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 |
|---|---|---|---|---|
494,227 | <blockquote>
<p>If I had $6$ feet of fencing could I fence a region that has area $3$
square feet?</p>
</blockquote>
<p>So, I must show that there is a curve in the plane of my fencing that has length $6$ feet that bounds the region of area. </p>
<p>How can I prove this? </p>
| Brian M. Scott | 12,042 | <p>HINT: For a given perimeter, a disk has the largest area. Does a disk with a circumference of $6$ have area $\ge 3$, or not?</p>
|
138,091 | <p>I am trying to compute an explicit formula using Mathematica for the following multinomial expression:</p>
<blockquote>
<p>\begin{equation} \sum_{n_{1}+n_{2}+...+n_{M}=N}^{M} {N \choose
n_{1},n_{2},...,n_{M }} \cdot n_{i} = ? \end{equation}</p>
</blockquote>
<p>where $i={1,2,...,M}$ and using </p>
<pre><code>... | JimB | 19,758 | <p>Given that it doesn't matter which index ($i$) one picks, here's a brute force algebraic approach:</p>
<p>\begin{align*}
\sum_{n_1+n_2+\cdots+n_M=N}n_1\binom{N}{n_1,n_2,\cdots,n_M}&=\sum_{n_1=0}^N n_1 \sum_{n_2+n_3+\cdots+n_M=N-n_1}\binom{N}{n_1,n_2,\cdots,n_M}\\
&=\sum_{n_1=0}^N n_1 \sum_{n_2+n_3+\cdots+n_... |
2,262,094 | <p>In Sheldon Axler's <em>Linear Algebra Done Right</em> Page 35, he gave a proof of "Length of linearly independent list <=length of spanning list"(see reference below) by using a process of adding a vector from independent list to the spanning list and then cancels a vector form the spanning list to form a new spa... | Community | -1 | <p>I found the above answer by @Tengu to be correct, but unsatisfying; it seemed like just a cheap trick to ensure that the <span class="math-container">$u_i$</span> came first, and therefore we must remove one of the <span class="math-container">$w_i$</span>.</p>
<p>Allow me to present a different perspective. Since <... |
3,793,581 | <p>I can solve this integral in a certain way but I'd like to know of other, simpler, techniques to attack it:</p>
<p><span class="math-container">\begin{align*}
\int _0^{\frac{\pi }{2}}\frac{\ln \left(\sin \left(x\right)\right)\ln \left(\cos \left(x\right)\right)}{\tan \left(x\right)}\:\mathrm{d}x&\overset{ t=\sin... | Z Ahmed | 671,540 | <p>Let
<span class="math-container">$$I=\int _0^{\frac{\pi }{2}}\frac{\ln (\sin \left(x\right))\ln (\cos \left(x\right))}{\tan \left(x\right)}\:\mathrm{d}x$$</span>
Let <span class="math-container">$\ln \cos x=-t$</span>, then we'll have:
<span class="math-container">$$I=-\frac{1}{2}\int_{0}^{\infty} t \ln(1-e^{-2t}) d... |
903,656 | <p>An urn has $2$ balls and each ball could be green, red or black. We draw a ball and it was green, then it was returned it to the urn. What is the probability that the next ball is red? </p>
<p>My attempt: I think it is just a probability of $1/4$ because we have 4 colors in total but on the other hand I think i ne... | spunkets | 170,680 | <p>The probability of drawing any one particular color in the original draw is 1/3. In that situation there are 2 copies of the same 3 color possibility. In that problem, there is a 1/2 probability of choosing one of the 2 sets of 3 colors. The probably is 2 * 1/2 * 1/3 =1/3. </p>
<p>In the second situation there i... |
894,159 | <p>I was assigned the following problem: find the value of $$\sum_{k=1}^{n} k \binom {n} {k}$$ by using the derivative of $(1+x)^n$, but I'm basically clueless. Can anyone give me a hint?</p>
| André Nicolas | 6,312 | <p>Imagine tossing a fair coin $n$ times. Then the mean number of heads is
$$\sum_0^n k\binom{n}{k}\frac{1}{2^n}.\tag{1}$$
We compute the mean another way. Let random variable $X_i$ be $1$ if we get a head on the $i$-th toss, and $0$ otherwise. Then the number $Y$ of heads is given by
$$Y=X_1+X_2+\cdots +X_n,$$
and th... |
2,303,106 | <p>I was looking at this question posted here some time ago.
<a href="https://math.stackexchange.com/questions/1353893/how-to-prove-plancherels-formula">How to Prove Plancherel's Formula?</a></p>
<p>I get it until in the third line he practically says that $\int _{- \infty}^{+\infty} e^{i(\omega - \omega')t} dt= ... | user121330 | 178,570 | <p>Supposing that $\omega = \omega'$, it's clear that we don't expect the integral to converge - the Dirac's Delta function is infinite at zero:</p>
<p>\begin{equation}\tag{1}
\int_{-\infty}^\infty e^{i ( \omega - \omega )t} dt=\int_{-\infty}^\infty 1\, dt = \infty
\end{equation}</p>
<p>In the case that $\omega \neq ... |
2,809,686 | <p>Let S={1,2,3,...,20}. Find the probability of choosing a subset of three numbers from the set S so that no two consecutive numbers are selected in the set.
"I am getting problem in forming the required number of sets."</p>
| true blue anil | 22,388 | <p>$\underline{Another\; approach}$</p>
<p>For any chosen subset of $3$, there will be $17$ left unchosen, and the three chosen must have come from any $3$ of $18$ gaps (including ends) marked with an uparrow,</p>
<p>$\uparrow\bullet\uparrow\bullet\uparrow\bullet\uparrow\bullet\uparrow\bullet\uparrow\bullet\uparrow\... |
409,220 | <p>$$f(x,y)=6x^3y^2-x^4y^2-x^3y^3$$
$$\frac{\delta f}{\delta x}=18x^2y^2-4x^3y^2-3x^2y^3$$
$$\frac{\delta f}{\delta y}=12x^3y-2x^4y-3x^3y^2$$
Points, in which partial derivatives ar equal to 0 are: (3,2), (x,0), (0,y), x,y are any real numbers. Now I find second derivatives
$$\Delta_1=\frac{\delta f}{\delta x^2}=36xy^2... | colormegone | 71,645 | <p>If we go back to the function and its first and second derivatives, we see that they can be factored as</p>
<p>$$ f(x, \ y) \ = \ x^3 \ y^2 \ ( \ 6 \ - \ x \ - \ y \ ) \ \ , $$</p>
<p>$$ f_x \ = \ x^2 \ y^2 \ ( \ 18 \ - \ 4x \ - \ 3y \ ) \ \ , \ \ f_y \ = \ x^3 \ y \ ( \ 12 \ - \ 2x \ - \ 3y \ ) \ \ , $$</p>
<p... |
3,154,407 | <p>I would like to define a function whose domain is any multiset of real numbers and image is a real number.</p>
<p>To my understanding, the domain of a function that can be applied on any set of real numbers is the power set <span class="math-container">$\mathcal{P}(\mathbb{R})$</span>. Is it correct? If yes, is the... | benlaug | 655,808 | <p>In the article <em>Mathematics of Multisets</em> by A. Syropoulos, this is defined as follows:</p>
<blockquote>
<p>Assume that <span class="math-container">$A$</span> is a set, then <span class="math-container">$\mathcal{P}^A$</span> is the set of all multisets which have <span class="math-container">$A$</span> a... |
2,619,131 | <p>How one can prove the following inequality?</p>
<p>$$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74\geq 0$$ </p>
<p>I plotted the graph on Wolfram Alpha and found that the inequality seems to hold. I was unable to represent the polynomial as a sum of squares. </p>
<p>It looks quite boring to ap... | Michael Rozenberg | 190,319 | <p>For $x<0$ it's obvious.</p>
<p>But for $x\geq0$ we obtain:
$$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74=$$
$$=(x^3-x^2-x+1)(58x^7+16x^6+85x^5+85x^4+207x^3+47x^2)+$$
$$+287x^4-138x^3-103x^2-20x+74>0,$$
where $$287x^4-138x^3-103x^2-20x+74=$$
$$=(16x^2-4x-5)^2+(31x^4-10x^3+x^2)+(40x^2-60x+... |
760,767 | <p>I don't understand the last part of this proof:</p>
<p><a href="http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup" rel="nofollow">http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup</a></p>
<p>where they say: $p \nmid \left[{N : P \cap N}\right]$, t... | user141421 | 141,421 | <p>Your first formula is incorrect. For instance, consider the line $y=x$ and you want to get the length of the line segment from $x=0$ to $x=1$. The length is $\sqrt2$, and the equation in polar coordinates is $\theta=\dfrac{\pi}4$. If we use the first formula, we get the length to be $0$.</p>
<p>In terms of $x$, $y$... |
2,498,628 | <p>This was a question in our exam and I did not know which change of variables or trick to apply</p>
<p><strong>How to show by inspection ( change of variables or whatever trick ) that</strong></p>
<p><span class="math-container">$$ \int_0^\infty \cos(x^2) dx = \int_0^\infty \sin(x^2) dx \tag{I} $$</span></p>
<p>Co... | Guy Fsone | 385,707 | <h2><Here is what I found</h2>
<p>Employing the change of variables <span class="math-container">$2u =x^2$</span> We get <span class="math-container">$$I=\int_0^\infty \cos(x^2) dx =\frac{1}{\sqrt{2}}\int^\infty_0\frac{\cos(2x)}{\sqrt{x}}\,dx$$</span> <span class="math-container">$$ J=\int_0^\infty \sin(x^2) dx=\fra... |
3,792,954 | <p>For vector space <span class="math-container">$V$</span> and <span class="math-container">$v \in V$</span>, there is a natural identification <span class="math-container">$T_vV \cong V$</span> where <span class="math-container">$T_vV$</span> is the tangent space of <span class="math-container">$V$</span> at <span cl... | Ivan | 157,467 | <p>One proof uses the addition formula for the hyperbolical tangent function:</p>
<p><span class="math-container">$$\tanh(a+b)=\frac{\tanh(a)+\tanh(b)}{1+ \tanh(a)\tanh(b)}$$</span></p>
<p>where <span class="math-container">$a,b \in \mathbb{R}$</span>.</p>
<p><span class="math-container">$\tanh$</span> maps from <span ... |
2,022,566 | <p>How can I calculate the below limit?
$$
\lim\limits_{x\to \infty} \left( \mathrm{e}^{\sqrt{x+1}} - \mathrm{e}^{\sqrt{x}} \right)
$$
In fact I know should use the L’Hospital’s Rule, but I do not how to use it.</p>
| haqnatural | 247,767 | <p>Using the fact that $\lim _{ x\rightarrow 0 }{ \frac { { e }^{ x }-1 }{ x } } =1\\ \\ $ we can write </p>
<p>$$\lim _{ x\rightarrow \infty }{ \left( e^{ \sqrt { x+1 } }-e^{ \sqrt { x } } \right) } =\lim _{ x\rightarrow \infty }{ { e }^{ \sqrt { x } }\left( e^{ \sqrt { x+1 } -\sqrt { x } }-1 \right) } =\lim... |
64,925 | <p>Suppose $G$ is a group and $V$ an irreducible representation of $G$. One has that $V\otimes V\cong \Lambda^2(V)\oplus Sym^2(V)$. It is well-known that if the trivial representation appears as a subrepresentation of $\Lambda^2(V)$ then $V$ is of quaternionic type; while if the trivial representation appears as a subr... | darij grinberg | 2,530 | <p>The trivial representation appears in $\wedge^2 V$ if and only if the representation $V^{\ast}$ has a $G$-invariant alternating bilinear form (because $\wedge^2 V\cong\wedge^2\left(\left(V^{\ast}\right)^{\ast}\right)$ is isomorphic to the $G$-module of all alternating bilinear forms on $V^{\ast}$, and $G$-invariant ... |
114,909 | <p>What is known about normal subgroups of $SL_2(\mathbb{C}[X])$? Can one hope for a congruence subgroup property, i.e. that every (non-central) normal subgroup contains the kernel of the reduction modulo some ideal of $\mathbb{C}[X]$?</p>
| Jim Humphreys | 4,231 | <p>[EDIT] These groups have been studied for a long time from various viewpoints, so there is a long paper-trail. I'd emphasize however that working over the complex numbers is usually similar to working over an arbitrary infinite field.
Finite fields on the other hand occur more often in arithmetic contexts. </p>
<p... |
2,330,514 | <p>Prove if n is a perfect square, n+2 is not a perfect square</p>
<blockquote>
<p>Assume n is a perfect square and n+2 is a perfect square (proof by
contradiction)</p>
<p>There exists positive integers a and b such that $n = a^2$ and $n + 2= b^2$</p>
<p>Then $a^2 + 2 = b^2$</p>
<p>Then $2 = b^2-a^2... | dxiv | 291,201 | <p>Alternative direct proof: if $n=k^2$ for $k \ge 1$ then $(k+1)^2 = k^2 + 2k + 1\gt k^2 + 2 = n+2 \gt k^2\,$ so $n+2$ lies strictly between squares of consecutive numbers, thus cannot be a perfect square.</p>
|
44,771 | <p>A capital delta ($\Delta$) is commonly used to indicate a difference (especially an incremental difference). For example, $\Delta x = x_1 - x_0$</p>
<p><strong>My question is: is there an analogue of this notation for ratios?</strong></p>
<p>In other words, what's the best symbol to use for $[?]$ in $[?]x = \dfrac... | Luboš Motl | 10,599 | <p>The best symbol to use is $\exp\Delta\log$:
$$\exp\Delta\log x = \exp(\log x_1-\log x_0) = \frac{x_1}{x_0}.$$
The point is that this operation isn't "qualitatively different" from $\Delta$, so it may be reduced to $\Delta$. So far, I haven't used any new symbols but if you want some multiplicative new creative symbo... |
3,372 | <p>Have any questions first proposed on Mathoverflow attracted enough interest from experts in their field that solving them would be considered a significant advance?</p>
<p>I don't want to count problems that are known (or strongly suspected) to be at least as hard as some previously described problem, unless the ve... | Joseph O'Rourke | 6,094 | <p>Here is a snapshot of the site analytics to which Martin refers:
<hr />
<a href="https://i.stack.imgur.com/zbrKj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zbrKj.png" alt="Traffic"></a>
<hr />
So, roughly $40,000$ page views per day (in the summer).
Might be interesting to know what happened ... |
1,053,506 | <p>I had thought that the ultra-metric property was just a rule that someone made up, that if applied shows some bizarre behavior. I however came across these notes: <a href="http://www.math.harvard.edu/~thorne/all.pdf" rel="nofollow">Lecture notes</a> and it seems that the ultra-metric property is actually derived fro... | David Holden | 79,543 | <p>you write:</p>
<p><b> it seems as if the ultra-metric or non-Archimedean metric is actually derived from the Archimedean metric as applied to p-adic numbers?</b></p>
<p>it would be more appropriate to say that the ultra-metric on $\mathbb{Q}$ is derived from <i>the same definitions</i> as the more intuitive Archim... |
3,712,128 | <p>I've been reading about combinatorial games, specifically about positions in such games can be classified as either winning or losing positions. However, what I'm not sure about now is how I can represent draws using this: situations where neither player wins or loses. Do I use a winning position or losing position ... | hdighfan | 796,243 | <p><span class="math-container">$z=0$</span> and <span class="math-container">$x=2y+1$</span> gives the equation of a line; <span class="math-container">$x=2y+1$</span> is clearly a line in the <span class="math-container">$xy$</span>-plane, and <span class="math-container">$z=0$</span> forces us to stay in this plane.... |
87,948 | <p>Let $\mu_t, t \geq 0,$ be a family of probability measures on the real line. One can assume whatever one wishes about them, although typically they will be continuous in some topology (usually at least the topology of weak convergence of measures), and they will be absolutely continuous with respect to Lebesgue meas... | Fabrice Baudoin | 48,356 | <p>The following result was proved in </p>
<p>Kellerer, H.G. (1972) Markov-Komposition und eine Anwendung auf Martingale. Math. Ann., 198,
99–122.</p>
<p>Let $p(y, t)$ be a family of marginal densities, with finite first moment, such that
for $s , t$ the density at time $t$ dominates the density at time $s$ in the conv... |
101,098 | <p>I apologize in advance because I don't know how to enter code to format equations, and I apologize for how elementary this question is. I am trying to teach myself some differential geometry, and it is helpful to apply it to a simple case, but that is where I am running into a wall.</p>
<p>Consider $M=\mathbb{R}^2$... | Dylan Moreland | 3,701 | <p>$\newcommand{\bR}{\mathbf{R}}$We can view the tangent space of $\bR^2$ at a point $P$ as the space of all <a href="http://en.wikipedia.org/wiki/Tangent_space#Definition_via_derivations">derivations</a> at $P$; that is, $T_P(\bR^2)$ is the set of linear maps $X\colon C^\infty(\bR^2) \to \bR$ satisfying a Leibniz rule... |
1,076,974 | <p>Does anyone know how I can determine the equation of the 3D object below? (Maybe there's a program that can do it?) I am looking for a formula to define this 3D object, but am having trouble finding one. </p>
<p>(If you can imagine the 2D object you see revolved about the x-axis, that is the 3D object I'm referring... | Alex Silva | 172,564 | <p>If you are looking for an object similiar to this of your figure, trace in spherical coordinates (see <a href="http://en.wikipedia.org/wiki/File:3D_Spherical_2.svg" rel="nofollow">http://en.wikipedia.org/wiki/File:3D_Spherical_2.svg</a>)
$$r = a\cdot sin(\phi)cos(\theta),$$ where $a$ is a positive constant, $-\frac{... |
681,737 | <p>What is the simplest way we can find which one of $\cos(\cos(1))$ and $\cos(\cos(\cos(1)))$ [in radians] is greater without using a calculator [pen and paper approach]? I thought of using some inequality relating $\cos(x)$ and $x$, but do not know anything helpful.
We can use basic calculus. Please help. </p>
| user44197 | 117,158 | <p>Because $\infty$ is not part of the real numbers. So set
$$
K_\alpha = [\alpha, \infty)
$$</p>
|
1,600,597 | <p>I'm currently going through Spivak's calculus, and after a lot of effort, i still can't seem to be able to figure this one out.</p>
<p>The problem states that you need to prove that $x = y$ or $x = -y$ if $x^n = y^n$</p>
<p>I tried to use the formula derived earlier for $x^n - y^n$ but that leaves either $(x-y) = ... | Community | -1 | <p>The factorization of $x^k-y^k$ is known to be $(x-y)(x^{k-1}+x^{k-2}y+x^{k-3}y^2+\cdots y^{k-1})$, as you can verify by direct multiplication (all but two terms cancel in pairs).</p>
<p>Then $x^{2k}-y^{2k}=(x^2-y^2)(x^{2k-2}+x^{2k-4}y^2+x^{2k-6}y^4+\cdots y^{2k-2})$.</p>
<p>As all terms have an even exponent in th... |
926,804 | <p>Is there a word for the quality of a number to be either positive or negative? Consider this question:</p>
<p><em>What's the ... (sign/positivity/negativity, but a word that could describe either) of number <strong>x</strong>?</em></p>
<p>Also, is there an all-encompassing word for the sign put in front of a numbe... | 43Quintillionaire | 567,812 | <p>It does help to have more specific technical terms like polarity, rather than sign, in the same way that it helps sailors to use port and starboard rather than left and right. The word sign IS correct, and it doesn't cause confusion like "left" would to sailors, but its many other common usages make it less valuable... |
3,069,262 | <p>Given some quadrilateral <span class="math-container">$Q \subset \mathbb R^2$</span> defined by the vertices <span class="math-container">$P_i = (x_i,y_i), i=1,2,3,4$</span> (you can assume they are in positive orientation), is there a function <span class="math-container">$f: \mathbb R^2 \to \mathbb R^2$</span> tha... | Aphelli | 556,825 | <p>Use the addition formula and simplify the <span class="math-container">$\cosh(\tanh^{-1}(\cdot))$</span>, <span class="math-container">$\sinh(\tanh^{-1}(\cdot))$</span>, your integrand becomes <span class="math-container">$2\cosh(3x)(1-3x^2)+4x\sinh(3x)$</span>.</p>
<p>Note that <span class="math-container">$2\cosh... |
268,152 | <p>I often see proofs, that claim to be <em>by induction</em>, but where the variable we induct on <em>doesn't</em> take value is $\mathbb{N}$ but only in some set $\{1,\ldots,m\}$.</p>
<p>Imagine for example that we have to prove an equality that encompasses $n$ variables on each side, where $n$ can <em>only</em> ran... | Nameless | 28,087 | <p>This "form" of induction is called <a href="http://en.wikipedia.org/wiki/Finitistic_induction" rel="nofollow">finite or finitistic induction</a>. If we want to prove a property $p$ holds $\forall x\in A$ where $A$ is a finite set $A=\left\{x_1,x_2,...,x_n\right\}$, then we just have to check if $p(x_1),p(x_2),...,p(... |
2,405,905 | <p>Let $R$ be a commutative semi-local ring (finitely many maximal ideals) such that $R/P$ is finite for every prime ideal $P$ of $R$ ; then is it true that $R$ is Artinian ring ? From the assumed condition , we get that $R$ has Krull dimension 0 ; so it is enough to ask : Is $R$ a Noetherian ring ? From the semi-loc... | rschwieb | 29,335 | <p>Take $V=\oplus_{i=1}^\infty F_2$ and form the ring </p>
<p>$$
R= \left\{\begin{bmatrix}a&v\\0&a\end{bmatrix}\middle|\,a\in F_2, v\in V\right\}
$$</p>
<p>It isn't noetherian because the image of $V$ contains infinite ascending chains of ideals. It's also local (with residue field $F_2$) and $0$-dimensional.... |
615,275 | <p>So I'm making a star Ship bridge game where the game is rendered using a 2-D Cartesian grid for positioning logic. The player has only the attributes of position and an arbitrary look-at angle (currently degrees). A "view-port" determines if a planet is within the angular difference of $45^\circ$ so that it can rend... | Sammy Black | 6,509 | <p>The inequality
$$
\sin \left( x - \frac{\pi}{3} \right) > \frac{\sqrt{2}}{2}
$$
implies that
$$
\frac{\pi}{4} < x - \frac{\pi}{3} < \frac{3\pi}{4}.
$$
Adding $\frac{\pi}{3}$ to all three expressions yields
$$
\frac{7\pi}{12} < x < \frac{13\pi}{12}.
$$
If you impose the initial restriction, then the up... |
398,176 | <p>I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume):</p>
<p>$$V = \int\limits_a^b {A(x)dx}$$</p>
<p>But this doesn't seem to work with the square. Since the size of the area of the square is $x^2$ then $A(x) = {x^2}$, then: </p>
<p>$$V = \int\limits... | vadim123 | 73,324 | <p>The real problem here is not the endpoints of your integral, it's that the function you are integrating is not constant with respect to the variable of integration. A cube has the same cross section everywhere, while in your original integral the cross section is bigger at the ends than in the middle. See @respons... |
398,176 | <p>I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume):</p>
<p>$$V = \int\limits_a^b {A(x)dx}$$</p>
<p>But this doesn't seem to work with the square. Since the size of the area of the square is $x^2$ then $A(x) = {x^2}$, then: </p>
<p>$$V = \int\limits... | marty cohen | 13,079 | <p>Actually, you have two errors there:</p>
<p>The minor one is that you seem to want a cube of side $2r$,
since your integral goes from $-r$ to $r$.</p>
<p>The major error, as others have said, is that you are finding the volume of a pyramid, not a cube. Actually, since you are integrating from $-r$ to $r$,
you are ... |
398,176 | <p>I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume):</p>
<p>$$V = \int\limits_a^b {A(x)dx}$$</p>
<p>But this doesn't seem to work with the square. Since the size of the area of the square is $x^2$ then $A(x) = {x^2}$, then: </p>
<p>$$V = \int\limits... | Euro Micelli | 78,887 | <p>Can you tell what your proposed solution represents? I think it's far more interesting (and important) for you to be able to look at the integral, and be able tell what it means.</p>
<p>If you think of $x^2$ in terms of the area of a square of sides $x$, then your proposed integral calculates the volume of a solid ... |
730,929 | <p>Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements.</p>
<p>I am not sure how to do this one, but furthermore, what does $a$ being algebraic over $F$ of degree $n$ mean? Does it mean the polynomial ... | Robert Lewis | 67,071 | <p>$a \in E$ is algebraic of degree $n$ over $F$ if there is a polynomial $f(x) \in F[x]$ with $\deg f = n$ and $f(a) = 0$ and there is no non-trivial polynomial $g(x) \in F[x]$ with $\deg g < \deg f$ and $g(a) = 0$. It is easy to see we may assume $f(x)$ to be monic, that is the leading coefficient of $f(x)$ is $1... |
2,508,508 | <p>Let $x_1$ be in $R$ with $ x_1>1$, and let $x_{k+1}=2- \frac{1}{x_k}$ for all $k$ in $N$. Show that the sequence $(x_k)_k$ is monotone and bounded and find its limit.</p>
<p>I am not sure how to start this problem.</p>
| John Griffin | 466,397 | <p>Use induction to show that $x_k > 1$ for every $k$. This shows that the sequence is bounded from below.</p>
<p>Next show that the sequence is decreasing by applying induction to prove $x_{k+1}\le x_k$ for every $k$.</p>
<p>Then the monotone convergence theorem implies that the sequence has a limit, say $x:=\lim... |
3,521,224 | <p>Let <span class="math-container">$(U_1,U_2,...) , (V_1,V_2,...)$</span> be two independent sequences of i.i.d. Uniform (0, 1) random variables.
Define the stopping time
<span class="math-container">$N = \min\left(n\geqslant 1\mid U_n \leqslant V^2_n\right)$</span>.</p>
<p>Obtain <span class="math-container">$P(N ... | Math1000 | 38,584 | <p>For <span class="math-container">$0<v<1$</span> we have <span class="math-container">$$\mathbb P(V_1^2\leqslant v) = \mathbb P(V_1\leqslant \sqrt v) = \sqrt v$$</span> and hence <span class="math-container">$V_1$</span> has density <span class="math-container">$f_{V_1}(v)=\frac12 v^{-\frac12}\mathsf 1_{(0,1)}(... |
1,633,810 | <p>For which $a \in \mathbb{R}$ is the integral $\int_1^\infty x^ae^{-x^3\sin^2x}dx$ finite?</p>
<p>I've been struggling with this question. Obviously when $a<-1$ the integral converges, but I have no idea what happens when $a\ge -1 $.</p>
<p>Any help would be appreciated</p>
| Mark Viola | 218,419 | <p>Fix $0<\delta <\pi/2$. We can write the integral of interest $I(a)$ as</p>
<p>$$\begin{align}
I(a)&=\int_1^\infty x^ae^{-x^3\sin^2(x)}\\\\
&=\int_1^{\pi-\delta} x^a e^{-x^3\sin^2(x)}\,dx+\sum_{n=1}^\infty \int_{n\pi-\delta}^{n\pi+\delta}x^a e^{-x^3\sin^2(x)}\,dx+\sum_{n=1}^\infty \int_{n\pi+\delta}^{... |
1,927,394 | <blockquote>
<p>Number of all positive continuous function <span class="math-container">$f(x)$</span> in <span class="math-container">$\left[0,1\right]$</span> which satisfy <span class="math-container">$\displaystyle \int^{1}_{0}f(x)dx=1$</span> and <span class="math-container">$\displaystyle \int^{1}_{0}xf(x)dx=\alp... | Sangchul Lee | 9,340 | <p>By the Cauchy-Schwarz inequality,</p>
<p>$$ \alpha^2 = \bigg( \int_{0}^{1} x f(x) \, dx \bigg)^2 \leq \bigg( \int_{0}^{1} f(x) \, dx \bigg)\bigg( \int_{0}^{1} x^2 f(x) \, dx \bigg) = \alpha^2. $$</p>
<p>Since the inequality is saturated, it reduces to an equality. This implies that $f(x)$ is a constant multiple of... |
2,645,948 | <p>I was studying neighbourhood methods from Overholt's book of Analytic Number theory(P No 42). There to estimate $Q(x)=\sum_{n \leq x}\mu^2(n)$ they have used a statement that </p>
<p>$$\sum_{j^2\leq x} \mu(j)\left[\frac x {j^2}\right]=x\sum_{j\leq \sqrt x}\frac {\mu(j)} {j^2}+ O(\sqrt x).$$</p>
<p>I am not getting... | Ethan Splaver | 50,290 | <p>Writing $M(x)=\sum_{n\leq x}\mu(n)$ and then applying Abel's summation formula gives us that:</p>
<p>$$Q(x)=\sum_{j^2\leq x} \mu(j)\left\lfloor\frac x {j^2}\right\rfloor=\sum_{n\leq \sqrt{x}} \mu(n)\left(\frac{x}{n^2}-\left\{\frac{x}{n^2}\right\}\right)=x\sum_{n\leq \sqrt{x}}\frac{\mu(n)}{n^2}-\sum_{n\leq \sqrt{x}}... |
937,064 | <p>The title pretty much says it all:</p>
<p>If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?</p>
<p><em>Edit:</em> Let me attempt to be a little more precise:</p>
<p>Suppose you have a proposition. Furthermore, suppose that assuming the proposition is false... | layman | 131,740 | <p>Yes. This is what is known as a <em>proof by contradiction.</em> When you want to prove a statement $P$ implies a statement $Q$ (i.e., you want to prove $P \implies Q$ is true), you always start by assuming $P$ is true.</p>
<p>Then, if you want to proceed by contradiction, you suppose $Q$ is false. Usually, if $... |
937,064 | <p>The title pretty much says it all:</p>
<p>If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?</p>
<p><em>Edit:</em> Let me attempt to be a little more precise:</p>
<p>Suppose you have a proposition. Furthermore, suppose that assuming the proposition is false... | MJD | 25,554 | <p>Certainly not. Suppose you have a red box and a green box, exactly one of which contains a treasure, and the following two statements about the boxes:</p>
<ol>
<li>The treasure is in the green box.</li>
<li>Exactly one of these statements is true.</li>
</ol>
<p>Assuming that the treasure is in the green box result... |
2,989,494 | <p>I am trying to derive properties of natural log and exponential just from the derivative properties.</p>
<p>Let <span class="math-container">$f : (0,\infty) \to \mathbb{R}$</span> and <span class="math-container">$g : \mathbb{R} \to \mathbb{R}$</span>.
Without knowing or stating that <span class="math-container">$f... | Fimpellizzeri | 173,410 | <p>You can do it 'by contradiction'.
If both <span class="math-container">$x<50$</span> and <span class="math-container">$y<50$</span>, then <span class="math-container">$x+y < 50 + 50 \implies x+y < 100$</span>, which contradicts our initial hypothesis of <span class="math-container">$x+y \geqslant 100$</s... |
2,348,131 | <p>In our class, we encountered a problem that is something like this: "A ball is thrown vertically upward with ...". Since the motion of the object is rectilinear and is a free fall, we all convene with the idea that the acceleration $a(t)$ is 32 feet per second square. However, we are confused about the sign of $a(t)... | stity | 285,341 | <p>Since it's a free fall, the acceleration is :
$$\vec{a}(t) = \vec{g}$$</p>
<p>Since it is rectilinear you get :</p>
<p>$$a(t) = \vec{a}(t).\vec{z} =\vec{g}.\vec{z}$$</p>
<p>So if $\vec{g}$ and $\vec{z}$ have the same sign, i.e. downward, you get $a(t) = g$.</p>
<p>And if $\vec{g}$ and $\vec{z}$ have opposite sig... |
2,822,355 | <p>this is a problem from one of the former exams from ordinary differential equations.</p>
<p>Find a solution to this equation:</p>
<p>$$x''''+6x''+25x=t\sinh t\cdot \cos(2t)$$</p>
<p>of course the only problem will be to find a particular solution, since the linear part is very simple to solve. My question is how ... | Aleksas Domarkas | 562,074 | <p>With free CAS Maxima solution of $\;x''''+6x''+25x=t\,\sinh(t)\, \cos(2t)\;$ is
$$x=\frac{t\, \left( 5 t+3\right) \, {e^{t}} \sin{\left( 2 t\right) }}{1600}-\frac{t\, \left( 5 t-3\right) \, {e^{-t}} \sin{\left( 2 t\right) }}{1600}\\-\frac{t\, \left( 20 t-33\right) \, {e^{t}} \cos{\left( 2 t\right) }}{3200}-\frac{t\... |
3,171,152 | <p>Let gamma 1 be a straight line from -i to i and let gamma 2 be the semi-circle of radius 1 in the right half plane from -i to i.</p>
<p>Evaluate</p>
<p><span class="math-container">$$\int_{\gamma_1}f(z)dz$$</span></p>
<p>and <span class="math-container">$$\int_{\gamma_2}f(z)dz$$</span></p>
<p>where f(z)=complex ... | bsbb4 | 337,971 | <p><span class="math-container">$f(z) = \bar{z}$</span> doesn't satisfy the Cauchy Riemann equations so it's not holomorphic. Therefore we can't assume that integrals along different paths give the same value as Cauchy's integral theorem fails.</p>
|
5,263 | <p>I recently tried to edit an old question <a href="https://mathoverflow.net/questions/39428/x-th-moment-method">x-th moment method</a> that had got bumped to the front page for other reasons. The post had an equation that was meant to be, and maybe at one point was, struck through, but it no longer is. That is, the... | Calvin Khor | 70,388 | <p>Yes, there is <code>\require{cancel}\cancel{1+1=2}</code><span class="math-container">$$\require{cancel}\cancel{1+1=2}$$</span> and <code>\require{enclose}\enclose{horizontalstrike}{1+1=2}</code><span class="math-container">$$\require{enclose}\enclose{horizontalstrike}{1+1=2}$$</span> There are some other options i... |
2,280,133 | <p><em>I need help to understand, some steps of the proof of this theorem.</em> </p>
<p><strong>(Kolmogorov-M. Riesz-Fréchet)</strong> Let $\mathcal{F}$ be a bounded set in $L^p(\mathbb{R}^N)$ with $1\leq p < \infty$. Assume that </p>
<p>\begin{equation}
\lim\limits_{|h|\longrightarrow 0 }\|\tau_hf-f\|_p=0 \ ... | MathRock | 417,393 | <p>$$\int|f(x-y)-f(x)|\rho_n(y)dy=\int|f(x-y)-f(x)|{(\rho_n(y))}^{1/p}{(\rho_n(y))^{1-1/p}}dy$$ Then use the Holder's inequality and $\int \rho_n=1$, get the result.</p>
|
253,584 | <p>Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=J_h^\mathtt{T}(x)J_h(x)\in\mathbb{R}^{n\times n}$ and $B(x):=J_h(x)J_h(x)^\mathtt{T}\in\mathbb{R}^{m\times m}$.</p>
... | Zoltan Zimboras | 12,897 | <p>I think Theorem 6.8 on page 122 in <a href="http://www.maths.ed.ac.uk/~aar/papers/kato1.pdf" rel="nofollow">Kato: Perturbation Theory for Linear Operators</a> may help (at least for the question concerning the eigenvalues of the symmetric $A$ and $B$ matrices).</p>
<p>Theorem:
Assume that $T(x)$ is a symmetric and ... |
1,552 | <p>Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current methods factor an arbitrary 200 digit number in a few hours (days? months? or what?).
Can current methods certify that an a... | TonyK | 767 | <blockquote>
<p>Closely related: what is the smallest known composite which has not been factored?</p>
</blockquote>
<p>As others have pointed out, this question can't be answered, even if you understand it as 'What is the smallest known composite none of whose factors are known?' (I can easily generate an enormous ... |
1,238,783 | <p>I am currently in high school where we are learning about present value. </p>
<p>I struggle with task like these: Say you get 6% interest each year, how much interest would that be each month?</p>
| Peter Webb | 60,051 | <p>This question has two possible answers. It depends on whether you are using compound or simple interest.</p>
<p>Debernardi's answer is correct, but I suspect its not the answer they want. The answer they want is 0.5% per month. I'll explain why.</p>
<p>First you should know the difference. Say you earned 10% inter... |
534,500 | <p>Is it true that if a sequence of random matrices $\{X_n\}$ converge in probability to a random matrix $X_n\overset{P}{\to}X$ as $n\to\infty$ that the elements $X_n^{(i,j)}\overset{P}{\to} X^{(i,j)}$ $\forall i,j$ also, or are there additional conditions required?</p>
<p>I think I have proved this using the norm $\|... | Carlos Llosa | 584,478 | <p>Yes. Let <span class="math-container">$e_k$</span> be a unit basis vector with zeroes everywhere except at the k-th position. Then <span class="math-container">$\forall i,j$</span>,
<span class="math-container">$$
X_n^{(i,j)} = e_i' X_n e_j \overset{P}{\to} e_i' X e_j = X^{(i,j)}.
$$</span></p>
|
58,947 | <p>Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?</p>
<p><strong>Edit:</strong> To rule out the case where $X$ has infinite genus, perhaps one could add the hypothesis that the topological space... | JHM | 20,516 | <p>Useful references for your question are Robert Brooks' "Platonic surfaces" and Dan Mangoubi's "Conformal Extension of Metrics of Negative Curvature" (both on arxiv). </p>
<p>I emailed Luca Migliorini requesting his paper. He told me it was basically his undergraduate thesis, published in a defunct italian journal... |
923,235 | <p>Let $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$
be a matrix of complex numbers. Find the characteristic polynomial $\chi_A(t)$ of $A$ and compute $\chi_A(A)$.</p>
<p>I just wanted to confirm that I did this correctly.</p>
<p>Tha answer I have is:
$$\chi_A(t)= \det\begin{pmatrix}a-t&b\\c&d-t\end{pmatri... | Joonas Ilmavirta | 166,535 | <p>No, your thinking does not seem right.
Try thinking of the following problem: If you have a polynomial $p(x)$, how to make sense of $p(A)$ for a square matrix $A$?</p>
<p>You found that the characteristic polynomial is
$$
\chi_A(t)
=
(ad-bc)-(a+d)t+t^2.
$$
Now we can plug $A$ in this polynomial (not in the determin... |
500,323 | <p>As a relative beginner trying to understand math more deeply, I'm trying to learn more about the mathematical laws (the laws of the operations $+, -, \times, \div$)</p>
<p>For example, I know the basic laws (the ones that are just taken to be true) -- the commutative, associative, and distributive laws. What area o... | hmakholm left over Monica | 14,366 | <p>The rules you speak about are the subject of <strong>elementary algebra</strong>.</p>
<p>It can be a bit difficult to find places where they are derived in rigorous detail, because elementary algebra is usually taught to children who don't particularly care for mathematical rigor, and so many textbooks will focus o... |
1,095,870 | <p>How can I solve the following inequality?</p>
<blockquote>
<p>$$\frac{\cos x -\tan^2(x/2)}{e^{1/(1+\cos x)}}>0$$</p>
</blockquote>
| Gonate | 195,844 | <p>I'll post what I have so far so that it may give you an idea of what to do. It's not complete and I'll try and finish it of when I have more time.</p>
<p>$e^{1/(1+\cos x)}$ will always be positive, so we only have to worry about the numerator.</p>
<p>Using the Weierstrass substitution, as @Lucian suggested, $t=\ta... |
2,831,130 | <p>Cauchy's induction principle states that:</p>
<blockquote>
<p>The set of propositions $p(1),...,p(n),...$ are all valid if: </p>
<ol>
<li>$p(2)$ is true.</li>
<li>$p(n)$ implies $p(n-1)$ is true.</li>
<li>$p(n)$ implies $p(2n)$ is true.</li>
</ol>
</blockquote>
<p>How to prove Cauchy's induct... | drhab | 75,923 | <p>There is a more elegant way to prove this. Making use of linearity of expectation we find:$$\mathbb E(B_1+\cdots+B_n)=\mathbb EB_1+\cdots+\mathbb EB_n=np$$ since $\mathbb EB_i=P(B_i=1)=p$ for every $i\in\{1,\dots,n\}$.</p>
<p>Linearity of expectation also works if the $B_i$ are not independent. So this route is not... |
3,125,958 | <p>I've been asked to consider this parabolic equation. </p>
<p><span class="math-container">$ 3\frac{∂^2u}{∂x^2} + 6\frac{∂^2u}{∂x∂y} +3\frac{∂^2u}{∂y^2} - \frac{∂u}{∂x} - 4\frac{∂u}{∂y} + u = 0$</span></p>
<p>I calculated the characteristic coordinates to be <span class="math-container">$ξ = y - x, η = x$</span>. T... | Mostafa Ayaz | 518,023 | <p><strong>Hint</strong></p>
<p>We have<span class="math-container">$${\partial^2 u\over \partial x^2}={\partial^2 u\over \partial \eta^2}+{\partial^2 u\over \partial ξ^2}-2{\partial^2 u\over \partial \eta\partial ξ}\\{\partial u\over \partial x}={\partial u\over \partial \eta}-{\partial u\over \partial ξ}\\{\partial ... |
3,576,979 | <p>Been working on this for some time now but have no idea if it's correct! Any hints are appreciated.</p>
<p>Recall the Fibonacci sequence: <span class="math-container">$f_1 = 1$</span>, <span class="math-container">$f_2 = 1$</span>, and for <span class="math-container">$n \geq 1$</span>, <span class="math-container"... | sedrick | 537,491 | <p>As J.W. Tanner mentioned, it's not true that <span class="math-container">$$\left(\frac{5}{4} \right)^k+ \left(\frac{5}{4} \right)^{k-1} > \left(\frac{5}{4} \right)^k\left(\frac{5}{4} \right)^k$$</span></p>
<p>(consider for example <span class="math-container">$k=3$</span> then <span class="math-container">$\fra... |
15,162 | <p>First off: I barely have any set theoretic knowledge, but I read a bit about cardinal arithmetic today and the following idea came to me, and since I found it kind of funny, I wanted to know a bit more about it.</p>
<p>If $A$ is the set of all real positive sequences that either converge to $0$ or diverge to $\inft... | Asaf Karagila | 622 | <p>I think that there are a handful of points that might need clarification here.</p>
<ul>
<li>There is no "set of all infinite cardinals", the family of all infinite cardinals is a proper class, and if it were a set it would imply the <a href="http://en.wikipedia.org/wiki/Burali-Forti_paradox" rel="noreferrer">Burali... |
2,017,818 | <p>Find three distinct triples (a, b, c) consisting of rational numbers that satisfy $a^2+b^2+c^2 =1$ and $a+b+c= \pm 1$.</p>
<p>By distinct it means that $(1, 0, 0)$ is a solution, but $(0, \pm 1, 0)$ counts as the same solution.</p>
<p>I can only seem to find two; namely $(1, 0, 0)$ and $( \frac{-1}{3}, \frac{2}{3}... | John Fisher | 387,114 | <p>$\frac{6}{7}, \frac{3}{7}, -\frac{2}{7}$ is the third solution.</p>
|
119,722 | <p>For a hyperplane arrangement $\mathcal{A}$ over a vector space $V$, we define its intersection poset, $L(\mathcal{A})$, as the set of all nonempty intersections of hyperplanes in $\mathcal{A}$ ordered by reverse inclusion. The empty intersection, $V$ itself, is the unique minimal element of $L(\mathcal{A})$.</p>
<p... | Rabee Tourky | 26,674 | <p>Chapters 4 and 8 of <em>Oriented Matroids
By Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, Gunter M. Ziegler</em>
reviews the big face lattice of oriented matroids, and when that is realizable as a hyperplane arrangement. Chapter 4 is self contained and I think you can skip to chapter 8 from th... |
38,252 | <p>I have a quadrilateral ABCD.
I want to find all the points x inside ABCD such that
$$angle(A,x,B)=angle(C,x,D)$$</p>
<p>Is there a known formula that gives these points ?</p>
<p><strong>Example:</strong></p>
<p>ABCD is a rectangle.
Let $x_1=mid[A,D]$ and $x_2=mid[B,C]$.
The points x are those lying on the line t... | JDH | 413 | <p>I believe that you are looking for ideas from the <a href="http://en.wikipedia.org/wiki/Perfect_set_property">Cantor Bendixson theorem</a>. </p>
<p>The main idea of the proof is the <em>Cantor-Bendixson derivative</em>. Given a closed set $X$, the derived set $X'$ consists of all limit points of $X$. That is, o... |
38,252 | <p>I have a quadrilateral ABCD.
I want to find all the points x inside ABCD such that
$$angle(A,x,B)=angle(C,x,D)$$</p>
<p>Is there a known formula that gives these points ?</p>
<p><strong>Example:</strong></p>
<p>ABCD is a rectangle.
Let $x_1=mid[A,D]$ and $x_2=mid[B,C]$.
The points x are those lying on the line t... | Seirios | 36,434 | <p>In fact, you can prove directly that any countable compact space $X$ is metrizable:</p>
<p>Let $A$ be a family of continuous functions from $X$ to $\mathbb{R}$ and let $e : \left\{ \begin{array}{ccc} X & \to & \mathbb{R}^A \\ x & \mapsto & (f(x)) \end{array} \right.$. If the family $A$ distinguishes... |
136,340 | <p>I defined the following functions</p>
<pre><code>CreatorQ[_] := False;
AnnihilatorQ[_] := False;
CreatorQ[q] := True;
AnnihilatorQ[p] := True;
CreatorQ[J[n_]] /; n < 0 := True;
AnnihilatorQ[J[n_]] /; n > 0 := True;
</code></pre>
<p>and when I ask for</p>
<pre><code>Assuming[r < 0, CreatorQ[J[r]]]
</code... | Pillsy | 531 | <p>The problem is that the pattern-matching in <code>CreatorQ</code> doesn't have any sort of knowledge of the assumption you're making about <code>r</code>, so the <code>Condition</code> won't fire. You can, as you commented, get around this by just redefining <code>CreatorQ</code> to return the inequality, which will... |
348,614 | <p>Is the following claim true: Let <span class="math-container">$\zeta(s)$</span> be the Riemann zeta function. I observed that as for large <span class="math-container">$n$</span>, as <span class="math-container">$s$</span> increased, </p>
<p><span class="math-container">$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} ... | JS Music | 150,164 | <p>Your summand is symmetric with respect to <span class="math-container">$k$</span> and <span class="math-container">$i$</span>:</p>
<p><span class="math-container">$$f(n,s) = \frac{1}{n}\sum_{k = 1}^n \sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\operatorname{lcm}(k,i)}\bigg)^s$$</span></p>
<p>We can sum along skew diag... |
2,881,020 | <p>We're trying to figure out something, and things aren't adding up. The senario is that you make $\$ 50,000$ a year. Every year you get a $15\%$ bonus of that income, which then gets added to your next year's income. So, the first year, you get $\$ 7,500$ which then makes your base $\$ 57,500$ the next year. When we ... | callculus42 | 144,421 | <p>You start with a bonus of 7500. Then every year the bonus decreases about $85\%$. But you carry over the bonuses of the previous years. That means that at the first year we have $b_1=7500$. And in the second year $b_2=7500+0.15\cdot 7500=8625$ and so on:</p>
<p>$b_3=7500+0.15\cdot 7500+0.15^2\cdot 7500$</p>
<p>..... |
2,176,081 | <p>I am trying to compute </p>
<blockquote>
<p>$$ \int_0^\infty \frac{\ln x}{x^2 +4}\,dx,$$</p>
</blockquote>
<p>which I find <a href="https://math.stackexchange.com/questions/2173289/integrating-int-0-infty-frac-ln-xx24-dx-with-residue-theorem/2173342">here</a>, without complex analysis. I am consistently getting ... | Olivier Oloa | 118,798 | <p>There is a mistake when writing, with $u=2t$, that
$$
- \int_0^\infty \frac{\ln(t)}{1+4t^2} dt=-\frac{1}{2} \int_0^\infty \frac{\ln(2u)}{1+u^2}\:du
$$ since $t=u/2$, <strong>we don't have</strong> $\ln (t)=\ln (2u).$ </p>
|
162,655 | <p>Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat? </p>
<p>I am especially interset in the case of Lorentzian Manifold whose sign signature is (- ,+ ,+ , + ). Of course, the example is not constricted in L... | José Figueroa-O'Farrill | 394 | <p>All riemannian manifolds with holonomy contained in $SU(n) \subset SO(2n)$, $Sp(n) \subset SO(4n)$, $G_2 \subset SO(7)$ and $Spin(7) \subset SO(8)$ are Ricci-flat. There are plenty of non-flat examples; e.g., those with holonomy <em>precisely</em> those groups.</p>
<p>In the Lorentzian setting, you could consider ... |
162,655 | <p>Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat? </p>
<p>I am especially interset in the case of Lorentzian Manifold whose sign signature is (- ,+ ,+ , + ). Of course, the example is not constricted in L... | Igor Khavkine | 2,622 | <p>Here's another explicit reference, on top of Ben Crowell's more general comment. The following paper discusses explicit examples of "pp-wave" spacetimes (Lorentzian, solving vacuum Einstein equations) that are geodesically complete: <em>Causal structures of pp-waves</em> by Veronika E. Hubeny and Mukund Rangamani (<... |
73,383 | <p>The problem is:
$$\displaystyle \lim_{(x,y,z) \rightarrow (0,0,0)} \frac{xy+2yz+3xz}{x^2+4y^2+9z^2}.$$</p>
<p>The tutor guessed it didn't exist, and he was correct. However, I'd like to understand why it doesn't exist.</p>
<p>I think I have to turn it into spherical coordinates and then see if the end result depen... | Community | -1 | <p>The limit for problems like these do not exist since the limit depends on the direction you approach. For the problem you have mentioned, say you approach $(0,0,0)$ along the direction $y = m_y x$ and $z = m_z x$, where $m_y$, $m_z$ are some constants, then we get $$\displaystyle \lim_{(x,y,z) \rightarrow (0,0,0)} \... |
1,342,069 | <p>In the <a href="https://en.wikipedia.org/wiki/Forgetful_functor" rel="nofollow">forgetful functor Wikipedia article</a> I read that </p>
<blockquote>
<p>"[Forgetful] Functors that forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may be indistinguishable ... | Martin Brandenburg | 1,650 | <p>If $\mathcal{C},\mathcal{D}$ are categories, then the projection functor $\mathcal{C} \times \mathcal{D} \to \mathcal{C}$ (which "forgets" the second coordinate) is not faithful (unless $\mathcal{D}$ is thin or $\mathcal{C}$ is empty).</p>
|
733,280 | <p>I cannot understand why $\log_{49}(\sqrt{ 7})= \frac{1}{4}$. If I take the $4$th root of $49$, I don't get $7$.</p>
<p>What I am not comprehending? </p>
| MPW | 113,214 | <p>No, you get $\sqrt 7$ as you should.</p>
<p>$$\log_{49}\sqrt 7 =\log_{49}7^{\frac12}=\frac12\log_{49}7=\frac12\cdot\frac12=\frac14$$</p>
|
1,779,088 | <blockquote>
<p>Prove
$$\sum_{i=1}^n i^{k+1}=(n+1)\sum_{i=1}^n i^k-\sum_{p=1}^n\sum_{i=1}^p i^k \tag1$$
for every integer $k\ge0$. </p>
</blockquote>
<p>By principle of induction,</p>
<p>$$\sum_{i=1}^n i = n(n+1)- \sum_{p=1}^n p$$
$$2\sum_{i=1}^n i = n(n+1)$$
$$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$
$\implies$$(... | Community | -1 | <p>In the RHS, a term $j^k$ is taken $n+1$ times in the first sum and $n-j+1$ times in the second, hence it contributes $(n+1-n+j-1)j^k=j^{k+1}$ to the total.</p>
<hr>
<p>$$\left|\begin{matrix}
1\cdot1^k\\
2\cdot2^k\\
3\cdot3^k\\
4\cdot4^k
\end{matrix}\right|=\left|\begin{matrix}
1^k&1^k&1^k&1^k&1^k\\... |
57,988 | <p>I am a programmer/analyst with limited (and pretty rusty) knowledge of math.</p>
<p>"Just for the heck of it" I have decided to try my hand at <a href="http://spectrum.ieee.org/automaton/robotics/artificial-intelligence/you-you-can-take-stanfords-intro-to-ai-course-next-quarter-for-free" rel="nofollow">Stanford's i... | Mathemagician1234 | 7,012 | <p>The best books I know that fit your criteria Marino are: </p>
<p><em>Introduction To Probability Theory</em> by Hoel, Port and Stone (make sure you get the 1971 edition;the later editions are terrible) This is the text I learned probability from under the very sure hand of Stefan Ralescu. I think you'll find it's e... |
341,648 | <p>I'm trying to understand what a tableaux ring is (it's not clear to me reading Young Tableaux by Fulton).</p>
<p>I studied what a monoid ring is on Serge Lang's Algebra, and then I read about modules, modules homomorphism. I'm trying to prove what is stated at page 121 (S. Lang, Algebra) while talking about algebra... | Community | -1 | <p>Consider $$f(z) = \log(1-e^{2 \pi zi }) = \log(e^{\pi zi}(e^{-\pi zi}-e^{\pi zi})) = \log(-2i) + \pi zi + \log(\sin(\pi z))$$
Then we have
\begin{align}
\int_0^1 f(z) dz & = \log(-2i) + \dfrac{i \pi}2 + \int_0^1 \log(\sin(\pi z))dz\\
& = \int_0^1 \log(\sin(\pi z))dz + \log(-2i) + \log(i)\\
& = \log(2) + ... |
18,686 | <p>Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:</p>
<p>$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,</p>
<p>where $\sim$ denotes "homeomorphic to". Obviously, $0 \leq \dim(B) \leq k$.</p>
<p>I have three questions: Given a $B \sub... | François G. Dorais | 2,000 | <p>The Cantor set satisfies $\dim(C^k) = 1$ for all $k$. You can easily find homeomorphic copies of the Cantor set with positive measure (e.g. at the $n$-th step remove every middle $3^n$-th instead of every middle third).</p>
|
1,305,257 | <p>I do not understand how to use the following information: If $f$ is entire, then </p>
<p>$$\lim _{|z| \rightarrow \infty} \frac{f(z)}{z^2}=2i.$$</p>
<p>So if $f$ is entire, it has a power series around $z_0=0$, so $f(z)=\Sigma_{n=0}^\infty a_nz^n$, and then we get </p>
<p>$$\lim _{|z| \rightarrow \infty} \frac{\S... | P Vanchinathan | 28,915 | <p>This is true for any polynomial $f(x)$ of even degree with positive coefficient for the highest degree term. Check that as $x\to\pm \infty$ the polynomial values $f(x)\to +\infty$. That means there exist an $N>0$ such that
for $a\in [-N, N], b\not\in [-N,N]$, we have $f(a)<f(b)$. Now the (global) minimum for ... |
1,305,257 | <p>I do not understand how to use the following information: If $f$ is entire, then </p>
<p>$$\lim _{|z| \rightarrow \infty} \frac{f(z)}{z^2}=2i.$$</p>
<p>So if $f$ is entire, it has a power series around $z_0=0$, so $f(z)=\Sigma_{n=0}^\infty a_nz^n$, and then we get </p>
<p>$$\lim _{|z| \rightarrow \infty} \frac{\S... | AlexR | 86,940 | <p>You can't just write $f(x) = x^4 g(x)$ because $f(0)$ is defined and $g(0)$ isn't. Instead, you can simply prove
$$\lim_{x\to\pm\infty} f(x) = \infty$$
(because the $x^4$ term dominates all others in the limit)<br>
With that settled, by the definition of these limits you can find some $R>0$ such that
$$f(x) > ... |
977,956 | <p>Can you help me solve this problem?</p>
<blockquote>
<p>Simplify: $\sin \dfrac{2\pi}{n} +\sin \dfrac{4\pi}{n} +\ldots +\sin \dfrac{2\pi(n-1)}{n}$.</p>
</blockquote>
| Yiyuan Lee | 104,919 | <p>The sum of all $n$th roots of unity (for $n > 1$) is zero. See <a href="http://en.wikipedia.org/wiki/Root_of_unity#Summation" rel="nofollow">here</a> for the proof. Its imaginary part is also zero. That is,</p>
<p>$$\sum_{k = 0}^{n - 1} \sin \frac{2\pi k}{n} = 0$$</p>
<p>Now simply subtract $\sin 0 = 0 $ from b... |
3,287,424 | <p>I have a function <span class="math-container">$$f(z)=\begin{cases}
e^{-z^{-4}} & z\neq0 \\
0 & z=0
\end{cases}$$</span></p>
<p>I have to show cauchy riemann equation is satisfied everywhere. I have shown that it isn't differentiable at <span class="math-container">$z=0$</span>. </p>
<p>Usually I will hav... | Nitin Uniyal | 246,221 | <p>As the question asks to use Cauchy-Riemann equations so either you convert it to get <span class="math-container">$u$</span> and <span class="math-container">$v$</span> in <span class="math-container">$x$</span> and <span class="math-container">$y$</span>; or use polar coordinates <span class="math-container">$r$</... |
134,673 | <p>I need to show that an automorphism of $S_n$ which takes transpositions to transpositions is an inner automorphism.</p>
<p>I thought it could be done by showing that such automorphisms form a subgroups $H\le Aut(S_n)$, that $Inn(S_n)\subset H$ and that they have the same number of elements. The number of inner auto... | Mark Bennet | 2,906 | <p>I think there is an easier way of looking at this (now edited to get the argument correct - and also simpler, thanks to Jyrki's comment). An inner automorphism of $S_n$ is equivalent to a permutation of the underlying set on which $S_n$ acts. Let's choose our generators to be a permutation $a=(1 2)$ and the n-cycle ... |
3,844,256 | <p>How can one prove the following deduction? Assume we know the following result.</p>
<p><span class="math-container">$$ \frac{1}{2}\arctan\left( \frac{y}{x+1} \right) + \frac{1}{2}\arctan\left( \frac{y}{x-1} \right) - \arctan\left( \frac{y}{x} \right) = c$$</span></p>
<p>Then, it is claimed that this is equivalent to... | Daron | 53,993 | <p>Here are some hints. First prove that for any connected subset <span class="math-container">$C \subset X$</span> the closure of is also connected. Then show <span class="math-container">$A \cup \{p\}$</span> is the closure of <span class="math-container">$A$</span>. To do so consider whether <span class="math-contai... |
143,274 | <p>I am trying to find the derivative of $\sqrt{9-x}$ using the definition of a derivative </p>
<p>$$\lim_{h\to 0} \frac {f(a+h)-f(a)}{h} $$</p>
<p>$$\lim_{h\to 0} \frac {\sqrt{9-(a+h)}-\sqrt{9-a}}{h} $$</p>
<p>So to simplify I multiply by the conjugate</p>
<p>$$\lim_{h\to0} \frac {\sqrt{9-(a+h)}-\sqrt{9-a}}{h}\cdo... | David Mitra | 18,986 | <p>You made a mistake when doing the multiplication upstairs:</p>
<p>When multiplying
$$
\Bigl( \color{maroon}{\sqrt{9-(a+h)} }- \color{darkgreen}{\sqrt {9-a}}\ \Bigr)\Bigl(\color{maroon}{\sqrt{9-(a+h)} }+\color{darkgreen}{ \sqrt {9-a}}\ \Bigr),
$$
you are using the rule
$$
(\color{maroon}a-\color{darkgreen}b)(\color{... |
666,461 | <p>The function $f(x)=x+\log x$ has only one root on $(0,\infty)$ which is in $(0,1)$.</p>
<p>Using the Intermediate value theorem: $f$ is continuous on $(0,\infty)$ and $f(0)=0+\log(0)=-\infty<0$ and $f(1)=1+\log(1)=1>0$. So there exists an $x$ such $f(x)=0$.</p>
<p>But how to show that this $x$ is the only ro... | guest | 125,454 | <p>$f(x)=x+log(x)=log(e^xx)$ the question of finding the roots of the function $f(x)$ is the same question of finding the set of solutions of the equation $xe^x=1$. Let $g(x)=xe^x-1$ and $g(x)$ have two roots on $(0,1)$, say a and b. Since $g(a)=g(b)=0$, $g$ is continuous on $[a,b]$ and $g$ is differentiable on $(a,b)$... |
1,663,838 | <p>Show that a positive integer $n \in \mathbb{N}$ is prime if and only if $\gcd(n,m)=1$ for all $0<m<n$.</p>
<p>I know that I can write $n=km+r$ for some $k,r \in \mathbb{Z}$ since $n>m$</p>
<p>and also that $1=an+bm$. for some $a,b \in \mathbb{Z}$</p>
<p>Further, I know that $n>1$ if I'm to show $n$ is... | fleablood | 280,126 | <p>This should be trivial.</p>
<p>If n is prime it has no factors but 1 and n. So gcd (n,m) can only equal 1 or n. If gcd(n,m) = n then that means n|m. So m $\ge$ n. So if m < n then gcd (n,m)=1.</p>
<p>That's one way.</p>
<p>If gcd (n,m)=1 for all m < n, then no number less than n divides n (other than 1).... |
3,354,990 | <p>I have points and limits of a function and even the shape of the function and I'm looking for the function, something that very interesting for me how could I control the curve of the function?</p>
<p>(1) <span class="math-container">$\lim\limits_{x \to inf} f(x) = 1 $</span></p>
<p>(2) <span class="math-container... | DSaad | 169,718 | <p>I found two answers after a lot of experimenting with variables: </p>
<p><span class="math-container">$\left(\frac{\left(e^{2\cdot e\cdot c\cdot x}\right)-1}{\left(e^{2\cdot e\cdot c\cdot x}\right)+1}\right)$</span></p>
<p><span class="math-container">$\left(\frac{1-\left(e^{-2\cdot e\cdot c\cdot x}\right)}{1+\lef... |
2,418,171 | <p>I would like to see an explicit example of two smooth isometric embeddings (in the Riemannian sense) $i_1,i_2:\mathbb{R}^2 \to \mathbb{R}^3$ such that there is no isometry $\varphi:\mathbb{R}^3\to \mathbb{R}^3$ suct that $i_2=\varphi\circ i_1$.</p>
<p>(I take here $\mathbb{R}^2,\mathbb{R}^3$ to be endowed with the ... | Chris Culter | 87,023 | <p>Sure, how about taking one image to be $\{x,y,0\}$ and the other $\{x,y,\sin x\}$?</p>
|
2,418,171 | <p>I would like to see an explicit example of two smooth isometric embeddings (in the Riemannian sense) $i_1,i_2:\mathbb{R}^2 \to \mathbb{R}^3$ such that there is no isometry $\varphi:\mathbb{R}^3\to \mathbb{R}^3$ suct that $i_2=\varphi\circ i_1$.</p>
<p>(I take here $\mathbb{R}^2,\mathbb{R}^3$ to be endowed with the ... | levap | 32,262 | <p>Take a unit speed curve $\alpha \colon \mathbb{R} \rightarrow \mathbb{R}^3$ whose image lies in the $xy$ plane and consider the map $\varphi \colon \mathbb{R}^2 \rightarrow \mathbb{R}^3$ given by</p>
<p>$$ \varphi(s,t) = \alpha(s) + te_3 $$</p>
<p>where $e_3 = (0,0,1)$. The image of $\varphi$ is a cylinder over $\... |
905,685 | <p>Let the balls be labelled $1,2,3,..n$ and the boxes be labelled $1,2,3,..,n$. </p>
<p>Now I want to find, </p>
<ul>
<li><p>What is the expected value of the minimum value of the label among the boxes which are non-empty </p></li>
<li><p>What is the expected number of boxes with exactly one ball in them? </p></li>
... | Graham Kemp | 135,106 | <h2>A</h2>
<p>To expectation of the minimum used label, $K$, we first measure the probability that all the balls being among the top $n-k$ boxes. That is, that the minimum label will be greater than some value $k$.</p>
<p>In the total space each of $n$ balls has a choice of $n$ boxes ($n^n$). In the restricted space... |
1,599,467 | <p>Here $f$ is a non-zero linear functional on a vector space $X$. I can show this true for one direction, </p>
<blockquote>
<p>Let $x_1, x_2 \in x + N(f)$</p>
<p>$\implies x_1 = x + y_1, \quad x_2 = x + y_2$, where $y_1, y_2 \in N(f).$</p>
<p>Then $f(x_1) = f(x) + f(y_1) = f(x) = f(x) + f(y_2) = f(x_2)$.... | symplectomorphic | 23,611 | <p>Might as well make my comment an answer:</p>
<p>If $f(x)=f(y)$, then $0=f(x)-f(y)=f(x-y)$ by linearity, so $x-y$ is in the null space. Hence $x$ and $y$ are mapped to the same element of the quotient space.</p>
|
95,314 | <p>To evaluate this type of limits, how can I do, considering $f$ differentiable, and $ f (x_0)> 0 $</p>
<p>$$\lim_{x\to x_0} \biggl(\frac{f(x)}{f(x_0)}\biggr)^{\frac{1}{\ln x -\ln x_0 }},\quad\quad x_0>0,$$</p>
<p>$$\lim_{x\to x_0} \frac{x_0^n f(x)-x^n f(x_0)}{x-x_0},\quad\quad n\in\mathbb{N}.$$</p>
| Davide Giraudo | 9,849 | <p>For the first: for $x\neq x_0$, since $f(x)>0$ in a neighborhood of $x_0$
\begin{align*}
\frac{f(x)}{f(x_0)}^{\frac 1{\ln x-\ln x_0}}&=\exp\left(\frac {\ln f(x)-\ln f(x_0)}{\ln x-\ln x_0}\right)\\
&=\exp\left(\frac {\ln f(x)-\ln f(x_0)}{x-x_0}\frac{x-x_0}{\ln x-\ln x_0}\right),
\end{align*}
and taking the... |
1,336,344 | <p>Given a matrix A of a strongly $k$ regular graph G(srg($n,k,\lambda,\mu$);$\lambda ,\mu >0;k>3$). The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$.
$A_x$ is the symmetric matrix of the graph $(G-x)$, where $C$ is the symmetric matrix of the graph created by vertices of ... | Michael | 179,940 | <p><strong>Proposed Solution:</strong></p>
<p>Arrangement of $G$: $A$ is the matrix of graph $G$ where $|G|=|A|=n$. Based on the adjacency of last vertex($n$ th vertex), A can be divided in to 4 sub-matrices where 2 are symmetric matrices($C,D$) and 2 are non square matrices($E^{T}, E$). One symmetric matrix, say, $D$... |
1,810,729 | <blockquote>
<p>Let $G$ be a group generated by $x,y$ with the relations $x^3=y^2=(xy)^2=1$. Then show that the order of $G$ is 6.</p>
</blockquote>
<p><strong>My attempt:</strong> So writing down the elements of $G$ we have $\{1,x,x^2,y,\}$. Other elements include $\{xy, xy^2, x^2y\}$ it seems I am counting more th... | Kushal Bhuyan | 259,670 | <p>Since $y^2=1$ so $xy^2=x$. So only one of them counts. Thus $6$ elements. The given relation is basically resembles to the structure of $D_3$, the dihedral group of order $6.$</p>
|
4,212,181 | <p>A uniform cable that is 2 pounds per feet and is 100 feet long hangs vertically from a pulley system at the top of a building (and the building is also 100 feet tall).</p>
<p>How much work is required to lift the cable until the bottom end of the cable is 20 feet below the top of the building?</p>
<p><span class="ma... | Oğuzhan Kılıç | 481,167 | <p>There are many ways you can solve this problem, let's first do it in your way. Note that i'm not bothering with units. Because of feet, pound. :( not nice. Also let me note that we must talk about the <span class="math-container">$g$</span> (gravitation constant) because if we don't say anything about it answer wşll... |
4,389,441 | <p>I'm dealing with a sample problem where I want to work out the probability of a fair coin toss landing <em>heads</em> and a fair die roll landing <em>6</em>. We are then told that <strong>at least</strong> one of those events has happened.</p>
<p>Why is the probability of this not as simple as <em>P(C)P(D) = 0.083</... | FOE | 1,018,997 | <p>It is clear that <span class="math-container">$P(C \cap D)= P(C)P(D)$</span>. However in this case you are not asking that question , your question here is:
<span class="math-container">$P(C \cap D \mid \text{ at least one of the events happened} )$</span> . In this new probability space the events are not independe... |
3,421,858 | <p><span class="math-container">$\sqrt{2}$</span> is irrational using proof by contradiction.</p>
<p>say <span class="math-container">$\sqrt{2}$</span> = <span class="math-container">$\frac{a}{b}$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are positive integers. </... | Randall | 464,495 | <p><span class="math-container">$b$</span> is selected to the smallest positive integer so that <span class="math-container">$b\sqrt{2}$</span> is an integer. One such <span class="math-container">$b$</span> exists by assumption, so just pick the smallest.</p>
<p>Next, <span class="math-container">$b^*\sqrt{2} = b(\s... |
3,421,858 | <p><span class="math-container">$\sqrt{2}$</span> is irrational using proof by contradiction.</p>
<p>say <span class="math-container">$\sqrt{2}$</span> = <span class="math-container">$\frac{a}{b}$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are positive integers. </... | Alberto Saracco | 715,058 | <p>Usually the proof goes along this lines.</p>
<p>Suppose by contradiction <span class="math-container">$\sqrt{2}=\frac{a}{b}$</span> with <span class="math-container">$a,b\in\mathbb N$</span>. We can assume the fraction to be reduced, in particular <span class="math-container">$a,b$</span> are not both even. </p>
<... |
703,031 | <p>In a sequence of integers, $A(n)=A(n-1)-A(n-2)$, where $A(n)$ is the $n$th term in the sequence, $n$ is an integer and $n\ge3$,$A(1)=1$,$A(2)=1$, calculate $S(1000)$, where $S(1000)$ is the sum of the first $1000$ terms.</p>
<p>How to approach these type of questions? Which topics should I study?</p>
| Barry Cipra | 86,747 | <p>A generally good way to approach problems like this is to "experiment" with the formulas to see what they say. That is, plug in numbers and calculate:</p>
<p>$$\begin{align}
A(1)&=1 & S(1)&=A(1)=1\\
A(2)&=1 & S(2)&=S(1)+A(2)=1+1=2\\
A(3)&=A(2)-A(1)=1-1=0 & S(3)&=S(2)+A(3)=2+0=2\... |
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