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 |
|---|---|---|---|---|
3,458,406 | <p>Define <span class="math-container">$f(n)=(2n)^2 + 1,n \in \mathbb{N}$</span></p>
<p>From <span class="math-container">$1$</span> to <span class="math-container">$10^7$</span> there's <span class="math-container">$15$</span> numbers that <span class="math-container">$f(n)$</span> is prime, <span class="math-contain... | Empy2 | 81,790 | <p>There are <span class="math-container">$40$</span> possible final two digits -those that end <span class="math-container">$2, 3, 7$</span> and <span class="math-container">$8$</span><br>
The chance that one of the <span class="math-container">$40$</span> appears four times is <span class="math-container">$$40{15\cho... |
535,533 | <p>My confusion is how do we define : $\sin (x)$ for $x\in \mathbb{R}$.</p>
<p>I only know that $\sin(x)$ is defined for degrees and radians..</p>
<p>Suddenly, I have seen what is $\sin (2)$.. </p>
<p>I have no idea how to interpret this when not much information is given what $2$ is... </p>
<p>does this mean $2$ r... | Arthur | 99,272 | <p>You are using radians in your case. The most common definition of the sine is $\sin(x) := \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} x^{2n+1}$ though, which coincides with the sine in radians as you know it.</p>
|
1,857,196 | <p><strong>Question :</strong>
Let $f(x) = \sum^n_{k=0}c_kx^k$ be a polynomial function then prove that if $f(x) = 0$ for $n+1$ distinct real values, then every coefficient $c_k$ in $f(x)$ is $0$ , thus $f(x) = 0$ for all real values of $x$.</p>
<p><strong>What I think :</strong>
My problem is that I have no other ori... | learning_math | 350,148 | <p>It is actually possible for a polynomial of degree $n$ to have more than $n$ roots. For example, consider the ring $R=\mathbb{Z}/25\mathbb{Z}$. Consider the polynomial $f(x)=x^2\in R[x]$. Then $f$ has $5$ roots, namely, $\overline{0}, \overline{5}, \overline{10}, \overline{15}, \overline{20}$ in $R$.</p>
<p>A polyn... |
1,448,213 | <p>In other words, consider $A_n$, the alternating group of the $n$-th symmetrical group $S_n$, is it true that
$$A_n=\{a^2\mid a\in A_n\}$$?
I tested for $S_3$ and it seemed to hold. If it is true, it will be very helpful to me for solving another problem. </p>
| Shervin | 273,520 | <p>Show that $\Vert Lf\Vert\le\alpha$ for all $\Vert f \Vert=1$. </p>
<p>Because </p>
<p>$1\ge \pm f, \forall x \in [0,1]$ </p>
<p>we have that </p>
<p>$1\pm f\ge 0$, </p>
<p>thus, </p>
<p>$L(1\pm f)\ge 0$. </p>
<p>By linearity </p>
<p>$-L1\le Lf\le L1$, </p>
<p>that is </p>
<p>$|Lf|\le L1, \forall x \in [0,1... |
3,806,122 | <p>I tried using Chinese remainder theorem but I kept getting 19 instead of 9.</p>
<p>Here are my steps</p>
<p><span class="math-container">$$
\begin{split}
M &= 88 = 8 \times 11 \\
x_1 &= 123^{456}\equiv 2^{456} \equiv 2^{6} \equiv 64 \equiv 9 \pmod{11} \\
y_1 &= 9^{-1} \equiv 9^9 \equiv (-2)^9 \equiv -512... | Stinking Bishop | 700,480 | <p><span class="math-container">$y_1$</span> should've been the inverse of <span class="math-container">$8\pmod{11}$</span>, not of <span class="math-container">$9\pmod{11}$</span>, so <span class="math-container">$y_1=7$</span>.</p>
<p>Similarly, <span class="math-container">$y_2$</span> should've been the inverse of ... |
135,252 | <p>Evaluate $\displaystyle \lim_{n \to +\infty}\sum_{k=n+1}^{2n}\frac{1}{k}$. What are the ways of counting such things? My last topic in school was Riemann integral, can I use it here?</p>
| Adam Rubinson | 29,156 | <p>The sum equals (the sum of 1/k from k=1 to 2n) minus (the sum of 1/k from k=n+2 to 2n). Then take the limit.</p>
|
376,861 | <p>A knot can be represented with a <a href="http://katlas.math.toronto.edu/wiki/MorseLink_Presentations" rel="nofollow noreferrer">Morse link presentation</a>, as a combination of cups, caps and crossings (which is not uniquely determined by the knot, of course):</p>
<p><a href="https://i.stack.imgur.com/47mxu.png" re... | M. Ozawa | 46,903 | <p>I understood your question.
I think it is true. First we isotope a Morse position to a bridge position without zig-zag moves. Then we have a bridge position of the trivial knot, which has been proved to be unique up to bridge isotopies by Otal.
Hence we have the trivial knot diagram in a bridge position.</p>
<p><em>... |
4,521,774 | <p>In many posts on MSE, it is discussed that Cauchy sequences can't be defined in General topological spaces and in a typical topology book it is discussed what converging sequences are, but, what I don't understand is, why, on an abstract level, does convergence generalize even without a metric while cauchy-ness does... | Lee Mosher | 26,501 | <p>The first abstract idea to come to terms with is this:</p>
<blockquote>
<p>The following statement is <em><strong>false</strong></em>: For every topological space <span class="math-container">$X$</span>, for any two metrics <span class="math-container">$d,d'$</span> that generate the topology on <span class="math-co... |
2,258,139 | <p>A natural number $n>1$ is called <em>good</em> if$$n \mid 2^n+1.$$ For example, $n=3$ is good, as $3 \mid 2^3+1=9$. Prove that if $N_1$ and $N_2$ are good, then:</p>
<ul>
<li>$\mathrm{lcm}(N_1,N_2)$ and $\gcd(N_1,N_2)$ are good,</li>
<li>$N_1\cdot N_2$ is good. </li>
</ul>
<p>This seems pretty difficult for me.... | SiXUlm | 58,484 | <p>Let $n_1,n_2$ be two good numbers. Denote: $k = LCM(n_1,n_2)$ and $d = (n_1,n_2)$. Then we have simple relation: $dk = n_1n_2$.</p>
<p>For the first part, @Aaron has already done. The LCM part can be done more simple using the following property: if $a | x$ and $b | x$ then $LCM(a,b) | x$. In our case, $n_1 | 2^{n_... |
4,235,480 | <p><span class="math-container">$~ \mathbb{N} \cup \left\{ 0 \right\} ~~ \leftarrow~~ \text{The set of integers each of which is greater or equal than zero} ~$</span></p>
<p>I want to know or create the alternative(s) of set of <span class="math-container">$~ \mathbb{N} \cup \left\{ 0 \right\} ~$</span></p>
<p>As I w... | David A. Craven | 804,921 | <p>Here is a proof that does not use the sizes of the classes, just that it has four classes. Let <span class="math-container">$x_i$</span> be an element from class <span class="math-container">$C_i$</span>. We start by inserting the trivial character, the action of <span class="math-container">$\chi_2$</span> on <span... |
2,571,395 | <p>I recently reached got a nice answer from my <a href="https://math.stackexchange.com/questions/2567486/integrating-int-x-1x-2-sqrt12at-ax-1x-22-dt">previous question</a> but I quickly that the problem would be unreasonable unless $x_1$ is not a variable and always holds some value, preferably 0, which simplifies the... | Jack D'Aurizio | 44,121 | <p>Let us assume to have a function $L$ defined in terms of $a,b>0$ as
$$ L(a,b) = \frac{1}{a}\int_{0}^{ab}\sqrt{1+t^2}\,dt \stackrel{t\mapsto au}{=}\int_{0}^{b}\sqrt{1+a^2 u^2}\,du. \tag{0}$$
Geometric interpretation: $\int_{x_0}^{x_1}\sqrt{1+f'(u)^2}\,du$ is the length of the graph of $f(x)$ over the interval $[x_... |
1,458,144 | <p><a href="https://i.stack.imgur.com/pJcCp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pJcCp.png" alt="enter image description here"></a></p>
<p>I have this function and I'm trying to write a program to compute it as n approaches 100. The problem is it overflows once it reaches around 50. The h... | John | 7,163 | <p>Well, $(a+b)(a-b)$ is $a^2 - b^2$, so perhaps you treat the two terms as $a^2$ and $b^2$ to get:</p>
<p>$$x_{n+1} = 2^{n+1}((1+2^{-n}x_n)^{1/4} + 1)((1+2^{-n}x_n)^{1/4} - 1)$$</p>
<p>Maybe this helps?</p>
|
4,286,296 | <p>I am trying to prove the following claim:</p>
<blockquote>
<p>Let <span class="math-container">$ 0\leq n \in \Bbb Z$</span> and suppose that there exists a <span class="math-container">$k \in \Bbb Z$</span> such that <span class="math-container">$n=4k+3$</span>.
Prove or disprove: <span class="math-container">$\sqr... | lone student | 460,967 | <blockquote>
<p><strong>Statement:</strong></p>
<p>Let <span class="math-container">$a,b,k\in\mathbb Z^{+}$</span>, where <span class="math-container">$\gcd (a,b)=1$</span> and if <span class="math-container">$4k+3=\frac{a^2}{b^2}$</span>, then <span class="math-container">$b^2=1$</span> or <span class="math-container"... |
137,435 | <p>Consider the braid group on n strands given in the usual Artin presentation. Then add extra relations: each Artin generator has order d. For example, if d=2, one recovers the symmetric group. I would like to know what the order of the group is for arbitrary n and d. Even knowing the name of such groups would be help... | David Mitra | 18,986 | <p>Assuming $A+B=\{a+b\mid a\in A, b\in B\}$:</p>
<p>$A=\{\,1,2,3,\ldots\,\}$ and $B=\{ \,-1 +{1\over2}, -2 +{1\over3} ,-3+{1\over4},\ldots\,\}$. The sum contains $\{\,{1\over2},{1\over3},{1\over4},\ldots\,\}$ but not its limit point $0$.</p>
|
396,297 | <p>Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$?</p>
<p>I've calculated that the recurrence relation for this integral is:</p>
<p>$\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} _{0} - n \cdot \int_0 ^{1} x^{n-1} e^x dx$</p>
<p>So if we let $I_n = \int_0 ^{1} x^n e^x \ dx$, we get ... | Community | -1 | <p>Following the Ishan Banerjee comment let $t=x^n$ hence $dx=\frac{1}{n}t^{\frac{1}{n}-1}dt$ and then
$$ (2n+1) \int_0 ^{1} x^n e^x dx=\frac{2n+1}{n}\int_0^1t^{1/n}e^{t^{1/n}}dt\to2e$$
by using the dominated convergence theorem.</p>
|
396,297 | <p>Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$?</p>
<p>I've calculated that the recurrence relation for this integral is:</p>
<p>$\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} _{0} - n \cdot \int_0 ^{1} x^{n-1} e^x dx$</p>
<p>So if we let $I_n = \int_0 ^{1} x^n e^x \ dx$, we get ... | Did | 6,179 | <p>Let $I_n=\int\limits_0^1x^n\mathrm e^x\mathrm dx$. By integration by parts, $(n+1)I_n=\left.x^{n+1}\mathrm e^x\right|_0^1-I_{n+1}=\mathrm e-I_{n+1}$. Now, $0\leqslant I_{n+1}\leqslant I_n$ hence $(n+1)I_n\leqslant\mathrm e\leqslant(n+2)I_n$. </p>
<p>This is enough to show that
$$
\left(2-\frac3n\right)\cdot\mathrm... |
396,297 | <p>Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$?</p>
<p>I've calculated that the recurrence relation for this integral is:</p>
<p>$\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} _{0} - n \cdot \int_0 ^{1} x^{n-1} e^x dx$</p>
<p>So if we let $I_n = \int_0 ^{1} x^n e^x \ dx$, we get ... | Mhenni Benghorbal | 35,472 | <p>A <a href="https://math.stackexchange.com/questions/375950/asymptotic-for-the-integral-involving-exponential/375970#375970">related technique</a>. You can use integration by parts technique by letting $u=e^{x}$ which leads to</p>
<p>$$ I_n = \left( 2\,n+1 \right) \left( {\frac {{{\rm e}}}{n+1}}-{\frac {{
{\rm e}}}... |
239,720 | <p>If $A$ is unital C$^*$-algebra, is it true that the multiplier algebra of $A \otimes \mathcal{K} $ is $ A \otimes \mathcal{B}(\mathcal{H})$? Where $\mathcal{K}$ is C$^*$-algebra of compact operators on the Hilbert space $\mathcal{H}$.</p>
| Ulrich Pennig | 3,995 | <p>The fact stated in the answer by vap is proven in the paper "<a href="http://www.sciencedirect.com/science/article/pii/0022123673900360" rel="noreferrer">Multipliers of C*-algebras</a>" by Akemann, Pedersen and Tomiyama (see Theorem 3.3, I guess). Moreover, they prove in Theorem 3.8 that multiplier algebras are not ... |
2,379,955 | <p>Assume I want to minimise this:
$$ \min_{x,y} \|A - x y^T\|_F^2$$
then I am finding best rank-1 approximation of A in the squared-error sense and this can be done via the SVD, selecting $x$ and $y$ as left and right singular vectors corresponding to the largest singular value of A.</p>
<p>Now instead, is possible t... | mathreadler | 213,607 | <p>You already have solutions with CVX (convex optimization), but in fact you can solve this using simple ordinary linear least squares and representing matrix multiplication with Kronecker products. Let $M_E$ represent multiplication by $E$ (from the right) and $v_A,v_C,v_x$ be the respective vectorization of $A,C,x$ ... |
136,264 | <p>I have a question concerning the stability analysis for a kind of differential equation taking the form
$$\dot x=Ax+Bw,$$
where $A\in \mathbb{R}^{n \times n}$, $B\in \mathbb{R}^{n \times m}$ are constant matrices and
$w \in \mathbb{R}^m$ is a normal random variable, i.e., $w\sim \mathcal{N}(0,W)$ with $W$ ... | 27hel27 | 47,623 | <p>You can also check out <a href="http://zbmath.org/journals/" rel="nofollow">http://zbmath.org/journals/</a> for details on the journal content. It doesn't give you a ranking though but you see at a glance, who published in the journal you are interested in or what topics are represented in the articles.</p>
|
2,872,701 | <blockquote>
<p>Let $K\subset N\subset M$ be $R-$submodules where $R$ is a commutative ring with unity. If $K$ is a direct summand of $M$ then show that $K$ is a direct summand of $N$. Further, if $N/K$ is a direct summand of $M/K$ then show that $N$ is a direct summand of $M$.</p>
</blockquote>
<p>It is easy to sho... | Matthé van der Lee | 75,745 | <p>Taking $K = N$, it would follow that $N$ is always a direct summand of $M$ - which is false in general (unless $R$ is semisimple, i.e. absolutely projective, i.e. all $R$-modules are projective; for example a field).</p>
|
4,466,733 | <p><span class="math-container">$$\frac{df(x)}{dx}=f(x+5)$$</span>
I am unable to solve this kind of integration using high school mathematics.
Please help.</p>
| emacs drives me nuts | 746,312 | <p>Just an ansatz or guess: Because differentiating <span class="math-container">$a^x$</span> gives a multiple of <span class="math-container">$a^x$</span>, and also <span class="math-container">$a^{x+\mathrm{const}}$</span> is a multiple of <span class="math-container">$a^x$</span>: Try <span class="math-container">$... |
2,548,177 | <p>I'd like to define <code>sumdiv</code> in Maple such that this:</p>
<pre><code>with(numtheory);
f:=x->x^2;
sumdiv(f(d)*mobius(100/d), d=1..100);
</code></pre>
<p>would do a sum on all divisors <code>d</code> of $100$.</p>
<p><strong>How to do such a sum over divisors in Maple?</strong></p>
<p>Here's what I've... | Brethlosze | 386,077 | <p>In Matlab, <code>D</code> is a vector containing all the divisors, for any <code>n</code>:</p>
<pre><code>n=10;
k=1:n;
D=K(rem(n,k)==0);
s=sum(D)
</code></pre>
<p>Edit: The sum of a function <code>f</code> over the divisors of <code>n</code>. Note the <code>.</code> operator before the <code>^</code> and <code>*</... |
2,573,487 | <p>I have given this set</p>
<blockquote>
<p>$$ M = \{ x \in [1,2]\times [3,4] ~|~ x\in\mathbb{Q}^2 \} \subset \mathbb{R}^2 $$</p>
</blockquote>
<p>First I have to identify the boundary $\partial M$ and then tell if it is open or closed.</p>
<p>I think that $$ \partial M = \{ (x,y) ~|~ x\not\in\mathbb{Q}^2, 1\leq ... | José Carlos Santos | 446,262 | <p>Since $\overline M=[1,2]\times[3,4]$ and $\mathring M=\emptyset$, the boundary of $M$ is $\overline M\setminus\mathring M=[1,2]\times[3,4]$, which is a closed set.</p>
|
1,878,975 | <p>X is for continuous random variable and it's nonnegative.
Then this is the formula.</p>
<p>$$E(X)=\int_0^\infty(1-F(x))dx$$</p>
<p>Does anyone know the proof?
I appreciate any help.</p>
| egreg | 62,967 | <p>Consider
$$
u=\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{1}{x^2}}}>
\sqrt{\frac{1}{2}+\frac{1}{2}}=1
$$
Assuming your substitutions are correct, the integral becomes
$$
\sqrt{2}\int\frac{1}{1-u^2}\,du=
\frac{\sqrt{2}}{2}\int\left(\frac{1}{1-u}+\frac{1}{1+u}\right)\,du=
\frac{\sqrt{2}}{2}\log\left|\frac{1+u}{1-u}\... |
2,170,501 | <p>How do you prove the sum of two monotone sequences is also monotone? </p>
<p>Here is my thought process: </p>
<p>Let $a_n$ and $b_n$ be two monotone increasing sequences. Then $\forall n \in N$, $a_n \leq a_{n+1}$ and $b_n \leq b_{n+1}$. Adding both inequalities you get $a_n + b_n \leq a_{n+1} + b_{n+1}$. Therefo... | Adren | 405,819 | <p>Consider the sequences defined by :</p>
<p>$$a_n=3n+(-1)^n$$</p>
<p>and</p>
<p>$$b_n=-2n$$</p>
<p>It is readily seen that $(a_n)$ is increasing, since for all $n\in\mathbb{N}$ : $a_{n+1}-a_n=3+2(-1)^{n+1}>0.$</p>
<p>Obviously, $(b_n)$ is decreasing.</p>
<p>And finally $(a_n+b_n)$ is known to be non monotoni... |
233,075 | <p>I am trying to solve the following equation in the Natural Numbers, with the condition <span class="math-container">$a\ge1$</span>, <span class="math-container">$b\ge1$</span>, and <span class="math-container">$r\ge3$</span>:</p>
<p><span class="math-container">$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + ... | bbgodfrey | 1,063 | <p>The excellent second solution by Roman, with <code>R</code> slightly modified, produces</p>
<pre><code>R = HornerForm[(a (3 + a) (-12 + 5 a) + 3 (9 - 14 b) b)/
((-1 + a) a (3 + a) - 3 (-1 + b) b)]
With[{s = 10^4}, Do[If[Divisible[a (3 + a) (-12 + 5 a) + 3 (9 - 14 b) b,
(-1 + a) a (3 + a) - 3 (-1 + b) b] &am... |
1,921,879 | <blockquote>
<p>Find all positive integers $n$ for which $\dfrac{x^n + y^n + z^n}2$ is a perfect square, whenever $x$, $y$, and $z$ are integers such that $x + y + z = 0$.</p>
</blockquote>
<p>I don't even know where to start.</p>
| Dietrich Burde | 83,966 | <p>I have no complete answer, but a start (as you wanted to know where to start). For $n=1$ we have that $(x^1+y^1+z^1)/2=0$ is a perfect square for all $x,y,z$ with $x+y+z=0$. So we may assume $n\ge 2$. Now choose, say, $(x,y,z)=(1,1,-2)$. Then $x+y+z=0$ and
$$
\frac{x^n+y^n+z^n}{2}=\frac{2+(-2)^n}{2}.
$$
This can nev... |
3,912,734 | <p>My text book in linear algebra - out of the blue - claims that:</p>
<p><span class="math-container">$|\lambda u|=|\lambda||u|$</span></p>
<p>Where u is a vector and <span class="math-container">$\lambda$</span> is a constant.</p>
<p>I would understand if || were used to denote absolute numbers, but in this book, || ... | Aryaman Maithani | 427,810 | <p>It is part of the definition of an inner product <span class="math-container">$\langle \;,\; \rangle$</span> on a complex vector space <span class="math-container">$V$</span> that</p>
<ul>
<li><span class="math-container">$\langle \lambda v, w\rangle = \lambda\langle v, w\rangle,$</span> and</li>
<li><span class="ma... |
377,925 | <p>This post comes from the suggestion of <a href="https://mathoverflow.net/users/18698/joel-moreira">Joel Moreira</a> in a <a href="https://mathoverflow.net/questions/377706/an-alternative-to-continued-fraction-and-applications#comment958452_377706">comment</a> on <a href="https://mathoverflow.net/q/377706/34538">An a... | katago | 114,101 | <blockquote>
<p><span class="math-container">$u_{0} \in \mathbb{Q} \quad u_{n+1}=\left[u_{n}\right]\left(u_{n}-\left[u_{n}\right]+1\right)$</span>, then <span class="math-container">$\{u_{n}\}_{n=1}^{+\infty}$</span> reach in integer. <span class="math-container">$\quad (*)$</span></p>
</blockquote>
<p>This can be prov... |
84,254 | <p>For me, a simplicial groupoid is a simplicial object in ${\mathbf{Grpd}}$. I am more general than Goerss-Jardine in this definition.</p>
<p>Do you have examples simplicial groupoids that occur in nature? Here's what I have got:</p>
<ol>
<li>Given a simplicial group $G$ acting on a simplicial set $X$, the action gr... | none | 20,161 | <p>I think it is very doable. I took a class like that as an undergrad. The textbook was Enderton's "Introduction to Mathematical Logic". It did enough Hilbert-style proof theory to get up to the incompleteness theorem, then discussed models, interpretations, Tarski's definition of truth, etc. It seemed great at th... |
84,254 | <p>For me, a simplicial groupoid is a simplicial object in ${\mathbf{Grpd}}$. I am more general than Goerss-Jardine in this definition.</p>
<p>Do you have examples simplicial groupoids that occur in nature? Here's what I have got:</p>
<ol>
<li>Given a simplicial group $G$ acting on a simplicial set $X$, the action gr... | user729424 | 5,698 | <p>Kenneth Kunen recently wrote a wonderful introduction to Mathematical Logic, called "The Foundations of Mathematics" (ISBN: 978-1-904987-14-7), published in 2009. The book's only prerequisite is the mathematical maturity that an Introduction to Analysis course would provide, so it sounds like your students would b... |
924,555 | <p>My homework question:</p>
<blockquote>
<p>From the order axioms for $\mathbb{R}$, show that $0 < 1$. [<em>Hint:</em> From the field axioms, $0 \not=1$. By the trichotomy property, either $0<1$ or $4<0$. Assuming $1 < 0$, get $0 < -1$. Now use Exercise 4.]</p>
</blockquote>
<p>Exercise 4 from my te... | orangeskid | 168,051 | <p>How about proving that $x^2 >0$ whenever $x \ne 0$ and then noticing that $1 = 1^2$.</p>
<p>So, let $x \ne 0$. Then $x > 0$ or $x < 0$. If $x>0$ then $x^2 = x\cdot x > 0$. If $ x < 0$ then adding $-x$ to both terms we get $0 < -x$ and therefore $0 < (-x)(-x) = x^2$. </p>
<p>One should also ... |
225,253 | <p>Simple question, I just cannot find something that explains it right out and to the point without giving a huge confusing explanation. The question that I am struggling with is to determine a limit of a function if it exists.</p>
<blockquote>
<p>Find: <span class="math-container">$$\lim_{x\to2}{f(x)},$$</span></p>
<... | Sidd Singal | 37,043 | <p>The answer is $4$. The limit asks for the value of a function as it <em>approaches</em> some $x$ value, not the exact value. There can be a hole at $x=2$ and your answer would still be valid. </p>
<p>For a more technical answer, take the following definition of a limit: $$\forall \varepsilon \gt 0 \: \exists \delta... |
2,153,421 | <p>I want to find curvatures and torsions for the following curves but get stuck with their natural parametrizations ($s$ is natural if $|\dot{\gamma}(s)| = 1$). Can anyone help me?</p>
<p>(a) $e^t(\cos t,\sin t,1)$</p>
<p>(b) $(t^3+t,t^3-t,\sqrt{3}t^2)$</p>
<p>(c) $3x^2+15y^2=1, z=xy$</p>
<p><strong>Update</strong... | Ng Chung Tak | 299,599 | <p>$(a)$
\begin{align*}
\mathbf{\dot{r}}(t) &= e^{t}(\cos t-\sin t, \sin t+\cos t,1) \\
|\mathbf{\dot{r}}(t)| &= e^{t}\sqrt{(\cos t-\sin t)^2+(\sin t+\cos t)^2+1} \\
&= e^{t}\sqrt{3} \\
s &= \int_{0}^{t} |\mathbf{\dot{r}}(t)| \, dt \\
&= \sqrt{3}(e^{t}-1) \\
t &= \ln \left( 1+\frac{s... |
388,766 | <p>I need to show that every element in $\Bbb Z/p\Bbb Z$ can be written as a sum of two squares. The case $p=2$ is trivial and $0$ is always $0^2 + 0^2$. So all I have to do is show that every element of $(\Bbb Z/p\Bbb Z)^\times$ (the group of units) can be expressed as a sum of two squares. The question hints that I s... | André Nicolas | 6,312 | <p>The following is another version of the argument by Jyrki Lahtonen. Beside the numbers $1,2,\dots, p-1$ write QR for quadratic residue and NR for quadratic non-residue. So we write QR beside $1$. </p>
<p>Note that we write each of QR and NR $\frac{p-1}{2}$ times. So at some time, QR is followed by NR. This means th... |
2,340,487 | <p>I was trying to compute this limit:
$$\lim_{x \to 0}\lim_{y \to 0} (x+y)\sin{\frac{x}{y}}$$</p>
<p>And this is my solution:
$$\lim_{x \to 0}\lim_{y \to 0}|(x+y)\sin{\frac{x}{y}}|\leq\lim_{x \to 0}\lim_{y \to 0} |(x+y)|=0$$</p>
<p>So I got the limit 0.</p>
<p>The answer was different. I have no idea what is wrong ... | robjohn | 13,854 | <p>Note that
$$
\lim_{y\to0}(x+y)\sin\left(\frac xy\right)
$$
is indeterminate for each non-zero $x$. Thus, there is no way to compute
$$
\lim_{x\to0}\lim_{y\to0}(x+y)\sin\left(\frac xy\right)
$$
Whereas,
$$
\lim_{x\to0}(x+y)\sin\left(\frac xy\right)=0
$$
for each non-zero $y$, so
$$
\lim_{y\to0}\lim_{x\to0}(x+y)\sin\l... |
4,458,863 | <p>Let <span class="math-container">$z_1,\;z_2,\;z_3\;$</span> be complex number such that <span class="math-container">$|z_1|=|z_2|=|z_3|=|z_1+z_2+z_3|=2\;\;$</span>. If <span class="math-container">$|z_1-z_3|=|z_1-z_2|\; \;$</span> and <span class="math-container">$z_2 \neq z_3.\; \; $</span> Then Find value of <span... | Claude Leibovici | 82,404 | <p>In the same spirit as @Dan, at least for small <span class="math-container">$x$</span>, use
<span class="math-container">$$\sqrt{\cosh (x)}=1+\sum_{n=1}^\infty \frac {(-1)^{n+1}}{2^n \,(2n)!}\,a_n\,x^{2n}$$</span> the first <span class="math-container">$a_n$</span> being
<span class="math-container">$$\{1,1,19,559,2... |
3,355,544 | <p><a href="https://i.stack.imgur.com/oYf7f.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/oYf7f.jpg" alt="enter image description here"></a></p>
<p>Given the following equations:</p>
<p><span class="math-container">$$x=t^2$$</span></p>
<p><span class="math-container">$$y=t^2$$</span></p>
<p><sp... | Quanto | 686,284 | <p>Rotate the <span class="math-container">$xyz$</span>-coordinates by 45 degrees around the <span class="math-container">$z$</span> axis to the <span class="math-container">$uvz$</span>-coordinates, which is equivalent to </p>
<p><span class="math-container">$$x=u+v, \>\>\> y=u-v$$</span></p>
<p>Rewrite the... |
544,779 | <p>So here are the contextual statements:
1) Maya either listens to music or does her homework. If she listens to music she feels happy.If she does her homework she feels unhappy. Therefore she will not do her homework while listening to music.</p>
<p>Let P be the statement "Maya listens to Music".
Q "Maya does homewo... | Community | -1 | <p>In defining the topology on the one point compactification, one needs to claim that $X\setminus K \cup \{\infty\}$ be open. But if $X$ is not Hausdorff, $K$ might not be closed. Thus it might be more natural to assume that $X$ is Hausdorff. </p>
<p>There are a lot of compactification, as long as $f(X)$ is dense in ... |
2,269,917 | <p>I've come across this question in a university exam paper. It's causing me a huge headache due to the fact that it goes from one vector space to another ($\def\R{\Bbb R}\R^3 \to \R^2$), otherwise it would be fairly standard. If anyone could shed some light on what I'm missing it would be much appreciated.</p>
<p>A4... | SEWillB | 441,390 | <p>I am assuming since you are a university student that you know the rank-nullity theorem- and hopefully it's proof! We take a basis for the kernel of the linear map, call the map $T$ between vector spaces $V$ and $W$;<br>
Say $\{v_1,...,v_m\} $ is a basis for $ker(T)$<br>
Extend to a basis for $V$, say $\{v_1,...,v_m... |
649,379 | <p>I'm on the final part of my project, where I have to prove the Noether-Lasker Theorem (or copy out the following proof and "fill in the gaps"). I'm looking for an explanation of what's going on at a macro-level. I think I could follow the proof, but I don't understand how it proves what it says it proves. I've alrea... | rschwieb | 29,335 | <p>After proving an ideal can be expressed at least one way as an intersection of primary ideals, the next logical step is to make that expression as "tight" as possible by eliminating redundancy.</p>
<p>The same idea applies to generating sets of vector spaces. After you find one generating set, then you can remove r... |
3,202,955 | <blockquote>
<p><strong>Note:</strong> Please do not give a solution; I would prefer guidance to help me complete the question myself. Thank you.</p>
</blockquote>
<hr>
<p>I am having trouble understanding and finding the continuous and residual spectrum. I am working through the following problem:</p>
<p>Let <spa... | Disintegrating By Parts | 112,478 | <p>A quick overview might be helpful, even though it does not really fit your requirement for an answer.</p>
<p><span class="math-container">$T_{\alpha}$</span> is a bounded normal operator. That rules out all but continuous and point spectrum. Every <span class="math-container">$\alpha_j$</span> is in the point spect... |
13,030 | <p>At work, we were discussing when is it the best time to change to winter tires for bikes and/or cars.</p>
<p>Using <code>WeatherData[]</code> and <code>DateListPlot[]</code>, it was fairly straightforward for me to create the diagram below:</p>
<p><img src="https://i.stack.imgur.com/Y5wNT.png" alt="Mean temperatur... | J. M.'s persistent exhaustion | 50 | <p>If my understanding of the question is correct, this does the job:</p>
<pre><code>cityTemp = WeatherData["Stockholm", "MeanTemperature",
{{1977, 7, 1}, {2011, 12, 31}, "Day"}];
(* proportions; I ignore February 29 for this *)
props = Array[(Count[#, {_List, _?Negative}]/Length[#]) &[Case... |
1,449,776 | <p>I have always known that $a^n=a*a*a*.....$(n times)</p>
<p>Then what exactly is the meaning if $a^0$ and why will it be equal to $1$?</p>
<p>I have checked it in the internet but everywhere the solution is based on the principle that $a^m*a^n=a^{m+n}$ and when $n=0$ it will be $a^m$ and clearly $a^0$ is equal to $... | Russ H | 273,811 | <p>You are correct in that $a$ is repeated zero times.</p>
<p>$a^n = a*a*a*... = 1 * a*a*a*...$</p>
<p>And so $a^0 = 1$ when $a$ is repeated zero times</p>
|
1,217,557 | <p>I was tasked with drawing the contour lines of $ z = \sqrt{xy} $, which I find a bit problematic since I can see no way in which one can plot (by hand, and not with wolfram and others....) the $ z = \sqrt{xy} $ graph in $R^2( x-, y- $ projection} to begin with for this surface...</p>
<p>How can one draw this conto... | MissMonicaE | 227,754 | <p>You're right, but I often see people using "linear" to mean "polynomial of degree 1" outside of linear algebra contexts, especially in introductory e.g. calculus courses. In fact, I say this to my calc tutoring students <em>all the time</em> and I didn't realize it's actually wrong until now.</p>
|
2,852,550 | <p>I was wondering, Is there Cauchy sequence that are not bounded ? Of course, in complete spaces is not possible. I have a theorem that says that if $(A,d)$ is not complete, then $(\bar A,d)$ is complete. But are there spaces s.t. indeed for all $\varepsilon>0$, there is $N$ s.t. $d(x_n,x_{m})<\varepsilon$ for a... | Florian R | 511,528 | <p>No, that cannot happen. Suppose that $(x_n)_{n \geq 0}$ is a Cauchy sequence in some metric space. By definition, there exists an integer $N \geq 0$ such that for all $n,m \geq N$ we have $d(x_n,x_m) \leq 1$. Thus, all the points $\{x_n ~|~ n \geq N\}$ are contained in the ball of radius 1 around $x_N$. By adding th... |
374,881 | <p>I'd like to know how I can recursively (iteratively) compute variance, so that I may calculate the standard deviation of a very large dataset in javascript. The input is a sorted array of positive integers.</p>
| Did | 6,179 | <p>Recall that, for every $n\geqslant1$,
$$
\bar x_n=\frac1n\sum_{k=1}^nx_k,
$$
and
$$
\bar\sigma^2_n=\frac1n\sum_{k=1}^n(x_k-\bar x_n)^2=\frac1n\sum_{k=1}^nx_k^2-(\bar x_n)^2.
$$
Hence simple algebraic manipulations starting from the identities
$$
(n+1)\bar x_{n+1}=n\bar x_n+x_{n+1},
$$
and
$$
(n+1)(\bar\sigma^2_{n+1}... |
688,782 | <p>$$a_n=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right) \cdots
\left(1-\frac{1}{n^2}\right)
$$</p>
<p>I have proved that this sequence is decreasing. However I am trying to figure out how to find its limit. </p>
| Cameron Williams | 22,551 | <p>Hint: try defining $b_n = \ln(a_n)$ (which is well-defined) and see what limit this goes to. Then use a certain exponential function to see what $\lim_n a_n$ is.</p>
|
19,996 | <p>In 1556, Tartaglia claimed that the sums<br>
1 + 2 + 4<br>
1 + 2 + 4 + 8<br>
1 + 2 + 4 + 8 + 16<br>
are alternative prime and composite. Show that his conjecture is false. </p>
<p>With a simple counter example, $1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255$, apparently it's false. However, I want to prove it in gen... | Gottfried Helms | 1,714 | <p>Although Ross's proof is much shorter and elegant I show my exercising because it uses a slightly different logic. </p>
<p>Assume, at some <em>n</em> we have $\small 2^n-1 = p_0 $ where $\small p_0 $ is prime. We use only the fact, that <em>n</em> must then be odd (otherwise it would factor into $\small (2^{n/2... |
1,841,882 | <p>By Cayley's theorem, we know that for any finite group $G$, there exists $N \in \mathbb{N}$ such that $G$ is isomorphic to a subgroup of $S_N$, the symmetric group on $N$ letters. Can we prove that for every finite group $G$ there is some symmetric group $S_N$ such that $G$ is isomorphic to a $normal$ subgroup of $S... | wnoise | 7,882 | <p>In general (i.e. for $N \neq 4$), the only normal subgroups of $S_N$ are $S_N$ itself, $A_N$, and $1.$ Therefore no, because most $G$ will not map to one of these. ($S_4$ has an additional normal subgroup, the Klein $4$-group hiding in it.)</p>
|
2,521,710 | <p>I am trying to do a proof for convergence. But I am stuck in my proof not getting any further... What is missing to finish that proof?</p>
<p>$$a_n = \frac{1}{(n+1)^2}$$
Show that: $$\lim_{n \to \infty}a_n=0$$</p>
<p>Let $e > 0$ and $\forall n \ge n_0 = \lceil \frac{1}{\sqrt{\epsilon}}\rceil+1 \in \mathbb Z^+... | copper.hat | 27,978 | <p>You are working too hard.</p>
<p>Look for a simpler upper bound before computing an $\epsilon$.</p>
<p>Note that if $n \ge 1$ then $0 \le a_n = {1 \over (1+n)^2} \le {1 \over 1+n} \le { 1\over n}$.</p>
<p>Choose $n \ge \max(1, {1 \over \epsilon})$.</p>
|
2,521,710 | <p>I am trying to do a proof for convergence. But I am stuck in my proof not getting any further... What is missing to finish that proof?</p>
<p>$$a_n = \frac{1}{(n+1)^2}$$
Show that: $$\lim_{n \to \infty}a_n=0$$</p>
<p>Let $e > 0$ and $\forall n \ge n_0 = \lceil \frac{1}{\sqrt{\epsilon}}\rceil+1 \in \mathbb Z^+... | Peter Szilas | 408,605 | <p>And one more:</p>
<p>$|a_n| = |\dfrac{1}{(n+1)^2}| \lt |\dfrac{1}{n^2}| \le |\dfrac{1}{n}|.$</p>
<p>Let $\epsilon \gt 0$ be given.</p>
<p>There is a $n_0$ such that $n_0 \gt 1/\epsilon.$
(Archimedes).</p>
<p>For $n \ge n_0 :$</p>
<p>$|a_n| \lt |\dfrac{1}{n}| \le \dfrac{1}{n_0} \lt \epsilon$.</p>
|
2,774,792 | <p>How would you go about proving the recursion
$$T(n) = T\left(\frac n4\right) + T\left(\frac{3n}4\right) + n$$is $\mathcal O(n\log n)$ using induction?</p>
<p>Thanks!</p>
| drhab | 75,923 | <p>Your answer is wrong.</p>
<p>The probability of "exactly $0$ the same" is indeed $\frac36\frac36\frac36=\frac{27}{216}$ as you suggest. </p>
<p>But the probability of "exactly $1$ the same" equals:$$\left[\frac16\frac16\frac16+3\frac16\frac16\frac36+3\frac16\frac36\frac36\right]3=\frac{111}{216}$$</p>
<p>If e.g. ... |
191,307 | <pre><code>roots = NSolve[Sin[2z] == Cos[Sin[z]] + 1 && -2π <= Re[z] <= 2π && -2π <= Im[z] <= 2π, z];
w = z\.roots;
ListPlot[{Re[w], Im[w]}, Ticks -> {{-2π, -π, 0, π, 2π}, {-2π, -π, 0, π, 2π}}]
</code></pre>
<p>In this code, <code>Ticks</code> command is not working properly, that i... | kglr | 125 | <p>Using Chip Hurst's example data:</p>
<pre><code>SeedRandom[1234];
pts = RandomReal[{0, 1}, {1000, 3}];
</code></pre>
<h3>SmoothKernelDistribution + DensityPlot3D</h3>
<pre><code>pdf[x_, y_, z_] := PDF[SmoothKernelDistribution[pts, MaxExtraBandwidths -> 0,
MaxMixtureKernels -> All], {x, y, z}]
DensityPlo... |
275,310 | <p>I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated</p>
| Dmitri Zaitsev | 59,225 | <p>An affine function between vector spaces is <em>linear</em> if and only if it fixes the origin.</p>
<p>In the simplest case of scalar functions in one variable,
<em>linear</em> functions are of the form $f(x)=ax$ and <em>affine</em> are $f(x)=ax +b$, where $a$ and $b$ are arbitrary constants.</p>
<p>More generall... |
118,701 | <p>I have two vectors of 134 elements each ($mu$, and $gt$). $mu$ contains Integers, and $gt$ contains machine precision Reals. I execute the following simple expression multiple times without changing either mu or gt:
$$
(mu/2*gt).gt
$$
I will get one of two different results: $88474.52216839303$ or $88474.52216839301... | Daniel Lichtblau | 51 | <p>It's not a bug and it's not so uncommon. For an explanation have a look <a href="http://blog.nag.com/2011/02/wandering-precision.html" rel="nofollow noreferrer">here</a>. This and some related issues also appear in <a href="http://forums.wolfram.com/mathgroup/archive/2011/Mar/msg00974.html" rel="nofollow noreferrer"... |
510,732 | <p>I am trying to think of a case where this is not true:</p>
<p>$f(n) = O(g(n))$ and $f(n) \neq \Omega(g(n))$, does $f(n) = o(g(n))$?</p>
<p>I suspect that it has to do with the varying $c$ and $n_{0}$ constants but am not completely sure. </p>
<p>Thanks!</p>
| Jonas Meyer | 1,424 | <p>Consider $f(n) = n+(-1)^nn$ and $g(n) = n$.</p>
|
1,530,874 | <p>Is there a case where a function $f$ that is not differentiable at $0$ and a function $g$ that is differentiable at $0$ where $f+g$ is differentiable at $0$?</p>
| C. Falcon | 285,416 | <p>One has: $$f=f+g-g.$$
Then, if $f+g$ and $g$ are differentiable at the origin, $f$ is differentiable at the origin with: $$f'(0)=(f+g)'(0)-g'(0).$$
Therefore, the answer to your question is no.</p>
|
3,580,293 | <blockquote>
<p>The value of <span class="math-container">$$\lim\limits_{x \rightarrow \infty} \left(5^x + 5^{3x}\right)^{\frac{1}{x}}$$</span> is...</p>
</blockquote>
<p>My approach :</p>
<blockquote>
<p><span class="math-container">$$\lim\limits_{x \rightarrow \infty} \left(5^x + 5^{3x}\right)^{\frac{1}{x}}$$</... | Axion004 | 258,202 | <p>From your last step</p>
<p><span class="math-container">$$\left(1 + 5^{2x}\right)^{1/x}=\Big[5^{2x}(1+5^{-2x})\Big]^{1/x}=\Big(5^{2}\Big)\Big(1+5^{-2x}\Big)^{1/x}$$</span></p>
<p>then take the limit as <span class="math-container">$x\to\infty$</span> and the above expression will approach <span class="math-contain... |
281,717 | <p>Suppose that $\beta \mathbb{R}$ is Stone–Čech compactification of $\mathbb{R}$. What is the closure of $\mathbb{Q}$? </p>
| Rustyn | 53,783 | <p>It means your first assumption. </p>
<p>Setting the derivative equal to $0$, we obtain:<br></p>
<p>$3x^2 - 6x = 0 \Rightarrow$<br>
$x(3x - 6)=0 \Rightarrow$ <br></p>
<p>$x=0,$ or $x=2$</p>
<p>$f(2) = 8$, $f(0) = 12$</p>
<p>Now we test end points,</p>
<p>$f(4) = 64 - 48 + 12 = 28$ <br>
$f(-2)= -8 -12 + 12 = -8... |
3,929,703 | <p>so, we know how to solve if the question was only <span class="math-container">$5$</span> different rings in <span class="math-container">$4$</span> different fingers, which is <span class="math-container">$4^5$</span>. but what if internal order of rings within the finger matters, is this counted in this answer or ... | user2661923 | 464,411 | <p>For me, this is simply a take off my shoes and count problem.</p>
<p>Type of distribution : <br>
Explanation <br>
Number of ways <br></p>
<p><span class="math-container">$5,0,0,0 :$</span><br>
<span class="math-container">$4$</span> ways of choosing which finger, <span class="math-container">$5!$</span> ways of perm... |
881,572 | <p>A relation $R$ on the set of real numbers can be thought of as a subset of the $xy$ plane. Moreover an equivalence relation on $S$ is determined by the subset $R$ of the set $S \times S$ consisting of those ordered pairs $(a,b)$ such that $ a \sim b$. </p>
<p>With this notation explain the geometric meaning of the ... | dioid | 165,958 | <p>Your description of reflexivity is correct.</p>
<p>For symmetry it means that the subset $R$ is "symmetric" around the line $y = x$, this means that for any point $(a, b)\in R$ its mirror point $(b, a)\in R$ (it's the point you get by doing reflection in the line $y=x$), i.e. either none of the two points $(a, b)$ ... |
881,572 | <p>A relation $R$ on the set of real numbers can be thought of as a subset of the $xy$ plane. Moreover an equivalence relation on $S$ is determined by the subset $R$ of the set $S \times S$ consisting of those ordered pairs $(a,b)$ such that $ a \sim b$. </p>
<p>With this notation explain the geometric meaning of the ... | M. Vinay | 152,030 | <p>First, you have to define clearly <em>how</em> you are representing a relation between two real numbers graphically.</p>
<p>For example, if $a \sim b$, are you plotting the point $(a, b)$ on the graph? Then it would make sense to plot each point $(a, a)$ for every $a \in S$ (but it will not make up the entire line ... |
3,742,517 | <p>Let <span class="math-container">$f(x)=x^3-1$</span>. To approximate the root <span class="math-container">$x^{\star}=1$</span>, we consider the sequence <span class="math-container">$(x_n)$</span> that we get if we apply Newton's method with <span class="math-container">$x_0>0$</span>. Show that the sequence con... | Marjan van den Akker | 805,370 | <p>I think that indeed the small value of epsilon is the problem. My suggestion is to change the objective to <span class="math-container">$M \sum x_i + \sum y_i$</span>, where <span class="math-container">$M$</span> is a big number (at least big enough to prioritize <span class="math-container">$\sum x_i$</span> over ... |
28,955 | <p>I need to crack a stream cipher with a repeating key.</p>
<p>The length of the key is definitely 16. Each key can be any of the characters numbered 32-126 in ASCII.</p>
<p>The algorithm goes like this:</p>
<p>Let's say you have a plain text:</p>
<p>"Welcome to Q&A for people studying math at any level and pr... | Alon Amit | 308 | <p>Take the two encrypted messages and XOR them with each other. You'll get the XOR of the two original, unencrypted messages since the identical keys cancel out. Deciphering this just requires patience and a good understanding of the encoding (what exactly is being XORed - the ASCII values of the letters? Some other b... |
28,955 | <p>I need to crack a stream cipher with a repeating key.</p>
<p>The length of the key is definitely 16. Each key can be any of the characters numbered 32-126 in ASCII.</p>
<p>The algorithm goes like this:</p>
<p>Let's say you have a plain text:</p>
<p>"Welcome to Q&A for people studying math at any level and pr... | castarco | 4,976 | <p>If you want to decrypt these texts, a good method is the old "Kasysky's method":</p>
<p>First, you have to know the frequencies of the characters in your plain text (if you know the language, it's easy). Then, search repeated characters and measure the space between them. If you make the greatest common divisor of ... |
942,651 | <p>Let $n,r,k$ be non negative integers such that $r,k\leq n$.</p>
<p>If $r^{n-r}=k^{n-k}$, when is this true other than $r=k$ ?</p>
<p>For example it is holds for $n=6,r=2,k=4$.</p>
| user153012 | 153,012 | <p>We have that $n,r,k$ be non negative integers such that $r,k\leq n$.</p>
<p>First of all $(r,k,n)=(0,0,n)$, where $n \neq 0$ is a trivial solution for $r^{n-r} = k^{n-k}$.</p>
<p>For further solutions, we have</p>
<p>$$\begin{align}
r^{n-r} & = k^{n-k} \\
(n-r) \cdot \ln r & = (n-k) \cdot \ln k \\
n &... |
32,849 | <p>I am trying to simulate a signal that randomly increases its phase, so far I have tried two thing but neither worked. I usually use matlab but I want to learn some <em>Mathematica</em> so I thought I would try this in <em>Mathematica</em>.</p>
<p>My first try was</p>
<pre><code>times = Table[i, {i, 0, 2, 0.05}];
f... | PlatoManiac | 240 | <p>One way will be to use <code>Nest</code> or something like that...</p>
<pre><code>tmax = 20;
times = Table[i, {i, 0, tmax, 0.05}];
fi = With[{dist = .8}, (*here you can control how big jump is possible between phases*)
NestList[RandomReal[{#, # + dist}] &,RandomReal[],-1 + Length@times]];
phaseIncrese =Tra... |
32,849 | <p>I am trying to simulate a signal that randomly increases its phase, so far I have tried two thing but neither worked. I usually use matlab but I want to learn some <em>Mathematica</em> so I thought I would try this in <em>Mathematica</em>.</p>
<p>My first try was</p>
<pre><code>times = Table[i, {i, 0, 2, 0.05}];
f... | ybeltukov | 4,678 | <p>You can use <code>Sin[t + phase[t]]</code> where <code>phase[t]</code> is <code>Interpolation</code> of a random process.</p>
<p>This random process can be implemented by <code>Accumulate</code></p>
<pre><code>tmin = 0;
tmax = 10;
dt = 0.1;
phase = Interpolation[
Transpose@{ Range[tmin, tmax, dt],
Accumu... |
43,743 | <p>An alternative title is: When can I homotope a continuous map to a smooth immersion?</p>
<p>I have a simple topology problem but it's outside my area of expertise and I worry may be rather subtle. Any help would be appreciated.</p>
<p>The set-up is the following:
Let $M$ be some (closed say) $n$ dimensional mani... | Johannes Ebert | 9,928 | <p>There is a general strategy for these kind of problems, which sometimes helps (the ''h-principle''): separate the homotopical and smooth aspects of the problem. Setup: $f:N \to M$ a map of smooth manifolds, $dim (N) < dim (M)$, $f|_{\partial N}$ is an immersion. </p>
<p>Step 1: if your $f$ is going to be homotop... |
1,134,510 | <p>Regarding My Background I have covered stuff like </p>
<p>1.Single Variable Calculus</p>
<p>2.Multivariable Calculus (Multiple Integration,Vector Calculus etc) (Thomas Finney)</p>
<p>3.Basic Linear Algebra Course (Containing Vector spaces,Linear Transformation)</p>
<p>4.Ordinary Differential Equation</p>
<p>5.R... | Community | -1 | <p>Another way to think is -
No of ways of selecting one alphabet out of 26 is $\binom {26}{1}$.
Then the number of ways of arranging it in 3 places out of 7 is $\binom {7}{3}$.
Then we are left with 25 alphabets and we have to select one. This can be done in $\binom{25}{1}$ ways. And we have only one way of arranging ... |
304 | <p>Per <a href="http://blog.stackoverflow.com/2010/07/moderator-pro-tempore/">this post on the SO/SE blog</a> (which, curiously, does not include math.SE in its graphic list), it looks like the admins will choose moderators pro tempore at about 7 days into the public beta. In the roughly 24 hours that we've been in pu... | Isaac | 72 | <p><strong>NO</strong> we do not need moderators pro tempore now</p>
|
130,914 | <p>I dont know how to proceed with solving $$\sum_{i=1}^{n}i^{k}(n+1-i).$$ Please give advise.</p>
| bspk | 28,987 | <p>$$\sum_{i=1}^{n}i^{k}(n+1-i)$$ </p>
<p>is same as </p>
<p>$$\sum_{i=1}^{n}i(n+1-i)^{k}$$</p>
<p>which looks like some combination of Eulerian number.</p>
|
431,690 | <p>As far as I know, for any $A$:
$$\mathbf{x}^{T}A\mathbf{y}=0;\forall\mathbf{x},\mathbf{y}\in R^n\Rightarrow A=0$$</p>
<p>Does it mean that
$$\mathbf{x}^{T}A\mathbf{x}=0;\forall\mathbf{x}\in R^n\Rightarrow A=0$$</p>
<p>The condition of the first claim $\forall\mathbf{x},\mathbf{y}\in R^n$ implies that we could take... | Doctor Dan | 84,274 | <p>$x^T A x = 0 \; \forall x => (x, Ax) = 0 \; \forall x$, hence $A$ can be an orthogonal matrix. This is what A in the @Daniel Fisher's counterexample is.</p>
|
4,194,611 | <p>Let <span class="math-container">$A \in \mathbb{R}^{m \times n}$</span>.</p>
<p><span class="math-container">$\forall i \in \mathbb{N} \; [ x_i \in \mathbb{R}^n \; \mbox{and} \; x_i \geq \textbf{0}] $</span></p>
<p>Assume that the sequence <span class="math-container">$Ax_1, Ax_2, ...$</span> converges to <span clas... | Mark Saving | 798,694 | <p>Come up with a basis <span class="math-container">$b_1, b_2, ..., b_k$</span> for the image of <span class="math-container">$A$</span>. Given some <span class="math-container">$y$</span> in the image of <span class="math-container">$A$</span>, there are unique <span class="math-container">$w_1, w_2, ..., w_k$</span>... |
4,194,611 | <p>Let <span class="math-container">$A \in \mathbb{R}^{m \times n}$</span>.</p>
<p><span class="math-container">$\forall i \in \mathbb{N} \; [ x_i \in \mathbb{R}^n \; \mbox{and} \; x_i \geq \textbf{0}] $</span></p>
<p>Assume that the sequence <span class="math-container">$Ax_1, Ax_2, ...$</span> converges to <span clas... | Mathews Boban | 948,760 | <p><strong>Claim 1</strong>: Let <span class="math-container">$\{ v_1, .. v_k\}$</span> be a linearly independent subset of <span class="math-container">$\mathbb{R}^n$</span>. Let <span class="math-container">$\{c_{i1}, c_{i2},\; ..,\; c_{ik}\}_{i = 1}^{\infty}$</span> be reals.
<span class="math-container">$$ (c_{i1}v... |
367,669 | <p><img src="https://i.stack.imgur.com/zQFyC.jpg" alt="enter image description here"></p>
<p>This is probably a very simple questions but I am not clear on Möbius transformations and how to solve this problem. I'd appreciate if somebody can point me towards a method to do these sort of questions or a webpage that expl... | Zen | 72,576 | <p>How about this:</p>
<ol>
<li>{x<sub>1</sub>: x<sub>1</sub> is prime OR 1}$\to 1$.</li>
<li>{x<sub>2</sub>: $x_2=2\cdot p$, for prime p>2}$\to 2$.</li>
<li>{x<sub>3</sub>: $x_3=3\cdot p$, for prime p>3}$\to 3$.</li>
<li>{x<sub>4</sub>: $x_4=4\cdot p$, for prime p>4}$\to 4$
etc.</li>
</ol>
<p>Sorry, I was a little ... |
3,716,619 | <p>Evaluating
<span class="math-container">$$\lim_{x\to 0}\left(\frac{\pi ^2}{\sin ^2\pi x}-\frac{1}{x^2}\right)$$</span>
with L'Hospital is so tedious. Does anyone know a way to evaluate the limit without using L'Hospital? I have no idea where to start.</p>
| Community | -1 | <p>Well, you've got your answer, and it's a good one, I'd use series expansions <em>always</em> in such case, but then, the answerer couldn't know you've ever heard of those expansions, and some of your comments show you aren't too familiar with them. That's why SE encourages sharing information about your mathematical... |
1,098,253 | <p>I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.</p>
| Anurag A | 68,092 | <p><strong>Hint</strong></p>
<p>Try the function $f(x)=\arctan{x} - x$. It's derivative is $\frac{-x^2}{1+x^2}$. Now use the monotonicity to get the required inequality.</p>
|
83,565 | <p>I am learning Mathematica on the fly, one of my tasks is to find the variance of white noise. I followed the tutorial for finding white noise by using the code:</p>
<pre><code>WN = WhiteNoiseProcess[NormalDistribution[0, 10]];
data = RandomFunction[WN, {0, 10000}];
</code></pre>
<p>I know I can use the following c... | ubpdqn | 1,997 | <p>It is valuable to look at the properties of these complex objects, e.g. in your example: </p>
<pre><code>data["Properties"]
</code></pre>
<p>To do your own variance:</p>
<pre><code>val = First@data["ValueList"];
Variance[val]
Total[(val - Mean[val])^2]/(Length[val] - 1)
</code></pre>
<p>You can compare results o... |
3,059,833 | <blockquote>
<p>If the equation <span class="math-container">$2^{2x} + a*2^{x+1} + a + 1=0$</span> has roots of opposite sign then the exhaustive values of a are?</p>
</blockquote>
<p>I tried taking <span class="math-container">$2^x = t$</span>. But then didn't know what to do.</p>
<p>The equation became, <span cla... | Michael Rozenberg | 190,319 | <p>The hint:</p>
<p>Solve the following system.
<span class="math-container">$$1^2+2a\cdot1+a+1<0$$</span> and
<span class="math-container">$$a+1>0.$$</span>
The first inequality says that <span class="math-container">$1$</span> is placed between <span class="math-container">$2^{x_1}$</span> and <span class="mat... |
3,059,833 | <blockquote>
<p>If the equation <span class="math-container">$2^{2x} + a*2^{x+1} + a + 1=0$</span> has roots of opposite sign then the exhaustive values of a are?</p>
</blockquote>
<p>I tried taking <span class="math-container">$2^x = t$</span>. But then didn't know what to do.</p>
<p>The equation became, <span cla... | zipirovich | 127,842 | <p>So far, so good! Now: if <span class="math-container">$x>0$</span>, then <span class="math-container">$t=2^x>1$</span>; and if <span class="math-container">$x<0$</span>, then <span class="math-container">$t=2^x<1$</span>. So for your new equation <span class="math-container">$t^2+2at+a+1=0$</span> you wa... |
1,452,121 | <p>I very well know that every open ball is an open set. and that every open set need not be an open ball. But illustrate me some counter example.</p>
| mkausp | 243,800 | <p>The most simple examples of open sets, which are not balls, in every metric space are $\emptyset$ and the space itself, which are open.</p>
<p>Even if you consider those sets to be balls with radius $0$ or $\infty$, respectively, you can take the union of two or more open balls, which is not necessarily an open bal... |
3,401,630 | <p>I am trying to prove the inequality
<span class="math-container">$$\frac{1}{n}-\frac{1}{(n+1)^2}>\frac{1}{n+1}\quad \forall \ n>1.$$</span> How would I go about doing this? I've tried solving it on my own but my final answer is <span class="math-container">$1>0$</span>. </p>
| Allawonder | 145,126 | <p>Your inequality is equivalent to <span class="math-container">$$\frac{1}{n}-\frac{1}{(n+1)^2}-\frac{1}{n+1}>0.$$</span> Simplifying LHS gives <span class="math-container">$$\frac{1}{n}-\frac{1}{(n+1)^2}-\frac{n+1}{(n+1)^2}=\frac1n-\frac{n+2}{(n+1)^2}=\frac{(n+1)^2-n(n+2)}{n(n+1)^2}.$$</span> Since the denominator... |
3,689,096 | <p>This was the Question:- Find all positive integers <span class="math-container">$n$</span> such that <span class="math-container">$\varphi(n)$</span> divides <span class="math-container">$n^2 + 3$</span></p>
<p>What I tried:-</p>
<p>I knew the solution and explanation of all positive integers <span class="math-con... | SB1729 | 466,737 | <p>First, we observe that <span class="math-container">$n$</span> can't be even. If it were even, <span class="math-container">$n^2+3$</span> would be odd and hence <span class="math-container">$\varphi(n)$</span> couldn't divide <span class="math-container">$n^2+3$</span> as <span class="math-container">$\varphi(n)$</... |
28,892 | <p>I was searching on MathSciNet recently for a certain paper by two mathematicians. As I often do, I just typed in the names of the two authors, figuring that would give me a short enough list. My strategy was rather dramatically unsuccessful in this case: the two mathematicians I listed have written 80 papers toget... | Victor Protsak | 5,740 | <p>This is a frivolous item solely to demonstrate the pitfalls of running MathSciNet searches and working with large datasets:</p>
<p>Type "Wang and Zhang" in the author field and get a list of 2417 items. Li and Wang are close contenders with 2300 total. I wouldn't venture a guess how many collaborations that represe... |
3,968,508 | <p>All I have got so far is that <span class="math-container">$a$</span> must divide <span class="math-container">$2002$</span>... Could anyone share some ideas? I mean is there any method rather than trial and error?</p>
| QED | 864,951 | <p>This isn't an answer, but it's an idea on how you could further narrow down your options</p>
<p>You can try taking (mod a) of either side to get <span class="math-container">$-1\equiv2001($</span>mod <span class="math-container">$a)$</span> (I'm assuming that's how you got a divides 2002).</p>
<p>Taking (mod 3) of e... |
3,968,508 | <p>All I have got so far is that <span class="math-container">$a$</span> must divide <span class="math-container">$2002$</span>... Could anyone share some ideas? I mean is there any method rather than trial and error?</p>
| quasi | 400,434 | <p>At most one of <span class="math-container">$a,a+1$</span> is a multiple of <span class="math-container">$3$</span>.</p>
<p>
If exactly one of <span class="math-container">$a,a+1$</span> is a multiple of <span class="math-container">$3$</span>, then
<span class="math-container">$$
a^{n+1}-(a+1)^n
$$</span>
would not... |
85,343 | <p>I am looking for a reference to study classical (i.e., not quantized) Yang-Mills theory. </p>
<p>Most of the sources I find focus on mathematical aspects of the theory, like Bleecker's book <em>Gauge theory and variational principles</em>, or Baez & Muniain's <em>Gauge fields, knots and gravity</em>.</p>
<p>Bu... | Liviu Nicolaescu | 20,302 | <p>Have you tried the book "the Geometry of physics" by Th. Frankel?</p>
|
85,343 | <p>I am looking for a reference to study classical (i.e., not quantized) Yang-Mills theory. </p>
<p>Most of the sources I find focus on mathematical aspects of the theory, like Bleecker's book <em>Gauge theory and variational principles</em>, or Baez & Muniain's <em>Gauge fields, knots and gravity</em>.</p>
<p>Bu... | Qfwfq | 4,721 | <p>Maybe you can have a look to Nakahara's <em>Geometry, Topology and Physics</em>, or is it too elementary for your purposes?</p>
|
1,808,881 | <p>I am struggling with finding all roots of unity in $\mathbb{Q}(i)$. I know that if $a+bi$ is a root of unity in $\mathbb{Q}(i)$, then $a^2+b^2=1$, and I know how to find all $a, b \in \mathbb{Q}$ that satisfy that equation. However, I do not know how to filter out the roots of unity. I think there should be a somewh... | Pipicito | 93,689 | <p>Let $\xi \in \mathbb{Q}(i)$ be a root of unity. We know that there exists $n\in \mathbb{N}$ such that $\xi$ is $n$-primitive. So, let's use the suggestive notation $\xi_n = \xi$. Now the cyclotomic extension $\mathbb{Q}(\xi_n) / \mathbb{Q}$ is a subextension of $\mathbb{Q}(i) / \mathbb{Q}$. By multiplicativity of th... |
374,619 | <p>In <a href="https://math.stackexchange.com/a/373935/752">this recent answer</a> to <a href="https://math.stackexchange.com/q/373918/752">this question</a> by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$</p... | Ross Millikan | 1,827 | <p>There is an infinite family of solutions coming from the idea $(2+\sqrt 3)^3=26+15\sqrt 3$. We can form $(2+\sqrt 3)^{3n}$ and find another solution. The next one is $(2+\sqrt 3)^6=1351+780 \sqrt 3$ and $(1351+780\sqrt 3)^{(1/3)}+(1351-780\sqrt 3)^{(1/3)}=14$ There is a recurrence, if $(a,b)$ is a solution, the n... |
374,619 | <p>In <a href="https://math.stackexchange.com/a/373935/752">this recent answer</a> to <a href="https://math.stackexchange.com/q/373918/752">this question</a> by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$</p... | Ian Mateus | 17,751 | <p>The solutions are of the form $\displaystyle(p, q)= \left(\frac{3t^2nr+n^3}{8},\,\frac{3n^2t+t^3r}{8}\right)$, for any rational parameter $t$. To prove it, we start with $$\left(p+q\sqrt{r}\right)^{1/3}+\left(p-q\sqrt{r}\right)^{1/3}=n\tag{$\left(p,q,n,r\right)\in\mathbb{N}^{4}$}$$
and cube both sides using the iden... |
234,409 | <p>I'm trying to obtain the coordinates of the border of the continents. I need this information to be ordered such that when I do, for example,</p>
<pre><code>ListLinePlot[data]
</code></pre>
<p>It does not yield a messed up image, as happens for disordered points. Initially I was trying by highlighting points on imag... | flinty | 72,682 | <p>I'm assuming you want points from the Mercator projection because you're trying to plot in 2D with <code>ListLinePlot</code>? Otherwise you'd be asking for the polygon wrapped on the sphere.</p>
<pre><code>(* get the GeoPosition points from the polygon *)
points=Flatten[First@First[Entity["GeographicRegion"... |
685,918 | <p>I'm doing exercises in Real Analysis of Folland and got stuck on this problem. I don't know how to calculate limit with the variable on the upper bound of the integral. Hope some one can help me solve this. I really appreciate.</p>
<blockquote>
<blockquote>
<p>Show that $\lim\limits_{k\rightarrow\infty}\int_0... | Yiorgos S. Smyrlis | 57,021 | <p>Let
$$
f_k(x)=\left\{\begin{array}{ccc}
\left(1-\frac{x}{k}\right)^{k}x^n & \text{if} & x\in [0,k], \\ \\
0 & \text{if} & x>k.
\end{array}
\right.
$$
Then
$$
0\le f_k(x)\le \mathrm{e}^{-x}x^n,
$$
for all $k$ and $x\ge 0$, and $\lim_{k\to\infty}f_k(x)=\mathrm{e}^{-x}x^n$, for all $x\ge 0$.</p>
<p... |
1,744,698 | <p>How to show that the characteristic polynomials of matrices A and B are $\lambda^{n-1}(\lambda ^2-\lambda -n)=0$ and $\lambda^{n-1}(\lambda^2+\lambda-n)=0$ respectively by applying elementary row or column operations.</p>
<p>$A=\begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
1 & 0 & 0 & \cdots... | openspace | 243,510 | <p>About first matrix. </p>
<p>Lets take the bottom line. As we know :
$$\det{A} = \sum{(-1)^{i+j}\cdot a_{i,j}\cdot M_{i,j}},$$ so we got:
$$S_{n+1} = a_{n+1,n+1}\cdot (-1)^{2n+2}\cdot S_{n} + (-1)^{n+2}a_{n+1,1}S'_{n},$$ where $S'_{n}= (-1)^{1 + n} \lambda^{n-1}(-1)^{n-1}$, because of :</p>
<p>$S'_{n} = \begin{bmat... |
542,951 | <p>I am trying show that the function $f:[0,1]\to \mathbb{R}$ defined by $f(x)=\sin \dfrac{1}{x}$ if $x\neq 0$ and $f(0)=0$ possesses IVP. Though it looks easy, but I am not getting any clue how to start with. Any help would be appreciated.</p>
| Dan Rust | 29,059 | <p>Show that there exists a subset $A$ of $(0,1]$ such that $f(A)=f([0,1])$ and such that $f|_A$ ($f$ restricted to the domain $A$) is a continuous function. You may then apply the intermediate value theorem to $f|_A$.</p>
<p>Note that the above proves that $f$ has the property that you mentioned in the comments, but ... |
1,851,084 | <p>I have to solve the following problem:
find the matrix $A \in M_{n \times n}(\mathbb{R})$ such that:
$$A^2+A=I$$ and $\det(A)=1$.
How many of these matrices can be found when $n$ is given?
Thanks in advance.</p>
| BDS | 59,605 | <p>Consider the Jordan Canonical Form for $A$; that is, $A = PJP^{-1}$ for some invertible $P$ and block diagonal matrix $J$ whose blocks are either diagonal or Jordan (same entry on the diagonal, have $1$s on the the diagonal above the main diagonal, and $0$s elsewhere).</p>
<p>Then, the equation reduces to $J^2 + J ... |
377,266 | <p>My question is very direct:</p>
<blockquote>
<p>What are the motivations for the name "jet"(subjet, superjet) in the context of viscosity solutions for second order fully nonlinear elliptic PDE?</p>
</blockquote>
<p>The definition of which can be seen in Crandall, Ishii, Lions:</p>
<p><em>Crandall, Michael... | asymptotic | 85,716 | <p>As I understand it, the word "jet" is meant to evoke the idea of a "spray" of curves through a point, or more accurately, their equivalence classes up to kth order contact</p>
|
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