qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,243,952 | <p>I'm trying to prove the following identity:</p>
<blockquote>
<p>For each <span class="math-container">$$i \in \mathbb{N}, \sum_{k=1}^{i} \binom{n-k}{n-i} = \frac{n \cdot \binom{n-1}{n-i}}{n+1-i}$$</span></p>
</blockquote>
<p>My attempt: I've tried to prove that <span class="math-container">$(n+1-i) \cdot \sum_{k... | Explorer | 630,833 | <p><span class="math-container">\begin{align}
\sum_{k=1}^i\binom{n-k}{n-i}
&= \sum_{k=1}^i\frac{(n-k)!}{(n+1-i)!(i-k-1)!}\left(\frac{n+1-i}{i-k}\right)\\
&=\sum_{k=1}^i\frac{(n-k)!}{(n+1-i)!(i-k-1)!}\left(\frac{n+1-k}{i-k}-1\right)\\
&= \sum_{k=1}^i\binom{n+1-k}{n+1-i}-\binom{n-k}{n+1-i}\\
&=\binom{n}{n... |
3,243,952 | <p>I'm trying to prove the following identity:</p>
<blockquote>
<p>For each <span class="math-container">$$i \in \mathbb{N}, \sum_{k=1}^{i} \binom{n-k}{n-i} = \frac{n \cdot \binom{n-1}{n-i}}{n+1-i}$$</span></p>
</blockquote>
<p>My attempt: I've tried to prove that <span class="math-container">$(n+1-i) \cdot \sum_{k... | epi163sqrt | 132,007 | <p>Here is a combinatorial proof using lattice paths. At first we note the right hand side can be written as
<span class="math-container">\begin{align*}
\frac{n \cdot \binom{n-1}{n-i}}{n+1-i}=\binom{n}{n+1-i}
\end{align*}</span>
We therefore want to show
<span class="math-container">\begin{align*}
\binom{n}{n+1-i}&... |
726,541 | <p>Is the subspace of rational numbers in the usual space of real numbers compact?</p>
<p>I'm not exactly sure what this is asking. Is this asking if I can generate a cover using a finite amount of sets from the rational numbers to cover all of the real numbers? Apparently, the answer is yes, but I'm not sure I could ... | Henno Brandsma | 4,280 | <p>A subspace $A$ of $\mathbb{R}$ is compact, means that for <em>every</em> family of open subsets of $\mathbb{R}$ that covers $A$, we can find a <em>finite</em> subfamily that also covers $A$.</p>
<p>In $\mathbb{R}$ we know that $A$ compact implies that $A$ is bounded: we can take the family $(-n,n)$ for $n \in \math... |
151,188 | <p>Let $G$ be a $p$-adic linear reductive group. For an irreducible admissible smooth representation $\pi$ of $G$, there is a distribution $\Theta(\pi)$, called the character distribution, attached to $\pi$. It is given by integration against a locally integrable function which is locally constant on the regular set of... | Ulrik Buchholtz | 2,004 | <p>You may be interested in Feferman's <em>Unfolding Program</em>, which gives a predicative closure to any schematic theory such as PA or ZFC. The unfolding of PA has the same strength as ATR$_0$, and Feferman conjectured that the unfolding of ZFC would have the strength of ZFC plus a range of small large cardinals, c... |
4,072,215 | <blockquote>
<p>Give an <span class="math-container">$\delta-\epsilon$</span> proof that the function <span class="math-container">$f:\mathbb{R}^3\to\mathbb{R}$</span> definded by <span class="math-container">$$f(x,y,z)=x^2y+2xz^2$$</span> is continuous at <span class="math-container">$(1,1,1)$</span>.</p>
</blockquote... | vitamin d | 870,871 | <p>If <span class="math-container">$f=\Theta(g)$</span>, then by the limit definition</p>
<p><span class="math-container">$$0 < \liminf_{x \to \infty} \left|\frac{f(x)}{g(x)}\right| \le \limsup_{x \to \infty} \left|\frac{f(x)}{g(x)}\right|< \infty.$$</span></p>
<p>Note that if we choose <span class="math-containe... |
767,382 | <p>We are given a bag of marbles containing 6 blue marbles, 7 red marbles, and 5 yellow marbles. We select 4 marbles without replacement. How can we calculate the expected value of the number of blue, red and yellow marbles taken?</p>
<p>If we select 1 marble, then it becomes much easier: 6/18 blue marbles, 7/18 red m... | rlartiga | 93,314 | <p>Hint:</p>
<p>Be $X_1$ the result of the first selection which can be $B,R,Y$. Then:
$$P(X_1=B)=\frac{6}{18}$$
$$P(X_1=R)=\frac{7}{18}$$
$$P(X_1=Y)=\frac{5}{18}$$
Then :
$$E(Number of Blue Balls In The First Select)=P(X_1=B)\cdot 1+P(X_1=R)\cdot 0 + P(X_1=Y)\cdot 0=P(X_1=B)$$</p>
<p>Be $X_2$ the result of the secon... |
957,891 | <p>if sets are created using this rule $a \times 2^m$ where $a$ is an odd number e.g</p>
<p>$a=1 {1,2,4,8,16,32,64}$</p>
<p>$a=3 {3,6,12,24,48,96}$</p>
<p>$a=99 {99}$</p>
<p>hence there are $50$ such sets. Will every number from $1-100$ inclusive appear in exactly one of these sets? If so why?</p>
| Jasser | 170,011 | <p>A function cannot be continuous and discontinuous at the same point.</p>
<p>Yes the function is discontinuous which is right as per your argument.</p>
<p>I think the question wanted to convey this..</p>
<p>It has a jumped discontinuity which means if the function is assigned some value at the point of discontinui... |
3,935,057 | <p>Since A is countable and B is uncountable then this would be undefined as B is larger than A right? Or am I missing something here?</p>
| Community | -1 | <p><span class="math-container">$A \setminus B$</span> is defined even if <span class="math-container">$B$</span> is not a subset of <span class="math-container">$A$</span>. It's the set of everything that is in <span class="math-container">$A$</span> but not in <span class="math-container">$B$</span>.</p>
<p>Since <sp... |
18,931 | <blockquote>
<p>(Application of the rank-nullity theorem) Suppose $S,T: V\to V$ are linear transformations of a finite dimensional vector space $V$, and that the composition $ST\colon V\to V$ is invertible. Show that then $T$ is one-to-one, and deduce that it is invertible. Show also that $S$ is invertible. Deduce th... | Arturo Magidin | 742 | <p>In general, it's possible for a composition to be invertible and neither function to be invertible: take $f\colon\{a\}\to\{1,2\}$ given by $f(a)=1$, and $g\colon\{1,2\}\to\{x\}$ given by $g(1)=g(2)=x$. Neither $f$ nor $g$ are invertible ($f$ is not onto, $g$ is not one-to-one) but $gf\colon\{a\}\to\{x\}$ is clearly ... |
18,931 | <blockquote>
<p>(Application of the rank-nullity theorem) Suppose $S,T: V\to V$ are linear transformations of a finite dimensional vector space $V$, and that the composition $ST\colon V\to V$ is invertible. Show that then $T$ is one-to-one, and deduce that it is invertible. Show also that $S$ is invertible. Deduce th... | alpha.Debi | 20,863 | <p>You can pass to a matrix representation on the same basis of V as domain and codomain.
Then the matrix of the composition is the product of the two matices.
The product of two square matrices is invertible if and only if each of them is invertible.
After this you know that a linear transformation is invertible if an... |
1,746,213 | <p>I'm trying to prove that $$\lim_{x \rightarrow \infty} x^ne^{-x}=0$$</p>
<p>So far I have said:</p>
<p>Let $\epsilon>0.$ </p>
<p>Then $$|x^ne^{-x}-0|=|x^ne^{-x}| \leq |x^n...|$$</p>
<p>I'm not sure how to bound $e^{-x}$ though. How can I do this?</p>
| Tsemo Aristide | 280,301 | <p>Hint: For $x>0$ $x^ne^{-x}={{x^n}\over e^x}$= ${{x^n}\over{\sum_m{x^m\over m!}}}$ $\leq $ ${x^n\over{{x^{n+1}\over{(n+1)!}}}}$</p>
|
2,481,743 | <p>Squaring both sides of $\sqrt{x} - \sqrt{2-2x} = 1$ and rearranging
I arrive at the quadratic $9x^2 - 10x + 1 = 0$ which has solutions $x=1/9$ and $x=1$.
I don't understand why $x=1$ fits the original equation but $x=1/9$ doesn't (left hand side gives $-1$).</p>
| Community | -1 | <p>Let $x=u^2$, so that $\sqrt x=u$ and $u\ge0$.</p>
<p>Then
$$\sqrt{2-2u^2}=u-1$$</p>
<p>which implies $u\ge1$, which implies $2-2u^2\le0$.</p>
<p>We needn't go further, the only solution is $u=1$.</p>
|
1,169,314 | <p>We have that:
$$\begin{align}
\lim_{x \to 2} x^2 &= 4\\
D_{x = 2}\left(x^2\right) &= 4
\end{align}$$</p>
<p>In both cases the result is $4$. So limit and derivative are always the same?<br>
If not, what is the relation (if any exists)?</p>
| Eric Tressler | 26,785 | <p>$\lim_{x \to 3} x^2 = 9$ and $\left(\frac{d}{dx} x^2\right)_{x=3} = 6$, which are not the same. When you mention "limit", you are really talking about the value of the function $f(x) = x^2$; in general, this is <strong>not</strong> the same as the value of its derivative.</p>
<p>However, derivatives and limits are ... |
3,239,860 | <p>This question is taken from book: Advanced Calculus: An Introduction to Classical Analysis, by Louis Brand. The book is concerned with introductory real analysis.</p>
<p>I request to help find the solution. </p>
<blockquote>
<p>If <span class="math-container">$n$</span> is a positive integer, find all roots of t... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$1=r\cos t,\dfrac xn=r\sin t$</span></p>
<p>Using <a href="http://mathworld.wolfram.com/deMoivresIdentity.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/deMoivresIdentity.html</a></p>
<p><span class="math-container">$ r^n(\cos nt+i\sin nt)=r^n(\cos nt-i\sin nt)$</span... |
332,974 | <p>So I've searched around the web for a few hours now, as</p>
<p>(i)
$\mathbf A = \begin{pmatrix}2&1\\0&2\end{pmatrix}$</p>
<p>The characteristic polynomial is $(\lambda-2)^2=0$, so $\lambda=2$, repeated.
A bit of algebra convinces us that $b=(1,0)^t$ is an eigenvector.
Setting $(\mathbf A-\lambda \mathbf I)... | Amzoti | 38,839 | <p>$\mathbf A = \begin{pmatrix}1&2&0\\0&1&3/4\\0&0&1\end{pmatrix}$</p>
<p>You should get an Eigensystem as follows:</p>
<p>$$\lambda_1 = 1, v_1 = (1,0,0)$$</p>
<p>$$\lambda_2 = 1, v_2 = (0, 1/2, 0)$$</p>
<p>$$\lambda_3 = 1, v_3 = (0, 0, 2/3)$$</p>
<p>Next, we can write the <a href="http://e... |
272,025 | <p>Let $A(\sigma)$, $\sigma > 0$ be an operator that acts on bounded continuous functions $f$ on $\mathbb{R}$ by the rule
$$
(A(t)f)(x) = \int\limits_{\mathbb{R}} f(y)\frac{1}{\sqrt{2 \pi t}}\exp\left( -\frac{(x-y)^2}{2t} \right)dy,
$$
and let $(A(0)f)(x) = f(x)$. Then $(A(t)f)(x) \to f(x)$ when $t \to +0$. But i... | Ittay Weiss | 30,953 | <p>The answer is a mixture of yes and no. $\sigma$-algebras serve the purpose of defining the collection of sets you <em>want</em> to measure. In measure theory with AC one may naively wish to measure all subsets (say of $\mathbb R$) but that is impossible and thus one must consider a $\sigma$-algebra strictly smaller ... |
123,524 | <p>So today in my final for number theory I had to prove that the Fermat numbers ($F_n=2^{2^n}+1$) are coprime. </p>
<p>I know that the standard proof uses the following: $F_n=F_1\cdots F_{n-1}+2$ and then the $\gcd$ divides $2$, and it cannot be two, and hence the numbers are coprime.</p>
<p>However, he asked us to ... | lab bhattacharjee | 33,337 | <p>Let $A_r=a^{2^r}+1,$</p>
<p>$A_{n+t}=a^{2^{n+t}}+1=(a^{2^n})^{2^t}+1=(A_n-1)^{2^t}+1\equiv2\pmod{A_n}$ for integer $t\ge1$</p>
<p>$\implies(A_{n+t},A_n)=(2,A_n)$</p>
<p>If $a$ is even, $A_r$ will be odd if $r\ge1$</p>
<p>$\implies(2,A_n)=1$ for $n\ge1$</p>
|
703,151 | <p>Let $(X,\leq_X )$ and $(Y,\leq_Y)$ be two well-ordered sets, such that the discrete space of $X$ is the same as the order topology on $(X,\leq_X)$ and the same goes for $Y$. I'd like to prove that $(X,\leq_X)\cong (Y,\leq_Y) $. It seems correct, but I'm having trouble on where to start from.</p>
<p>Another thing is... | user642796 | 8,348 | <p>For the first part, as Henno Brandsma mentions in his comment, (up to order-isomorphism) the only well-ordered sets which are discrete under the order topology are the finite ones (<em>i.e.</em>, $0 < 1 < \cdots < n-1$ for some natural number $n$), and $\mathbb{N}$ itself. That each of these are discrete u... |
683,780 | <p>How do i find the area enclosed within the $3$ points when the plane intersects the $x, y$ and $z$ axis and </p>
<p>The plane is $ax+by+cz=1$</p>
<p>Need help on solving this as it has been bothering me for some time</p>
| Mark Bennet | 2,906 | <p>The points are $X=(\frac 1a,0,0), Y=(0,\frac 1b,0), Z=(0,0,\frac 1c)$</p>
<p>You can either compute the lengths of the sides of the triangle using pythagoras, and then the area using <a href="http://en.wikipedia.org/wiki/Heron%27s_formula" rel="nofollow">Heron's formula</a>. Or compute the magnitude of the vector p... |
3,195,385 | <p>On p. 39 of James and Liebeck's book "Representations and Characters of Groups" The term FG module is defined for F a field, G a group V a vectors space and an action of G on V is defined. </p>
<p>Where is the module in all this? My understanding is that a module is a vector space with multiplication by a ring (i... | sunfishstanford | 536,961 | <p>The answer by Alex Ortiz is really great and was very helpful for me, but I think it’s possible to more directly address the original question: why does a <span class="math-container">$FG$</span>-module (for example, as defined by James & Liebeck) qualify as a module?</p>
<p>The key to a module is to have the sc... |
3,101,176 | <p>Suppose you have to pay <span class="math-container">$15$</span> to roll a dice repeatedly until the first <span class="math-container">$6$</span> comes up. Then you will get paid as many dollars as the square of the no of rolls <span class="math-container">$X^2$</span>.</p>
<p>I have a equation as follows:
<span c... | jvdhooft | 437,988 | <p>Your calculation of <span class="math-container">$E(X^2)$</span> is absolutely correct, it indeed equals <span class="math-container">$66$</span>. You indeed need to be able to roll the die at least four times to make profit, which means that you need to not throw a <span class="math-container">$6$</span> three time... |
706,287 | <p>I programmed up a little algorithm that goes like this:</p>
<ol>
<li>Fix two positive, real numbers, call them $\alpha$ and $\beta$.</li>
<li>Generate a new, random, real number, $x \in [0,1]$ </li>
<li>Set $\alpha$ = $x\alpha\beta$,</li>
<li>Repeat 2 and 3 until $\alpha$ is equal to the smallest float supported (I... | ShreevatsaR | 205 | <p>For a fixed $\beta$, your algorithm can be stated as follows:</p>
<ul>
<li>Pick $\alpha_0$ arbitrarily.</li>
<li>For each $n \ge 1$, generate a uniformly random real number $X_n$ in $[0, 1]$ and set $\alpha_n = X_n\alpha_{n-1}\beta$.</li>
</ul>
<p>Thus each $\alpha_n$ is got by multiplying the previous $\alpha_{n-... |
2,582,863 | <p>Bounty ending tomorrow:</p>
<p>Consider the following function defined by two infinite integrals:
$$
F(\epsilon) = \int_0^\infty
\frac{q^3 (1+q)e^{-2q}}{q^4+\epsilon^4} \, \mathrm{d} q
+ \int_0^\infty \frac{\, G_q \, e^{-q}}{q^2(q^4+\epsilon^4)} \, \mathrm{d} q
$$
where
$$
G_q := G \left( \left[ \left[\frac{1}... | Community | -1 | <p>This is a nonlinear equation. This can however be converted into linear form using the substitution $a_n = b_n^2$. Now the equation becomes: $$a_{n+2} - 7a_{n+1} + 12a_n = (5n^2+3)4^n$$</p>
<p>Hope you can take it from here.</p>
<hr>
<p>Now, the solution to the homogenous part is: $a^{\{h\}} = c_1(3)^n + c_2(4)^n... |
775,220 | <p>I started a discussion with some friends trying to understand: if I roll 1 dice of 10(1d10) and 1 dice of 4(1d4) 20 times, and do it again 1000 times. Why the d4 have the probability to score the average(50) or more, more times than the d10???</p>
<p>I also wrote a java code to try to understand that take a look:</... | Graham Kemp | 135,106 | <blockquote>
<p>Random generator = new Random();</p>
</blockquote>
<p>This is your problem right here. A Rookie programming mistake. </p>
<p>Algorithmic random number generators are not truly random, they are really algorithms that generate a <em>fixed</em> but random-looking sequence of numbers.</p>
<p>Since yo... |
775,220 | <p>I started a discussion with some friends trying to understand: if I roll 1 dice of 10(1d10) and 1 dice of 4(1d4) 20 times, and do it again 1000 times. Why the d4 have the probability to score the average(50) or more, more times than the d10???</p>
<p>I also wrote a java code to try to understand that take a look:</... | user2357112 | 91,416 | <p>The probability that a number of dice rolls produces at least the average can be divided into two components:</p>
<ol>
<li>The probability that the total is exactly the average. Call this probability $X_4$ for the D4 rolls and $X_{10}$ for the D10s.</li>
<li>The probability that the total is more than the average. ... |
970,578 | <p>Solve the following equation: $\arctan x + \arctan (x^2-1) = \frac{3\pi}{4}$.</p>
<h2>What I did</h2>
<p>Let $\arctan x = \alpha, \arctan(x^2-1) = \beta$, $\qquad\alpha+\beta = \frac{3\pi}{4}$</p>
<p>$\tan(\alpha+\beta) = \tan(\frac{3\pi}{4}) = -1$</p>
<p>$$\frac{\tan\alpha + tan\beta}{1-\tan\alpha\tan\beta} = ... | lab bhattacharjee | 33,337 | <p>Liek <a href="https://math.stackexchange.com/questions/138310/show-that-2-tan-12-pi-cos-1-frac35/583359#583359">Show that $2\tan^{-1}(2) = \pi - \cos^{-1}(\frac{3}{5})$</a>, </p>
<p>$$\arctan x+\arctan(x^2-1)=\begin{cases} \arctan\left(\dfrac{x+x^2-1}{1-x(x^2-1)}\right) &\mbox{if } x(x^2-1)<1 \\\pi+ \arctan\... |
1,054,169 | <p>Find all real functions $f:\Bbb R\rightarrow\Bbb R$ so that $f(x^2)+f(2y^2)=[f(x+y)+f(y)][f(x-y)+f(y)]$, for all real numbers $x$ and $y$.</p>
<p>$f(x)=x^2$ is the only solution I think.</p>
<p>So far I have got:</p>
<p>$f(x^2)=[f(x)]^2+2f(x)f(0)+[f(0)]^2-f(0)$</p>
<p>$f(2y^2)=4[f(y)^2]-f(0)$</p>
<p>And by putt... | Neutral Element | 43,491 | <p>There are three non-symmetries between the LHS and RHS of your equation: $x\to-x$
, $y\to-y$, and $x^{2}\leftrightarrow2y^{2}$.</p>
<p>From $y\to-y$ after you subtract and make $x=0$ you get $f^{2}(y)=f^{2}(-y)$ for every y (see barto's comment).</p>
<p>From $x\to-x$ after you subtract and make $y=0$ you get $f(0... |
2,940,288 | <p>Suppose that <span class="math-container">$f:[0,1]\to \mathbb{R}$</span> is a continuous function and that <span class="math-container">$f(c)=\min\{f(x):0 \le x \le 1\}$</span> for some <span class="math-container">$c\in(0,1)$</span> . Prove that <span class="math-container">$f(x)=f(y)$</span> for some <span class=... | DeepSea | 101,504 | <p>"I don't understand what you saying – Inverse Problem". OK. I try to express his thought as best as I can here so you can understand his proof. To make it easier, put <span class="math-container">$p = f(c)$</span>, and assume <span class="math-container">$q=f(0) < f(1) = r$</span>. Since <span class="math-conta... |
204,384 | <p>Being far from an expert in the subject I was wondering if people can hint towards a modern exposition of the developments in the last 10 years ? Or if not then suggest some sub-subjects in CFT that are kind of central and popular right now ? Thanks!</p>
| Matthias Wendt | 50,846 | <p>I am sufficiently far away from the field that "I keep hearing this a lot" really means something - so probably higher-dimensional class field theory should be on the list of current trends in class field theory. The general goal is to express the abelianization of the étale fundamental group of a variety $X/\mathbb... |
1,187,894 | <p><img src="https://i.stack.imgur.com/LFtIh.jpg" alt="enter image description here"></p>
<p>I thought I knew how to solve this kind of problems, but I really don't know how to start...</p>
| Disintegrating By Parts | 112,478 | <p>Laplace transforms were derived in a very strange way by Oliver Heaviside, who is considered by many to be the Father of modern Electrical Engineering. He created 'operator' methods for solving ordinary differential equations. (The 'D' operator was Heaviside's notation, and the algebraic method was his, including us... |
3,402,072 | <p>I was watching a youtube video on epsilon-delta proofs when I came across this argument. We wanted to show <span class="math-container">$\lim_\limits{x \to 5} \sqrt{x+4} = 3$</span>. He started the proof of by :
<span class="math-container">$|\sqrt{x+4}-3|<\epsilon \implies -\epsilon <\sqrt{x+4}-3<\epsilo... | Ameet Sharma | 122,510 | <p>We can let</p>
<p><span class="math-container">$\epsilon_2 = min\{3,\epsilon\}$</span></p>
<p>Then if we complete our proof using <span class="math-container">$\epsilon_2$</span>, we know it works for <span class="math-container">$\epsilon$</span> since <span class="math-container">$\epsilon_2 \le \epsilon$</span>... |
132,134 | <p>The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen in <a href="http://en.wikipedia.org/wiki/Temperley-Lieb_algebra" rel="nofollow">the Wikipedia article</a>.</p>
<block... | Bruce Westbury | 3,992 | <p>First I don't think you have stated the defining relations correctly. [AK: you were quite right, and I've edited the question.]</p>
<p>It sounds as though you should start with the work of Richard Green.
The following papers seem relevant.</p>
<p><a href="http://arxiv.org/abs/q-alg/9712018">http://arxiv.org/abs/q-... |
1,172,576 | <p>I've seen that the <a href="http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation">Cesaro sum</a> is $1/2$ but haven't been able to find the steps for figuring that out.</p>
| Andrew D. Hwang | 86,418 | <p>Here's a fairly detailed sketch: The partial sums of the original series are $s_{2k-1} = 1$, $s_{2k} = 0$, for $k \geq 1$. The averaged partial sums of <em>this</em> sequence are
$$
t_{n} = \frac{1}{n} \sum_{k=1}^{n} s_{k}
= \begin{cases}
\frac{m}{2m-1} & \text{if $n = 2m-1$ is odd,} \\
\frac{m}{2m} = \fra... |
1,585,908 | <p>I am currently self-studying complex analysis in preparation for my course that will commence next year and I am stuck with this question as shown below:</p>
<p>In the complex plane, draw a picture of $$S=\{z \in \mathbb{C} : \vert z-1 \vert + \vert z+1 \vert = 2\}.$$</p>
<p>I was told that by squaring both sides,... | Théophile | 26,091 | <p>The trick is the identity $|z|^2 = z\bar z$. Squaring both sides of your equation,
$$|z-1|^2 + 2|z-1||z+1| + |z+1|^2 = 4.$$</p>
<p>Now rearrange this and use the identity above to get
$$(z-1)(\overline{z-1}) + (z+1)(\overline{z+1}) + 2|z^2-1| = 4.$$</p>
<p>But $\overline{z-1} = \bar z-1$, so $(z-1)(\overline{z-1})... |
2,663,192 | <p>Is it possible, based on a given turnover rate per year (in %), to calculate the average time (in years) an employee stays in an organization?</p>
<p>I have looked at the Coupon Collector's problem:</p>
<p><a href="https://en.wikipedia.org/wiki/Coupon_collector's_problem" rel="nofollow noreferrer">https://en.w... | Especially Lime | 341,019 | <p>I don't think this has anything to do with the coupon collector problem.</p>
<p>Say the company has $100$ employees. After $1$ year they have employed $100$ people for a total of $100$ person-years. After $2$ years they have employed $105$ people for a total of $200$ person years. After $n$ years they have employed... |
3,485,208 | <p>the reverse of the number <span class="math-container">$129$</span> is <span class="math-container">$921$</span> and these add to <span class="math-container">$1050$</span> which is divisible by <span class="math-container">$30$</span>. How many three-digit numbers have the property that, when added to their reverse... | Yly | 253,140 | <p>Write the number as <span class="math-container">$abc $</span> and use the divisibility rules for 10 and 3. From the rule for 10, we know that the last digit of <span class="math-container">$abc + cba$</span> is 0, hence <span class="math-container">$a+c=10$</span>. Thus the options are <span class="math-container... |
1,560,549 | <p>Approximate $3+ \displaystyle \sum_{x = 2}^{999}\dfrac{3(1000-x)}{1000+x}$. It may help to know that $\ln 2 = 0.69$. </p>
<p>I was thinking of doing the integral test to approximate this but I am unsure if this would work.</p>
| Ross Millikan | 1,827 | <p>Since the problem says "approximate" and does not give an acceptable error, I would just say $0$ and be done. You are probably expected to pull out the $3$ in the sum and say $$\sum_{i=2}^{999}\frac{1000-i}{1000+i}\approx \int_2^{1000}\frac{1000-i}{1000+i}di$$ Can you see why this is a reasonable approximation? Ca... |
637,017 | <p>$d_1(x,y)=d(x,y)+|f(x)-f(y)|$</p>
<p>Suppose $f(x)=1$ if $x\in\mathbb{Q}$ and 0 otherwise for a metric space = $\mathbb{R}$.</p>
<p>With this function I am shocked that $d_1$ is a metric. I do not doubt my proof though.</p>
<p>It's still very strange, for example consider $d_1(0,p)\le1$, what p satisfy this? p=1 ... | Rob Silversmith | 31,753 | <p>One way to visualize the space is this. Suppose you have two parallel roads, distance 1 mile apart. One road represents the rational numbers, the other represents the irrational numbers. You can cross between the two roads, but only perpendicular. (This stretches the road analogy, I'll have to think of something a b... |
19,096 | <p>I've got $8n^2 \lt 64n\log(n)$ and I need to find the $n$ range if $n\gt 0$ to satisfy the inequality.</p>
| Listing | 3,123 | <p>$n$ suggests that it might be a natural number.</p>
<p>Therefore you could use</p>
<p>$8 n^2 < 64 n log(n)$</p>
<p>$\Longleftrightarrow 0 < 8 log(n) - n$</p>
<p>You can easily see that $n$ grows much faster than $log(n)$. Therefore you can see that for $n > 26$ there are no integer solutions. Also the c... |
426,579 | <p>I was doing some work with diagonalization of a matrix $A$ in order to find a matrix $P$ such that $\,P^{-1}AP\,$ was diagonal. In order to that I set $\;\lambda I_{n}=0\;$ and found the characteristic polynomial and its roots.</p>
<p>When I factored my characteristic polynomial I obtained $\;\lambda^2(\lambda-2),\... | amWhy | 9,003 | <p>Absolutely yes: It is very possible $\lambda = 0$. Zero is allowed. </p>
<p>You may be mixing up what you know about eigenvectors --- the zero <strong><em>vector</em></strong> cannot be an <strong><em>eigenvector</em></strong>. </p>
<p>But for an <strong><em>eigenvalue</em></strong> $\lambda$, it is certainly po... |
264,887 | <p>I have a question about the integral of <span class="math-container">$\ln x$</span>.</p>
<p>When I try to calculate the integral of <span class="math-container">$\ln x$</span> from 0 to 1, I always get the following result.</p>
<ul>
<li><span class="math-container">$\int_0^1 \ln x = x(\ln x -1) |_0^1 = 1(\ln 1 -1)... | saz | 36,150 | <p>Consider the probability space $([0,1],\mathcal{B}[0,1],\lambda|_{[0,1]})$ (where $\mathcal{B}$ denotes the borel-$\sigma$-algebra, $\lambda|_{[0,1]}$ the lebesgue measure on $[0,1]$) and</p>
<p>$$X_n(\omega) := \frac{1}{n} \qquad (\omega \in [0,1])$$</p>
<p>Then we know that $X_n \to 0=:X$ almost surely and $$\ma... |
1,324,045 | <p>Here $(x_i) \in \ell^2$. All I have on paper right now is
$$\rVert Tx\rVert = \sqrt{\sum_{i=1}^{\infty} \frac{x_i ^2}{i^2}}$$</p>
<p>and I'm not sure about what's next. Please help. </p>
| Carsten S | 90,962 | <p>One: o. Two: oo. Both together: ooo. Those are three. (This is of course essentially the same answer as Voyska's, but I wanted to put it slightly less formally.)</p>
|
72,220 | <p>I am a graduate student thinking about some stuff.</p>
<p>1) If I have an idea for a result but it is not yet complete, is it advisable to mention it to others (especially more experienced people)? Is it possible that he/she will take it, develop it quicker than I do and publish it? How should one behave in these s... | Scott Carter | 36,108 | <p>A number of things can happen. I have heard stories of each of them.
(1) The senior person may have already solved this problem, but not written the solution down. Either she did not think it valuable, or she was thinking about using the solution for a more broad problem. In this case, she will keep her mouth shut.
... |
72,220 | <p>I am a graduate student thinking about some stuff.</p>
<p>1) If I have an idea for a result but it is not yet complete, is it advisable to mention it to others (especially more experienced people)? Is it possible that he/she will take it, develop it quicker than I do and publish it? How should one behave in these s... | Jim Conant | 9,417 | <p>I think the risks of a senior mathematician stealing your idea are pretty slight. The most likely scenario is that he or she will see something in the idea that you don't see, which might end up in some profitable joint research. Either that, or they'll be too busy with their own projects. I wouldn't worry too much.... |
56,688 | <p>Please help me to prove tha
$$\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$$ converges and its limit is $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{1}{n}$.</p>
| amine | 14,409 | <p>You can use the mean value theorem to prove first that
$$\frac{1}{n+1}<\ln(n+1)-\ln n<\frac{1}{n}$$
then, $\ln(n+1)<1+\ldots+\frac{1}{n}<1+\ln n$, and finally
$$\ln(n+1)-\ln n<\gamma_n<1$$
which implies the convergence of $\gamma_n$ and its limit $\gamma\in[0,1]$.</p>
<p>For the second part, I am ... |
56,688 | <p>Please help me to prove tha
$$\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$$ converges and its limit is $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{1}{n}$.</p>
| GEdgar | 442 | <p>Do we have a "proof without words" question here? I don't find it. There is a lovely diagram found in certain calculus texts that does this for us. Maybe I will animate it, for fun.</p>
|
1,031,337 | <p>What might be meant by $\left\{0,1,2\right\}^{\mathbb{Z}}$?</p>
<p>I do not know what this notation means-</p>
| Simon S | 21,495 | <p>In general $A^B$ is used to denote the set of functions $B \rightarrow A$.</p>
<p>Hence in your case, $$\{ 0, 1,2 \}^{\mathbb{Z}} = \{ f : \mathbb{Z} \rightarrow \{ 0, 1, 2 \} \}$$ This set can also be thought of as 'sequences' indexed by all integers (positive, zero and negative) of $0, 1$ and $2$.</p>
|
610,423 | <p>How can one construct a universal prefix-free Turing machine (TM)? By a universal prefix-free TM, I mean a prefix-free TM $U$ (that is, a TM whose domain is prefix-free) such that for every prefix-free partial recursive function $f$, there feasibly exists $\rho_f$ such that $U(\rho_fx) = f(x)$ for any $x$.</p>
<p>... | hmakholm left over Monica | 14,366 | <p>Given input $\rho x$, start by simulating $T_\rho$ on input $x$ until it halts, counting steps. (If it doesn't halt, then just diverge).</p>
<p>Let $n$ be the number of steps it takes for $T_\rho$ to halt on $x$.</p>
<p>Now enumerate all strings $y$ that are either prefixes of $x$ or have $x$ as a prefix and are a... |
1,490,480 | <p>Please help me with the following question: </p>
<p>Let $A$ be $m \times n$, $B$ be $n \times p$ matrices with $\text{rank}(A)=m$, $\text{rank}(B)=p$, where $p < m < n$.<br>
What are conditions such that $\text{rank}(AB)=\text{rank}(B)$???</p>
<p>Thanks!</p>
| Peter Franek | 62,009 | <p>No, $\begin{pmatrix} 1 & -2 & 0\\ 0 & -2 & 4\end{pmatrix}\begin{pmatrix} 4 & 0 & 0 \\ 0 & 2 & 0\\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}1\\ 1 \\ 1\end{pmatrix}=(0)$. </p>
<p>This is a simple counter-example for $(p,m,n)=(1,2,3)$.</p>
|
3,226,962 | <p>I know that if <span class="math-container">$f$</span> is a polynomial and <span class="math-container">$f(a)=0,f'(a)=0...,f^{(k)}(a)\neq 0$</span>, then <span class="math-container">$a$</span> is a root of multiplicity <span class="math-container">$k$</span>. Does this work for a differentiable function that is not... | Clayton | 43,239 | <p>Statements like this should be interpreted as: <span class="math-container">$f(x)$</span> has a double root (or triple root, etc.) at <span class="math-container">$x_0$</span> if <span class="math-container">$f(x)=(x-x_0)^2g(x)$</span> where <span class="math-container">$g(x_0)\neq0$</span> and <span class="math-con... |
2,978,752 | <p>I am wondering how to use the properties of conditional expectations to show that:</p>
<p><span class="math-container">$$
\mathbb{E}[\mathbb{E}[Y|Z,X,W]|X] = \mathbb{E}[\mathbb{E}[\mathbb{E}[Y|Z,X,W]|Z,X]|X]
$$</span></p>
<p>for random variables <span class="math-container">$Y,Z,X,W$</span>? It appears to me that ... | KingofHeadbutt | 474,648 | <p>If F is a subalgebra of G, then E[E[X|F]|G]=E[E[X|G]|F]=E[X|F].</p>
<p>First, E[E[X|F]|G]=E[X|F] because E[X|F] is G-measurable.</p>
<p>To prove that E[E[X|G]|F]=E[X|F], for all F-measurable event A, E[E[E[X|G]|F]1_A]=E[E[E[X|G]1_A|F]]=E[E[X|G]1_A]=E[E[X1_A|G]]=E[X1_A].</p>
<p>since E[E[X|G]|F] is F-measurable, i... |
3,519,833 | <p>I'm trying to find the value of <span class="math-container">$x$</span> in the following formula.</p>
<p><span class="math-container">$$\frac{1-(1-x)^n}{x} = y$$</span></p>
<p>How would the formula look isolating <span class="math-container">$x$</span>?</p>
<p>Thanks!</p>
| SlipEternal | 156,808 | <p>Let's try to turn this into a polynomial equation. By the Binomial Theorem:</p>
<p><span class="math-container">$$(1-x)^n = \sum_{k=0}^n\dbinom{n}{k}(-1)^kx^k$$</span></p>
<p>Simplifying:
<span class="math-container">$$\dfrac{1-(1-x)^n}{x} = \sum_{k=1}^n\dbinom{n}{k}(-1)^{k+1}x^{k-1}$$</span></p>
<p>Thus:
<span c... |
2,410,736 | <p>If we have the product of 3 primes,
A<em>B</em>s
two of them large: A, B
and one small: s
We don t know the product of A<em>B,
(Given that if we knew A</em>B we couldn t factorize it, because of the size and the computational complexity of the problem.)
can we find the small number s through factorization?
Is t... | bof | 111,012 | <p><a href="https://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras" rel="nofollow noreferrer">Stone's representation theorem</a> says that every Boolean algebra $B$ is isomorphic to the algebra of clopen sets of a certain compact Hausdorff space called its Stone space. To get the counterexa... |
3,395,874 | <p>In Sets and Groups by Green excesses 1 for Chapter 4 reads:</p>
<p>"1 Show that there are <span class="math-container">$n^{n^2}$</span> different binary operations on a set <span class="math-container">$A$</span> of order <span class="math-container">$n$</span>. Find all sixteen binary operations on <span class="ma... | IrbidMath | 255,977 | <p>Tensor properties </p>
<p><span class="math-container">$(m_1+m_2\otimes n)=(m_1\otimes n)+(m_2\otimes n)$</span> </p>
<p>Hence
<span class="math-container">$M\otimes 0=(M\otimes 0+0)=(M\otimes 0 )+ (M\otimes 0)$</span> cancel and we are done </p>
|
2,425 | <p>The limit is $$\lim_{x \to \infty} \left[ {x^{x+1} \over (x+1)^x} - { (x-1)^x\over x^{x-1}}\right]$$</p>
<p>Experimentally, this limit appears to converge to ${1 \over e}$, but I can't figure out how to solve it.</p>
| Bill Dubuque | 242 | <p>Making the change of variables <span class="math-container">$\; z = 1/x \;$</span> and collecting together like exponents yields</p>
<p><span class="math-container">$\quad\quad\quad\quad \displaystyle
\lim_{z\to 0^+} {\frac{(1+z)^{\large -1/z} - (1-z)^{\large 1/z}}{z}}\; = \;\lim_{z\to 0^+} {\frac{f(-z)-f(z)}{z}} $... |
2,425 | <p>The limit is $$\lim_{x \to \infty} \left[ {x^{x+1} \over (x+1)^x} - { (x-1)^x\over x^{x-1}}\right]$$</p>
<p>Experimentally, this limit appears to converge to ${1 \over e}$, but I can't figure out how to solve it.</p>
| Blue | 409 | <p><b>HINT</b> You can write the difference as $x \left( y^{-x} - z^x \right)=x\left(1-(yz)^x\right)/y^z$, where $y$ and $z$ are rational functions of $x$ such that the limiting value of $y^x$ (and $z^x$) is well-known, finite, and non-zero. Now focus on finding the limit of $x\left(1-(yz)^x\right)$.</p>
<p>(Restricti... |
1,311,858 | <p>I'm struggling with an induction problem here. </p>
<p>I have to prove that $2^{2^n}- 6$ (two to the power of two to the power of $n$ minus six) is divisible by $10$.</p>
<p>I already figured some steps and I also figured it's only divisible for n being an even number. No idea how to prove it though.</p>
<p>I got... | Lucian | 93,448 | <p><span class="math-container">$\displaystyle\int_L^Lf(x)~dx=0~$</span> if <span class="math-container">$~f(L)~$</span> is bounded, since it represents the area of a finite straight line </p>
<p>segment; i.e., <span class="math-container">$~\displaystyle\int_0^0\sin\bigg(\frac1x\bigg)~dx=0.~$</span> So, unless you ca... |
321,032 | <p>Take any convex set <span class="math-container">$A\subset\mathbb{R}^n$</span> which contains a neighborhood of the origin and let <span class="math-container">$f_A$</span> be the associated <em>Minkowski functional</em>
<span class="math-container">$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$</span>
which... | Piotr Hajlasz | 121,665 | <blockquote>
<p><em>Existence of minimizers should not be a severe issue...</em></p>
</blockquote>
<p>The proof of the existence follows form the Arzela-Ascoli theorem, but the proof is not entirely obvious. This is Proposition 1.1 in:</p>
<p><strong>E. Giusti</strong>, <em>Direct methods in the calculus of variati... |
4,114,479 | <p>I would like to make and <strong>visualize</strong> graph library of <span class="math-container">$n$</span> vertices that each vertex can only have maximum of four edges. The reason behind this is that the results will be used for designing chemical libraries where each vertex is a carbon and a carbon can only form... | gilleain | 26,904 | <p>For just alkanes, the suggestions in the comments are perfectly fine (nAUTy, for example). If you want more then I would suggest one of the programs on this list:</p>
<p><a href="https://fiehnlab.ucdavis.edu/projects/seven-golden-rules/molecular-isomer-generator" rel="nofollow noreferrer">https://fiehnlab.ucdavis.ed... |
4,114,479 | <p>I would like to make and <strong>visualize</strong> graph library of <span class="math-container">$n$</span> vertices that each vertex can only have maximum of four edges. The reason behind this is that the results will be used for designing chemical libraries where each vertex is a carbon and a carbon can only form... | IV_ | 292,527 | <p><a href="https://en.wikipedia.org/wiki/Chemical_graph_generator" rel="nofollow noreferrer">Wikipedia: Chemical graph generator</a></p>
<p><a href="https://en.wikipedia.org/wiki/List_of_straight-chain_alkanes" rel="nofollow noreferrer">Wikipedia: List of straight-chain alkanes</a></p>
<p>Computer algebra systems like... |
2,467,608 | <p>For a vector-valued function $f:c\in E\subseteq \Bbb R^n\to\Bbb R^m$, I want to prove that the linear transformation $T$ such that $\displaystyle\lim_{h\to 0}\frac{f(c+h)-f(c)-T(h)}{|h|}=0$ is unique.</p>
<p>My attempt:
Suppose $T_1,T_2:\Bbb R^n\to\Bbb R^m$ such that $\displaystyle\lim_{h\to 0}\frac{f(c+h)-f(c)-T_... | Randall | 464,495 | <p>Suppose $T$ and $S$ are two correct derivatives. Write $U = T-S$. Then
$$
|U\mathbf{h}| \leq |f(\mathbf{c}+\mathbf{h})-f(\mathbf{c})-T\mathbf{h}| + |f(\mathbf{c}+\mathbf{h})-f(\mathbf{c})-S\mathbf{h}|
$$
should convince you that $\lim_{\mathbf{h} \to \mathbf{0}} \frac{|U\mathbf{h}|}{|\mathbf{h}|} = 0$. Fixing $\... |
54,474 | <p>I have been doing some experiments on classes of random matrices, and it seems (visually) that the distribution of eigenvalue spacings is consistent with GOE or GUE or GSE. Unfortunately, to test this, I would like to (a) be able to generate GOE/GUE/GSE spacing variates (I suppose I COULD generate very large Gaussia... | Andrei MF | 7,482 | <p>I suggest you to take a look at the paper "How to Generate Random Matrices from the Classical
Compact Groups" by F. Mezzadri, Notices AMS 54 (5), 592-604 (2007). Can be downloaded freely from here:
<a href="http://www.google.es/url?sa=t&source=web&cd=1&ved=0CBcQFjAA&url=http%3A%2F%2Fwww.ams.org%2Fnot... |
868,643 | <p>How do you integrate the following by using Euler's formula, without using integration by parts? <span class="math-container">$$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}d\theta$$</span></p>
<p>I did integrate it by parts, by writing the <span class="math-container">$3$</span> in the numerat... | Timothy Shoaf | 164,613 | <p>By remembering that $e^{i\theta}=\cos(\theta) + i\sin(\theta)$ it is then easy to see that </p>
<p>$$\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}$$</p>
<p>and</p>
<p>$$\sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}$$</p>
<p>Perform your substitutions, and carry out your integration. Does that help?<... |
150,029 | <p>I want to show the below statement:</p>
<blockquote>
<p>$\lim_{n\rightarrow \infty} \dfrac{n+1}{2^{n\cdot n!}} = 0$</p>
</blockquote>
<p>I can see that it is true, because the part $2^{n\cdot n!}$ will be greater than $n+1$, when $n\rightarrow \infty$. However I cannot accept this argument, because it isn't form... | Community | -1 | <p>Note that your informal argument is wrong too. $(2n+1)$ is greater than $(n+1)$, but</p>
<p>$$ \lim_{n \to +\infty} \frac{n+1}{2n+1} = \frac{1}{2} \neq 0 $$</p>
<p>What you mean was not that $2^{n \cdot n!}$ is merely greater, but some superlative statement about how much greater it is.</p>
<p>Formally, you're tr... |
150,029 | <p>I want to show the below statement:</p>
<blockquote>
<p>$\lim_{n\rightarrow \infty} \dfrac{n+1}{2^{n\cdot n!}} = 0$</p>
</blockquote>
<p>I can see that it is true, because the part $2^{n\cdot n!}$ will be greater than $n+1$, when $n\rightarrow \infty$. However I cannot accept this argument, because it isn't form... | Alex Becker | 8,173 | <p>You want to show that for any $\epsilon>0$, there is some $N$ such that
$$n\geq N\implies \left|\frac{n+1}{2^{n\cdot n!}}\right|<\epsilon.$$
In order to do so, we can use the fact that the real numbers are <em>Archimendian</em>, meaning that for any $\epsilon>0$ we have some natural number $m$ such that $m... |
3,395,435 | <p>Say I have sets of <span class="math-container">$n$</span> points <span class="math-container">$(x_i,y_i) \in \mathbb{R}^{2}$</span>. In a typical regression analysis, we find the equation of a "best-fit" line <span class="math-container">$y=mx+b$</span>. By construction the system has no exact solution and so there... | Pebeto | 605,486 | <p>Your derivation is correct but you made a slight mistake. You have <span class="math-container">$$A^\top A =[n],$$</span> and therefore the inverse is just <span class="math-container">$[1/n],$</span> then</p>
<p><span class="math-container">$$\hat{x}=\left(A^\top A\right)^{-1} A^\top \pmatrix{y_1\cr y_2\cr \vdots\... |
4,150,014 | <p>How can I prove this statement?</p>
<p><span class="math-container">$$0<\frac{\ln(\frac{1}{\ln2})}{\ln2}<1$$</span></p>
<p>I have tried using the fact that <span class="math-container">$1=\frac{1}{\ln e}<\frac{1}{\ln 2}$</span>.</p>
<p>I have also tried writing it as <span class="math-container">$\log_2(\fr... | A-Level Student | 764,058 | <p>The statement is equivalent to showing that <span class="math-container">$$\ln 2>\frac{1}{2}\iff2\ln 2>1\iff\ln4>1.$$</span>
Since this is clearly true, since <span class="math-container">$2<e<3$</span>, we now know that <span class="math-container">$$\ln 2>\frac{1}{2}.$$</span>
This means that <sp... |
238,853 | <p>I want to get the difference between the following two regions, an annulus and a crown:
<a href="https://i.stack.imgur.com/Uz8bZ.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Uz8bZ.png" alt="enter image description here" /></a>
<a href="https://i.stack.imgur.com/fKODA.png" rel="noreferrer"><img src="http... | Tim Laska | 61,809 | <p>It looks like you have introduced non-manifold geometry into your model. I explain non-manifold in more detail in my answer <a href="https://mathematica.stackexchange.com/a/234105">here</a>. Even full-featured CAD packages will fail to create non-manifold geometry, as shown here:</p>
<p><a href="https://i.stack.imgu... |
930,121 | <p>Alice rolls a 12 sided die (the faces labeled 1 through 12) and Bob rolls a 20 sided die (the faces labeled 1 through 20). After seeing their roll (but not the other person's roll), each person can choose to roll again. Bob wins if his final roll is strictly greater than Alice's final roll. What is the probability t... | almagest | 172,006 | <p>HINTS</p>
<p>[more hints added, assuming downvote means my first attempt does not make it easy enough]</p>
<p>I am assuming that A's die is 1-12 and B's is 1-20 (not stated in the question as I write this).</p>
<p>An elementary point to get started. If your die has 12 sides, you have a half-chance of getting 1-6.... |
3,353,608 | <p>Prove by induction that:</p>
<p><span class="math-container">$64 \mid 3^{2n + 2} - 8n - 9$</span>, <span class="math-container">$n > 0$</span></p>
<p>I've tried to manipulate <span class="math-container">$P(k + 1)$</span> and haven't come up with anything like <span class="math-container">$P(k)$</span>.</p>
<p... | José Carlos Santos | 446,262 | <p>Note that<span class="math-container">\begin{align}3^{2(k+1)+2}-8(k+1)-9&=9\times3^{2k+2}-8k-9-8\\&=8(3^{2k+2}-1)+3^{2k+2}-8k-9.\end{align}</span>Now, you are assuming that <span class="math-container">$64\mid3^{2k+2}-8k-9$</span>. So, all that remains to be proved is that <span class="math-container">$64\mi... |
3,353,608 | <p>Prove by induction that:</p>
<p><span class="math-container">$64 \mid 3^{2n + 2} - 8n - 9$</span>, <span class="math-container">$n > 0$</span></p>
<p>I've tried to manipulate <span class="math-container">$P(k + 1)$</span> and haven't come up with anything like <span class="math-container">$P(k)$</span>.</p>
<p... | fleablood | 280,126 | <p>You know <span class="math-container">$64\mid3^{2n+2} -8n -9$</span>.</p>
<p>You <em>want</em> to show <span class="math-container">$64\mid3^{2(n+1) + 2} - 8(n+1) - 9$</span></p>
<p>So it suffices to show that <span class="math-container">$3^{2n + 4} - 8(n+1) - 9 = (3^{n+2} -8n -9) + 64M$</span> for some integer <... |
152,070 | <p>Help!</p>
<p>I am writing a package with a function which takes lots and <em>lots</em> of different options which are <code>String</code>s. When used, the front end autocompletion is suggesting them in <code>Symbol</code> form. If the resulting code is run, it is added to the <code>General`</code> context. Obser... | jkuczm | 14,303 | <p>You can use undocumented <code>"OptionNames"</code> property of <code>SyntaxInformation</code> to customize option names suggestions (and <a href="https://mathematica.stackexchange.com/a/98478/14303">coloring of invalid option names</a>).</p>
<pre><code>MyFunc // ClearAll
MyFunc // Options = {"MyOpt" -> True};
M... |
478,422 | <p>The question asks to prove that when 33 rooks are placed on an $8 \times 8$ chessboard that there are a total of 5 rooks that aren't attacking each other.</p>
<p>What I know:</p>
<ol>
<li>64 squares</li>
<li>Rooks attack in straight lines</li>
<li>at least 1 row must have more than 5 rooks</li>
<li>at least 1 colu... | KeithS | 23,565 | <p>I suppose the problem here is to prove the statement is true regardless of their arrangement.</p>
<p>Your points about the rules are correct, but I would first correct you that 32 pieces could be placed on a chessboard with only four on any given rank or file (by placing all pieces on the white squares or all the b... |
3,977,672 | <p>I have seen in many examples that using the fundamental definition of derivative only gives the correct results while checking for differentiability at a point. But why do we get wrong results while we directly apply the rules of differentiation and taking the limit at the point. Here is an example</p>
<p><span clas... | epi163sqrt | 132,007 | <p><strong>Hint:</strong> Note that in order that <span class="math-container">$y=y(x)$</span> is differentiable at <span class="math-container">$x=1$</span> it has to be continuous at <span class="math-container">$x=1$</span>. Here we have
<span class="math-container">\begin{align*}
\lim_{x\to 1^{-}}y(x)=\lim_{x\to 1^... |
18,771 | <p>Show that if a manifold $M$ is contractible, then every vector bundle over $M$ is equivalent to the trivial bundle.</p>
<p><em>Got this as homework but I'm kind of lost in the hole vector bundle subject, so if anyone could give some hints to this one as well as to understanding the subject, it would be highly appre... | Mariano Suárez-Álvarez | 274 | <p>Show that if $f,g:X\to Y$ are two maps between manifolds which are homotopic, and if $E$ is a vector bundle on $Y$, then $f^{-1}(E)$ and $g^{-1}(E)$ are isomorphic vector bundles on $X$. (You don't really need that both $X$ and $Y$ be manifolds: you can get by if they are just spaces and $X$ is paracompact...)</p>
... |
1,740,307 | <p>I know that $\mathbb{Q}[X,Y]$ is not a PID. Does this imply that the quotient ring $\mathbb{Q}[X,Y]/\langle X^2+Y^2-1 \rangle$ is not a PID?</p>
| user26857 | 121,097 | <p>No, it doesn't! </p>
<p>$R=\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is not a PID, but for other reasons. Actually it is not even a UFD for $x$ is irreducible, but not prime.</p>
|
2,657,804 | <p>I have a software test which identifies if there is a virus inserted into a computer. However, there is 5% probability of false alarm (there is a 5% chance that the test will be positive when the virus is not inserted).</p>
<p>I run the test and comes out positive.</p>
<p>How can I conclude about whether the syste... | Ethan Bolker | 72,858 | <p>@KevInLi 's answer is correct, and I've upvoted it. I want to add to his observation that you don't have enough information.</p>
<p>Intuitively speaking, if virus infection is (very) rare then (most) many positive test results will be false positives. If infection is common then they are more likely to be true posi... |
918,624 | <p>I am trying to refresh my memory with linear algebra. I am confused with these set of questions:</p>
<blockquote>
<ol>
<li><p>Show that $a=\begin{pmatrix}
1 \\
i \\
-1 \\
-i
\end{pmatrix},$ $b=\begin{pmatrix}
1 \\
-1 \\
1 \\
-1
\end{pmatrix}, $$\ c=\begin{pmatrix}
1 \\
-i \\
-1 \\... | voldemort | 118,052 | <p>You forgot a very important thing:</p>
<p>In $\mathbb{C^n}$ inner product between $(x_1,x_2,\cdots,x_n)$ and $(y_1,y_2,\cdots,y_n)$ is given by $\sum_{i=1}^{n}x_i\overline{y_i}$.</p>
<p>More generally, the inner product between two functions is $\int_{-\pi}^{\pi} f(x)\overline{g(x)} dx$.</p>
<p>Use these to get t... |
2,899,798 | <p>My work so far:</p>
<p>Letting $z = x+iy$, where $x,y$ are real,</p>
<p>then $\Im(z) = y > 0$</p>
<p>$y+1 > 0$</p>
<p>$1/(y+1) > 0$</p>
<p>But I am not sure how to proceed. I also attempted to multiply $-1/(z+1)$ by its conjugate, but that led to $(-x-1+iy)/((x+1)^2 + y^2)$, which I feel is the wrong w... | Robert Lewis | 67,071 | <p>$z = x + iy, \; x, y \in \Bbb R, \; y > 0; \tag 1$</p>
<p>$z + 1 = (1 + x) + iy; \tag 2$</p>
<p>$\dfrac{1}{z + 1} = \dfrac{1}{(1 + x) + iy}$
$= \dfrac{(1 + x) - iy}{(1 + x)^2 + y^2} = \dfrac{1 + x}{(1 + x)^2 + y^2} -i\dfrac{y}{(1 + x)^2 + y^2}; \tag 3$</p>
<p>$-\dfrac{1}{z + 1} = -\dfrac{1 + x}{(1 + x)^2 + y^2... |
3,480,368 | <p>Let <span class="math-container">$f$</span> be a weakly increasing step function. <span class="math-container">$g(x)=ax$</span> be a weakly increasing linear function.</p>
<p>Let <span class="math-container">$h=f+g$</span>. What is <span class="math-container">$h^{-1}$</span>?</p>
<p><span class="math-container">$... | Charles Hudgins | 647,305 | <p>It looks like what you've called <span class="math-container">$h^{-1}$</span> is a function such that <span class="math-container">$h^{-1} \circ h = 1$</span>. Sometimes, such functions are called retractions. </p>
<p>As the first comment on the OP pointed out, we shouldn't really use <span class="math-container">$... |
269,763 | <p>Well, I was not quite aware that addition of points is not defined in metric spaces but is defined only on linear spaces and others. </p>
<p>Could anyone elaborate why is this?
Is the addition of intervals (closed or open) defined?</p>
<p>I am confused as to how addition of points is undefined on metric space but... | Community | -1 | <p>In the definition of a vector space, there is the operation of vector addition where two vectors are added. In the definition of a metric space, there is no addition. Whether the addition of intervals is defined depends on whether we have defined such a thing.</p>
<p>The same set can become different metric spaces ... |
269,763 | <p>Well, I was not quite aware that addition of points is not defined in metric spaces but is defined only on linear spaces and others. </p>
<p>Could anyone elaborate why is this?
Is the addition of intervals (closed or open) defined?</p>
<p>I am confused as to how addition of points is undefined on metric space but... | Hagen von Eitzen | 39,174 | <p>In metric spaces, we have a notion of distance because by definition, a metric space is a space with a metric. But a priori there is no notion of addition. Simply because it does not appear as part of the concept.</p>
<p>In abelian groups, we have a notion of addition, again by the very definition of the concept of... |
33,821 | <p>The following differentiated implicitly with respect to $\theta$:</p>
<p>$3x = \tan \, \theta $</p>
<p>The book says </p>
<p>$3 dx = \sec^2 \theta \, d \theta$</p>
<p>One could start the calculation like this (I think):</p>
<p>$ \frac{d}{d \theta}3x = \frac{d}{d \theta} \tan \, \theta $</p>
<p>From there I'm n... | J126 | 2,838 | <p>This is differential notation. If $y=f(x)$, then it is customary to write $dy=f'(x)dx$. What this says is that a small change in $x$, $dx$, produces an approximate change on $y$ of $f'(x)dx$. In your instance you have $x=f(\theta)$ and so $dx=f'(\theta)d\theta$. The concept comes into play when you t... |
2,768,590 | <p>If f is uniformly differentialable on $(a,b)$ then $f'$ is continuous on $(a,b)$? </p>
<p>I just knew the definition </p>
<p>A function $f:(a,b)\to R$ is said to be uniformly differentialable iff f is differntiable on $(a,b)$, and for each $\epsilon>0$ there is $\delta>0$ such that $0<|x-y|<\delta$ an... | RRL | 148,510 | <p>Hint:</p>
<p>For $x < z < y$, $|x-y| < \delta \implies |x-z|, |y-z| < \delta$ and</p>
<p>$$f'(x) - f'(y) \\ = f'(x) - \frac{f(x) - f(z)}{x-z} + \frac{f(x) - f(z)}{x-z} - f'(z) + f'(z) - \frac{f(z) - f(y)}{z-y} + \frac{f(z) - f(y)}{z-y} - f'(y) $$</p>
|
4,405,311 | <p>I am trying to solve the below problem.</p>
<blockquote>
<p>Suppose that five <span class="math-container">$1$</span>s and four <span class="math-container">$0$</span>s are arranged around a circle. Form a new circle by placing a <span class="math-container">$0$</span> between any two unequal adjacent numbers and a ... | Jaap Scherphuis | 362,967 | <p>I think your argument is just fine - to get all zeros you will need the previous state to be fully alternating between 0 and 1, and that is not possible with an odd-sized circle.</p>
<p>Here is another proof. If you go around the circle once, counting the number of times it changes from 0 to 1 and also the number of... |
1,787,867 | <p>Prove that $\dfrac{2\cos x}{\cos 2x + 1 }= \sec x$. </p>
<p>So far I have:
$\dfrac{2\cos x}{\cos 2x + 1 }= \dfrac 1 {\cos x}$</p>
<p>Where do I go from here?</p>
| Michael Hardy | 11,667 | <p>It seems to be a widespread practice for students to begin attempted proofs of trigonometric identities by writing things like this:
$$\frac{2\cos x}{\cos 2x + 1 }= \sec x$$
$$\frac{2\cos x}{\cos 2x + 1 }= \frac 1 {\cos x}$$</p>
<p>That's ok in scratchwork, but the finished proof should go like this:
$$
\sec x = \f... |
151,134 | <p>I am having trouble with notations like 1 cm$^{-3}$, especially since I am converting them between compound units.</p>
<p>Is there a way to express 1 cmcm$^{-3}$ without writing the negative exponent?</p>
<p>The particular queston I was asked was:</p>
<blockquote>
<p>The EC standard for lead in the atmosphere i... | Will Orrick | 3,736 | <p>I believe the quoted passage contains an error.</p>
<blockquote>
<p>The EC standard for lead in the atmosphere is 2 µg/m$^{−3}$. Express this value in scientific notation in g/cm$^{−3}$, to an appropriate number of significant figures.</p>
</blockquote>
<p>It should either be 2 µg m$^{−3}$ or 2 µg/m$^3$, both of... |
2,896,287 | <p>I was looking at the area under $y={1\over x}$, and randomly I wondered if there was some point at which the areas on either side were equal.</p>
<p>I set about finding the answer using this method:
$$\\
\lim_{a\to0^+} {\int_a^p{dx\over x}}=\lim_{b\to\infty} {\int_p^b{dx\over x}}\\
\lim_{a\to0^+}\ln p - \ln a=\lim_... | Anastassis Kapetanakis | 342,024 | <p>We have that $\lim_{a\to0^+} {\int_a^p{dx\over x}}$ diverges and $\lim_{b\to\infty} {\int_p^b{dx\over x}}$ diverges too. So, since these two areas tend to $\infty$, you cannot compare those two. Hence you cannot find a point where these two areas are equal.</p>
<p>From the indeterminate that you found you cannot co... |
173,508 | <p>I am having an issue with the following complex analysis problem. I am suppose to
find the coefficients of $z^{-1}$, $z^{-2}$ and $z^{-3}$ in the Laurent series for
$\displaystyle \frac{1}{\sin z}$ around $z_0 = 0$ which is valid for $2\pi < |z| < 3\pi$.</p>
<p>One way I thought of doing it was to say
$$ ... | copper.hat | 27,978 | <p>Just grind away:</p>
<p>$((0+0)+(0+3)+(0+6)+(0+9))+((2+0)+(2+3)+(2+6)+(2+9))+((4+0)+(4+3)+(4+6)+(4+9)) = 78$</p>
|
173,508 | <p>I am having an issue with the following complex analysis problem. I am suppose to
find the coefficients of $z^{-1}$, $z^{-2}$ and $z^{-3}$ in the Laurent series for
$\displaystyle \frac{1}{\sin z}$ around $z_0 = 0$ which is valid for $2\pi < |z| < 3\pi$.</p>
<p>One way I thought of doing it was to say
$$ ... | Alex Moore-Niemi | 266,193 | <p>I hope maybe a related presentation familiar to programmers helps readers "see" the double summation better. With double summation, I imagine a 2d matrix with i=rows 0,1,2 and j = columns 0,1,2,3:</p>
<p><span class="math-container">\begin{matrix}
& j=0 & j=1 & j=2 & j=3 \\
i=0 & 0 &am... |
173,508 | <p>I am having an issue with the following complex analysis problem. I am suppose to
find the coefficients of $z^{-1}$, $z^{-2}$ and $z^{-3}$ in the Laurent series for
$\displaystyle \frac{1}{\sin z}$ around $z_0 = 0$ which is valid for $2\pi < |z| < 3\pi$.</p>
<p>One way I thought of doing it was to say
$$ ... | Myo Nyunt | 828,003 | <p>If in a double summation there involves a finite sum, then the order of the summation can be interchanged.</p>
<p>Indeed,</p>
<p><span class="math-container">$$∑_{i=0}^2\;∑_{j=0}^3 \;(2i+3j)=∑_{i=0}^2\;∑_{j=0}^3\;2i+∑_{i=0}^2\;∑_{j=0}^3\;3j$$</span>
<span class="math-container">$$=∑_{j=0}^3\;∑_{i=0}^2\;2i+∑_{i=0}^2\... |
28,878 | <p>I defined a function in Mathematica. It is a function of n, so f[n_]:=, but in the definition, I used a sum, $\sum_{k=0}^n$. So, the $k$ is just an index variable for the sum and no $k$ shows up in the final answer. As I was using this function I tried evaluating f[k-1] and got a weird answer, 0. I finally figur... | Mr.Wizard | 9,909 | <p>An alternative to <code>Module</code> or <code>Block</code> is to use <a href="http://reference.wolfram.com/mathematica/ref/character/FormalK.html" rel="nofollow">Formal Symbols</a>. This allows for the cleanest code.</p>
<p>One may still run into trouble depending on how you choose to use Formal symbols elsewhere... |
1,302,879 | <p>There is a question in our analysis book and I have lots of problem with it. It says that:</p>
<p>set</p>
<p>$$
f(x)= \begin{cases}
0 \quad & x\leq0\\
\sin(\frac{\pi}{x}) \quad & x>0
\end{cases}
$$</p>
<p>$$
g(x)=
\begin{cases}
0 \quad & x\leq0\\
1 \quad & x>0.
\end{cases}
$$</p>
<p>... | Breaking Bioinformatics | 218,401 | <p>For $f(x)$, use the Taylor expansion of sine about zero to express $$\sin \left( \frac{\pi}{x} \right)$$ as an infinite sum. From there, integrate across the sum to give a function for the antiderivative. </p>
<p>As for your expression for $G$, the problem is that $g$ is not defined at zero. Given that an antideriv... |
130,345 | <p>I am trying to implement this function and solve for parameters: <a href="https://i.stack.imgur.com/VLfdJ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VLfdJ.jpg" alt="enter image description here"></a></p>
<p>Now as many can see it is a maximum likelihood estimation:
I have this data: </p>
<... | cyrille.piatecki | 38,721 | <p>Here is an other approach but still with bad results --- I have looked some years ago to the estimation of this kind of sde and I think it is normal</p>
<pre><code> step := 0.01
tt := 120
data = RandomFunction[
OrnsteinUhlenbeckProcess[0, 4, 5, 13], {0, tt, step}][[2]][[1,
1]];
\[DoubleStruckCapitalL][μ_, η_,... |
2,496,658 | <p>Let <span class="math-container">$\mathcal{H}$</span> be a complex finite dimensional Hilbert space (<span class="math-container">$\dim\mathcal{H}=d$</span>). Let <span class="math-container">$A\in M_d(\mathbb{C})$</span>. </p>
<blockquote>
<p>How can I show without using spectral considerations that
<span clas... | daw | 136,544 | <p>Let me denote <span class="math-container">$\rho:=\lim \|A^n\|^{1/n}$</span>. Then <span class="math-container">$|\lambda|\le\rho$</span> for all eigenvalues. Moreover, there is an eigenvalue <span class="math-container">$\lambda$</span> of <span class="math-container">$A$</span> such that <span class="math-containe... |
85,024 | <p>There are two results famously associated with Cantor's celebrated diagonal argument. The first is the proof that the reals are uncountable.</p>
<p><img src="https://i.stack.imgur.com/v9G8r.png" alt="Cantor's Diagonal Argument"></p>
<p>This clearly illustrates the namesake of the diagonal argument in this case... | Asaf Karagila | 622 | <p>Diagonalization is a common method in mathematics. Essentially it means "write it in an infinite matrix and then walk along a coordinate line which approaches infinity on both axes".</p>
<p>The "Cantor diagonal argument" says write the numbers in a matrix, and take the $n$-th number's $n$-th digit, and change it.</... |
460,881 | <p>Can we simplify the following surface integral in some way, or say anything about the properties it must have?
$$
\oint_{\partial\mathbb{B}\left(\vec{0},R\right)} \left|\vec{U}\left(\vec{r}\right) \times \hat{r} \right|^2 \;ds\left(\vec{r}\right)
$$
$\vec{U}$ is a continuous vector field the value of which depends ... | achille hui | 59,379 | <p>For simplicity of discussion, let us assume your $\vec{J}$ has only one relevant length scale $R_s$, i.e.</p>
<p>$$\left|\frac{\partial \vec{J}(\vec{a},t)}{\partial \vec{a}}\right| \sim~ O\left(\frac{|\vec{J}(\vec{a},t)|}{R_s}\right)$$
If that is the case, then $\vec{J}$ has only two relevant time scales $T_s$ and ... |
3,232,582 | <p>I was wondering if I could get some help understanding the following fact in an academic paper. The setup is as follows: </p>
<p>An open subset of <span class="math-container">$\Omega \subset \mathbb R^k$</span> is an <em>open convex cone</em> if it is closed with respect to taking linear combinations of its elemen... | Tsemo Aristide | 280,301 | <p>Hint: the fact that <span class="math-container">$G_x(\Omega) $</span> preserves a bounded open <span class="math-container">$U$</span> set implies that the modulus of its elements is <span class="math-container">$1$</span> (to see this consider an open ball <span class="math-container">$B$</span> containing <span c... |
1,630,335 | <p>Show that the list $(1,2), (3,5)$ is a basis for $\mathbb R^2$.</p>
<p>In order to show it is a basis, I have show that the list is linearly independent and that the list spans $\mathbb R^2$. </p>
<p>So I understand how to show linear independence (you simply set up a system of equations). However, I am uncertain ... | Community | -1 | <p>There's an easier way to show that it is a basis of $R^2$. </p>
<p>There's a sentence that states that if you can show a a group of vectors are linearly independent, and the dimension of the group is equal to the dimension of the space, then that group is a basis for the space.</p>
|
255,807 | <p>With <code>x</code>, <code>y</code>, <code>z</code> all positive, how can one simplify the following expression:</p>
<pre><code> f[x_,y_,z_]=(Sqrt[1 + x - y -
Sqrt[(-1 + x + y)^2 + 4 z^2]] (-1 + x + y -
Sqrt[(-1 + x + y)^2 + 4 z^2]))/(2 Sqrt[2] z Sqrt[
2 - (2 (-1 + x + y))/(-1 + x + y + Sqrt[(-1 + x + y)^... | Alexei Boulbitch | 788 | <p>Try the following:</p>
<p>Step 1:</p>
<pre><code>red = Reduce[
1 + x - y - Sqrt[(-1 + x + y)^2 + 4 z^2] > 0 &&
2 - (2 (-1 + x + y))/(-1 + x + y + Sqrt[(-1 + x + y)^2 + 4 z^2]) >
0]
(* y < 1 && ((0 < x <=
1 - y && (-Sqrt[x - x y] < z < 0 ||
0 &l... |
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