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 |
|---|---|---|---|---|
4,365,859 | <p>My professor had given us a problem. It goes like -</p>
<blockquote>
<p>Find equation of circle passing through intersection of circle <span class="math-container">$S=x^2+y^2-12x-4y-10=0$</span> and <span class="math-container">$L:3x+y=10$</span> and having radius equal to that of circle <span class="math-container"... | Math Lover | 801,574 | <p>After changing the coordinates, in effect you are rotating <span class="math-container">$x^2 + (y-a)^2 = a^2$</span> around x-axis.</p>
<p>The circle is <span class="math-container">$x^2 + y^2 = 2 ay$</span></p>
<p><span class="math-container">$ \displaystyle y' = \frac{x}{a-y}$</span></p>
<p><span class="math-conta... |
403,977 | <blockquote>
<p>Let $u_n>u_{n+1}>0$ for all $n \in \Bbb N$, and suppose that $u_2+u_4+u_8+u_{16}+\dots$ diverges. Prove that $\sum_{n=1}^{\infty}\frac{u_n}{n}$diverges.</p>
</blockquote>
<p>Please provide me a hint or a full solution.</p>
| Bill Kleinhans | 73,675 | <p>Suppose $ u_n = (\log n)^{-1}$</p>
|
224,474 | <p>I want to generate the following matrix for any n:</p>
<pre><code> Table[A[i, j], {i, 0, n}, {j, 0, n}];
</code></pre>
<p>Where,</p>
<pre><code> A ={{0,1,0},{0,0,2},{0,0,0}} when n=2;
A={{0,1,0,0},{0,0,2,0},{0,0,0,3},{0,0,0,0}} when n=3;
A={{0,1,0,0,0},{0,0,2,0,0},{0,0,0,3,0},{0,0,0,0,4},{0,... | kglr | 125 | <p>You can also use <a href="https://reference.wolfram.com/language/ref/SparseArray.html" rel="noreferrer"><code>SparseArray</code></a> + <a href="https://reference.wolfram.com/language/ref/Band.html" rel="noreferrer"><code>Band</code></a>:</p>
<pre><code>ClearAll[sA]
sA[n_] := SparseArray[Band[{1, 2}, Automatic] ->... |
3,926,580 | <p>I'm trying to prove with squeeze theorem that the limit of the following series equals 1:</p>
<p><span class="math-container">$$\frac{1+\sqrt{2}+\sqrt[3]{3}+...+\sqrt[n]{n}}{n}$$</span></p>
<p>For the left side of the inequality I did:</p>
<p><span class="math-container">$$\frac{1+\sqrt{1}+\sqrt[3]{1}+...+\sqrt[n]{1... | robjohn | 13,854 | <p>As shown in <a href="https://math.stackexchange.com/a/3926707">this answer</a>, the Binomial Theorem says that for <span class="math-container">$n\ge1$</span>,
<span class="math-container">$$
\begin{align}
1\le n^{1/n}
&\le1+\sqrt{\frac2n}\tag{1a}\\
&\le1+\frac{2\sqrt2}{\sqrt{n}+\sqrt{n-1}}\tag{1b}\\[3pt]
&a... |
2,211,461 | <p>Given that $f$ is an integrable function on $X$ and $\{E_k\}_{k=1}^\infty$ where each $E_k$ is a measurable set such that $\lim_{k\rightarrow \infty} \mu(E_k) = 0$</p>
<p>Can we show that $$\lim_{k\rightarrow \infty} \int_{E_k} fd\mu = 0$$ </p>
<p>I want to prove like this:
$$|\int_{E_k} fd\mu| \leq sup|f|\cdot \m... | mathworker21 | 366,088 | <p>Let $f_k = f\chi_{E_k}$. Note $f_k \to 0$ almost everywhere since $\mu(E_k) \to 0$. Also, $|f_k| \le |f|$ which is integrable, so Dominated Convergence Theorem gives you the result.</p>
|
3,826,152 | <p>How can I verify this inequality by using induction?</p>
<p><span class="math-container">$$\frac1{2n}\le \frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}$$</span></p>
<p>I understand the principle of induction, but I'm struggling with how my teacher is solving this problem. I Highlighted the area I'm stuck ... | Fawkes4494d3 | 260,674 | <blockquote>
<p>In this screenshot from your image call the LHS and RHS of the first inequality (which is a consequence of the induction hypothesis) <span class="math-container">$B$</span> and <span class="math-container">$C$</span> respectively. The first inequality is stating that <span class="math-container">$\boxed... |
144,550 | <p>Cross post <a href="http://community.wolfram.com/groups/-/m/t/1120587?p_p_auth=9d8Myocf" rel="nofollow noreferrer">here</a></p>
<hr>
<p>A word path means the last character of the last word is same to the first character of the next word.And the path have no duplicated word.If I have $55$ words,I can find it with ... | Gustavo Delfino | 251 | <p>You are clearly a beginner. Experienced Mathematica users run away in panic when presented so many <code>For</code> loops. Using <code>Do</code> is better than nesting <code>For</code> and <code>Table</code> is better. In general you never <code>Print</code> something you want to export.</p>
<p>These steps may help... |
144,550 | <p>Cross post <a href="http://community.wolfram.com/groups/-/m/t/1120587?p_p_auth=9d8Myocf" rel="nofollow noreferrer">here</a></p>
<hr>
<p>A word path means the last character of the last word is same to the first character of the next word.And the path have no duplicated word.If I have $55$ words,I can find it with ... | george2079 | 2,079 | <p>another approach:</p>
<pre><code>B[n_, b_, l_, m_, d_, p_] := l + m - d - p + n
{##, If[(amax =
NArgMax[{B[n, Sequence@##], n >= 1}, Element[n, Integers]]) >
0, Sequence @@ {amax, B[amax, Sequence@##]},
Sequence @@ {0, 0}]} & @@@
Tuples[{
Range[2, 3], Range[6, 7]... |
4,551,543 | <p>You have ten coins, each with a bias for head drawn from a uniform distribution [0,1]. For example, a coin with a 0.7 bias means that the probability of flipping a head is 0.7. You are allowed 100 flips and you are free to choose any coins to flip. Each time you flip a head, you get $1. How much money would you pay ... | Bafs | 846,919 | <p>If you are asking for the optimum strategy, I would also be curious to hear from others. My first instinct would be to identify the coins by number, and flip Coin #1 until it came up tails. Then I would move on to each subsequent coin, flipping until I get the first tails. This would establish a very shaky estimate ... |
957,604 | <p>Let <strong>X</strong> and <strong>Y</strong> have the joint probability density function given by:</p>
<p>f(x,y)=$\frac{1}{4}exp{\frac{-(x+y)}{2}}$, x>0 and y>0</p>
<p>(a) Find Pr(x<1,y>1)</p>
<p>(b) Find Pr(y<$x^2$)</p>
<p>This is how I tackled (a):</p>
<p>Pr(x<1,y>1)=$\int\int exp{\frac{-(x+y)}{2}}$... | Jared | 138,018 | <p>a) What are all of the possibilities for transferring to bowl II?</p>
<ol>
<li> Both chips are white (what is the probability)?
<li> Both chips are red (what is the probability)?
<li> One chip is red and one is white (what is the probability)?
</ol>
<p>From each of those possibilities you will end up (in bowl II) ... |
95,168 | <p>I try to prove the following</p>
<p>$$\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$$</p>
<p>with $r \geq 3$ and $r \in \mathbb{P}$. Do I have to make in induction over $r$ or any better ideas?</p>
<p>Any help is appreciated.</p>
| Did | 6,179 | <p>Let $x_n=2^{-n}{2n\choose n+1}$. Then $x_{n+1}=x_n\frac{(2n+1)(n+1)}{n(n+2)}\gt x_n$ for every $n\geqslant1$ and $x_2=1$, hence $x_n\geqslant1$ for every $n\geqslant2$. </p>
<p>Since $\varphi(r)\geqslant2$ for every integer $r\geqslant3$, the estimate above implies that the desired inequality holds for every (not n... |
3,995,954 | <p>Today during a Calc I exam we(me and my classmates) have been ask to prove <span class="math-container">$$-a\cdot e\cdot \ln(x)\le x^{-a}$$</span>, <span class="math-container">$\forall x>0$</span> and <span class="math-container">$\forall a\in \mathbb{R}$</span>. But, noneone in the room has known how to do it. ... | Tito Eliatron | 84,972 | <p><strong>HINT</strong></p>
<p>Try to find the maximum of <span class="math-container">$f(t):=\frac{\ln(t)}{t}$</span> and recall that your inequality is equivalent to
<span class="math-container">$$ \frac{\ln(x^{-a})}{x^{-a}}\le\frac{1}{e}.$$</span></p>
|
3,787,576 | <p><span class="math-container">$$
\mbox{Prove}\quad
\int_{0}^{1}{\mathrm{d}x \over
\left(\,{x - 2}\,\right)\,
\sqrt[\Large 5]{\,x^{2}\,\left(\,{1 - x}\,\right)^{3}\,}\,}
=
-\,{2^{11/10}\,\pi \over \,\sqrt{\,{5 + \,\sqrt{\,{5}\,}}\,}\,}
$$</span></p>
<ul>
<li>Being honest I havent got a clue where to start. I dont thi... | Zenix | 711,043 | <h3>Hint:</h3>
<p>Substitute <span class="math-container">$x \rightarrow\frac{1}{x-1}$</span>. We'll get: <span class="math-container">$$-\int_0^\infty \dfrac{x^{-3/5}dx}{(2x+1)}$$</span>
Can you continue from here?</p>
|
114,831 | <p>$$A_n=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}$$
Try to prove $$\lim_{n \to \infty}n(\ln 2-A_n) = \frac{1}{4}$$</p>
<p>I try to decompose $\ln 2$ as $$\ln(2n)-\ln(n)=\ln\left(1+\frac{1}{2n-1}\right)+\dots+\ln\left(1+\frac{1}{n}\right)\;,$$ but I can't continue, is that right?</p>
| Michael Hardy | 11,667 | <p>Letting $f(x)=1/x$ we have
$$
\int_1^2 f(x) \; dx = \log_e 2
$$
and
$$
\frac 1 n \left( f\left(1+\frac 1 n\right) + f\left(1+\frac 2 n\right) + f\left(1+\frac 3 n\right) + \cdots + f\left(1+\frac n n\right) \right) \to \int_1^2 f(x) \; dx\text{ as }n\to\infty,
$$
so
$$
\frac 1 n \left( \frac{n}{n+1} + \frac{n}{... |
217,711 | <p>Set
$$
g(x)=\sum_{k=0}^{\infty}\frac{1}{x^{2k+1}+1} \quad \text{for} \quad x>1.
$$</p>
<p>Is it true that
$$
\frac{x^{2}+1}{x(x^{2}-1)}+\frac{g'(x)}{g(x)}>0 \quad \text{for}\quad x>1?
$$
The answer seems to be positive. I spent several hours in proving this statement but I did not come up with anyth... | Fedor Petrov | 4,312 | <p>I am not sure at all, please double check.</p>
<p>Denote $u(t)=(1+t)^{-1}$, it is a decreasing convex function on $[0,1]$, but $u(e^s)$ is concave on $(-\infty,0]$. As Neil Strickland notes, what we have to prove is that $(x-1/x)g(x)$ increases, or, if we denote $y=1/x$, that $(1/y-y)g(1/y)$ decreases. We have $$(1... |
1,933,410 | <p>I'm reading a book about differential geometry and there's a part where he talks about the standar spherical coordinates on $S^2$, which are given by:</p>
<p>$$x:(0,2\pi)\times (0,\pi)\to S^2, x(\theta, \phi) = (\cos\theta\sin\phi,\sin\theta\sin\phi, \cos\phi)$$</p>
<p>Shouldn't it be in $S^3$?</p>
| Alexis Olson | 11,246 | <p>No. <a href="https://en.wikipedia.org/wiki/Sphere" rel="nofollow">$S^2$</a> is the ordinary sphere embedded in three dimensions. $S^3$ would have $4$ coordinates in Cartesian form.</p>
<p>It is called $S^2$ since it is a $2D$ surface in the sense that any point on it can be described by two coordinates ($\theta, \p... |
2,866,768 | <p>Question: Use the trigonometric identity $\cos(2A)=1-2\sin^2(A)$ to show that $$\sin\left(\frac{\pi}{12}\right)=\sqrt{\frac{1}{2} - \frac{\sqrt{3}}{4}}$$</p>
<p>What are good strategies to figure out this question in Particular?</p>
| LJFan | 580,348 | <p>Use the trigonemetric identity, we get $\sin{A}=\pm\sqrt{\frac{1-\cos{2A}}{2}}$</p>
<p>Substitute $A=\frac{\pi}{12}$ in it, we can easily find that $\sin A>0$. Therefore $\sin{\frac{\pi}{12}}=\sqrt{\frac{1-\cos{\frac{\pi}{6}}}{2}}=\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}=\sqrt{\frac{1}{2}-\frac{\sqrt{3}}{4}}$</p>
|
3,005,591 | <p>I wish to find the integers of <span class="math-container">$a,b,c$</span> and <span class="math-container">$d$</span> such that:
<span class="math-container">$$225a + 360b +432c +480d = 3$$</span>
which is equal to:
<span class="math-container">$$75a + 120b +144c+ 160d =1$$</span></p>
<p>I know I have to use the... | Mohammad Riazi-Kermani | 514,496 | <p>First divide both sides by <span class="math-container">$3$</span> to get <span class="math-container">$$ 75a+120b+144c+160d=1$$</span></p>
<p>Now let <span class="math-container">$$a=b=c=x$$</span></p>
<p>Your equation changes to <span class="math-container">$$339x+160d=1$$</span></p>
<p>You can solve this one f... |
3,005,591 | <p>I wish to find the integers of <span class="math-container">$a,b,c$</span> and <span class="math-container">$d$</span> such that:
<span class="math-container">$$225a + 360b +432c +480d = 3$$</span>
which is equal to:
<span class="math-container">$$75a + 120b +144c+ 160d =1$$</span></p>
<p>I know I have to use the... | fleablood | 280,126 | <p>TO solve <span class="math-container">$75a+120b+144c+160d=1$</span></p>
<p>You can always you Euclidean Algorithm to solve <span class="math-container">$75A + 120B = \gcd(75,120)=15$</span></p>
<p>And to solve <span class="math-container">$120\beta + 144\gamma = \gcd(120,144) = 24$</span></p>
<p>And to solve <spa... |
3,005,591 | <p>I wish to find the integers of <span class="math-container">$a,b,c$</span> and <span class="math-container">$d$</span> such that:
<span class="math-container">$$225a + 360b +432c +480d = 3$$</span>
which is equal to:
<span class="math-container">$$75a + 120b +144c+ 160d =1$$</span></p>
<p>I know I have to use the... | Steven Alexis Gregory | 75,410 | <p>Here is a description of a program that I wrote to solve such problems. It is definitely not optimized.</p>
<p>I start by considering the equalities</p>
<p><span class="math-container">\begin{array}{r}
160 &= 160(1) &+& 144(0) &+& 120(0) &+& 75(0) \\
144 &= 160(0) &+& ... |
2,640,143 | <h2>Problem</h2>
<p>Compute following:</p>
<p>$$ \int_{1}^{2} \int_{0}^{z} \int_{0}^{y+z}\frac{1}{(z+y+x)^3}dxdydz $$</p>
<h2> Attempt to solve </h2>
<p>Now i could probably solving this by opening $(z+y+x)^3$ parenthesis and then solving 3 separate integrals involving variables $x,y,z$. Opening these parenthesis w... | Mauro ALLEGRANZA | 108,274 | <p><a href="https://en.wikipedia.org/wiki/Logarithm#History" rel="nofollow noreferrer">Logarithm</a> was invented around 1600 by <a href="https://en.wikipedia.org/wiki/John_Napier" rel="nofollow noreferrer">John Napier</a> to simplify difficult calculations: a multiplication can be reduced to addition. </p>
<p>It was ... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Jose Brox | 1,234 | <p>The Laplace transform to systematically solve homogeneous ODE's with constant coefficients by transforming them in polynomial equations (and then transforming the solution back).</p>
<p><a href="http://en.wikipedia.org/wiki/Laplace_transform#Example_.231%3A_Solving_a_differential_equation" rel="nofollow">Solving a ... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Qiaochu Yuan | 290 | <p>If you're a combinatorialist and you want to know the asymptotics of a sequence $a_n$ with a nice generating function $A(z) = \sum_{n \ge 0} a_n z^n$, the very first thing you should do is find out if $A$ is meromorphic, since then one can analyze the asymptotics of $a_n$ using its poles. Even if $A$ isn't meromorp... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Harrison Brown | 382 | <p>Set theory provides by far the easiest proof of the existence of transcendental numbers -- just show that the algebraic numbers and the integers can be put into 1-1 correspondence, but the real numbers and the integers can not. Liouville's proof isn't too hard, but it's nowhere near as elegant.</p>
|
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | gowers | 1,459 | <p>For me personally, the first time I "got" Fourier analysis was when I understood how it could be used to prove Roth's theorem on arithmetic progressions (that any dense set of integers contains an arithmetic progression of length 3). It can be proved in several ways, but when you see this way you immediately realize... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Christopher Creutzig | 2,933 | <p>Proving the termination of <a href="http://en.wikipedia.org/wiki/Goodstein%27s_theorem" rel="nofollow">Goodstein sequences</a> (a problem in natural numbers) via arithmetic on infinite ordinals.</p>
|
2,159,092 | <p>$\sum _{k=1}^{n}\mathrm{sin}kz=\frac{\mathrm{sin}\frac{n+1}{{2}}\cdot \mathrm{sin}\frac{nz}{2}}{\mathrm{sin}\frac{z}{2}}$
Proof for $z\neq 0$</p>
| Community | -1 | <p>The main idea connecting logic to set theory is the idea that <em>propositions are subsets</em>.</p>
<p>On the one hand, if you are interpreting propositional logic in some (set-theoretic) domain $D$ of discourse, then each proposition gets interpreted as a subset of $D$, and connectives are operations on subsets.<... |
3,104,058 | <p>I need to make a part of a program in java that calculates a circle center. It has to be a circle through a given point that touches another circle, and the variable circle center has the possibility to move over a given line.</p>
<p><a href="https://i.stack.imgur.com/RQgBC.png" rel="nofollow noreferrer"><img src="... | amd | 265,466 | <p>You haven’t mentioned this explicitly, but from the illustration is appears that you’re only interested in <em>externally</em> tangent circles. If the radius of the circle is <span class="math-container">$r$</span>, then the points you are looking for satisfy <span class="math-container">$|PA|-|PB|=r$</span>. Set <s... |
1,562,907 | <p>I want to know whether $$\mathbb{Q}[x]/(x^3-1)$$ is a field or not. Is it as simple as determining if $x^3-1$ is irreducible in $\Bbb Q$?</p>
<p>But since it has roots , $x=1$ wouldn't this imply it is reducible and hence not a field? Or is there much more to it I am not considering?</p>
| George V. Williams | 54,806 | <p><strong>Hint</strong>: let $y = g(x)$, so $x = g^{-1}(y)$. Can you use your equation to solve for $g^{-1}(y)$?</p>
|
1,562,907 | <p>I want to know whether $$\mathbb{Q}[x]/(x^3-1)$$ is a field or not. Is it as simple as determining if $x^3-1$ is irreducible in $\Bbb Q$?</p>
<p>But since it has roots , $x=1$ wouldn't this imply it is reducible and hence not a field? Or is there much more to it I am not considering?</p>
| marwalix | 441 | <p>Start with $f(x)=g(x)-1$ and apply $f^{-1}$ to get</p>
<p>$$x=f^{-1}(g(x)-1)$$</p>
<p>Now set $y=g(x)$ this gives $x=g^{-1}(y)$ and replace in the above to get</p>
<p>$$g^{-1}(y)=f^{-1}(y-1)$$</p>
|
607,324 | <blockquote>
<p><span class="math-container">$d \in \mathbb{Z}$</span> is a square-free integer (<span class="math-container">$d \ne 1$</span>, and <span class="math-container">$d$</span> has no factors of the form <span class="math-container">$c^2$</span> except <span class="math-container">$c = \pm 1$</span>), and ... | Stephen Montgomery-Smith | 22,016 | <p>I assume that $R[X]$ means the polynomials over a ring $R$.</p>
<p>So using Taylor's series:
$$ f(X+t) = \sum_{k=0}^\infty \frac{t^k}{k!} f^{(k)}(X) = \sum_{k=0}^\infty \frac{t^k}{k!} D^k f(X) = \exp(tD) f(X) .$$
Note that the sum is actually finite since $D^k f(X) = 0$ for $k$ large enough.</p>
<p>The proof of Ta... |
853,735 | <p>Can anyone show, step-by-step, how the expression on the LHS can be turned into the expression on the RHS?</p>
<p>$x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$</p>
| Community | -1 | <p>Assuming all the numbers are positive, we in fact have
$$ x^a y^b \le a^a b^b (a+b)^{-(a+b)} (x+y)^{a+b} \tag{$\ast$} $$
with equality if and only if $x/y = a/b$.</p>
<p>This is a standard application of the AM/GM inequality, as follows:
\begin{align*}
&\left(\frac{x}{x+y}\cdot\frac{a+b}{a}\right)^{a/(a+b)}
\le... |
3,634,490 | <p>Let <span class="math-container">$N \geq 1$</span> be an integer.</p>
<p>Let <span class="math-container">$X$</span> be a standard <span class="math-container">$\mathbb{R}^N$</span> Gaussian vector (all components are <span class="math-container">$\mathcal{N}(0, 1)$</span> and i. i. d.).</p>
<p>Let <span class="ma... | charmd | 332,790 | <p>For some <span class="math-container">$n$</span> (<span class="math-container">$1$</span>, <span class="math-container">$2$</span> and all multiples of <span class="math-container">$4$</span> if you admit a certain conjecture; all powers of <span class="math-container">$2$</span> if you don't), we can find explicite... |
2,683,731 | <p>If a relation is not symmetric shall we say that it is anti symmetric?
In a quiz show's preliminary round i had this question looks easy but i couldn't answer.can anyone answer?</p>
| Bram28 | 256,001 | <p>No. Anti-symmetry means that if $R(x,y)$ and $R(y,x)$ then $x=y$. In other words, for anti-symmetry you cannot have both $R(a,b)$ and $R(b,a)$ for different $a$ and $b$. But that is not the negation of symmetry. </p>
<p>For example, for $R$ to be not anti-symmetric, we could have both $R(a,b)$ and $R(b,a)$, but to... |
2,683,731 | <p>If a relation is not symmetric shall we say that it is anti symmetric?
In a quiz show's preliminary round i had this question looks easy but i couldn't answer.can anyone answer?</p>
| user061703 | 515,578 | <p>You need to note about the logic (the "logic" tag should be added).</p>
<p>A relation $R$ over the set $X$ is <strong>symmetric</strong> if and only if this statement is true: "For all $a$ and $b$ in $X$, $a$ is related to $b$ $\Leftrightarrow$ $b$ is related to $a$."</p>
<p>A relation $R$ over the set $X$ is <str... |
4,415,254 | <p>When is <span class="math-container">$n!>x^n$</span>? assuming that x is a fixed positive. I know we can take the log of both sides and use the following formula:</p>
<p><span class="math-container">$$n\log(x) = \log(x^n) < \log(n!) = \sum_{i = 1}^n\log(i)$$</span>.
But this still gives me difficulty trying to... | Paolo Leonetti | 45,736 | <p>Use that (see <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow noreferrer">here</a>) for <span class="math-container">$n\ge 1$</span>:
<span class="math-container">$$
n!>\sqrt{2\pi n}\left(\frac{n}{e}\right)^ne^{\frac{1}{12n
+1}}.
$$</span>
Hence it is enough to check that
<span cl... |
3,631,714 | <p>If <span class="math-container">$f: \mathbb{R}^n \to \mathbb{R} $</span> is a function on the Schwartz space <span class="math-container">$\mathcal{S} (\mathbb{R}^n) $</span> we define the Fourier transform of <span class="math-container">$f$</span> as the function
<span class="math-container">$$ \hat f(\xi)= \int_{... | Community | -1 | <p>I think that there is a misunderstanding of how math is taught in our country, it's not as intensive as you think. JEE is an exam which is taken by million students but only <span class="math-container">$11,000$</span> get selected. So if you require books of <strong>JEE Advanced</strong> standard then better go for... |
1,515,817 | <p>I conjecture that in a consecutive sequence of $n$ natural numbers all greater than $n$, there exists at least one number which is not divisible by any prime number less than or equal to $n/2$.</p>
<p>Can any one prove or disprove this?</p>
| Eric Naslund | 6,075 | <p>The conjecture is false. This problem is directly related to finding large gaps between primes, and the methods of Erdos, Rankin and others.</p>
<p>Define the <em>Jacobsthal function</em> $j(q)$ to be the largest gap between consecutive reduced residues modulo $q$, that is the largest gap between elements that are ... |
2,229,532 | <p>I have a circle which has a triangle inscribed in it.</p>
<p>The circle radius R = 4</p>
<p>The triangle ABC vertices divide circle into 3 arcs in 1:2:3 ratio</p>
<p>Find the perimeter and area of triangle.</p>
<p>Can you guys help me with this one? </p>
| ThoughtOfGod | 418,784 | <p>One can observe that the arc lengths can be shown as parts of 6, for 1+2+3 is equal to 6. Once this has been observed, one can see the triangle has a side that of which is a diameter, and another as a radius.</p>
<p>It may be easier to think of it as parts of a hexagon divided into 6 equal parts.</p>
<p>Because an... |
1,401,661 | <blockquote>
<p>Dice are cubes with pips (small dots) on their sides, representing
numbers 1 through 6. Two dice are considered the same if they can be
rotated and placed in such a way that they present matching numbers on
the top, bottom, left, right, front, and back sides.</p>
<p>Below is an example of t... | André Nicolas | 6,312 | <p>Let us sit down at a table. Whatever die we have, it can be put on the table with the $1$ face down. </p>
<p>There are then $5$ possibilities for the up face (only one is legal, the face opposite $1$ is always $6$, but we will not worry about that). We make a choice for the up face, like $3$, count the dice that ha... |
553,103 | <p>consider the series of function $\sum f_n$ with $f_n(x)=\frac{x}{x^2+n^2}$.
It is easy to see that there is pointwise convergence on $\mathbb{R}$ (to a function that we'll call $f$) but not normal convergence. I want to know whether there is uniform convergence towards $f$ or not. </p>
<p>I tried to disprove unifo... | Khosrotash | 104,171 | <p><img src="https://i.stack.imgur.com/8jDVf.jpg" alt="enter image description here">
if you fcator 4cos^2x from 4cos^2x+sin^2x
you will have an arctan form integral
at the end
note that period of the function is pi
so your problem reduce to 4*integral(0 -->pi/2) f(x)</p>
|
510,672 | <p>My prof taught us that during Gaussian Elimination, we can perform three elementary operations to transform the matrix:</p>
<p>1) Multiple both sides of a row by a non-zero constant
2) Add or subtract rows
3) Interchanging rows</p>
<p>In addition to those, why isn't removing zero rows an elementary operation? It d... | Community | -1 | <p>Aside: which row operations are called elementary is purely a matter of convention: we can pick any operations we want to be called the elementary ones. Those three are the ones we chosen.</p>
<hr>
<p>The problem is that row operations are formal things. e.g. there is a row operation called "multiply the first row... |
3,131,791 | <p>I read this:</p>
<blockquote>
<p><strong>Definition 1.1</strong>. <em>The</em> complex projective line <span class="math-container">$\mathbb{CP}^1$</span> <em>(or just</em> <span class="math-container">$\mathbb{P}^1$</span><em>) is the set of all ordered pairs of complex numbers</em> <span class="math-container">... | Matt Samuel | 187,867 | <p>What about <span class="math-container">$F(x) =x+1$</span>? Is it mystifying that <span class="math-container">$F(-1)=0$</span> but <span class="math-container">$F(2\cdot - 1)\neq 0$</span>? The property homogenous polynomials have is that scaling the coordinates scales all coefficients of the polynomial uniformly, ... |
2,501,450 | <p>I was trying for a while to prove non-existence of theu following limit:</p>
<p>$$\lim_{n\to\infty}(-1)^n\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}$$</p>
<p>Unfortunately, with no results.</p>
<p>My hope was to show, that:</p>
<p>$$\lim_{n\to\infty}\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}\not=0$$</p>
<p>But showing that wa... | avz2611 | 142,634 | <p>let $\sqrt[n]{5}=x$
$$\lim_{n\to\infty}\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}=\lim_{n\to\infty}\frac{2^n+4n+6}{2^n(x-1)}=\lim_{n\to\infty}\frac{(2^n+4n+6)\cdot(1+x+x^2+\cdots+x^{n-1})}{2^n(x^n-1)}$$
now substitute $x$ and get your answer to be non-zero</p>
|
594,482 | <p>If a,b and c are elements of a ring, does the equation ax+b=c always have a solution x? If it does, must the solution be unique? </p>
| Dietrich Burde | 83,966 | <p>About the existence of a solution: for the ring $\mathbb{Z}$ there exists a solution to $ax=b$ if and only if $a\mid b$. For the ring $M_n(K)$ the matrix equation $AX=B$ is a special case of the Sylverster equation, i.e., $AX+XC=B$. See the discussion here: <a href="https://math.stackexchange.com/questions/179413/ex... |
1,216,576 | <p><img src="https://i.stack.imgur.com/p4xbH.png" alt="enter image description here"></p>
<blockquote>
<p>$ABC$ and $BDE$ are two equilateral triangles such that $D$ is the midpoint of
$BC$. If $AE$ intersects $BC$ at $F$, show that $Area(\triangle BFE)=2Area(\triangle FED)$</p>
</blockquote>
<p><strong>Solution... | Tae Hyung Kim | 94,401 | <p>This is the well known Vandermonde's identity. There's a rather nice proof of a generalization of it. Consider the generating functions $(1 + x)^n\cdot(1 + x)^m$ and $(1 + x)^{n + m}$. Then obviously the $r$th degree term must be the same for both generating functions. Expand the first one to get
$$ \left(1 + \bino... |
118,985 | <p>Here's a question I came across:</p>
<blockquote>
<p>count the number of permutations of 10 men, 10 women and one child,
with the limitation that a man will not sit next to another man, and a
woman will not sit next to another woman</p>
</blockquote>
<p>What I tried was using the inclusion-exclusion principl... | Community | -1 | <p>Note that $\frac{x^2}{e^{x}} < \frac1{x^2}$ for $x>9$. Hence, split the integral from $2$ to $9$ and then from $9$ to $\infty$ and argue why both are finite.</p>
|
2,797,039 | <p><strong>1)</strong> Let $\{f_n\}$ be a sequence of nonnegative measurable functions of $\mathbb R$ that converges pointwise on $\mathbb R$ to $f$ integrable. Show that</p>
<p>$$\int_{\mathbb R} f = \lim_{n\to \infty}\int_{\mathbb R}f_n \Rightarrow \int_{E} f = \lim_{n\to \infty}\int_{E}f_n $$</p>
<p>for any measu... | fleablood | 280,126 | <p>Interesting question.</p>
<p>Assume the die roll is $m$. There are ${m \choose 2}$ positions that the two jacks can occupy. And three are $4*3$ possible jacks they can be. (I'm distinguishing order; I suspect the math will be easier-- I may be wrong, but so long as I am consistent it doesn't matter). There $\fr... |
2,804,086 | <p>A distribution is an element of the continuous dual space of some function space. Let us take the Schwartz space $\mathcal{S} := \mathcal{S}(\mathbb{R}^n)$ just as an example. A distribution $\phi \in \mathcal{S}'$ is then a map
$$ \phi: \mathcal{S} \rightarrow \mathbb{C}.$$</p>
<p>My question is this: how do I int... | C. Dubussy | 310,801 | <p>When you have a $C^{\infty}$ map $f : \mathbb{R}^n \to \mathbb{R}^m$ which is proper (it is automaticaly the case with a $C^{\infty}$ bijection like translations), you can define the pushforward of a distribution by $f$. If $u$ is a distribution on $\mathbb{R}^n$ then, $f_{!}u$ is a distribution on $\mathbb{R}^m$ d... |
2,804,086 | <p>A distribution is an element of the continuous dual space of some function space. Let us take the Schwartz space $\mathcal{S} := \mathcal{S}(\mathbb{R}^n)$ just as an example. A distribution $\phi \in \mathcal{S}'$ is then a map
$$ \phi: \mathcal{S} \rightarrow \mathbb{C}.$$</p>
<p>My question is this: how do I int... | goblin GONE | 42,339 | <p>The trouble is that given a map $f : X \rightarrow Y$, the notation $\phi(f(x))$ should really mean "$\phi$ pulled back across $f$." But of course, distributions don't pull back in a natural way. Therefore I think the notation $\phi(f(x))$ only makes sense if $f$ is an invertible function, in which case it really me... |
2,365,120 | <p>I'm doing some revision on indices and surds. How do you simplify </p>
<blockquote>
<p>$(xy^2)^p\sqrt{x^q} $</p>
</blockquote>
<p>Bit confused because my textbook says the answer is </p>
<blockquote>
<p>$x^{p+q/2}y^{2p}$</p>
</blockquote>
<p>I can understand simplifying but only when it's the same base. I'm ... | Siong Thye Goh | 306,553 | <p>Hint:</p>
<p>Great you can handle when things are in the same base.</p>
<p>Simplify $$x^p \sqrt{x^q}$$</p>
<p>and simplify $$(y^2)^p$$</p>
<p>Remark:
Note that </p>
<p>$$(1/2)q = q/2$$</p>
<p>but $$(1/2)q \neq \frac{1}{2q}$$</p>
|
903,831 | <p>One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in parallel to ring of integers, I think converting a polynomial ring to a numerical (i.e. integer) ring is useful.</p>
... | rschwieb | 29,335 | <p>The requirements are a little vague, but it sounds like you'd like to find a function from a polynomial ring $R[X]$ into the set $\Bbb Z$, and then equip $\Bbb Z$ with a potentially unusual addition and multiplication to make the map a ring isomorphism, thereby "representing the ring $R[X]$ with integers." </p>
<p>... |
2,661,718 | <p>My basic understanding is that each time
I have 30% chance of winning the prize
so between 3 and 4 tries I should win it</p>
<p>cause .3 +.3 +.3 = 90% as I need to win only once with 3 try
.3+.3+.3+.3 = 120% with 4 try</p>
<p>I do remember a formula saying 1-(0.7)^3 = 65.7% chance but I dont remember what that ... | egreg | 62,967 | <p>It can be easier if you set $1+x=t$, so the inequality becomes
$$
(t-1)^2\ge t(\log t)^2
$$
If we consider the function
$$
f(t)=(t-1)^2-t(\log t)^2
$$
defined for $t>0$, we have
$$
\lim_{t\to0}f(t)=1,\qquad \lim_{t\to\infty}f(t)=\infty
$$
Moreover
$$
f'(t)=2(t-1)-(\log t)^2-2\log t
$$
and
$$
f''(t)=2-2\frac{\log ... |
294,383 | <p>Evaluate :
$$\int_{0}^{+\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x}$$</p>
| Felix Marin | 85,343 | <p>\begin{align}
?
&\equiv
{1 \over 4}\int_{-\infty}^{\infty}
{x - \sinh\left(x\right) \over x^{2}\sinh\left(x\right)}\,{\rm d}x
=
{1 \over 4}\sum_{n = 1}^{\infty}2\pi{\rm i}
\lim_{x \to {\rm i}\,n\,\pi}
{\left\lbrack x - \sinh\left(x\right)\right\rbrack\left(x - {\rm i}n\pi\right)
\over
x^{2}\sinh\left(x\right)}... |
686,167 | <p>I'm working on an assignment where part of it is showing that $S_k=0$ for even $k$ and $S_k=1$ for odd $k$, where</p>
<blockquote>
<p>$$S_k:=\sum_{j=0}^{n}\cos(k\pi x_j)= \frac{1}{2}\sum_{j=0}^{n}(e^{ik\pi x_{j}}+e^{-ik\pi x_{j}}) $$</p>
<p>Here $x_j=j/(n+1)$.</p>
</blockquote>
<p>So, working through the al... | Gabriel Romon | 66,096 | <p>Reminders first:</p>
<p>$$\forall w,z \in \mathbb C, \overline{\left(\frac{w}{z}\right)}=\frac{\overline w}{\overline z}$$
$$\forall w,z \in \mathbb C, \overline{w+z}=\overline w +\overline z$$
$$\forall a\in \mathbb R, \overline{e^{ia}}=e^{-ia},$$
$$\forall z \in \mathbb C, z+ \overline z=2 \Re( z)$$
Apply these r... |
2,148,979 | <p>I want to find $A$ such that $$A\sum \limits_{n=m}^{\infty}1/n^{3}=1$$</p>
<p>for any natural value of $m$.</p>
| Simply Beautiful Art | 272,831 | <p>In terms of the <a href="https://en.wikipedia.org/wiki/Hurwitz_zeta_function" rel="nofollow noreferrer">Hurwitz zeta function</a>:</p>
<p>$$A=\frac1{\zeta(3,m)}$$</p>
<p>There is no further closed form, unless you allow</p>
<p>$$A=\frac1{\zeta(3)-\sum_{n=1}^{m-1}\frac1{n^3}}$$</p>
<p>Approximations may be done w... |
2,752,421 | <p>a) Consider the equation $x_1 + x_2 + x_3 + x_4 = 35$. How many different solutions does this equation have
if all the variables must be positive integers? Enter the exact numeric answer.</p>
<p>b) Suppose that a license plate consists of three letters followed by three digits. How many different license plates st... | Brian Tung | 224,454 | <p>(a) Almost. The restriction is that all variables must be <em>positive</em> integers, so we can rewrite this as</p>
<p>$$
y_1+y_2+y_3+y_4 = 31
$$</p>
<p>with $y_i = x_i-1$ being a non-negative integer for all $i$. Then stars-and-bars gives us $C(34, 31) = C(34, 3)$.</p>
<p>(b) Almost. As Tony Jacobs points out... |
3,592,368 | <p>I managed to solve this question using trigonometry. But I wondered if there'd be anyway of doing it using only synthetic geometry. Here it is.</p>
<blockquote>
<p>Let <span class="math-container">$ABC$</span> be a right isosceles triangle of hypotenuse <span class="math-container">$AB$</span>. Let also <span class=... | Blue | 409 | <p><a href="https://i.stack.imgur.com/Tzett.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Tzett.png" alt="enter image description here"></a></p>
<p><span class="math-container">$$|\triangle PBC| = \frac12|CP||BQ| = \frac12k^2$$</span></p>
|
1,369,428 | <p>I am trying to solve the following equation;
$$\int_{-1}^{1}e^{i(x+a\cos x)} \, \mathrm{d}(\cos x)$$ or $$\int_{0}^{\pi}e^{i(x+a\cos x)} \sin x \, \mathrm{d}x$$</p>
<p>I tried this in <em>Wolfram Alpha</em>, but it says that answer cannot be obtained.</p>
| Josh Broadhurst | 249,961 | <p>If you leave out the (what I assume is a) constant $a$ in your original problem and type it into WolframAlpha, it tells you that the solution involves the Bessel function of the first kind ($J_1$) and gives the following answer
$$ (\pi J_1(1)-2 \cos(1)+2 \sin(1))i \approx 1.9848i$$ </p>
<p>If you set $a=2$ as the i... |
2,073,794 | <p>Let's consider the following variant of Collatz (3n+1) : </p>
<p>if $n$ is odd then $n \to 3n+1$</p>
<p>if $n$ is even then you can choose : $n \to n/2$ or $n \to 3n+1$</p>
<p>With this definition, is it possible to construct a cycle other than the trivial one, i.e., $1\to 4 \to 2 \to 1$?</p>
<p>Best regards</p>... | florence | 343,842 | <p>$$7\to 22$$
$$22\to11$$
$$11\to34$$
$$34\to17$$
$$17\to52$$
$$52\to26\to13$$
$$13\to40$$
$$40\to20\to10\to5$$
$$5\to16$$
$$16\to8\to4\to2$$
$$2\to 3\cdot2+1=7$$</p>
|
2,073,794 | <p>Let's consider the following variant of Collatz (3n+1) : </p>
<p>if $n$ is odd then $n \to 3n+1$</p>
<p>if $n$ is even then you can choose : $n \to n/2$ or $n \to 3n+1$</p>
<p>With this definition, is it possible to construct a cycle other than the trivial one, i.e., $1\to 4 \to 2 \to 1$?</p>
<p>Best regards</p>... | meriton | 71,993 | <p>There are quite a few such cycles. Here's the list of all cycles of length $\le 30$ starting from any number $< 10^5$ that never exceed $2^{63}$:</p>
<pre><code>[2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4]
[2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 13, 40, 20, 10, 5, 16, 8, 4]
[2,... |
78,461 | <p>I'm trying to solve the following problem:</p>
<p>For a vector $v$ of length $c$, $\min \frac{\sum v[i]^4}{(\sum v[i]^2)^2}$ subject to $\sum v[i] = N$.</p>
<p>I can solve this numerically for a given $c$ using the following command: </p>
<pre><code>Minimize[{Total[z^4]/Total[z^2]^2, Total[z] == N}, z]
</code></... | 2012rcampion | 21,750 | <p>You're like 99% of the way there. You just need to tell Mathematica that <code>z</code> is a vector by feeding it the components:</p>
<pre><code>With[{z = Array[v, 3]},
Minimize[{Total[z^4]/Total[z^2]^2, Total[z] == n}, z]]
</code></pre>
<p>For dimensions <code>1</code> and <code>2</code> the answers are as exp... |
1,671,933 | <p>In the question <a href="https://math.stackexchange.com/questions/1502484/the-application-of-nimbers-to-nim-strategy">on nimbers</a>, the original poster asks for the meaning of Nimber <a href="https://en.wikipedia.org/wiki/Nimber#Multiplication" rel="nofollow noreferrer">multiplication</a> in the context of imparti... | E. Z. L. | 811,580 | <p><em>Winning Ways</em> Volume 3, Chapter 14, considers coin-turning games. One such game is <em>Turning Corners</em>, where a move is to turn over the four corners of a rectangle with horizontal and vertical sides, but the North-East most coin needs to be turned from heads to tails. A sample move shows that it define... |
3,916,261 | <p>In the book Optimal Transport for Applied Mathematicians, Theorem 1.37, the authors states an inequality, but I'm having trouble seeing how one shows that the inequality is indeed true.</p>
<p><strong>Definition</strong>: Given a function <span class="math-container">$c:X\times Y \to \mathbb R$</span>, we say that <... | Sidharth Ghoshal | 58,294 | <p>It is NOT wrong to say <span class="math-container">$\sin(x): \mathbb{R} \rightarrow \mathbb{R}$</span> since [-1,1] is in <span class="math-container">$\mathbb{R}$</span> but perhaps the you could say "it is not honest" after we prove that <span class="math-container">$\sin(x)$</span> only takes on value... |
966,267 | <p>Original PDE
$$T_t=\alpha T_{xx}$$
I need to solve this equation numerically and analytically and compared them. I've already done the numerical part. But I need to solve it analytically now. </p>
<p>Given the initial condition
$$T(x,0)=sin(\frac{\pi x}{L})$$
where $$L=1$$</p>
<p>I would like to find the exact sol... | Alex Zorn | 73,104 | <p>Two comments.</p>
<p>First, you're integral computation looks something like this:</p>
<p>$$2\int_0^1\sin(\pi x)\sin(n\pi x)\, dx = \int_0^1 \cos((n - 1)\pi x) - \cos((n+1)\pi x)\, dx$$</p>
<p>Now, it's tempting to write:</p>
<p>$$\int \cos((n-1)\pi x)\, dx = \frac{\sin((n-1)\pi x)}{\pi(n-1)} + C$$</p>
<p>But o... |
2,668,779 | <p>The exercise consists of showing that the function $f(x,y)=x^4 + y^4$ has a global minimum and maximum under the constraint $x^4 + y^4 - 2xy = 2$.</p>
<p>In the solution to the exercise, it it follows that the constraint is compact if we can show that $\lim_{x^2 + y^2 \rightarrow \infty} x^4 + y^4 - 3xy - 2 \righta... | StackTD | 159,845 | <blockquote>
<p>How else can a set of vectors that matches this condition not be a vector space?</p>
</blockquote>
<p>Recall that there is another condition: besides being non-empty (or containing the zero vector) and being closed under scalar multiplication, it also needs to be closed under <em>addition</em>.</p>
... |
828,668 | <p>Let $G$ be a finite group. If there exists an $a\in G$ not equal to the identity such that for all $x\in G$,$\phi(x) = axa^{-1}=x^{p+1} $ is an automorphism of $G$ then $G$ is a $p$-group.</p>
<p>This is what I have so for.</p>
<p>The order of $a$ is $p$ since $\phi(a) = a= a^pa\rightarrow a^p=e$ therefore the $o... | Jack Schmidt | 583 | <p>Here is a different proof from Dan Shved's:</p>
<p><strong>Lemma:</strong> In any such group $G$, $a$ normalizes every subgroup, and $a$ lies in every Sylow $p$-subgroup.</p>
<blockquote class="spoiler">
<p> <em>Proof:</em> Let $H$ be a subgroup. Then for $x \in H$, $axa^{-1}=x^{p+1} \in H$, so $aHa^{-1} \leq H$... |
1,726,540 | <p>How should $x^{\frac{1}{x}}$ be differentiated? I know the answer is
$$\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$$</p>
<p>but I do not understand how to get there.</p>
<h2>Attempt at solution.</h2>
<p>I believe the following is true:</p>
<p>$$
\begin{aligned}\frac{d}{dx}x^u&=ux^{u-1}\cdot u^\prime\\
\frac{d}{dx}a^... | Plutoro | 108,709 | <p><strong>Hint:</strong>
Let $y=x^{1/x}$. Now take the natural log of both sides.
$$\ln(y)=\ln(x^{1/x})=\frac{1}{x}\ln(x).$$ Now you can differentiate both sides and solve to find $y'$. I'll even do the left side for you:
$$\frac{d}{dx}\ln(y)=\frac{y'}{y}=\frac{y'}{x^{1/x}}.$$</p>
|
3,998,484 | <p>For the purpose of clarity, I do not want anybody to prove this statement, I am just looking to get some help translating it into the contrapositive using <em>symbolic logic</em>. Right now I have the following translation:</p>
<hr />
<p>Original Statement:</p>
<p><span class="math-container">$$(\exists k \in \mathb... | jlammy | 304,635 | <p>Your answer for the kernel is wrong -- <span class="math-container">$(a,9)$</span> and <span class="math-container">$(a,18)$</span> are also in the kernel, so <span class="math-container">$$\ker\alpha=\mathbb Z_9\times\{0,9,18\}\cong\mathbb Z_9\times\mathbb Z_3.$$</span>
So by the isomorphism theorem, <span class="m... |
2,636,679 | <p>For which values of x do the vectors $(x,0,x + 1),(1,x,1),(0,x,−x)$ form a basis for R3?</p>
<p>Is 1 a correct answer to this. The vectors would each be linearly independent, however this seems too simple and I'm guessing theres an error somewhere.</p>
<p>If 1 works then it would seem any real number would work si... | TryAgain | 520,267 | <p>How did you find this value?</p>
<p>You can also "glue" the vectors together to form a matrix (dimension 3x3), and try to find its determinant. You will obtain a polynom in $x$ and the zeros of this polynom are the values for which the three vectors are linearly dependant.</p>
|
2,636,679 | <p>For which values of x do the vectors $(x,0,x + 1),(1,x,1),(0,x,−x)$ form a basis for R3?</p>
<p>Is 1 a correct answer to this. The vectors would each be linearly independent, however this seems too simple and I'm guessing theres an error somewhere.</p>
<p>If 1 works then it would seem any real number would work si... | mvw | 86,776 | <p>A basis of $\mathbb{R}^3$ must have three linear independent vectors, so the equation
$$
\lambda_1 (x,0,x+1)^t + \lambda_2 (1,x,1)^t + \lambda_3 (0,x,−x)^t = (0,0,0)^t
$$
is allowed to have only the solution $(\lambda_1, \lambda_2, \lambda_3) = (0,0,0)$. In matrix form
$$
A \lambda = 0 \iff \\
\begin{pmatrix}
x &... |
3,025,375 | <p>What is<span class="math-container">$$\lim_{n→∞}n^3(\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt2)?$$</span>So it is<span class="math-container">$$\lim_{n→∞}\frac{n^3(\sqrt{n^2+\sqrt{n^4+1}})^2-(n\sqrt{2})^2}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}=\lim_{n→∞}\frac{n^3(n^2+\sqrt{n^4+1}-2n^2)}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}.$$</span... | Community | -1 | <p><strong>The expedite way:</strong></p>
<p><span class="math-container">$$\sqrt{1+\sqrt{1+n^{-4}}}=\sqrt{1+1+\dfrac12n^{-4}+o(n^{-4})}=\sqrt2\sqrt{1+\dfrac14n^{-4}+o(n^{-4})}=\sqrt2\left(1+\dfrac18n^{-4}+o(n^{-4})\right)$$</span></p>
<p>and the limit is</p>
<p><span class="math-container">$$\frac{\sqrt2}8.$$</span... |
3,699,640 | <p>How to show <span class="math-container">$x^2=20$</span> has no solution in <span class="math-container">$2$</span>-adic ring of integers <span class="math-container">$\mathbb{Z}_2$</span> ?</p>
<p>What is the general criterion fora solution of <span class="math-container">$x^2=a$</span> in <span class="math-contai... | Community | -1 | <p>It seems like your attempt probably works.</p>
<p>A slightly more general version of Result 2 that could help you is that squares in the <span class="math-container">$2$</span>-adic integers are of the form <span class="math-container">$2^{2k}u$</span> where <span class="math-container">$u\in \mathbb Z_2^*$</span> ... |
3,699,640 | <p>How to show <span class="math-container">$x^2=20$</span> has no solution in <span class="math-container">$2$</span>-adic ring of integers <span class="math-container">$\mathbb{Z}_2$</span> ?</p>
<p>What is the general criterion fora solution of <span class="math-container">$x^2=a$</span> in <span class="math-contai... | Mathmo123 | 154,802 | <p>The other answers do a good job of answering your more general question, but I want to address the specific case of <span class="math-container">$x^2 = 20$</span> in the simplest way possible.</p>
<p>Remember how Hensel's lemma works: we perform an iterative procedure to find a solution to <span class="math-contain... |
1,234,320 | <p>It is given that $4 x^4 + 9 y^4 = 64$.</p>
<p>Then what will be the maximum value of $x^2 + y^2$?</p>
<p>I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .</p>
| Jean-Claude Arbaut | 43,608 | <p>Without using Lagrange multipliers</p>
<p>It's equivalent to find $x$ that maximizes</p>
<p>$$f(x)=x^2+\frac23\sqrt{16-x^4}$$</p>
<p>You have</p>
<p>$$f'(x)=2x-\frac13\frac{4x^3}{\sqrt{16-x^4}}$$</p>
<p>Hence $f'(x)=0$ iff $x=0$ or</p>
<p>$$\frac{2x^2}{\sqrt{16-x^4}}=3$$
$$2x^2=3\sqrt{16-x^4}$$
$$4x^4=9(16-x^4... |
3,707,132 | <p>I believe my proof of this simple fact is fine, but after a few false starts, I was hoping that someone could look this over. In particular, I am interested in whether there is an alternate proof.</p>
<blockquote>
<p>For a real number <span class="math-container">$a$</span> and non-empty subset of reals <span cla... | zkutch | 775,801 | <p><span class="math-container">$$\sup\left\lbrace x + y\right\rbrace = \sup \left\lbrace x \right\rbrace + \sup \left\lbrace x \right\rbrace $$</span>
<span class="math-container">$$\inf\left\lbrace x + y\right\rbrace = \inf \left\lbrace x \right\rbrace + \inf \left\lbrace x \right\rbrace $$</span>
For non negative nu... |
760,216 | <p>The sewing pattern on a basketball is composed of two great circles and a single curve that intersects each great circle twice. Does this curve have a name? Are there any parametric descriptions of the curve?</p>
<p>The same question can apply to the sewing pattern on a tennis ball.</p>
<p>Answers about why this s... | Will Jagy | 10,400 | <p>I don't have a basketball here to compare. However, I can say that the seams on a baseball and on a tennis ball are similar to the intersection of <a href="http://www.indiana.edu/~minimal/maze/enneper.html" rel="nofollow noreferrer">Enneper's minimal surface</a> with a sphere centered at the, well, the origin of all... |
4,183,734 | <p>I just started trigonometry, and I came across this problem. I know that this problem would be very simple with a calculator, but without one, I'm lost. How would you determine if <span class="math-container">$\sin 4$</span> is bigger than <span class="math-container">$\sin 3$</span> in negativity or positivity? How... | fleablood | 280,126 | <p>For values close to <span class="math-container">$\pi$</span> (or close to <span class="math-container">$0$</span>) we have <span class="math-container">$|\sin (\pi \pm k)| = |\sin k|$</span>.</p>
<p>And if <span class="math-container">$|k|$</span> is closer to <span class="math-container">$0$</span> (but less than ... |
457,654 | <p>This question is a continuation of: <a href="https://math.stackexchange.com/questions/456118/finding-a-set-of-values-with-the-binomial-theorem">Finding a set of values with the binomial theorem</a></p>
<p>For $n \in \mathbb N$ and the function $p(x) = \left(x + \frac 1x\right)^{n}$.</p>
<p>By the binomial theorem... | Alex Youcis | 16,497 | <p>Hint: Euler's theorem comes from the fact that $\#(\mathbb{F}_p^\times)=p-1$. The minimality coming from the fact that $\mathbb{F}_p^\times$ is cyclic--as is the unit group of any finite field. Now, assume that $m$ is irreducible. Then, $k=\mathbb{F}[x]/(m(x))$ is a field. Moreover, it's easy to see that $\dim_{\mat... |
39,184 | <p>Hi everybody,</p>
<p>Reading about Geometry Processing, I have realized that people in this area are very interested in regular vertices(degree=6) rather than irregular ones.</p>
<p>Can anybody give me reasons why this is an important property?
Suppose that we can have a mesh with all regular vertices (genus 1) bu... | Darsh Ranjan | 302 | <p>Some geometric algorithms work better around regular vertices. For example, Loop subdivision is a procedure to turn a triangular mesh into a smooth surface (i. e., a "subdivision surface"). The amount of smoothness varies, however: IIRC, the normal vector to the surface is continuous everywhere, but the curvature is... |
2,572,428 | <p>$$z^2=3+4i ,(z=x+iy)$$
Seems easy? That's what I thought!</p>
<p>I get a system of equations that I can't solve:
$x^2-y^2=3$ and $2xiy=4i$ which I then get $x=\frac{2}{y}$</p>
<p>I can't solve the system of equations?</p>
| Alex Zorn | 62,875 | <p>Taking norms, you have: $|z|^2 = |3 + 4i|$, so $x^2 + y^2 = 5$. Combining this with $x^2 - y^2 = 3$ yields $2x^2 = 8$, hence $x = \pm 2$. </p>
|
2,738,023 | <p>We know we can check if two vectors are 'orthogonal' by doing an inner product.</p>
<p>$a*b=0$</p>
<p>tells us that these two vectors are orthogonal</p>
<p>here comes the question:</p>
<p>if there a way to compute if they are 'parallel'? i.e., they are pointing at the same direction. </p>
| user | 505,767 | <p>Note that two vectors $\vec v_1,\vec v_2\neq \vec 0$ are parallel $$\iff \vec v_1=k\cdot \vec v_2$$</p>
<p>for some $k\in \mathbb{R}$ and this condition is easy to check component by component.</p>
<p>For vectors in $\mathbb{R^2}$ or $\mathbb{R^3}$ we could check the condition by cross product.</p>
<p>More in gen... |
1,004,837 | <p>The Warsaw circle is defined as a subset of $\mathbb{R}^2$：
$$\left\{\left(x,\sin\frac{1}{x}\right): x\in\left(0,\frac{1}{2\pi}\right]\right\}\cup\left\{(0,y):-1\leq y\leq1\right\}\cup C\;,$$
where $C$ is the image of a curve connecting the other two pieces. </p>
<p>A map from Warsaw circle to a single point space ... | Jason DeVito | 331 | <p>Let $W$ denote the Warsaw circle. By collapsing the interval piece down to a point, the quotient space is homeomorophic to $S^1$. This gives a map $f:W\rightarrow W/{\sim} \cong S^1$.</p>
<p>I claim this map is not null homotopic. Believing this for a second, note that for any contractible space $X$ any continuo... |
1,004,837 | <p>The Warsaw circle is defined as a subset of $\mathbb{R}^2$：
$$\left\{\left(x,\sin\frac{1}{x}\right): x\in\left(0,\frac{1}{2\pi}\right]\right\}\cup\left\{(0,y):-1\leq y\leq1\right\}\cup C\;,$$
where $C$ is the image of a curve connecting the other two pieces. </p>
<p>A map from Warsaw circle to a single point space ... | Matteo Doni | 557,505 | <p>Let <span class="math-container">$W$</span> denote the Warsaw circle and <span class="math-container">$C$</span> is the space as in the question. By collapsing <span class="math-container">$C$</span> down to a point we find the map <span class="math-container">$f:W\rightarrow W/C \cong S^1$</span>.</p>
<p><span cla... |
3,287,067 | <p>I'm reading a solution to the following exercise:</p>
<p>"Assume that <span class="math-container">$\lim_{x\to c}f\left(x\right)=L$</span>, where <span class="math-container">$L\ne0$</span>, and assume <span class="math-container">$\lim_{x\to c}g\left(x\right)=0.$</span> Show that <span class="math-container">$\li... | J. W. Tanner | 615,567 | <p><span class="math-container">$\cos$</span> is an even function, but <span class="math-container">$$\int_{-\pi}^{\pi} \cos x dx =\sin x |_{-\pi}^{\pi}=0$$</span></p>
|
3,495,779 | <p>The question is:
Prove that if graph <span class="math-container">$G$</span> is a <strong>Connected Planar Graph</strong> where each region is made up of at least <span class="math-container">$k\,(k\ge3)$</span> edges, then it must satisfy:</p>
<p><span class="math-container">$e\ge\frac{k(v-2)}{k-2}$</span></p>
<p... | Acccumulation | 476,070 | <p>Generally, when you're using induction to prove a fact about relationship between a dependent variable and an independent variable, the induction is on the independent variable. Another generalization is that if you're proving an inequality, the induction is likely on the "less than" side. The expression <span class... |
2,376,315 | <p>So I'm trying to solve the problem irrational ^ irrational = rational. Here is my proof
Let $i_{1},i_{2}$ be two irrational numbers and r be a rational number such that $$i_{1}^{i_{2}} = r$$
So we can rewrite this as $$i_{1}^{i_{2}} = \frac{p}{q}$$
Then by applying ln() to both sides we get $$i_2\ln(i_1)... | Reese Johnston | 351,805 | <p>When writing a proof - any proof - the first and most important step is to decide <em>what you are proving</em>. If the thing you end up with is not exactly the thing you were proving, you're not done with the proof.</p>
<p>In this case, it's not clear to me what you're trying to prove - are you trying to prove tha... |
310,651 | <p>$\mathbb{Z}_{30}$* $= \{1,7,11,13,17,19,23,29\}$</p>
<p>The number of elements is $8$ and $8$ is not prime, therefore $\mathbb{Z}_{30}$* is not cyclic.</p>
<p>and the generators are $7,11,13,17,19,23,29$.</p>
<p>can anyone correct me please?</p>
| Julien | 38,053 | <p>Note that $30=2\cdot 3\cdot 5$.</p>
<p>By the Chinese remainder theorem,</p>
<p>$$
\mathbb{Z}/30\mathbb{Z}^*=\mathbb{Z}/2\mathbb{Z}^*\times \mathbb{Z}/3\mathbb{Z}^*\times \mathbb{Z}/5\mathbb{Z}^*=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}.
$$</p>
<p>So you're right, this is not cyclic, but not for the re... |
2,919,096 | <p>I am trying to solve a system of two second-order ODEs. After separating them, I obtained a fourth-order independent ODE as illustrated below. I wonder if there is a specific technique to solve it.</p>
<p>$$y^{(4)}+\frac{a_1}{x} y^{(3)}+\frac{a_2}{x^2}y^{(2)}+a_3y^{(2)}+\frac{a_4}{x}y^{(1)}+a_5y=0$$</p>
| Ernie060 | 592,621 | <p>A power series method might do the trick. Substitute
$$ y = \sum_{k=0}^\infty c_k x^k$$
in the differential equation $$x^2 y^{(4)} + a_1 x y^{(3)} + \cdots + a_5 x^2 y = 0.$$
You will obtain a series in which the coefficients are difference equations in the unknowns $c_k$. For concrete constants $a_n$, this differen... |
2,919,096 | <p>I am trying to solve a system of two second-order ODEs. After separating them, I obtained a fourth-order independent ODE as illustrated below. I wonder if there is a specific technique to solve it.</p>
<p>$$y^{(4)}+\frac{a_1}{x} y^{(3)}+\frac{a_2}{x^2}y^{(2)}+a_3y^{(2)}+\frac{a_4}{x}y^{(1)}+a_5y=0$$</p>
| Przemo | 99,778 | <p>One technique for solving linear higher order ODEs is to seek solutions in the form <span class="math-container">$y(x) = y_1(x) \cdot y_2(x)$</span> where <span class="math-container">$y_{1,2}$</span> are solutions of some lower order ODE (input ODE), solutions that are known. In order to guarantee success of this m... |
2,652,675 | <p>Given that A = $\begin{bmatrix} 2 & 1 \\ -5 & -4 \end{bmatrix} $ and B = $\begin{bmatrix} 3 & -1 \\ -1 & 0 \end{bmatrix} $ </p>
<p>Find a 2 X 2 matrix C such that $CA= B$</p>
<p>I multiply both sides by $A^{-1}$ </p>
<p>Since $A^{-1}A = I $ </p>
<p>$ CI = BA^{-1}$ </p>
<p>Since $CI = IC = C$ </p... | confused_wallet | 390,931 | <p>Note that the definition for the topology generated by $\mathscr{B}$ is the set of <em>all</em> subsets $U$ of $X$ with the stated property. It follows that the topology generated is unique.</p>
|
11,435 | <p>I know what they are and how to define them. They have to serve some purpose - else why include them? But I never used them and all examples on this site and the docs never show a practical use of <code>UpValues</code>.</p>
<p>Can someone give me an example to see when they are actually needed?</p>
| rm -rf | 5 | <p>Yes, <code>UpValues</code> are certainly useful in that you can bind definitions and custom behaviours to the symbol rather than the operator. For instance, I can define (a simple, silly example):</p>
<pre><code>g /: g[x_] + g[y_] := g[x] g[y]
</code></pre>
<p>to actually multiply the two when I add them. This def... |
4,159,337 | <p>How would you show
<span class="math-container">$$\log_2 3 + \log_3 4 + \log_4 5 + \log_5 6 > 5?$$</span></p>
<p>After trying to represent the mentioned expression in the same base, a messy expression is created.
The hint mentioned that the proof can take help of quadratic equations.</p>
<p>Could you provide some... | Barry Cipra | 86,747 | <p>Converting to logs of a single base and using AGM, we have</p>
<p><span class="math-container">$$\begin{align}
\log_23+\log_34+\log_45+\log_56
&={\log3\over\log2}+{\log4\over\log3}+{\log5\over\log4}+{\log6\over\log5}\\
&\ge4\sqrt[4]{{\log3\over\log2}\cdot{\log4\over\log3}\cdot{\log5\over\log4}\cdot{\log6\ove... |
1,606,023 | <blockquote>
<p>Given that $U_{n+1} = U_n-U_n^2$ and $0 < U_0 < 1$, show that $0 < U_n \leq 1/4$ for all $n \in \mathbb{N}^*$.</p>
</blockquote>
<p>Given that $S_n$ is the sum of $\frac{1}{1-U_k}$ from k=0 to n</p>
<p>Calculate $S_n$ in function of n </p>
<p>I see that $\frac{1}{1-U_{n}}=\frac{1}{U_{n+1}}... | zu7 | 303,898 | <ol>
<li>Show that $0<U_{n+1}<U_n$ for all $n$</li>
<li>Show that $U_1<1/4$ for any choice of $0<U_0<1$</li>
</ol>
|
2,121,611 | <p>Compute a quadrature of</p>
<p>$\int_c^d\int_a^b f(x,y)dxdy$</p>
<p>using the Simpson rule and estimate the error.</p>
<p>So the Simpson rule says </p>
<p>$S(f) = (b-a)/6(f(a)+4f((a+b)/2) +f(b))$</p>
<p>So i get </p>
<p>$\int_c^d(b-a)/6(f(a)+4f((a+b)/2) +f(b))dy$</p>
<p>Is that even correct? How do I go on?<... | Alexander | 806,961 | <p>A C implementation for applying Simpson's Rule towards solving double integrals can be found here if you are interested.
<a href="https://github.com/adel314/Small-C-Plus--for-TMS9900" rel="nofollow noreferrer">Simpson integration technique for
evaluating double integrals</a></p>
<p>It can be also represented in the... |
764,437 | <p>I am trying to prove that $\mathbb{Z}[i]\cong \mathbb{Z}[x]/(x^2+1)$.</p>
<p>My initial plan was to use the first isomorphism theorem. I showed that there is a map $\phi: \mathbb{Z}[x] \rightarrow \mathbb{Z}[i]$, given by $\phi(f)=f(i).$ This map is onto and homorphic. The part I have a question on is showing th... | Ellya | 135,305 | <p>When quotient out an ideal, we consider what happens to the ring when all the elements in the ideal are considered as identity elements.</p>
<p>Now if $x^2+1=0\Rightarrow x=\pm i$ let us take the "+" root.</p>
<p>$\mathbb{Z}[x]/(x^2+1)=\{f\in\mathbb{Z}[x]\,|x^2+1=0\}=\{f\in\mathbb{Z}[x]\,|x=i\}=\{a+bi|a,b\in\mathb... |
1,191,740 | <p>$a_{n+1} - a_n = 3n^2 - n$ ;$a_0=3$</p>
<p>I need help for solving the particular solution.
Based on a chart in my textbook if you get $n^2$ the particular solution would be
$A_2n^2 + A_1n + A_0$ and $n$ has the particular solution of $A_1n+A_0$.<br>
So given $3n^2 - n$, my first thought was that if the equation w... | DeepSea | 101,504 | <p><strong>hint</strong>: $a_n = (a_n-a_{n-1})+(a_{n-1}-a_{n-2})+\cdots +(a_2-a_1)+(a_1-a_0)+a_0$</p>
|
7,064 | <p>The <a href="http://en.wikipedia.org/wiki/Long_division" rel="nofollow">Wikipedia article</a> on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation when I moved to the US. However, I did enjoy s... | Arturo Magidin | 742 | <p>I use the same one that is used in the U.S.; it is the one that is (or at least, was up until the 80s) taught in Mexico. I had some classmates in high school who had emigrated form Argentina: they used the one described in the Wikipedia article as "European", not the one described as "Latin America". (I'll also note... |
2,421,421 | <blockquote>
<p>Evaluate
$$ \lim_{x\to \pi/2} \frac{\sqrt{1+\cos(2x)}}{\sqrt{\pi}-\sqrt{2x}}$$ </p>
</blockquote>
<p>I tried to solve this by L'Hospital's rule..but that doesn't give a solution..appreciate if you can give a clue.</p>
| farruhota | 425,072 | <p>Alternatively:
$$\lim_{x\to \pi/2} \frac{\sqrt{1+\cos(2x)}}{\sqrt{\pi}-\sqrt{2x}}=\lim_{x\to \pi/2} \frac{\sqrt{1+\cos(2x)}}{\sqrt{\pi}-\sqrt{2x}}\cdot \frac{\sqrt{1-\cos(2x)}}{\sqrt{1-\cos(2x)}}\cdot \frac{\sqrt{\pi}+\sqrt{2x}}{\sqrt{\pi}+\sqrt{2x}}=$$
$$\lim_{x\to \pi/2} \frac{|\sin{(2x)|}\cdot(\sqrt{\pi}+\sqrt{2x... |
954,281 | <p>Given a complex measure: $\mu:\Sigma\to\mathbb{C}$.</p>
<p><img src="https://i.stack.imgur.com/YS26y.jpg" alt="Variation Measure"></p>
<p>Consider its decomposition into positive measures:
$$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$</p>
<blockquote>
<p>Does it split i... | C-star-W-star | 79,762 | <p>Consider the Radon-Nikodym derivatives:
$$\mu_\alpha(E)=\int_Eu_\alpha\mathrm{d}|\mu|$$
They give rise to the constraints:
$$u_\alpha\geq0:\quad|u|=1\implies u_\alpha=\chi_{A_\alpha}$$</p>
<blockquote>
<p>So the above can only hold for: $\mathrm{im}u\subseteq\{1,i,-1,-i\}$</p>
</blockquote>
<p>As an example when... |
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