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 |
|---|---|---|---|---|
4,245,475 | <p>Evaluate <span class="math-container">$ \, \displaystyle \int _{0}^{4} \frac{1}{(2x+8)\, \sqrt{x(x+8)}}\, dx. $</span>
<br /><br /><span class="math-container">$My\ work:-$</span><br />
by completing the square and substitution i.e. <span class="math-container">$\displaystyle \left(\begin{array}{rl}x+4 & = 4\sec... | egreg | 62,967 | <p>Let <span class="math-container">$x\in V$</span>, <span class="math-container">$x\ne0$</span>. Then <span class="math-container">$\{x\}$</span> is a linearly independent set, so there exists a basis <span class="math-container">$\mathscr{B}$</span> of <span class="math-container">$V$</span> such that <span class="ma... |
1,817,542 | <p><strong>Problem:</strong> Let $(X, Y)$ be uniformly distributed on the unit disk $\{ (x,y) : x^2 + y^2 \le 1\}$. Let $R = \sqrt{X^2 + Y^2}$. Find the CDF and PDF of $R$.</p>
<p><strong>Attempted Solution:</strong> First note that $r \in R = \sqrt{X^2 + Y^2}$ represents a point on $\mathbb{R}^2$ with radius $r$ abou... | Community | -1 | <p>As the random variable is uniformly distributed, the probability of $R$ not exceeding a given $r$ is proportional to the enclosed area.</p>
<p>$$P(R\le r)\propto r^2.$$</p>
<p>As the probability is exactly $1$ for the radius $r=1$, the constant of proportionality is $1$.</p>
<p>For $r\le1$,</p>
<p>$$F_R(r)=r^2,\... |
1,817,542 | <p><strong>Problem:</strong> Let $(X, Y)$ be uniformly distributed on the unit disk $\{ (x,y) : x^2 + y^2 \le 1\}$. Let $R = \sqrt{X^2 + Y^2}$. Find the CDF and PDF of $R$.</p>
<p><strong>Attempted Solution:</strong> First note that $r \in R = \sqrt{X^2 + Y^2}$ represents a point on $\mathbb{R}^2$ with radius $r$ abou... | Em. | 290,196 | <p>One of the reasons I love probability is that there are usual a million ways to do a problem.</p>
<p>I provide a heuristic alternative.</p>
<p>We recognize that we are interested in the event $R\in dr$. In words, this means we want the radius to fall in an infinitesimal annulus, with infinitesimal width $dr$, and ... |
1,302,990 | <p>I want to ask basic question. In our mathematics classes ,while teaching the Fourier series and transform topic,the professor says that when the signal is periodic ,we should use Fourier series and Fourier transform for aperiodic signals.</p>
<p>My question is can't we use Fourier transform formula in case of per... | Kishan | 247,422 | <p>Fourier Series (FS) exists only for periodic signals.</p>
<p>Fourier Transform (FT) is derived from FS, i.e. FT is the envelope of the FS. Thus as the frequency domain became more finer, time domain enlarges making it a aperiodic signal.</p>
<p>So when FT is applied on a periodic signal, the result is just, T*X(k)... |
306,011 | <p>Does anyone have a proof for $$\int_0^{\infty}\frac{\sin(x^2)}{x^2}\,dx=\sqrt{\frac{\pi}{2}}.$$
I tried to get it from contour integrating $$\frac{e^{iz^2}-1}{z^2},$$ but failed.
Thanks.</p>
| Kaster | 49,333 | <p>I'll extend <strong>@sos440</strong>'s answer a bit.
$$
I = \int_o^\infty \frac {\sin x^2}{x^2}dx = -\left. \frac{\sin x^2}x \right|_0^\infty + 2\int_0^\infty \cos x^2 dx \\
\left .\frac{\sin x^2}x \right |_0^\infty = \lim_{x \rightarrow \infty} \frac {\sin x^2}x - \lim_{x \rightarrow 0} \frac {\sin x^2}x = 0-\lim \... |
3,754,548 | <p>Suppose there is a strictly convex continuous function <span class="math-container">$f$</span>: <span class="math-container">$R^n$</span> <span class="math-container">$\rightarrow$</span> <span class="math-container">$R$</span>.</p>
<p>Is the supremum of <span class="math-container">$f$</span> always infinity? How c... | Paul | 138,918 | <p>The tabular method will work, in just the same way as the traditional by parts method works. The last row in the table is the integral still to be done. You will find that the third row in your table (the integral still to do) is a multiple of the original integral. If you call the original integral <span class="mat... |
1,252,414 | <p>In rectangle ABCD below, points F and G lie on segment AB such that AF = FG = GB and E is the midpoint of segment DC. Also, segment AC intersects segment EF at H and segment EG at J. The area of rectangle ABCD is 70. Find the area of triangle AHF.</p>
<p>(Note: This question has been slightly changed from the origi... | the_candyman | 51,370 | <p>Let's say that $AB=3x$, $CD=2y$ and $BC=h$. Then, $$3xh = 2yh = 70,$$and
$$x = \frac{2y}{3}.$$
The triangles $AFH$ and $HEC$ are similar and have area:</p>
<p>$$S_{AFH} = \frac{x h_1}{2}, S_{HEC} = \frac{y h_2}{2}$$</p>
<p>where $h_1$ and $h_2$ are the heights of the 2 triangles, with $h_1+h_2 = h$ and $h_1 = \fra... |
1,698,039 | <p>Alright, so let's say I have $$\frac{x^{-6}}{-x^{-4}}$$ The answer is $\dfrac{1}{x^2}$, but why isn't it $\dfrac{1}{-x^2}$?</p>
| Tae Hyung Kim | 94,401 | <p>Let's think about what negative exponents really mean. Consider the following list
$$2^3 = 8$$
$$2^2 = 4$$
$$2^1 = 2$$
Notice that every time we decrease an exponent, we divide by 2. Continuing this list, we have that
$$2^3 = 8$$
$$2^2 = 4$$
$$2^1 = 2$$
$$2^0 = 1$$
$$2^{-1} = 1/2$$
$$2^{-2} = 1/4$$
Rather interestin... |
655,378 | <p>I'm new to discrete mathematics and was wondering whether the following functions are one to one:</p>
<p>$$f(x) = x - 1$$
$$f(x) = x^2 + 1$$</p>
<p>The reason I stand by this is because for the first equation:</p>
<p>$$x - 1 = y - 1\\x = y$$</p>
<p>and for the second one:</p>
<p>$$x^2 +1 = y^2 +1\\x = y$$</p>
| RE60K | 67,609 | <p>$$\frac{dy}{dx} = \frac{1}{x\cos(y) +\sin(2y)}=\frac{1}{x\cos(y) +2\sin(y)\cos y}\\
\frac{d(\sin y)}{dx}=\frac{1}{x+2\sin y}$$
Then $$\frac{dx}{d(\sin y)} -x=2\sin y\\xe^{\int -d(\sin y)}=\int 2\sin ye^{\int -d(\sin y)}d(\sin y)\\xe^{-\sin y}=2\int \sin ye^{-\sin y}d(\sin y)=-2e^{-\sin y}(\sin y+1)+c\\x+2\sin y+2=ce... |
2,814,703 | <p>I am reading <a href="https://en.wikipedia.org/wiki/Lower_limit_topology" rel="nofollow noreferrer">lower limit topology</a> on Wikipedia, which states that the lower limit topology </p>
<blockquote>
<p>[...] is the topology generated by the basis of all half-open intervals $[a,b)$, where a and b are real numbers... | qualcuno | 362,866 | <p>One can write </p>
<p>$$
(a,b) = \bigcup_{n\geq 1}\left[a+\frac{b-a}{2n},b\right)
$$</p>
<p>One inclusion is straighforward. For the non trivial one, pick $t \in (a,b)$, and $n$ large enough such that $t > a +\frac{b-a}{2n}$, so that $t \in \left[a+ \frac{b-a}{2n},b\right)$.</p>
|
3,710,018 | <p>Problem:
Find prime solutions to the equation
<strong><span class="math-container">$p^2+1=q^2+r^2$</span></strong></p>
<p>I welcome you to post your own solutions as well</p>
<p>I have found a <em>strange solution</em> which I can't understand why it works(or what's the math behind it.) Here it is through examples... | Edward Porcella | 403,946 | <p>This seems to hold much more broadly than just for primes.</p>
<p>Let<span class="math-container">$$r^2-1=ab$$</span>where <span class="math-container">$b>a$</span>, and let <span class="math-container">$p=\frac{b+a}{2}$</span> and <span class="math-container">$q=\frac{b-a}{2}$</span>.</p>
<p>Then the equation <s... |
3,808,575 | <p>Assuming I have the statement ∀x(∀y¬Q(x,y)∨P(x)), can I pull the universal quantifier ∀y out of the parenthesis? Meaning, is this statement equivalent to ∀x∀y(¬Q(x,y)∨P(x)) ?</p>
<p>An approach I tried so far:</p>
<ol>
<li>∀x((∃y Q(x,y) ) => P(x)). (original eq.)</li>
<li>∀x((∀y¬Q(x,y))∨P(x)) (... | Graham Kemp | 135,106 | <p>The original expression: <span class="math-container">$\forall x~((\exists y~Q(x,y))\to P(x))$</span> says "For any <span class="math-container">$x$</span> it holds that if some <span class="math-container">$y$</span> satisfies <span class="math-container">$Q(x,y)$</span>, then <span class="math-container">$P(x... |
3,682,987 | <blockquote>
<p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span> be positive real numbers. What is the smallest possible value of <span class="math-container">$(a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)$</span>?</p>
</... | Calvin Lin | 54,563 | <p><strong>Hint:</strong> Do you know why <span class="math-container">$ ( a + b+ c) ( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} ) \geq 9 $</span>? </p>
<blockquote class="spoiler">
<p> Arithmetic Mean - Harmonic Mean.</p>
</blockquote>
<p>Hence, conclude that <span class="math-container">$ 2 ( a + b + c) ( \frac{1... |
3,684,917 | <p>Let <span class="math-container">$C_{1}$</span> and <span class="math-container">$C_{2}$</span> be polytopes in <span class="math-container">$\mathbb{R}^{n}$</span> such that
<span class="math-container">$C_{1}=conv\left( V\right) $</span> with <span class="math-container">$V$</span> being a set of vertices. If
<s... | Jotadiolyne Dicci | 780,918 | <p>With the help of the answers you sent, I was able to find a form that seems rather simple to me. Then I'm not 100% sure it works all the time.</p>
<p>Let <span class="math-container">$S$</span> be the number of integers congruent to <span class="math-container">${n}\pmod p$</span> in the interval <span class="math-... |
1,419,897 | <blockquote>
<p><strong>Theorem:</strong> Let $A$ be a bounded infinite subset of $\mathbb{R}^l$. Then
it has a limit point.</p>
</blockquote>
<p>So this is the Euclidean version of the Bolzano-Weierstrass theorem, the thing is that I was trying to prove it by induction, but it doesn't help because in the case $l=... | Kitegi | 120,267 | <p>Proving it for $l=1$ is enough to get the rest.</p>
<p>Let $X_n = (x_{1,n},\dots,x_{l,n})$ be an injective infinite sequence of points in the set $A$.</p>
<p>$x_{1,n}$ is bounded</p>
<p>So there is a subsequence s.t $x_{1,\phi_1(n)}$ converges</p>
<p>Now consider $X_{\phi_1(n)} = (x_{1,\phi_1(n)},\dots,x_{l,\phi... |
909,228 | <p>I'm trying to find a closed form for the following sum
$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$
where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.</p>
<p>Could you help me with it?</p>
| Ali Shadhar | 432,085 | <p><strong>Alternative approach:</strong></p>
<p>first we start with proving the following equality that appeared as Problem 11921 in The American Mathematical Monthly 2016 proposed by <strong>Cornel Ioan Valean</strong>:
<span class="math-container">\begin{equation*}
S=\ln^22\sum_{n=1}^{\infty}\frac{H_n}{(n+1) 2^{n+... |
658,761 | <p>Sorry if this question is too basic.</p>
<p>We can suppose that we have a matrix that reduces to the identity matrix in reduced row-echelon form. My question is fairly simple: Can we multiply one or more columns by a constant and still be able to reduce the matrix to the identity?</p>
<p>I'd like a proof of this... | heropup | 118,193 | <p>In <em>Mathematica</em> (version 6 or newer)</p>
<pre><code>Plot3D[x/(1 - y), {x, -1, 1}, {y, -1, 1},
RegionFunction -> Function[{x, y}, x^2 + y^2 < 1]]
</code></pre>
|
658,761 | <p>Sorry if this question is too basic.</p>
<p>We can suppose that we have a matrix that reduces to the identity matrix in reduced row-echelon form. My question is fairly simple: Can we multiply one or more columns by a constant and still be able to reduce the matrix to the identity?</p>
<p>I'd like a proof of this... | Mikasa | 8,581 | <p>In Maple, you can use the following codes:</p>
<pre><code>[> with(plots):
plot3d(x/(1-y), x = -1 .. 1, y = -sqrt(1-x^2) .. sqrt(1-x^2), axes = boxed, filled = true,numpoints = 1000,color=green);
</code></pre>
<p><img src="https://i.stack.imgur.com/ysKzx.png" alt="enter image description here"></p>
|
2,030,116 | <p>How can i prove that $\sqrt[12]{2}$ is irrational number? </p>
<p>I'm trying: </p>
<p>$$\sqrt[12]{2} = \frac{p}{q}$$ where $p$, $q$ are integers</p>
<p>it follows that :</p>
<p>$$p^{12} = 2q^{12} $$</p>
<p>What is argument of irrationality in this case?
From what we know that the right-hand side has an even n... | Fred | 380,717 | <p>If $\sqrt[12]{2} = \frac{p}{q}$, then $\sqrt[]{2}=(\sqrt[12]{2})^6=\frac{p^6}{q^6}$.</p>
<p>Your turn !</p>
|
2,509,308 | <p>My friend asked me a so called modified version of <a href="https://math.stackexchange.com/questions/96826/the-monty-hall-problem#">Monty Hall problem</a> in his opinion.
But I find the description a bit spooky and maybe someone here can enlighten
us with what is the problem with the description of the problem, or m... | Doug M | 317,162 | <p>If the host has no inside information, then with no new information there should be nothing to be gained by switching.</p>
<p>However you can model the outcomes explictly.</p>
<p>scenario 1. You choose the right card (P = 1/3). The host reveals a goat probability (always).</p>
<p>scenario 2. you choose the wron... |
105,857 | <p>Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Is $\text{SL}_2(\mathcal{O})$ generated by elementary matrices? If it isn't, is there any other natural generating set for it?</p>
<p>The usual argument shows that this is true for $\mathcal{O} = \mathbb{Z}$ (or, more generally, a Euclidean d... | Luc Guyot | 84,349 | <p>L. N. Vaserstein's theorem [2] asserts that if <span class="math-container">$R$</span> is a Dedekind ring of arithmetic type with infinitely many units then <span class="math-container">$SL_2(R) = E_2(R)$</span> holds. In other words <span class="math-container">$R$</span> is a <span class="math-container">$GE_2$</... |
1,123,050 | <p>This is the same problem asked here. - <a href="https://math.stackexchange.com/questions/1105927/next-step-to-take-to-reach-the-contradiction">Next step to take to reach the contradiction?</a>
Here is it again.</p>
<p><img src="https://i.stack.imgur.com/onqzq.png" alt="enter image description here"></p>
<p>I under... | nelv | 136,824 | <p>Think of it physically: each measure assigns different <em>weights</em> to given sets: consider for example the particular case $d\mu=df(x)=f'(x)dx$ for a well behaved $f(x)$. Here you can really see the difference between the "ordinary" measure $dx$, which does not care about the location of the set, and $f'(x)dx$,... |
1,351,350 | <p>Assume that probability of $A$ is $0.6$ and probability of $B$ is at least $0.75$. Then how do I calculate the probability of both $A$ and $B$ happening together?</p>
| lulu | 252,071 | <p>You can't calculate that from the information given. </p>
<p>As an illustration: Imagine you choose a value N at random from 1, ..., 20. at one extreme, suppose A is the event {N ≤ 12} and B is the event {N ≤ 15}. Then A $\Rightarrow$ B and the probability of both is .6.
Now suppose A is the event {N ≤ 12} but ... |
4,442,223 | <p>How does one show this?
<span class="math-container">$$
\exp(-x) \sum_{k=0}^\infty x^k \frac{(k+m)!}{(k!)^2} = L_m(-x) m!,
$$</span> where <span class="math-container">$m$</span> is a positive integer, and <span class="math-container">$L_{m}(x)$</span> is the <span class="math-container">$m$</span>th order Laguerre ... | Paul Frost | 349,785 | <p>The definition you found in <a href="https://mathworld.wolfram.com/SimplicialComplex.html" rel="nofollow noreferrer">Wolfram MathWorld</a> describes simplicial complexes as a collection <span class="math-container">$K$</span> of simplices in some <span class="math-container">$\mathbb R^n$</span> such that</p>
<ol>
<... |
1,749,730 | <p>What is the maximum number of faces of totally convex solid that one can "see" from a point? </p>
<p>...and, more importantly, how can I ask this question better? (I'm a college student with little experience in asking well formed questions, much less answering them.) </p>
<p>By "see" I mean something like this: y... | Vincent | 332,815 | <p>Here are some definitions (in dimention 3, but you can easily generalize):</p>
<p><em>Definition</em> : Given a finite number of points with coordinates $ P_1 = (x_1,y_1,z_1), .., P_n = (x_n,y_n,z_n) $, a convex solid is the convex hull of these points, i.e. all the points of the space defined as $\sum_{1\leq i \le... |
898,683 | <p>Given a pool of 30 balls (5 of each color). When drawing 8 balls without replacement, what is the probability of getting at least one of each color?</p>
<p>Related: <a href="https://math.stackexchange.com/questions/897730/probability-of-drawing-at-least-one-red-and-at-least-one-green-ball">Probability of drawing at... | André Nicolas | 6,312 | <p><strong>Edit:</strong> The question has changed. We leave the answer to the original question, which specified $6$ draws. We append a sketch for the new question.</p>
<hr>
<p>We want the probability that the balls all have different colours. Imagine (it makes no difference) that we draw the balls one at a time. </... |
898,683 | <p>Given a pool of 30 balls (5 of each color). When drawing 8 balls without replacement, what is the probability of getting at least one of each color?</p>
<p>Related: <a href="https://math.stackexchange.com/questions/897730/probability-of-drawing-at-least-one-red-and-at-least-one-green-ball">Probability of drawing at... | Thomas Andrews | 7,933 | <p>The quickest approach is to count ways to get $6$ colors on $8$ balls. There are essentially two cases, $\langle 3,1,1,1,1,1\rangle$ and $\langle 2,2,1,1,1,1\rangle$. There are $\binom{6}{1}\binom{5}{3}\binom{5}{1}^5=187,500$ ways to get the first case. The $\binom{6}{1}$ counts the number of ways of choosing one c... |
973,035 | <p>I'm wondering whether there is an invertible function $f: \mathbb{R} \to \mathbb{R}$ such that $f(-1)=0$, $f(0)=1$ and $f(1)=-1$. I think it's not but I'm missing a real proof.</p>
<p>The easiest would be to show such a function cannot be injective... But I don't see how? I don't see any other way of starting this ... | Tomasz Kania | 17,929 | <p>There is no invertible function like that which is <em>continuous</em>. Indeed, $f(0)=1$ and $f(1)=-1$ so by the <a href="http://en.wikipedia.org/wiki/Intermediate_value_theorem" rel="nofollow">Intermediate Value Theorem</a>, there is $\xi \in (0,1)$ such that $f(\xi)=0 = f(-1)$.</p>
<p>There are discontinuous bije... |
172,292 | <p>I am trying to find the residue of the function $$f(z)=(z+1)^2e^{3/z^2}$$
at $z=0$.
I tried the following in Mathematica</p>
<pre><code>Residue[(z+1)^2*Exp[3/z^2],{z,0}]
</code></pre>
<p>which remains unevaluated. Computing this by hand gives the value of $6$. What is going on?</p>
<p>I’ve noticed that Mathemati... | Akku14 | 34,287 | <p>Or integrate around zero</p>
<pre><code>Integrate[(z + 1)^2 Exp[3/z^2], {z, 1, I, -1, -I,
1}]/(2 Pi I) // Simplify
(* 6 *)
</code></pre>
|
2,347,820 | <p>What is the solution to $\log_{10} x -x=2?$</p>
<p>I have tried to solve it but I couldn't. I've got to $x^x =200$.</p>
| 高田航 | 407,845 | <p>$f(x)=log_{10}(x)-x$ remains completely within the fourth quadrant, so $f(x)$ will never be equal to $2$. </p>
|
1,709,713 | <p>How do you make the jump from:</p>
<p>$$\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$$</p>
<p>To:</p>
<p>$$\frac{25^{21}-4^{21}}{25^{21}-4(25^{20})}$$</p>
| Henricus V. | 239,207 | <p>$$ \frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}
= \frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}} \frac{25^{21}}{25^{21}}
= \frac{25^{21}-(\frac{4}{25})^{21}25^{21}}{25^{21}-\frac{4}{25}25^{21}}
= \frac{25^{21}-4^{21}}{25^{21}-4 \cdot 25^{20}}
$$</p>
|
1,896,008 | <p>Is the following statement correct: </p>
<blockquote>
<p>If $A$ and $B$ are closed subsets of $[0,\infty)$, then $A+B=\{x+y:x \in A,y \in B\}$ is closed in $[0,\infty)$.</p>
</blockquote>
| Piquito | 219,998 | <p>HINT.-The addition in $\Bbb R$ is continuous (from $\Bbb R^2$ to $\Bbb R$). You have, for example two ways to prove $A+B$ is closed.</p>
<p>1) If $\{z_n\}$ is a sequence convergent to $z$ and contained in $A+B$ then $z\in A+B$.</p>
<p>2) The complement of $A+B$ is easily proved to be open (using also the continuit... |
1,621,269 | <p>I have tried everything in my knowledge and no, I cannot state it.
I have tried a factorizor online which tells me that it is not factorizable hence irreducible. But I cannot reason why.</p>
<p>I looked at Eisenstein's criteria but obviously, there is no prime $q$ that fits the criteria so this is useless.</p>
<p>... | Alex | 48,061 | <p>Let $x = y+1$. Then this equation becomes:</p>
<p>$y^5+3y^2+9y+3$. Use Eisenstein to show this is irreducible. Therefore, the original is also irreducible.</p>
|
489,907 | <p>I've only got the following parts of a triangle:</p>
<ul>
<li>Line A to B </li>
<li>Line B to C</li>
</ul>
<p>And optionally the Line from A to C if needed?</p>
<p>I'm trying to get the point X</p>
<p><a href="https://i.stack.imgur.com/f6leO.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/f6le... | Mufasa | 49,003 | <p>OK, then working off your diagram above and assuming AX is perpendicular to BC, you can proceed as follows:<br/>
Let's call the co-ordinates of point A $(x_a, y_a)$, point B $(x_b, y_b)$, point C $(x_c, y_c)$ and point X $(x, y)$.<br/>
For line BC we can write: $\frac{y-y_c}{x-x_c}=\frac{y_b-y_c}{x_b-x_c}=m$ where $... |
2,670,082 | <p>I have quite an interesting infinite totient sum. My task is to evaluate</p>
<p>$\sum_{n=1}^{\infty} \frac{\phi(n)}{5^n +1}.$</p>
<p>The problem is that I have no idea how to go from here as I have never seen such a problem before. The usual techique of writing $n$ and $\phi(n)$ in terms of the prime factorization... | achille hui | 59,379 | <p>Notice for any prime <span class="math-container">$p$</span> and integer <span class="math-container">$k \ge 0$</span>, we have</p>
<p><span class="math-container">$$\varphi(p^k) = \begin{cases}p^k - p^{k-1}, & k > 0\\ 1,& k = 0\end{cases}
\quad\implies\quad \sum_{\ell=0}^k \varphi(p^\ell) = p^k
$$</span>... |
515,659 | <p>Question is to check :</p>
<p>For any real number $c$, the polynomial $x^3+x+c$ has exactly one real root .</p>
<p>the way in which i have proceeded is :</p>
<p>let $a$ be one real root for $x^3+x+c$ i.e., we have $a^3+a+c=0$</p>
<p>i have seen that $(x-a)(x^2+ax+(a^2+1))=x^3+x-a^3-a$</p>
<p>But, $a^3+a+c=0$. S... | Hagen von Eitzen | 39,174 | <p>You essentially split off the linear factor belnging to a real root and showed per determinant that the remaining quadratic has no real root.
This is fine but does not readily generalize to higher degrees.
Instead, it is probably easier to show that $f$ is injective: Assume $a,b$ are two real roots, i.e. $f(a)=f(b)=... |
515,659 | <p>Question is to check :</p>
<p>For any real number $c$, the polynomial $x^3+x+c$ has exactly one real root .</p>
<p>the way in which i have proceeded is :</p>
<p>let $a$ be one real root for $x^3+x+c$ i.e., we have $a^3+a+c=0$</p>
<p>i have seen that $(x-a)(x^2+ax+(a^2+1))=x^3+x-a^3-a$</p>
<p>But, $a^3+a+c=0$. S... | Mark Bennet | 2,906 | <p>It is a bit unclear what you want to generalise. $x^3+x+c$ is a strictly increasing function of $x$ - a strictly increasing (or decreasing) polynomial of odd order with real coefficients will always have a single real root.</p>
<p>Or you may be saying that the polynomial can be written as $p(x)=(x-a)q(x)$ where $q(... |
515,659 | <p>Question is to check :</p>
<p>For any real number $c$, the polynomial $x^3+x+c$ has exactly one real root .</p>
<p>the way in which i have proceeded is :</p>
<p>let $a$ be one real root for $x^3+x+c$ i.e., we have $a^3+a+c=0$</p>
<p>i have seen that $(x-a)(x^2+ax+(a^2+1))=x^3+x-a^3-a$</p>
<p>But, $a^3+a+c=0$. S... | Michael Hoppe | 93,935 | <p>$(x^3+x+1)'=3x^2+1>0$ for all real $x$.</p>
|
80,918 | <p>Could anyone please tell me what could be the math function to get the number of zeros in given decimal representation of numbers? I scratched my head on Combination and Permutation but couldn't come up with generic answer. The number length can be up to 1000 digits, so you can represent a number as a String.</p>
<... | Dima | 44,950 | <p>The approach is correct, but the answer is wrong. There are 145 zeros from 1 to 751:
9 zeros from 1 to 99, 120 zeros from 100 to 699 (20 x 6) and 16 zeros from 700 to 759</p>
<p>Total: 9+120+16 = 145.</p>
|
1,100,906 | <p>What would be the highest power of two in the given expression?</p>
<p>$32!+33!+34!+35!+...+87!+88!+89!+90!\ ?$</p>
<p>I know there are 59 terms involved. I also know the powers of two in each term. I found that $32!$ has 31 two's. If we take 32! out of every term the resulting 59 terms has 2 odd terms and 57 even... | Milo Brandt | 174,927 | <p>You've simply stopped calculating one step too soon! You've shown that that expression, with $32!$ divided out is divisible by $2$, but if you just checked whether it was divisible by $4$, you would see that it is indeed not. In particular, the divided expression would be
$$\frac{32!}{32!}+\frac{33!}{32!}+\frac{34!}... |
2,280,666 | <p>Let AD be the altitude corresponding to the hypotenuse BC of the right triangle ABC. The circle of diameter AD intersects AB at M and AC at N shown. Prove $\frac{BM}{CN}$= $\bigg(\frac{AB}{AC}\bigg)^{3}$.</p>
<p>So far I have...</p>
<p>The power of B is $BD^{2}=(BM)(BA)$</p>
<p>The power of C is $CD^{2}=(CN)(CA)$... | Martín Forsberg Conde | 249,492 | <p><strong>Hint</strong>: What can you say about triangle $BMD$ given that A forms a right angle? Try to express $\frac{BM}{CN}$ in terms of the relations of sides that you know.</p>
<p>__</p>
<p>__</p>
<p>__</p>
<p>__</p>
<p>__</p>
<p><strong>Solution</strong>: The angle in M of $AMD$ is right, since it sees a d... |
189,074 | <p>the function f defined by $f(x)=(x^3+1)/3$ has three fixed points say α,β,γ where
$-2<α<-1$, $0<β<1$, $1<γ<2$.
For arbitrarily chosen $x_{1}$, define ${x_{n}}$ by setting $x_{n+1}=f(x_{n})$
If $α<x_{1}<γ$, prove that $x_{n}\rightarrow β$ as $n \rightarrow \infty$</p>
<p>I think I must prove ... | copper.hat | 27,978 | <p>This is not meant to be an answer, but I can't add a picture to comments or another answer...</p>
<p><img src="https://i.stack.imgur.com/WwgrH.png" alt="enter image description here"></p>
|
564,378 | <p>Suppose we start with a rational number $a_0$, and define $a_{n+1}=2a_n^2-1$ for $n\geq 0$. For what $a_0$ will it be the case that $a_i=a_j$ for some $i\neq j$?</p>
<p>We can start with something like $a_0=1$, then $a_1=1$ so $a_0=a_1$.</p>
<p>If $a_0=0$, we get $0, -1, 1, 1, \ldots$</p>
<p>Likewise if $a_0=-1$,... | DonAntonio | 31,254 | <p>In general</p>
<p>$$a_{n+1}:=2a_n^2-1=2a_m^2-1=:a_m\iff a_n=\pm a_m$$</p>
<p>If we choose $\;n\;$ to be the minimal index s.t. $\;a_{n+1}=a_{m+1}\;$ , for some $\;m\neq n\;$ , the above means that</p>
<p>$$a_n=-a_m\iff 2a_{n-1}^2-1=-2a_{m-1}^2+1\iff a_{n-1}^2+a_{m-1}^2=1\ldots$$</p>
<p>Try to take it from here.<... |
1,550,603 | <p>I got confirmed from a graduate school starting from next year and I will major algebraic geometry.</p>
<p>Until now, I have never thought that I study little things than others with my age. However, I heard that <strong>some</strong> of my colleagues already studied Hartshorne at least once and quite a few of them... | paul garrett | 12,291 | <p>Your questions in order:</p>
<p>Yes, it is highly advantageous to <em>be</em> <em>exposed</em> <em>to</em> more sophisticated ("graduate") subjects as early as one can tolerate it.</p>
<p>Not clear that one should "study" them.</p>
<p>Yes, some people do "read ahead". I myself found it very helpful.</p>
<p>I wou... |
3,527,919 | <p>I've tried to prove this property of Bessel function but I don't seem to be going anywhere</p>
<p><span class="math-container">$$\sqrt{\frac 12 \pi x} J_\frac 32 (x) = \cfrac{\sin x}{x} - \cos x$$</span></p>
<p>I have tried substituting <span class="math-container">$\frac 32$</span> for <span class="math-container... | Jean Marie | 305,862 | <p>Three methods :</p>
<p>1) By using the general (recurrence) formula :</p>
<p><span class="math-container">$$J_{\nu-1}(x)+J_{\nu+1}(x)=\frac{2 \nu}{x}J_{\nu}(x)$$</span></p>
<p>(formula 2.4 p. 13 of this excellent <a href="https://www.math.ust.hk/~machiang/150/Intro_bessel_bk_Nov08.pdf" rel="nofollow noreferrer">d... |
1,023,575 | <p>How would one factor a number, say $9+4\sqrt{2}$ in $\mathbb{Z}[\sqrt{2}]$?</p>
<p>This is what I've attemped to do:
$$(a_1+b_1\sqrt{2})(a_2+b_2\sqrt{2}) $$
$$a_1a_2+a_1b_2\sqrt{2}+a_2b_1\sqrt{2}+2b_1b_2$$
Thus,
\begin{eqnarray}
a_1a_2+2b_1b_2&=&9 \\
a_1b_2+a_2b_1 &=& 4.
\end{eqnarray}</p>
<p>But t... | laerne | 192,873 | <p>Not all numbers can be factored in $\mathbb{Z}[\sqrt{2}]$, it has also $\mathbb{Z}[\sqrt{2}]$-prime numbers, like $3$ or $\sqrt{2}$. Besides, other numbers have many factorisation. For example numbers of $\mathbb{z}$ like $60$ which have at least all the $\mathbb{Z}$ like factorisation, like $10\cdot6$ or $12\cdot... |
211,705 | <p>I am given a table of possible <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> values that can be generated in a casino. In the game, both are generated with each turn.</p>
<p><img src="https://i.stack.imgur.com/G0nLn.jpg" alt="enter image description here" /></p>
<blockquote>... | Ross Millikan | 1,827 | <p>For a) you need to multiply by $8$, as you are paying $8X_1$</p>
<p>For b) you are doing the right thing. The only way to simplify it is to find some pattern. For example, in c) if $X_1$ and $X_2$ were independent (they are not) you could just multiply the expectations.</p>
|
3,440,873 | <p>I do not how to solve this, can such equation even exist? For the root to lie on the y intercept, the line would have to pass through origin, which means one root will be 0, breaking down the whole the thing. Am I missing something here?</p>
| Dietrich Burde | 83,966 | <p>The German wikipedia entry <a href="https://de.wikipedia.org/wiki/Intervall_(Mathematik)" rel="nofollow noreferrer">here</a> explains the notations of open, half-open, closed intervals. It agrees with the <a href="https://en.wikipedia.org/wiki/Interval_(mathematics)" rel="nofollow noreferrer">English entry</a>.</p>
... |
1,159,860 | <p>If $$f:[a,b]\times [c,d] \to \mathbb{R}$$ is continuous and $f_{y}$ is continuous, let $$F(x,y)=\int_{a}^{x} f(t,y)dt.$$ </p>
<ol>
<li>Find $F_x$ and $F_y$.</li>
<li>If $G(x)=\int_{a}^{g(x)}f(t,x)dt$, find $G'(x)$</li>
</ol>
<p>My try: </p>
<p>For (1) $$F(x+h,y)-F(x,y)=\int_{a}^{x+h} f(t,y)dt-\int_{a}^{x}f(t,y)dt... | JMP | 210,189 | <p>$P(A-B)$ means the probability that $A$ happens and $B$ doesn't. As $P(A)=a$ and $P(¬B)=1-b$, the answer is $a(1-b)$.</p>
|
3,668,101 | <p>I know that if <span class="math-container">$n \bmod k \le k-1$</span> then this sum is converge then it has finite sum, I just guess it's <span class="math-container">$\ln(k)$</span> because when <span class="math-container">$k=1$</span> sum is <span class="math-container">$0=ln(1)$</span>. I really don't know how ... | HOANXA | 760,747 | <p>My idea is:</p>
<p><span class="math-container">$\sum \frac{nmodk}{n(n+1)}=\frac{1}{1.2}+\frac{2}{2.3}+...+\frac{k-1}{(k-1)k}+...=\frac{1}{1}-\frac{1}{2}+\frac{2}{2}-\frac{2}{3}+...$</span>
<span class="math-container">$=(\frac{\:1}{1\:}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{k-1}+\frac{k-1}{k})+(\frac{\:1}{k+1\:}+\f... |
2,346,489 | <p>I am studying the Proposition: Let $D$ be a Dedekind domain, $F$ its field of fractions, $E$ a finite dimensional extension field of $F$ and $D'$ the subring of $E$ of $D$ integral elements. Assume that $E/F$ is a finite separable field extension. Then $D'$ is a finitely generated $D$-module.</p>
<p>I have to show ... | Community | -1 | <p>If I understand correctly your question, here is a sort of answer that you might like.</p>
<p><strong>Hints and facts</strong>:</p>
<p>Because of the situation we have (which is really good in fact since the rings we have very <em>good</em> structures) the extension $D \subset D^{\prime}$ is integral, and $D$ is n... |
2,136,411 | <p>According to the Theorem 12.7 of the book Analytic Nymber Theory by Apostol, $$\zeta(1-s) = 2(2\pi)^{-s} \Gamma(s) \cos \big(\frac{\pi s}{2}) \zeta(s)$$ which results in (as the book also says) that $\zeta(-2n) =0$ for $n=1,2,3, \dots$, the so-called trival zeros of $\zeta(s)$. </p>
<p>But how on earth $\zeta(-2n) ... | Start wearing purple | 73,025 | <p>In the same spirit, we have
$$2^0+2^1+2^2+2^3+\ldots =-1.$$
The seeming paradox is that the sum on the left is <em>defined</em> as the analytic continuation of the series $\sum_{k=0}^{\infty}z^k=\frac{1}{1-z}$ outside its original domain of convergence $|z|<1$.</p>
|
3,258,642 | <blockquote>
<p>If the roots of quadratic equation <span class="math-container">$$x^2 − 2ax + a^2 + a – 3 = 0$$</span>
are real and less than <span class="math-container">$3$</span>, find the range of <span class="math-container">$a$</span>.</p>
</blockquote>
<p>The roots are <span class="math-container">$a... | Bernard | 202,857 | <p>You don't have to calculate the roots, nor square them – only use high-school theorems on the sign of quadratic polynomials:</p>
<ol>
<li>The polynomial <span class="math-container">$p(x)=x^2 − 2ax + a^2 + a – 3$</span> has <em>real</em> roots, so its reduced discriminant is non-negative:
<span class="math-containe... |
3,275,732 | <p>How can I solve it without using matrix? I tried it to solve by using systems. But I have no idea how deal with "<span class="math-container">$0$</span>"</p>
| awkward | 76,172 | <p>Since we have the values of the polynomial at successive integers (-1,0,1 and 2), one way to find the interpolating polynomial is to use a table of finite differences:
<span class="math-container">$$\begin{matrix}
-2\\
&3\\
1 & &-2\\
&1 & &8\\
2 & &6\\
&7\\... |
1,860,459 | <blockquote>
<p>Prove that $4k < 2^k$ by induction.</p>
</blockquote>
<p>It holds for $k = 5$. Assume $ k = n + 1 $. Then</p>
<p>$4(n+1) < 2^{(n+1)}$</p>
<p>$4n + 4 < 2^n * 2$</p>
<p>$2n + 2 \leq 2^n$</p>
<p>Now I just need to show that</p>
<p>$2n + 2 \leq 4n$</p>
<p>$n + 1 \leq 2n$</p>
<p>$1 \leq n$... | Researcher314 | 165,956 | <p>You got the basis step correct by checking that $4n<2^n$ for $n=5$.</p>
<p>Next, you must prove that $4N<2^N \implies 4(N+1) < 2^{N+1}$ for $N\geq5$</p>
<p>When trying to solve this problem, simplify the right-hand-side of this implication to a form that is easily comparable to the left-hand-side.</p>
<p... |
1,860,459 | <blockquote>
<p>Prove that $4k < 2^k$ by induction.</p>
</blockquote>
<p>It holds for $k = 5$. Assume $ k = n + 1 $. Then</p>
<p>$4(n+1) < 2^{(n+1)}$</p>
<p>$4n + 4 < 2^n * 2$</p>
<p>$2n + 2 \leq 2^n$</p>
<p>Now I just need to show that</p>
<p>$2n + 2 \leq 4n$</p>
<p>$n + 1 \leq 2n$</p>
<p>$1 \leq n$... | Bill Dubuque | 242 | <p><strong>Hint</strong> $ $ Equivalently we seek to prove that $\,f(n) = 2^n/(4n) > 1\,$ for all $\,n\ge 5.$</p>
<p>Note $\,f(5)>1\,$ and $\,f(n\!+\!1)/f(n) = 2n/(n\!+\!1) \ge 1\,$ for $\,n\ge 5\,$ so $\,\color{#c00}{f(n\!+\!1) \ge f(n)}\,$ </p>
<p>Hence the induction reduces to a trivial one: $ $ an $\rm\colo... |
1,860,459 | <blockquote>
<p>Prove that $4k < 2^k$ by induction.</p>
</blockquote>
<p>It holds for $k = 5$. Assume $ k = n + 1 $. Then</p>
<p>$4(n+1) < 2^{(n+1)}$</p>
<p>$4n + 4 < 2^n * 2$</p>
<p>$2n + 2 \leq 2^n$</p>
<p>Now I just need to show that</p>
<p>$2n + 2 \leq 4n$</p>
<p>$n + 1 \leq 2n$</p>
<p>$1 \leq n$... | fleablood | 280,126 | <p>You made lots of little mistakes but your biggest mistake is that you inductions step, doesn't actually do any inducing.</p>
<p>A proper induction step always goes like this:</p>
<p>=======</p>
<p>We assume that something is true for a specific $k = n$</p>
<p>$\implies$</p>
<p>Then something is true for $k = n+... |
4,071,619 | <blockquote>
<p>There are two German couples, two Japanese couples and one unmarried person. If all 9 persons are two be interviewed one by one then the total number of ways of arranging their interviews such that no wife gives an interview before her husband is?</p>
</blockquote>
<p>I tried using the string method, bu... | John Omielan | 602,049 | <p>You can use induction to prove</p>
<p><span class="math-container">$$\frac{10^n-1}{9n} \tag{1}\label{eq1A}$$</span></p>
<p>is an integer when <span class="math-container">$n = 3^k$</span>. The base case of <span class="math-container">$k = 0$</span> gives <span class="math-container">$3^k = 1$</span>, with \eqref{eq... |
551,662 | <p>I am reading "What Is Mathematics? An Elementary Approach to Ideas and Methods"
And I am stuck here, I don't get it. I have posted a screen shot underlining what my doubt is..</p>
<p>I dont get it when the author says while the pythagoras theorem is : $a^2 + b^2 = c^2$
and then he says $x=a/c$ and $y=b/c$
and then... | Hagen von Eitzen | 39,174 | <p>You replaced one of the two $a$s making up $a^2$ with $cx$. Try replacing both (and similarly for the two $b$'s in $b^2$ with $cy$)</p>
|
3,997,968 | <p>I'm trying to figure out how to get the point x = 3 :
What's given here are the points S and G .
(Assuming the 2 angles are equal)
<a href="https://i.stack.imgur.com/COFMn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/COFMn.png" alt="enter image description here" /></a></p>
<p>Apparently, we can... | Narasimham | 95,860 | <p>Slightly different notation for co-ordinates.</p>
<p>To arrive at that equation equate slopes of straight lines <span class="math-container">$(SI, IG2)$</span> because after reflection the three points now are in a straight line.</p>
<p>( Reflection law requires that incidence and reflected angles be equal).</p>
<p... |
147,742 | <p>Is this statement true?</p>
<blockquote>
<p>$L:V\to V$ is a linear map with eigenvalue (not necessarily the only one) $a$. Suppose $(L-aI)^{m+1}(v)=0$ where $m$ is the power of the $(x-aI)$ term in the minimal polynomial of $L$. Then $(L-aI)^m(v)=0$ also.</p>
</blockquote>
<p>Some thoughts: </p>
<p>So this is e... | DonAntonio | 31,254 | <p>But $\,L-aI\,$ cannot have nullspace equal to zero as it is a singular map: $$\exists\,0\neq v\in V\,\,s.t.\,\,Lv=av\Longleftrightarrow (L-aI)(v)=0\,$$ and this seems to disprove your idea.</p>
<p>Now, if $\,Tv=0\,$ for some lin. transf., then $\,T^n(v)=0\,,\,\forall n\in\mathbb{N}\,$ , so both $\,(L-aI)^m\,\,,\,\,... |
147,742 | <p>Is this statement true?</p>
<blockquote>
<p>$L:V\to V$ is a linear map with eigenvalue (not necessarily the only one) $a$. Suppose $(L-aI)^{m+1}(v)=0$ where $m$ is the power of the $(x-aI)$ term in the minimal polynomial of $L$. Then $(L-aI)^m(v)=0$ also.</p>
</blockquote>
<p>Some thoughts: </p>
<p>So this is e... | Simon Markett | 30,357 | <p>The statement is true. In general you have
$$Ker(L-aI)\subseteq Ker(L-aI)^2\subseteq\cdots $$
This sequence stabilises, i.e. there is an $m$ such that
$$Ker(L-aI)^i=Ker(L-aI)^{i+1}$$
for all $i\geq m$. This $m$ is precisely the $m$ which is the exponent of $x-a$ in the minimal polynomial. In particular you have
$$K... |
1,229,227 | <p>Does it make sense to write $\frac{d}{dx}u(x,t)$ or can one only write $\frac{\partial}{\partial x}u(x,t)$?</p>
| hyperkahler | 188,593 | <p>Actually, the $\frac{dy}{dx}$ maybe considered as a quotient of two differentials, since $dy= y' \cdot dx$, whereas $\frac{\partial y}{\partial x}$ is an indivisible symbol.</p>
<p>According to this, it's convinient to write such type of equations $\frac{d(f(g(x))}{dx}= \frac{\partial{f}}{\partial{g}} \cdot \frac{\... |
1,229,227 | <p>Does it make sense to write $\frac{d}{dx}u(x,t)$ or can one only write $\frac{\partial}{\partial x}u(x,t)$?</p>
| GPerez | 118,574 | <p>For convention's sake, I'll pretend you asked about $\frac{d}{dt}$ and $\frac{\partial}{\partial t}$. It's the same exact question this way, and you'll see the reason for the change in notation afterwards.</p>
<p>Getting to the point, say you have a function of two variables $$u:\Bbb R^2\to\Bbb R^k\atop (x,t)\mapst... |
503,358 | <p>I remember I saw this question somewhere in Lang's undergraduate real analysis.</p>
<blockquote>
<p>Given any real number $\ge0$, show that it has a square root.</p>
</blockquote>
| user66733 | 66,733 | <p>It depends on how you define a real number. If you use Dedekind cuts, then you should show that the set $\{ x \in \mathbb{Q}^+: x^2<2\}$ is a Dedekind cut. If you use Cauchy sequences to define a real number, you can prove that the the sequence that is obtained from Newton's method is Cauchy:</p>
<p>$$ a_{n+1} =... |
982,780 | <p>I have the following system of <span class="math-container">$M$</span> linear equations in <span class="math-container">$N$</span> unknowns.</p>
<p><span class="math-container">$$
\begin{bmatrix}
3 & 0 & 1 & 0 & -1 & -3 & 2\\
1 & 2 & 0 & 4 & 0 & 0 & -1\\
1 & 1 &a... | Harald Hanche-Olsen | 23,290 | <p>I would try this: Given a family of closed subsets where any finite intersection is nonempty, add all such finite intersections to the family, so you have a family of nonempty, closed subsets which is closed under finite intersections. Any $F$ in the family has a smallest and a largest element. Consider the supremum... |
982,780 | <p>I have the following system of <span class="math-container">$M$</span> linear equations in <span class="math-container">$N$</span> unknowns.</p>
<p><span class="math-container">$$
\begin{bmatrix}
3 & 0 & 1 & 0 & -1 & -3 & 2\\
1 & 2 & 0 & 4 & 0 & 0 & -1\\
1 & 1 &a... | Andreas Blass | 48,510 | <p>If the conclusion fails, then there is, in $X$, a first element $\beta$ such that $]\leftarrow,\beta]$ is not compact. Let $C$ be an open cover of $]\leftarrow,\beta]$ with no finite subcover, and let $U$ be an element of $C$ that contains $\beta$. Of course, $U$ doesn't include all of $]\leftarrow,\beta]$, so, by ... |
1,282,419 | <p>Let $\Delta\subset\Bbb C$ be the open unitary disk. Let $\varphi:\Delta\to\Bbb R$ defined as follows: $\varphi(z)=1$ if $\Re z\ge0$, $\varphi(z)=0$ otherwise. So $\varphi$ is upper semicontinous.</p>
<p>In order to prove $\varphi$ is NOT subharmonic, I've to find a compact subset $K\Subset\Delta$ and a real valued ... | zhw. | 228,045 | <p>Another approach is through the mean value inequality: If $\varphi$ is subharmonic, then</p>
<p>$$1=\varphi(1/4) \le (1/2\pi)\int_0^{2\pi}\varphi(1/4 + (1/2)e^{it})\,dt.$$</p>
<p>But it's clear the last integral is $<1,$ contradiction.</p>
|
2,476,973 | <p>A fair six-sided die carries $1$ on one face, $2$ on two of its faces, and<br>
$3$ on the remaining three faces. </p>
<p>Suppose the die is rolled twice, and let $X$ be the random variable ’total score'. Find the probability distribution of $X$.</p>
| Graham Kemp | 135,106 | <p>The six-sided die has <code>1</code> on one face, <code>2</code> on two faces, and <code>3</code> on three faces. That gives the support and priobabilities for an individual roll.</p>
<p>The die is rolled twice, and the result of each roll added to give $T$. Thus let $T=T_1+T_2$, with $T_1,T_2$ being ... |
214,766 | <p>Is there an efficient way to check a number x and remove all prime factors in the number which are less than some n? For example for n = 200:</p>
<pre><code>x=88984589931961415442566827779929187431222364934742868664124547963532933
FactorInteger[x]
{{29, 2}, {31, 1}, {37, 2}, {269, 1}, {271,
1}, {3420047160553... | kglr | 125 | <p>The first list:</p>
<pre><code>listsa = GatherBy[Join @@ (Thread /@ list), Last];
Column[listsa]
</code></pre>
<p><a href="https://i.stack.imgur.com/b5H55.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/b5H55.png" alt="enter image description here"></a></p>
<p>The second list:</p>
<pre><code>l... |
3,078,097 | <blockquote>
<p>Why it is impossible to split the natural numbers into two sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span> such that for distinct elements <span class="math-container">$m, n \in A$</span> we have <span class="math-container">$m + n \in B$</span> and vice-versa?... | Julián C. Cano | 731,543 | <p><strong>Theorem [Schur, 1916]</strong>. For any partition of the set of positive integers in finite pieces, there are integers <span class="math-container">$x$</span> and <span class="math-container">$y$</span> with <span class="math-container">$x \neq y$</span> such that the set <span class="math-container">$\{x,y,... |
25,853 | <p>With regard to an undergraduate statistics course, I am developing a standardized list of point deductions with the TAs (doctoral students) so that graders are consistent in what they are taking off intermediate points for. For example, most problems are 10 points total, and my proposed point deductions for interme... | guest | 20,674 | <p>I advise being less intricate and put less load on the graders. I personally would go with all, half or zero credit for every question.</p>
<p>*All is correct answer (and some reasonable explication, not an essay, but also not a bare number.</p>
<p>*Half shows some decent knowledge of the process, but founders part... |
25,853 | <p>With regard to an undergraduate statistics course, I am developing a standardized list of point deductions with the TAs (doctoral students) so that graders are consistent in what they are taking off intermediate points for. For example, most problems are 10 points total, and my proposed point deductions for interme... | TomKern | 15,671 | <p>When designing a grading system for a question on an exam, I would identify the key skills being tested on that question. This can involve looking at other questions on the exam: if there are other questions that test a skill, I like to weigh it less unless it's a vital skill. A good exam is actually created startin... |
564,360 | <p>Lets take the example, if we take the expression $\frac{X!}{y_1!\cdot y_2!\cdots y_n!} $as long as summation $S=y_1+y_2+...y_n$ is less than or equals $X$, the remainder is always $0$. Thats How the permutation of $X$ things where there is $y_1$ things same , $y_2$ things same works. My question is, why does this ha... | Arthur | 15,500 | <p>Not a full answer, but an outline of how I convinced myself of this fact:</p>
<p>Prime factorization is the key word. The key result is that if $y_1 + y_2 + \cdots y_n \leq x$ then for any prime $p$ the power of $p$ in the factorization of $x!$ is at least as high as in $y_1!y_2!\cdots y_n!$.</p>
<p>Once you've co... |
782,507 | <p>So here's a somewhat incoherent question.</p>
<p>To define characteristic classes in the Chern–Weil way, one takes a curvature form $\Omega$ on a vector bundle $E \to M$ and an invariant polynomial $f$ on $\mathrm{GL}(\mathrm{rk }(E),\mathbb R)$, and then forms the cohomology class $c = \Big[f\big(\!\frac 1{2... | Mikhail Katz | 72,694 | <p>The point is to think of dual integer lattices. This can be seen already in the case of the unit circle. The class in cohomology dual to the fundamental homology class will be $d\theta$ divided by $2\pi$ since the circle has length $2\pi$. On the unit 2-sphere the curvature is $1$ but the total area is $4\pi$, whic... |
782,507 | <p>So here's a somewhat incoherent question.</p>
<p>To define characteristic classes in the Chern–Weil way, one takes a curvature form $\Omega$ on a vector bundle $E \to M$ and an invariant polynomial $f$ on $\mathrm{GL}(\mathrm{rk }(E),\mathbb R)$, and then forms the cohomology class $c = \Big[f\big(\!\frac 1{2... | R.S. | 151,441 | <p>The inclusion $\mathbb{Z}\hookrightarrow\mathbb{R}$ induces a homomorphism $\check{H}^2(M;\mathbb{Z})\longrightarrow \check{H}^2(M;\mathbb{R})$ and under the identification $\check{H}^2(M;\mathbb{R})\cong H_{dR}^2(M;\mathbb{R})$ one has a homomorphism $\check{H}^2(M;\mathbb{Z})\longrightarrow H_{dR}^2(M;\mathbb{R})$... |
8,658 | <p>$f(x) = \frac{1}{\cos x}$</p>
<p>$f'(x) = \frac{\sin(x)}{\cos^2(x)}$</p>
<p>$f''(x) = \frac{2\sin^2(x)+\cos^2(x)}{\cos^3(x)}$</p>
<p>$f^{(3)}(x) = \frac{6\sin^3(x)+5\cos^2(x)\sin(x)}{cos^4(x)}$</p>
<p>$\vdots$</p>
<p>$f^{(n)}(x) = \frac{ ?}{cos^{n+1}(x)}$</p>
<p>Some of these are easy: <a href="http://darkwing... | Qiaochu Yuan | 232 | <p>That depends on what you mean by "easy." As far as the examples in the .pdf you link to, I claim that the following are true for any reasonable definition of "easy":</p>
<ul>
<li>It is easy to compute the iterated derivatives of powers and logarithms. </li>
<li>It is easy to compute the iterated derivatives of so... |
8,658 | <p>$f(x) = \frac{1}{\cos x}$</p>
<p>$f'(x) = \frac{\sin(x)}{\cos^2(x)}$</p>
<p>$f''(x) = \frac{2\sin^2(x)+\cos^2(x)}{\cos^3(x)}$</p>
<p>$f^{(3)}(x) = \frac{6\sin^3(x)+5\cos^2(x)\sin(x)}{cos^4(x)}$</p>
<p>$\vdots$</p>
<p>$f^{(n)}(x) = \frac{ ?}{cos^{n+1}(x)}$</p>
<p>Some of these are easy: <a href="http://darkwing... | MarkV | 2,515 | <p>You might want to check out Faa di Bruno's formula: <a href="http://mathworld.wolfram.com/FaadiBrunosFormula.html" rel="nofollow">http://mathworld.wolfram.com/FaadiBrunosFormula.html</a> </p>
<p>In your case you have $f(x) = g(h(x))$ where $g(x) = 1/x$ and $h(x) = \cos(x)$. The answer is not simple unfortunately,... |
33,543 | <p>Let $M$ be a filtered module over a filtered algebra $A$, and suppose $gr(M)$ is flat over $gr(A)$, where $gr$ means the associated graded module and algebra, respectively.</p>
<p>What can one say in general about the flatness of $M$ over $A$, or with relevant assumptions (for instance in the above, we should assum... | Emerton | 2,874 | <p>Dear David,</p>
<p>See Prop. 1.2 of <a href="http://arxiv.org/abs/math/0206056" rel="nofollow">Algebras of p-adic distributions and admissible representations</a> by Peter Schneider and Jeremy Teitelbaum, for one such result:</p>
<p>Proposition 1.2. Suppose that $gr^{\bullet} R$ and $gr^{\bullet} A$ are left noeth... |
320,348 | <p>I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$</p>
<p>I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively things like $${2n+2\choose n+1}={2n+2\over n+1}{2n+1\choose n}=2\cdot {2n+1\over n+1}{2n\choose n},$$</p>
<p... | Bartek | 23,371 | <p>After Martin Sleziak and Marc van Leeuwen's comments, I've found <a href="https://math.stackexchange.com/a/219938/23371">this inductive proof</a> of Vandermonde's identity. (On this very site.)</p>
|
1,250,258 | <p>It's been 10 years since my last math class so I'm very rusty. How would I go about proving
$$3^n < n!$$
where $n \geq 7$?</p>
<p>I understand that factorials grow faster than set values with a variable exponent. Just not sure how to start proving it mathematically.</p>
| Belgi | 21,335 | <p>Since $2187=3^{7}<7!=5040$ then for $n>7$
$$
3^{n}=3^{7}\cdot3^{n-7}<7!\cdot8\cdot9\cdot\ldots\cdot n=n!
$$</p>
<p>since $8,9,\dots n>3$ </p>
|
122,770 | <p>Given that $A$ is an open set in $\mathbb R^n$ and $f:A \to \mathbb R^n$ is differentiable, and its derivative is non-singular at every point in $A$, prove that $f(A)$ is open in $\mathbb R^n$</p>
<p>Note $f$ is differentiable, <em>not</em> continuously differentiable. </p>
| azarel | 20,998 | <p>By the inverse function theorem for each $x\in A$ there exists open sets $x\in U$ and $f(x)\in V$ so that $f|_U:U\to V$ is a diffeomorphism. So in particular $f(U)=V$ hence $f(x)\in V\subset f(A)$. </p>
|
2,014,067 | <p>Please advise how to integrate this function. I think I need to use parts to do this, but I can't seem to get to the right answer. I know what the final solution is supposed to be, but I can't figure out where I'm going wrong in my effort to get there.</p>
<blockquote>
<p>$$\int \ln (1+2x)\:dx$$</p>
</blockquot... | parsiad | 64,601 | <p><strong>Hint</strong>: what happens when you substitute $u \equiv 1+2x$? </p>
|
2,014,067 | <p>Please advise how to integrate this function. I think I need to use parts to do this, but I can't seem to get to the right answer. I know what the final solution is supposed to be, but I can't figure out where I'm going wrong in my effort to get there.</p>
<blockquote>
<p>$$\int \ln (1+2x)\:dx$$</p>
</blockquot... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may integrate by parts as follows
$$
\int \ln(1+2x)\:dx=\frac{1+2x}2\:\ln(1+2x)-\int \frac{1+2x}2\cdot\frac{(1+2x)'}{(1+2x)}\:dx=\:?
$$</p>
|
223,162 | <p>I have a modulus function that looks like this $f(x) = 2x+3 \bmod b$, and i have to show that $x = y$ to prove that the function is $1-1.$</p>
<p>I know that mod functions can't be algebraically manipulated like regular function, so I was wondering if it was even possible or if I was just wasting my time scratching... | André Nicolas | 6,312 | <p>The number $b$ you are given must be <strong>odd</strong>. </p>
<p>We have $2x+3 \equiv 2y+3 \pmod {b}$ iff $(2x+3)-(2y+3)$ is divisible by $b$ iff $2(x-y)$ is divisible by $b$. </p>
<p>If $b$ is odd, this is true iff $x-y$ is divisible by $b$, that is, iff $x\equiv y \pmod{b}$.</p>
<p>Thus if $b$ is odd, our fu... |
223,162 | <p>I have a modulus function that looks like this $f(x) = 2x+3 \bmod b$, and i have to show that $x = y$ to prove that the function is $1-1.$</p>
<p>I know that mod functions can't be algebraically manipulated like regular function, so I was wondering if it was even possible or if I was just wasting my time scratching... | Bill Dubuque | 242 | <p>It is much more insightful to consider the following generalization.</p>
<p><strong>Theorem</strong> $\ $ The following are equivalent for integers $\rm\:c,\, m\,$.</p>
<p>$(1)\rm\ \ \ gcd(c,m) = 1$<br>
$(2)\rm\ \ \ c\:$ is invertible $\rm\,(mod\ m)$<br>
$(3)\rm\ \ \ x\to cx+d\:$ is $\:1$-$1\:$ $\rm\,(mod\ m)$<br>... |
2,055,878 | <p>If$$ x^2+x+1=0$$ find the value of $$8x^{282}+1799x^{183}+87x^{51}+124x^{-3}+1$$</p>
<p>Solving this equation gives imaginary solutions. </p>
<p>Is there an easy way to do this ?</p>
| DonAntonio | 31,254 | <p>Hint:</p>
<p>Observe that such a $\;x\;$ fufills $\;x^3=1\;$ ...</p>
|
1,410,586 | <blockquote>
<p>For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m - \log k| < \log n$. Find the sum of all possible values of the product $ mn$.</p>
</blockquote>
<p><strong>HINTS ONLY!</strong></p>
<p>Obviously, converting ... | user137794 | 137,794 | <p>Perhaps it will help to see</p>
<p>$$|\log m-\log k| < \log n \implies \log m-\log n < \log k < \log m+\log n \implies \log\frac{m}{n} < \log k < \log mn$$</p>
<p>Where $\log$ is monotonic.</p>
|
2,206,247 | <p><strong>Question:</strong> Consider the following non linear recurrence relation defined for $n \in \mathbb{N}$:</p>
<p>$$a_1=1, \ \ \ a_{n}=na_0+(n-1)a_1+(n-2)a_2+\cdots+2a_{n-2}+a_{n-1}$$</p>
<p>a) Calculate $a_1,a_2,a_3,a_4.$</p>
<p>b) Use induction to prove for all positive integers that:</p>
<p>$$a_n=\dfra... | Stefano | 387,021 | <p>Since $a$ and $b$ are non negative, this is equivalent to</p>
<p>$$\sqrt{x^2+y^2} \le |x|+|y|, $$</p>
<p>which has a nice geometric interpretation if you think of right triangles.</p>
|
64,643 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/4467/a1-2-is-either-an-integer-or-an-irrational-number">$a^{1/2}$ is either an integer or an irrational number</a> </p>
</blockquote>
<p>I know how to prove $\sqrt 2$ is an irrational number. Who can tell ... | Aryabhata | 1,102 | <p>A continued fraction proof of the irrationality of $x = \sqrt{3} - 1$, from which the irrationality of $\sqrt{3}$ follows. (A continued fraction proof of $\sqrt{2}$ can be found here: <a href="https://math.stackexchange.com/questions/5/how-can-you-prove-that-the-square-root-of-two-is-irrational/16526#16526">How can ... |
3,244,866 | <p>How can i prove that <span class="math-container">$$2^n\not \in O(n^2)$$</span> by formal definition and not using limits?</p>
<p>With:</p>
<p><a href="https://i.stack.imgur.com/id9gx.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/id9gx.png" alt="enter image description here"></a></p>
| auscrypt | 675,509 | <p>Like you noted, <span class="math-container">$n$</span> odd doesn't work.</p>
<p>For <span class="math-container">$n$</span> even, note that looking only at non-negative integers is sufficient. We have, for <span class="math-container">$n\ge 4$</span>,</p>
<p><span class="math-container">$$\left(n^3+\frac{n}{2}\ri... |
3,244,866 | <p>How can i prove that <span class="math-container">$$2^n\not \in O(n^2)$$</span> by formal definition and not using limits?</p>
<p>With:</p>
<p><a href="https://i.stack.imgur.com/id9gx.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/id9gx.png" alt="enter image description here"></a></p>
| Mike | 544,150 | <p>The only value that works is <span class="math-container">$n=2$</span>.</p>
<p>Indeed, if there is another value that works, then there exists integers <span class="math-container">$n > 2$</span> and <span class="math-container">$k>0$</span> such that</p>
<p><span class="math-container">$$n^6 + n^4 +1 = (n^3... |
695,648 | <p>How can I count the numbers of $5$ digits such that at least one of the digits appears more than one time? </p>
<p>My thoughts are:<br>
I count all the possible numbers of $5$ digits: $10^5 = 100000$. Then, I subtract the numbers that don't have repeated digits, which I calculate this way: $10*9*8*7*6$ $= 30240 $. ... | user76568 | 74,917 | <p>It is helpful to look at the negation of your requirement.<br>
No digit is ever repeated for $10 \cdot 9 \cdot 8 \cdot 7 \cdot 6=30240$ numbers.<br>
The total number of $5$ digit numbers is $10^5=100000$. </p>
<p>So the number of $5$ digits numbers with at least $1$ digit repeating more than once is:
$$10^5 - 10 \... |
695,648 | <p>How can I count the numbers of $5$ digits such that at least one of the digits appears more than one time? </p>
<p>My thoughts are:<br>
I count all the possible numbers of $5$ digits: $10^5 = 100000$. Then, I subtract the numbers that don't have repeated digits, which I calculate this way: $10*9*8*7*6$ $= 30240 $. ... | Community | -1 | <p><strong>Helpful question:</strong> In how many ways can we make or arrange five digits so that there is no repetition?</p>
<p>We know that there are 10 possible digits to choose from for the first digit, 0 - 10. However, we can't have the first digit be a 0, else we get a four-digit number. So, we're down to 9. For... |
4,519,106 | <p>After I learned about the existence of such a concept as a contrapositive, I always try to translate any statements into a contrapositive. And every time I fail. I haven't found a general technique for this yet. I think that if I know the statement and its contrapositive form, it will give me a better understandin... | Nitin Uniyal | 246,221 | <p><span class="math-container">$\sim(A\subset B)\equiv A\not\subset B$</span> rather than <span class="math-container">$B\subseteq A$</span>.</p>
|
79,869 | <p>Let <span class="math-container">$(X,\mu,\mathcal{F})$</span> be a probability space. The paper <em><a href="http://projecteuclid.org/euclid.aoms/1177693405" rel="nofollow noreferrer">Equiconvergence of Martingales</a></em> by Edward Boylan introduced a pseudometric on sub-<span class="math-container">$\sigma$</spa... | Bill Johnson | 2,554 | <p>Take a sequence $A_n$ of independent sets of measure $1/2$. Given two different subsets $B$ and $C$ of natural numbers, suppose WLOG that there is an $n$ in $B\sim C$. Now $\mu(A_n\Delta A) = 1/2$ for all sets $A$ which are independent of $A_n$, so the distance from the sigma algebra generated by $(A_n)_{n\in B}$ ... |
1,918,408 | <p>Let $C(B)$ be a $\infty$-order polynomial: $$ C(B) = \sum_{k=0}^\infty \alpha_k B^k$$</p>
<p>Show that $$C(B) = C(1) + (1-B)C^*(B)$$ where $C^*(B)$ is a another $\infty$-order polynomial.</p>
<p>This comes from the prove of the Engle-Granger Representation Theorem in their <a href="http://www.uta.edu/faculty/crowd... | ajotatxe | 132,456 | <p>When $\inf B<0$, $|\inf B|\ge \sup B$, $\sup A>0$ and $\sup A\ge |\inf A|$.</p>
|
1,807,479 | <blockquote>
<p>I recently took a test and was confused about a question. I feel that
the answer is B. Could anyone please elucidate it. Thanks!</p>
</blockquote>
<p>The point $(−4, 3)$ is on the terminal side of angle $\theta$ as sketched below. Find $\cos\theta$.</p>
<p><a href="https://i.stack.imgur.com/BiOiI.... | Bernard | 202,857 | <p>The correct answer is A. The hypotenuse in the following figure has length $5$ by Pythagoras:</p>
<p><a href="https://i.stack.imgur.com/xlJyj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xlJyj.png" alt="enter image description here"></a></p>
|
2,464,890 | <p>Here is link to some limit questions:</p>
<p><a href="https://i.stack.imgur.com/2rM9f.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2rM9f.png" alt="Example" /></a>
Can anyone explain how has answers were derived? In (a), how can we cancel out <span class="math-container">$(x-2)$</span>? And how ... | Bernard | 202,857 | <p>They simply use the following rule to simplify fractions when there's a common factor in the numerator and the denominator:
$$\frac{A\color{blue}\not\! C}{B\color{blue}\not\! C}=\frac AB,$$
and they determine the limit of the simplified fraction.</p>
|
108,010 | <p>It is not necessarily true that the closure of an open ball $B_{r}(x)$ is equal
to the closed ball of the same radius $r$ centered at the same point $x$. For a quick example, take $X$ to be any set and define a metric
$$
d(x,y)=
\begin{cases}
0\qquad&\text{if and only if $x=y$}\\
1&\text{otherwise}
\end{case... | Mathecm | 544,686 | <blockquote>
<p>More general, we can prove that for any normed space <span class="math-container">$(V,\|\cdot\|)$</span>, if <span class="math-container">$x_0\in V$</span> and <span class="math-container">$R>0$</span>, then <span class="math-container">$$\overline{B(x_0;R)}^{\|\cdot\|}=B[x_0;R],$$</span> where <sp... |
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