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 |
|---|---|---|---|---|
3,728,707 | <p>I would like to solve this linear diophantine equation:
<span class="math-container">$$
40x_1+296x_2+945x_3+2048x_4+4500x_5+8640x_6=616103
$$</span>
All the answers have to be an integer number in the interval <span class="math-container">$\{[10] \cup [29,95]\}$</span>.</p>
<p>As a first step, I started by finding a... | Jan Eerland | 226,665 | <blockquote>
<p>Not a 'real' answer, but it was too big for a comment.</p>
</blockquote>
<p>I wrote and ran some Mathematica code:</p>
<pre><code>In[1]:=FullSimplify[
Solve[{40*x1 + 296*x2 + 945*x3 + 2048*x4 + 4500*x5 + 8640*x6 ==
616103, 1 <= x1 <= x2 <= x3 <= x4 <= x5 <= x6 <= 1000}, {x1, x... |
192,391 | <p>I know that this is sometimes the case, but that some matrices are not tensors.
So what is the intuitive and specific demands of a matrix to also be a tensor?
Does it need to be quadratic, singular or something else?</p>
<p>Some sources I read seem to suggest that all rank 2 matrices are tensors while other just cl... | Jim Belk | 1,726 | <p>This question doesn't have a single good answer, because there isn't a universally agreed upon definition of "tensor" in mathematics. In particular:</p>
<ol>
<li><p>Tensors are sometimes defined as multidimensional arrays, in the same way that a matrix is a two-dimensional array. From this point of view, a matrix... |
3,232,998 | <p>In Velleman's How to Prove It, the strategy given for proving goal of the form <span class="math-container">$P \lor Q$</span> goes like this:</p>
<blockquote>
<p>If <span class="math-container">$P$</span> is true, then clearly <span class="math-container">$P \lor Q$</span> is true. Now suppose <span class="math-c... | 雨が好きな人 | 438,600 | <p><em>Any</em> statement in <em>any</em> proof is redundant if you consider it ‘too obvious’.</p>
<p>In this case, maybe you consider it obvious that <span class="math-container">$P \implies P \lor Q$</span> (and I wouldn’t blame you). It does follow directly from the definition of <span class="math-container">$\lor$... |
3,232,998 | <p>In Velleman's How to Prove It, the strategy given for proving goal of the form <span class="math-container">$P \lor Q$</span> goes like this:</p>
<blockquote>
<p>If <span class="math-container">$P$</span> is true, then clearly <span class="math-container">$P \lor Q$</span> is true. Now suppose <span class="math-c... | matt | 401,452 | <p>I think that those are two different ways of proving the same thing. If we want to prove that P or Q is true, we can prove that P is true, or we can prove that Q is true. We could also assume P is false and show that it implies Q is true. Then we would have proven the disjunction as it is logically equivalent to the... |
1,294,792 | <p>Can a 2-dimensional sphere be decomposed into two disjoint homeomorphic subspaces? If yes, can these subspaces be non-discrete / connected / have some other good properties?</p>
| Chappers | 221,811 | <p>You can actually prove this using Fourier series: consider the Fourier series of $\cos{\alpha x}$ on $[-\pi,\pi]$. The function is continuous, and differentiable except at the endpoints, so the Fourier series converges to the function everywhere. Some easy integration will show you that the Fourier coefficients are ... |
1,342,951 | <p>I was thinking what function I should compare it to. If I say whether a function is smaller or bigger than this one, then I must prove that. I was thinking of (x+1)^2 but I realized that this converges so that won't prove the integral diverges. So, what function should I use?</p>
| Michael Hardy | 11,667 | <p>It would be circular only if you were to take the equality $P(A,B) = P(A \cap B) = P(A \mid B)P(B)$ to be a definition. So don't do that.</p>
|
47,728 | <p>Function <code>CreateTemporary</code> can generate a unique file name like</p>
<pre><code>file = CreateTemporary[]
(* "C:\Documents and Settings\kkouptsov\Local \Settings\Temp\m-263a0380-3a03-49ac-95cb-d21390b2d3fd" *)
</code></pre>
<p>How do we generate a unique file name such as above example? The reason I don't... | halirutan | 187 | <p>If it is merely about creating something unique, why don't you use the current time to build a readable file-name which is unique?</p>
<pre><code>DateString[{"file", "Year", "-", "MonthNameShort",
"Day", "-", "Hour24", "Minute", "Second", ".txt"}]
(* "file2014-May13-013029.txt" *)
</code></pre>
<p>Since it cont... |
786,187 | <p>I can't seem to figure this one out:</p>
<p>$\int \frac{t^5}{\sqrt{t^2+2}}dt$</p>
<p>$t^5 / \sqrt{t^2+2}$</p>
<p>I know I need to substitute t for a trigonometric function, $\tan$ I think</p>
<p>Any hints are greatly appreciated!</p>
| David H | 55,051 | <p>Alternatively, you don't necessarily need trigonometric substitution to solve this problem.</p>
<p>Substituting $u=t^2+2$,</p>
<p>$$\int \frac{t^5}{\sqrt{t^2+2}} dt = \int \frac{(u-2)^2}{2\sqrt{u}}du\\
=\int\frac12u^{3/2}du-\int 2\sqrt{u}\,du+\int\frac{2}{\sqrt{u}}du\\
=\frac15u^{5/2}-\frac43 u^{3/2}+4\sqrt{u}+con... |
193,355 | <p>I am taking a calculus course, and in one of the exercises in the book, I am asked to find the limits for both sides of $\sqrt{x}$ where $x \to 0$.</p>
<p>Graph for sqrt(x) from <a href="http://www.wolframalpha.com/input/?i=sqrt%28x%29" rel="nofollow noreferrer">WolframAlpha</a>:<br>
<img src="https://i.stack.imgur... | gerardlouw | 29,729 | <p>The answers to your questions depend on whether you are working with $\mathbb{R_{\geq 0}}$ or $\mathbb{R}$ as your domain. $\sqrt{x}$ is really not the same function in these two cases.</p>
<p>With a domain of $\mathbb{R_{\geq 0}}$:</p>
<ul>
<li>Clearly $\lim_{x \to 0^+} \sqrt{x} = 0$. However $\lim_{x \to 0^-} \s... |
2,612,538 | <p>I want to find the Eigenvalues and Eigenvectors of the following matrix $$A=\begin{pmatrix}0&1\\-3&0\end{pmatrix}.$$</p>
<p>I'm able to calculate the Eigenvalues, which are simply the zero points of A's characteristic polynom $\lambda^2 + 3$:<br>
$$\lambda_{1,2} = \pm i\sqrt{3}\\$$ </p>
<p>But how do I cal... | amd | 265,466 | <p>I suggest checking your input to Wolfram|Alpha since neither vector that you’ve listed in your question is an eigenvector of $A$. For instance, $A[i\sqrt3,1]^T=[1,-3i\sqrt3]^T$, which is not a scalar multiple of $[i\sqrt3,1]^T$.</p>
<p>That aside, you should expect to end up with $0=0$ when you combine the two equa... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| S. P. Forman | 236,021 | <p>He meant that when you think you understand math, you have in fact fallen under the sway of an <em>illusion</em>. This illusion of understanding is brought on by familiarity ("getting used to it") coupled with intellectual laziness. People get used to doing math and having it work and they forget that no one knows... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| Len Margolin | 869,981 | <p>Poincare once wrote that all of mathematics is one plus one equals two, and a lot of definitions. Speaking as a physicist, I believe the most important thing to understand about mathematics is whether those definitions are useful in describing reality (i.e., observations). I don't believe mathematicians are similar... |
522,179 | <p>Everyone says that the dot product is interpreted as the projection of A onto B (if you are dot producting A and B), but isn't that length just equal to |A|$\cos \left( \theta \right)$? Why does the dot product have an extra |B|?</p>
| Aaron Golden | 37,535 | <p>If we didn't account for the length of $B$ then the order of the operands to the dot product would matter. $A$ dot $B$ would not be equal to $B$ dot $A$, and that would be irritating.</p>
<p>I suppose you could think of taking the dot product of $A$ and $B$ as going in three steps. First scaling $B$ to be unit le... |
2,153,580 | <p>I'm trying to prove that $K((x))/K(x)$ is not an algebraic extension, where $K$ is a field of characteristic $p>0$, and $K((x))$ is the field of fractions of $K[[x]]:=\{\sum_{i=0}^{\infty}a_ix^i\mid a_i\in K\}$.</p>
<p>To make it more concrete, I'm considering the case $K=\mathbb{Z}/p\mathbb{Z}$ and $x=p$, so th... | Sungjin Kim | 67,070 | <p>Maybe showing that a certain element is transcendental is not easy. I think that about the same or more work is required as to prove that $e^p\in\mathbb{R}$ is transcendental. </p>
<p>One way to get around is to use the cardinality comparison. This way is not constructive, but we can prove that there is $\tau\in\ma... |
65,865 | <p>There are so many different optimization algorithms out there, and lots of research going on. However, I have difficulties to find good comparison between them, and all articles / books / papers seem to evaluate them in different ways.</p>
<p>Isn't there some set of standard functions to run "standardized" benchmar... | H. Vabri | 410,305 | <p>Due to optimisation problems varying considerably you might want to creating a library of test problems that somewhat represent the problems you are trying to solve, optimising these using multiple optimisation algorithms and comparing their performance. For that an increasingly accepted method to compare optimisati... |
779,328 | <p>I am almost a beginner in the topics of Fourier transform. So, I am asking this question here.</p>
<p>Let <span class="math-container">$n=3$</span> and let <span class="math-container">$\mu_t$</span> denote surface measure on the sphere <span class="math-container">$|x|=t$</span>. Then how do we show that
<span cla... | Giuseppe Negro | 8,157 | <p>It is easy if we use Kirchhoff's formula for the solution to the following wave equation:
\begin{equation}
\begin{cases}
(\partial^2_t-\Delta)u=0, & x\in \mathbb{R}^3,\ t\in \mathbb{R}\\
u(0, x)=0 \\
u_t(0, x)=f
\end{cases}
\end{equation}
Kirchhoff's formula says that
\begin{equation}
u(t,x)=\frac{1}{4\pi t} \... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | guest | 13,465 | <p>The simple answer I would give is that there is a good chance they go into the business world at SOME TIME in the future of their life. Perhaps they will get an MBA at some point to help them move up. And you'd laugh at how deer in the headlights the English major types were with finance math. And how the STEM ma... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | Lawnmower Man | 13,468 | <h1>Citizenship</h1>
<p>The health of a democracy depends on the electorate being thoughtful and <em>informed</em>. And said citizens can only be informed if they understand the information around them (e.g., the news). Very few policies boil down to: "A is morally right and B is morally wrong." Those issues were m... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | Reese Johnston | 10,081 | <p>Math, taught well, teaches modes of thought, not facts. A person who is exposed to a wider variety of modes of thought will be more prepared to critically consider new ideas. It's the same reason we teach history and literature - I can't honestly say that I've had reason to use my middle-school knowledge of Macbeth ... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | A. I. Breveleri | 13,476 | <p>Many people think that the clientele of our public school system is <em>the students</em>. Others act as if the clientele were the <em>parents</em> of the students. </p>
<p>Those people are wrong. The client is the <em>citizenry of his state and nation</em>, who need the electorate to be educated and informed, in o... |
473,972 | <p>Let $X$ be an infinite set. Prove that there is a bijective function $f: X \rightarrow X$ with the property that for every $x \in X$ and all $n > 0$: $f^n(x) \neq x$.</p>
<p>I've tried to proved this by considering a bijective function $g: \mathbb{Z} \times X \rightarrow X$ in a certain way (by the composition ... | Community | -1 | <p><strong>Hint</strong>: By changing the order of integration:</p>
<p>$$\int_0^1 \int_x^1 e^{x/y} dy dx = \int_0^1 \int_0^y e^{x/y} dx dy$$</p>
<p>Now the inside integral can be evaluated by</p>
<p>$$\int_0^y e^{x/y} dx = y e^{x/y} |_0^y = y e^1 - y e^0$$</p>
|
72,994 | <p>I am not able to solve this question from chapter "Similar & Congruent Triangles" in my book.</p>
<p>Can some one help to calculate <strong>AC</strong>?</p>
<p><img src="https://i.stack.imgur.com/5ZTn9.jpg" alt="HERE">.</p>
| Ajay Kumar | 375,350 | <p><a href="https://i.stack.imgur.com/XCTvV.png" rel="nofollow noreferrer">enter image description here</a> using triangle similarity
here the triangle 1 is similar to the triangle 2, also the triangle 2 is similar to the triangle 3</p>
|
2,932,510 | <p><a href="https://i.stack.imgur.com/9kTCs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9kTCs.png" alt="enter image description here"></a>
Apostol tells the reader to verify the last sentence.
I've verified the backward implication (<span class="math-container">$a_kx+b_k=0\implies \text{equality}... | Siong Thye Goh | 306,553 | <p>The input to <span class="math-container">$f$</span> is <span class="math-container">$2x+6$</span>,</p>
<p>hence <span class="math-container">$$y = -2(2x+6)^4 - 4$$</span></p>
<p>We can also write it as </p>
<p><span class="math-container">$$y = -2(2(x+\color{red}3))^4 - 4$$</span></p>
<p>since <span class="math... |
123,675 | <p>What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the theory I would like to cover, however, I would love some more nice examples and applications. Of course, I have some ideas... | Andrej Bauer | 1,176 | <p>Here is one that is not too grand.</p>
<p>Suppose you want to embed some (essentially small) category of spaces into a cartesian-closed category, so as to extend it to a <a href="http://ncatlab.org/nlab/show/convenient+category+of+topological+spaces" rel="nofollow">convenient category</a>. Then you can use a gros t... |
4,090,775 | <p>Let <span class="math-container">$\mu$</span> be the Lebesgue measure and, <span class="math-container">$E$</span> and <span class="math-container">$F$</span> compact subsets of <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$E\subset F$</span> and <span class="math-container... | DanielWainfleet | 254,665 | <p>There seems to be a new style of teaching measure theory without mentioning <span class="math-container">$inner$</span> measure. Let <span class="math-container">$m^o$</span> denote outer measure.</p>
<p>For <span class="math-container">$S\subseteq \Bbb R,$</span> let <span class="math-container">$C(S)$</span> be th... |
2,315,599 | <p>I am trying to evaluate the integral<br>
$\int (\frac{1}{z(\exp(z)-1)}dz$ $\mathbb{C}:|z|=1\}$</p>
<p>using Laurent theorem. $0$ is a singularity point and both denominators becomes $0$ when $z=0$. So it's hard to find a method to evaluate this using Residue theorem. Could someone please show me how to solve thi... | Deepak | 151,732 | <p>On that "special day", the girl walks for an extra $12 - 3 = 9$ minutes to get to the new meeting point. The tram has travelled for $3$ minutes less than usual to get to that new meeting point. The distance of the new meeting point from the usual one is the same for both. Therefore the tram is travelling $\frac 93 =... |
2,482,083 | <p>I want to know if the following is a true statement:</p>
<blockquote>
<p>If $A$ is real a symmetric and $V$ is orthogonal then $V^TAV=D$ where $D$ is a diagonal matrix. </p>
</blockquote>
<p>To prove it I can say </p>
<p>$$(V^TAV)^T=V^TA^TV = V^TAV.$$
So $V^TAV$ is itself symmetric. So if I know $V^TAV$ is tri... | user334639 | 221,027 | <p>It is false: take $V$ to be the identity and $A$ to be symmetric but not diagonal.</p>
<p>Then $D=V^TAV=A$ so $D$ is not diagonal.</p>
|
2,810,755 | <p>I'm trying to set up a double integral on $e^{-(x+y)}$ and the range listed is: $0 < x < y < \infty.$</p>
<p>I'm interpreting this as $0 < x < y$ for $x$'s range and $x < y < \infty$ as $y$'s range. I've put those bounds on my two integrals and proceeded with the mission. However, $y$ is stil... | tien lee | 557,074 | <p>Inductive Hypothesis:-</p>
<p>Assume $P(k)$ is true</p>
<p>$1\cdot2+2\cdot3+3\cdot4+.....+\sum _{ n=1 }^{ k }{ n(n+1) } =\frac{k(k+1)(k+2)}{3}$</p>
<p>Now show that Inductive Hypothesis is true. Show that $P(k+1)$ is true
$$\sum _{ n=1 }^{ k+1 }{ n(n+1) } =\frac{k+1(k+1+1)(k+1+2)}{3}=\frac{k+1(k+2)(k+3)}{3}$$
Now... |
118,658 | <p>I have set of files in the format 20160615-1-0.asc, 20160615-2-0.asc,... 20160615-5-0.asc etc. The following commands can List the file names at once which can be used to import later. </p>
<pre><code> files = Range[1, 5];
c = "D:\\20160615\\20160615-" <> ToString[#] <> "-0.asc" & /@files
</cod... | m_goldberg | 3,066 | <p>It seems that a simple modification of your code will work. </p>
<pre><code>fileList[prefix_String, m_Integer, n_Integer] :=
With[{dash = "-"},
prefix <> dash <> ToString[#] <> dash <> ToString[n] <> ".asc"] & /@ Range[m]
</code></pre>
<p>Then</p>
<pre><code>fileList["D:\\... |
90,965 | <p>Im wondering if there's an existing literature on this binary operation involving graphs wherein you identity $n$ vertices from one graph with $n$ vertices from the other such that the resulting structure is still a graph (no loops and multiple edges). For instance, given two paths $[a,b,c]$ and $[d,e,f]$, letting $... | Marc Olschok | 22,141 | <p>Since I cannot add comments, I put this as an answer.
I do not know about specific graph-theoretical literature about the
operation you describe, but it seems that it can be exhibited as
a pushout in a suitable category.</p>
<p>For simplicity, I suppose you consider only undirected graphs.</p>
<p>Take the category... |
90,965 | <p>Im wondering if there's an existing literature on this binary operation involving graphs wherein you identity $n$ vertices from one graph with $n$ vertices from the other such that the resulting structure is still a graph (no loops and multiple edges). For instance, given two paths $[a,b,c]$ and $[d,e,f]$, letting $... | Felix Goldberg | 22,051 | <p>Have a look at this:</p>
<p><a href="http://www.win.tue.nl/~hholst/Research%20slides/minimum%20rank.pdf" rel="nofollow">http://www.win.tue.nl/~hholst/Research%20slides/minimum%20rank.pdf</a></p>
<p>I think that what van der Holst calls <em>Separations</em> is what you need.</p>
|
3,556,109 | <p>Suppose that <span class="math-container">$f : (0, 2] \to \mathbb{R}$</span> is a continuous function and that <span class="math-container">$\lim_{x \to 0}f(x) = L$</span> for some <span class="math-container">$L \in \mathbb{R}$</span>.
Define <span class="math-container">$g : [0, 2] \to \mathbb{R}$</span> via <spa... | Leander Tilsted Kristensen | 631,468 | <p>"Clearly the former implies the latter via the triangle inequality." No, it is actually the other way around. The triangle inequality states
<span class="math-container">$$|\sum_i x_iy_i| \leq \sum_i |x_i y_i|,$$</span></p>
<p>so clearly Hölders inequality implies Cauchy-Schwarz and it would seem that Hölders is th... |
2,559,639 | <p>In the book of <em>General Topology</em> by Munkres, on page 104, it is given that</p>
<p><a href="https://i.stack.imgur.com/Pv9MU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Pv9MU.png" alt="enter image description here"></a></p>
<p>However, as far as I know, the lower limit topology <span c... | Angina Seng | 436,618 | <p>$(a,b)=\bigcup_{n=1}^\infty[a+1/n,b)$</p>
|
40,669 | <p><a href="http://en.wikipedia.org/wiki/Wang_tile">Wang tiles</a> are (by Wikipedia): "equal-sized squares with a color on each edge which can be arranged side by side (on a regular square grid) so that abutting edges of adjacent tiles have the same color; the tiles cannot be rotated or reflected."</p>
<p>The usual d... | blatmalt | 562,720 | <p>There is also the newer work of Kari "On the Undecidability of the Tiling Problem", which shows the undecidability of the domino problem by a reduction to the immortality problem for affine functions.</p>
<p>It shows the undecidability of the domino problem in the hyperbolic plane which was not shown by Robinson.
... |
2,204,834 | <p>I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is
$$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of 8, you can say that$$5^{n+1}+2\cdot 3^n+1=8\cdot m \hspace{1cm}\forall m\i... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>If $a_n=(2b+1)^n$</p>
<p>$$a_{m+2}-a_m=8(2b+1)^m\cdot\dfrac{b(b+1)}2$$ which is multiple of $8$ as $b(b+1)$ is even.</p>
|
1,057,429 | <p>In mathematics statistics. I'm a bit confused by the terminology <em>normal approximation</em>. What is it? Is it just something you say when you approximate, for example the normal distribution? </p>
| j4GGy | 186,141 | <p>By the <a href="http://en.wikipedia.org/wiki/Spectral_theorem" rel="nofollow">finite-dimensional spectral theorem</a> there exists an orthogonal matrix $C$ (i.e. $C^-1 = C^T$ such that $C^TAC = D = diag(d_1,\ldots, d_n)$ is a diagonal matrix. Let $e_k \in \mathbb{R}^n$ denote the vector, which $k$-th coordinate is $... |
3,048,340 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is a convex function and differentiable everywhere on <span class="math-container">$(0, \infty)$</span> and satisfies
<span class="math-container">$$\lim_{x \rightarrow\infty}{f(x)}=A$$</span>
where <span class="math-container">$A$</span> is an arbit... | JimmyK4542 | 155,509 | <p><strong>Hint</strong>: You've successfully come up with a contradiction if <span class="math-container">$f'(x_0) > 0$</span> for any <span class="math-container">$x_0 \in (0,\infty)$</span>. Therefore, <span class="math-container">$f'(x) \le 0$</span> for all <span class="math-container">$x \in (0,\infty)$</span>... |
1,887,315 | <p>$e$ is one of the most important numbers in our universe, it is everywhere. When I try to find out why the most common explanation is some reverse-engineered physics or finance problem. But these are just one off examples of why $e$ <em>is</em> important, they fall short of illuminating the origin of its significanc... | Wavelet | 336,350 | <p>$e^{\lambda x}$ is the solution to the equation $Df=\lambda f$. This is equivalent to saying that $e^{\lambda x}$ is an eigenfunction of the differentiation operator D for any value $\lambda \in \mathbb{R}$. Now how ubiquitous is differentiation to physics?</p>
|
7,900 | <p>This semesters I am teaching a course to 12-13 year old kids in which they are supposed to learn the basics of spreadsheet usage.</p>
<p>I am having difficult in coming up with fun / interesting exercises to teach them the basic functions, such as average, max, minimum and the like. One of the biggest problems is h... | Sue VanHattum | 60 | <p>I used Excel to solve many of the problems I encountered in <a href="https://projecteuler.net/" rel="nofollow">Project Euler</a>. I bet they'd find some of those problems interesting and accessible.</p>
|
3,562,324 | <blockquote>
<p>Find all positive integers x such that <span class="math-container">$x^4-8x+16$</span> is a full square.</p>
</blockquote>
<p>I presented it as <span class="math-container">$(x-4)^2+x^4-x^2$</span> and found that <span class="math-container">$x = +1$</span> satisfy the condition. Then I equalized it ... | J. W. Tanner | 615,567 | <p><strong>Hint:</strong></p>
<p>For <span class="math-container">$x>2$</span>, <span class="math-container">$(x^2-1)^2<x^4-8x+16<(x^2)^2$</span>.</p>
|
2,626,454 | <blockquote>
<p>I used Wolfram Alpha to try and compute the integral $$\int_0^{\pi/2}(1+\cos2x)^{-0.5}\,dx$$ and I'm told that the integral does not converge. However, if I try
$$\lim_{a\to0.5^-}\int_0^{\pi/2}\frac{dx}{(1+\cos2x)^a}$$
by substituting in values of $a$ close to $0.5$ e.g $a\in\{0.49,0.499,0.4999...... | mickep | 97,236 | <p>In the interval you are looking at
$$
\frac{1}{(1+\cos 2x)^a}=\frac{1}{2^a\cos^{2a}x}.
$$
Hence, the integral you are looking at is
$$
\frac{1}{2^a}\int_0^{\pi/2}\frac{1}{\cos^{2a}x}\,dx=\frac{1}{2^{a+1}}B(1/2,1/2-a)
=\frac{1}{2^{a+1}}\sqrt{\pi}\frac{\Gamma(1/2-a)}{\Gamma(1-a)}.
$$
Here $B$ denotes the Beta function... |
2,328 | <p>There is an interesting discussion taking place at MO over the suitability of this question: <a href="https://mathoverflow.net/questions/210089/complex-structure-on-s6-gets-published-in-journ-math-phys">Complex structure on $S^6$ gets published in Journ. Math. Phys</a>. The question is about the correctness of a pap... | The Masked Avenger | 35,626 | <p>I think Joseph O'Rourke meant to say something else. I will guess that something like
the following is intended.</p>
<p>MathOverflow is not a forum for judging, critiquing, or refereeing preprints or published
articles. MathOverflow is for answering specific questions in mathematics, including
but not limited to ... |
607,353 | <p>So I've recently been looking at the <a href="http://www.photographyblogger.net/stereographic-projection-photography-100-little-planets/" rel="nofollow">Tiny Planet</a> images.</p>
<p>I've been googling a few things to try and find out how images are converted from normal to a tiny planet. Some phone apps, as well ... | Logan Keefe | 115,380 | <p>The article you have linked has "Stereographic Projection" in the title. <a href="https://en.wikipedia.org/wiki/Stereographic_projection#Definition" rel="nofollow">Here</a> is a link to the relevant part of the Wikipedia article. It is difficult to tell solely from looking at the images, but it seems as if the origi... |
2,760,113 | <p>Suppose $f'$ is continuous on $[a,b]$ and $\epsilon >0$ is given.
Prove that $\exists\; \delta>0$ such that $$\left\lvert \frac{f(t)-f(x)}{t-x}-f'(x)\right\rvert<\epsilon$$</p>
<p>$\forall \;0<|t-x|<\delta, a\le x \le b ,a\le t\le b$.</p>
<p>How to solve this problem since using compactness argumen... | Mark Pineau | 432,691 | <p>Following from my hint I gave in the comments:</p>
<p>$f'$ is continuous on the compact interval $[a,b]$, and thus is uniformly continuous on $[a,b]$. </p>
<p>$\therefore \ \forall \epsilon>0$, $\exists \ \delta>0$, such that for all $x,t\in [a,b]$, we have </p>
<p>$$0<|x-t|<\delta\implies|f'(x)-f'(t... |
1,820,853 | <p>How to derive the Gregory series for the inverse tangent function?
Why is Gregory series applicable only to the set $ [-\pi/4,\pi/4] $ ?</p>
| user353587 | 353,587 | <p>I think the most straightforward way is to yes the expansions of <strong>log(1+x) and log(1-x)</strong>
<strong>log(1+x)=x - x^2/2 + x^3/3....and so on.</strong></p>
<p><strong>log(1-x)=-x-x^2/2-x^3/3....and so on.</strong></p>
<p>Now subtract the two series to get
<strong>log[(1+x)/(1-x)]=
2[x+x^3/3+x^5/5+x^7/7..... |
2,338,797 | <p>A bag contains $3$ red balls and $4$ blue balls. Find the probability of picking, at random, two balls of </p>
<p>a) the same color </p>
<p>b) different colors</p>
| Zack Miodownik | 450,075 | <p>The probability of picking 2 blues balls (lol) is 4/7 times 3/6 which is 2/7. The probability of picking 2 red balls is 3/7 times 2/6 which is 1/7. Thus the probability of picking two same colored balls is 3/7 (The sum of the two). Thus the probability of picking different colored balls is 1-3/7 which is 4/7.</p>
|
2,338,797 | <p>A bag contains $3$ red balls and $4$ blue balls. Find the probability of picking, at random, two balls of </p>
<p>a) the same color </p>
<p>b) different colors</p>
| BruceET | 221,800 | <p>Answers by @Mandar, @Zack, and @David are in agreement, clear, and correct (+1's).</p>
<p>I just wanted to show you the role of the 'hypergeometric distribution' in solving
problems such as this. Let $X$ be the number of red balls in two draws without replacement. Then
$$P(X = k) = \frac{{3 \choose k}{4 \choose {2-... |
1,544,789 | <blockquote>
<p>Calculate $\int_0^{\infty} \frac{x^{2N+1}}{a+x^{-b}} e^{-c x^2} dx$ where $a,c > 0$ and $b>1$.</p>
</blockquote>
<p>The best I could do about this integral is to find an upperbound for it:</p>
<p>$\int_0^{\infty} \frac{x^{2N+1}}{a+x^{-b}} e^{-c x^2} dx \leq \int_0^{\infty} \frac{x^{2N+1}}{x^{-... | mrprottolo | 84,266 | <p>The reason is that the integral you evaluated doesn't give the area of the quadrilateral $R$. Indeed, called $A$ its area, you have
$$\int\int_R dx dy=A$$
and
$$\int\int_R x dx dy\neq \int\int_R dx dy.$$</p>
|
267,161 | <p>I would like to write <code>x^5 + 1/x^5</code> in terms of <code>x^2 + 1/x^2</code>, <code>x^3 + 1/x^3</code> and <code>x + 1/x</code> or express <code>x^6 + 1/x^6</code> in terms of <code>x^2 + 1/x^2</code> and <code>x^4 + 1/x^4</code> etc.</p>
<p>How can I do that in <em>Mathematica</em>?</p>
| Daniel Lichtblau | 51 | <p>@Artes lready showed how one might use <code>Solve</code> or <code>Eliminate</code> for this task. Since it involves algebraic rewriting of a rational expression, a standard method from computational algebra involves Groebner bases. Technical-sounding, yes, but also quite powerful.</p>
<p>The idea is to write the in... |
625,395 | <blockquote>
<p>Let $f:\mathbb{R}^3 \to \mathbb{R}$ be a smooth function rapidly
decreasing to zero as $|(x,y,z)| \to \infty$, and let $D(t)$ denote
$$D(t)=\left\{(x,y,z)\in \mathbb{R}^3 \mid ax+by+cz \leq t\right\},$$
where $a,b,c\in \mathbb{R}$. Compute $$\frac{d}{dt}
\int_{D(t)}f(x,y,z)\,dx\,dy\,dz.$$</p>
<... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displayst... |
3,575,867 | <p>I need to prove that <span class="math-container">${n\choose 0} < {n\choose 1} < \ldots < {n\choose (n+1)/2}$</span> (for every <span class="math-container">$n$</span>)
I tried to do it with induction but I got stuck.</p>
| Paras Khosla | 478,779 | <p><strong>Hint</strong>:</p>
<p>By definition <span class="math-container">${n\choose k}=\frac{n!}{k!(n-k)!}$</span>, where <span class="math-container">$n, k \in \mathbb{N}$</span> and <span class="math-container">$n\ge k $</span>. Use this definition assuming the proposition to be true and show that it satisfies th... |
3,575,867 | <p>I need to prove that <span class="math-container">${n\choose 0} < {n\choose 1} < \ldots < {n\choose (n+1)/2}$</span> (for every <span class="math-container">$n$</span>)
I tried to do it with induction but I got stuck.</p>
| SomeStrangeUser | 68,387 | <p>Note that <span class="math-container">$\binom{n}{k}=\frac{n-k+1}{k}\cdot\binom{n}{k-1}$</span>. Hence, the inequality is true as long as <span class="math-container">$n-k+1>k$</span>, or rearranging: <span class="math-container">$\frac{n+1}{2}>k$</span>.</p>
|
4,181,235 | <p>The topology on <span class="math-container">$\mathcal D$</span> is the locally convex topology induced by seminorms like <span class="math-container">$\|\partial_\alpha f\|_\infty$</span>, where <span class="math-container">$\alpha=(d_1,\ldots,d_n) $</span> and <span class="math-container">$\partial_\alpha=\partial... | Aaron Hendrickson | 271,783 | <p>This is not the most direct approach but it does show where hypergeometric functions appear as mentioned by @DrSonnhardGraubner.</p>
<hr />
<p>We write
<span class="math-container">$$
\begin{align}
\sum_{j=0}^n\binom{a+j}{b}
&=\sum_{j=0}^n\frac{\Gamma(a+1+j)}{\Gamma(b+1)\Gamma(a-b+1+j)}\\
&=\frac{\Gamma(a+1)... |
2,880,377 | <p>As the title states, I need to find the limit for $x\left(x + 1 - \frac{1}{\sin(\frac{1}{1+x})}\right)$ as $x \rightarrow \infty$, as part of a larger proof I am working on.</p>
<p>I believe the answer is 0. I think that to start, I can show that $\frac{1}{\sin(\frac{1}{1+x})} \rightarrow x + 1$ for large x. By loo... | user1337 | 62,839 | <p>According to Taylor's theorem with remainder, we have
$$\sin t=t-\frac{t^3}{6}+R_3(t) $$
where $|R_3(t)| \leq \frac{t^4}{24}$. Taking $t=\frac{1}{x+1}$ and plugging in your expression gives
$$\frac{x (x+1) \left(6 R_3(t) (x+1)^3-1\right)}{6 R_3(t) (x+1)^3+6 x (x+2)+5}=\frac{-x^2+o(x^2)}{6x^2+o(x^2)} $$
which approa... |
2,226,307 | <p>Let <span class="math-container">$c\in \mathbb{R}$</span> and <span class="math-container">$f:\mathbb{R}\to \mathbb{R}$</span> is continuous at <span class="math-container">$c$</span>. If for every positive <span class="math-container">$\delta$</span>, there is a point <span class="math-container">$y\in (c-\delta, c... | Tsemo Aristide | 280,301 | <p>Take $\delta=1/n$, there exists $y_n\in (c-1/n,c+1/n)$ with $f(y_n)=0$, $lim_ny_n=c$ implies $f(c)=lim_nf(y_n)=0$ since $f$ is continue.</p>
|
849,510 | <p>Let $f:\mathbb{R}\to\mathbb{R}$ be a $T$-periodic function, that is $f(t+T)=f(t)$ for all $t\in \mathbb{R}$. Assume that
$$\int_0^{+\infty}|f(s)|ds<+\infty.$$
Now if we assume in addition that $f$ is continuous, my intuition tells me that we must have necessarily $f=0$, is this correct ? </p>
| Community | -1 | <p><strong>Hint</strong>: </p>
<p>$$\int_0^{\infty} |f(s)| ds = \sum_{k = 0}^{\infty} \int_{kT}^{(k + 1)T} |f(s)| ds$$</p>
|
25,412 | <p>This is hopefully a simpler version of this previous unanswered <a href="https://mathematica.stackexchange.com/questions/23496/solving-recursion-relations-using-mathematica">question</a> of mine. </p>
<p>Let me just focus on the two expressions $F_2^{(s)}$ and $F_3^{(s)}$ given in A.3 and A.4 of page 19 of <a href... | TransferOrbit | 3,236 | <p>Based on the assumption that those F functions produce scalar results…</p>
<p>If you just want to manipulate the vectors as entities without considering their components, then you can enter the function for F2s</p>
<pre><code>f2s[q_, k1_] := (5/
14) + (3 (Norm[k1])^2)/(28 (Norm[q])^2) + (3 Norm[
k1]^2)/(28 (Nor... |
2,509,583 | <p>How is intermediate value theorem valid for $\sin x$ in $[0,\pi]$?</p>
<p>It has max value $1$ in the interval $[0,\pi]$ which doesn't lie between values given by $\sin0$ and $\sin\pi$.</p>
| José Carlos Santos | 446,262 | <p>Let $f\colon[a,b]\longrightarrow\mathbb R$ be a continuous function and let $y\in\mathbb R$. The intermediate value theorem says that <em>if</em> $f(a)\geqslant y\geqslant f(b)$ or if $f(a)\leqslant y\leqslant f(b)$, then there is a $c\in[a,b]$ such that $f(c)=y$. But it says <em>nothing</em> if $y$ lies outside th... |
2,662,007 | <p>I'd like to show that the group given by presentation $G=\langle a,b \ | \ a^2b^2 \rangle$ is not isomorphic to any of $\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$, or $\mathbb{Z} \times \mathbb{Z}$.</p>
<p>For $G \ncong \mathbb{Z}$, I note that $G$ is not free because it has a relator in its presentation.</p>
<p>For $G... | Jeremy Rickard | 88,262 | <p>The empty $G$-set is the initial object.</p>
<p>The coproduct of $G$-sets is the disjoint union.</p>
<p>Call a $G$-set “indecomposable” if it is not the coproduct of two non-empty $G$-sets, so an indecomposable $G$-set is just a transitive one.</p>
<p>An epimorphism of $G$-sets is just a surjective map of $G$-set... |
247,622 | <p>Find the dimension of the vector subspace in $\mathbb{R}^{4}$ spanned by the columns </p>
<p>$$
\begin{pmatrix}
1 & 1 & 2 & 4 \\
t & 3 & 2 & 8 \\
0 & 0 & 2 & 2 \\
0 & 0 & 0 & 0
\end{pmatrix}
$$</p>
<p>(discussion over real $t$).</p>
<p>Thanks :) </p>
| kahen | 1,269 | <p>Basic facts about <a href="https://en.wikipedia.org/wiki/Net_%28mathematics%29" rel="nofollow">nets</a>: A function $f: X\to Y$ is continuous at $x \in X$ if and only if for every net $(x_\alpha)$ in $X$ converging to $x$ we have that $f(x_\alpha) \to f(x)$; and a set $A \subseteq X$ is closed if and only if for eve... |
247,622 | <p>Find the dimension of the vector subspace in $\mathbb{R}^{4}$ spanned by the columns </p>
<p>$$
\begin{pmatrix}
1 & 1 & 2 & 4 \\
t & 3 & 2 & 8 \\
0 & 0 & 2 & 2 \\
0 & 0 & 0 & 0
\end{pmatrix}
$$</p>
<p>(discussion over real $t$).</p>
<p>Thanks :) </p>
| user46128 | 46,128 | <p>Let $G=\{x \in X : f(x)=x\}$.
We want to prove that $G$ is closed, hence that $G^c$ is open.</p>
<p>So take $y\in G^c$.
Since X is Hausdorff we can find a two open sets U and V so that:
$y\in U$, $f(y)\in V$ and $U\cap V= \emptyset$.
Since f is continuous we can find a Set $V'$ s.t. $f(V')\subset V$.
The set $W=V' ... |
247,622 | <p>Find the dimension of the vector subspace in $\mathbb{R}^{4}$ spanned by the columns </p>
<p>$$
\begin{pmatrix}
1 & 1 & 2 & 4 \\
t & 3 & 2 & 8 \\
0 & 0 & 2 & 2 \\
0 & 0 & 0 & 0
\end{pmatrix}
$$</p>
<p>(discussion over real $t$).</p>
<p>Thanks :) </p>
| Julian Kuelshammer | 15,416 | <p>Since there are several answers addressing the first question, let me just give an example for the second one. A good example for a T1-space that is not T2 is $\mathbb{N}$ with the cofinite topology, i.e. the closed sets are the finite ones. Then take the map $f:\mathbb{N}\to \mathbb{N}$, given as $f(x)=\begin{cases... |
59,105 | <p>Let $n$ be a positive integer. $\; $ Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be everywhere <a href="http://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative">Frechet differentiable</a>.
<br>
Let $S$ be a subset of $\mathbb{R}^n$ with <a href="http://en.wikipedia.org/wiki/Lebesgue_measure">Lebesgue measure</a> zero.</... | George Lowther | 1,321 | <p>Yes. In fact, if $f$ has a Frechet derivative $Df$ on a measurable set $S\subseteq{R}^n$ then
$$
\begin{align}
\mu(f(S))\le\int_S\left\vert{\rm det} Df\right\vert\,d\mu&&(1)
\end{align}
$$
where $\mu$ is the Lebesgue measure on $\mathbb{R}^n$. In particular, if $\mu(S)=0$ then the right hand side is zero... |
1,598,255 | <p>Consider the system
$$
\dot{x}=y,\qquad\dot{y}=x+x^2-y.
$$
It has two equilibria, namely $(0,0)$ and $(-1,0)$.</p>
<p>I would like to linearize the system in both equilibria.</p>
<p>My start is to set $\Delta x=x-x_0,\qquad \Delta y=y-y_0$. Then
$$
\dot{\Delta x}=\dot{x}-\dot{x_0}=y-y_0=\Delta y
$$
and
$$
\dot{\De... | Lutz Lehmann | 115,115 | <p>No, that is wrong. $x_0$ and $y_0$ are constants, only $Δx$ and $Δy$ are functions of $t$. Thus
\begin{align}
(Δx)˙=\frac d{dt}(x_0+Δx(t))&=y_0\quad+\quad Δy
\\ \\
(Δy)˙=\frac d{dt}(y_0+Δy(t))
&=(x_0+Δx)+(x_0+Δx)^2−(y_0+Δy)
\\
&=x_0+x_0^2-y_0\quad+\quad (1+2x_0)Δx-Δy\quad+\quad Δx^2
\end{align}</p>
|
221,137 | <p>What's the difference between Fourier transformations and Fourier Series? </p>
<p>Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?</p>
| Vitali Pom | 692,929 | <p>I see you’re asking probably about the most “meaningful two ones” that are seemingly differ, but yet comparable if you compare them one to one and these are discrete time signal’s fourier transform and discrete time signal’s fourier series, while both signals are periodic. </p>
<p>I’ll explain it in the simplest wa... |
1,148,624 | <p>This is for teaching math. I'm wondering if someone knows some striking near-equalities between simple arithmetic expressions. I vaguely remember that such things exist (e.g., numbers that look alike out 10 digits, but that are really different), but I don't remember where I saw them or what they are. I think there ... | Asinomás | 33,907 | <p>$e^{\sqrt{163}\pi}=262537412640768743.99999999999925$ is almost an integer. It was conjectured for a long while this number could in fact be an integer.</p>
<p>You can find more on this here:</p>
<p><a href="https://www.youtube.com/watch?v=DRxAVA6gYMM" rel="nofollow noreferrer">https://www.youtube.com/watch?v=DRx... |
2,948,673 | <p>I was thinking about the squeeze theorem here. We can denote the <span class="math-container">$\left[\frac{1}{|x|}\right] =n$</span>, and then try something like:</p>
<p><span class="math-container">$$x^2(1+2+3+...+(n-1)+(n-1)) \leq x^2 \frac{n(n+1)}{2} \leq x^2(1+2+3+...+(n-1)+(n+1))$$</span></p>
<p>But I don't k... | ajotatxe | 132,456 | <p>Given that
<span class="math-container">$$\left[\frac1{|x|}\right]=n$$</span>
we have
<span class="math-container">$$n\le\frac1{|x|}<n+1$$</span>
Therefore
<span class="math-container">$$\frac1n\ge|x|>\frac1{n+1}$$</span>
and thus,
<span class="math-container">$$\frac1{(n+1)^2}<x^2\le\frac1{n^2}$$</span></... |
313,597 | <p>Let $V$ be a finite-dimensional vector space with an ordered base $\beta$ over a field $F$.</p>
<p>Let $x\in V$ an $[x]_{\beta}$ be a coordinate vector of $x$ relative to $\beta$.</p>
<p>Is this an element of $M_{n\times 1}(F)$ or $F^n$?</p>
<p>Since we usually treat them as a matrix, i think it should be defined... | Berci | 41,488 | <p>You are absolutely correct.</p>
<p>However, -- and if you read some specific text, then it's probably written somewhere explicitly -- it is usual to <em>identify</em> $F^n$ with the <em>culomn vectors</em>, i.e. $n\times 1$ matrices. As you say, $F^n\ne M_{n\times 1}(F)$, but they are isomorphic. It is also usual t... |
1,221,368 | <p>An inheritance is divided between three people in the ratio $4:7:2$. If the least amount received is $\$\ 2300$ calculate how much the other two people received?</p>
| Mufasa | 49,003 | <p>If the amount is distributed in the ratio $4:7:2$ then you know the person who got the least got:$$\frac{2}{4+7+2}=\frac{2}{13}$$of the total amount.</p>
<p>Use this fact to arrive at the answer.</p>
|
103,358 | <p>I am trying to illustrate some simple ideas with exponents. I can manually express something like $5^4$ as $5 \cdot 5 \cdot 5 \cdot 5$, but wondered how to get <em>Mathematica</em> to do that for me.</p>
<p>I found the example below in the documentation, but can't figure out how to "massage" it to work with a num... | Zhi-Bang Yao | 84,879 | <p>I have just solved this problem by turning off the automatic updates in the preferences. Hope it will work for you too!</p>
|
650,381 | <p>the definition of variance is $V(X) = E((X-E(X))^2 )$</p>
<p>For a discrete random variable: if we have put $Y = g(X)$ , where $g$ is a real function</p>
<p>$E(Y) = E(g(X)) = \sum\limits_{k} g(k)p_X(k)$ , where $p_X(k) = P(X = k) $ </p>
<p>how can i use this formula to show that:</p>
<p>$V(X) = \sum\limits_{... | jaranna | 119,935 | <p>@Amzoti: Could you tell me where is my mistake: $A= \frac{i}{3} \begin{bmatrix} 1&-2&1\\-2&1&1\\1&1&-2\end{bmatrix} = \begin{bmatrix} \frac{i}{3}&\frac{-2i}{3}&\frac{i}{3}\\\frac{-2i}{3}&\frac{i}{3}&\frac{i}{3}\\\frac{i}{3}&\frac{i}{3}&\frac{-2i}{3}\end{bmatrix} \\ A_{... |
359,816 | <p>Let $(f_n)$ and $(g_n)$ be sequences of nonnegative function in $L^1(\mathbb R)$, for which
$$
f_n \to 0, \\
g_n \to 0,
$$
almost everywhere.
Show
$$
\int_A \frac{2f_n g_n}{1+f_n^2+g_n^2} \to 0,
$$
when $A \subset \mathbb R$ is a set of finite measure.</p>
<p>Define
$$
h_n = \frac{2f_n g_n}{1+f_n^2+g_n^2}.
$$
I kno... | Davide Giraudo | 9,849 | <p><strong>Hint:</strong> Egorov's theorem. Where there is uniform convergence, no problem, and on "the small set", the integrand is bounded by $1$.</p>
|
2,850,435 | <p>The equation is
$$
\int f(x) = \frac{1}{f'(x)}
$$
I saw it on a shirt so I thought it is probably a joke, but I couldn't get it. So I tried to solve it. On the first look I thought this is never true. Then I thought of differentiating the whole thing.</p>
<p>$$
f(x) = - \frac{f''(x)}{f'(x)^2}
$$ </p>
<p>This is wh... | Exit path | 161,569 | <p>Since pointed maps $X \wedge Y \to Z$ are precisely pointed maps $X \times Y \to Z$ that send the wedge sum of $X$ and $Y$ to the base point in $Z$, by the adjunction we know that such maps correspond to certain maps $X \to Maps(Y,Z)$. Can you figure out which these are?</p>
<p>These adjunctions work for all reason... |
1,425,961 | <p>I am about to take a course in undergraduate Algebra which is using Artin's Algebra. I have been studying from it and haven't really appreciated the combination of text and problems (granted, I haven't gone too deep into the text). </p>
<p>Now, this is the text for the class, so I would expect a nice compliment f... | JeffW89 | 200,928 | <p>Dummit and Foote has a more standard format, and covers a wider variety of topics, but it's extremely dry (and feels very "text-booky" to me). I didn't like it much. Take a look at Vinberg instead. Takes a very similar approach to Artin, but it's more readable in my opinion and has a more manageable number of pro... |
1,808,019 | <p>I'm trying to better understand how the following group action looks like on a torus. The group $\mathbb{R}$ acts on the torus $C\times C=\{(e^{ix},e^{iy}):x,y\in \mathbb{R}\}$ ($C$ being the unit circle in $\mathbb{C}$) via $$t\cdot (e^{ix},e^{iy})\mapsto (e^{i(x+t)},e^{iy})$$</p>
<p>I am able (I think) to visuali... | Behrouz Maleki | 343,616 | <p>for $n=1$ we have
$${{(x+y)}^{\,1}}=\ 1\ =\sum\limits_{i=0}^{1}{\left( \begin{matrix}
1 \\
0 \\
\end{matrix} \right)}\,{{x}^{1-i}}{{y}^{i}}=\left( \begin{matrix}
1 \\
0 \\
\end{matrix} \right)\,{{x}^{1}}+\left( \begin{matrix}
1 \\
1 \\
\end{matrix} \right)\,{{y}^{1}}=x+y$$</p>
<p>for $n=k$ ... |
1,320,659 | <p>Is the ratio test applicable for testing convergence of infinite products?</p>
<p>In other words, consider the sequence $(a_i)_{i=1}^\infty$ of non-zero real numbers.</p>
<p>Also, consider the product $\displaystyle\mathcal P=\prod_{k=1}^\infty a_k$</p>
<p>Are the following statements true?</p>
<blockquote>
<p... | Zach Stone | 38,565 | <p>With products, $a_k\to 1$ is the right assumption, otherwise stupid things happen. Just as in series, we usually just assume $a_k \to 0$. So in this context,
$$
\lim_{k\to\infty}\left|\frac{a_{k+1}}{a_{k}}\right| =1
$$
should happen every time.</p>
<p>It's worth noting that
$$\prod_{k=1}^{\infty}a_k$$
converges i... |
2,666,390 | <p>How do I show that for $x>0$:
$$\frac{x-(x^2+1)\arctan(x)}{x^2(x^2+1)} < 0$$</p>
<p>I tried to do it somehow using the fact that $$\frac{\arctan(x)}{x} < 1$$ but still didn't figure it out...</p>
| Jack D'Aurizio | 44,121 | <p>The question is equivalent to proving $\arctan(x)>\frac{x}{x^2+1}$ for any $x>0$, but this is trivial since
$$ \arctan(x)=\int_{0}^{x}\frac{dt}{t^2+1}> \int_{0}^{x}\frac{dt}{x^2+1}=\frac{x}{x^2+1}.$$</p>
|
389,912 | <p>I'm writing a paper and want to cite some references to efficiently prove that over any field <span class="math-container">$k$</span> of characteristic zero, every irreducible representation of a product of symmetric groups, say</p>
<p><span class="math-container">$$ S_{n_1} \times \cdots \times S_{n_p} $$</span></p... | Geoff Robinson | 14,450 | <p>A positive answer to the first question for <span class="math-container">$S_{n}$</span> follows from the work of Alfred Young. He proved that every irreducible representation of <span class="math-container">$S_{n}$</span> over <span class="math-container">$\mathbb{C}$</span> is afforded by an explicit <span class="m... |
3,720,856 | <p><strong>Question:</strong>
Prove that <span class="math-container">$\sin(nx) \cos((n+1)x)-\sin((n-1)x)\cos(nx) = \sin(x) \cos(2nx)$</span> for <span class="math-container">$n \in \mathbb{R}$</span>.</p>
<p><strong>My attempts:</strong></p>
<blockquote>
<p>I initially began messing around with the product to sum iden... | J.G. | 56,861 | <p>The left-hand side is<span class="math-container">$$\begin{align}&\sin nx(\cos nx\cos x-\sin nx\sin x)-(\sin nx\cos x-\cos nx\sin x)\cos nx\\&=(\cos^2nx-\sin^2nx)\sin x\\&=\cos 2nx\sin x.\end{align}$$</span></p>
|
2,744,172 | <p>(Note: This question is tangentially related to this <a href="https://math.stackexchange.com/q/3609304">later one</a>.)</p>
<p>Let <span class="math-container">$$\sigma(x) = \sum_{d \mid x}{d}$$</span> denote the <em>sum of divisors</em> of <span class="math-container">$x \in \mathbb{N}$</span>, where <span class=... | Servaes | 30,382 | <p>Let <span class="math-container">$k$</span> and <span class="math-container">$m$</span> be odd integers greater than <span class="math-container">$1$</span> such that
<span class="math-container">$$(m+1)\sigma(k)=2mk.$$</span>
Then clearly <span class="math-container">$\sigma(k)<2k$</span>, and hence the above is... |
2,563,779 | <p>Let $(X,d)$ be a metric space. If every subset of $X$ is bounded, does it mean that the space itself is bounded?</p>
| José Carlos Santos | 446,262 | <p>Of course. After all, $X$ is a subset of itself. On the other hand, if $X$ is bounded, every subset of $X$ is bounded too.</p>
|
4,552,988 | <p>I want to prove by diagonalization that the set of surjective total computable functions from N to N is not recursively enumerable. I know that the result is trivial using Rice's theorem, but I am trying to prove it only by a direct diagonalization argument. However, supposing that we can enumerate the functions of ... | insipidintegrator | 1,062,486 | <p>Now you have (after the edit(s)) that <span class="math-container">$\displaystyle f(0)=\int_0^0\frac{dt}{\sqrt{1+t^4}}=0$</span> so <span class="math-container">$f^{-1}(0)=g(0)=0$</span> so <span class="math-container">$g’(0)=\sqrt{1+(g(0))^4}=1.$</span></p>
|
4,552,988 | <p>I want to prove by diagonalization that the set of surjective total computable functions from N to N is not recursively enumerable. I know that the result is trivial using Rice's theorem, but I am trying to prove it only by a direct diagonalization argument. However, supposing that we can enumerate the functions of ... | xpaul | 66,420 | <p>This is what you have to do next. Let <span class="math-container">$x=0$</span> in
<span class="math-container">$$f'(g(x))g'(x)=1.$$</span>
Then you have
<span class="math-container">$$ f'(g(0))g'(0)=1. $$</span>
Noting <span class="math-container">$g(0)=0$</span> and <span class="math-container">$f'(0)=\frac1{\sqrt... |
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | JTP - Apologise to Monica | 64 | <p>I only became aware of τ a few weeks ago. It's clever, and in theory, might simplify the teaching a bit. If one could wave a wand and rewrite every text book, and change every pi key on a calculator to tau, I'm in. Even though it would be tough for me to rewire my brain away from the pi related equations I've come t... |
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | Eddie | 307 | <p>I feel as though a lot of this misses the point. Teaching the unit circle and trig functions in terms of "one turn" makes a lot more intuitive sense to young students than do arbitrary variables tau or pi. It just so happens that Tau is equivalent to one turn. Once students grasp the concept of the unit circle, then... |
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | Benoît Kloeckner | 187 | <p>After coming back to this question, it strikes me that in fact the best thing might be to explicitly teach both. Not to accommodate the advantages of $\tau$ with the entrenchment of $\pi$, but simply because it makes clearer that pi is a convention, has a definition, and is not just some magical formula given to us.... |
1,833,495 | <p>I tried to solve this like this.</p>
<p>$x=1,y=1$ is solution.</p>
<p>And Let $x=a y=b\, (a\geq 1,b\geq2)$</p>
<p>Then, $11$ can divide $11^b = 10^a+1$</p>
<p>so $10^a = 10 \pmod{11}$ but order of $11(10) = 2$.</p>
<p>Then there is contradiction.</p>
<p>Can I solve this problem while I make $x=a y=b\, (a\geq2 ... | Joss | 338,303 | <p>If you take logarithms in both sides you get $x=log(11^y-1)$ so you have a solution for each $y$ greater than $0$. This works in case you aren´t looking for integer solutions. </p>
|
2,922,290 | <p>Lemma 2.2</p>
<p>Let $(X,d)$ be a compact metric space and suppose that $U$ is a clopen subset of $X$. </p>
<p>There exists $\delta >0 $ s.t $d(x,y) \geq \delta $ $\forall x \in U $ and $y\in X-U $</p>
<p>Proof attempt</p>
<p>Suppose that it is not true then for all $\delta >0 $ $\exists x \in U $ and $\e... | mathworker21 | 366,088 | <p>Take $y \in U^c$. For each $n \ge 1$, let $E_n = \{x \in U : d(x,y) \ge \frac{1}{n}\}$. Each $E_n$ is closed and $\cap_{n \ge 1} E_n = \emptyset$. By compactness (and the fact that $E_n \subseteq E_m$ for $n \ge m$), some $E_N \not = \emptyset$.</p>
<p>For $n \ge 1$, let $F_n = \{y \not \in U : \exists x \in U, d(x... |
3,511,565 | <p>I am in the middle of an exercise and I am stuck in a final step in which I want to show that <span class="math-container">$y-x^2 \notin \langle x(x-1), (x-1)y\rangle$</span> in <span class="math-container">$\mathbb{C}[x,y]$</span>. My first thought was that <span class="math-container">$x-1$</span> does not divide ... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/YY0aH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YY0aH.png" alt="enter image description here"></a></p>
<p>The property checks out
with
a straightforward proof using
complex numbers representation, with
the circumcenter at the origin,</p>
<p><span class=... |
204,003 | <p>I have a sequence of numbers
like 1,7,22,45,12,96,21,45,65,36,85,14,51,16,18,17,16....65...</p>
<blockquote>
<p>IS there any formula to check whether the sequence is random or not ?</p>
</blockquote>
<p>In my case</p>
<ol>
<li>odd numbers are not random since previous+2 </li>
<li>even numbers are not random <... | Godot | 38,875 | <p>First of all you have to tell what "random" means...</p>
<p>But in the algorithmic context there is a nice definition.</p>
<p>To be more specific: a number $\alpha\in(0,1)$ is algorithmically random if for every $n$ the size of the shortest program giving you the first $n$ bits of $\alpha$ is at least $n$ bits lon... |
2,588,271 | <blockquote>
<p>Show that $\mathcal{L}\{t^{1/2}\}=\sqrt{\pi}/(2s^{3/2}), \: s>0$</p>
</blockquote>
<p>By the definition of Laplace transform we get:</p>
<p>$$\mathcal{L}\{t^{1/2}\} = \int_0^\infty t^{1/2}e^{-st} \, dt = \{x = \sqrt{st} \} = \dfrac{2}{s^{3/2}} \int_0^\infty e^{-x^2} x^2 \, dx. $$</p>
<p>A known ... | Fimpellizzeri | 173,410 | <p><strong>Hint:</strong> Consider integration by parts on</p>
<p>$$
\left(
x\,e^{-x^2}
\right)
\cdot x
$$</p>
|
2,588,271 | <blockquote>
<p>Show that $\mathcal{L}\{t^{1/2}\}=\sqrt{\pi}/(2s^{3/2}), \: s>0$</p>
</blockquote>
<p>By the definition of Laplace transform we get:</p>
<p>$$\mathcal{L}\{t^{1/2}\} = \int_0^\infty t^{1/2}e^{-st} \, dt = \{x = \sqrt{st} \} = \dfrac{2}{s^{3/2}} \int_0^\infty e^{-x^2} x^2 \, dx. $$</p>
<p>A known ... | Tob Ernack | 275,602 | <p>Rewrite the integral as $$\int\limits_0^\infty-\frac{x}{2} \left(-2xe^{-x^2}\right)dx$$ and use integration by parts with $f(x) = -\frac{x}{2}$ and $g(x) = e^{-x^2}$ to obtain $$\left. -\frac x 2 e^{-x^2}\right|_0^\infty + \frac{1}{2} \int\limits_0^\infty e^{-x^2}dx = \frac{\sqrt{\pi}}{4}$$</p>
|
474,260 | <p>We know that irrational number has not periodic digits of finite number as rational number.<br>
All this means that we can find out which digit exist in any position of rational number.<br>
But what about non-rational or irrational numbers?<br>
For example:<br>
How to find out which digit exists in Fortieth position... | Foo Barrigno | 73,411 | <p>In some cases, yes. See <a href="http://en.wikipedia.org/wiki/Spigot_algorithm" rel="nofollow">spigot algorithms</a> and digit extraction algorithms. In most cases however, no. Most irrational numbers have no such algorithm.</p>
|
1,970,082 | <p>How would you go about proving that:</p>
<p>$\lim_{x\to2^-}\frac{1}{x-2}= -\infty$
(i.e. x approaching 2 from left hand side)</p>
<p>Been struggling to make much headway on this. Not sure how I'm supposed to use Epsilon in this case. So far, I have: <br/>
Given any negative number M there exists $\delta$ s.t. $f(x... | Alexis Olson | 11,246 | <p>For a given $\epsilon$, having $\frac{1}{q} \ge \epsilon$ means that $q$ must be bounded above by $q \le \frac{1}{\epsilon}$ so you only have finitely many $q$. Now since all the $\frac{p}{q}$ must be in a bounded interval, you can only have finitely many $p$ as well (or else the ratio would grow arbitrarily large).... |
190,905 | <p>I want to make two general graphics something like the following.</p>
<p><a href="https://i.stack.imgur.com/1SNYt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1SNYt.png" alt="enter image description here"></a></p>
<p>I want to illustrate the approach to the function by means of a polynomial. ... | kglr | 125 | <pre><code>f[x_] := 1 + x + Sin[10 Sqrt[1 + x]];
g[x_] := 1 + x + Cos[10 Sqrt[2 + x]];
xx = x /. NSolve[{f'[x] == g'[x], 0 <= x <= 8}, x];
ticks = MapIndexed[{#, Style[ Subscript["x", ToString@ #2[[1]]], 16]} &, xx];
Plot[{f[x], g[x]}, {x, 0, 3 Pi}, AxesLabel -> {x, y},
MeshFunctions -> {f'[#] - g'[#... |
14,167 | <p>I randomly pick a natural number <em>n</em>. Assuming that I would have picked each number with the same probability, what was the probability for me to pick <em>n</em> before I did it?</p>
| Anne van Rossum | 63,110 | <p>It would require a definition of a probability distribution with an infinitesimal probability mass at every natural number. Dirac delta functions of infinitesimal length. </p>
<p>In (Bayesian) statistics, functions that relax the axiom of "summation to one" are known as <a href="https://en.wikipedia.org/wiki/Prior_... |
1,073,730 | <p>Is it possible to prove the proposition</p>
<p>if $P \land Q \Rightarrow R$</p>
<p>then</p>
<p>$P \land R \Rightarrow \lnot Q $</p>
<p>or in other words</p>
<p>($P \land Q \Rightarrow R) \Rightarrow (P \land R \Rightarrow \lnot Q )$</p>
| parsiad | 64,601 | <p>Consider $P,Q,R=1$. The first statement is true and the second statement is...</p>
|
1,496,676 | <p>let's $F(x)= x'Ax + a'x $</p>
<p>where $x'Ax$ is a quadratic form and $a'$ is defined as a vector. </p>
<p>$$A:= \left[ \begin{matrix} 6 &1&1 \\ 1&2&0 \\ 1&0&4\end{matrix}\right] $$</p>
<p>Does there exist a global minimum point (absolute min) for this function $F(x)$?</p>
<p>note: I know... | John Alexiou | 3,301 | <p>Isn't minimum or maximum were the gradient is zero (vector)?</p>
<p>$$ \frac{{\rm d}F(x)}{{\rm d}x} = 2 A x + a' =0 $$</p>
<p>So the unique solution </p>
<p>$$ x = - \frac{1}{2} A^{-1} a' $$</p>
<p>if it exists must be a minimum or maximum. </p>
<h2>Edit 1</h2>
<p>So for your example suppose $a'=(a_1,a_2,a_3)$... |
708,138 | <p>Show that the maximum of a convex function on a convex polytope occurs at one of the extreme points of the polytope. I am really stuck here and don't know where to begin.</p>
| Jimmy R. | 128,037 | <p>Hint: For the following reasoning use</p>
<ol>
<li>The definition of convexity of a function</li>
<li>The fact that the convex polytope over which the function is maximized contains all convex combinations of points in it. In fact every point in the convex polytope can be expressed as a convex combination of it's e... |
2,027,888 | <h2>Exercise</h2>
<p>If $H$ is the Heaviside function, prove, using the definition below, that $\lim \limits_{t \to 0}{H(t)}$ does not exist.</p>
<hr>
<h2>Definition</h2>
<blockquote>
<p>Let $f$ be a function defined on some open interval that contains the number $a$, except possible $a$ itself. Then we say that ... | Doug M | 317,162 | <p>$\forall$ means "for all" or "for any"</p>
<p>$\exists$ means "there exists"</p>
<p>For any $\delta > 0$ there exist a $t$ such that $|t|<\delta$ and $|H(t) - L|>\epsilon$</p>
<p>What are we doing.</p>
<p>If the limit exist. Then:</p>
<p>For any $\epsilon > 0$ there exist a $\delta>0$ such that w... |
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