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 |
|---|---|---|---|---|
1,268,103 | <p>Definition: Let $X$ be a topological space and let $\sim_C$ be the equivalence relation on $X$ defined by $x \sim_C y$ if $x$ and $y$ lie in a connected subset of $X$. The components of $X$ are the equivalence classes of the equivalence relation $\sim_C$. </p>
<p>Question: Prove that each component of $X$ is a clos... | Community | -1 | <p>I think I came up with a simpler proof: That components are the largest connected subsets of $X$; I choose one of them, say $C$. Since C is connected in X, thus so is $Cl(C)$. Thus $C=Cl(C)$ and then $C$ is closed.</p>
|
2,831,968 | <p>$\DeclareMathOperator{\var}{var}$It is just a general question I could not get my mind around.</p>
<p>Assume that $E[X]= 20$ and $\var[X]= 5$, then$$
E[1.2X]= 1.2·E[X]= 1.2×20= 24= 20 + 4 = E[X] + E[0.2X],\\
\var[1.2X]= 1.44·\var[X]= 1.44×5= 7.2.
$$
For$$
\var[1.2X]= \var[X + 0.2X]= \var[X] + \mathord{??} = \var[0.... | preferred_anon | 27,150 | <p>The rule $$\text{Var}(X+Y) = \text{Var}(X) + \text{Var} (Y)$$
is only valid if $X$ and $Y$ are independent. $X$ and $aX$ are never independent (this should be intuitively clear - if you know $X$ then you know $aX$) except in the case where $\text{Var}(X) = 0$, i.e. when $X$ is a point distribution. </p>
|
2,831,968 | <p>$\DeclareMathOperator{\var}{var}$It is just a general question I could not get my mind around.</p>
<p>Assume that $E[X]= 20$ and $\var[X]= 5$, then$$
E[1.2X]= 1.2·E[X]= 1.2×20= 24= 20 + 4 = E[X] + E[0.2X],\\
\var[1.2X]= 1.44·\var[X]= 1.44×5= 7.2.
$$
For$$
\var[1.2X]= \var[X + 0.2X]= \var[X] + \mathord{??} = \var[0.... | farruhota | 425,072 | <p>Use the <a href="https://en.wikipedia.org/wiki/Variance" rel="nofollow noreferrer">basic property</a> of the variance: $\text{Var}[aX]=a^2\text{Var}[X]$. </p>
<p>Hence:</p>
<p>$$\begin{align}\text{Var}[(1+a)X]&=(1+a)^2\text{Var}[X]=\\
&=\text{Var}[X]+(2a+a^2)\text{Var}[X]=\\
&=\text{Var}[X]+\text{Var}[... |
24,939 | <p>Sorry, I'm just starting to learn mathematica.</p>
<p>I have the following two-argument function:</p>
<pre><code>h[{x_, y_}] := x ^ y
</code></pre>
<p>When I do the following:</p>
<pre><code>Map[h, {{1, 2}, {2, 2}, {3, 2}}]
</code></pre>
<p>I get the expected output:</p>
<pre><code>{1, 4, 9}
</code></pre>
<p>... | amr | 950 | <p>For educational purposes, here's a couple other ways to do this:</p>
<pre><code>Power @@@ {{1, 2}, {2, 2}, {3, 2}}
</code></pre>
<p></p>
<pre><code>Power[Sequence @@ #] & /@ {{1, 2}, {2, 2}, {3, 2}}
</code></pre>
<p></p>
<pre><code>Cases[{{1, 2}, {2, 2}, {3, 2}}, List[x__] :> Power[x]]
</code></pre>
<p>... |
1,678,004 | <p>I am stuck on a word problem about picking teams. I thought it would be very simple, but to my surprise, I could not solve it. So here's the problem..</p>
<p>Andrea, Melissa, and Carol are in a class of 27 girls. The teacher chooses
students at random to make up teams of three. What is the probability that Andrea... | robjohn | 13,854 | <p>There are $\frac{27!}{3!^9}$ ways to make up $9$ teams of $3$ numbered $1$-$9$.</p>
<p>There are $9\frac{24!}{3!^8}$ ways to make up $9$ teams of $3$ numbered $1$-$9$ with Andrea, Melissa, and Carol on the same team. This counts $9$ teams that Andrea, Melissa, and Carol could be on, times the number of ways to make... |
2,490,654 | <p>$$\int_0^1\sqrt\frac x{1-x}\,dx$$
I saw in my book that the solution is $x=\cos^2u$ and $dx=-2\cos u\sin u\ du$.<br>
I would like to see different approaches, can you provide them?</p>
| Jack D'Aurizio | 44,121 | <p>Here it is an entirely different approach, through Fourier-Legendre series expansions.<br>
By the generating function for shifted Legendre polynomials, for any $x\in(0,1)$ we have:</p>
<p>$$ \sqrt{x}=\sum_{n\geq 0}\frac{2(-1)^n}{(1-2n)(2n+3)}\,P_n(2x-1), $$
$$ \frac{1}{\sqrt{1-x}}=\sum_{n\geq 0} 2\,P_n(2x-1), $$
he... |
2,490,654 | <p>$$\int_0^1\sqrt\frac x{1-x}\,dx$$
I saw in my book that the solution is $x=\cos^2u$ and $dx=-2\cos u\sin u\ du$.<br>
I would like to see different approaches, can you provide them?</p>
| Jack Tiger Lam | 186,030 | <p>Let <span class="math-container">$u = \sqrt{1-x}$</span></p>
<p>Then the integral transforms into:</p>
<p><span class="math-container">$$2 \int_0^1 \sqrt{1-u^2} \, \mathrm{d} u = \int_{-1}^1 \sqrt{1-u^2} \, \mathrm{d} u$$</span></p>
<p>Which has a nice geometric interpretation.</p>
|
2,993,166 | <p>Suppose I have an operator valued function, <span class="math-container">$\omega\mapsto A(\omega)$</span>; for each <span class="math-container">$\omega$</span>, <span class="math-container">$A(\omega):X\to Y$</span>, is a bounded linear operator with <span class="math-container">$X$</span> and <span class="math-con... | nonuser | 463,553 | <p>You can try with candidates for integer roots, that are <span class="math-container">$\pm1,\pm2,\pm3$</span> and <span class="math-container">$\pm 6$</span>. If non of them works then you still have the Cardano formulas.</p>
|
4,052,683 | <p>If <span class="math-container">$\gcd(a,b,c) = 1$</span> and <span class="math-container">$c = {ab\over a-b}$</span>, then prove that <span class="math-container">$a-b$</span> is a square. <span class="math-container">$\\$</span><br />
Well I tried expressing <span class="math-container">$a=p_1^{a_1}.p_2^{a_2} \cdot... | Especially Lime | 341,019 | <p>Suppose <span class="math-container">$p\mid\gcd(a,b)$</span>, where <span class="math-container">$p$</span> is prime. Let <span class="math-container">$p^r$</span> be the highest power dividing <span class="math-container">$a$</span>, and <span class="math-container">$p^s$</span> be the highest power dividing <span ... |
4,052,683 | <p>If <span class="math-container">$\gcd(a,b,c) = 1$</span> and <span class="math-container">$c = {ab\over a-b}$</span>, then prove that <span class="math-container">$a-b$</span> is a square. <span class="math-container">$\\$</span><br />
Well I tried expressing <span class="math-container">$a=p_1^{a_1}.p_2^{a_2} \cdot... | Will Jagy | 10,400 | <p>the best behaved indefinite ternary quadratic form is <span class="math-container">$y^2 - zx.$</span> It takes little to prove that, demanding <span class="math-container">$x>0$</span> and <span class="math-container">$\gcd(x,y,z) = 1,$</span></p>
<p><span class="math-container">$$ x = u^2, \; \; y=uv, \; \; z = ... |
128,656 | <p><img src="https://i.stack.imgur.com/AyYxe.jpg" alt=""Put the alphabet in math..."" /></p>
<p><strong>variable</strong>: A symbol used to represent one or more numbers.</p>
<p>Or alternatively: A symbol used to represent any member of a given set.</p>
<p>High school students are justifiably confused by the... | Prasad G | 25,314 | <p>First expression has only one variable which is h and it is fixed.</p>
<p>Second expression has two variables which are h and C. Here h and C are depended from one another. values of h will be changed when values of C change or values of C will be changed when values of h change. That's why h and C have so many va... |
128,656 | <p><img src="https://i.stack.imgur.com/AyYxe.jpg" alt=""Put the alphabet in math..."" /></p>
<p><strong>variable</strong>: A symbol used to represent one or more numbers.</p>
<p>Or alternatively: A symbol used to represent any member of a given set.</p>
<p>High school students are justifiably confused by the... | user 7269591 | 43,115 | <p>The duality embodied in the definition of a variable as:</p>
<p>"A symbol used to represent one or more numbers."</p>
<p>Is such that we don't have to make the distinction between an "unknown specific quantity" and a "varying" quantity.</p>
|
3,321,863 | <p>The eigenvalues of a positive-definite matrix are guaranteed to be <span class="math-container">$> 0$</span>; but does anyone know of sufficient conditions when they will also all be <span class="math-container">$\le 1$</span>?</p>
| Klaus | 635,596 | <p>There are certainly many possible answers to this question, including the obvious one given by @Kavi Rama Murthy. Here is a slightly less obvious one: Gershgorin's circle theorem implies that if the sum of the absolute values of the entries in each row does not exceed <span class="math-container">$1$</span>, then th... |
731,624 | <p>For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$ ?</p>
<p>Motivation for this question rose from an intuitive question that when I draw a virtual continuous straight line between $1$ and $2$ and virtually pick a point... | Dan Shved | 47,560 | <p>No. You can have a set $\mathbb{X}$ with <a href="http://en.wikipedia.org/wiki/Total_order" rel="nofollow">linear order</a> such that $\mathbb{R} \subsetneq \mathbb{X}$ and the restriction of that order from $\mathbb{X}$ to $\mathbb{R}$ coincides with the standard order on $\mathbb{R}$.</p>
<p><strong>Example 1.</s... |
3,504,777 | <p>Here's what I initially started with:</p>
<blockquote>
<blockquote>
<p>Find a 2x2 non zero matrix <span class="math-container">$A$</span>, satisfying <span class="math-container">$A^2=A$</span>, and <span class="math-container">$A\neq I$</span>.</p>
</blockquote>
</blockquote>
<p>I understand that this is ... | user1551 | 1,551 | <blockquote>
<p><em>The observation here, is that in a <span class="math-container">$2$</span>x<span class="math-container">$2$</span> matrix, which has <span class="math-container">$4$</span> entries, knowing any pair of entries other than the one along the main diagonal helps us determine other entries.</em></p>
</... |
2,115,170 | <p>Well that's the question I am trying to solve. I did check it for a few $q$ and it seems to hold. However, I'm not sure how I would go about proving this. I actually cannot figure out where to start. I tried adding and subtracting $2q$ to make a perfect square. I think I might have to use mod 10 in this to make the ... | zipirovich | 127,842 | <p>First of all, if $q$ is odd, then $q^2+1$ is ... and therefore can't be prime. Thus we only need to consider even values of $q$. All even numbers end with $0$, $2$, $4$, $6$, or $8$. And apparently, we only need to explain why the last digit can't be $2$ or $8$. Hint: if the last digit of $q$ is $2$ or $8$, what is ... |
1,714 | <p>Say I have a function $f(x)$ that is given explicitly in its functional form, and I want to find its Fourier transform[1]. If $f$ is too complicated to have an analytic expression for $\hat f(k)$, how do I obtain it numerically?</p>
<p>The naive and stupid way, which I currently use, is evaluating the integral for ... | J. M.'s persistent exhaustion | 50 | <p>There is the function <code>NFourierTransform[]</code> (as well as <code>NInverseFourierTransform[]</code>) implemented in the package <code>FourierSeries`</code>. The function, as with the related kernel functions, takes a <code>FourierParameters</code> option so you can adjust computations to your preferred normal... |
3,260,068 | <p>I've tried a couple of things trying to solve this problem but I get no answer.</p>
<p>These are one of the few things I know about “Gcd” and division:</p>
<p>If <span class="math-container">$a\mid b$</span> and <span class="math-container">$a \mid c$</span>, then <span class="math-container">$a \mid b \cdot x + c... | Duns | 375,866 | <p><strong>My attempt</strong>: Perhaps you can finish it<br>
<span class="math-container">$\nabla_z ||P^{1/2}z||_2 = \frac{P^{T/2}P^{1/2} z}{||P^{1/2}z||_2}$</span>. Now you want optimize
<span class="math-container">$$ \min \frac{z^T P^{T/2}P^{1/2}}{||P^{1/2}z||_2} v \quad \text{s.t.} ||v||^2_2 = 1,
$$</span>
whic... |
3,260,068 | <p>I've tried a couple of things trying to solve this problem but I get no answer.</p>
<p>These are one of the few things I know about “Gcd” and division:</p>
<p>If <span class="math-container">$a\mid b$</span> and <span class="math-container">$a \mid c$</span>, then <span class="math-container">$a \mid b \cdot x + c... | GingerBreadMan | 370,828 | <p><span class="math-container">$$ \Delta x_{\textrm{nsd}} = \textrm{argmin} \{ \nabla f(x)^Tv \mid \space\space\space \vert\vert v \vert\vert_{P} \le 1 \} $$</span> </p>
<p><span class="math-container">$$ = \textrm{argmin} \{ \nabla f(x)^Tv \mid \space\space\space\vert\vert P^{1/2}v \vert\vert_{2} \le 1 \} $$</span> ... |
2,800,416 | <p>this is my first question and I don't quite understand how do I confront this equation:</p>
<p>$z^2+i\sqrt{32}z-6i=0$</p>
<p>I tried using the quadratic formula but it doesn't seem to give me a correct answer, any help will be much obliged.</p>
<p>Thank you! :)</p>
| Shirish Kulhari | 458,802 | <p>Hint: Let $z = a + ib$, separate the equation into the real and imaginary parts and equate both parts to $0$.</p>
|
27,233 | <p>A finitely presented, countable discrete group $G$ is amenable if there is a finitely additive measure $m$ on the subsets of $G \backslash${$e$} with total mass 1 and satisfying $m(gX)=mX$ for all $X\subseteq G \backslash${$e$} and all $g \in G.$ </p>
<p>A countable discrete group $G$ is inner amenable if there is ... | bah | 23,535 | <p>Is there a non inner amenable locally compact group [map]group</p>
|
2,204,642 | <p>I have a hard time believing that there can exist a bijection $f:\Bbb R^2\to \Bbb R$.</p>
<p>I just cannot get around my intuition that a one-to-one map of a one-dimensional space (or interval) must also be one-dimensional.</p>
<p>I am not a mathematician. I am interested in the topic since it has potential releva... | zhw. | 228,045 | <p>This might help clear things up: Let $B$ denote the set of binary sequences, i.e., those sequences $(a_n)$ for which $a_n \in \{0,1\}$ for each $n.$ I'm hoping you are familiar with the fact that $B$ and $\mathbb R$ have the same cardinality. In this cleaner setting, where we don't have to worry about decimal expani... |
965,819 | <p>So I am doing a question were I have the set
column matrix 1 = (3, -8, 1) and column matrix 2 = (6, 2, 5)
and the question is asking if this is either a bases for R2 or R3.
Can I just say that since the matrix is not a square matrix (nxn) it cannot be a bases for R3 since it is not invertible -- it's not invertible... | Brian M. Scott | 12,042 | <p>You’ve nearly answered the first question; all that remains is to verify the triangle inequality, which boils down to showing that for all $x,y,z\in\Bbb X$,</p>
<p>$$|f(x)-f(y)|\le|f(x)-f(z)|+|f(z)-f(y)|\,$$</p>
<p>which should be very straightforward. For the actual write-up, I suggest first proving that $d$ sepa... |
4,028,717 | <p><span class="math-container">$$\dfrac{\sqrt[3]{3}}{\sqrt[3]{1}+\sqrt[3]{2}}=\sqrt[3]{\sqrt[3]{2}-1}$$</span></p>
<p>I could not multiply by the conjugate since it is a cube root. Can you show me a way to simplify it?</p>
<p>Thanks!</p>
| Joshua Wang | 773,061 | <p>Cubing both sides, we wish to prove:</p>
<p><span class="math-container">$$\sqrt[3]{2}-1 = \bigg(\frac{\sqrt[3]{3}}{1 + \sqrt[3]{2}}\bigg)^{3}$$</span></p>
<p>We have:</p>
<p><span class="math-container">$$\bigg(\frac{\sqrt[3]{3}}{1 + \sqrt[3]{2}}\bigg)^{3} = \frac{3}{1 + 3\sqrt[3]{2} + 3\sqrt[3]{2}^{2} + 2}=\frac{1... |
450,370 | <blockquote>
<p>Use Lagrange multipliers to determine the shortest distance from a point $\,x \in R^n\,$ to a plane $\{y\mid b^Ty = c\}.$</p>
</blockquote>
<p>I don't even know where to start!</p>
| pitchounet | 61,409 | <p>To use Lagrange multipliers, you must write this problem as a constrained minimisation problem. Let $\Vert \cdot \Vert$ be the norm associated to the canonical dot product in $\mathbb{R}^{n}$ and $\mathcal{P} = \lbrace y \in \mathbb{R}^{n}, \, {}^t b y = c \rbrace$. If I'm not mistaken, your problem is :</p>
<p>$$ ... |
975,759 | <p>Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$?</p>
<p>I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their preimage? My problem here would be to check <strong>all</strong> closed sets and the closeness of their preimage so ... | Mateus Sampaio | 101,351 | <p>No it is not. Consider the sequence $(f_n)\in C^1[0,\pi]$, given by
$$f_n(x)=\frac{\sin nx}{n}$$
Then, $\lim_{n\to\infty}f_n=0$, but $Df_n=\cos nx$ does not converge to zero, as $n\to\infty$.</p>
|
975,759 | <p>Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$?</p>
<p>I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their preimage? My problem here would be to check <strong>all</strong> closed sets and the closeness of their preimage so ... | wisefool | 51,807 | <p>I'll assume you consider the topologies induced by the $\mathcal{C}^1$-norm:
$$\|f\|_{\mathcal{C}^1([a,b])}=\sup_{[a,b]}|f| + \sup_{[a,b]}|f'|$$
and by the sup norm.
If you consider the topology of the uniform convergence on both spaces, then the answer is no, as pointed out in other answers.</p>
<p>The operator $f... |
39,441 | <p>In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because m∠ADC=m∠BAE I just dont know how to approach the problem and i would love some help
<img src="https://i.stack.imgur.com/Bkdb... | Isaac | 72 | <p><em>edit</em> Given that this was an NCTM calendar problem, I doubt that my solution is the intended one, but I've added some more detail to flesh it out.</p>
<ol>
<li>Label the point of intersection of $\overline{AE}$ and $\overline{CD}$ as $X$.</li>
<li>Use the technique of <a href="http://en.wikipedia.org/wiki/M... |
39,441 | <p>In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because m∠ADC=m∠BAE I just dont know how to approach the problem and i would love some help
<img src="https://i.stack.imgur.com/Bkdb... | André Nicolas | 6,312 | <p>This solution will be essentially the same as the one given by @Isaac, except for the appeal to <a href="http://en.wikipedia.org/wiki/Ceva%27s_theorem" rel="nofollow">Ceva's Theorem</a>. While reading the words, please look at a diagram: the whole thing would be much easier to explain at a blackboard by pointing!</... |
159,851 | <p>If Im not wrong when we use the command <code>N</code> we are using float point numbers in machine precision, right? And the machine precision depends on the bits of the CPU where Mathematica is installed. Then, by example, irrational numbers are truncated at some place and there is some error associated to the magn... | george2079 | 2,079 | <p>compare against a higher precision calculation</p>
<pre><code>n = 100000;
ListPlot[Log[10,
Abs[Accumulate@Tan[N[Range[n]]] -
Accumulate@Tan[N[Range[n], 1000]]]]]
</code></pre>
<p><a href="https://i.stack.imgur.com/mCCi2.png" rel="noreferrer"><img src="https://i.stack.imgur.com/mCCi2.png" alt="enter image d... |
3,232,174 | <blockquote>
<p>Andy has a cube of edge length 8 cm. He paints the outside of the cube red and then divides the cube into smaller cubes, each of edge length 1 cm. Andy randomly chooses one of the unit cubes and rolls it on a table. If the cube lands so that an unpainted face is on the bottom, touching the table, what... | Sanket Agrawal | 579,054 | <p>Assuming that for each cube, there is an equal probability <span class="math-container">$\frac{1}{6}$</span> for each of the six faces to come at the bottom after the roll.</p>
<p>Now, for a cube of length <span class="math-container">$8$</span> cm painted at the surface and then divided into cubes each of length <... |
21,290 | <p>Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$? </p>
<p>I am looking for elementary example (so there should be a proof of transcendence that does not use any big machinery).</p>
| Qiaochu Yuan | 290 | <p>If $k$ has characteristic zero, then $\displaystyle e^t = \sum_{n \ge 0} \frac{t^n}{n!}$ is certainly transcendental over $k[t]$; the proof is essentially by repeated formal differentiation of any purported algebraic relation satisfied by $e^t$. </p>
<p><strong>Edit:</strong> Let me fill in a few details. Given ... |
21,290 | <p>Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$? </p>
<p>I am looking for elementary example (so there should be a proof of transcendence that does not use any big machinery).</p>
| Gerry Myerson | 3,684 | <p>How about $\sum t^{n!}$? Doesn't a "sea-of-zeroes" argument show it can't be algebraic? </p>
|
21,290 | <p>Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$? </p>
<p>I am looking for elementary example (so there should be a proof of transcendence that does not use any big machinery).</p>
| maks | 3,882 | <ol>
<li>For characteristic 0, <a href="http://arxiv.org/abs/1003.2221" rel="noreferrer">http://arxiv.org/abs/1003.2221</a></li>
<li>For characteristic p, <a href="http://arxiv.org/abs/0810.3709" rel="noreferrer">http://arxiv.org/abs/0810.3709</a></li>
</ol>
|
1,238,430 | <p>I'm working on a project that involves that set $P = \{\{n_1, \ldots, n_k\} \mid k \in \mathbb{N}, n_i \in \mathbb{N} \text{ and } n_1 + \cdots +n_k = n\}$ of all integer partitions of a number $n$. Is there a standard notation for this?</p>
<p>I know that $p(n)$ is commonly used to denote the <em>number</em> of i... | user76284 | 76,284 | <p>I would simply <a href="https://mathoverflow.net/questions/344464/notation-for-the-set-of-all-injections-from-a-into-b/344590#344590">overload</a> the <span class="math-container">$p$</span> notation to refer to the <em>set</em> of partitions when its argument is a set. That way you have <span class="math-container"... |
278,486 | <p>I'm just curious because I was trying to come up with a weird function with weird discontinuities. Then I thought</p>
<p>$$f(x)=\prod_{n=1}^\infty \dfrac{1}{1-\frac{1}{nx}}$$</p>
<p>So what's this product?</p>
| Clive Newstead | 19,542 | <blockquote>
<p>This was an answer to the OP's first question which has since been edited, which asked about the product $\displaystyle \prod_{n=1}^{\infty} \dfrac{n}{nx-1}$.</p>
</blockquote>
<p>Consider the possible values of $x$ in cases.</p>
<ul>
<li><p>If $|x|>1$, then for sufficiently large $n$, you'll hav... |
2,776,763 | <p>Consider two sequences $\{b_n\}_{n\in \mathbb{N}}$ and $\{a_n\}_{n\in \mathbb{N}}$. Suppose that
$$
(*) \hspace{1cm} \lim_{n\rightarrow \infty} (b_n+a_n)=L<\infty
$$</p>
<p>Does this imply
$$
\exists \lim_{n\rightarrow \infty} b_n \text{ and it is finite}
$$
$$
\exists \lim_{n\rightarrow \infty} a_n \text{ and i... | Nick | 490,822 | <p>Let $\theta$ denote the (counterclockwise) angle from $\vec{v}$ to $\vec{u}$. From your diagram, you can see that $\hat{u}$ is a multiple of $\vec{v}$. Note that $\frac{1}{||\vec{v}||}\vec{v}$ is a unit vector with the same direction hence $\vec{v}$ is a scalar multiple of $\frac{1}{||\vec{v}||}\vec{v}$ also. In fa... |
748,013 | <p>Definition: $f(X)=${$f(x)|x\in X$}, "$|A|$"represents the number of elements in the set A.
<br/>In the title, $f:S\to T$, "$iff$" means "if and only if".$S$and $T$ are finite sets.
Two definitions for being onto:
<br/>1.If for every element $ t$ in $T$, there exists some $s\in S$, such that $f(s)=t$, then it's onto... | 5xum | 112,884 | <p>I think the point of the author is simply saying that you can understand $\mathbb R^n$ as a set of $n$-tuples, but also as a set of $n$-rows or $n$-columns. It doesn't matter how you look at it, all three views are isomorphic.</p>
|
39,802 | <p>Suppose $f:X \to Y$ is a function of sets. Then we can take the quotient $X/\text{~}$ by identifying $x \text{~} y$ if and only if $f(x)=f(y)$. Now suppose instead that $f:X \to Y$ is a map of simplicial sets. I want to emulate this homotopically, by adding a 1-simplex between $x$ and $y$ if there is a 1-simplex fro... | Tim Porter | 3,502 | <p>This answer is perhaps a gloss on David's one. It is often useful to replace taking a quotient by forming the equivalence relation as a groupoid. Thus the initial situation you describe has the classical equivalence-relation-from-a-function form. This will work in any category with pullbacks as it is the pullback o... |
4,325,440 | <p>I need to compute <span class="math-container">$$I:=\int_{-\infty}^{\infty} \frac{\sin(ax)}{x(\pi^2-a^2x^2)}dx$$</span>
(<span class="math-container">$a>0$</span>)</p>
<p>Probably there is a way to compute it with residue theorem.</p>
<p>My thoughts:</p>
<ul>
<li>The singularity at <span class="math-container">$x... | Claude Leibovici | 82,404 | <p><span class="math-container">$$I=\int\frac{\sin (a x)}{x \left(\pi ^2-a^2 x^2\right)}\,dx=\int \frac{\sin (t)}{(t-\pi)t(t+\pi) }\,dt$$</span>
<span class="math-container">$$\frac{1}{(t-\pi)t(t+\pi) }=\frac 1 {\pi^2}\Bigg[\frac{1}{2 (t-\pi )}+\frac{1}{2 (t+\pi )}-\frac{1}{t} \Bigg]$$</span></p>
<p>Consider
<span clas... |
4,325,440 | <p>I need to compute <span class="math-container">$$I:=\int_{-\infty}^{\infty} \frac{\sin(ax)}{x(\pi^2-a^2x^2)}dx$$</span>
(<span class="math-container">$a>0$</span>)</p>
<p>Probably there is a way to compute it with residue theorem.</p>
<p>My thoughts:</p>
<ul>
<li>The singularity at <span class="math-container">$x... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
917,849 | <p>I tried following but then I got stuck</p>
<p>$676 = 26*26$ </p>
<p>$12^{39} + 14^{39}$ is divisible $26$ for sure since $a^n + b^n$ is divisible by $(a+b)$ when $n$ is odd. But what to do next?</p>
| jyothika1 | 172,825 | <p>$$\left((39-12)^{39}+(39-14)^{39}\right) \div 26 \times 26$$
so $$\left(27^{39}+25^{39}\right) \div 26$$
so $$1^{39}+(-1)^{39}$$</p>
|
63,315 | <p>I am trying to identify a sequence related to the von Mangoldt function matrix. Since I believe/conjecture that the columns in the matrix have period lengths as in this sequence b:</p>
<pre><code>b = Table[Product[Exp[MangoldtLambda[n]], {n, 1, k}], {k, 1, nn}];
</code></pre>
<p>and since "b" grows rather fast, th... | Fred Daniel Kline | 973 | <p>Denominators of first sequence: <a href="http://oeis.org/A003418" rel="nofollow noreferrer">A003418</a>, which are symmetric sequences of greatest divisors <= ii.</p>
<p>The denominators of the differences are these greatest divisors.</p>
<p>See <a href="https://math.stackexchange.com/q/886041/28555">here</a> and... |
3,220,543 | <p>A natural number <span class="math-container">$n$</span> is said to be a good number if and only if <span class="math-container">$4$</span> times the sum of its digits equals the original number. We have to find out the sum of all such good numbers.</p>
<p>I'm 99% sure the only such good natural numbers are <span ... | Jerry Chang | 518,781 | <p><strong>Hint:</strong> Write <span class="math-container">$$n = \sum_{k=1}^{m}a_k10^k,$$</span>
where <span class="math-container">$n$</span> is an <span class="math-container">$m$</span>-digits number.
Then <span class="math-container">$$\sum_{k=1}^{m}a_k10^k - \sum_{k=1}^{m}a_k\cdot4 = 0.$$</span> What happened if... |
3,271,419 | <p>What are the last 2 digits of <span class="math-container">$2017^{2017}$</span>?</p>
<p>Notice that
<span class="math-container">$$2017 (2017) = 2017 ( 2000 + 10 + 7) = (....000) + (....70) + (2017 \times 7)$$</span>
so the last two digits of <span class="math-container">$2017^{2}$</span> are the last two digits o... | AlvinL | 229,673 | <p>As we're interested in the last two digits, it suffices to compute this modulo <span class="math-container">$100$</span>. Hence
<span class="math-container">$$
2017 ^{2017} \equiv 17^{17} \equiv 77 \pmod{100}.
$$</span>
The first equivalence holds by Euler's theorem (see the <em>mothertopic</em> mentioned by J. Lah... |
1,157,306 | <p>Is the following system has any positive integer solution $(x,y,u,v)$?
$$\begin{cases} x^2+y^2=u^2\\ x^2-y^2=v^2 \end{cases}$$
I can prove that any pair of these integers can be relatively prime, but I couldn't find any solution. Any hint?</p>
<p>Thanks in advance!</p>
| Greg Martin | 16,078 | <p>I believe there are no (nontrivial) solutions. Note that in any solution, the numbers $v^2,x^2,u^2$ would be in arithmetic progression with common difference $y^2$. One can parametrize all three-term arithmetic progressions of squares (see for example <a href="http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/3s... |
102,976 | <p>Is there a way to evaluate a string containing RPN in Mathematica?</p>
<p>SE thinks this question is too short, so let me expand on it.
Do you know of any function, that provides the following functionality?</p>
<pre><code>EvalRPN["5 4 + 3 /"]
</code></pre>
<blockquote>
<p>3</p>
</blockquote>
<p>Or even symbol... | Edmund | 19,542 | <p>This will take a few seconds to run the first time in a session that an operator is used since it needs to download the <code>"WolframLanguageSymbol"</code> <code>Entity</code> for that operator. After that subsequent calls in the session with the entity will be fast since it is in the cache.</p>
<p><code>getMathOp... |
102,976 | <p>Is there a way to evaluate a string containing RPN in Mathematica?</p>
<p>SE thinks this question is too short, so let me expand on it.
Do you know of any function, that provides the following functionality?</p>
<pre><code>EvalRPN["5 4 + 3 /"]
</code></pre>
<blockquote>
<p>3</p>
</blockquote>
<p>Or even symbol... | Pankaj Sejwal | 1,561 | <p>You can find this approach easier,</p>
<pre><code>l={"a", "b", "+", "c", "/"}
Polish[q_List] :=
ReplaceRepeated[
q, {s___, a_, b_, c_String,
d___} /; (c == "+" || c == "/" || c == "*" ||
c == "/") :> {s, {a, c, b}, d}] //. {a___, {b_String, c_,
d_String}, e___} :> {a, b <> c &... |
56,337 | <p>Suppose I have the following list:</p>
<pre><code>list = {a, b, c, d}
</code></pre>
<p>I want to generate this result:</p>
<pre><code>{{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}}
</code></pre>
<p>What could be the shortest way?</p>
<p>The list elements can... | Timothy Wofford | 9,391 | <p>Here is a straightforward implementation...</p>
<pre><code>Table[Table[f[list[[i]], list[[j]]], {j, i, Length@list}], {i, Length@list}]
</code></pre>
<p>Here is my flattened table:
If <code>list</code> is a sorted list of unique elements</p>
<pre><code>list = {a, b, c, d}
g[a_, b_] := f @@ Sort@{a, b};
Union@Flat... |
56,337 | <p>Suppose I have the following list:</p>
<pre><code>list = {a, b, c, d}
</code></pre>
<p>I want to generate this result:</p>
<pre><code>{{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}}
</code></pre>
<p>What could be the shortest way?</p>
<p>The list elements can... | Mr.Wizard | 121 | <p>I don't believe anyone has posted exactly this formulation:</p>
<pre><code>MapIndexed[#[[#2[[1]] ;;]] &, Outer[f, #, #]] &
</code></pre>
<p>Not terribly efficient but the question asked for shortest, not fastest.</p>
<hr>
<h3>Argument</h3>
<p>Although not optimal my method is both <em>more efficient</em... |
704,517 | <p>Let $D \subset \mathbb{C}$ be a discrete subset and let $f : D \mapsto \mathbb{C}$ be a function. Show that $f$ is continuous.</p>
<p>What's the best way to do this? I was thinking a proof by contradiction since a direct proof seems a little tricky...</p>
<p>Definitions i am using:</p>
<p>$D \subset \mathbb{C}$ i... | Jeff Snider | 119,951 | <p>Let $E:=f[D]$ be the range of $f$, so that $E\subset\mathbb{C}$.</p>
<p>Under $f$, the preimage of any open set $U\subset E$ is a set in $D$. That is, $$f^{-1}[U]\subset D.$$ All sets in $D$ are open, thus $f^{-1}[U]$ is open. Hence $f$ is continuous.</p>
<p>As a reference, <a href="https://math.stackexchange.c... |
126,251 | <p>Suppose one has a finite number of distances $d_1,\ldots,d_k$ on the Euclidean plane all of which metricize the usual Euclidean topology.</p>
<p>Define for each pair of points $x$ and $y$ in the plane
$$d(x,y) = \inf\left\lbrace d_{i_1}(x_0,x_1) + \cdots d_{i_l}(x_{l-1},x_l) \right\rbrace$$
where the infimum is tak... | Denis | 21,059 | <p>Unless I'm missing something, the answer is almost trivially no:</p>
<p>For all i, $d_i$ takes values in $\mathbb{R}^+$, so the $d(x,y)=0$ if and only if there is a decomposition such that for all $j$, $d_{i_j}(x_{j-1},x_j)=0$, which happens if and only if $x_{j-1}=x_j$. Therefore, if $d(x,y)=0$, then $x=x_0=x_1=\d... |
3,034,766 | <p>So is all it's saying that if there are two functions that have the same derivatives for every single <span class="math-container">$x$</span> in the interval, then <span class="math-container">$f(x) = g(x) + \alpha$</span>, means that the second function is just the exact same as <span class="math-container">$f(x)$<... | MPW | 113,214 | <p>Yes, because the difference <span class="math-container">$f-g$</span> must then have zero derivative everywhere and therefore be constant. This means the graph of <span class="math-container">$f$</span> is the graph of <span class="math-container">$g$</span> shifted vertically.</p>
|
2,224,980 | <p>Apologies if this kind of question is not allowed here - if so please delete it.</p>
<p>I was just wondering if anyone could recommend a book on mathematical analysis that is interesting enough to sit down and read for enjoyment alone? Something not written in the style of a textbook?</p>
<p>All the best.</p>
| Tim kinsella | 15,183 | <p>I remember Visual Complex Analysis by Tristan Needham being enjoyable for armchair reading.</p>
|
2,224,980 | <p>Apologies if this kind of question is not allowed here - if so please delete it.</p>
<p>I was just wondering if anyone could recommend a book on mathematical analysis that is interesting enough to sit down and read for enjoyment alone? Something not written in the style of a textbook?</p>
<p>All the best.</p>
| Samuel Bird | 134,225 | <p>An Introduction to Mathematical Analysis by Burkill is nice and concise, yet still flows very well and has a few good exercises. Only covers basic analysis though, nothing beyond first or maybe second year undergrad. I also like that it is smaller so you can carry it around to read on a bench or on a train.</p>
|
2,224,980 | <p>Apologies if this kind of question is not allowed here - if so please delete it.</p>
<p>I was just wondering if anyone could recommend a book on mathematical analysis that is interesting enough to sit down and read for enjoyment alone? Something not written in the style of a textbook?</p>
<p>All the best.</p>
| Stella Biderman | 123,230 | <p><a href="http://rads.stackoverflow.com/amzn/click/1493927116" rel="nofollow noreferrer">Understanding Analysis</a> is an awesome book in my opinion. It's concise and highly readable, but manages to not sacrifice too much rigor when doing so. I think it's a great book for someone who is interested in getting their ha... |
230,997 | <p>How would I go about proving this: Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1</p>
| Michael Joyce | 17,673 | <p>Hint: $\gcd(a,b)$ divides both $a$ and $b$, so if you divide both sides of the equation by $\gcd(a,b)$ ...</p>
|
2,344,259 | <blockquote>
<p>$$\int_0^\infty \frac{x^2}{x^4+1} \; dx $$</p>
</blockquote>
<p>All I know this integral must be solved with beta function, but how do I come to the form $$\beta (x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}\;dt \text{ ?}$$</p>
| Jack D'Aurizio | 44,121 | <p>$$\begin{eqnarray*}I=\int_0^{+\infty}\frac{x^2\,dx}{x^4+1}\stackrel{\text{parity}}{=} \frac{1}{2} \int_{-\infty}^{+\infty}\frac{dx}{x^2+\frac{1}{x^2}}&=&\frac{1}{2} \int_{-\infty}^{+\infty}\frac{dx}{\left(x-\frac{1}{x}\right)^2+2} \\[8pt]
(\text{by Glasser's Master Theorem})&=&\int_0^{+\infty}\frac{d... |
4,008,196 | <p>Wikipedia contains the following figure (to be found, e.g. <a href="https://en.wikipedia.org/wiki/Monoid" rel="nofollow noreferrer">here</a>) in order to visualize the relations between several algebraic structures. I highlighted a part that I find especially interesting.</p>
<img src="https://i.stack.imgur.com/18HU... | Michael Kinyon | 444,012 | <p>It should be pointed out that the accepted answer's definition of "has division" is not how the term is used in quasigroup theory or universal algebra. A magma <span class="math-container">$(M,\cdot)$</span> (or binar or groupoid in the old noncategorical sense) is said to be a <em>division magma</em> if f... |
2,110,681 | <p>I recently started a Discrete Mathematics course in college and I am having some difficulties with one of the homework questions. I need to learn this, so please guide me through at least two steps to get the ball rolling. </p>
<p>The question reads: Show that if $A$ and $B$ are sets, then: $(A \cap B) \cup (A \cap... | Julio Maldonado Henríquez | 406,412 | <p>The idea is to achieve get close $B$ and $B^c$. Then we use distributive property: $(A\cap B)\cup(A\cap B^c)=A\cap (B\cup B^c)=A\cap X=A $,
with $X$ the universe</p>
|
821,758 | <blockquote>
<p>Find the mean of $a, a+d, a+2d, a+3d,\dots,a+nd$</p>
</blockquote>
<p>I have no idea what to do in this question but i have tried the following:
$$mean\ \bar{x}= \frac{(a)+(a+d)+(a+2d)+(a+3d)+\cdots+(a+nd)}{n+1} $$</p>
<p>This is obvious step we do while dealing with mean.Now i don't know what do bu... | Ross Millikan | 1,827 | <p>Now distribute to make the numerator $xa+yd$. How many copies of $a$ do you have? What is the total of the coefficients of $d$?</p>
|
499,587 | <p>$$M = \left(\begin{smallmatrix} a_1 & a_2 & a_3 & a_4\\ b_1 & b_2 & b_3 & b_4\\ a_1 & c_2 & b_2 & c_4\\ a_4 & d_2 & b_3 & c_4\\ b_1 & c_2 & a_2 & e_4\\ b_4 & d_2 & a_3 & e_4\end{smallmatrix}\right)$$ All of the equations equal to 26; augmented, ... | Oleg567 | 47,993 | <p>As I see, there are $960$ solutions ($80$ different solutions, ignoring rotation and mirror-transformation).</p>
<p>For example, $2$ of them: </p>
<p><img src="https://i.stack.imgur.com/KVpwS.png" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/vWyOd.png" alt="enter image descriptio... |
3,838,943 | <p>Let <span class="math-container">$z_n$</span> be a Blaschke sequence in <span class="math-container">$\mathbb{D}$</span> and let <span class="math-container">$B$</span> be the Blaschke product defined by <span class="math-container">$$B(z)=z^m\prod_{n=1}^{\infty}\frac{|z_n|}{z_n}\frac{z_n-z}{1-\bar{z}_nz}$$</span>
I... | Oliver Díaz | 121,671 | <p>From mu comment:</p>
<p>By brute force
<span class="math-container">$$
B'(z)= mz^{m-1}\prod_{n\in\mathbb{N}}\frac{|z_n|}{z_n}\frac{z_n-z}{1-\overline{z_n}z_n} + z^m\sum_{n\in\mathbb{N}}\frac{|z_{n}|}{z_{n}}\frac{(\bar{z}_{n}z-1)+(z_{n}-z)\bar{z}_{n}}{(1-\bar{z}_{n}z)^{2}}\prod_{k\neq n}\frac{|z_k|}{z_k}\frac{z_k-z}... |
1,308,045 | <p>consider the set $X = \{20, 30, 40, 50, 60, 70\}$ and the mean $\bar{x} = 45$ then $\sum^6_{i=1}(x_i-\bar{x})^2 = 1750 = \sum^6_{i=1}x_i^2 - 6\bar{x}^2$.</p>
<ol>
<li>How would I transform the first term by hand to the second. What are the exact steps?</li>
<li>Does this transformation always lead to the same resul... | Barry | 90,638 | <p>We can derive what $a$ is based on the equality:</p>
<p>$$\sum_i(x_i-a)^2=\sum_i x_i^2 - na^2$$
$$\sum_i((x_i-a)^2-x_i^2)= - na^2$$
$$\sum_i{-2x_ia + a^2}= -na^2$$
$$na^2 - 2a\sum_ix = -na^2$$
$$2na^2=2a\sum_ix$$</p>
<p>So either $a=0$ or $a=\frac{\sum{x}}{n} = \bar{x}$.</p>
|
442,043 | <p>Assume I have a non-empty finite set $S$ with $x=|S|$. I want to divide the set $S$ into subsets $S_1, S_2, .., S_n$ (<em>Edit:</em> Yes, $S = \cup S_i$, and I'm embarrassed that I forgot to include that) such that: </p>
<ul>
<li>$ |S_i| = y, \forall 1 \le i \le n$ (The cardinality of each subset is fixed) </li>... | Felix Goldberg | 53,608 | <p>To add to what @hardmath had written, Fisher's inequality guarantees $n \leq x$.</p>
<p>See, e.g., <a href="http://lovelace.thi.informatik.uni-frankfurt.de/~jukna/EC_Book/sample/172-173.pdf" rel="noreferrer">here</a> for a statement and a proof.</p>
|
1,573,141 | <p>Consider the following equality:</p>
<p>$$x=(1-2)(1+2+4)+(2-3)(4+6+9)+(3-4)(9+12+16)+....+(49-50)(2401+2450+2500)$$</p>
<p>Solve for $x$.</p>
<p>The only thing I noticed is the first part like $(1-2)$,$(3-4)$ gives us $-1$ but then I just don't see what the trick behind this problem is.</p>
| Adhvaitha | 228,265 | <p>Each term is of the form
$$(n-(n+1))(n^2 + (n^2+n) + (n+1)^2) = -(3n^2+3n+1)$$
Hence,
$$x = - \sum_{n=1}^{m} (3n^2+3n+1) = - m(m^2+3m+3)$$
where $m=49$.</p>
|
1,037,972 | <p>The morse code is made up of marks called dots and dashes."Q", for example is (--,--).Is it possible to make up such a code so that every letter of the alphabet is represented by at most three marks?</p>
<p>i have tried this question as follows
with 3 marks we can form 2+4+8=14 letters
my answer is coming correct ... | mvw | 86,776 | <p>Your input alphabet is $\Sigma = \{ A, \ldots, Z \}$ having $\lvert \Sigma \rvert = 26$ different symbols. </p>
<p>The alphabet used for transmission is $\Sigma_T = \{s_. , s_- \}$ having $\lvert \Sigma_T\rvert = 2$ different symbols and you want to form words of length $1$, $2$ and $3$.
$$
L \subset \Sigma_T^1 \c... |
1,999,162 | <p>Assume that $\{x_n\}$ is a sequence of real numbers and $a_n=\frac{x_1+\dots+x_n}{n}$ . </p>
<p>a) Prove that $\displaystyle \liminf_{n \to\infty} x_n \le \liminf_{n \to\infty} a_n \le \limsup_{n \to\infty} a_n \le \limsup_{n \rightarrow \infty} x_n$. </p>
<p>b) Give an example such that all of the limits writte... | yoyostein | 28,012 | <p>Partial answer for (a):</p>
<p>First assume that <span class="math-container">$(x_n)$</span> is a bounded squence.</p>
<p>Let <span class="math-container">$L=\limsup_{n\to\infty}x_n<\infty$</span>. By definition of <span class="math-container">$\limsup$</span>, there exists <span class="math-container">$K$</span>... |
1,999,162 | <p>Assume that $\{x_n\}$ is a sequence of real numbers and $a_n=\frac{x_1+\dots+x_n}{n}$ . </p>
<p>a) Prove that $\displaystyle \liminf_{n \to\infty} x_n \le \liminf_{n \to\infty} a_n \le \limsup_{n \to\infty} a_n \le \limsup_{n \rightarrow \infty} x_n$. </p>
<p>b) Give an example such that all of the limits writte... | NerdOnTour | 961,598 | <p>After @yoyostein's answer has hopefully helped you in tackling exercise a), I want to add a hint for b) and c):</p>
<p><strong>Hint for b):</strong> In my opinion, the difficult part about it is to have the middle inequality <span class="math-container">$\liminf a_n < \limsup a_n$</span>. To achieve it, it may be... |
55,933 | <p>I am afraid this post may show my naivety. At a recent conference, someone told me that there are some arguments in computability theory that don't relativize. Unfortunately, this person (who I think may be an MO regular) couldn't give me any examples off-hand.</p>
<p>My naive understanding is that one can take a... | Andrew Marks | 6,151 | <p>Early in the history of recursion theory, the realization that all known proofs in the subject could be relativized in the manner you indicate led Hartley Rogers to make what is called the homogeneity conjecture.</p>
<p>Let $\mathcal{D}$ be the structure of the Turing degrees with the partial order of Turing reduci... |
3,567,002 | <p>I want to factorize such equation:
<span class="math-container">$$b^2c + bc^2 +a^2c+ac^2+a^2b+ab^2 +2abc$$</span> into product of linear factors.<br>
May I know is there any quick way/trick to do so?I am very confused with such equation. </p>
<p>Hello users, my main point here is what to think when we looking at f... | Michael Rozenberg | 190,319 | <p>If <span class="math-container">$a=-b$</span> we'll get <span class="math-container">$0$</span>...</p>
<p>Thus, it's just <span class="math-container">$$(a+b)(a+c)(b+c).$$</span>
Try to get it.</p>
|
3,567,002 | <p>I want to factorize such equation:
<span class="math-container">$$b^2c + bc^2 +a^2c+ac^2+a^2b+ab^2 +2abc$$</span> into product of linear factors.<br>
May I know is there any quick way/trick to do so?I am very confused with such equation. </p>
<p>Hello users, my main point here is what to think when we looking at f... | Will Jagy | 10,400 | <p>There is a nice article on this, Gary Brookfield (2016) <em>Factoring Forms</em>, The American Mathematical Monthly,
volume 123 number 4, pages 347-362. The main theorem was proved by Aronhold in 1849. </p>
<p>The general theorem, easy to state, is that a homogeneous cubic factors completely over the complexes if a... |
1,660,116 | <p>I would like to get a closed form of $A_n(x)$ if verifies the following recurrence relation</p>
<p>$$A_n(x)=\frac{d}{dx}\left(\frac{A_{n-1}(x)}{a-\cos x}\right)\,\,\,\text{and}\,\,\,A_0(x)=1.$$</p>
<p>Really I need to know the general term of $A_n(0)$. </p>
<p>Any ideas or suggestions are welcome.</p>
| ppooppii | 303,449 | <p>The first few terms:</p>
<ul>
<li>$A_0(x)=1$</li>
<li>$\displaystyle A_1(x)=\frac{-\sin x}{(a-\cos x)^2}$</li>
<li>$\displaystyle A_2(x)=3A_1^2(x)+\frac{-\cos x}{(a-\cos x)^3}$</li>
<li>$\displaystyle A_3(x)=-15A_1^3(x)+10A_1(x)A_2(x)+\frac{\sin x}{(a-\cos x)^4}$</li>
<li>$\displaystyle A_4(x)=105A_1^4(x)-105A_1^2(... |
4,582,099 | <p>Find limit of the given function:</p>
<p><span class="math-container">$$\lim_{x\rightarrow0} \frac{(4^{\arcsin(x^2)} - 1)(\sqrt[10]{1 - \arctan(3x^2)} - 1)}{(1-\cos\tan6x)\ln(1-\sqrt{\sin x^2})}
$$</span></p>
<p>I tried putting 0 instead of x
<span class="math-container">$$\lim_{x\rightarrow0} \frac{(4^{\arcsin(x^2)... | Enrico M. | 266,764 | <p>As the comment suggests, you shall use Taylor expansion up to <span class="math-container">$O(3)$</span>, that is:</p>
<p><span class="math-container">$$4^{\arcsin(x^2)} \approx 1+x^2 \log (4)+O\left(x^4\right)$$</span></p>
<p><span class="math-container">$$\sqrt[10]{1- \arctan(3x^2)} \approx 1-\frac{3 x^2}{10}+O\le... |
1,711,266 | <p>The circle inscribed in the triangle $ABC$ touches the sides $BC$ , $CA$ , and $AB$ in the points $A_1,B_1,C_1$ respectively. Similarly the circle inscribed in the triangle $A_1B_1C_1$ touches the sides in $A_2,B_2,C_2$ respectively, and so on. If $A_nB_nC_n$ be the $n^{th}$ $\triangle$ so formed, Prove its angles ... | nbubis | 28,743 | <p>The triangles created in such a way are known as <a href="http://mathworld.wolfram.com/ContactTriangle.html" rel="nofollow">"Contact Triangles"</a>.</p>
<p>If I understand correctly, the result you are looking for has been published. to quote form the above MathWorld link:</p>
<blockquote>
<p>Beginning with an a... |
384,628 | <p>i have searched through internet, but found only paid articles. Need to understand how Petersen graph can be contracted to K33, it says what through deleting the central vertex of 3-symmetric drawig. But what is 3-symmetric drawing?</p>
| MJD | 25,554 | <p>$\mathbb R$ is the set of real numbers. $[0,1]$ is the set of real numbers that are between 0 and 1, inclusive.</p>
<p>“$f(x) = x^2$” means that this is the function where, if you put in $x$, where $x$ is a real number between 0 and 1, you get out the real number $x^2$. </p>
|
3,395,549 | <p>Consider the topological space <span class="math-container">$R^n$</span> with the standard topology. Let <span class="math-container">$A$</span> be any affine subspace. Prove that <span class="math-container">$A$</span> is a closed subset of <span class="math-container">$R^n$</span>.</p>
<p>If I recall things corre... | Matematleta | 138,929 | <p>In your post, you are looking for a function to use to conclude. Here is a sketch of an approach: it is enough to show that subspaces are closed, because affine spaces are translations of these, and the function <span class="math-container">$\vec x\mapsto \vec x+\vec u$</span> for fixed <span class="math-container">... |
3,395,549 | <p>Consider the topological space <span class="math-container">$R^n$</span> with the standard topology. Let <span class="math-container">$A$</span> be any affine subspace. Prove that <span class="math-container">$A$</span> is a closed subset of <span class="math-container">$R^n$</span>.</p>
<p>If I recall things corre... | Walt | 550,810 | <p>Let <span class="math-container">$V$</span> be a subspace of <span class="math-container">$\mathbb{R}^n$</span>. Take <span class="math-container">$\{v_1,...,v_p\}$</span> to be a basis of <span class="math-container">$V$</span>. We can extend this to a basis of <span class="math-container">$\mathbb{R}^n$</span> den... |
1,449,845 | <p>So the biquadratic equation is $x^4+(2-\sqrt3)x^2+2+\sqrt3=0$. Let $a_1,a_2,a_3,a_4$ be its roots. So we have to find the value of $(1-a_1)(1-a_2)(1-a_3)(1-a_4)$ . <br>
<strong>My attempt:</strong> <br>
So of we put $x^2=t$, and let the roots of the new quadratic equation be $a_1,a_2$. So we get that $a_1=-a_3;a_2=-... | Mark Bennet | 2,906 | <p>You are confusing which are the roots. You have that $$t^2+(2-\sqrt 3)t+(2+\sqrt 3)=0$$</p>
<p>The roots of this equation are $a_1^2$ and $a_2^2$ and you want to find $$(1-a_1^2)(1-a_2^2)=1-(a_1^2+a_2^2)+a_1^2a_2^2$$</p>
<p>Now reading off the equation for $t$ you get $a_1^2+a_2^2=\sqrt 3-2$ and $a_1^2a_2^2=2+\sqr... |
1,822,362 | <p><a href="https://i.stack.imgur.com/jOOyV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jOOyV.jpg" alt="enter image description here"></a></p>
<p>I'm not sure if I understand this question and was wondering if anyone could provide any insight to an answer. The only thing I can think of adding is... | callculus42 | 144,421 | <p>A special case of integer variables are binary variables. The second constraint can be reformulated as</p>
<p>$y_1\cdot x_2+\frac{1}{3}\cdot y_2\cdot x_2+\frac{1}{4}\cdot y_3\cdot x_2+\frac{1}{10}\cdot y_4\cdot x_2=1$</p>
<p>$y_1, y_2, y_3, y_4\in \{0,1 \}$</p>
|
1,123,979 | <p>I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would break ice for my future cases.</p>
<p>case 1: Evaluate: $\int x\sqrt{x+2}dx $</p>
<p>case 2: Evaluate: $\int \frac{x^... | graydad | 166,967 | <p><strong>Hint for 1:</strong> If $u = x+2$ then $du = dx$ and $x = u-2$. Hence your first integral becomes $$\int x\sqrt{x+2}dx = \int(u-2)\sqrt{u}\space du$$ Remember this is single variable calculus, so it doesn't make sense to evaluate an integral like $\int x\sqrt{u}\space dx$ where both $x$ and $u$ are variable... |
143,696 | <p>Does anyone know if there is a way to scale a graph from the command Periodogram?
More specifically in terms of the y axis. I am wondering if there is a way that I can make the highest value on the y axis of these graphs to be "1" while still keeping the ratio in comparison to the highest value of all the other valu... | kglr | 125 | <p>You can post-process the output of <code>Periodogram</code> to rescale the <code>Line</code>s:</p>
<pre><code>ClearAll[postProcessF]
postProcessF[d__, o1 : OptionsPattern[]][scale_: {0, 1}, imgsize_: 300, style_: Blue] :=
Graphics[Periodogram[d, o1][[1]] /.
Line[x_] :> {style, Line[Transpose[{x[[All, 1]], ... |
51,596 | <p>Some time ago, I asked <a href="https://math.stackexchange.com/q/42276/8271">this</a> here. A restricted form of the second question could be this:</p>
<blockquote>
<p>If $f$ is a function with continuous first derivative in $\mathbb{R}$ and such that $$\lim_{x\to \infty} f'(x) =a,$$ with $a\gt 0$, then $$\lim_{x... | Pushpak Dagade | 6,094 | <p>I will try to prove is in a different way which can be much simpler - using visualization. </p>
<p>Imagine how will a function look if it has a constant, positive slope -<br>
A straight line, with a positive angle with the positive x axis.<br>
Although this can be imagined, I am attaching a simple pic -<br>
<img s... |
2,060,793 | <p>I need to know wich answer is right</p>
<p><a href="https://i.stack.imgur.com/QcxdH.jpg" rel="nofollow noreferrer">https://i.stack.imgur.com/QcxdH.jpg</a></p>
<p>I tried to solve it using recursivity but I didn't get any one of them</p>
<ul>
<li>$y_1=\sqrt{x}$ </li>
<li>$y'=\frac{1}{2\sqrt{x}}=\frac{1}{2y_1}$</li... | Community | -1 | <p>$$y^2-y=x$$ then $$(2y-1)y'=1.$$</p>
|
1,303,868 | <p>Theorem:
$$\left\langle a^k \right\rangle = \left\langle a^{\gcd(n,k)}\right\rangle$$
Let G be a group and $$ a \in G$$ such that $$|a|=n$$
Then:</p>
<p>$$\left\langle a^k \right\rangle = \left\langle a^{\gcd(n,k)}\right\rangle$$</p>
<p>The proof begins by letting d = gcd(n,k) such that d is a divisor of k so ther... | fkraiem | 118,488 | <p>Given an element $a$ of a group $G$, $\langle a\rangle$ is by definition <em>the smallest subgroup of $G$ which contains $a$</em>. This means in particular that if $H$ is a subgroup of $G$ which contains $a$, then $\langle a\rangle \subseteq H$.</p>
<p>Here, you want to prove two statements of this form (for ease o... |
55,022 | <p>Why does <code>DiscretizeGraphics</code> seems to work on one <code>GraphicsComplex</code> and not the other? Here is an example that works:</p>
<pre><code>v = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
p1 = Graphics[GraphicsComplex[v, Polygon[{1, 2, 3, 4}]]];
DiscretizeGraphics[p1]
</code></pre>
<p>But this does not</p>... | user21 | 18,437 | <p>In the end this is a bug and I filed that.</p>
<p>Now, what is going on: If you extract the coords and polygons from the <code>GraphicsComplex</code> and try to set up a <code>MeshRegion</code> you get a warning:</p>
<pre><code>gc = First@
ParametricPlot3D[{Cos[t], Sin[u], Sin[t]}, {u, 0, 2 Pi}, {t, 0,
2 ... |
27,679 | <p>I have a graph here and I want to be able to manipulate the 4 parameters that are in the Block command. I want to watch the graph change as I change the parameters. </p>
<p>I've done this before, but the Block command that Mathematica suggested throws me off a little bit. </p>
<p>Here is the code: </p>
<pre><cod... | Stefan | 2,448 | <p><strong>Memoize NDSolve</strong></p>
<p>If you want to convince with a Manipulation it must be smooth. But I'm sorry to say, that if you naively choose all the parameters and use them for <code>Manipulate</code> this is frustrating slow.</p>
<p>The thing I'm trying here is to memoize the output of <code>NDSolve</c... |
3,667,925 | <p>I can solve for this case <span class="math-container">$x_{1} + x_{2} + x_{3} + x_{4} \leq 6$</span>.But with the zero I don't know.What changes?</p>
| MathRoc | 761,306 | <p>The function
<span class="math-container">\begin{align*}
g_n(x)& :=\frac{\text{d}}{\text{d}x}\left\{\left(1-x^2\right)P_n^{'}(x)\right\}\\
& =\left(1-x^2\right)P_n^{''}(x)-2xP_n^{'}(x)\tag{1}
\end{align*}</span>
is clearly a polynomial of degree <span class="math-container">$n$</span>. Then
<span class="math... |
3,667,925 | <p>I can solve for this case <span class="math-container">$x_{1} + x_{2} + x_{3} + x_{4} \leq 6$</span>.But with the zero I don't know.What changes?</p>
| WA Don | 542,712 | <p>This is another approach that does not explicitly use integration. Use <span class="math-container">$ D $</span> to stand for <span class="math-container">$ d/dx$</span>.</p>
<p>First, apply Leibniz rule for the <span class="math-container">$n+2$</span> derivative of a product,</p>
<p><span class="math-container"... |
3,000,207 | <p>I have two matrices:</p>
<p><span class="math-container">$A = \begin{bmatrix}
1 &2 \\
-1 &4
\end{bmatrix}
$</span></p>
<p>and </p>
<p><span class="math-container">$B = \begin{bmatrix}
0 &-1 \\
2 &3
\end{bmatrix}
$</span></p>
<p>I need to find a monic cubic polynomial <span class="math-container">... | Calum Gilhooley | 213,690 | <ol>
<li><p>In a homogeneous inequality, this doesn't matter; and <span class="math-container">$\ldots$</span></p></li>
<li><p><span class="math-container">$\ldots$</span> there is little difference between the two solutions in this respect, because if you take <span class="math-container">$a, b, c$</span> from <em>Pro... |
3,000,207 | <p>I have two matrices:</p>
<p><span class="math-container">$A = \begin{bmatrix}
1 &2 \\
-1 &4
\end{bmatrix}
$</span></p>
<p>and </p>
<p><span class="math-container">$B = \begin{bmatrix}
0 &-1 \\
2 &3
\end{bmatrix}
$</span></p>
<p>I need to find a monic cubic polynomial <span class="math-container">... | Michael Rozenberg | 190,319 | <p>Another solution.</p>
<p>We need to show that:
<span class="math-container">$$\frac{3}{16}-\sum_{cyc}\frac{1}{(2a+b+c)^2}\geq0$$</span> or
<span class="math-container">$$\sum_{cyc}\left(\frac{3}{16a(a+b+c)}-\frac{1}{(2a+b+c)^2}\right)\geq0$$</span> or
<span class="math-container">$$\sum_{cyc}\frac{(3(2a+b+c)^2-16a(... |
1,257,144 | <p>Is it true that $ f(\bar{z})=\overline {f(z)}$,
Where z is complex?</p>
<p>I think it holds when $f(z)$ is holomorphic since we have $f(z)=p(x,y)+iq(x,y)=p(z,0)+iq(z,0)$
Any help...</p>
| mrf | 19,440 | <p>It is true if (and only if) $f$ is real-valued on the real axis.</p>
<p>For simplicity, assume that $f$ is holomorphic on the whole plane. Let $g(z) = \overline{f(\bar z)}$. It follows from Cauchy-Riemann's equations that $g$ is also holomorphic everywhere, see for example <a href="https://math.stackexchange.com/qu... |
3,694,053 | <p>Considering the set of real numbers:</p>
<p><span class="math-container">$$A = \left\{\ln\left(\frac{2n+\sqrt{n}}{2n-\sqrt{n}}\right): n \in \mathbb{N} \right\}.$$</span></p>
<p>I must prove that <span class="math-container">$0$</span> is the greatest lower bound of <span class="math-container">$A$</span>.</p>
<p... | Andrei | 331,661 | <p>The problem is that from <span class="math-container">$\pi/2$</span> to <span class="math-container">$\pi$</span>, <span class="math-container">$$\cos\theta=-\sqrt{1-\sin^2\theta}$$</span> Note the minus sign in front of the square root.</p>
|
9,672 | <p>This might be a somewhat silly and inconsequential question, but it's aroused my curiosity. One has the theorem in commutative algebra that the integral closure of a domain $A$ in its field of fractions $Q(A)$ is the intersection of all the valuation subrings of $Q(A)$ containing $A$. This naturally leads to the imp... | Ilya Nikokoshev | 65 | <p>Congratulations, you've just discovered the definition of <strong><a href="http://en.wikipedia.org/wiki/Normal_variety" rel="nofollow">normal variety</a></strong> — precisely the one for which locally all rings are integral closures in their fields of fractions.</p>
<p>This definition leads to several ineters... |
872,693 | <p>Let $\mu$ be a Borel measure supported on $\mathbb{Q} \subset \mathbb{R}$. Must $\mu$ be a sum of Dirac measures?</p>
| Ilya | 5,887 | <p>I guess you mean infinite sums. The following statement holds true, which you may think of more general (or not). If $A$ is a countable set, then any positive, say $\sigma$-finite measure defined on $2^A$ is a sum of Dirac measure. This follows directly from $\sigma$-additivity of the measure. Consider any such meas... |
164,683 | <p>I am trying to figure out the elements of the $2\times 2$ matrix</p>
<p>$$B=A_nA_{n-1}A_{n-2}\cdots A_1,\;\; n=1,2,3,\ldots$$</p>
<p>where</p>
<p>$$A_k=\begin{bmatrix}a-2k&-k(k-1)-b\\1&0\end{bmatrix},$$</p>
<p>with $a,b>0$ fixed. I wrote the following script to see what's going on:</p>
<pre><code>$As... | J. M.'s persistent exhaustion | 50 | <p>Here is a method that gets you a good way forward:</p>
<p>Set up the recursions and initial conditions first:</p>
<pre><code>Transpose[Thread /@
Thread[Array[C[##][k] &, {2, 2}] ==
{{a - 2 k, -k (k - 1) - b}, {1, 0}}.Array[C[##][k-1] &, {2, 2}]]]
{{C[1, 1][k] == (a - 2 k) C[1, 1]... |
2,622,311 | <p>Just need a bit of help with this.</p>
<p>Number Theory: Show for all positive integers σ(2n)>2σ(n)</p>
<p>Sigma being the function of total numbers of factors including 1 and itself from number theory</p>
<p>I know a good starting point would be to consider n=2^r m, but I'm really stuck on how to apply this to a... | Donald Splutterwit | 404,247 | <p>$ \sigma(n)$ is the sum of the divisors of $n$.
\begin{eqnarray*}
\sigma(n)= \sum_{ d \mid n} d.
\end{eqnarray*}
Now if $d \mid n $ then $ 2d \mid 2n$ (and there could be other values that divide $2n$) so
\begin{eqnarray*}
\sigma(2n) \geq \sum_{ 2d \mid 2n} 2d = 2\sum_{ d \mid n} d = 2 \sigma(n).
\end{eqnarray*}</... |
2,427,747 | <p>I want to prove that $2^{n+2} +3^{2n+1}$ is divisible by $7$ for all $n \in \mathbb{N}$ using proof by induction.</p>
<p>Attempt</p>
<p>Prove true for $n = 1$</p>
<p>$2^{1+2} + 3^{2(1) +1} = 35$</p>
<p>35 is divisible by 7 so true for $n =1$</p>
<p><em>Induction step</em>: Assume true for $n = k$ and prove true... | DJohnM | 58,220 | <p>Yet another view:</p>
<p>Take your $n=k$ version and double it. Then add $21(3^{2k})$ and you have your $k+1$ expression. Both things you're adding are divisible by $7$</p>
|
344,119 | <p>$$\int_{0}^{\pi/2}\int_{x}^{\pi/2}\frac{\cos y}{y} \, \operatorname{d}\!y\, \operatorname{d}\!x$$</p>
<p>$\iint_R2xy^2 \, \operatorname{d}\!A$ where R is the right half of the unit circle</p>
| Cameron Buie | 28,900 | <p>That first one doesn't converge. Note that if it did, $$\begin{align}\int_0^{\pi/2}\frac{\cos y}y\,dy &= \int_0^{\pi/3}\frac{\cos y}y\,dy+\int_{\pi/3}^{\pi/2}\frac{\cos y}y\,dy\\ &\geq \int_0^{\pi/3}\frac{\cos y}y\,dy\\ &\geq \frac12\int_0^{\pi/3}\frac1y\,dy\\ &=\frac12\lim_{a\to 0^+}\int_a^{\pi/3}\f... |
3,764,423 | <p>Let <span class="math-container">$X$</span> be a set and show that the following function defines a metric.
<span class="math-container">$f(x, y) = (0 \text{ if } x = y \text{ and } 1 \text{ if } x \neq y)$</span></p>
<p>I'm especially having trouble with the symmetry and triangle inequality steps. Thanks so much!</... | Henry | 6,460 | <p>One approach is to use an expansion of <a href="https://en.wikipedia.org/wiki/Harmonic_number" rel="nofollow noreferrer">harmonic numbers</a> where <span class="math-container">$H(n)= \sum \limits_{i=1}^n \frac1n = \log_e(n) + \gamma + \frac{1}{2n}+O(n^{-2})$</span></p>
<p>You then have <span class="math-container">... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.