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 |
|---|---|---|---|---|
2,485,109 | <blockquote>
<p>Proof by induction that $1^2 + 2^2 + 3^2 +.......+ n^2 = \frac{1}{6}\cdot n\cdot (1+n)\cdot (1+2n)$</p>
</blockquote>
<p>I tried showing that </p>
<p>$1^2 + 2^2 + 3^2 +.......+ (k+1)^2 = \frac{1}{6}\cdot (k+1)\cdot (1+(k+1))\cdot (1+2(k+1))$</p>
<p>By using the left side:</p>
<p>$1^2 + 2^2 + 3^2 +... | Aderinsola Joshua | 395,530 | <p>How many pair of positive integers are solutions to the equation.<br>
$$5X + 7Y = 1234$$
because $1234$ is even, and only the sum of two odd numbers or two even numbers is even, and also because $5$ and $7$ are both odd.<br>
Therefore $X$ and $Y$ can be both odd or both even; if $X$ was odd, $Y$ would be odd and vic... |
1,730,445 | <p>I am checking whether the limit is true or not. $z$ is complex number
\begin{equation}
\lim_{z \rightarrow 0} z\sin(\frac{1}{z})=0
\end{equation}
I found Laurent series of $z\sin(\frac{1}{z})$ which is $\sum_{n=0}^{\infty} (\frac{1}{z})^{2n}\frac{1}{(2n+1)!}(-1)^n$ </p>
<p>the series is not defined $z =0$.</p>
... | Henricus V. | 239,207 | <p>Casorati-Weierstrass theorem implies that the limit does not exist. In fact, approaching from $i\mathbb{R}$ gives
$$ \lim_{h \to 0} ih \sin \frac{1}{ih} = -\lim_{h \to 0} h \sinh \frac{1}{h} = -\infty
$$
By l'Hopital.</p>
|
2,442,297 | <blockquote>
<p>Ryan has been given a salary increase of $7.39\%$. The salary increase is for the value of €$4231$.</p>
<p>His salary is now $x$. Solve for $x$.</p>
</blockquote>
<p>My head is saying </p>
<p>$$\begin{align}
4231 / 7.39 &= 572 \\
572 * 100 &= 57,200
\end{align}$$</p>
<p>is not correct,... | Community | -1 | <p>As someone who's been told to go into accounting, you forgot to take markup versus margin into account. The 7.39% is a markup on the old amount (or about a 6.88% margin on the new amount) you didn't successfully find 100% of the original amount and forgot to add the markup back in you needed to find 107.39% of the ... |
3,727,772 | <p><span class="math-container">$Y_1,Y_2,\ldots,Y_{n+1}$</span> be non-empty subsets of <span class="math-container">$\{1,2,3\ldots,n\}$</span>. Prove that there exists non-empty disjoint subsets <span class="math-container">$A_1$</span> and <span class="math-container">$A_2$</span> of <span class="math-container">$\{1... | lhl73 | 797,186 | <p>Let <span class="math-container">$e_i$</span> for <span class="math-container">$i = 1, \ldots, n$</span> be the standard basis for <span class="math-container">$\Bbb{R}^n$</span>. That is, <span class="math-container">$e_i$</span> is the <span class="math-container">$n$</span>-tuple where the <span class="math-conta... |
223,754 | <p>Suppose $f$ and $g$ are such that $f(g(x)) = 1$. Does this imply that $g(f(x))$ is constant?</p>
| Community | -1 | <p>NO.</p>
<p>$$f(x) = \begin{cases} 1 & x \geq 0 \\ 0 & x < 0 \end{cases}$$</p>
<p>$$g(x) = x^2, \,\,\, \forall x \in \mathbb{R}$$</p>
<p>We then have $$f(g(x)) = f(x^2) = 1, \,\,\,\, \forall x \in \mathbb{R}$$
whereas $$g(f(x)) = f(x)^2 = \begin{cases} 1 & x \geq 0 \\ 0 & x < 0 \end{cases}$$<... |
223,754 | <p>Suppose $f$ and $g$ are such that $f(g(x)) = 1$. Does this imply that $g(f(x))$ is constant?</p>
| Community | -1 | <p>No, $\forall x \in I$ and $I' = f^{-1}(1) $ \ $ g(x)$
$$f(g(x)) = 1$$
$$f(g(f(x))) = 1$$
$$g(f(x)) \in f^{-1}(1) = g(x) \cup I'$$</p>
|
1,620,175 | <p>Given metric space $M = (\mathbb{R}^2, d)$ where $d = \operatorname{max}\{|x_1 - y_1|, |x_2 - y_2|\}$, how can one measure distance from some arbitrary point $X$ to the line $y = 3$, let's say?</p>
<p>How it can be done, by use of the identity
$$\operatorname{max}\{x, y\} = \frac{1}{2}(x + y+ |x - y|)\text{?}$$</p>... | Rob Arthan | 23,171 | <p>Hint: the "circles" of radius $r$ about $X$ in $M$ are squares with edges parallel to the axes that are centred on $X$ and have sides of length $r/2$. See <a href="https://en.wikipedia.org/wiki/Uniform_norm" rel="nofollow">https://en.wikipedia.org/wiki/Uniform_norm</a> or <a href="https://en.wikipedia.org/wiki/Norm_... |
2,002,672 | <p>I am ok with finding a power series for $\tanh^{-1}{z}$ since I can just see that since $\tanh iz = i\tan z$, then $\tanh^{-1}z = \frac1i \tan^{-1}{iz}$ and use the power series for $\tan$ to get:</p>
<p>$$\tanh^{-1}{z} = \sum^\infty_{k=0} \frac{z^{2k+1}}{2k+1}$$</p>
<p>But then I am asked to deduce that:</p>
<p>... | Alex M. | 164,025 | <p>In the case of $\sin$, $\cos$, $\exp$, $\tan$ etc., there is nothing to prove because most mathematicians use those power series as their <em>definitions</em>, meaning that those functions are analytic by definition.</p>
<p>Composition of analytic functions is again analytic on the subsets where it may be performed... |
239,097 | <p>Recently, I want to buy some 30xx series GPUs which are used to do deep learning jobs, so does anyone know whether Mathematica's <code>NetTrain</code> supports 30xx series GPUs?</p>
| Fidel I. Schaposnik | 45,839 | <p>At the time of writing this (Feb. 6, 2021), Mathematica 12.2 doesn't support these cards, see for example:</p>
<p><a href="https://mathematica.stackexchange.com/questions/237234/extremely-long-gpu-initialization-times-on-mathematica-12-2-and-rtx3090">Extremely Long GPU Initialization Times on Mathematica 12.2 and RT... |
263,359 | <p>Let $H$ be a monoid, and denote by $H^\times$ and $\mathcal A(H)$, respectively, the <em>set of units</em> (or <em>invertible elements</em>) and the <em>set of atoms</em> (or <em>irreducible elements</em>) of $H$ (an element $a \in H$ is an atom if $a \notin H^\times$ and $a = xy$ for some $x, y \in H$ implies $x \i... | Salvo Tringali | 16,537 | <p><a href="https://mathoverflow.net/a/263362/16537">Benjamin Steinberg's example</a> is neat and brilliant. Here is a different construction, which, though blatantly complicate, could help with related questions (I had had it in mind for days, but for some reason I couldn't make it work until minutes ago...).</p>
<p>... |
2,844,170 | <p>First off, sorry if this is a basic question or one that has been asked before. I really don't know how to phrase it, so it's a hard question to google.</p>
<p>I'm looking for a function that will generate a line similar to the one below</p>
<pre><code> __/
__/
__/
/
</code></pre>
<p>I'm pretty good at... | zachThePerson | 408,153 | <p>John Wayland's answer was good, and worked on the project I needed, but after playing around on a different project I came up with a more "tweakable," function that does the same thing, but allows changing of parameters.</p>
<p>D is the duration of one cycle, T is the amount of time the transition should t... |
2,970,053 | <p>I've been trying to understand this problem for hours but not getting it. HELP!!!</p>
<p>The correct answer is <span class="math-container">$\frac{2}{3}$</span>, but I don't know why this is the correct answer.</p>
<p>Thank you in advance for your help!</p>
| trancelocation | 467,003 | <p>Note that the symmetry argument of M. Wind produces an elegant solution. </p>
<p>Here, I present a solution "on combinatorical foot":</p>
<ul>
<li>First, arrange the <span class="math-container">$37$</span> other books: <span class="math-container">$\color{blue}{37!}$</span></li>
<li>These <span class="math-contai... |
2,205,776 | <p>I am pretty sure there is an easy Counter example But i do not find One right Now.</p>
| Fly by Night | 38,495 | <p>Your question is in-correct, there is no counter-example.</p>
<p>For a sequence of integers $(n_k)$ to converge, you need: for any $\varepsilon >0 $, there is an integer $K >0$, such that $k \ge K \ \ \implies \ \ \left|n_k-n_K\right| < \varepsilon $.</p>
<p>If, for example, $\varepsilon = \frac{1}{4}$, t... |
672,412 | <p>I am reading an e-book called <a href="http://www.ldsinsight.org/">To Infinity and Beyond</a> by Dr. Kent A Bessey. In the book the author makes the claim that Georg Cantor made a discovery "where half of a pie is as large as the whole".</p>
<p>In talking about it, he seems to claim that because half a pie can be b... | Peteris | 124,006 | <p>The pies are a bad analogy, since they aren't infinite. However, maybe the following analogy would be useful and a different perspective from other answers:</p>
<p>Suppose a genie gives you a magic cup of beer that's always full, no matter how much you pour out or drink of it. Now, let's suppose that you get anoth... |
706,514 | <p>I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic? </p>
| Dan Rust | 29,059 | <p>Let $X'=X\sqcup Y$ where $Y$ is a space with cardinality greater than $X$ and make the basepoint of $X'$ still lie inside the copy of $X$. Then $\pi_1(X')=\pi_1(X)$ but they are not homeomorphic because of different cardinalities.</p>
|
3,399,276 | <p>If the problem is to write the following with simplified polynomials</p>
<p><span class="math-container">$$\frac{x^2 + 5x + 6}{x^2+1}$$</span></p>
<p>Is it possible to do this problem with synthetic division? If so, how?</p>
<p>I've tried googling, finding on youtube, even plug this in Wolfram Alpha, no helpful r... | Dr. Sonnhard Graubner | 175,066 | <p>To find all zeros of <span class="math-container">$$\frac{x^2+5x+6}{x^2+1}$$</span> you have to solve the equation <span class="math-container">$$x^2+5x+6=0$$</span>
Or this here
<span class="math-container">$$\frac{x^2+5x+6}{x^2+1}=1+{\frac {5\,x+5}{{x}^{2}+1}}$$</span>
It is <span class="math-container">$$\frac{x^... |
796,787 | <p>Given a Lipschitz function $g$ (i.e. $|g(x) - g(y)| \leq L |x - y|, \forall x, y \in dom(g)$), and an function $f$ integrable on $[a, b]$, how do we prove $g \circ f$ is integrable on $[a, b]$, preferably using Darboux integrals/sums?</p>
<p>Let us assume $g, f$ have appropriate domains and codomains.</p>
| EPS | 133,563 | <p><strong>Hint</strong>: A function is (Riemann) integrable if and only if its set of discontinuity has measure zero.</p>
|
209,044 | <p>Please help me deal with this kind of question about ODEs.
My codes are as follows</p>
<pre><code>m = 100;
a = D[x[t], {t, 2}];
t1up = 2 x''[t] + 1/2 (490 + 34 x''[t] + 2 (490 + 50 x''[t]));
t1down = 490 + 53 x''[t];
t1 = Piecewise[{{t1up, x'[t] >= 0}, {t1down, x'[t] < 0}}]
equa00 = t1 == m*a
t0 = 50;
s1 = ND... | xinxin guo | 4,756 | <p>Changing the last line to: </p>
<pre><code>s1 = NDSolve[{equa00, x[0] == 1, x'[0] == 1}, x, {t, 0, 50}, SolveDelayed -> True]
</code></pre>
<p>or</p>
<pre><code>s1 = NDSolve[{equa00, x[0] == 1, x'[0] == 1}, x, {t, 0, 50},
Method -> {"EquationSimplification" -> "Residual"}]
</code></pre>
<p>seems h... |
85,228 | <p>I wonder if the periodic paths of a lightray trapped between two nonconcentric circles,
each perfectly reflecting, are known.
The behavior of such rays seems chaotically complicated. For example, left below the highlighted
initial ray has slope <span class="math-container">$\frac{1}{12}$</span>, while that to the r... | Igor Rivin | 11,142 | <p>Following up on @Ian's comments: the magic words are "monotone twist maps of the annulus" and "Aubry-Mather theory". googling either of the above (or looking in Katok-Hasselblad, who have a whole chapter on the subject) is your best bet.</p>
|
4,602,464 | <p>I've been asked to provide context to my original question, so here's the context:</p>
<p>The rectangle in the problem below represents a pool table whose "pool table light" cannot be easily moved, but CAN easily be rotated. No portion of the pool table's perimeter can be too close to a wall in order for t... | Nick Castillo | 683,736 | <p>These are subtly different. All one forms on a given (assume finite dimensional vector space) are linear maps from vectors to the field. On a manifold each point has a vector space associated with it (the tangent space <span class="math-container">$T_pM$</span>) upon which one can define one forms as above. The c... |
3,257,387 | <p>denote <span class="math-container">$(a_n)$</span> as a sequence. If the consecutive difference tend to <span class="math-container">$0$</span> as <span class="math-container">$n$</span> tends to infinity, do it mean that <span class="math-container">$(a_n)$</span> converges? I feel like it should mean so but I am s... | Floris Claassens | 638,208 | <p>No, consider
<span class="math-container">$$a_{n}=\sum^{n}_{i=1}\frac{1}{i}.$$</span>
Clearly <span class="math-container">$a_{n+1}-a_{n}=\frac{1}{n+1}$</span> tends to <span class="math-container">$0$</span>, but <span class="math-container">$a_{n}$</span> diverges to infinity.</p>
|
2,487,866 | <p>Gödel's <a href="https://en.wikipedia.org/wiki/Constructible_universe" rel="nofollow noreferrer">constructible universe</a> seems to have some attractive properties. Sets are constructed in a very regular, easy-to-understand way, and one has a definite answer on certain major set theoretic questions, such as the gen... | Michael Weiss | 79,741 | <p>I think a realist philosophy accounts for much of the interest. (Of course, $L$ itself has been scrutinized thoroughly over the years.) Many (most?) set theorists take a so-called Platonist attitude: they believe that the universe of sets "really exists" in some sense. Quoting from Drake's <em>Set Theory: an Introdu... |
1,424,561 | <p>I know that $\sum_0^\infty \frac{\lambda^x} {x!} = e^\lambda$, but I'm having a really difficult time dealing with the extra $x$.</p>
| Laurent Duval | 257,503 | <p>The additional $x$ (from your initial knowledge) should lead you to think about derivatives with respect to $\lambda $. Remember a polynomial differentiates as:
$$ \left(x^n\right)' = nx^{n-1}$$. So you can rewrite in a close form:
$$ \sum_{x=0}^\infty \frac{x\lambda^x}{x!} = \lambda\sum_{x=0}^\infty \frac{x\lambda^... |
1,424,561 | <p>I know that $\sum_0^\infty \frac{\lambda^x} {x!} = e^\lambda$, but I'm having a really difficult time dealing with the extra $x$.</p>
| Dilip Sarwate | 15,941 | <p>Very inelegantly and in schoolboy fashion we can write</p>
<p>\begin{align}
\sum_{x=0}^\infty x\frac{\lambda^x}{x!}
&= 0\frac{\lambda^0}{0!} + 1\frac{\lambda^1}{1!} + 2\frac{\lambda^2}{2!}
+ 3\frac{\lambda^3}{3!} + 4\frac{\lambda^4}{4!} + \cdots\\
&= 0 + \lambda +\frac{\lambda^2}{1!} + \frac{\lambda^3}{2!} ... |
549,254 | <blockquote>
<p>$$\frac{1}{3}=.33\bar{3}$$ </p>
</blockquote>
<p>is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio? </p>
<p>I understand that $\pi$ is irrational because it extends infinitely <em>... | Paramanand Singh | 72,031 | <p>I believe the fundamental problem (or confusion) here is that OP finds it difficult to believe that a rational number, which is a ratio of two finite integers, can have a representation which is infinite. This confusion is primarily due to the fact that most people try to think of a number and its representation as ... |
99,750 | <p>Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).</p>
<p>I would like a reference for the description of the algebra $End_{G^F}( \mathbb{C}[G^F/U^F] )$. More precisely, I'd ... | Ian Agol | 1,345 | <p>There's a nice construction of the $E_8$ Lie algebra due to Borcherds based on methods from vertex operator algebras, but with no understanding of vertex algebras needed. See <a href="http://math.berkeley.edu/~anton/written/LieGroups/LieGroups.pdf">p. 152 of these notes</a> from a course by Borcherds and others. See... |
2,903,105 | <p>A Fourier transform of function R into function Q is defined as:
$$Q(\underline{k}) = \int_{}^{}R(\underline{x}) e^{-i\underline{k}·\underline{x}} \mathrm{d}\underline{x}.$$
where I've underlined $\underline{x}$ and $\underline{k}$ to denote that they are a set of vectors.</p>
<p>Suppose that $R(\underline{x})$ is ... | leonbloy | 312 | <p>This is not totally rigorous, but for Physics it should be enough :-)</p>
<p>In general, if $Q(\underline{k}) $ is the transform of $R(\underline{x})$, then the transform of $R(\underline{x} + a \underline{u})$ where $ \underline{u}=(1,1,1)$ and $a$ is some scalar, is
$$Q'(\underline{k})=\exp(i 2 \pi a \, \underli... |
3,542,374 | <p>I have the matrix of stacked constraints</p>
<p><span class="math-container">$$\begin{bmatrix} x_1^2 & x_1y_1 & y_1^2 & x_1 & y_1 & 1 \\ x_2^2 & x_2y_2 & y_2^2 & x_2 & y_2 & 1 \\ x_1^2 & x_3y_3 & y_3^2 & x_3 & y_3 & 1 \\ x_4^2 & x_4y_4 & y_4^2 &... | Jonas Linssen | 598,157 | <p>It means that your matrix has rank 5, so its null space has dimension <span class="math-container">$6-5=1$</span>. This means that you have exactly one nonzero solution <span class="math-container">$c$</span> with norm/magnitude/length 1 and whose first nonzero entry is positive. Any other solution is a multiple of ... |
646,596 | <p>I need to calculate the described squared circle... with only given the length of the side (a).</p>
<p><img src="https://i.stack.imgur.com/VZ8Yn.png" alt="enter image description here"></p>
<p>I need to calculate the area of described squared circle. How exactly is it done with only a side given?</p>
| John | 7,163 | <p>If the length of the side of the square is $a$ then its diagonal is $\sqrt{2}a$, which is also the diameter of the circle. The radius of the circle is half this: $a/\sqrt{2}.$ This makes the area of the circle $\pi a^2 / 2.$ </p>
|
925,558 | <p>How do I prove that the inductive sequence $y_{n+1}= \dfrac {2y_n + 3}{4}$ is bounded? $y(1)=1$</p>
<p><strong>Attempt:</strong> Let us assume that the given sequence is unbounded. : </p>
<p>Then, $y_{n+1} \rightarrow \infty$ either for some finite $n$ or when $n \rightarrow \infty$</p>
<blockquote>
<p>CASE $... | egreg | 62,967 | <p>An unbounded sequence need not diverge to $\infty$, even if its values are positive.</p>
<hr>
<p>If the sequence converges, then its limit $l$ satisfies
$$
l=\frac{2l+3}{4}
$$
so $l=3/2$. We could try proving that $y_n\le 3$ for all $n$. The base case is clear, so assume that $y_n\le 3$ and try to prove that $y_{n... |
3,671,675 | <blockquote>
<p>Question: Suppose <span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span> is a twice differentiable function with <span class="math-container">$f(0)=1$</span>, <span class="math-container">$f'(0)=0$</span> and satisfies <span class="math-container">$f''(x)-5f'(x)+6f(x)\ge 0$</span> for every <... | Aditya Dwivedi | 697,953 | <p>Separate the terms as <span class="math-container">$\ {f^{\prime\prime}}-3{f^{\prime}}-2(f^{\prime} - 3f ) \ge 0$</span></p>
<p>now multiplying by <span class="math-container">$ e^{-2x}$</span> and integrating from 0 to x we get <span class="math-container">$f^{\prime} \ - \ 3f \ge -3e^{2x}$</span></p>
<... |
3,671,675 | <blockquote>
<p>Question: Suppose <span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span> is a twice differentiable function with <span class="math-container">$f(0)=1$</span>, <span class="math-container">$f'(0)=0$</span> and satisfies <span class="math-container">$f''(x)-5f'(x)+6f(x)\ge 0$</span> for every <... | N.Quy | 600,377 | <p>I have no idea about following your proof by contradiction. However, I have another proof.</p>
<p>This problem has the same idea as a simple problem: Let <span class="math-container">$f :\mathbb{R} \to \mathbb{R}$</span> be continuously differentiable such that <span class="math-container">$f(0)=1$</span> and <span... |
1,127,480 | <p>Consider the set $S = \{x \in \mathbb R: x < \frac2x\}$. Determine the value of $sup$ $S$ (if it exists). </p>
<p>Here is my attempt:<br>
Firstly, $S = \{x \in \mathbb R: 0 < x < \sqrt 2$ $\lor$ $x < -\sqrt2 \}$.<br>
Since $1 \in S$, $S \ne \emptyset$.<br>
Also, $\forall x \in S$, $x < \sqrt2$ and s... | Misto | 211,919 | <p>You shall prove that the supremum is not greater nor less than $\sqrt 2$. ie by supposing so in both cases and getting a contradiction. </p>
|
185,097 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/1/different-kinds-of-infinities">Different kinds of infinities?</a> </p>
</blockquote>
<p>Today I got to know that two infinity can be compared, But I want to know how is this possible? infinity will be in... | Thomas | 26,188 | <p>The question about several types of infinity has come up several times before. As linked to in the comments about, you can read through the question and answer given in <a href="https://math.stackexchange.com/questions/1/different-kinds-of-infinities">What Does it Really Mean to Have Different Kinds of Infinities?</... |
2,062,357 | <p>Let $\sum_{i=1}^\infty a_i$ be an absolute convergent series and let $\sum_{i=1}^\infty b_i$ be a sub-series of it, i.e.
$$b_j=c_j\cdot a_j\quad (c_j\in\{0,1\}),\qquad\forall j\in\mathbb N$$
We can say that $\sum_{i=1}^\infty b_i$ is absolutely convergent as well. So its limit, say $L$, is a real number.</p>
<p>Now... | Brian M. Scott | 12,042 | <p>Let $\mathscr{A}$ be the family of sets. Your conditions amounts to saying that the partial order $\langle\mathscr{A},\supseteq\rangle$ is a directed forest. The example in your picture is even an arborescence (directed tree):</p>
<pre><code> * light brown
... |
944,965 | <p>Question: whats the order of the element a=33 in Z60 (under modular addition)?</p>
<p>Answer: $\langle 33 \rangle= \{33,6,39,12,45,18,51,24,57,30,3,36,9,42,15,48,21,54,27,0,\}$. Therefore, $\text{order}(33)=20$</p>
<p>Whats the inverse of $33$ in $\mathbb{Z}_{60}$ (under modular addition)?</p>
<p>Im struggling wi... | paw88789 | 147,810 | <p>The inverse of $33$ in this additive group is $-33$, but in $\mathbb{Z}_{60}$, we have $-33\equiv 27\pmod{60}$</p>
|
36,625 | <p>Say $L\mathbb{C}^\times$ is the loop group of smooth maps $S^1 \to \mathbb{C}^\times$. There is a submonoid $L_{poly}\mathbb{C}^\times$ of loops that look like $w_0 + w_1z +w_2z^2 + \cdots + w_nz^n$ where $z = e^{i\theta}$ (as Andrew notes below this is not a group because its not closed under taking inverses). Eq... | David Corwin | 1,355 | <p>0) Under most reasonable notions of dimension, the (real) dimension would be $2n$. For a polynomial up to scaling is the same as an element of $\mathbb{C}^n$ (i.e. a collection of $n$ roots). We can mod out by scaling by a positive real factor, giving something of real dimension $2n-1$. When doing this modding out, ... |
36,625 | <p>Say $L\mathbb{C}^\times$ is the loop group of smooth maps $S^1 \to \mathbb{C}^\times$. There is a submonoid $L_{poly}\mathbb{C}^\times$ of loops that look like $w_0 + w_1z +w_2z^2 + \cdots + w_nz^n$ where $z = e^{i\theta}$ (as Andrew notes below this is not a group because its not closed under taking inverses). Eq... | Andrew Stacey | 45 | <p>(This is perhaps more of a comment than an answer, though I hope it also gives you some ideas where to look further.)</p>
<p>I have difficulty with the fourth word of the second sentence: "subgroup". The space that you describe is not a subgroup of $L \mathbb{C}^\times$. It is a submonoid, but is not closed under... |
1,116,202 | <p>I am taking a class in which knowledge of gradients is a prerequisite. I am familiar with gradients but don't have too much experience, so I am having trouble understanding the following example.</p>
<p>$\theta, x \in \mathbb R^d$.</p>
<p>Define $\nabla J(\theta) = \begin{pmatrix} \frac{\partial}{\partial \theta_1... | Amitai Yuval | 166,201 | <p>There are a few alternative ways to think of it, which may make it simpler to understand. One which is quite similar to the above calculation is$$\nabla(\theta\cdot x)=\nabla\left(\sum\theta_ix_i\right)=\left(\begin{array}{c}\frac{\partial}{\partial\theta_1}\sum\theta_ix_i\\\vdots\\\frac{\partial}{\partial\theta_d}\... |
2,355,246 | <p>The optimisation problem [PROBLEM 1]:
$$\max \sum_{t=T} x_t $$
subject to:
$$\ (1) x_t = (1+\alpha) x_{t-1}\ \forall t ..T$$
$$\ t=\{ 0..T\} , x_0= given$$
$$\ (e.g) x_1 = (1+\alpha) x_0\ $$
$$\ (e.g) x_2 = (1+\alpha) x_1\ $$
$$\ (e.g) x_3 = (1+\alpha) x_2\ $$ </p>
<p>$$\ $$</p>
<p>$$\ (2) x_t \ge 0 $$<... | Daron | 53,993 | <p>Just consider the upper-right quadrant of the unit circle. </p>
<p>I presume you are happy to admit there are infinitely many points in the interval $[0,1]$.</p>
<p>Each $x \in [0,1]$ is the $x$-coordinate of a point on the upper-right quadrant. </p>
<p>The $y$-value of that point is is $y= \displaystyle \sqrt{1-... |
440,744 | <p>The vector space dimension of the cohomology group of the <span class="math-container">$2$</span>-plane Grassmannian <span class="math-container">$\mathrm{Gr}_{2,n}$</span> is given by the number of tuples <span class="math-container">$(\lambda_1,\lambda_2)$</span> satisfying
<span class="math-container">$$
n - 2 \g... | Ben Webster | 66 | <p>This is a special case of geometric Satake. I have to admit, I'm really struggling to find a place where this theorem is stated in an elementary-ish statement that makes this clear and am totally failing. By <a href="https://arxiv.org/pdf/math/0401222.pdf" rel="noreferrer">Mirkovic-Vilonen, Thm. 7.3</a> (and the d... |
1,283,516 | <p>Given a triangle $ABC$, make it a point $D$ on the side $AB$. Show that $\overline {CD}$ is smaller than the length of one of the sides $BC$ and $AC$.</p>
<p>Ideas? The triangular inequality will not.</p>
<p>I wanted to try the theorem of the exterior angle and then apply a preposition that says "If two angles of ... | Jack D'Aurizio | 44,121 | <p>We may assume without loss of generality that $CA\geq CB$. Then the triangle $ABC$ lies inside the circle $\Gamma$ with centre $C$ and radius $CA$. Assuming $CD>CA$, $D$ lies outside $\Gamma$, but that leads to a contradiction, since a circle is a convex set and the segment $AB$ lies inside $\Gamma$.</p>
<p>Anot... |
1,283,516 | <p>Given a triangle $ABC$, make it a point $D$ on the side $AB$. Show that $\overline {CD}$ is smaller than the length of one of the sides $BC$ and $AC$.</p>
<p>Ideas? The triangular inequality will not.</p>
<p>I wanted to try the theorem of the exterior angle and then apply a preposition that says "If two angles of ... | grand_chat | 215,011 | <p>The triangle inequality--in Euclidean vector space--does work. Think of $A$, $B$, $C$, $D$ as vectors in $\mathbb R^d$, so that the Euclidean norm $||A-C||$ is the length of the side $AC$, and $||D-C||$ is the length of segment $DC$, and so on. By definition of $D$, we have $D=\lambda A + (1-\lambda) B$ for some $\l... |
2,273,477 | <p><strong>Bessel's Inequality</strong></p>
<p>Let $(X, \langle\cdot,\cdot\rangle )$ be an inner product space and $(e_k)$ an orthonormal sequence in $X$. Then for every $x \in X$ : $$ \sum_{1}^{\infty} |\langle x,e_k\rangle |^2 \le ||x||^2$$
where $\| \cdot\|$ is of course the norm induced by the inner product. </p>
... | Disintegrating By Parts | 112,478 | <p>Bessel's inequality is equality for some $x$ iff $x$ is in the closed linear span $\overline{[\{ e_k \}]}$ of the orthonormal elements $\{ e_n \}$. You don't have to assume that $\sum_k \langle x,e_n\rangle e_n$ converges to $x$, but that is the final conclusion if $x \in \overline{[\{ e_k \}]}$. This conclusion fol... |
349,212 | <p><strong>Context:</strong> I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts like Noether's theorem etc. in a mathematically rigorous way since the standard physics literature ... | Pedro Lauridsen Ribeiro | 11,211 | <p>I do not know if it is good form for MO to cite one's own papers when answering a question, but I will take the chance. This matter is addressed in quite a bit of detail in my joint paper with Romeo Brunetti and Klaus Fredenhagen, </p>
<ul>
<li>R. Brunetti, K. Fredenhagen, P. L. Ribeiro, <em>Algebraic Structure of ... |
2,367,788 | <p>Need to find the total numbers out of all 6 digit numbers where a digit is repeated exactly 4 times in the number. </p>
<p>Eg. 111122, 111123 is valid
But 111121 is not valid.</p>
| Henno Brandsma | 4,280 | <p>Let's construct all such numbers, starting from the left:
The first digit can be chosen in $9$ ways, as $0$ is forbidden.</p>
<p>case 1: this is also the repeated digit: then we can pick $5 \choose 3$ remaining places where that same digit will appear. And finally we are left with picking out of $9$ remaining digit... |
1,393,311 | <blockquote>
<p>Let $f:[0,1]\to[0,1]$ be a continuous function. Define $h:(0,1)\to[0,1]$ such that, $$h(x)=f(x)-\left\lfloor f(x)\right\rfloor$$Is $h$ continuous? Here $\left\lfloor x\right\rfloor$ is the <a href="https://www.google.co.in/url?sa=t&rct=j&q=&esrc=s&source=web&cd=7&cad=rja&ua... | Dominik | 259,493 | <p>Yes, $A\setminus B^\circ = A \cap (B^\circ)^c$ is compact since it is the intersection of a compact and a closed set. You could even drop any assumptions about B, as long as A is compact.</p>
|
2,826,327 | <p>$$\lim_{x\rightarrow \infty }\left ( 1+\frac{3}{x+2} \right )^{3x-6}$$</p>
<p>I've tried to factor and simplfy the expression. I got:</p>
<p>$${\left ( 1+\frac{3}{x+2} \right )^{\frac{1}{x+2}}}^{3({x^2-4})}$$</p>
<p>I set $x$ to $1/t$ I get:</p>
<p>$${\left ( 1+\frac{3}{\frac{1}{t}+2} \right )^{\frac{1}{\frac{1}... | Federico | 327,744 | <p>To get the result notice that $3x-6 = 9 \cdot \frac{x+2}{3} - 12$ then you limit is
$$
\lim_{x \to +\infty} \left[ \left( 1 + \frac{1}{\frac{x+2}{3}} \right)^{\frac{x+2}{3}} \right]^9 \cdot \left( 1 + \frac{3}{x+2} \right)^{-12}.
$$
The second term goes to $1$, while the term inside the brackets goes to $e$. Therefo... |
2,826,327 | <p>$$\lim_{x\rightarrow \infty }\left ( 1+\frac{3}{x+2} \right )^{3x-6}$$</p>
<p>I've tried to factor and simplfy the expression. I got:</p>
<p>$${\left ( 1+\frac{3}{x+2} \right )^{\frac{1}{x+2}}}^{3({x^2-4})}$$</p>
<p>I set $x$ to $1/t$ I get:</p>
<p>$${\left ( 1+\frac{3}{\frac{1}{t}+2} \right )^{\frac{1}{\frac{1}... | Doug M | 317,162 | <p>$\lim_\limits{x\to\infty}\left(1+\frac {3}{x+2}\right)^{3x-6}$</p>
<p>$y = x+2$</p>
<p>$\lim_\limits{y\to\infty}\left(1+\frac {3}{y}\right)^{3y-12}\\
\lim_\limits{y\to\infty}\left(1+\frac {3}{y}\right)^{3y}\lim_\limits{y\to\infty}\left(1+\frac {3}{y}\right)^{-12}$</p>
<p>Let't attack these separately
$\lim_\limit... |
1,163,001 | <p>I need help to prove this congruence: </p>
<p>$$ 3^n -4(2^n) + 6(1^n) + (-1)^n \equiv 0 \pmod {24} $$</p>
<p>I have tried to used Euler's Theorem on the powers of 2 and 3 individually but now I'm stuck.</p>
| Brian M. Scott | 12,042 | <p>As revised, it's still not true for $n=0$. However, it is true for all positive integers $n$, so assume that $n\ge 1$. </p>
<p>HINT: Try proving by induction that $3^n\equiv 3\pmod{24}$ when $n$ is odd, and $3^n\equiv 9\pmod{24}$ when $n$ is even. Then discover and prove a similar result for $4(2^n)$. Then put the ... |
1,691,009 | <p>I am struggling with a particular concept, I'll lay it out how I think will best allow for an answer:</p>
<p>Let $A =$ $\Bbb Z$ be the set of all integers and the universe of discourse. </p>
<pre><code>Let B, be the set of even numbers
Let C, be the set of odd numbers
Let D, be the set of positive numbers
Let E... | gordon sung | 249,633 | <p>1) $(A-B)$ is $\{x\in \mathbb Z| x\in A \cap x \notin B\} \iff \{x\in \mathbb Z| x\in A \cap x \notin 2\mathbb Z\}$ </p>
<p>This means the element $x$ found in the set is an odd integer.</p>
<p>2) $C \cap D$ means integers that are both positive and odd $\iff$ positive odd integers</p>
<p>3) $\overline{(D \cup B... |
711,922 | <p>I know there are well-ordered sets that are not countable.</p>
<p>Suppose you are given an uncountable, well-ordered set $S$.</p>
<p>Isn't it possible to provide a bijection $f:\mathbb{N} \rightarrow S$ as following?</p>
<p>$S$ is well-ordered, so it has the smallest element, say $s_1$. $S \setminus$ {$s_1$} is a... | 5xum | 112,884 | <blockquote>
<p>Suppose you are given an uncountable, well-ordered set $S$.</p>
<p>Isn't it possible to provide a bijection $f:\mathbb{N} \rightarrow S$</p>
</blockquote>
<p>No. $\mathbb N$ is countable. You will end up with a set of elements $\{s_1,s_2,s_3,\dots\}$, but there will exist elements you have not c... |
516,501 | <p>Find limit: $x_n=\dfrac{1+\frac12+...+\frac1{2^n}}{1+\frac14+...+\frac1{4^n}}$</p>
<p>as $n \rightarrow \infty$</p>
<p>My "intuition" says that it should be $\frac34$ but I don't know how to proove it with rigour.</p>
| drhab | 75,923 | <p>Hint:have a look at $\frac{1+\frac{1}{2}+\cdots+\frac{1}{2^{n}}}{1+\frac{1}{4}+\cdots+\frac{1}{4^{n}}}\times\frac{2-1}{4-1}=\frac{\left(1+\frac{1}{2}+\cdots+\frac{1}{2^{n}}\right)\left(2-1\right)}{\left(1+\frac{1}{4}+\cdots+\frac{1}{4^{n}}\right)\left(4-1\right)}$ You will 'loose' quite some terms.</p>
|
3,133,798 | <blockquote>
<p>Find an asymptotic expansion at order <span class="math-container">$6$</span> of <span class="math-container">$f(x) = \int_x^{x^2} \frac{\mathrm{d}t}{\sqrt{1+t^4}}$</span></p>
</blockquote>
<p>I don't know how to proceed. I think I need to do a change of variable yet I don't know which one. I tried <... | marty cohen | 13,079 | <p>You want
<span class="math-container">$f(x) = \int_x^{x^2} \frac{\mathrm{d}t}{\sqrt{1+t^4}}
$</span>
to order 6.</p>
<p>Being as general as possible,
let
<span class="math-container">$f(x)
= \int_{x^a}^{x^b} (1+t^u)^v dt
$</span>.</p>
<p>Then,
using the generalized binomial theorem,</p>
<p><span class="math-cont... |
618,288 | <p>I know it's been answered before (at least to the case with $n$ different eigenvalues) but I didn't find a proof for the general case, and I would like some help with this question.</p>
<p>We are given linear transforms $S,T: V\to V$ where $V$ is some vector space.</p>
<p>We are given that $S$ and $T$ commute, $ST... | user44197 | 117,158 | <p>You are almost there...</p>
<p>You have shown that the eigenspace of $S$ corresponding to $\lambda$ is invariant under $T$.
If $\lambda$ is an eigenvalue of multiplicity 1, then $Tv$ is a multiple of $v$ and hence is an eigenvalue of $T$.</p>
<p>If this is not the case, all you can say is the invariance of the eig... |
618,288 | <p>I know it's been answered before (at least to the case with $n$ different eigenvalues) but I didn't find a proof for the general case, and I would like some help with this question.</p>
<p>We are given linear transforms $S,T: V\to V$ where $V$ is some vector space.</p>
<p>We are given that $S$ and $T$ commute, $ST... | Community | -1 | <p>The result which's known for two (or more) diagonalizable matrices that commute is that are simultaneous diagonalizable, that's they are diagonalizable in the same basis. Let's prove it by induction on the dimension $\dim E$ (the result is trivial if $\dim E=1$):</p>
<p>If $S$ or $T$ is an homothetie then the resul... |
618,288 | <p>I know it's been answered before (at least to the case with $n$ different eigenvalues) but I didn't find a proof for the general case, and I would like some help with this question.</p>
<p>We are given linear transforms $S,T: V\to V$ where $V$ is some vector space.</p>
<p>We are given that $S$ and $T$ commute, $ST... | Marc van Leeuwen | 18,880 | <p>The result is <strong>not true</strong>. The linear transformations of a space of dimension$~3$ with, with respect to a fixed basis, have diagonal matrices with diagonal entries $(0,0,1)$ respectively $(0,1,1)$ satisfy all hypotheses, but they do not have an eigenspace in common.</p>
|
863,375 | <p>$f(t) = t^3 -t^2 +t + 7$.</p>
<p>Just made it up, but looking through previous tests, they come up a lot when trying to find eigenvalues. How would I easily factorize this or make it=0?</p>
<p>Wow, thanks for the quick and thorough answers. </p>
<p>The one I'm looking at is t^3 -t^2 -t +1</p>
<p>What is the fast... | Dennis Gulko | 6,948 | <p>The only general way is the Cardano-Tartaglia formula: <a href="http://en.m.wikipedia.org/wiki/Cubic_function" rel="nofollow">http://en.m.wikipedia.org/wiki/Cubic_function</a> </p>
<p>In most exam questions, it does not require that. You just have to 'guess' one root. You can do that by knowing that if a rational ... |
863,375 | <p>$f(t) = t^3 -t^2 +t + 7$.</p>
<p>Just made it up, but looking through previous tests, they come up a lot when trying to find eigenvalues. How would I easily factorize this or make it=0?</p>
<p>Wow, thanks for the quick and thorough answers. </p>
<p>The one I'm looking at is t^3 -t^2 -t +1</p>
<p>What is the fast... | Peter Woolfitt | 145,826 | <p>The example you have given only has $1$ real root and that is </p>
<p>$$t = \frac{1}{3}\left[1-\frac{2^\frac{2}{3}}{(3 \sqrt{267}-49)^\frac{1}{3}}+(2 (3 \sqrt{267}-49))^\frac{1}{3}\right]$$</p>
<p>as <a href="http://www.wolframalpha.com/input/?i=t%5E3-t%5E2%2Bt%2B7" rel="nofollow">computed by Wolfram|Alpha</a>. In... |
3,583,600 | <blockquote>
<p>Define a branch for <span class="math-container">$\sqrt{1+\sqrt{z}}$</span> and show it is analytic.</p>
</blockquote>
<p>I defined a branch <span class="math-container">$(-\pi, \pi)$</span>, and so that means the function <span class="math-container">$\sqrt{1+\sqrt{z}}$</span> is analytic on <span c... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$1+\sqrt z= -t$</span> with <span class="math-container">$t \geq 0$</span> then <span class="math-container">$z=(1-t)^{2}$</span>. The range of <span class="math-container">$(1-t)^{2}$</span> on <span class="math-container">$[0,\infty)$</span> is <span class="math-container">$[0,\inft... |
237,464 | <p>Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.</p>
<p>Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$
$$f(a;b):=\prod\limits_{k=1}^\infty f_k(a;b)$$
QUESTIONs: </p>
<p>(1) Does $f(a;b)=1$ have any solution with $a\neq 0$?</p>
<p>(2) If yes: Sing... | Iosif Pinelis | 36,721 | <p>Note that for any $a\in[-\frac12,\frac12]\setminus\{0\}$ and $b\in[0,\infty)$ one has $f_k(a;b)-1\sim a/k$ as $k\to\infty$. So, the product diverges to $0$ if $a<0$ and to $\infty$ if $a>0$. Thus, the equation $f(a;b)=1$ has no solutions with $a\neq 0$. </p>
<hr>
<p>Details on $f_k(a;b)-1\sim a/k$ as $k\to\i... |
289,757 | <p>I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician would say I have not heard a lot of math, for myself I would say that I... | Laura Balzano | 60,172 | <p>Before I was in academics, I was a software engineer for a signal processing company. I also didn't "do" a lot of math. However, I had to know how to incorporate pre-written mathematical algorithms into my own code, and I had to use my strong logic to write and debug good code. </p>
<p>Don't discount the fact that ... |
289,757 | <p>I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician would say I have not heard a lot of math, for myself I would say that I... | Michaela Light | 32,656 | <p>Math has been useful to me in software business for </p>
<ul>
<li>the ability to organize complex systems analysis into parts (aka
modules or lemmas) </li>
<li>to be able to keep working on a very complex problem when the end is not in sight</li>
<li>to have intuition when results don't look right</li>
<li>to be ve... |
2,447,850 | <p>So I have to prove 2 things:</p>
<ol>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R, x>0$. </p></li>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R$. </p></li>
</ol>
<p>For #1, I kn... | Peter Szilas | 408,605 | <p>A little bit of support for Koto:</p>
<p>For every $x \in \mathbb{R}$ the exponential series </p>
<p>$\star)$ $\exp(x): = \sum_{n=0}^{\infty}\dfrac{x^n}{n!}$ converges absolutely.</p>
<p>Hence: $\lim_{n \rightarrow \infty} |\dfrac{x^n}{n!}| = 0$ for $x \in \mathbb{R}$.</p>
<p>Proof of $\star$):</p>
<p>Ratio tes... |
3,741,122 | <p>Recently I've tried to find the difference between partial differentiation and total differentiation.
I've heard the total derivative is defined on single value functions, while the partial derivative by contrast is defined on multivariate functions.
My problem is, that total differentiation is used on multivariate ... | Christian Blatter | 1,303 | <p>In linear algebra you learn about linear transformations <span class="math-container">$A:\>{\mathbb R}^n\to{\mathbb R}^m$</span>. This <span class="math-container">$A$</span> is a single letter, but the full information content of <span class="math-container">$A$</span> is encased in an <span class="math-containe... |
1,051,357 | <p>$$
\begin{bmatrix}
-1&&2&&-3&&4\\
5&&0&&2&&-2\\
2&&1&&1&&2\\
0&&0&&3&&-2\\
\end{bmatrix}
$$</p>
<p>I wanted to confirm that if I use $(w, x,$$ y, z)$ would the image of the vector be:</p>
<pre><code>-w + 2x - 3y + 4z
5w... | datalava | 187,292 | <p>There are many things that can tell you whether or not a matrix is invertible (see <a href="http://en.wikipedia.org/wiki/Invertible_matrix#The_invertible_matrix_theorem" rel="nofollow">The Invertible Matrix Theorem</a>.) </p>
<p>One of these is whether the matrix has full rank. The reduced row echelon form can tell... |
4,029,725 | <p>In the book by Artificial Intelligence by Norvig and Russel, I came across following problem:</p>
<blockquote>
<p>Prove if correct: <span class="math-container">$(A ∧ B) \models (A ⇔ B).$</span></p>
</blockquote>
<p>I quickly interpreted <span class="math-container">$\models$</span> as <span class="math-container">$... | ryang | 21,813 | <blockquote>
<p><strong>(a)</strong> After reading Graham's answer, I felt that the truth table above is proving the "tautology" <span class="math-container">$(A ∧ B) \implies (A ⇔ B)$</span>, but not <span class="math-container">$(A ∧ B) \models (A ⇔ B)$</span>.<br />
<strong>(b)</strong> To prove <span clas... |
830,599 | <p>The function $f$ is defined as follows:
$$f(x):=\sum_{j=1}^{\infty} \frac{x^j}{j!} e^{-x}$$</p>
<p>It's easy to see that $f(0)=0$. But I am interested in the value
$$\lim_{x \rightarrow 0^+} f(x).$$</p>
<p>Even <a href="https://www.wolframalpha.com/input/?i=lim_%28x-%3E0%29+%28sum_%28i%3D1%29%5Einfinity+x%5Ej%2F%... | Michael Albanese | 39,599 | <p><strong>Hint:</strong> Note that $$f(x) = \sum_{j=1}^{\infty}\frac{x^j}{j!}e^{-x} = e^{-x}\sum_{j=1}^{\infty}\frac{x^j}{j!} = e^{-x}\left(\sum_{j=0}^{\infty}\frac{x^j}{j!}-1\right).$$</p>
|
13,829 | <p>I was trying to understand the notion of a connection. I have heard in seminars that a connection is more or less a differential equation. I read the definition of Kozsul connection and I am trying to assimilate it. So far I cannot see why a connection is a differential equation. Please help me with some clarificati... | Jonas Meyer | 1,424 | <p>Another method: You could write it as the sum of the integrals on the intervals $\left[\sqrt{\frac{\pi}{2}+k\pi},\sqrt{\frac{\pi}{2}+(k+1)\pi}\right]$, and make a substitution $u=x^2$ to bound the integral on such an interval below by $\frac{1}{\sqrt{\frac{\pi}{2}+(k+1)\pi}}$. (I'm ignoring the interval $\left[0,\s... |
2,439,324 | <blockquote>
<p>Does there exist an uncountable set $A \subset \mathbb{R}$, such that for every $a \in A$ and every $\epsilon>0$, $(a-\epsilon,a+\epsilon)\not\subset A$?</p>
</blockquote>
<p>I am not sure what the answer is, but I am having trouble trying to construct such a set. Any hints?</p>
| Community | -1 | <p><a href="https://en.wikipedia.org/wiki/Cantor_set" rel="nofollow noreferrer">This</a> totally disconnected set is also an example.</p>
|
73,675 | <p>Dear all,</p>
<p>I'm seeking a reference for a claim made in lecture 8 of Jacob Lurie's chromatic homotopy theory notes (<a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf" rel="noreferrer">http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf</a>). More particularly, Theorem 6 of this lecture s... | S. Carnahan | 121 | <p>MIT OpenCourseWare has some <a href="http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/" rel="nofollow">notes from a course</a> that Lurie taught in 2007. I believe <a href="http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-top... |
3,351,012 | <p><span class="math-container">$\int e^{2\theta}\sin(3\theta)d\theta$</span> seems to be leading me in circles. The integral I get when I use integration by parts, <span class="math-container">$\int e^{2\theta}\cos(3\theta)d\theta$</span> just leads me back to <span class="math-container">$\int e^{2\theta}\sin(3\theta... | Gabe | 449,093 | <p>Take your examples together, <span class="math-container">\begin{align*}
\int e^{2\theta}\sin(3\theta)d\theta
&=\frac{1}{2}e^{2\theta}\sin(3\theta)- \frac{3}{2} \left(\frac{1}{2}e^{2\theta}\cos(3\theta)+ \frac{3}{2} \int e^{2\theta}\sin(3\theta)d\theta\right)
\end{align*}</span>
Substituting the integral for a v... |
3,351,012 | <p><span class="math-container">$\int e^{2\theta}\sin(3\theta)d\theta$</span> seems to be leading me in circles. The integral I get when I use integration by parts, <span class="math-container">$\int e^{2\theta}\cos(3\theta)d\theta$</span> just leads me back to <span class="math-container">$\int e^{2\theta}\sin(3\theta... | farruhota | 425,072 | <p>Another method is to predict the answer:
<span class="math-container">$$\int e^{2x}\sin(3x)dx=Ae^{2x}\sin (3x)+Be^{2x}\cos (3x)+C \Rightarrow \\
e^{2x}\sin (3x)=2Ae^{2x}\sin (3x)+3Ae^{2x}\cos (3x)+2Be^{2x}\cos (3x)-3Be^{2x}\sin (3x) \Rightarrow \\
\begin{cases} 2A-3B=1\\ 3A+2B=0\end{cases}\Rightarrow A=\frac2{13};B=... |
2,385,152 | <p>Given a matrix A, e.g.
$$
A=\begin{bmatrix}
a_{11}& a_{12} & a_{13} \\
a_{21}& a_{22} & a_{23} \\
a_{31}& a_{32} & a_{33} \\
\end{bmatrix}
$$
eliminating the row and the column corresponding $a_{21}$ results in a smaller matrix
$$
B=\begin{bmatrix}
a_{12} & a_{13} \\
a_{32} & a_{... | Dave | 334,366 | <p>This is a type of submatrix that one obtains by removing a select number of rows and columns. In your case, you eliminated the second row and first column, so a common notation is that this is a $(2,1)$-submatrix of $A$. Minors and cofactors are obtained through determinants of certain submatrices. Check out this <a... |
1,019,408 | <p>Usually, $f$ denotes a function, $f(x)$ is an image of $x$ under $f$. But what's $f(X)$ if $X$ is a set? </p>
<p>edit: Please, disregard the body of this question. I had to put something here to be able to post the question. </p>
| Ashwin Ganesan | 157,927 | <p>If $f(S)=S$, we say $f$ fixes the set $S$ setwise. If $f(s)=s$, for all $s \in S$, then we say $f$ fixes the set $S$ pointwise. The terminology is intuitive and makes it clear that the latter condition is stronger than and implies the former. But the former does not imply the latter; for eg, take $f$ to be the bi... |
516,244 | <p>My professor gave us this example on her notes:</p>
<p>$$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$</p>
<p>So I know we're supposed to find the partial fraction, which ends up being</p>
<p>$$\left(\frac{3}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}=
\frac{1}{n}-\frac{1}{n+3}\right)$$</p>
<p>So ba... | Caleb Stanford | 68,107 | <p>Heron's formula is inefficient; there is in fact a direct formula. If the triangle has one vertex at the origin, and the other two vertices are $(a,b)$ and $(c,d)$, the formula for its area is
$$
A = \frac{\left| ad - bc \right|}{2}
$$</p>
<p>To get a formula where the vertices can be anywhere, just subtract the c... |
516,244 | <p>My professor gave us this example on her notes:</p>
<p>$$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$</p>
<p>So I know we're supposed to find the partial fraction, which ends up being</p>
<p>$$\left(\frac{3}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}=
\frac{1}{n}-\frac{1}{n+3}\right)$$</p>
<p>So ba... | Michael Hoppe | 93,935 | <p>Now here's a solution which works in any vector space with an inner product: take the half of the root of the Gram-Determinant of two sides of the triangle, that is
<span class="math-container">$$\frac12\sqrt{\det\begin{pmatrix}\langle b-a,b-a\rangle&
\langle b-a,c-a\rangle\\
\langle b-a,c-a\rangle & \langle... |
1,743,465 | <p>If $a=(1,2,3,4,5)$ is an example of a vector in $\mathbb R^5$, what could be an example of a vector in $\mathbb C^5$ ? is it $a=(1,2,3,4,i5)$ ?</p>
<p>Also, $x=a+ib$ is $2$ dimensional, can a complex number be one dimensional? like when $a=0$ or $b=0$, but if $b=0$ then it is a real number, so can we say that all... | Surb | 154,545 | <p>$$(1,i,1+i,3+2i,7i,\pi+\sqrt 2i)$$</p>
|
482,030 | <p>I'm reading a proof of the irrationality of <span class="math-container">$\sqrt 2$</span>. In a step it states that <span class="math-container">$2d^2=n^2$</span> implies that <span class="math-container">$n$</span> is multiple of <span class="math-container">$2$</span>. How?</p>
| Ömer | 55,199 | <p>square of even numbers are even, so $n$ is even</p>
|
4,020,412 | <p>I was reading about ODE variable separation solving and the book says that assuming that a function can be expressed as the product of two single-variable functions loses generality, which I understand. I cannot, however, prove that it does. For example: the function <span class="math-container">$\sqrt{x+t}$</span> ... | mathcounterexamples.net | 187,663 | <p>If <span class="math-container">$f(x,t) = \sqrt{x+t} = X(x) \cdot Y(t)$</span> for all <span class="math-container">$x, t \ge 0$</span> then</p>
<p><span class="math-container">$$X(x)\cdot Y(0) = \sqrt{x} \text{ and } X(0) \cdot Y(t) = \sqrt t$$</span></p>
<p>But <span class="math-container">$X(0) \cdot Y(0) = \sqrt... |
918,509 | <p>Let $X$ be a random variable with pdf $f$. I would like to know why:</p>
<p>$$\operatorname{E} [X] = \int_\Omega X \, \mathrm{d}P = \int_\Omega X(\omega) P(\mathrm{d}\omega)= \int_{-\infty}^\infty x f(x)\, \mathrm{d}x . $$</p>
<p>I mean I don't get it why it is all equal and what the notation in third term mean... | almagest | 172,006 | <p>Define $a(n,1)=n(1+2+\dots+(n+1))=n(n+1)(n+2)/2$. Then define recursively $$a(n,k)=n\sum_{r=1}^{n+1}a(r,k-1)$$ Then your sequence is $a(1,1),a(1,2),a(1,3),\dots$. So entering that into Mathematica, we get $3, 15, 105, 945, 10395, 135135,\dots$. Checking with oeis.org we find these are just the numbers $1\times3\time... |
507,133 | <p>I came across a problem in a book that asked us to find the first number $n$ such that $\phi(n)\geqslant 1,000$ it turns out that the answer is 1009, which is a prime number. There were several questions of this type, and our professor conjectured that it will always be the next prime. However, no one has been able ... | Will Jagy | 10,400 | <p>Initial response: this should follow from a fairly mild conjecture on <a href="http://en.wikipedia.org/wiki/Prime_gap" rel="noreferrer">prime gaps</a>, that there for a prime $p \geq 127,$ there is always a prime $q$ with $p < q < p + \sqrt p.$ The last known bad one is $113,$ as $\sqrt {113} \approx 10.63,$ t... |
2,066,765 | <p>Are there any examples of subspaces of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?</p>
| Orest Bucicovschi | 378,410 | <p>As an observation, every subspace generated by countably many linearly independent vectors (that is, a subspace of dimension $\aleph_0$) inside a Banach space is not closed, since a Banach space cannot have (algebraic) dimension $\aleph_0$. ( if it were closed, if would be itself a Banach space).</p>
|
1,940,837 | <p>I don't understand the difference between $\bigoplus_{i=1}^n M_i$ and $\prod_{i=1}^n M_i$ where $\{M_i\}_{i=1,...,n}$ is a collection of $R-$module. When $I$ is not finite, $$\prod_{i\in I}M_i=\{(x_i)_{i\in I}\mid x_i\in I\}$$ and $$\bigoplus _{i\in I}M_i=\{(x_i)_{i\in I}\mid x_i=0\text{ except for finite number of ... | Zelos Malum | 197,853 | <p>For finite cases they are identical and either and both can be used interchangably without problem. The reason why the notation exists is because for the infinite case they are NOT the same. One is the universal product construct, the other is the coproduct. Both of them garantuees the existence of a homomorphism of... |
2,452,135 | <p>I need to prove the set
$A=\left \{ \left ( r\cos t,r\sin t \right ) \mid 0\leq t\leq \theta ,r\geq 0\right \}$ is closed. I have tried to use the theorems for continuous functions, but I always into troubles cause the domain is not bounded. </p>
| Mariam | 486,284 | <p>hint: a set is closed if its complementary is open.</p>
|
2,452,135 | <p>I need to prove the set
$A=\left \{ \left ( r\cos t,r\sin t \right ) \mid 0\leq t\leq \theta ,r\geq 0\right \}$ is closed. I have tried to use the theorems for continuous functions, but I always into troubles cause the domain is not bounded. </p>
| Alex Provost | 59,556 | <p>Pick any point $p \in \mathbb{R}^2 \setminus A$. (If such a point does not exist, then $A = \mathbb{R}^2$ is closed.) Let $d > 0$ be the smallest distance between this point and the rays $t = 0$, $t = \theta$. This choice of $d$ ensures that the distance between $p$ and any point in the sector $A$ is at least $d$... |
73,219 | <p>For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a simply connected Riemann surfaces is conformally equivalent to one of the following:</p>
<ul>
<li>Riemann Sphere $\math... | Sam Nead | 1,650 | <p>NEW ANSWER:</p>
<p>As there has been much confusion on this point (some of it mine...): </p>
<blockquote>
<p>Definition: A Riemannian 2-manifold $S$ is of <em>hyperbolic type</em> if the universal cover of $S$ is conformally equivalent to the open unit disk, $D$. </p>
</blockquote>
<p>On the other hand we have<... |
180,495 | <blockquote>
<p>Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all $x,y\in\mathcal{H}$. Prove that $T$ is a bounded directly from the uniform boundedness principle and not the closed grap... | Nate Eldredge | 822 | <p>Hint: Consider the family of linear functionals $f_x$ defined by $f_x(y) = \langle Tx,y \rangle$, as $x$ ranges over the unit ball of $\mathcal{H}$.</p>
|
766,765 | <blockquote>
<p>Let $f:[0,\infty]\rightarrow \mathbb{R}$ a bounded function in each bounded interval. If $\lim_{x\rightarrow \infty} [f(x+1)-f(x)]=L$ then prove that $\lim_{x\rightarrow \infty}\displaystyle\frac{f(x)}{x}=L$</p>
</blockquote>
<p>I appreciate any hint to solve this problem. </p>
<p>Thanks a lot!</p>
| user21820 | 21,820 | <p>For any $a > 0$:</p>
<p>  Let $y$ be such that $f(x+1) - f(x) \in L + [-a,a]$ for any $x \ge y$</p>
<p>  Then $f(y+t+k) - f(y+t) \in k ( L + [-a,a] )$ for any $k \in \mathbb{N}$ and $t \in [0,1]$</p>
<p>  Also $f(y+t)$ is bounded over all $t \in [0,1]$</p>
<p>  Thus ${\large \frac{f(y+t+k)... |
2,602,438 | <p>For an embedded software implementation I would like to compute,</p>
<p>$S(b) = \sum_{i=1}^{N}log( x_i - b )$,</p>
<p>for various values of $b$. Here $x_i$ is an array of fixed numbers.</p>
<p>Is there a fast way to do this without having to recompute the sum?</p>
<p>--</p>
<p>I tried looking at the Taylor expa... | Community | -1 | <p>$x^2\not \ge 0\implies $ $x $ is not a real number...</p>
|
184,241 | <p>I am having issues with the code:</p>
<pre><code>CC = p0^2/(p0 - L)
Dbuy = (1 - CC)*(L/p0) + (CC - p0)*Log[CC/p0]
Dwait = (1 - p1)*(L/(p1 - p0))
Dlock = (1 - p1)*(1 - L/p0 - L/(p1 - p0)) + (p1 - CC)*(1 - L/p0) - L*Log[(p1 - p0)/(CC - p0)]
RevLocking1 = Table[NMaximize[{x*Dbuy*p0 + x*Dwait*0.4*p1 + x*Dlock*(p0 + 0.4... | Carl Woll | 45,431 | <p>You can use the undocumented <code>AllowedHeads</code> option of <a href="http://reference.wolfram.com/language/ref/Transpose" rel="noreferrer"><code>Transpose</code></a> to do this:</p>
<pre><code>tr = Transpose[listData, AllowedHeads -> {Association, List}]
</code></pre>
<blockquote>
<p><|"Input" -> {1, ... |
64,491 | <p>I need to change the ticks in the xaxis for representation pH values, but the ticks are not dispersed along the range of xaxis. Another question: why the vertical line defined with Epilog is not visible?</p>
<pre><code>Kb1 := Kw/Ka1
Kb2 := Kw/Ka2
oh := Kw/x
f[x_] := Cs Kb2 oh/(Kb2 oh + oh^2 + Kb1 Kb2)
ff[x_] = Simp... | ubpdqn | 1,997 | <p>In version 10,</p>
<pre><code>LogLinearPlot[ff[x], {x, 10^-15, 15 10^-10}, Frame -> True,
FrameTicks -> {{Automatic,
Automatic}, {Table[{10^j, -j}, {j, -15, -9}], None}},
FrameLabel -> {"pH", "[HA]"},
GridLines -> {{{1.725 10^-10, Red}}, None}]
</code></pre>
<p><img src="https://i.stack.imgu... |
3,072,198 | <p>How can one explain that <span class="math-container">$$\frac{d}{dx}\left(\int_0^x{\cos(t^2+t)dt}\right) = \cos(x^2+x)$$</span>
Without solving the integral?</p>
<p>I know it's related to the fundamental theorem of calculus, but here we have a derivative with respect to <span class="math-container">$x$</span>, whil... | Parcly Taxel | 357,390 | <p>Say <span class="math-container">$f(t)=\cos(t^2+t)$</span> and <em>an</em> antiderivative is <span class="math-container">$F(t)$</span>. The integral in question is, by the fundamental theorem of calculus,
<span class="math-container">$$F(x)-F(0)$$</span>
<span class="math-container">$F(0)$</span> is a constant and ... |
3,552,264 | <p>How to find function <span class="math-container">$f(x)$</span> that has continuous derivative on <span class="math-container">$[0,2]$</span> satisfies the following conditions:</p>
<ol>
<li><span class="math-container">$f(2) = 3$</span></li>
<li><span class="math-container">$\displaystyle \int_0^2 [f'(x)]^2 dx = 4... | Fred | 380,717 | <p>We have, by Cauchy - Schwarz:</p>
<p><span class="math-container">\begin{align}
23 & =\int_0^2 x^3f'(x) \, dx \le \left(\int_0^2 x^6 dx\right)^{1/2} \cdot \left(\int_0^2 (f'(x))^2 \, dx \right)^{1/2} \\
& =2 \left(\int_0^2 x^6 \, dx\right)^{1/2} \le 2 \left(\int_0^2 2^6 \, dx \right)^{1/2} =\sqrt{2} \cdot 16... |
286,989 | <p>There is a very simple formulation for the character of irreducible representations of $S_n$ evaluated on an n-cycle, i.e. that it is 0 on all non-hook partitions, and $(-1)^m$ on hooks. Is there an analogous computation for irreducible characters of $B_n$, the hyperoctahedral group, evaluated on signed 2n-cycles? T... | Richard Stanley | 2,807 | <p>In general, if $(\lambda,\mu)$ is a bipartition of $n$, then
$$ \prod_i(p_{\lambda_i}(x)+p_{\lambda_i}(y))\cdot\prod_j
(p_{\mu_j}(x)-p_{\mu_j}(y)) = \sum_{(\alpha,\beta)}
\chi^{\alpha,\beta}(\lambda,\mu)s_\alpha(x)s_\beta(y), $$
where $(\alpha,\beta)$ ranges over all bipartitions of $n$ and ... |
53,596 | <p>It's "well known" that, for any weight $k$ and level $N$, the space $S_k(\Gamma_1(N))$ of cusp forms of that weight and level has a basis in which all the Hecke operators act by matrices with entries in $\mathbb{Z}$; consequently all the Hecke eigenvalues are algebraic numbers (indeed algebraic integers).</p>
<p>I ... | Socky | 33,127 | <p>Let $S = S_{\mathbf{Q}} = M_{13}(\Gamma_1(N),\mathbf{Q})$, and $S_{\mathbf{C}} = S \otimes \mathbf{C}$ denote the corresponding space of modular forms over $\mathbf{C}$.</p>
<p>Let $V \subset S \times S$ be the subspace cut out by pairs of forms $(A,B)$ satisfying the following equation:</p>
<p>$$A \cdot E_{12} = ... |
2,975,505 | <p>Assume that g : <strong>R</strong> → <strong>R</strong> is a bijection and define f : <strong>R</strong> → <strong>R</strong> by</p>
<p>f(x) = 2g(x) + 1.</p>
<p>Determine, with proof, whether f is a bijection.</p>
<p>My opinion:</p>
<p><em>I know that a function should either be increasing or decreasing in inter... | user29418 | 386,180 | <p>The continuous function <span class="math-container">$h(x) = \pi + \frac{1}{2}\sin \left ( \frac{x}{2}\right) - x $</span> is positive at 0 and negative at <span class="math-container">$2\pi$</span>, the interval being <span class="math-container">$[ \pi, -\pi ]$</span>, so by the Intermediate Value Theorem there is... |
970,872 | <p>There is a common brain teaser that goes like this:</p>
<p>You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: if you light one end of the rope, it will take exactly one hour to burn all the way to the other end. But it d... | Ralph D. Jeffords | 826,037 | <p>Here is my proof that no matter how much the rate of burning varies over the time interval <span class="math-container">$[0, 1]$</span> as long as the time to burn when lit at one end is 1 hour then lighting the rope simultaneously at both ends will consume the rope after 1/2 hour.</p>
<p>Formally, the amount the ro... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Beren Sanders | 1,148 | <p>Terry Tao wrote on this subject; I think you will be happy with his <a href="http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/" rel="noreferrer">conclusions</a>.</p>
<p>An excerpt: "The number of interesting mathematical research areas and problems to work on is vast – far mo... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | G. Jay Kerns | 10,351 | <p>You don't have to be Michael Jordan to play basketball.</p>
|
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