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 |
|---|---|---|---|---|
405,772 | <p>I encountered a conformal mapping on the complex plane:$$z\rightarrow e^{i\pi z}$$
and I am not sure about where it does send the point at infinity. If I could say something along the lines: $$\text{Im}(\infty) = \infty$$
Then it would map it to the origin but there is still a voice in my head saying that this equal... | Cameron Buie | 28,900 | <p>If we allow $z$ to approach the point at infinity along the positive imaginary axis, we find that $e^{i\pi z}$ tends to $0$. Approaching along the negative imaginary axis, we find that $e^{i\pi z}$ gets big without bound. It turns out that we can make $e^{i\pi z}$ tend toward anything we like, just by allowing $z$ t... |
2,184,593 | <p>By Cauchy's criterion of limit (not sequencial criterion), show that $$\lim_{x\to 0}(\sin{\frac{1}{x}}+x\cos{\frac{1}{x}})$$ does not exist.</p>
<p>Cauchy's criterion of limit </p>
<p>$\lim_{x\to c}f(x)=l$ iff for every $\epsilon>0$, there exists $\delta$ such that $$|f(x_2)-f(x_1)|<\epsilon$$ for $0<|x_1... | Juniven Acapulco | 44,376 | <p>One part of Cauchy's Criterion says that</p>
<blockquote>
<p><strong>RESULT:</strong> If $\exists \epsilon>0$ such that $\forall \delta >0$, we can find $x_1,x_2$ satisfying $0<|x_1-a|<\delta$ and $0<|x_2-a|<\delta$ but $|f(x_2)-f(x_1)|\geq \epsilon$ then $\lim_{x\to a}f(x)$ does not exist. </p>... |
4,613,214 | <p>I have to do a large modulo but my answer is incorrect.<br />
I am given:<br />
<span class="math-container">$$ 111^{4733} \mod 9467 $$</span></p>
<ul>
<li>9467 prime</li>
<li>111 and 9467 are coprime</li>
<li>Also note that 4733*2=9466<br />
So we can Apply Euler's theorem</li>
</ul>
<p><span class="math-container"... | Giorgos Giapitzakis | 907,711 | <p>In modulo arithmetic, fractional powers are not well defined. For example, <span class="math-container">$1^{1/2}$</span> can just as easily be <span class="math-container">$1 \pmod{9467}$</span> or <span class="math-container">$9466 \pmod{9467}$</span>. In this case, it's best to notice that you are asked to calcula... |
37,013 | <p><strong>Question:</strong> What are some interesting or useful applications of the Hahn-Banach theorem(s)?</p>
<p><strong>Motivation:</strong> Most of the time, I dislike most of Analysis. During a final examination, a question sparked my interest in the Hahn-Banach theorem(s). One of my favorite things to do is... | AD - Stop Putin - | 1,154 | <p>How about the Wiener Tauberian theorem: </p>
<p><strong>Theorem (N. Wiener 1932).</strong> For $f\in L^1(\mathbb{R})$, let $X= \operatorname{span}\{f_t:t\in\mathbb{R}\}$ (that is the linear subspace spanned by the translates of $f$). Then the closure of $X$ in $L^1$ is $L^1$ if and only if the Fourier transform of ... |
230,971 | <p>At the moment I use <code>Length[ DeleteDuplicates[ array ] ] == 1</code> to check whether an array is constant, but I'm not sure whether this is optimal.</p>
<p>What would be the quickest way to test whether an array consists of equal elements?</p>
<p>What if the elements would be integers?</p>
<p>What if they are ... | kglr | 125 | <p><code>Statistics`Library`ConstantVectorQ</code> is quite fast.</p>
<p>Using Sjoerd's input examples:</p>
<pre><code>const = ConstantArray[1, 100000];
nonconst = Append[const, 2];
nonconst2 = Prepend[const, 2];
t11 = Statistics`Library`ConstantVectorQ@const // RepeatedTiming;
t21 = CountDistinct[const] == 1 // Repe... |
2,086,006 | <p>You have $7$ boxes in front of you and $140$ kittens are sitting side-by-side inside the
boxes, $20$ in each box. You want to take some kittens as your pets. However the
kittens are very cowardly. Each time you chose a kitten from a box, the kittens that
are in that box to the left of it go to the box in the left, t... | Olivier Oloa | 118,798 | <p>One may write, as $n \to \infty$,
$$
\begin{align}
a_n&=n\left(e^{\frac{\large\ln(ea)}n}-e^{\large\frac{\ln(a)}n} \right)
\\&=n\left(e^{\large\frac{1+\ln(a)}n}-e^{\large\frac{\ln(a)}n} \right)
\\&=n \cdot e^{\large\frac{\ln(a)}n}\left(e^{\large\frac{1}n}-1 \right)
\\&= e^{\large\frac{\ln(a)}n}\cdot\f... |
2,520,768 | <p>How would I approach this problem? </p>
<p>Let $(a, b, c) \in \mathbb{Z^3}$ with $a^2 + b^2 = c^2$. Show that:
$$
60 \,\mid\, abc
$$</p>
| user502959 | 502,959 | <p>If one of them is 0, then the product is 0, divisible by 60.</p>
<p>Assume that $abc\neq 0$, WLOG $a,b,c>0$.</p>
<p>Then $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$ for some $m,n\in\mathbb{N}$ (well known thing, <a href="https://en.wikipedia.org/wiki/Pythagorean_triple" rel="nofollow noreferrer">https://en.wikipedia.org/... |
2,595,247 | <p>What is equation of circle when two lines y=x and y=x-4 are tangent to a circle at (2,2) and (4,0) respectively.</p>
| Ѕᴀᴀᴅ | 302,797 | <p>If it's a Lebesgue integral, it's apparent by the dominated convergence theorem. If it's a Riemann integral, it can be proved by showing$$
\lim_{n \to \infty} \int_0^{A} x^{n + 1} f'(x) \,\mathrm{d}x = 0
$$
for every $0 < A <1$.</p>
|
3,355,542 | <p>Let <span class="math-container">$f \in L^{1} [0,1]$</span> such that for all smooth function <span class="math-container">$h: [0,1] \to \mathbb R$</span> with <span class="math-container">$h(0) = h(1) = 0$</span> one has <span class="math-container">$\int_{0}^{1} f(t) h'(t) = 0$</span>. Prove that <span class="mat... | David C. Ullrich | 248,223 | <p><strong>Hehe:</strong> If <span class="math-container">$n$</span> is a non-zero integer then <span class="math-container">$e^{2\pi int}=h_n'(t)$</span>. So <span class="math-container">$\hat f(n)=0$</span> for <span class="math-container">$n\ne0$</span>, hence <span class="math-container">$f=\hat f(0)$</span> almos... |
203,464 | <p>I would like to exclude the point <code>{x=0,y=0}</code> in the function definition</p>
<pre><code>f = Function[{x, y}, {x/(x^2 + y^2), -(y/(x^2 + y^2))}]
</code></pre>
<p>So far I tried <code>ConditionalExpression</code>and <code>/;</code> without success.</p>
<p>Thanks!</p>
| kglr | 125 | <p>You can use <a href="https://reference.wolfram.com/language/ref/Outer.html" rel="nofollow noreferrer"><code>Outer</code></a> or <a href="https://reference.wolfram.com/language/ref/Tuples.html" rel="nofollow noreferrer"><code>Tuples</code></a> as follows:</p>
<pre><code>Join @@ Outer[List @* Plus, {a, b, c}, {d, e, ... |
4,359,372 | <p>My question is: Does there exist <span class="math-container">$x_n$</span> (<span class="math-container">$n\geq 0$</span>) such that <span class="math-container">$x_n$</span> is a bounded and divergent sequence with <span class="math-container">$$x_{n+m}\leq (x_n+x_m)/2$$</span> for all <span class="math-container">... | user2661923 | 464,411 | <p>Extending the answer of bjcolby15:</p>
<p>It is assumed that all of the <em>weights</em> are only allowed on one side of the scale and the object to be weighed is on the other side of the scale.</p>
<p>Any positive integer can be expressed in base <span class="math-container">$2$</span> format. In such a format, e... |
298,284 | <p>Let $\zeta(M,s)$ be the Minakshisundaram-Pleijel zeta function, which encodes the eigenvalues of the Laplace-Beltrami operator. Where can I find a proof or reference of the following identity? If $M$ is a surface:
$$\zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA$$</p>
<p>Where $K $ is the Gaussian Curvature.</p>
| Sylvain JULIEN | 13,625 | <p>If you read French, it seems Marcel Berger adresses this question in his review article in Development of Mathematics 1950-2000, Birkhaüser.</p>
|
298,284 | <p>Let $\zeta(M,s)$ be the Minakshisundaram-Pleijel zeta function, which encodes the eigenvalues of the Laplace-Beltrami operator. Where can I find a proof or reference of the following identity? If $M$ is a surface:
$$\zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA$$</p>
<p>Where $K $ is the Gaussian Curvature.</p>
| Alex M. | 54,780 | <p>Look for "A Panoramic View of Riemannian Geometry" by Marcel Berger: you will find what you are looking for in sub-subchapter 9.7.2 "Great Hopes" (pages 421-422). Taking a look at 1.8.5 "Second Way: the Heat Equation" (page 100) where things are done for surfaces with boundary embedded in Euclidean spaces will impro... |
1,048,526 | <p>I'm trying to bound the quantity
<span class="math-container">$\langle \nabla \Psi(x),\bar{x}-x \rangle$</span> above, with the bound depending on <span class="math-container">$\|x-\bar{x}\|$</span> and perhaps also of <span class="math-container">$\|x-y\|$</span> for fixed (but not varying) points <span class="math... | xel | 418,533 | <p>I think the property you are looking for is Lipschtiz continuity of the function $\Psi$, as this is equivalent to bounded subgradients.</p>
|
2,501,518 | <p>$\begin{pmatrix}
a \\
b
\end{pmatrix}
\begin{pmatrix}
a \\
b
\end{pmatrix}^T
\begin{pmatrix}
C & D \\
D^T & E
\end{pmatrix}=
\begin{pmatrix}
I_m & 0\\
0 & I_n
\end{pmatrix}
$</p>
<p>$a$ and $b$ are vectors with length $m$ and $n$ respectively. C has dimension $m$ by $m$ and E $n$ by $n$. </p>
<p>... | Dustin G. Mixon | 442,087 | <p>There are no solutions in the case where $A$ and $B$ are square. By 1 and 4, $A^{-1}=A^TC+B^TD^T$ and $B^{-1}=A^TD+B^TE^T$. This implies that $A$, $B$, and their inverses all have full rank, as do any product of these. But 2 and 3 report $BA^{-1}=AB^{-1}=0$, a contradiction.</p>
|
3,858,362 | <p>Solve <span class="math-container">$$\dfrac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=0.$$</span>
We have <span class="math-container">$D_x:\begin{cases}x^2-5x+4\ge0\\x^2-5x+4\ne0\end{cases}\iff x^2-5x+4>0\iff x\in(-\infty;1)\cup(4;+\infty).$</span> Now I am trying to solve the equation <span class="math-container">$x^3-4... | QED | 91,884 | <p><span class="math-container">$$\frac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=\frac{(x-4)(x-2)(x+2)}{\sqrt{(x-4)(x-1)}}==\frac{\sqrt{x-4}(x-2)(x+2)}{\sqrt{x-1}}$$</span>
The fraction will be <span class="math-container">$=0$</span> if the numerator is <span class="math-container">$0$</span> and the denominator is not <span ... |
3,858,362 | <p>Solve <span class="math-container">$$\dfrac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=0.$$</span>
We have <span class="math-container">$D_x:\begin{cases}x^2-5x+4\ge0\\x^2-5x+4\ne0\end{cases}\iff x^2-5x+4>0\iff x\in(-\infty;1)\cup(4;+\infty).$</span> Now I am trying to solve the equation <span class="math-container">$x^3-4... | user2661923 | 464,411 | <p>Alternate approach</p>
<p>My algebra abilities are limited. I will show you how I would attack the
problem.</p>
<p>Given <span class="math-container">$$\frac{f(x)}{\sqrt{g(x)}} = 0$$</span></p>
<p>where (if I understand correctly) <span class="math-container">$x$</span> may be any real number</p>
<p>then my first s... |
3,536,671 | <p>I have the following mathematical operations to use: Add, Divide, Minimum, Minus, Modulo, Multiply and Round.</p>
<p>With these I need to get a number, run it through a combination of these and return 0 if the number is negative or equal to 0 and the number itself if the number is greater than 0.</p>
<p>Is that po... | Jaap Scherphuis | 362,967 | <p>Here is a slightly different way.</p>
<p>What you really want is <span class="math-container">$\max(0,x)$</span>, but you don't have the max function available. Fortunately however you do have the min function, and you can use the fact that <span class="math-container">$\max(a,b) = -\min(-a,-b)$</span>.</p>
<p>So ... |
4,280,426 | <blockquote>
<p>We have a bag with <span class="math-container">$3$</span> black balls and <span class="math-container">$5$</span> white balls. What is the probability of picking out two white balls if at least one of them is white?</p>
</blockquote>
<p>If <span class="math-container">$A$</span> is the event of first b... | Atticus Stonestrom | 663,661 | <p>Yes, it is the case that <span class="math-container">$\operatorname{cl}(A)=\mathbb{R}$</span>. Here is an argument that works in more general contexts: by definition, a subset <span class="math-container">$X\subseteq\mathbb{R}$</span> is closed if and only if either <span class="math-container">$X$</span> is counta... |
2,511,095 | <p>Let $p$ be an odd prime. We know that the polynomial $x^{p-1}-1$ splits into linear factors modulo $p$. If $p$ is of the form $4k+1$ then we can write
$$x^{p-1}-1=x^{4k}-1=(x^{2k}+1)(x^{2k}-1).$$
The theorem of Lagrange tells us that any polynomial congruence of degree $n$ mod $p$ has at most $n$ solutions. Hence we... | Dr. Sonnhard Graubner | 175,066 | <p>Setting $a=bt$ then we get
$$b^3+t^3+39b^3t-18=0$$
$$3b^3t^2+13b^3-5=0$$ eliminating $b^3$ then we get
$$5t^3-54t^2+195t-234=0$$
one solution is $t=3$ then you will get $a=3b$
plugging this in the given equation we get $$a=\frac{3}{2},b=\frac{1}{2}$$</p>
|
2,511,095 | <p>Let $p$ be an odd prime. We know that the polynomial $x^{p-1}-1$ splits into linear factors modulo $p$. If $p$ is of the form $4k+1$ then we can write
$$x^{p-1}-1=x^{4k}-1=(x^{2k}+1)(x^{2k}-1).$$
The theorem of Lagrange tells us that any polynomial congruence of degree $n$ mod $p$ has at most $n$ solutions. Hence we... | Ennar | 122,131 | <p><em>Disclaimer: This answer is probably not appropriate considering (algebra-precalculus) tag, but I'll write it anyway since it might be useful to others stumbling upon this question.</em></p>
<p>The system is easier to solve when considering original equation $$(a+b\sqrt{13})^3=18+5\sqrt{13}.$$</p>
<p>If we look... |
300,163 | <p>I need to integrate the $z/\bar z$ (where $\bar z$ is the conjugate of $z$) counterclockwise in the upper half ($y>0$) of a donut-shaped ring. The outer circle is $|z|=4$ and the inner circle is $|z|=2$. </p>
<p><strong>My method:</strong></p>
<p>$z/\bar z = e^{2i\theta}$ - which is entire over the complex plan... | Ron Gordon | 53,268 | <p>In the ccw direction, there are 4 contributions to this integral:</p>
<p>$$\begin{align}\oint_C dz \frac{z}{z^*} &= 4 \int_0^{\pi} d\theta \: e^{i 3 \theta} - 2\int_0^{\pi} d\theta \: e^{i 3 \theta} + \int_{-4}^{-2} dt + \int_2^4 dt\\ &= \frac{-8}{3 i} + \frac{4}{3 i} + 2 + 2 \\ &= 4 + i\frac{4}{3} \en... |
878,785 | <p>I know that the common approach in order to find an angle is to calculate the dot product between 2 vectors and then calculate arcus cos of it. But in this solution I can get an angle only in the range(0, 180) degrees. What would be the proper way to get an angle in range of (0, 360)?</p>
| MvG | 35,416 | <p><em>I'm adapting <a href="https://stackoverflow.com/a/16544330/1468366">my answer on Stack Overflow</a>.</em></p>
<h1>2D case</h1>
<p>Just like the <a href="http://en.wikipedia.org/wiki/Dot_product" rel="noreferrer">dot product</a> is proportional to the cosine of the angle, the <a href="http://en.wikipedia.org/wi... |
436,225 | <p><a href="http://en.wikipedia.org/wiki/Incidence_matrix">The incidence matrix</a> of a graph is a way to represent the graph. Why go through the trouble of creating this representation of a graph? In other words what are the applications of the incidence matrix or some interesting properties it reveals about its grap... | Lord Soth | 70,323 | <p>Because then one may apply matrix theoretical tools to graph theory problems. One area where it is useful is when you consider flows on a graph, e.g. <a href="http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces/graphs-networks-incidence-matrices/MIT18_06SCF11_Ses1.12su... |
2,506,279 | <blockquote>
<p>If $\lim_{x\to \infty}xf(x^2+1)=2$ then find
$$\lim_{x\to 0}\dfrac{2f'(1/x)}{x\sqrt{x}}=?$$</p>
</blockquote>
<p>My Try :
$$g(x):=xf(x^2+1)\\g'(x)=f(x^2+1)+2xf'(x^2+1)$$
Now what?</p>
| vinc17 | 459,608 | <p>The sigma notation is correct, and maybe the best one as it is non-ambiguous.</p>
<p>The <a href="https://en.wikipedia.org/wiki/Floating-point_arithmetic#Floating-point_numbers" rel="nofollow noreferrer">Wikipedia page</a> excluded $2^{-3}$, $2^{-5}$ and $2^{-6}$ because their corresponding bits were 0's: $0 × 2^{-... |
387,268 | <p>Let <span class="math-container">$A$</span> be an <span class="math-container">$N\times N$</span> nonnegative matrix with all diagonal entries equal to zero and such that there is <span class="math-container">$n_0$</span> such that all entries of <span class="math-container">$A^{n_0}$</span> are strictly positive. L... | Jochen Glueck | 102,946 | <p><em>Partial answer:</em> For the special case of self-adjoint matrices, the answer to (2) is <strong>yes</strong>. Funnily enough, this has nothing to do with the non-negativity of the matrix:</p>
<p><strong>Proposition.</strong> Let <span class="math-container">$A \not= 0$</span> be a self-adjoint complex <span cla... |
426,974 | <p>Suppose the dynamical system <span class="math-container">$(X,T)$</span> has only proper factors (i.e. not <span class="math-container">$(X,T)$</span> itself) of zero topological entropy. Does the system <span class="math-container">$(X,T)$</span> also have zero entropy?</p>
| Ronnie Pavlov | 116,357 | <p>This question is very related to the question of <strong>lowering topological entropy</strong>, introduced in ``Can one always lower topological entropy?'' by Shub and Weiss and then very nearly solved by Lindenstrauss in "Lowering topological entropy" and "Mean Dimension, Small Entropy Factors, and a... |
3,266,930 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a positive random variable on the <span class="math-container">$(\Omega,\mathscr{A},P)$</span>. Show that if <span class="math-container">$X\in L_p$</span> for <span class="math-container">$1<p<\infty$</span>.
Prove <span class="math-container">$\... | Bernard | 202,857 | <p><em>Euler's theorem</em> asserts that every element <span class="math-container">$a$</span> which is coprime to <span class="math-container">$n$</span>, i.e. which is a unit mod. <span class="math-container">$n$</span>, satisfies <span class="math-container">$\:a^{\varphi(n)}\equiv 1\mod n$</span>. </p>
<p>This doe... |
3,854,286 | <p>This was an exercise in my class, please help:</p>
<blockquote>
<p>Put <span class="math-container">$A = {\mathbb Q}[x,y]$</span> and <span class="math-container">$B = {\mathbb Q}[x,z]$</span>. Consider the morphism <span class="math-container">$f \colon A \to B$</span> of <span class="math-container">${\mathbb Q}$<... | David Holmes | 618,250 | <p>I don't follow Mindlack's proof, as I'm not sure what their <span class="math-container">$C$</span> is. The proof I had in mind when posting the question was the following:</p>
<p>For the justification that <span class="math-container">$$y \otimes 1 \neq x \otimes z,$$</span> could use the "truncated Koszul co... |
3,410,802 | <p>I was trying to prove that a surjective Endomorphism <span class="math-container">$f:A \to A$</span> of a noetherian ring is also injective. I would like to know why this argument is not correct?
<span class="math-container">$A/\rm{Ker}f \cong \rm{Im}f=A \Rightarrow Kerf=\{0\}$</span></p>
| Community | -1 | <p>It doesn't work because there isn't any general principle of ring theory according to which <span class="math-container">$R/I$</span> should be isomorphic to <span class="math-container">$R$</span> only if <span class="math-container">$I=0$</span>.</p>
<p>If <span class="math-container">$f$</span> were surjective b... |
3,410,802 | <p>I was trying to prove that a surjective Endomorphism <span class="math-container">$f:A \to A$</span> of a noetherian ring is also injective. I would like to know why this argument is not correct?
<span class="math-container">$A/\rm{Ker}f \cong \rm{Im}f=A \Rightarrow Kerf=\{0\}$</span></p>
| Claudius | 218,931 | <p>Your argument cannot be correct, since it doesn't use the noetherian hypothesis; this is because for a non-noetherian ring the statement is wrong:<br>
Consider the polynomial ring <span class="math-container">$A = \mathbb Z[x_1,x_2,x_3,\dotsc]$</span> in infinitely many variables. It is clearly not noetherian. Now, ... |
529,260 | <p>Let $V$ be a complex vector space of dimension $n$ with a scalar product, and let $u$ be an unitary vector in $V$. Let $H_u: V \to V$ be defined as</p>
<p>$$H_u(v) = v - 2 \langle v,u \rangle u$$</p>
<p>for all $v \in V$. I need to find the minimal polynomial and the characteristic polynomial of this linear operat... | Felix Marin | 85,343 | <p>$$
\left\langle \nu',H_{u}\left(\nu\right)\right\rangle
=
\left\langle \nu', \nu\right\rangle
-
2\left\langle \nu, u\right\rangle
\left\langle \nu', u\right\rangle
$$</p>
|
1,943,351 | <p>Good day,</p>
<p>In class we said that if a random variable <span class="math-container">$X-Y$</span> is independent of random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> then <span class="math-container">$X-Y$</span> is almost sure constant, i.e. there exists a <spa... | zhoraster | 262,269 | <p>Extending Landon Carter's answer,</p>
<blockquote>
<p>i.e. $f(X,Y)$ is almost sure constant it it is independent of $X$ and $Y$?</p>
</blockquote>
<p>No. Flip a coin twice, let $X_i = \mathbf{1}_{\text{H on $i$th flip}}$, $i=1,2$, $Z = \mathbf{1}_{X_1=X_2}$. Then $Z$ is independent of both $X_1$ and $X_2$, but i... |
1,506,726 | <p>Let $V$ is a finite dimensional vector space, and $H_{1}, H_{2}$ be subspaces of $V$ such that $V=H_{1}\oplus H_{2}$. Now $V/H_{1}$ is isomorphic to $H_{2}$.</p>
<p>If we replace the vector space $V$ by a group $G$ and consider $H_1$ ,a normal subgroup of $G$, then <strong>under what conditions</strong> can we say ... | Remy | 284,272 | <blockquote>
<p><strong>Lemma.</strong> If $H_1, H_2 \subseteq G$ are two subgroups, with $H_1$ normal, such that $H_1 \cap H_2 = 0$ and $H_1 H_2 = G$, then $G/H_1 \cong H_2$.</p>
</blockquote>
<p>Note that when $H_1$ is normal, the set $H_1 H_2$ is just the set of products $h_1 h_2$ with $h_1 \in H_1$ and $h_2 \in ... |
1,506,726 | <p>Let $V$ is a finite dimensional vector space, and $H_{1}, H_{2}$ be subspaces of $V$ such that $V=H_{1}\oplus H_{2}$. Now $V/H_{1}$ is isomorphic to $H_{2}$.</p>
<p>If we replace the vector space $V$ by a group $G$ and consider $H_1$ ,a normal subgroup of $G$, then <strong>under what conditions</strong> can we say ... | BCLC | 140,308 | <p>Different proof of Remy's Lemma:</p>
<ol>
<li><p>Observe <span class="math-container">$g \in G$</span> is uniquely <span class="math-container">$g=ab$</span> for some <span class="math-container">$(a,b) \in H_1 \times H_2$</span></p></li>
<li><p>Define the projection map onto <span class="math-container">$H_2$</spa... |
1,473,513 | <p>The motion of a pendulum is described by the differential equation</p>
<p><span class="math-container">$$ \ddot\theta +\frac gl \sin \theta = 0$$</span></p>
<p>if we integrate this equation with respect to <span class="math-container">$\theta$</span> we obtain</p>
<p><span class="math-container">$$ \frac 12 \dot... | Ron Gordon | 53,268 | <p>Multiply the equation through by $\dot{\theta}$:</p>
<p>$$\dot{\theta}\, \ddot{\theta} +\frac{g}{\ell} \dot{\theta} \sin{\theta} = 0$$</p>
<p>Integrate with respect to $t$.</p>
<p>$$\int dt \, \dot{\theta}\, \ddot{\theta} = \int d\dot{\theta} \, \dot{\theta} = \frac12 \dot{\theta}^2 + C$$</p>
<p>$$\int dt\, \... |
1,993,693 | <blockquote>
<p>$$\lim_{x \rightarrow +\infty} \frac{2^x}{x}$$ $$\lim_{x \rightarrow
\infty} \frac{x^{50}}{e^x}$$</p>
</blockquote>
<p>I don't really know how to solve this.</p>
<p>As for the first one, I know that $\lim_{x \rightarrow \infty} a^x=0$ , I supposed that helps...?</p>
<p>How do I solve these (prefer... | hamam_Abdallah | 369,188 | <p>Hint for the first.</p>
<p>taking logarithm we get</p>
<p>$$\lim_{x\to +\infty}(x\ln(2)-\ln(x))=$$</p>
<p>$$\lim_{x\to +\infty} x\left(\ln(2)-\frac{\ln(x)}{x}\right)=$$</p>
<p>$$+\infty$$</p>
<p>since $\lim_{x\to+\infty}\frac{\ln(x)}{x}=0$.</p>
<p>thus the first limit is $+\infty$.</p>
<p>the same approach gi... |
1,993,693 | <blockquote>
<p>$$\lim_{x \rightarrow +\infty} \frac{2^x}{x}$$ $$\lim_{x \rightarrow
\infty} \frac{x^{50}}{e^x}$$</p>
</blockquote>
<p>I don't really know how to solve this.</p>
<p>As for the first one, I know that $\lim_{x \rightarrow \infty} a^x=0$ , I supposed that helps...?</p>
<p>How do I solve these (prefer... | E.H.E | 187,799 | <p>Hint:
$$2^x=\sum_{k=0}^{\infty }\frac{(x\log 2)^k}{k!}$$
and
$$\frac{x^{50}}{e^x}=\frac{50!x^{50}}{x^{50}+50!(1+x+\frac{x^2}{21}+....\frac{x^{49}}{49!}+\frac{x^{51}}{51!}......)}$$</p>
|
404,472 | <p>Let $F$ and $F′$ be two finite fields with nine and four elements respectively.
How many field homomorphisms are there from $F$ to $F′$?</p>
| Jared | 65,034 | <p>Hint $1$: A homomorphism of fields is injective. Can you see why?</p>
<p>Hint $2$: Hint $1$ answers your question. Can you see why?</p>
|
123,918 | <p>Someone <a href="https://stackoverflow.com/questions/9851628/minimal-positive-number-divisible-to-n">asked this question</a> in SO:</p>
<blockquote>
<p><span class="math-container">$1\le N\le 1000$</span></p>
<p>How to find the minimal positive number, that is divisible by N, and
its digit sum should be equal to N.<... | Monoide | 27,545 | <p>If 10 doesn't divise N we can define the set $A = \{(n=x_1x_2\cdots x_i0) : \sum\limits_{k} x_k = N\}$ wich is a set containing at least one solution (if it exists) of the problem above (in particular the minimum one). The size of $A$ can be brutaly bounded by $10^N$. It is easy to construct all elements in $A$ by a... |
2,384,538 | <p>I am studying Linear Algebra Done Right, chapter 2 problem 6 states:</p>
<blockquote>
<p>Prove that the real vector space consisting of all continuous real valued functions on the interval $[0,1]$ is infinite dimensional.</p>
</blockquote>
<p><strong>My solution:</strong></p>
<p>Consider the sequence of functio... | Rodrigo A. Pérez | 88,190 | <p>Suppose $x^n = \sum\limits_{k=1}^{n-1} a_kx^k$. Consider what happens when $x$ is much larger than $|a_1|+\ldots+|a_{n-1}|$. By the triangle inequality, you will get a contradiction. Essentially, you are showing that $x^n$ grows faster than any degree $n-1$ polynomial...</p>
|
1,970,305 | <p>I have just begun reading through Section 3.2 of Hatcher's Algebraic Topology. While I reasonably understood the computations relating to the cup product, I was unsure of the purpose of the cup product. From what I knew, it does not help us to compute cohomology groups, given that we need the cohomology groups to co... | Curious Math Student | 333,346 | <p>I was able to figure it out. So I thought I would post it in case someone else searched for help on a similar problem!</p>
<p>Solve $6x+15y+10z = 53 \rightarrow 6x+5(3y+2z)=53.$ Let $w=3y+2z.$ Solve:</p>
<p>$$3y+2z=w\ (1)
\\6x+5w=53\ (2)$$</p>
<p>Solution to $(2):\ gcd(6,5) = 1.$ So forming a linear combo, get $6... |
2,280,052 | <p>Wolfram Alpha says:
$$i\lim_{x \to \infty} x = i\infty$$</p>
<p>I'm having a bit of trouble understanding what $i\infty$ means. In the long run, it seems that whatever gets multiplied by $\infty$ doesn't really matter. $\infty$ sort of takes over, and the magnitude of whatever is being multiplied is irrelevant. I.e... | Level River St | 137,034 | <p>Real numbers and imaginary numbers are different things. <code>∞</code> is different from <code>∞i</code>, just like infinity oranges and infinity bottles of juice are different things. There are operations that will convert one to the other, but that is beside the point.</p>
<p>I think a good way to see this is to ... |
1,317,610 | <p>Let $u = u(t,x)$ satisfy the PDE
$$
\frac{\partial u}{\partial t} = \frac{1}{2}c^2\frac{\partial^2 u}{\partial x^2} + (a + bx)\frac{\partial u}{\partial x} + f u,
$$
where $a,b,c,f \in \mathbb{R}$ are constant.</p>
<p>I'm aware of solution methods for when $c \propto x^2$ (so not constant) and $a = 0$, for which I ... | Brian Tung | 224,454 | <p>An alternative to Joffan's solution is to count up all the ways there could be exactly $k$ vowels (as suggested by André Nicolas). We then get</p>
<p>$$
N = \sum_{k=1}^8 \binom{8}{k} 5^k 21^{8-k}
$$</p>
<p>All the methods yield $N = 171004205215$, confirming the expression $26^8-21^8$ you originally derived.</p>
|
202,247 | <p>I'm working on some problem in algebraic geometry. I need a reference to the following result:</p>
<p>Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$
be a non zero polynomial. The complement manifold $\mathbb{C}^{h}\setminus\left\lbrace F=0\right\rbrace$ is a
nonempt... | Peter Michor | 26,935 | <p>$F$ is non zero, thus the complement is not empty. The regular part of the zero set, namely $\{z:F(z)=0, dF(z)\ne 0\}$, has complex codimension 1, thus real codimension 2; so the complement of this is connected. The singular part $\{z:F(z)=0, dF(z)=0\}$ is of higher codimension. There are only finitely many such par... |
1,255,334 | <p>A classmate and I are studying this following question from Stein-Shakarchi, Chapter 2, Exercise 12:</p>
<blockquote>
<p>Show that there are <span class="math-container">$f \in L^1(\mathbb{R}^d)$</span> and a sequence <span class="math-container">$\{f_n\}$</span> with <span class="math-container">$f_n \in L^1(\mathb... | zhw. | 228,045 | <p>In one dimension, consider in order </p>
<p>$$\chi_{[0,1]}, \chi_{[0,1/2]}, \chi_{[1/2,1]},\chi_{[0,1/3]},\chi_{[1/3,2/3]},\chi_{[2/3,1]}, \dots $$</p>
<p>This sequence $\to 0$ in $L^1,$ and pointwise nowhere.</p>
|
3,276,572 | <p>Let be <span class="math-container">$\lVert \cdot \rVert$</span> a matrix norm (submultiplicative).</p>
<p>Do we have for all matrices of determinant 1, the following lower bound:</p>
<p><span class="math-container">$$\lVert M \rVert \geq 1$$</span></p>
<p>I'm very confused and could not find any counterexample a... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$\lVert M\rVert<1$</span>, then, if <span class="math-container">$B$</span> is the closed unit ball, the volume of <span class="math-container">$M(B)$</span> will be smaller than the valume of <span class="math-container">$B$</span>. But that cannot happen because, sense, <span cla... |
3,276,572 | <p>Let be <span class="math-container">$\lVert \cdot \rVert$</span> a matrix norm (submultiplicative).</p>
<p>Do we have for all matrices of determinant 1, the following lower bound:</p>
<p><span class="math-container">$$\lVert M \rVert \geq 1$$</span></p>
<p>I'm very confused and could not find any counterexample a... | Theo Bendit | 248,286 | <p><a href="https://math.stackexchange.com/questions/2855044/why-is-the-norm-of-a-matrix-larger-than-its-eigenvalue/">The norm of a matrix is larger than its eigenvalues</a>. The determinant is the product of its eigenvalues. So, if the determinant of <span class="math-container">$M$</span> is <span class="math-contain... |
3,324,647 | <p>Say you have the following matrix A in <span class="math-container">$R^2 \rightarrow R^2$</span>:</p>
<p><span class="math-container">$
\begin{bmatrix}
7 & -10 \\
5 & -8
\end{bmatrix}
$</span></p>
<p>Thus the eigenvalues/eigenvectors are: 2 <span class="math-container">$\begin{bmatrix} 2 \\ 1 \end{bmatrix}... | angryavian | 43,949 | <p>You need to write <span class="math-container">$(2,3)$</span> as a linear combination of the eigenvectors.</p>
<p>In this case, <span class="math-container">$(2,3) = -(2,1) + 4 (1,1)$</span>, so
<span class="math-container">$$A \begin{bmatrix}2 \\ 3 \end{bmatrix} = - A \begin{bmatrix}2 \\ 1 \end{bmatrix} + 4 A \b... |
4,114,180 | <p>The Theorem is as follows:</p>
<p>For any numbers x and y, the following statements are true:</p>
<ol>
<li><span class="math-container">$|x|<y$</span> if and only if <span class="math-container">$-y<x<y$</span></li>
<li><span class="math-container">$|x|\leq{y}$</span> if and only if <span class="math-contai... | user0102 | 322,814 | <p>First, let us consider that <span class="math-container">$x\geq 1$</span>. Then it results that
<span class="math-container">\begin{align*}
2|x| - 3 \geq |x-1| & \Longleftrightarrow 2x - 3 \geq x - 1\\\\
& \Longleftrightarrow x \geq 2
\end{align*}</span></p>
<p>Thus the first solution set is given by <span c... |
1,946,881 | <p>Looking around I have found lots of material on continuous time Markov processes on finite or countable state spaces, say $\{0,1,\ldots,J\}$ for some $J\in\mathbb{N}$ or just $\mathbb{N}$. Similarly I have earlier worked with (discrete time) Markov chains on general state spaces, following the modern classic by Meyn... | Sean Roberson | 171,839 | <p>You've actually defined an <strong>identity element</strong>, not an inverse.</p>
<p>Anyway, in your second case, you're saying that the choice of $e$ is dependent upon the chosen $x$, so the identity is not unique. The first says you can find an $e$ for which any $x$ will yield $xe = ex = x$. This $e$ works for an... |
2,655,518 | <p>$2ac=bc$
find the ratio ( $K$ )
what is the ratio of their area?
<a href="https://i.stack.imgur.com/9NPRi.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9NPRi.png" alt="enter image description here"></a>I found out it is $2$ or $1/2$
is it true? </p>
<p>if the question isn't clear, make sure to... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>By <a href="https://en.wikipedia.org/wiki/Square_root#Properties_and_uses" rel="nofollow noreferrer">definition, in the real numbers,</a> $\sqrt{25}=+5$ (the <strong>positive</strong> number such that its square is $25$). </p>
<p>And the same for any root of even index.</p>
<p>Note that if we defin... |
1,802,497 | <p>If $f(x)=2 [x]+\cos x$
Then $f:R \to R$ is: </p>
<p>$(A)$ One-One and onto</p>
<p>$(B)$ One-One and into</p>
<p>$(C)$ Many-One and into</p>
<p>$(D)$ Many-One and onto</p>
<p>$[ .]$ represent floor function (also known as greatest integer function
)</p>
<p>Clearly $f(x)$ is into as $2[x]$ is an even integer an... | Eff | 112,061 | <p>You are right with respect to surjectiveness (it is not onto).</p>
<p><strong>Hint:</strong></p>
<p>For injectiveness (one to one), look in a neighbourhood around $x = 3\pi$ for example.</p>
|
1,802,497 | <p>If $f(x)=2 [x]+\cos x$
Then $f:R \to R$ is: </p>
<p>$(A)$ One-One and onto</p>
<p>$(B)$ One-One and into</p>
<p>$(C)$ Many-One and into</p>
<p>$(D)$ Many-One and onto</p>
<p>$[ .]$ represent floor function (also known as greatest integer function
)</p>
<p>Clearly $f(x)$ is into as $2[x]$ is an even integer an... | copper.hat | 27,978 | <p>Note that $\pi \in (3,4)$ hence $f$ has a strict local minimum on $[3,4]$ at
$\pi$. It follows that $f$ is not injective.</p>
<p>Note that if $x \ge 0$ then $f(x) >0$ and if $x< 0$ then $f(x) <0$. Hence $0$ is not in the range and so it is not surjective.</p>
|
1,077,284 | <p>I am trying to find the equation of a 3D surface as illustrated below. The boundaries of this surface is comprised of two planar elliptical arcs $AB$ and $AC$ as well as a 3D arc $BC$ which is a 3D curve on an elliptical surface described nicely in <a href="https://math.stackexchange.com/a/1075515/62050">this post</... | dindoun | 202,614 | <p>I found $\int\limits^1_0 t^5(t^2-1)^2 dt = \frac{1}{60}$ with the same use of the theorem than JimmyK4542</p>
|
1,463,419 | <p>A letter has come from exclusively LONDON or CLIFTON, but on the postmark only $2$ consecutive letters ''ON'' are found to be visible. What is the probability that the letter came from LONDON?</p>
<hr>
<p>This is a question of conditional probability. Let $A$ be the event that the letter has come from LONDON. Let... | Graham Kemp | 135,106 | <p>You wish to find the posterior probability that the word is <code>LONDON</code> given that the letters <code>ON</code> are visible. This is: $\mathsf P(A\mid B)$.</p>
<p>You should be able to evaluate the conditional probability that the letters <code>ON</code> are visible given that the word is <code>LONDON... |
46,905 | <p>I need to draw a set of curves on one graph (characteristics equations). As you can see they have exchanged x and y axes. My goal is to plot all those curves on one graph. Are there ways to do that? </p>
<pre><code>f[t_, t0_] := -(2 - 4/Pi*ArcTan[2])*Exp[-t]*(t - t0);
g[x_, x0_] := (x - x0)/(-(2 - 4/Pi*ArcTan[x + ... | qwerty | 5,861 | <p>Another way :</p>
<pre><code>f[t_, t0_] := -(2 - 4/Pi*ArcTan[2])*Exp[-t]*(t - t0);
g[x_, x0_] := (x - x0)/(-(2 - 4/Pi*ArcTan[x + 2]));
curveset1 = Show[Table[ Plot[f[t, t0], {t, 0, 1}, PlotRange -> {0, -0.3}],
{t0, 0, 1, 0.1}]] // First;
curveset2 = Show[Table[ Plot[g[x, x0], {x, 0, -0.3}, PlotRange -> {0,... |
4,278,763 | <p>Let <span class="math-container">$X_1,...$</span> be a sequence of independent and identically distributed random variables with mean <span class="math-container">$0$</span> and variance <span class="math-container">$\sigma^2$</span>. Let <span class="math-container">$S_n=\sum^n_{i=1}X_i$</span> and show that <span ... | angryavian | 43,949 | <ul>
<li>Prove that <span class="math-container">$S_{n+1}^2 = S_n^2 + 2X_{n+1} S_n + X_{n+1}^2$</span>.</li>
<li>Show that <span class="math-container">$E[S_{n+1}^2 \mid \mathcal{F}_n] = S_n^2 + \sigma^2$</span>.</li>
</ul>
|
4,278,763 | <p>Let <span class="math-container">$X_1,...$</span> be a sequence of independent and identically distributed random variables with mean <span class="math-container">$0$</span> and variance <span class="math-container">$\sigma^2$</span>. Let <span class="math-container">$S_n=\sum^n_{i=1}X_i$</span> and show that <span ... | James Anderson | 784,963 | <p>Firstly, <span class="math-container">$S_{n+1}^2=(S_n+X_{n+1})^2=S_n^2+2S_nX_{n+1}+X_{n+1}^2$</span></p>
<p>Then
<span class="math-container">\begin{align}
E[Z_{n+1}-Z_n|\mathcal{F}_n]&=E[S_n^2+2S_nX_{n+1}+X_{n+1}^2-(n+1)\sigma^2-(S_n^2-n\sigma^2)|\mathcal{F}_n]\\
&=E[2S_nX_{n+1}+X_{... |
1,765,530 | <p>How many $5$-digit numbers (including leading $0$'s) are there with no digit appearing exactly $2$ times? The solution is supposed to be derived using Inclusion-Exclusion.</p>
<p>Here is my attempt at a solution:</p>
<p>Let $A_0$= sequences where there are two $0$'s that appear in the sequence.</p>
<p>...</p>
<p... | Marko Riedel | 44,883 | <p>Suppose we ask about $N$ digit numbers including leading zeros where
no digit appears two times.</p>
<p><P>The species of set partitions with no two-elements sets is
$$\mathfrak{P}(\mathcal{U}\mathfrak{P}_{\ne 2, \ge 1}(\mathcal{Z})).$$</p>
<p>This yields the generating function</p>
<p>$${n\brace k}_{\ne 2} =
\... |
2,865,122 | <p><a href="http://math.sfsu.edu/beck/complex.html" rel="nofollow noreferrer">A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka</a> Exer 3.8</p>
<blockquote>
<p>Suppose <span class="math-container">$f$</span> is holomorphic in region <span class="math-container">$G$<... | trying | 309,917 | <p>In every point on a line in the plane the possible tangent vectors form a real monodimensional vector space. An holomorphic function has in every point of its domain a derivative map that is a complex linear, that is a roto-homothetic real transformation of the plane.</p>
<p>Turning attention to your problem: in ev... |
2,865,122 | <p><a href="http://math.sfsu.edu/beck/complex.html" rel="nofollow noreferrer">A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka</a> Exer 3.8</p>
<blockquote>
<p>Suppose <span class="math-container">$f$</span> is holomorphic in region <span class="math-container">$G$<... | zhw. | 228,045 | <p>Note that $f\overline {f}=1$ in $G.$ This implies $\overline {f} =1/f$ in $G.$ Hence $\overline {f}$ is holomorphic in $G.$ This implies both $f+\overline f= 2\text { Re } f$ and $f-\overline f=2i\text { Im } f$ are holomorphic in $G.$ By the remarks you made right after $(2)$ in your question, these functions are c... |
2,865,122 | <p><a href="http://math.sfsu.edu/beck/complex.html" rel="nofollow noreferrer">A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka</a> Exer 3.8</p>
<blockquote>
<p>Suppose <span class="math-container">$f$</span> is holomorphic in region <span class="math-container">$G$<... | BCLC | 140,308 | <p>i guess <a href="https://en.wikipedia.org/wiki/Open_mapping_theorem_(complex_analysis)" rel="nofollow noreferrer">open mapping theorem</a>, as suggested by Angina Seng, is <a href="https://academia.stackexchange.com/q/116019">inadmissible</a>, but anyhoo</p>
<p><span class="math-container">$f$</span> is either const... |
223,176 | <p>I made this problem: </p>
<p>$f(x)=e^{f^{\prime \prime}}$ </p>
<p>I have just been taught the first derivative, and was thinking about what if the derivative depended upon it own derivative. I understand that $e^x$ is its "own" derivative, but the problem I made I was thinking that the first derviative is not lo... | Berci | 41,488 | <p>Such kind of problems usually ask for the function $f$ itself (of course, then its derivative can be calculated, too). </p>
<p>And, such is called a <a href="http://en.wikipedia.org/wiki/Differential_equation" rel="nofollow">Differential equation</a>.</p>
|
2,860,360 | <p>It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book <em>Algebraic Number Theory</em>:</p>
<p>"<em>Example 7.20</em> For $K=\mathbb{Q}(\sqrt[3]{2} )$, the discriminant is 108, and $r_{2}=1$. So the Minkowski bound is $\approx 2... | RayDansh | 572,459 | <p>The probability at least one person succeeds out of $6$ equals $1$ minus the probability that all of the $6$ fail. So if the success rate is $p$, then the probability at least one person succeeds out of $n$ people is $1-(1-p)^n$. </p>
<p>Going to your example of $20$% success and $6$ people, we get $1-(1-0.20)^6=0... |
2,860,360 | <p>It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book <em>Algebraic Number Theory</em>:</p>
<p>"<em>Example 7.20</em> For $K=\mathbb{Q}(\sqrt[3]{2} )$, the discriminant is 108, and $r_{2}=1$. So the Minkowski bound is $\approx 2... | saulspatz | 235,128 | <p>If six people each try $1,000,000$ times, the total number of success is approximate $1,200,000.$ The success rate is approximately $$
\frac{1200000}{6000000}=.2$$</p>
<p>You seem to have overlooked the fact that there are six million trials.</p>
|
71,031 | <p>In 1976 Cappell and Shaneson gave some examples of knots in homotopy 4-spheres and for some time these examples were considered as possible counter-examples to the smooth 4-dimensional Poincare conjecture.</p>
<p>In a series of papers, Akbulut and Gompf have shown most of these Cappell-Shaneson knots actually are kn... | Min Hoon Kim | 21,694 | <p>I think the explicit embedding of Cappell-Shaneson knot is given in the following paper:</p>
<p>S. Akbulut and R. Kirby, A potential smooth counterexample to in dimension 4 to the Poincare conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture, Topology 24 (1985) 375--390. (See Figure 16 of that ... |
1,967,847 | <blockquote>
<p>A vector space $V$ is called <strong>finite-dimensional</strong> if there is a finite subset of $V$ that is a basis for $V$. If there is no such finite subset of $V$, then $V$ is called <strong>infinite-dimensional</strong>.</p>
<hr>
<p>We now establish some results about finite-dimensional ... | Faraad Armwood | 317,914 | <p>I don't understand why this question was downvoted since it is an honest one. For a vector space $V$, the existence of a finite basis is all you need to determine the dimension (which is unique). This is because you can show that if $A$ is a basis for $V$ and $|A| = n$ then any other basis $B$ for $V$ also have to h... |
2,963,886 | <p>What do these statements mean in discrete mathematics?</p>
<p><strong>Example 1:</strong> Let <span class="math-container">$P:\mathbb{Z}\times \mathbb{Z}\to \{T,F\}$</span>, where <span class="math-container">$P(x,y)$</span> denotes "<span class="math-container">$x+y=5$</span>".</p>
<p><strong>Example 2:</... | Bertrand Wittgenstein's Ghost | 606,249 | <p><strong>Example 1</strong>: The set of ordered pairs of integers such that the sum of the first and the second is 5. P(x,y)=xPy which means P is a 2-place predicate defined as x+y=5.</p>
<p><strong>Example 2</strong>: The second one is the Boolean definition of Predicate. It, in essence, says given a relationship P... |
1,572,593 | <p>Let $G$ a group, $H$ and $K$ two subgroups of G of finite order such that $H \cap K = \{1_G\}$. </p>
<p>I already show the first exercise which says that the cardinal of $HK$ is $|H||K|$.</p>
<p>The second exercise ask to deduce that if $|G|=pm$ where $p$ is a prime number and $p>m$, then $G$ has at most one su... | kccu | 255,727 | <p>First notice that if $p>m$, $p \nmid m$. So the $p$-Sylows of $G$ will have order $p$. The Sylow theorems tell you that the number of $p$-Sylows is congruent to $1$ modulo $p$ and is also a divisor of $m$. Using the fact that $p>m$, what are/is the possible number(s) of $p$-Sylows? Now consider conjugates of t... |
479,594 | <p>I was wandering which is the best way to generate various combinations of $x_i$ such that $$\sum\limits_{i=1}^7 x_i = 1.0$$</p>
<p>where $ x_i \in \{0.0, 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0\}$</p>
<p>I can generate these using brute-force, i.e checking through all $ 11^7$ combinations and only taking those whi... | Marko Riedel | 44,883 | <p>You could use generating functions. Put $$f(z) = \sum_{n\ge 1} a_n z^n.$$</p>
<p>Summing your recurrence for $n\ge 2$ and multiplying by $z^n,$ we get
$$ f(z) - \frac{1}{4} z = \frac{1}{4} \sum_{n\ge 2} n z^n
- \frac{1}{2} z \sum_{n\ge 2} z^{n-1} \sum_{k=1}^{n-1} a_k.$$</p>
<p>Simplify to obtain
$$ f(z) - \frac{1}... |
1,259,853 | <p>Why the derivative of $n^{1/n} = \sqrt[n]{n}$ is $n^{1/n} \left( \frac{1}{n^2} - \frac{\log(n)}{n^2}\right)$ (according to Maxima and other tools online)?</p>
<p>I have tried to applied the chain rule, but it comes something completely different:</p>
<p>$$\frac{1}{n} n^{\frac{1}{n} - 1} \cdot 1 = \frac{1}{n} n^\fr... | Kevin Church | 229,638 | <p>The previous answers have explained how to do the calculation correctly. I want to comment on why your method didn't work.</p>
<p>The reason you have the wrong answer is you haven't applied the chain rule correctly. You started with $f(n)=n^{1/n}$, and tried to apply the chain rule. Since your first calculation ... |
3,089,326 | <p>9 person randomly enter 3 different rooms. What is the probability that</p>
<p>a)the first room has 3 person?</p>
<p>b)every room has 3 person?</p>
<p>c)the first room has n person, second room has 3 persons, third room 2 persons?</p>
<p>What i want to know is that which probability techniques i need to use when... | Community | -1 | <p>Continuity of <span class="math-container">$f$</span> on <span class="math-container">$\mathbb{R}$</span> is obtained iff <span class="math-container">$\lim_{h\to 0} f(x+h)-f(x)=0$</span>. As I noted in my comment, this expression is equal to <span class="math-container">$\lim_{h\to 0} f(x)+f(h)-f(x)=\lim_{h\to 0}f(... |
2,915,685 | <blockquote>
<p>Let $X$ be Banach space and $Y$ be a normed vector space and suppose that I find a linear map $T \in L(X,Y)$. With the aid of $T$, I wonder under what condition I can conclude that $Y$ is also a Banach space.</p>
</blockquote>
<p>Is $T$ a linear isomorphism enough? I encounter this question when thin... | Community | -1 | <p>This is not true since if you take $\omega (1)= 1 , \omega (n) = 0 $ for $n>1$ then $\ell^p (\omega ) $ is not a Hausdorff topological vector soace.</p>
|
1,749,909 | <p>I need a help with somthing:
I need to tell if these two integrals are Convergence\Absolute convergence:</p>
<p>$\int _1^{\infty }\frac{\ln x}{\sqrt{x^3-x^2-x+1}}dx$, $\int _0^{\infty \:}\:\frac{\left(x^{\frac{1}{4}}+1\right)\cdot \sin\left(2\sqrt{x}\right)}{x}dx$
Now I compute this and I find that both converge. B... | CiaPan | 152,299 | <p>If you remind them the integration is somewhat like a 'continuous' analogue to a 'discrete' summation, it would become quite obvious the integral of a zero function should be zero, similar to a sum of zero terms sequence. And that the zero value of the integral corresponds to 'no area' of a degenerate figure 'betwee... |
3,493,519 | <p>Can I get a verification if this is the right way to approach this problem?</p>
<blockquote>
<p>Give an example of a linear map <span class="math-container">$T$</span> such that <span class="math-container">$\dim(\operatorname{null}T) = 3$</span> and <span class="math-container">$\dim(\operatorname{range}T) = 2$<... | Ben Grossmann | 81,360 | <p>Yes, your example is correct (though like the other answerer, I would tend to prefer something more specific).</p>
<p>Based on your previous post and your use of the word "thus", I assume that you're also trying to prove that the range and nullspace have the dimension you claim (even though such a step is ... |
207,807 | <p>Is there an explicit example of a non-commutative monoid $M$ such that for all elements $m,n \in M$ and $p \in \mathbb{N}$ we have $(m \cdot n)^p=m^p \cdot n^p$?</p>
<p>It suffices to construct a semigroup $H$ with an absorbing element $0$ such that $a^2=0$ for all $a$, because then $M := H \cup \{e\}$ will do the ... | Boris Novikov | 62,565 | <p>If this question is still interested for you:</p>
<p>Let $P(\le)$ be a poset. Define a multiplication on pairs $\{(a,b)|a\le b\}\subset P\times P$ with an extra zero by the rule:
$(a,b)(c,d)=(a,d)$ if $b=c$, otherwise $(a,b)(c,d)=0$.</p>
|
2,713,201 | <p>How would you work something like this out? </p>
<p>Are there similar problems to
$$\frac{d\left( (\cos(x))^{\cos(x)}\right)}{dx}$$
which could be worked out the same way?</p>
| MPW | 113,214 | <p>A generalized power rule is
$$(f^g)'=gf^{g-1}\cdot f' + f^g\log f \cdot g'$$</p>
<p>This generalizes the power rule and the exponential rule simultaneously. It is easily obtained by using implicit differentiation. But it is <em>extremely</em> easy to remember if you look at the components of the sum: when $g$ is co... |
3,443,094 | <blockquote>
<p>If <span class="math-container">$$\lim_{x\to 0}\frac{ae^x-b}{x}=2$$</span> the find <span class="math-container">$a,b$</span></p>
</blockquote>
<p><span class="math-container">$$
\lim_{x\to 0}\frac{ae^x-b}{x}=\lim_{x\to 0}\frac{a(e^x-1)+a-b}{x}=\lim_{x\to 0}\frac{a(e^x-1)}{x}+\lim_{x\to 0}\frac{a-b}{... | QC_QAOA | 364,346 | <p>Going off of your work, we know that <span class="math-container">$a=b$</span>. Thus, the question is: what <span class="math-container">$a$</span> makes</p>
<p><span class="math-container">$$\lim_{x\to 0}a\frac{e^x-1}{x}=2?$$</span></p>
<p>Using the Taylor Series for <span class="math-container">$e^x$</span>, we ... |
3,811,154 | <p>Prove that this integral is less than infinity. If <span class="math-container">$0<a<c$</span> and <span class="math-container">$0<b$</span>: <span class="math-container">$$\int_0^\infty \frac{|x|^a}{(x+b)^{c+1}} dx.$$</span></p>
<p>From inspection, because <span class="math-container">$a<c$</span> and <... | user58697 | 58,697 | <p>Notice that since <span class="math-container">$b > 0$</span>, the <span class="math-container">$\int_0^\infty \frac{|x|^a}{(x+b)^{c+1}} dx < \int_0^\infty \frac{|x + b|^a}{(x+b)^{c+1}} dx = \int_0^\infty \frac{1}{(x+b)^{c - a+1}} dx$</span>. Now, since <span class="math-container">$c > a$</span>, the last ... |
4,405,145 | <p>Given <span class="math-container">$x_1, x_2, x_3, x_4, x_5$</span> be independent standard normal random variable and <span class="math-container">$\bar x$</span> the sample mean <span class="math-container">$\bar x= (x_1 + x_2 + x_3 + x_4 + x_5)/5$</span>. Then <span class="math-container">$\Pr(\bar x\leqslant c)$... | user97357329 | 630,243 | <p><strong>A second solution by Cornel Ioan Valean</strong> (in large steps)</p>
<p>Let's start with the variable change <span class="math-container">$\displaystyle x\mapsto i\frac{1-\sqrt{x}}{1+\sqrt{x}}$</span> and with understanding the resulting integral as a PV integral, and then we have
<span class="math-containe... |
1,034,698 | <p>I have an assignment with the following question:</p>
<pre><code>Does an Orthogonal Matrix exist such that its first row consists of the
following values:
</code></pre>
<p>($1$/$\sqrt{3}$, $1$/$\sqrt{3}$, $1$/$\sqrt{3}$)</p>
<pre><code>If there is, find one.
</code></pre>
<p>I know I can solve this question wi... | Learnmore | 294,365 | <p>I think an easy way to solve this is as follows:</p>
<p>dim $(\mathbb R^3)/W$=dim $\mathbb R^3$-dim $W$</p>
<p>Now let $(a,b,c)\in W$ then $2a+3b-c=0$
so $c=2a+3b$</p>
<p>so a basis for $W$ is $\{(1,0,2)^t,(0,1,3)^t\}$ so dim$W$=2</p>
<p>So you get your answer.</p>
<p>Another way out is try the linear mapping ... |
3,583,879 | <blockquote>
<p>a) $P_5=11$$</p>
<p>b) <span class="math-container">$P_1+P_2+P_3+P_4+P_5 =26$</span></p>
</blockquote>
<p>For the first part
<span class="math-container">$$\alpha^5+\beta ^5$$</span>
<span class="math-container">$$=(\alpha^3+\beta ^3)^2-2(\alpha \beta )^3$$</span></p>
<p>I found the value of <... | Z Ahmed | 671,540 | <p>if <span class="math-container">$$x^2-x-1=0~~~(1)$$</span> has roots as <span class="math-container">$a,b$</span> then <span class="math-container">$P_k=a^k+b^k,P_0=2,P_1=a+b=1$</span> and
<span class="math-container">$$a^2-a-1=0~~~(2),~~ b^2-b-1=0~~~(3)$$</span>
Multiply Eq.(2) once by <span class="math-container"... |
2,805,312 | <p>Suppose $E$ is a vector bundle over $M, d^E$ a covariant derivative, $\sigma\in\Omega^p(E)$ and $\mu$ a q-form.</p>
<p>I have seen the following pair of formulae for wedge products:</p>
<p>$d^E(\mu\wedge \sigma)=d\mu\wedge\sigma+(-1)^q\mu\wedge d^E\sigma$</p>
<p>$d^E(\sigma\wedge\mu)=d^E\sigma\wedge\mu+(-1)^p\si... | mcwiggler | 450,808 | <p>You need an extra step of permuting forms to get $\mu$ all the way to the right in your second term, permuting past the one-form $d^Es$ changes the sign by a factor $(-1)^q$, leaving the correct sign $(-1)^p$ in your expression.</p>
|
33,993 | <p>I am given the parameters for a bivariate normal distribution ($\mu_x, \mu_y, \sigma_x, \sigma_y,$ and $\rho$). How would I go about finding the Var($Y|X=x$)? I was able to find E[$Y|X=x$] by writing $X$ and $Y$ in terms of two standard normal variables and finding the expectation in such a manner. I am unsure how t... | GWu | 8,829 | <p>First, the joint PDF $f(x,y)$ is obvious, just plug in your parameters. <a href="http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Bivariate_case" rel="nofollow">Bivariate Normal</a>.
Then you can find the marginal density for $X$, which gives you the conditional density of $Y$ given $X=x$:
$$f_{Y|X}(y|x... |
131,283 | <p>I came across a question which required us to find $\displaystyle\sum_{n=3}^{\infty}\frac{1}{n^5-5n^3+4n}$. I simplified it to $\displaystyle\sum_{n=3}^{\infty}\frac{1}{(n-2)(n-1)n(n+1)(n+2)}$ which simplifies to $\displaystyle\sum_{n=3}^{\infty}\frac{(n-3)!}{(n+2)!}$. I thought it might have something to do with pa... | Kirthi Raman | 25,538 | <p>If you know partial fractions, this should be
$$\frac{(n-3)!}{(n+2)!}=\frac{1}{4n}+\frac{1}{24(n-2)}+\frac{1}{24(n+2)}-\frac{1}{6(n-1)}-\frac{1}{6(n+1)}$$</p>
<p>And you might have to simplify the finite sum to get an expression like</p>
<p>$$\sum_{n=3}^{m}\frac{(n-3)!}{(n+2)!}=\frac{m^4+2m^3-m^2-2m-24}{96(m-1)m(... |
2,713,038 | <p>I've seen some solutions to this problem but I'm wondering what is incorrect about an argument like this:</p>
<p>$S = \{x \in \mathbb{R}^d: |x| = 1\}$, then $\delta S = \{x \in \mathbb{R}^d: |\delta x| = 1\}$, and so</p>
<p>\begin{align*}
\{ x \in \mathbb{R}^d: |\delta ||x| = 1\} & = \{ x \in \mathbb{R}^d: |x|... | p4sch | 530,357 | <p>$\delta S$ is for any $\delta>0$ a sphere with radius $\delta>0$. No matter how big $\delta >0$ is choosen, it does not become the set $\{0\}$.</p>
|
290,910 | <p>Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the "equator" of the octahedron eventually will. But for what reasons? How do the defining property of a geodesic - having zero geodesic curvature - apply to a sequence of edges?</p>
... | Joseph O'Rourke | 237 | <p>It so happens I drew the "equator of a dodecahedron" for one of my papers, so I can't resist including it here: <br />
<img src="https://i.stack.imgur.com/1fXvS.png" alt="Dodecahedron equator"><br /></p>
<p>Two points I'd like to make. First, a geodesic ... |
290,910 | <p>Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the "equator" of the octahedron eventually will. But for what reasons? How do the defining property of a geodesic - having zero geodesic curvature - apply to a sequence of edges?</p>
... | Hans-Peter Stricker | 1,792 | <p>There seems to be an almost trivial definition (which relies on a specific realization of a polyhedron): if the polyhedron is convex and inscribed into a sphere and the central projections of the edges onto the sphere sum up to one great arc or circle, then the edges are geodesic. (Note that each single edge is proj... |
1,248,068 | <p>Let $S$ be a set of cardinality $\aleph_1$. Consider the directed family $\mathcal{C}$ (here <em>directed</em> means <em>directed with respect to the inclusion</em>) of all countably infinite subsets of $S$. Suppose that</p>
<p>$$\mathcal{C} = \bigcup_{n=1}^\infty \mathcal{C}_n$$</p>
<p>for some families $\mathcal... | Guesta | 234,404 | <p>Improving on hot-queens result, note that this is true even if S is countably infinite. To show this identify S with rationals and note that some $C_n$ must contain uncountably many reals where we view a real x as the set of rationals less than x.</p>
|
45,771 | <p>Hi, it seems like a big field and I'm having trouble getting some solid/classic references to get me started.</p>
<p>If $U \subset \mathbb{R}^d$ is a bounded domain with, say, $C^2$-boundary $\partial U$ and $(S(t),t \ge 0)$ is the Dirichlet heat semigroup on $L^p(U)$ then $(S(t) f)(x) = \int_U G_U(t,x,y) f(y)\,dy$... | Suvrit | 8,430 | <p>Warning: the following response is that of a "googlist" not of an expert. </p>
<p>Although not classical (as per your request), perhaps the following two offer reasonable pointers? If you find these to be unhelpful, please let me know. I also had a longer listing available, if I find it, I will update my answer.</p... |
59,846 | <p>In "The New Book of Prime Number Records", Ribenboim reviews the known results on the degree and number of variables of prime-representing polynomials (those are polynomials such that the set of positive values they obtain for nonnegative integral values of the variables coincides with the set of primes). For exampl... | Charles | 1,778 | <p>I <a href="https://mathoverflow.net/questions/75637/is-there-a-two-variable-prime-representing-polynomial-in-the-sense-of-jones-sato">asked the same question at MathOverflow</a> (linking here) where I noted that, at least as of 1982, the problem was still open because even universal Diophantine equations were not kn... |
950,485 | <p>I have been trying to solve the following limit but am completely stuck.</p>
<p>$$\lim_{\alpha \rightarrow \infty} 1-\left( \frac{y+\alpha}{\alpha-1} \right)^{-\alpha}$$</p>
<p>I have tried inverting the ratio and came up with the following expression:</p>
<p>$$ 1 - \lim_{\alpha \rightarrow \infty} \left( 1-\frac... | Rogelio Molina | 87,320 | <p>You can try the $u =y + \alpha$ substitution, in this way you don't need L'Hopital's rule: </p>
<p>\begin{eqnarray}
\lim_{\alpha \to \infty} \left( 1- \frac{1+y}{y+\alpha} \right)^{\alpha} = \lim_{u \to \infty} \left( 1- \frac{1+y}{u} \right)^{u-y} = \lim_{u \to \infty} \left( 1- \frac{1+y}{u} \right)^{u}/ \lim_{... |
3,965,164 | <p>I know the standard and expanded forms of the equation of the circle in the simple 2d space,</p>
<p><span class="math-container">${(x-a)}^2+{(y-b)}^2=r^2$</span></p>
<p><span class="math-container">$x^2-2ax+y^2-2by=c$</span></p>
<p>So in 3d space what are the equations for a circle laying in an arbitrary plane,
and ... | azif00 | 680,927 | <p>For example, if <span class="math-container">$\mathbb Q = \{q_n\}_{n \in \mathbb N}$</span>, then <span class="math-container">$$\mathbb R \setminus \{q_0\} \supseteq \mathbb R \setminus \{q_0,q_1\} \supseteq \mathbb R \setminus \{q_0,q_1,q_2\} \supseteq \cdots$$</span> and <span class="math-container">$$\bigcap_{n ... |
3,298,516 | <p>I have trouble with understanding proof of next theorem:</p>
<blockquote>
<p>Let <span class="math-container">$X,Y \in L_{2} ( \Omega, P)$</span>. Then
<span class="math-container">$$ | \mathbb{E} (XY) | \le \sqrt{\mathbb{E} X^{2} \mathbb{E} Y^{2}} .$$</span></p>
<p>Proof:
Let <span class="math-container">$\Omega = ... | Dominik Kutek | 601,852 | <p>Step with equality <span class="math-container">$|\mathbb{E} (XY) | \leq \mathbb{E} | XY | = \sum_{k=1}^{n} ( | X (\omega_{k}) | | Y (\omega_{k}) | P({\omega_{k}}))$</span> isn't correct, since there (can be) more than <span class="math-container">$n$</span> elements in <span class="math-container">$\Omega$</span> f... |
3,298,516 | <p>I have trouble with understanding proof of next theorem:</p>
<blockquote>
<p>Let <span class="math-container">$X,Y \in L_{2} ( \Omega, P)$</span>. Then
<span class="math-container">$$ | \mathbb{E} (XY) | \le \sqrt{\mathbb{E} X^{2} \mathbb{E} Y^{2}} .$$</span></p>
<p>Proof:
Let <span class="math-container">$\Omega = ... | drhab | 75,923 | <p>This is not really an answer to your question, but is too much for a comment.</p>
<p>I think it is better just to drop this proof (which is dubious and at least not general).</p>
<hr>
<p>Observe that for every <span class="math-container">$t\in\mathbb R$</span> we have:<span class="math-container">$$\mathbb E(tX-... |
2,653,483 | <p>Let $a =111 \ldots 1$, where the digit $1$ appears $2018$ consecutive times.</p>
<p>Let $b = 222 \ldots 2$, where the digit $2$ appears $1009$ consecutive times.</p>
<p>Without using a calculator, evaluate $\sqrt{a − b}$.</p>
| Donald Splutterwit | 404,247 | <p>Let
\begin{eqnarray*}
N=\sum_{i=0}^{1008} 10^i =\underbrace{11\cdots 1}_{\text{1009 ones}} \\
M=10^{1009}+1
\end{eqnarray*}
note that $M-2=9N$ and<br>
\begin{eqnarray*}
a=NM \\
b=2N
\end{eqnarray*}
So $a-b=9N^2$ and
\begin{eqnarray*}
\sqrt{a-b} = 3N=\underbrace{\color{red}{33\cdots 3}}_{\text{1009 threes}}.
\end{... |
2,653,483 | <p>Let $a =111 \ldots 1$, where the digit $1$ appears $2018$ consecutive times.</p>
<p>Let $b = 222 \ldots 2$, where the digit $2$ appears $1009$ consecutive times.</p>
<p>Without using a calculator, evaluate $\sqrt{a − b}$.</p>
| Tiago Emilio Siller | 526,875 | <p>$a= \overbrace{11\ldots11}^{2018} = \dfrac{10^{2018}-1}{9}$</p>
<p>$b= \overbrace{22\ldots22}^{1009} = 2\cdot\dfrac{10^{1009}-1}{9}$</p>
<p>$\Rightarrow a-b = \dfrac{10^{2018}-1}{9}-2\cdot\dfrac{10^{1009}-1}{9} = \dfrac{10^{2018}-2\cdot10^{1009}+1}{9} = \left(\dfrac{10^{1009}-1}{3} \right)^2$</p>
<p>$\Rightarrow... |
2,209,438 | <p>I am trying to find this limit,</p>
<blockquote>
<p>$$\lim_{x \rightarrow 0} \frac{1}{x^4} \int_{\sin{x}}^{x} \arctan{t}dt$$</p>
</blockquote>
<p>Using the fundamental theorem of calculus, part 1,
$\arctan$ is a continuous function, so
$$F(x):=\int_0^x \arctan{t}dt$$
and I can change the limit to
$$\lim_{x \righ... | Paramanand Singh | 72,031 | <p>Note that by the Mean Value Theorem for integrals we have $$\int_{\sin x}^{x}\arctan t\,dt = (x - \sin x)\arctan c$$ for some $c$ between $x$ and $\sin x$. Then we have $$\lim_{x \to 0}\frac{1}{x^{4}}\int_{\sin x}^{x}\arctan t\,dt = \lim_{x \to 0}\frac{x - \sin x}{x^{3}}\cdot\frac{\arctan c}{x}$$ The first factor on... |
2,934,028 | <blockquote>
<p>A particle moves along the top of the
parabola <span class="math-container">$y^2 = 2x$</span> from left to right at a constant speed of 5 units
per second. Find the velocity of the particle as it moves through
the point <span class="math-container">$(2, 2)$</span>. </p>
</blockquote>
<p>So I is... | Peter Szilas | 408,605 | <p><span class="math-container">$(2y) dy/dx=2$</span>; <span class="math-container">$dy/dx =1/y$</span>.</p>
<p>Slope at <span class="math-container">$(2,2)$</span>: <span class="math-container">$dy/dx =1/2= \tan \alpha$</span>.</p>
<p><span class="math-container">$\cos \alpha =2/√5$</span>, <span class="math-contain... |
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