically normal fibers. Then supposedly the number of $\textbf{connected}$ components of the geometric fibers is constant. Why is this? Without some kind of vanishing of cohomology or information on the base, I don't see why this is true. <... | Qfwfq | 4,721 | <p>Define $$\phi(z):=\frac{1}{2\pi i}\int_{S^1}(\zeta-z)^{-1}\cdot \varphi(\zeta)d\zeta$$</p>
<p>for $|z|<1$, where $\varphi: S^1\to\mathbb{C}$ is a Peano curve (i.e. its image has nonempty interior), and $\phi(z):=\varphi(z)$ for $z\in S^1$. [<strong>Edit</strong>: this construction doesn't work because $\phi$, a... |
416,387 | <p>In the process of solving a DE and imposing the initial condition I came up with the following question.</p>
<p>I've reached the stage that</p>
<p>$$\ln y + C = \int\left(\frac{2}{x+2}-\frac{1}{x+1}\right)dx$$
$$\Rightarrow \ln y +C=2\ln|x+2|-\ln|x+1|$$
$$\Rightarrow y=A\frac{(x+2)^2}{|x+1|}.$$</p>
<p>Now I had a... | Ted Shifrin | 71,348 | <p>First of all, you should presumably have had
$$\ln |y| + C = 2\ln |x+2| - \ln |x+1|\,, \quad\text{leading to}\quad |y| = A\frac{(x+2)^2}{|x+1|}\,.$$
Since a solution curve through $(-4,-3)$ cannot cross the lines $x=-1$ and $y=0$, we infer that throughout $y<0$ and $x<-1$, so $|y|=-y$ and $|x+1|=-(x+1)$. Thus,... |
4,353,891 | <p>How to prove that <span class="math-container">$A=\{(x,y)\in \mathbb{R}^2: |x|+|y|^{1/2}<1\}$</span> is convex? I tried using the definition but couldn’t go far, since the second component involves square root(tried squaring, that made it complicated).</p>
<p>I plotted it in mathematica and the graph comes out to... | Zim | 793,815 | <p>It's easily seen <a href="https://www.desmos.com/calculator/hvwbfhhlav" rel="nofollow noreferrer">graphically</a> that this set is not convex.</p>
|
3,170,629 | <p>I am trying to define a sequence.
The first few terms of the sequence are:</p>
<p><span class="math-container">$2,5,13,43,61$</span></p>
<p>Not yet found other terms because I am working with paper and pen, no software.</p>
<p>Why the first term is <span class="math-container">$5$</span>?</p>
<p>Let be <span cla... | Community | -1 | <pre><code>`my(a=0,b=0);forprime(x=1,50000,y=x;while(isprime(y-primepi(y)),y-=primepi(y);b++);if(b>a,a=b;print(x));b=0)`
</code></pre>
<p>produces 14897 as the next one. Then no more below 500000. There's not too much to say except primes in the sequence will be primes at even indices after the first, simply becaus... |
3,170,629 | <p>I am trying to define a sequence.
The first few terms of the sequence are:</p>
<p><span class="math-container">$2,5,13,43,61$</span></p>
<p>Not yet found other terms because I am working with paper and pen, no software.</p>
<p>Why the first term is <span class="math-container">$5$</span>?</p>
<p>Let be <span cla... | lucasb | 654,079 | <p>Using <span class="math-container">$S$</span> to denote the sequence you're trying to define, one may do so in
terms of two auxiliary functions <span class="math-container">$N$</span> and <span class="math-container">$L$</span>, where <span class="math-container">$N$</span> assigns to every prime number <span class... |
2,976,613 | <p>I want to show, that <span class="math-container">$a:=\sum \limits_{n=0}^{\infty} \left(\dfrac{2n+n^3}{3-4n}\right)^n$</span> is not converging, because <span class="math-container">$\lim \limits_{n \to \infty}(a)\neq 0 \; (*)$</span>. Therefore, the series can't be absolute converge too.</p>
<p>Firstly, I try to s... | user | 505,767 | <p>Recall that </p>
<p><span class="math-container">$$\sum_0^\infty a_n <\infty \implies a_n \to 0$$</span></p>
<p>therefore if <span class="math-container">$a_n \not \to 0$</span> the series can’t converge.</p>
<p>In that case for <span class="math-container">$n\ge 3$</span> we have</p>
<p><span class="math-con... |
2,976,613 | <p>I want to show, that <span class="math-container">$a:=\sum \limits_{n=0}^{\infty} \left(\dfrac{2n+n^3}{3-4n}\right)^n$</span> is not converging, because <span class="math-container">$\lim \limits_{n \to \infty}(a)\neq 0 \; (*)$</span>. Therefore, the series can't be absolute converge too.</p>
<p>Firstly, I try to s... | Peter Szilas | 408,605 | <p>Let <span class="math-container">$n \ge 4$</span>.</p>
<p><span class="math-container">$|a_n|:=\left |\dfrac{n^3+2n}{4n-3}\right |^n \gt$</span></p>
<p><span class="math-container">$\left (\dfrac{n^3}{4n}\right )^n \ge n^n.$</span></p>
<p><span class="math-container">$\lim_{n \rightarrow \infty} |a_n| \not = 0$</... |
1,143,889 | <p>I am interested in counting the number of hyperedges in the complete, $t$-uniform hypergraph on $n$ vertices which intersect a specified set of $k$ vertices. This is trivial, the answer is:</p>
<p>$$\sum_{i=1}^t {k \choose i}{n-k \choose t-i}.$$</p>
<p>My questions is whether there is a nice simplification of thi... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\dsc}[1]{\displaystyle{\color{red}{#1}}}
\newcommand{\ex... |
813,121 | <p>I have to calculate this : $$ \lim_{x\to 0}\frac{2-x}{x^3}e^{(x-1)/x^2} $$ Can somebody help me?</p>
| Michael Hardy | 11,667 | <p>Letting $w=1/x$, we have
$$
\lim_{x\downarrow 0}\frac{2-x}{x^3}e^{(x-1)/x^2} = \lim_{w\to+\infty} \left(2 - \frac 1 w \right) w^3 e^{w^2\left(\frac 1 w - 1\right)} = \lim_{w\to+\infty} (2w^3 - w^2) e^{w-w^2}
$$
$$
= \lim_{w\to+\infty} \frac{2w^3-w^2}{e^{w^2-w}}.
$$
L'Hopital should handle that.</p>
<p>Maybe I'll po... |
173,745 | <p>I am trying to prove the following: 'If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces (that is, they have a countable dense subset), then the product metric space $X_1 \times X_2$ is separable.' It seems pretty straightforward, but I would really appreciate it if someone could verify that my proof works.<... | Paul | 17,980 | <p>I think it is not the answer for your question. It's better to see it as a comment after answering of Kevin Carlson. However I know there is a <a href="http://planetmath.org/HewittMarczewskiPondiczeryTheorem.html" rel="nofollow">result from Pondiczery, Hewitt and Marczewski</a> that (recently when I read a textbook ... |
2,237,963 | <p>One-point compactification of $S_{\Omega}$ is homeomorphic with $\bar S_{\Omega}$.</p>
<p>Let $X$ be a topological space. Then the One-point compactification of $X$ is a certain compact space $X^*$ together with an open embedding $c : X \to X^*$ such that the complement of $X$ in $X^*$ consists of a single point, ... | Calvin Lin | 54,563 | <p>Drafting behind Michael Rozenberg's clever answer, appealing to the concavity of <span class="math-container">$\sin$</span> on <span class="math-container">$[0, \pi]$</span> quickly reduces the problem to showing the inequality
<span class="math-container">$$2 \sin 1 > \frac{8}{5} .$$</span></p>
<p>Then, from Ma... |
175,576 | <p><a href="https://i.stack.imgur.com/wtjxT.png" rel="noreferrer"><img src="https://i.stack.imgur.com/wtjxT.png" alt="enter image description here"></a></p>
<p>This should work? But it is giving me the wrong t values</p>
| kglr | 125 | <pre><code>r[t_] := {6 Cos[ 2.9 Pi t], Cos[3.5 Pi t] + 5 t};
pp = ParametricPlot[r[t], {t, 0, 1} ];
pt = First @ Graphics`Mesh`FindIntersections[pp];
Show[pp, Graphics @ {Red, PointSize[Large], Point @ pt},
PlotLabel -> Style[Row[{"intersection = ", pt}], 16]]
</code></pre>
<p><a href="https://i.stack.imgur.com... |
175,576 | <p><a href="https://i.stack.imgur.com/wtjxT.png" rel="noreferrer"><img src="https://i.stack.imgur.com/wtjxT.png" alt="enter image description here"></a></p>
<p>This should work? But it is giving me the wrong t values</p>
| Akku14 | 34,287 | <p>NSolve can do the job, if you use rationalized parameters. Even Solve can do it, gives Root expression.</p>
<pre><code>r[t_] = {6 Cos[29/10 Pi t], Cos[35/10 Pi t] + 5 t};
eq1 = And @@ Thread[r[t1] == r[t2]] && 0 < t1 < t2 < 1
eq2 = eq1 // TrigToExp // Simplify
(* E^(-(29/10) I \[Pi] t2) + E^((... |
175,576 | <p><a href="https://i.stack.imgur.com/wtjxT.png" rel="noreferrer"><img src="https://i.stack.imgur.com/wtjxT.png" alt="enter image description here"></a></p>
<p>This should work? But it is giving me the wrong t values</p>
| Bob Hanlon | 9,362 | <pre><code>r[t_] := {6 Cos[29 Pi t/10], Cos[7 Pi t/2] + 5 t};
ParametricPlot[r[t], {t, 0, 1},
ColorFunction -> Function[{x, y, t}, ColorData["Rainbow"][t]],
PlotLegends -> BarLegend["Rainbow"]]
</code></pre>
<p><a href="https://i.stack.imgur.com/hRME1.png" rel="nofollow noreferrer"><img src="https://i.stack... |
4,320,849 | <p>I had this problem in an exam I recently appeared for:</p>
<blockquote>
<p>Find the range of
<span class="math-container">$$y =\frac{x^2+2x+4}{2x^2+4x+9}$$</span></p>
</blockquote>
<p>By randomly assuming the value of <span class="math-container">$x$</span>, I got the lower range of this expression as <span class="m... | Darshan P. | 992,496 | <p><span class="math-container">$$y = \frac{x^2 + 2x + 4}{2x^2 + 4x + 9} = \frac12 - \frac {1/2}{2x^2 + 4x + 9} \implies \frac {dy}{dx} = \frac {4x + 4}{\text{whatever}} \text{ Let } \color{green}{\frac{dy}{dx} = 0 \implies x = -1}, y(x = -1) = \color{blue}{\frac37} \text{ also } y(x\to \infty) = \color{blue}{\frac 12}... |
2,441,488 | <p>We are not unfamiliar to imaginary angles, so what can be $\tan(a+ib)$</p>
<blockquote>
<p>If$$\tan(a+ib)=x+iy$$then find $$x,y$$</p>
</blockquote>
<p>My attempt,</p>
<p>Let $x$ and $z$ be two complex bombers such that,
$$\tan(z)=x$$
$$z=\tan^{-1} (x)$$
$$z=\tan^{-1}(re^{i\theta}).$$
But this doesn't seem to ge... | Community | -1 | <p><strong>Hint:</strong></p>
<p>From the complex definitions of the trigonometric functions,</p>
<p>$$\tan z=\frac2{2i}\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}.$$</p>
<p>When you substitute $z=a+ib$,</p>
<p>$$\tan(a+ib)=\frac1i\frac{e^{-b+ia}-e^{b-ia}}{e^{-b+ia}+e^{b-ia}}=-i\frac{e^{-b+ia}-e^{b-ia}}{e^{-b+ia}+e^{b-ia... |
1,206,528 | <p>Find the matrix $A^{50}$ given</p>
<p>$$A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}$$ as well as for $$A=\begin{bmatrix} 2 & 0 \\ 2 & 1\end{bmatrix}$$</p>
<p>I was practicing some questions for my exam and I found questions of this form in a previous year's paper.</p>
<p>I don't know how to ... | Community | -1 | <p>An unorthodoxical but quick solution.</p>
<p>Let us compute the powers by recurrence:</p>
<p>$$
\begin{bmatrix} a_{n+1} & b_{n+1} \\ c_{n+1} & d_{n+1} \end{bmatrix}=
\begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}
\begin{bmatrix} a_n & b_n \\ c_n & d_n \end{bmatrix}
=\begin{bmatrix} 2a_n-c_n ... |
1,206,980 | <p>Here is the problem: if $v^1$, $v^2$, ..., $v^d$ is an orthonormal basis in $\mathbb{R}^d$, then show that
$$
||A - A\sum_{i = 1}^k v^i(v^i)^T ||^2_F = \sum_{i = k+1}^d||Av^i ||_2^2.
$$</p>
<p>I am having trouble understanding this step of a longer proof. The text that I got this from just says that this holds be... | Chappers | 221,811 | <p>Interchanging the order of integration allows us to write $T$ as a single integral:
$$ T(p(x))(t) = \int_0^t \int_0^y p(x) \, dx \, dy = \int_0^t p(x) \int_x^t \, dy \, dx = \int_0^t (t-x)p(x) \, dx $$
(the region of integration is $0<x<y<t$, which tells you what the new limits should be)
Now, given $p(x)=x... |
2,815,900 | <p>Consider the proposition $∀x (P (x) → Q (x))$, where $P (x)$ and $Q (x)$ are predicates on a domain $U$. Prove or disprove the following statement, justifying the answer.</p>
<p>If the statement $∀x (P (x) → Q (x))$ is FALSE then it's FALSE also $∀x P (x)$</p>
<p>How can I solve this exercise?</p>
| Prajwal Kansakar | 49,781 | <p>Consider the statement, </p>
<p>For all $x$, if $x^2\geq 0$, then $x\geq 0$.</p>
<p>with the domain consisting of real numbers. This is false. What about </p>
<p>For all $x$, $x^2\geq 0$?</p>
|
1,721,055 | <p>This afternoon I've been studying the pythagorean identities & compound angles. I've got a problem with a question working with 2 sets of compound angles:</p>
<blockquote>
<p>Solve, in the interval $0^\circ \leq \theta \leq 360^\circ$, $$\cos(\theta + 25^\circ) + \sin(\theta +65^\circ) = 1$$ </p>
</blockquote... | Doug M | 317,162 | <p>$\cos(\theta + 25) + \sin(\theta+65) = 1$</p>
<p>$\cos(\theta + 25) + \cos(25-\theta) = 1$</p>
<p>$\cos(\theta + 25) + \cos(\theta-25) = 1$</p>
<p>$2\cos\theta \cos(25) = 1$</p>
<p>$\cos \theta = (1/2) \sec 25$</p>
<p>$\theta$ ~ 57 degrees, 303 degrees</p>
|
1,571,083 | <p>Given the 2 terms
$$ \frac{k + a}{k + b}$$
and $$\frac{a}{b}$$
with $a, b, k \in \mathbb{R^+}$ and $a > b$</p>
<p>I want to show, that the first term is always bigger than the second one.</p>
<p><strong>My try</strong>
$$
\frac{k + a}{k + b} > \frac{a}{b} \\
\frac{k + b + (a - b)}{k + b} > \frac{b + ... | fredq | 297,080 | <h1>What my answer is about</h1>
<ul>
<li>I give the correct calculation to answer the question.</li>
<li>I point out that maybe the reason the event seems unlikely is because of the way we look at it.</li>
<li>I do some more calculation to show that similar events may be in fact common.</li>
</ul>
<h2>The correct ca... |
1,887,395 | <p>Let $D=\{Z\in\mathbb{C}:|z|<1\}$. Suppose $f:\overline D\to\mathbb{C}$ is continuous, holomorphic in $D$, and $f(\partial D)\subset \mathbb{R}$. Define $h:\mathbb{C}\to\mathbb{C}$ by
$$
h(z)=\begin{cases}
f(z) &\text{if}\ z\in \overline D\\
\overline{f(1/\overline z)} &\te... | Zubzub | 349,735 | <p>Some ideas here :
Write it as
$$
\sum_{n=0}^\infty \sum_{i=1}^n \frac{1}{i\ n} \ \ - \sum_{n=0}^\infty \sum_{i=1}^n \frac{1}{i(n+k)}
$$
Now let's explicitly write some terms of the positive sum and of the negative sum in a triangle-shape:
$$
\frac{1}{1}\frac{1}{1} + \\
\frac{1}{1}\frac{1}{2} + \frac{1}{2} \frac{1}{... |
4,295,652 | <p>In the integration formula <span class="math-container">$\int dx/x = log x + c$</span>, Is the log natural or log base 10? The formula appears in many problems and i just got a problem wrong for apparently using the wrong log. Could you please enlighten me about the right log to be used in integration.</p>
| Angel | 109,318 | <p>Consider the differential equation <span class="math-container">$y'=y$</span> with <span class="math-container">$y(0)=1$</span> on <span class="math-container">$\mathbb{R}$</span>. It can be proven that this has a unique solution. We <em>define</em> the exponential function on <span class="math-container">$\mathbb{R... |
1,429,000 | <p>Prove that for any integer $n$, the integer $n^2 + 7n + 1$ is odd.</p>
<p>I have $n=2k+1$ for some $k\in Z$</p>
<p>I really do not how to do this problem. any help in understanding would be greatly appreciated.</p>
| Robert Soupe | 149,436 | <p>There are only two scenarios here: $n$ is odd or $n$ is even. That alone is enough to determine whether $n^2 + 7n + 1$ is odd or even.</p>
<ul>
<li>If $n$ is odd, then so is $n^2$, as well as $7n$. Then $n^2 + 7n$ is even, because two odd numbers add up to an even number. But then there's the $+ 1$, which makes the... |
2,349,312 | <blockquote>
<p>Let $F$ be a subfield of the complex numbers. Let $A,B\in F^{n\times n}$ and $X\in F^{n\times 1}.$ If $A\ne B,$ then $AX\ne BX$ for every non-zero $X.$</p>
</blockquote>
<p>I am NOT sure if the above statement is right. I did some computation with random matrices and it seems true to me. This is the ... | Wuestenfux | 417,848 | <p>In an integral domain $R$, we have $ax=bx\Rightarrow a=b$ for all $a,b,x\in R$ with $x\ne 0$. To see this, write $(a-b)x=0$. Since $R$ has no zero divisors and $x\ne 0$ we must have $a-b=0$, i.e., $a=b$. But an $n\times n$ matrix ring is not an integral domain for $n\geq 2$.</p>
|
762,964 | <blockquote>
<p>Let <span class="math-container">$X$</span> be from a normal distribution <span class="math-container">$N(\theta,1)$</span>.</p>
<p>a) Find a sufficient statistic for <span class="math-container">$\theta$</span>.<br />
b) Is <span class="math-container">$S_n^2$</span> a
sufficient statistic for <span cl... | Michael Hardy | 11,667 | <p>$$
\sum_{i=1}^n (x_i - \theta)^2 = \left( \sum_{i=1}^n x_i^2 \right) -2\theta \left( \sum_{i=1}^n x_i \right) + n\theta^2
$$
Therefore
$$
\exp \left( \frac {-1} 2 \sum_{i=1}^n (x_i-\theta)^2 \right) = \underbrace{ e^{-n\theta^2/2}\cdot \exp\left( \theta\sum_{i=1}^n x_i \right)}_{\large\text{first factor}} \cdot \und... |
531,895 | <p>Assume K is a field that is not algebraically closed, then how to prove that for any m>0, there is a polynomial with m variables over K that possess a unique zero point?</p>
| Slade | 33,433 | <p>Take any $\alpha \in \overline{K}\setminus K$, let $m_\alpha$ be its minimal polynomial, and let $n$ be its degree.</p>
<p>For $m=2$, let $f(X,Y) = Y^n m_\alpha (X/Y)$. Then $f(X,Y)\in K[X,Y]$ and $f$ has only $(0,0)$ as a root in $K^2$.</p>
<p>For $m=3$, write $g(X,Y,Z) = f(f(X,Y),Z)$. If $g(a,b,c)=0$ for $a,b,... |
1,283,541 | <p>Find the least value of $f(x)=3^{-x+1} + e^{-x-1}$.</p>
<p>I tried to use the maxima/minima concept but it was of no use. Please help.</p>
| Giuseppe Negro | 8,157 | <p>I don't know if it is "algebraic", but you might start from the observation that the function
$$
\theta\in \mathbb{R}\mapsto e^{i\theta}\in \mathbb{C}
$$
is Lipschitz with constant $1$. This can be proved via the fundamental theorem of calculus:
$$
\left\lvert e^{i\theta+h}-e^{i\theta}\right\rvert=\left\lvert e^{i\t... |
1,080,572 | <p>I am having problem with this question , kindly please help me with this ,</p>
<p>Let $$S = \{x : x^{6} -x^{5} \leq 100\}$$</p>
<p>And $$T =\{x^{2} - 2x : x \in ( 0, \infty)\}$$</p>
<p>Then I have to show that set $S \cap T$ is closed and bounded. $S$ appears to be closed and bounded as it less than or equal to $... | drhab | 75,923 | <p>Hint on being closed:</p>
<p>$T$ can also be written as: $\{x^2-2x:x\in[0,\infty)\}$ since $0=2^2-2.2\in T$</p>
<p>So both sets $S$ and $T$ are preimages of closed sets under continuous functions.</p>
<p>Hint on being bounded.</p>
<p>If a set is bounded then so are its subsets.</p>
|
2,345,142 | <p>Can you use Skolemization to reduce a formula to the variables you want it to be about? I was trying to think of a nice algorithmic way to do it but only ended up having problems. </p>
<p>Say you have a formula $G$ in prenex normal form, i.e. something looking like this:</p>
<p>$$G \equiv Q_1x_1...Q_nx_n\;F$$</p>
... | Bram28 | 256,001 | <p>Unless I misunderstand you, this doesn't work.</p>
<p>Say you have $\forall x \exists y Loves(x,y)$ : Everyone loves someone.</p>
<p>By your method you get (again, if I understand you correctly):</p>
<p>$\forall x Loves(x,c_1)$</p>
<p>But that can't be right, for that would imply that there is something $c_1$ th... |
121,631 | <p>How to prove most simply that if a polyonmial $f$, has only real coefficients and $f(c)=0$, and $k$ is the complex conjugate of $c$, then $f(k)=0$?</p>
| Jacob Salomon | 71,482 | <p>I've long forgotten the details so maybe someone else can complete this. I also don't see how folks have formatted the conjugate with the overhead bar so I will use ~ for conjugate.</p>
<p>When I took complex variable, the proof was along the lines of:</p>
<p>Suppose z1 is indeed a zero of f(z).
Assume ~z1 is <st... |
282,780 | <p>Let $R = k[x_1, \ldots, x_n]$ for $k$ a field of characteristic zero and let $S \subset R$ be a graded sub-$k$-algebra (for the standard grading: $\deg x_i = 1$) such that $R$ is a free $S$-module of finite rank. Does this imply $S \cong k[y_1,\ldots,y_n]$?</p>
| Neil Strickland | 10,366 | <p>Note that $R$ and $S$ each have only one graded maximal ideal, so they are local in the graded sense, so most of the standard results for ungraded local rings are applicable. There is an obvious finite resolution of $k$ by modules that are finitely generated and free over $R$ and thus also over $S$. This implies t... |
292,780 | <p>I have to realize an NFA that recognizes the language of strings on the alphabet {a, b} ending with: bb, ba, baa. I thought that there must be the following states:</p>
<p>$q_0$: the string ends with bb.</p>
<p>$q_1$: the string ends with ba.</p>
<p>$q_2$: the string ends with baa.</p>
<p>Is right the definition... | Rick Decker | 36,993 | <p>As you seem to understand, the states in a FA can be designed to correspond to having seen a particular form of input. One solution (not with the minimal number of states, but easier to understand) uses five states:</p>
<ul>
<li>$q_0$: Processes any input and takes special action if a $b$ is seen. (start state)</li... |
136,570 | <p>Good day everyone. I was reading the more advanced lectures on complex analysis and encountered a lot of questions, concerning the determination of complex logarithm. As far I don't even understand the concept of it, but I'll do provide you with several practical questions concerning the topic. </p>
<p>First of all... | Dan | 1,374 | <p>By inverting <a href="http://en.wikipedia.org/wiki/Euler%27s_formula" rel="nofollow">Euler's formula</a> $e^{ix} = \cos{x} + i \sin{x}$, you get</p>
<p>$\log(r(\cos{\theta} + i \sin{\theta}))$</p>
<p>$ = \log{r} + \log(\cos{\theta} + i \sin{\theta})$</p>
<p>$ = \log{r} + i \theta$</p>
<p>The problem here is that... |
1,610,800 | <p>When dividing $f(x)$ by $g(x)$: $f(x)=g(x)Q(x)+R(x)$.
How to find the quotient $Q(x)$ and the remainder $R(x)$?
For example: $f(x)=\ 2x^4+13x^3+18x^2+x-4 \ $
, $g(x)=\ x^2+5x+2 \ $</p>
<blockquote>
<p>At first $g(x)= (x+4.56)(x+0.44)$ then we use synthetic division : $Q(x)=
\ 2x^2+3x-1 \ $ , $R(x)= \ 0.04x-1.98... | Alex | 12,133 | <p>Well
$$(x + 4.56)(x + 0.44) = x^2 + 5x + 2.0064 \neq x^2 + 5x + 2 = g(x), $$
so no, anything using that factorization couldn't tell you anything about $g(x)$. Your $Q(x)$ looks OK, but your $R(x)$ certainly doesn't satisfy $f(x) = Q(x)g(x) + R(x)$. Now, once you use $g(x)$ to get the correct $R(x)$ you will have an ... |
2,994,440 | <p><strong>Caution</strong>: Axiom of Completeness is not assumed here.</p>
<p><strong>Before reading my attempt, I want you to think up a proof of your own.</strong></p>
<p>Here is my attempt:</p>
<blockquote class="spoiler">
<p> Let <span class="math-container">$D$</span> be the set of all upper bounds of <span ... | Misha Lavrov | 383,078 | <p>You need to at some point invoke the definition of <span class="math-container">$\sup$</span> (or something that follows from it), because otherwise you're just talking about a symbol you know nothing about.</p>
<p>One definition of <span class="math-container">$\sup A$</span> is a real number <span class="math-con... |
2,994,440 | <p><strong>Caution</strong>: Axiom of Completeness is not assumed here.</p>
<p><strong>Before reading my attempt, I want you to think up a proof of your own.</strong></p>
<p>Here is my attempt:</p>
<blockquote class="spoiler">
<p> Let <span class="math-container">$D$</span> be the set of all upper bounds of <span ... | DanielWainfleet | 254,665 | <p>Let <span class="math-container">$U(A)$</span> be the set of upper bounds for <span class="math-container">$A.$</span> The definition of <span class="math-container">$\sup A$</span> is <span class="math-container">$\min U(A),$</span> which exists iff <span class="math-container">$U(A)$</span> has a least member.</... |
225,730 | <p>I am reading a paper "2-vector spaces and groupoid" by <a href="http://arxiv.org/abs/0810.2361" rel="noreferrer">Jeffrey Morton </a> and I need a help to understand the following.</p>
<p>Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor category from $X$ to $\mathbb{Vect}$. Suppose we have... | Bart Michels | 43,288 | <p>To expand on Qiaochu Yuan's answer: The Nakayama isomorphism is what you get when you take the "natural" isomorphism of <span class="math-container">$\mathbb C[G]$</span> with its left <span class="math-container">$\mathbb C[H]$</span>-dual in the sequence of isomorphisms
<span class="math-container">$$\begin{align*... |
2,777,051 | <p>When talking about operator semigroups, some books use the notation $(e^{tA})_{t \geq 0}.$ Doesn't this create an ambiguity? Why are results proved for this semigroup instead of taking a general semigroup, say $(T(t))_{t \geq 0}$? </p>
| Pedro | 70,305 | <p><em>When talking about operator semigroups, some books use the notation <span class="math-container">$(e^{tA})_{t \geq 0}.$</span> Doesn't this create an ambiguity?</em></p>
<p>No, because for these authors the notation <span class="math-container">$(e^{tA})_{t \geq 0}$</span> means the semigroup <span class="math-c... |
542,397 | <p>Consider the sequence $\{a_n\}_1^\infty$ such that $ a_n = $ ${n^m + 1}\over {n^{m+1} + 1}$ and $m \in \mathbb{R}$</p>
<p>EDIT: This is incorrect for $m < -1$, Then add the condition $m \geq -1$</p>
<p>I want to show this sequence is monotonically decreasing.<br>
This is not homework, I've seen many specific e... | N. S. | 9,176 | <p>$$\frac{n^m + 1}{n^{m+1} + 1} \geq \frac{(n+1)^m + 1}{(n+1)^{m+1} + 1} \Leftrightarrow \\
(n^m + 1)((n+1)^{m+1} + 1) \geq ((n+1)^m + 1)(n^{m+1} + 1)\Leftrightarrow \\
n^m (n+1)^{m+1} + n^m +(n+1)^{m+1} + 1\geq (n+1)^m n^{m+1} + (n+1)^m + n^{m+1} + 1 \Leftrightarrow \\
n^m (n+1)^{m} +n(n+1)^{m} \geq n^m (n-1) \Leftr... |
1,598,335 | <p>Find the transformation that takes $y=3^x$ to $y=\textit{e}^x$.
I have tried: </p>
<p>Let
$y=3^x$ to $y=e^{x'}$ </p>
<p>$$\log_{3}(y)=x\quad\text{hence}\quad\log_{3}(y)=\frac{\log_{e}(y)}{\log_{e}(3)}$$</p>
<p>$$x\log_{e}(3)=x'$$</p>
<p>Gives the transformation as dilate by $\log_e(3)$
And also this:</p>
<p>$$... | Narasimham | 95,860 | <p>Using short-hand $s,c$ for $\sin \theta, \cos \theta$
$$x= s/ \sqrt 2 + c/2; \, y = s/ \sqrt 2 - c/2 $$</p>
<p>Remember as formulas the results/identities:</p>
<p>$$ ( x+y)^2 +( x+y)^2 = 2(x^2 +y^2);\, ( x+y)^2-( x+y)^2 = 4 x y ; $$</p>
<p>$$ 3 (x^2 +y^2) = 3 s^2 + 3 c^2/2$$</p>
<p>$$ -2 x y = c^2/2 - s^2 $$... |
382,935 | <p>I reading my textbook and I don't understand the concept of distributions or number of solutions to an equation. It's explained that this problem is 1/4 types of sampling/distributions problems. An example is provided to illustrate:</p>
<blockquote>
<p>In how many ways can 4 identical jobs (indistinguishable ba... | user1527227 | 73,163 | <p>Let's say we have the following problem: Distribution of n identical objects into k boxes. </p>
<p>Let's consider a specific problem: How many ways can you put 4 identical balls into 2 boxes? We will draw a picture to understand the solution to this problem. Imagine 4 identical items are {<em>*</em>*}. If we ... |
1,654,354 | <p>Why is $A=\{(x_1,x_2,...,x_n)|\exists_{i\ne j}: x_i=x_j\}$ a null set?</p>
<p>This claim was shown in a solution I ran into, and I don't see how it holds. I try to follow the formal definition of nullity, which is having the possibility to be covered by a collection of open cubes, whose volume is as small as desire... | orangeskid | 168,051 | <p>HINT:</p>
<p>For $n\ge 2$ denote by $S_n$ the subset of $\mathbb{R}^n $ consisting of points with the first two coordinates equal. Then with the indentification $\mathbb{R}^n = \mathbb{ R}^2 \times \mathbb{R}^{n-2} $ we have
$$S_n = S_2 \times \mathbb{R}^{n-2} $$</p>
<p>Hence it is enough to show that $S_2 \su... |
1,529,731 | <p>I'm currently studying for the GRE and a specific type of question has me stumped. I have a two part question, but first here is one problem and my work:</p>
<p><strong>"If x is the remainder when a multiple of 4 is divided by 6, and y is the remainder when a multiple of 2 is divided by 3, what is the greatest poss... | Edem | 868,915 | <p>Multiples of 4 are: 4,8,12,16,20,24 etc.. and multiples of 2: 2,4,6,8,10,12,14 etc..
Divide the listed multiples of 4 by 6 to check the remainder by finding the remainder in each multiple. It is obvious that 6 goes into 16 with a remainder of 4 giving the value for X. Remember the question said the greatest possible... |
30,789 | <p>I'm trying to identify the frequencies in my time history samples, and I can see a frequency in the time history, but can't see it in its Fourier transform.
Here it is :</p>
<p>the sample data:</p>
<pre><code>dt = 0.01;(*0.01 second per sample*)
ls={7.18384,9.08503,7.13301,9.03243,7.23692,8.82911,7.48153,8.50053,7... | bill s | 1,783 | <p>Simon has hit the problem on the head (those are not real frequency components you see). But there is a way to discover what these frequencies are, by taking the envelope of the signal and transforming (DFTing) the envelope. </p>
<pre><code>data = ls - MeanFilter[ls, 5];
decay = 0.06; rise = 0.2;
filt[z_, u_] := Ma... |
1,232,439 | <p><strong>EDIT:</strong> This is for a production scheduling problem with quadratic production and linear inventory costs. </p>
<p>The goal is to \begin{equation*}
\max_{u} \int_{0}^{T} -(c_{1}u^{2} + c_{2}y) dt
\end{equation*}
subject to
\begin{align*}
y' &= u,\\
y(0) &= 0,\\
y(T) &=B.\\
\end{a... | Archaick | 191,173 | <p>First, we need to find the number of people in the room before Mr. L arrived, call it $N$. $N$ is the greatest integer such that ${N \choose 2} \leq 100$. This is because ${{N} \choose 2}+(N-1) < {{N+1} \choose 2}$. We find then that $N=14$ since ${14 \choose 2}+9=100$. That is, 5 people left the room, since Mr. ... |
84,977 | <p>In how many ways 3 flags of colors black, purple & yellow can be arranged at the corners of an equilateral triangle?</p>
| Quixotic | 2,109 | <p>This appears to me as a problem of <a href="http://mathworld.wolfram.com/CircularPermutation.html" rel="nofollow">circular permutation</a>. </p>
<p>Fix one flag of any color (say black) at a particular corner of the specified equilateral triangle, then the other two flags (purple and yellow) can be arranged in $2!$... |
2,226,150 | <blockquote>
<p>The following diophantine equation came up in the past paper of a Mathematics competition that I am doing soon: $$ 2(x+y)=xy+9.$$ </p>
</blockquote>
<p>Although I know that the solution is $(1,7)$, I am unsure as of how to reach this result. Clearly, the product $xy$ must be odd since $2(x+y)$ must b... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
4,373,262 | <p>Let G be a graph with order 9 so that the degree of each vertex is either 5 or 6. Prove
that there are either at least 5 vertices of degree 6 or at least 6 vertices of degree 5</p>
| Acccumulation | 476,070 | <p>By the Pigeonhole Principle, one of the degree classes has at least 5 members. If that degree class is 6, we're done. So assume there are at least 5 vertices with degree 5. We can't have an odd number of odd degrees, so there must be at least 6 vertices with degree 5.</p>
<p>If you're confused by the last statement,... |
504,997 | <p>I have a dynamic equation,
$$ \frac{\dot{k}}{k} = s k^{\alpha - 1} + \delta + n$$
Where $\dot{k}/k$ is the capital growth rate as a function of savings $s$, capital $k$, capital depreciation rate $\delta$, and population growth rate $n$.</p>
<p>I have been asked to find the change in the growth rate as $k$ increase... | baudolino | 67,409 | <p>The simplest way is via chain rule:
$$
\dfrac{\partial \dot{k}/k}{\partial k} = \dfrac{\partial \dot{k}/k}{\partial \ln{k}}\dfrac{\partial \ln{k}}{\partial k}
$$
...from which you can move things around to get your quantity of interest, e.g.
$$\dfrac{\partial \dot{k}/k}{\partial \ln{k}} = \dfrac{\partial \dot{k}/k}{... |
3,469,582 | <p>I have a feeling that <span class="math-container">$\lim_{x\rightarrow\infty}\frac{\log f(x^2)}{f(x)}=0$</span> for any positive monotone increasing function such that <span class="math-container">$\lim_{x\rightarrow\infty}f(x)=\infty$</span>, (in reality I am thinking mostly about super-exponential growth here but ... | Maximilian Janisch | 631,742 | <p>This is not always the case because it would imply that <span class="math-container">$$\exp(f(x))>f(x^2)$$</span> for all <span class="math-container">$x$</span> large enough, which can't be the case for all <span class="math-container">$f$</span> that satisfy your conditions. More precisely, it is not true if <s... |
8,756 | <p>I'd like to suggest that after a Question receives an Accepted Answer, some consideration be given to revising the title (if appropriate) to reflect what the real issue turned out to be.</p>
<p>It seems to me users often pick titles when first posting a question that are uninformative and are worth revisiting once ... | hardmath | 3,111 | <p>Searching meta.SO for "retitled questions" gives only eight hits (but some relevant ideas), while "editing title" returns more than 1,000.</p>
<p>One thread discusses <a href="https://meta.stackexchange.com/q/139970/155839">What do you look for in a question title?</a> and broaches the appropriateness of making cha... |
8,756 | <p>I'd like to suggest that after a Question receives an Accepted Answer, some consideration be given to revising the title (if appropriate) to reflect what the real issue turned out to be.</p>
<p>It seems to me users often pick titles when first posting a question that are uninformative and are worth revisiting once ... | Gerry Myerson | 8,269 | <p>There are many many questions with titles such as "combinatorics problem", "about a combinatorics problem", "doubt in a combinatorics problem", and so on. I have just retitled three of them. I'll stop there (for the time being). </p>
|
2,607,668 | <p>I am trying to prove/disprove $\operatorname{Arg}(zw)=\operatorname{Arg}(z)+\operatorname{Arg}(w)$. Apparently $\operatorname{Arg}(zw)=\operatorname{Arg}(z)+\operatorname{Arg}(w)+2k\pi$ where $k=0,1,\text{ or }-1$, but I have no idea why. I keep on finding that answer online. I am very lost on how to prove this stat... | Alvaro | 518,181 | <p>"but I have no idea why" ... Because exponential is periodic with period $2 \pi i$ over the complex plane. This is a picture from a Mathcad worksheet, but try this with your calculator:</p>
<p><a href="https://i.stack.imgur.com/7rHpu.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7rHpu.gif" alt... |
3,771,972 | <blockquote>
<p>If gcd<span class="math-container">$(m,n)=1$</span>, then <span class="math-container">$\phi(mn)=\phi(m)\phi(n)$</span>.</p>
</blockquote>
<p>My text book write that <span class="math-container">$f:\Bbb Z/_{(mn)}\Bbb Z\to\Bbb Z/_m\Bbb Z \times \Bbb Z/_n\Bbb Z$</span> by <span class="math-container">$f(... | Wuestenfux | 417,848 | <p>The first isomorphism refers to a ring isomorphism.</p>
<p>The second assertion refers to the mapping of the multiplicative groups of the involved rings. They are mapped accordingly.</p>
|
50,023 | <p>two problems from Dugundji's book page $156$. (I don't know why the system deletes the word hi in this sentence)</p>
<p>$1$. Let $X$ be a Hausdorff space. Show that:</p>
<p>a) $\bigcap \{F: x \in F , F \ \textrm{closed}\} = \{x\}$.</p>
<p>b) $\bigcap \{U : x \in U, U \ \textrm{open}\} = \{x\}$.</p>
<p>c) The ab... | Brian M. Scott | 12,042 | <p>Some style comments on 1(a): As user6312 says, you should just begin by supposing that $y \ne x$; starting with $y \notin \{x\}$ is an unnecessary complication. More important, what Hausdorffness gives you is not just disjoint open sets $U_x$ and $V_y$, but disjoint open sets $U_x$ and $V_y$ <strong>such that $x \in... |
1,461,484 | <p>For example, look at this sentence from Perko's text on dynamical system</p>
<p>"It follows from Cauchy Schwarz inequality that if $T \in L(R^n)$ is represented by the matrix $A$ with respect to the standard basis for $R^n$ $_\cdots$" pg 11</p>
<p>What does it mean for a $T: R^n \to R^n$ to be represented by a mat... | skyking | 265,767 | <p>The concept of matrix can be useful in other situation than for linear operators, and there exists linear operators that can not be represented by matrices.</p>
<p>The matrix is just an array of elements and are added and multiplied according to certain rules. For this it is only required that there is a meaningful... |
2,282,359 | <p>I am trying to calculate this limit:
$$\lim_{x \to \infty} x^2(\ln x-\ln (x-1))-x$$
The answer is $1/2$ but I am trying to verify this through proper means. I have tried L'Hospital's Rule by factoring out an $x$ and putting that as $\frac{1}{x}$ in the denominator (indeterminate form) but it becomes hopeless afterwa... | marty cohen | 13,079 | <p>$\begin{array}\\
\lim_{x \to \infty}x^2(\ln x-\ln (x-1))-x
&= \lim_{x \to \infty} (-x^2(\ln (1-1/x))-x)\\
&= \lim_{y \to 0} (-\frac1{y^2}(\ln (1-y))-\frac1{y})\\
&= \lim_{y \to 0} (-\frac1{y^2}((y+\frac{y^2}{2}+O(\frac1{y^3}))-\frac1{y})\\
&= \lim_{y \to 0} ((\frac1{y}+\frac{1}{2}+O(\frac1{y}))-\fra... |
2,282,359 | <p>I am trying to calculate this limit:
$$\lim_{x \to \infty} x^2(\ln x-\ln (x-1))-x$$
The answer is $1/2$ but I am trying to verify this through proper means. I have tried L'Hospital's Rule by factoring out an $x$ and putting that as $\frac{1}{x}$ in the denominator (indeterminate form) but it becomes hopeless afterwa... | hamam_Abdallah | 369,188 | <p>Using fact that
$$\ln (1+u)=u-\frac {u^2}{2}(1+\epsilon (u)) $$ with $\lim_0 \epsilon (u)=0$,
we get
$$x^2 \ln (\frac{x}{x-1})-x=$$</p>
<p>$$x^2\ln(1+\frac {1}{x-1})-x =$$
$$=x^2 (\frac {1}{x-1}-\frac{1}{2 (x-1)^2} (1+\epsilon (x)) -x$$
with $\lim_{+\infty}\epsilon (x)=0$,
$$=\frac {x}{x-1}-\frac {x^2}{2 (x-1)^2}\l... |
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