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 |
|---|---|---|---|---|
3,516,776 | <p>I've been trying to solve this for limit comparison test with <span class="math-container">$a_n=a^\frac{1}{n}+a^{-\frac{1}{n}}-2 , b_n= \frac{1}{n}$</span>,
but
<span class="math-container">$\frac{a_n}{b_n}\rightarrow\ln{a}(a^{\frac{1}{x}}-a^{-\frac{1}{x}})\rightarrow 0$</span>.
Any help appreciated.</p>
| Pythagoras | 701,578 | <p>Assume <span class="math-container">$a>0$</span>. Let <span class="math-container">$x_n=\frac 1{2n}.$</span> Then <span class="math-container">$$\sum_{n=1}^\infty \left(a^{1/n}+a^{-1/n}-2\right)=\sum_{n=1}^\infty(a^{x_n}-a^{-x_n})^2$$</span> <span class="math-container">$$=\sum_{n=1}^\infty a^{2x_n}(1-a^{-2x_n})^... |
2,519,620 | <blockquote>
<p><strong>Question :</strong> Three balls are to be randomly selected without replacement from an urn containing $20$ balls numbered $1$ through $20$. If we bet that at least one of the balls that are drawn hasa number as large as or larger than $17$, what is the probability that we win the bet?</p>
</b... | drhab | 75,923 | <p>For $i=1,2,3$ let $E_i$ denote the event that the number of the $i$-th drawn ball is smaller than $17$. </p>
<p>Then you are looking for: $$1-P(E_1\cap E_2\cap E_3)=1-P(E_1)P(E_2\mid E_1)P(E_3\mid E_1\cap E_2)=1-\frac{16}{20}\frac{15}{19}\frac{14}{18}$$</p>
|
2,519,620 | <blockquote>
<p><strong>Question :</strong> Three balls are to be randomly selected without replacement from an urn containing $20$ balls numbered $1$ through $20$. If we bet that at least one of the balls that are drawn hasa number as large as or larger than $17$, what is the probability that we win the bet?</p>
</b... | Kyky | 423,726 | <p>We need to find the probability of not getting a ball above $17$ first. In the urn, there are $4$ balls that are equal or larger than $17$ ($17,18,19,20$). Since there are $20$ balls, there are $20-4=16$ number of balls that are below $17$ in the urn. That means there is a $\frac{16}{20}$ chance that the first ball ... |
527,576 | <blockquote>
<p>Three men (out of 7) and three women (out of 6) will be chosen to
serve on a 7 member committee. In how many ways can the committee be
formed?</p>
</blockquote>
<p>I did 7C3 to get 35 men.</p>
<p>Then i did 6C3 to get 20 women.</p>
<p>Then i decide to add up 20 + 35 and get 55 but it is suggest... | Bill Cook | 16,423 | <p>The choices of men and women are made independently. Independent $\Longleftrightarrow$ Multiply.</p>
<p>Why? A concrete example might help.</p>
<p>I want to choose 1 fruit from $\{apple,orange,banana\}$ and 1 drink from $\{water,tea\}$. Then I have $3 \times 2 = 6$ choices: $(apple,water)$, $(orange,water)$, $(ban... |
129,788 | <blockquote>
<p>Let be A and B two events from the same sample set. If $\space P(A)+P(B)=1$, can one say that they are opposite events?</p>
</blockquote>
<p>In my thought:</p>
<p>$\space P(A)+P(B)=1$</p>
<p>$\space P(A)=1-P(B)$</p>
<p>So they are opposite events. But my book says no! It says that is not necessary... | Yang | 28,639 | <p>If B is the complementary event of A, then they satisfy the property:</p>
<p>$P(A)=1-P(B)$</p>
<p>But the reverse (your claim) is not true. The key thing is to realize that events are not necessarily mutually exclusive (<em>disjoint</em>, $P(A\cap B)=\emptyset$). </p>
<p>If they happen to be disjoint, then your c... |
4,644,186 | <p>Let m be a positive integer.Find the values of <span class="math-container">$$\sum_{k=0}^n \frac{{n\choose k }}{k+1}$$</span>. Leave your answer in terms of n where appropriate.</p>
<p>Remark. There is an alternative method for computing the sums described here: make use of integration.</p>
<p>I can only list out th... | Z Ahmed | 671,540 | <p><span class="math-container">$$\sum_{k=0}^{n}{n \choose k} x^k= (1+x)^n$$</span>
<span class="math-container">$$\implies \int_{0}^{1}\sum_{k=0}^{n} {n \choose k} x^k dx=\int_{0}^{1} (1+x)^n dx.$$</span>
<span class="math-container">$$\implies \sum_{k=0}^{n} \frac{{n \choose k}}{k+1}=\frac{2^{n+1}-1}{n+1}.$$</span></... |
3,059,571 | <p><span class="math-container">$$\lim_{x\to \frac\pi2} \frac{(1-\tan(\frac x2))(1-\sin(x))}{(1+\tan(\frac x2))(\pi-2x)^3}$$</span></p>
<p>I only know of L'hopital method but that is very long. Is there a shorter method to solve this?</p>
| lab bhattacharjee | 33,337 | <p>Set <span class="math-container">$\pi-2x=4y$</span> to find <span class="math-container">$$\lim_{y\to0}\dfrac{\tan y(1-\cos2y)}{(2y)^3}=\lim{...}\left(\dfrac{\sin y}y\right)^3\dfrac1{4\cos y}=?$$</span></p>
|
2,699,942 | <p>I am confused about one thing during the lecture. </p>
<p>Let $x_n = n$ and $A_n = \{x_k | k \ge n\} = \{n, n+1, n+2, ...\}$.</p>
<p>Then, $\inf A_n = n $, and $\sup A_n = \infty$. </p>
<p>My lecturer also said that $\lim\inf x_n = \lim\inf A _n=\lim n$. </p>
<p>My thinking is that $\{x_n\}_{n=1}^{\infty}=\{1, ... | user284331 | 284,331 | <p>Because we define $\liminf\limits_{n\rightarrow\infty}x_{n}=\lim\limits_{n\rightarrow\infty}\left(\inf\limits_{k\geq n}x_{k}\right)=\lim\limits_{n\rightarrow\infty}\left(\inf A_n\right)$.</p>
<p>It is not defined as $\lim\limits_{n\rightarrow\infty}\inf\{x_{k}:k=1,2,...\}$. Note that $\inf\{x_{k}: k=1,2,...\}$ is a... |
21,156 | <p>The title says it all, is there a way to get in contact which users who consistently post answers without using <span class="math-container">$\LaTeX$</span>? I've come across a user who does that and (as I had some free time) edited about 10-15 of his posts, some of his answers were barely readable; on each post I l... | nbubis | 28,743 | <p>I know of some experienced users on Math.SE who regularly post excellent answers, but due to physical disabilities or technological inexpertise have a lot of difficulty typing in $\LaTeX$.</p>
<p>I'm not saying all users have an actual problem, but for the sake of those who do, it's worth giving them the benefit of... |
21,156 | <p>The title says it all, is there a way to get in contact which users who consistently post answers without using <span class="math-container">$\LaTeX$</span>? I've come across a user who does that and (as I had some free time) edited about 10-15 of his posts, some of his answers were barely readable; on each post I l... | ASCII Advocate | 260,903 | <p>ASCII is more universal than LaTeX. Where's the problem?</p>
|
1,720,053 | <p>The PDF describes the probability of a random variable to take on a given value:</p>
<p>$f(x)=P(X=x)$</p>
<p>My question is whether this value can become greater than $1$?</p>
<p>Quote from wikipedia:</p>
<p>"Unlike a probability, a probability density function can take on values greater than one; for example, t... | drhab | 75,923 | <p>Your conception of <a href="https://en.wikipedia.org/wiki/Probability_density_function" rel="noreferrer">probability density function</a> is wrong.</p>
<p>You are mixing it up with <a href="https://en.wikipedia.org/wiki/Probability_mass_function" rel="noreferrer">probability mass function</a>.</p>
<p>If <span class=... |
1,720,053 | <p>The PDF describes the probability of a random variable to take on a given value:</p>
<p>$f(x)=P(X=x)$</p>
<p>My question is whether this value can become greater than $1$?</p>
<p>Quote from wikipedia:</p>
<p>"Unlike a probability, a probability density function can take on values greater than one; for example, t... | Shikhar Srivastava | 1,131,158 | <p>Here's an intuition:</p>
<p>Probability Density exists in the continuous space. Probability Mass exists in the discrete space.</p>
<p>The PDF <span class="math-container">$f(x)$</span> is the derivative of the CDF <span class="math-container">$F(x)$</span>:
<span class="math-container">$$ f(x) = \frac{d(F(X))}{d(x)}... |
2,258,557 | <p>Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$ where D $\in R$</p>
<p>I understand that a vertical straight line can be defined by the equation $z+\bar z= D$ because suppose $z =x+yi$ then $\bar z = x-yi$ Thus, $z+\bar z = x+yi+x-yi=2x$ which is an arbitrary vertic... | Community | -1 | <p>Hack the equation.</p>
<p>Substitute:</p>
<p>$$
\begin{cases}
z = x + iy \\
z_0 = x_0 + iy_0
\end{cases}
$$</p>
<p>Do some algebraic manipulations and you'll obtain the equation of a line.</p>
<p>Maybe what confuses you is that neither $z_0$ nor $D$ have a clear geometric interpretation in terms of intercepts... |
59,495 | <p>Suppose $K$ is an $n$-dimensional $C^2$ convex body in $\mathbb{R}^{n+1}$. We choose two distinct directions $z_0, z_1\in\mathbb{S^{n}}$. If $P_1$ and $P_2$ are the corresponding hyperplanes($z_0\perp P_1$ and $z_1\perp P_2$) and $K'$ is the projection of $K$ on $P_1\cap P_2$, what is the $Vol(K')$?
We know the supp... | Dick Palais | 7,311 | <p>There is a paper here:</p>
<p><a href="http://www.math.poly.edu/~alvarez/pdfs/crofton.pdf">http://www.math.poly.edu/~alvarez/pdfs/crofton.pdf</a></p>
<p>that develops a theory of "Gelfand Transforms" which in a sense made precise there is a generalization of both the Radon Transform and the Cauchy-Crofton formula.... |
4,450,169 | <p>Here is a (seemingly) simple problem in group theory. Given a non-elementary finite nilpotent group <span class="math-container">$N$</span>, show there exist <span class="math-container">$p \neq q$</span> primes such that <span class="math-container">$N$</span> has a quotient <span class="math-container">$\Bbb Z_{pq... | ancient mathematician | 414,424 | <p>(i) A finite nilpotent group is the direct product of its Sylow-subgroups.</p>
<p>(ii) The quotient of a finite <span class="math-container">$p$</span>-group by its Frattini subgroup is elementary abelian.</p>
<p>(iii) If the quotient of a finite group by its Frattini subgroup (the set of non-generators) is cyclic t... |
2,255,617 | <p>I am trying to learn how to do proofs by contradiction. The proof is,</p>
<p>"Prove by Contradiction that there are no positive real roots of $x^6 + 2x^3 +4x + 5$"</p>
<p>I understand that now I am attempting to prove that there is a positive real root of this equation, so I am able to contradict myself within the... | Community | -1 | <p>No. $\mathbb{Z} \to \mathbb{F}_2$ and $\mathbb{Z} \to \mathbb{Q}$ are counterexamples.</p>
|
21,372 | <blockquote>
<p>Let $ y = \min \{ (x + 6), (4 – x) \}$, then find $y$.</p>
</blockquote>
<p>How to solve this problem?</p>
| Bill Dubuque | 242 | <p><strong>HINT</strong> $\ $ For any continuous functions $\rm\:f,g,\:$ the intermediate value theorem implies that $\rm\ f-g\ $ will have constant sign between its roots. So you need only partition $\mathbb R$ by these roots and then evaluate the function at any test point of each interval to determine the sign on th... |
3,433,492 | <p>I know that a function can admitted multiple series representation (according to Eugene Catalan), but I wonder if there is a proof for the fact that each analytic function has only one unique Taylor series representation. I know that Taylor series are defined by derivatives of increasing order. A function has one an... | Φίλ λιπ | 494,571 | <p>I think this simple proof is sufficient. I'm going to do it in two cases, but really the first case is a special case of the second.</p>
<p>Suppose a function <span class="math-container">$f(x)$</span> has two taylor series representations.</p>
<p><span class="math-container">$$f(x)=\sum a_n x^n$$</span></p>
<p><spa... |
396,794 | <p>Let's say that a (right) module <span class="math-container">$M$</span> is <em>well complemented</em> if every non-zero submodule of <span class="math-container">$M$</span> has an indecomposable direct summand (by the way, is there a better or more standard name for this property?). For instance, every module of fin... | Luc Guyot | 84,349 | <p><strong>No</strong>, a <a href="https://en.wikipedia.org/wiki/Free_Boolean_algebra" rel="nofollow noreferrer">free Boolean algebra</a> <span class="math-container">$R$</span> on an infinite cardinal <span class="math-container">$\kappa$</span> (e.g., if <span class="math-container">$\kappa = \aleph_0$</span>, <span ... |
945,395 | <p>Let $a_1$ be real, and define $$a_{n+1}=\frac{2a_n^3}{1+a_n^4}$$ How can I prove that this $\{a_n\}$ to have limit. </p>
<p>I find it is hard to track. What I can do is just when $a_1=1$ then $a_n=1$; when $a_1=-1$, then $a_n=-1$; when $|a_1|<1$, $a_n\to 0$. When $|a_1|>1$, I have not find any idea.</p>
| Christian Blatter | 1,303 | <p>We are given the recursion
$$a_0:=a>0,\qquad a_{n+1}:=f(a_n)\quad(n\geq0)$$
with
$$f(x):={2x^3\over 1+x^4}\ .$$
The following figure shows the graph of $f$ for $x\geq0$:</p>
<p><img src="https://i.stack.imgur.com/8lDnT.jpg" alt="enter image description here"></p>
<p>Since
$$f(x)\geq0,\qquad x-f(x)={x(1-x^2)^2\... |
120,687 | <p>Consider the following code</p>
<pre><code>styles = {Red, Blue, {Red, Dashed}, {Blue, Dashed}}
pt1 = Plot[{x^2, 2 x^2, 1/x^2, 2/x^2}, {x, 0, 3}, Frame -> True,
PlotStyle -> styles, PlotLegends -> {"1", "2", "1", "2"}]
</code></pre>
<p>I would like the two red lines to carry the same label "1" and the two... | Michael E2 | 4,999 | <p>First, <code>NIntegrate[f1[x], {x, xmin, xmax}]</code> usually proceeds by constructing an <code>Experimental`NumericalFunction</code> from the expression for <code>f1[x]</code>. This will circumvent an attempt to memoize <code>f1</code> in the OP's manner, <code>f1[x_] := f1[x] =...</code>. One can prevent this b... |
12,927 | <p>The problem:</p>
<p><strong><em>Three poles standing at the points $A$, $B$ and $C$ subtend angles $\alpha$, $\beta$ and $\gamma$ respectively, at the circumcenter of $\Delta ABC$.If the heights of these poles are in arithmetic progression; then show that $\cot \alpha$, $\cot \beta$ and $\cot \gamma $ are in harmon... | Ross Millikan | 1,827 | <p>I would read it that the poles have a diameter>0. The pole at A has diameter so the angle seen from the circumcenter is $\alpha$. See if that works.</p>
|
2,884,785 | <p>Find all integral pairs (x,y) such that - $$( xy - 1)^2 = (x +1)^2 + ( y+1)^2$$</p>
<hr>
<p><strong>My Approach :</strong></p>
<p>I just expanded this equation and wrote it in another form - $$\frac{(xy+1)(xy-1)}{(x+y)}-2=x+y$$ and from this we can say that $(x+y)|(xy+1) \ \mathrm{or}\ (x+y)|(xy-1) $ . But i don'... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>$$x^2(y^2-1)-2x(1+y)-(y+1)^2=0$$</p>
<p>What if $y+1=0?$</p>
<p>Else $$x^2(y-1)-2x-(y+1)=0$$</p>
<p>Method$\#1:$</p>
<p>$$x=\dfrac{2\pm\sqrt{4+4(y^2-1)}}{2(y-1)}=\dfrac{1\pm y}{1+y}$$</p>
<p>Now $\dfrac{1-y}{1+y}=\dfrac2{1+y}-1$</p>
<p>$\implies1+y$ must divide $2$</p>
<p>Method $\#2:$</p>
<p>B... |
1,141,074 | <p>I need help with this integral: $$\int\frac{\sqrt{\tan x}}{\cos^2x}dx$$ I tried substitution and other methods, but all have lead me to this expression: $$2\int\sqrt{\tan x}(1+\tan^2 x)dx$$ where I can't calculate anything... Any suggestions? Thanks!</p>
| Emilio Novati | 187,568 | <p>$$
\int \dfrac{\sqrt{\tan x}}{\cos^2 x} dx =
\int \sqrt{\tan x} \; d(\tan x) = \dfrac{2}{3}\sqrt{(\tan x)^3} +C
$$</p>
|
62,790 | <p>Among several possible definitions of ordered pairs - see below - I find Kuratowski's the least compelling: its membership graph (2) has one node more than necessary (compared to (1)), it is not as "symmetric" as possible (compared to (3) and (4)), and it is not as "intuitive" as (4) - which captures the intuition, ... | Andreas Blass | 6,794 | <p>Kuratowski's definition arose naturally out of Kuratowski's idea for representing any linear order of a set $S$ in terms of just sets, not ordered pairs. The idea was that a linear ordering of $S$ can be represented by the set of initial segments of $S$. Here "initial segment" means a nonempty subset of $S$ closed... |
1,144,141 | <p>I have a question for my exam and I find it hard to understand.</p>
<p>I have to prove that the following formula is logically valid:</p>
<p><img src="https://i.stack.imgur.com/kdELq.jpg" alt="Example"></p>
<p>The professor told me to "push" all the symbols inside the brackets, and use the deduction theorem.</p>
... | Hagen von Eitzen | 39,174 | <p>Assume $\forall yp(y)$. Then $p(x)\to\forall yp(y)$ is true for any $x$.
If on th eother hand $\neg \forall yp(y)$, then $\exists y\neg p(y)$. Let $x$ be such an $y$ then again $p(x)\to\forall yp(y)$ is true, this time because the antecedent is false.</p>
|
3,772,923 | <p>My child's teacher raised a quesion in class for students who are interested to prove. The teacher says that the volume of a cube is the greatest among rectangular-faced shapes of the same perimeter and asks his students to prove this proposition.</p>
<p>I considered the relationship between the length of the sides ... | mjw | 655,367 | <p>If you mean by "perimeter" the sum of the edges, then yes, the cube is the maximal rectangular parallelepiped among those with the same "perimeter".</p>
<p>Let the edges have lengths <span class="math-container">$(a,b,c)$</span>.</p>
<p>Then the volume is <span class="math-container">$V=abc$</spa... |
295,545 | <p>The following figure depicts the paths from home to work. SAM never travels through the park when going to work.</p>
<p><img src="https://i.stack.imgur.com/IANqM.png" alt="enter image description here"></p>
| Mårten W | 58,780 | <p>What the author meant was that you want two vectors satisfying the equation of the plane (that is, they should lie in the plane). Since the equation of the plane is $x-y-z=0$, one can choose two of the coordinates, and then solve for the third.</p>
<p>In your case $a$ is obtained by choosing $x=0$ and $y=1$, and pl... |
7,981 | <p>I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?</p>
| vonjd | 346 | <p>In very layman's terms it states that there is some order in the distribution of the primes (which seem to occur totally chaotic at first sight). Or to say it like Shakespeare: "Though this be madness, yet there is method in 't."</p>
<p>If you want to know more there is a new trilogy about that topic where the firs... |
7,981 | <p>I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?</p>
| Community | -1 | <p>Here is a very simply description of the Riemann Hypothesis that requires nothing more than a 3rd grade education to understand:</p>
<p><a href="http://www.jstor.org/pss/2323497" rel="nofollow">http://www.jstor.org/pss/2323497</a></p>
<p>There is also a beautiful proof linking the Farey sequence of fractions to th... |
2,107,854 | <p>What is the limit when $n \to \infty$?</p>
<p>$$\lim_{n \to \infty} \frac{1}{n^4} \sum_{J=0}^{2n-1} J^3=?$$</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
9,437 | <p>I love the way that Mathematica allows me to type in of formulas. It is really easy to type complicated expressions with shortcuts on the keyboard. It would be great if I could use Mathematica completely to publish my articles. The biggest reason I don't already do this is:</p>
<p>I can't find the proper tutorial f... | NCSNY | 19,431 | <p>It is easier to create a notebook object and set options. For example, if I want a PDF with some plots and an equation and save it as a PDF in Landscape form (so its not cut off):</p>
<pre><code>newbook = CreateDocument[{plot1, plot2, plot3, equations1}]
SetOptions[newbook,
PrintingOptions -> {"PaperOrie... |
2,508,011 | <blockquote>
<p>find the <span class="math-container">$x$</span> :</p>
<p><span class="math-container">$$x^2(x-1)^2+x^2=8(x-1)^2$$</span></p>
</blockquote>
<hr />
<p>My Try :</p>
<p><span class="math-container">$$x^2(x-1)^2+x^2=8(x-1)^2\\ x^2(x^2-2x+1)+x^2=8(x^2-2x+1)\\x^4-2x^3+x^2+x^2=8x^2-16x+8\\x^4-2x^3-6x^2+16x-8=0... | C. Dubussy | 310,801 | <p>Hint : Try the divisors of $8$.</p>
|
2,508,011 | <blockquote>
<p>find the <span class="math-container">$x$</span> :</p>
<p><span class="math-container">$$x^2(x-1)^2+x^2=8(x-1)^2$$</span></p>
</blockquote>
<hr />
<p>My Try :</p>
<p><span class="math-container">$$x^2(x-1)^2+x^2=8(x-1)^2\\ x^2(x^2-2x+1)+x^2=8(x^2-2x+1)\\x^4-2x^3+x^2+x^2=8x^2-16x+8\\x^4-2x^3-6x^2+16x-8=0... | thesmallprint | 438,651 | <p>So it can be checked that your polynomial $$x^4-2x^3-6x^2+16x-8=0$$ has a factor of $(x-2)$. We check that this is so by the <a href="https://en.wikipedia.org/wiki/Factor_theorem" rel="nofollow noreferrer">factor theorem</a>. Then (and it should be checked by you too) that by carrying out the appropriate algebraic d... |
3,615,117 | <p>I want to find the intersection of the sphere <span class="math-container">$x^2+y^2+z^2 = 1$</span> and the plane <span class="math-container">$x+y+z=0$</span>. </p>
<p><span class="math-container">$z=-(x+y)$</span> that gives <span class="math-container">$x^2+y^2+xy= \frac 12$</span></p>
<p>How do I represent thi... | Ninad Munshi | 698,724 | <p>Let <span class="math-container">$\sqrt{2}x = u+v$</span> and <span class="math-container">$\sqrt{2}y=u-v$</span>. Then the resulting expression is</p>
<p><span class="math-container">$$3u^2 + v^2 = 1$$</span></p>
<p>which is the standard form of an ellipse and since the transformation is a pure rotation, the shap... |
2,274,736 | <p>I am finding particular subgroups of $Q_{12}$ and had a couple of questions about it.</p>
<p>$Q_{12}=\langle a,b:a^6=1,b^2=a^3,ba=a^{-1}b\rangle$</p>
<p>Firstly here is part of a solution I came across: </p>
<p>The first step is to establish the orders of the elements. So $1$ has order 1, $a^3$ has order 2, $a^2$... | caverac | 384,830 | <p>Recall that for a multinomial distribution the PDF is</p>
<p>$$
f(x_1, \ldots, x_k; n, \pi_1, \ldots, \pi_k) = \frac{n!}{x_1!\cdots x_k!}\pi^{x_1} \cdots \pi^{x_k} = {{n} \choose {x_1,\ldots,x_k}}\pi^{x_1} \cdots \pi^{x_k}
$$</p>
<p>with</p>
<p>$$
\sum_{i= 1}^k x_i = n
$$</p>
<p>In your case $k = 3$. The likelih... |
853,659 | <p>Evaluate the integral:</p>
<p>$$\int \frac{x^6}{x^4-1} \, \mathrm{d}x$$</p>
<p>After a lot of help I have reached this point:</p>
<p>$x^2 = Ax^3 - Ax + Bx^2 - B + Cx^3 + Cx^2 + Cx + C + Dx^3 - Dx^2 + Dx - D$</p>
<p>But now I don't really know how to solve for $A, B, C$, and $D$. Please help!</p>
| M. Strochyk | 40,362 | <p>Integrand can be transformed without long divison
$$\begin{gathered}\frac{x^6}{x^4 - 1}=\frac{x^6-1+1}{x^4 - 1}=\frac{(x^2-1)(x^4+x^2+1)}{x^4 - 1}+\frac{1}{x^4 - 1}=\\
=\frac{x^4+x^2+1}{x^2 + 1} + \frac{1}{x^4 - 1}=\\
=x^2+\frac{1}{x^2 + 1}+\frac{1}{2}\left(\frac{1}{x^2-1} - \frac{1}{x^2+1} \right)=\\
=x^2 + \frac{1... |
1,284,938 | <p>I was revising for one of my end of year maths exams, then I came across this example on how to find lines of tangents to ellipses outside the curve. Personally, I'd use differentiation and slopes to find such lines, but the lecturer does something simpler and more elegant.</p>
<p>The question is: "Find the equatio... | mickep | 97,236 | <p>If I understand your question correct, rewrite it as
$$
y'=2e^x-y,
$$
which is first order and linear,
$$
y'+y=2e^x,\quad\text{so}\quad D(e^x y)=2e^{2x},\quad\text{and hence}\quad e^xy=e^{2x}+C.
$$</p>
|
4,312,323 | <p>I get why <span class="math-container">$\sqrt{9} = \pm 3$</span>. But (at least I think) the ± is there because there's a certain ambiguity as to which number was squared to obtain <span class="math-container">$9$</span>.</p>
<p>Does that mean that if we remove the ambiguity <span class="math-container">$\sqrt{3^2} ... | 欲しい未来 | 772,122 | <p>When we have <span class="math-container">$\sqrt{x}$</span> it is usually assumed to be the <em><strong>principal square root operator</strong></em>, which means it returns only the positive root. We have
<span class="math-container">$$\sqrt{\cdot}: \mathbb R_{\geq 0} \to \mathbb R_{\geq 0}\\
x \mapsto \sqrt{x}$$</... |
23,911 | <p>I am teaching a course on Riemann Surfaces next term, and would <strong>like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties</strong> (over, say, $\mathbb{C}$). I am looking for examples that would be meaningful to 2nd year US g... | Henri | 5,659 | <p>I think there's some big difference concerning the metric approach too.</p>
<p>In fact, the Gram-Schmidt process (which is real analytic) enables us -in real differential geometry- to find some local orthonormal frames (for any hermitian bundle, and in particular for the tangent bundle), whereas in the holomorphic ... |
23,911 | <p>I am teaching a course on Riemann Surfaces next term, and would <strong>like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties</strong> (over, say, $\mathbb{C}$). I am looking for examples that would be meaningful to 2nd year US g... | Heinrich Hartmann | 5,714 | <p>For a closed analytic subset Z ⊂ S of a (say compact) complex manifold
with complement U=S-Z one has additivity of the (topological) Euler characteristic:</p>
<p>Χ(S)=Χ(Z)+Χ(U).</p>
<p>This is wrong for if S and Z are topological spaces or smooth manifolds.
Indeed, take for Z a point on a circle S... |
1,424,273 | <p>Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?</p>
<p>My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all $n\ge N$. And we know $0< a_n$ for all $n\in \mathbb{N}$, particularly $0<a_n$ for all $n\ge N$. Maybe by c... | Math1000 | 38,584 | <p>If $a_n>0$ for all $n$, then $\liminf_{n\to\infty} a_n \geqslant 0$, for if not, there would exist an $n$ such that $\inf_{k\geqslant n}a_k<0$. But the $\inf$ of a set of positive numbers cannot be negative, so $\liminf_{n\to\infty} a_n\geqslant0$, and because $a_n$ is convergent, $$\lim_{n\to\infty}a_n=\limin... |
1,424,273 | <p>Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?</p>
<p>My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all $n\ge N$. And we know $0< a_n$ for all $n\in \mathbb{N}$, particularly $0<a_n$ for all $n\ge N$. Maybe by c... | John Dawkins | 189,130 | <p>Using your notation, if $n\ge N$ then $|a_n-L|<\epsilon$. This absolute-value inequality if equivalent to the inequalities $a_n-\epsilon< L < a_n+\epsilon$. In particular (use only the left-hand inequality and take $n=N$),
$$
-\epsilon<a_N-\epsilon<L,
$$
Summarizing: $-\epsilon <L$ for each $\epsi... |
4,080,776 | <p>I am doing an individual study of an abstract algebra for number theory course online. I just started, so I hope my question just note come off as too trivial. The lecture notes state that the ring of <span class="math-container">$p$</span>-adic integers does not have a ring endomorphism.</p>
<h3>Questions:</h3>
<p>... | Torsten Schoeneberg | 96,384 | <p>Re Question 1) Yes, the identity <span class="math-container">$id: \mathbb Z_p \rightarrow \mathbb Z_p$</span> is of course a ring endomorphism.</p>
<p>Re Question 2) To show that there is no other, assume <span class="math-container">$f: \mathbb Z_p \rightarrow \mathbb Z_p$</span> is any ring endomorphism. We have ... |
2,355,852 | <p>Given are $m$ bins with equal probability of choosing one of them. Unknown number of balls $n$ is placed into the bins, and, at the end of placement, we observe number of empty bins $m_e$ and non-empty bins $m_{n}$.</p>
<p>Given $m$, $m_e$, $m_n$, what is the most likely number of balls $n$, which have been placed ... | Henry | 6,460 | <p>Presumably $m=m_e+m_n$. </p>
<p>The likelihood of seeing $m_n$ out of $m$ occupied is $\dfrac{S_2(n,m_n)\, m!}{m^n \;(m-m_n)!}$ where $S_2(x,y)$ is a <a href="https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow noreferrer">Stirling number of the second kind</a>. I do not know an easy ... |
2,853,989 | <p>I'm trying to demonstrate that $\left( 1+\frac1 n \right)^n$ is bigger than $2$. I have tried to prove that $\left( 1+\frac1 n \right)^n$ is smaller than $\left( 1+\frac1{n+1} \right)^{n+1}$ by expanding
$\left( 1+\frac1n \right)^n = \sum\limits_{i=0}^n \left( \frac{n}{k} \right) \frac{1}{n^k}$ and $\left( 1+\frac1... | mechanodroid | 144,766 | <p>\begin{align}
\left(1+\frac1n\right)^n &= \sum_{k=0}^n {n \choose k} \frac1{n^k} \\
&= \sum_{k=0}^n \frac{n(n-1)\cdots(n-k+1)}{k!n^k} \\
&= \sum_{k=0}^n \frac1{k!}\left(1-\frac1n\right)\left(1-\frac2n\right)\cdots \left(1-\frac{k-1}n\right)
\end{align}</p>
<p>so</p>
<p>\begin{align}
\left(1+\frac1{n+1}... |
2,853,989 | <p>I'm trying to demonstrate that $\left( 1+\frac1 n \right)^n$ is bigger than $2$. I have tried to prove that $\left( 1+\frac1 n \right)^n$ is smaller than $\left( 1+\frac1{n+1} \right)^{n+1}$ by expanding
$\left( 1+\frac1n \right)^n = \sum\limits_{i=0}^n \left( \frac{n}{k} \right) \frac{1}{n^k}$ and $\left( 1+\frac1... | Momo | 384,029 | <p>Another way is to prove first that your sequence is monotonically increasing like has been done here:</p>
<p><a href="https://math.stackexchange.com/questions/167843/i-have-to-show-1-frac1nn-is-monotonically-increasing-sequence">I have to show $(1+\frac1n)^n$ is monotonically increasing sequence</a></p>
<p>... and... |
2,853,989 | <p>I'm trying to demonstrate that $\left( 1+\frac1 n \right)^n$ is bigger than $2$. I have tried to prove that $\left( 1+\frac1 n \right)^n$ is smaller than $\left( 1+\frac1{n+1} \right)^{n+1}$ by expanding
$\left( 1+\frac1n \right)^n = \sum\limits_{i=0}^n \left( \frac{n}{k} \right) \frac{1}{n^k}$ and $\left( 1+\frac1... | copper.hat | 27,978 | <p>Let $f(x) = (1+x)^n$. Note that $f$ is convex for $x \ge 0$ and so
$f(x) \ge f(0)+f'(0) x = 1+xn$. Hence $f({1 \over n}) \ge 2$.</p>
|
12,204 | <p>A tag named <a href="https://math.stackexchange.com/questions/tagged/tricks" class="post-tag" title="show questions tagged 'tricks'" rel="tag">tricks</a> has recently been created in <a href="https://math.stackexchange.com/questions/616672/2-tricks-to-prove-every-group-with-an-identity-and-xx-identity-is-abe... | Martin Sleziak | 8,297 | <p>Just offering another possibility to vote for/against: </p>
<p>Keep the <a href="https://math.stackexchange.com/questions/tagged/tricks" class="post-tag" title="show questions tagged 'tricks'" rel="tag">tricks</a> tag, but make it a synonym of <a href="https://math.stackexchange.com/questions/tagged/proof-s... |
1,693,045 | <p>I know if $x=e^{\frac{2\pi i}{17}}$ then $x^{17}=1$ and $\Re(x)=\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>But how do I form a polynomial which has root $\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>I know you can consider de Moivre's theorem and expand the LHS using binomial theorem but that will take a long time.</p... | David C. Ullrich | 248,223 | <p>Probably the best way is to just show that the sum of two algebraic numbers is algebraic. This is not obvious, but if you look at it just right it's much easier than it seems at first.</p>
<p>Regard $\Bbb C$ as a vector space over $\Bbb Q$. Any linear-algebra concepts below refer to $\Bbb Q$-linear subspaces of $\B... |
1,693,045 | <p>I know if $x=e^{\frac{2\pi i}{17}}$ then $x^{17}=1$ and $\Re(x)=\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>But how do I form a polynomial which has root $\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>I know you can consider de Moivre's theorem and expand the LHS using binomial theorem but that will take a long time.</p... | Ron Gordon | 53,268 | <p>Consider that</p>
<p>$$\sin{\left ( \frac{9 \pi}{17} \right )} = \sin{\left ( \frac{8 \pi}{17} \right )} $$</p>
<p>or, letting $y = \frac{\pi}{17}$,</p>
<p>$$3 \sin{3 y} - 4 \sin^3{3 y} = 2 \sin{4 y} \cos{4 y} $$</p>
<p>or</p>
<p>$$9 \sin{y} - 120 \sin^3{y} + 432 \sin^5{y} - 576 \sin^7{y} + 256 \sin^9{y} \\= 8 ... |
4,178,548 | <p>I'm starting a Linear Algebra course and I'm a bit confused.</p>
<p>Say we have a vector
<span class="math-container">$x = \begin{pmatrix}
x_1\\
x_2\\
\end{pmatrix}$</span>, and another vector <span class="math-container">$y = \begin{pmatrix}
y_1\\
y_2\\
\end{pmatrix}$</span></p>
<p>When we have a matrix composed of... | Acccumulation | 476,070 | <p>There are contexts where matrices can be considered to be a vector of vectors. There are two types of vectors: row vectors and column vectors. A matrix can be viewed as either column vector where all the elements are row vectors, or a row vectors where all the elements are column vectors.</p>
<p>Thus <span class="ma... |
3,053,975 | <p><span class="math-container">$3^6-3^3 +1$</span> factors?, 37 and 19, but how to do it using factoring, <span class="math-container">$3^3(3^3-1)+1$</span>, can't somehow put the 1 inside </p>
| Oscar Lanzi | 248,217 | <p>Numbers having the form <span class="math-container">$n^2-n+1$</span> can never have prime factors congruent to <span class="math-container">$2$</span> modulo <span class="math-container">$3$</span>. Since <span class="math-container">$3$</span> is obviously out and any factorization must include a prime factor les... |
275,430 | <p>I'm trying to give an $\epsilon$-$\delta$ proof that the following function $f$ is continuous for $x\notin\mathbb Q$ but isn't for $x\in\mathbb Q$. </p>
<p>Let $f:\mathbb{A\subset R\to R}, \mathbb{A=\{x\in R| x>0\}}$ be given by:
$$
f(x) = \begin{cases}
1/n,&x=m/n\in\mathbb Q
\\
0,&x\notin\mathbb Q
\en... | Brian M. Scott | 12,042 | <p>HINTS: To show that $f$ is continuous at an irrational $x$, show for any $n\in\Bbb Z^+$ you can choose $\delta>0$ small enough so that the interval $(x-\delta,x+\delta)$ contains no fraction with a denominator $\le n$ in lowest term. Use the fact that between two rationals with denominator $m$ there is a gap of a... |
275,430 | <p>I'm trying to give an $\epsilon$-$\delta$ proof that the following function $f$ is continuous for $x\notin\mathbb Q$ but isn't for $x\in\mathbb Q$. </p>
<p>Let $f:\mathbb{A\subset R\to R}, \mathbb{A=\{x\in R| x>0\}}$ be given by:
$$
f(x) = \begin{cases}
1/n,&x=m/n\in\mathbb Q
\\
0,&x\notin\mathbb Q
\en... | Hagen von Eitzen | 39,174 | <p>For $x=\frac mn\in\mathbb Q$ select $\epsilon=\frac1{2n}>0$. Assume there is $\delta>0$ such that $|x-y|<\delta$ implies $|f(y)-f(x)|<\epsilon$.
Then especially $f(y)>\frac{1}{2n}$ for such $y$, which means that alls such $y$ are rational and have denominator $<2n$. Even without knowing that the i... |
2,292,520 | <p>I know that the logical negation of $$\neg(a \rightarrow b)= a \wedge \neg b $$ I am not clear what that means in the following simple setting:</p>
<p>So its clear that $$x\geq 2 \to x^2\geq 4.$$ Now I can write the logical negation of $a\to b$ as $a \wedge \neg b$, but what does that intuitively mean? </p>
<p>Sup... | Mirko | 188,367 | <p>Say $a$ is $x\ge2$ and $b$ is $x^2\ge14$ (and formally an universal quantifier should be involved as in @JMoravitz comment, i.e. $\forall x, x\ge2\to x^2\ge14$ ). Pick $x=3$, then $a$ is true, but $b$ is false. In other words, $x\ge2$ does not imply that $x^2\ge14$. Formally, the negation here is $\exists x, (x\ge2 ... |
1,804,042 | <p><strong>Edit:</strong> Here is the original problem; it is possible that my recurrence for the stationary distribution $\pi$ is incorrect.</p>
<blockquote>
<p>Consider a single server queue where customers arrive according to a Poisson process with intensity $\lambda$ and request i.i.d. $\mathsf{Exp}(\mu)$ servic... | Przemo | 99,778 | <p>The solution to the recurrence in question is definitely governed by the roots of the characteristic equation which reads:
<span class="math-container">\begin{equation}
\lambda z^2- \left( \lambda + \mu + \theta \right) z + \mu =0
\end{equation}</span></p>
<p>where </p>
<p><span class="math-container">\begin{eqnar... |
2,196,037 | <p>Let $E$ be a universal set and $\{A_{\alpha}\}_{\alpha \in J},$ for some index set $J$ be a family of subsets of $E.$</p>
<p>Prove that:
(a)$E-\bigcup_{\alpha \in J}A_{\alpha} = \bigcap_{\alpha \in J}($R$-A_{\alpha}).$</p>
<p>I do not know what is $R$ or it is a mistake in the question, Could anyone help me ? </p... | Axion004 | 258,202 | <p>\begin{eqnarray*}
(\bigcup_{i\in\Lambda}A_i)^c &=& \{x|x\notin \bigcup_{i\in\Lambda}A_i\} \\
&=& \{x|\forall i\in\Lambda,~~x\notin A_i\}\\
&=& \{x|\forall i\in\Lambda,~~x\in A_i^c\} \\
&=& \... |
3,554,891 | <p>Let's take a look back at this familiar "Law of cosines":</p>
<blockquote>
<p>Consider the triangle <span class="math-container">$\triangle ABC$</span>. Let <span class="math-container">$a = BC, b = AC, c = AB$</span>; <span class="math-container">$\angle A, \angle B, \angle C$</span> are the angles of the ... | Intelligenti pauca | 255,730 | <p>In quadrilateral <span class="math-container">$ABCD$</span>, in addition to <span class="math-container">$a = BC$</span>, <span class="math-container">$b = CD$</span>, <span class="math-container">$c = AB$</span>, <span class="math-container">$d = AD$</span>,
also set:
<span class="math-container">$$
AG=e,\quad CG=f... |
129 | <p>Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not sufficient. Are there any other special homotopical properties of manifolds?</p>
| John Klein | 8,032 | <p>Sean: this gives a Poincare space which is not homotopy equivalent to a closed manifold. the idea is that the Spivak fibration of the <span class="math-container">$5$</span> dimensional Poincare space doesn't lift to a stable vector bundle. One can prove this as follows: let <span class="math-container">$X^5$</span>... |
456,892 | <p>Find all solutions of $4\cos^2(x)-4\sin(x)-5=0$ in the interval $(6\pi, 8\pi)$.</p>
<p>I tried to work it out and got: $4y^2-4y -9 = 0$, but I can't figure out what $\cos x = $from there to finish the problem.</p>
| Robert Israel | 8,508 | <p>If $\sin(x) = y$, $\cos(x) = \pm \sqrt{1-y^2}$. But your equation for $y$ is not quite right.</p>
|
456,892 | <p>Find all solutions of $4\cos^2(x)-4\sin(x)-5=0$ in the interval $(6\pi, 8\pi)$.</p>
<p>I tried to work it out and got: $4y^2-4y -9 = 0$, but I can't figure out what $\cos x = $from there to finish the problem.</p>
| lab bhattacharjee | 33,337 | <p>As already found, $\sin x=\frac12$ which is $=\sin(-\frac\pi6)$</p>
<p>$$\implies x=n\pi+(-1)^n\left(-\frac\pi6\right)$$</p>
<p>If $n$ is even, $=2m$(say), </p>
<p>$x=2m\pi-\frac\pi6=(12m-1)\frac\pi6 $ and we need $6\pi<x<8\pi\implies 6\pi<(12m-1)\frac\pi6<8\pi \implies\frac{37}{12}<m<\frac{49}... |
1,441,624 | <p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span> and <span class="math-container">$c$</span> be elements of a group <span class="math-container">$G$</span>, how can I prove that <span class="math-container">$abc$</span> and <span class="math-container">$cba$</span> do not necessar... | Shaun | 104,041 | <p><strong>Hint:</strong> Try the dihedral group <span class="math-container">$D_4$</span> of eight elements.</p>
|
204,365 | <p>Consider a positive matrix <code>M</code> and a positive vector <code>b</code>, e.g.</p>
<pre><code>nn = 1000;
M = Table[RandomReal[{0, 100}], {i, 1, nn}, {j, 1, nn}];
b = Table[RandomReal[{0, 100}], {i, 1, nn}];
</code></pre>
<p>I would like to find a positive vector <code>X</code></p>
<pre><code>X = Array[x,... | Bob Hanlon | 9,362 | <p><strong>THIS IS AN EXTENDED COMMENT RATHER THAN AN ANSWER.</strong></p>
<p>It is inefficient to use <code>Table</code> to generate random numbers.</p>
<pre><code>Clear["Global`*"]
nn = 1000;
SeedRandom[10];
t1 = AbsoluteTiming[
{M1 = Table[RandomReal[{0, 100}], {i, 1, nn}, {j, 1, nn}],
b1 = Table... |
3,557,840 | <p>Find the quadratic polynomial <span class="math-container">$p(x)$</span> for given data points <span class="math-container">$$p(x_0)=y_0, p'(x_1)=y_1', p(x_2)=y_2 \text{ with } x_0 \neq x_2.$$</span></p>
<p><strong>My approach</strong></p>
<p>I tried the problem taking <span class="math-container">$p(x)=a+bx+c x^2... | Alain Remillard | 278,299 | <p>Starting with a generic quadratic polynomial, you could create three linear equations.
<span class="math-container">$$p(x)=a+bx+cx^2$$</span>
<span class="math-container">$$p'(x)=b+2cx$$</span>
Then
<span class="math-container">$$\begin{cases}y_0=a+bx_0+cx_0^2\\y_1'=b+2cx_1\\y_2=a+bx_2+cx_2^2\end{cases}$$</span>
Fin... |
292,948 | <p>I have been trying to prove that if $A$ is a closed set which is also an intersection of countably many open sets then $A$ is the zero set for some continuous real-valued function however have thus far failed. Is this even true?</p>
| Brian M. Scott | 12,042 | <p>It is not true.</p>
<p>John Thomas, <a href="http://www.jstor.org/stable/2317272"><em>A regular space, not completely regular</em></a>, Amer. Math. Monthly <strong>76</strong> (1969), 181-182, constructed a regular Hausdorff space $X$ with two points, $p$, and $q$, such that for each continuous $f:X\to\Bbb R$, $f(a... |
292,948 | <p>I have been trying to prove that if $A$ is a closed set which is also an intersection of countably many open sets then $A$ is the zero set for some continuous real-valued function however have thus far failed. Is this even true?</p>
| Martin | 49,437 | <p>Brian's answer covers the question fully. For fun, here's another example:</p>
<p>Bing's <a href="http://dx.doi.org/10.1090/S0002-9939-1953-0060806-9" rel="nofollow">irrational slope space</a> is a countable and connected Hausdorff space. </p>
<p>Now observe:</p>
<ol>
<li><p>If $f \colon X \to \mathbb{R}$ is cont... |
292,948 | <p>I have been trying to prove that if $A$ is a closed set which is also an intersection of countably many open sets then $A$ is the zero set for some continuous real-valued function however have thus far failed. Is this even true?</p>
| GEdgar | 442 | <p>Not even in completely regular Hausdorff spaces. In general we have
$$
\text{compact $G_\delta$}\qquad\Longrightarrow\qquad
\text{zero-set}\qquad\Longrightarrow\qquad
\text{closed $G_\delta$}
$$
but none reversible.</p>
|
218,915 | <blockquote>
<p>Prove that for any integer $n$, $\gcd (3n^2+5n+7, n^2+1)=1$ or $41$.</p>
</blockquote>
<p>The following answer is convoluted because I've intentionally created excess solutions. However, I can't figure out how to eliminate them! Anyone?</p>
<p>Let $$d=\gcd (3n^2+5n+7, n^2+1).$$
Then $$d|[(3n^2+5n+7)... | robjohn | 13,854 | <p>Suppose that
$$
(3n^2+5n+7,n^2+1)=(5n+4,n^2+1)\ne1\tag{1}
$$
then either
$$
(5n+4,n+i)=(4-5i,n+i)\ne1\tag{2}
$$
or
$$
(5n+4,n-i)=(4+5i,n-i)\ne1\tag{3}
$$
Since $4-5i$ is a Gaussian prime, $(2)\Rightarrow4-5i\,|\,n+i$. That is,
$$
\frac{n+i}{4-5i}=\frac{(4n-5)+(5n+4)i}{41}\in\mathbb{Z}[i]\tag{4}
$$
which is true if a... |
2,042,428 | <p>If I'm correct, hidden induction is when we use something along the lines of "etc..." in a proof by induction. Are there any examples of when this would be appropriate (or when it's not appropriate but used anyway)?</p>
| Sean Keeler | 395,712 | <p>First, here is an example of when this works. Let $X$ and $Y$ be Hausdorff spaces. This implies $X\times Y$ with the product topology is Hausdorff. Therefore any finite product of Hausdorff spaces is Hausdorff. The "hidden induction" is the idea that $X\times Y$ is a Hausdorff space, which implies any Hausdorff spac... |
2,042,428 | <p>If I'm correct, hidden induction is when we use something along the lines of "etc..." in a proof by induction. Are there any examples of when this would be appropriate (or when it's not appropriate but used anyway)?</p>
| Jack M | 30,481 | <p>Hidden induction happens a lot in cases where you go <em>backwards</em> from $n$ to $1$, using some kind of reduction argument. For example, the proof that every number can be written as a product of primes:</p>
<blockquote>
<p>Let $n$ be some number. If it's prime, then we're done. Otherwise it can be written as... |
1,309,728 | <p>I know what a 3x10 looks like, but I cannot seem to find a distinguishable pattern to extend it to a 3x14.</p>
<p>The 3x10 pattern I'm using looks like the one at the top right of figure 6 of <a href="http://faculty.olin.edu/~sadams/DM/ktpaper.pdf" rel="nofollow">this paper</a>.</p>
<p>Any help would be greatly ap... | user26857 | 121,097 | <p>Maybe you want to know more, namely: $$f(X,Y)\in(X-a,Y-b) \text{ iff } f(a,b)=0.$$</p>
<p>This solves instantly your question: $f(a,b)=0$ for $f(X,Y)=XY-1$ means $ab=1$.</p>
|
3,362,115 | <blockquote>
<p>Find the maximum value of <span class="math-container">$y/x$</span> if it satisfies <span class="math-container">$(x-5)^2+(y-4)^2=6$</span>.</p>
</blockquote>
<p>Geometrically, this is finding the slope of the tangent from the origin to the circle. Other than solving this equation with <span class="m... | Jan-Magnus Økland | 28,956 | <p>See <a href="https://en.wikipedia.org/wiki/Pole_and_polar" rel="nofollow noreferrer">Pole and polar</a>. </p>
<p>The polar line of the origin <span class="math-container">$-5x-4y+35=0$</span> intersects the circle as seen in the image below, giving the tangents from the origin to the circle.</p>
<p><a href="https:... |
3,362,115 | <blockquote>
<p>Find the maximum value of <span class="math-container">$y/x$</span> if it satisfies <span class="math-container">$(x-5)^2+(y-4)^2=6$</span>.</p>
</blockquote>
<p>Geometrically, this is finding the slope of the tangent from the origin to the circle. Other than solving this equation with <span class="m... | Cesareo | 397,348 | <p>The line <span class="math-container">$y = \lambda x$</span> and the circle <span class="math-container">$(x-5)^2+(y-4)^2= 6$</span> should intersect and <span class="math-container">$\max \frac xy =\max \lambda$</span> should be located at a tangency point hence solving for <span class="math-container">$x$</span></... |
3,362,115 | <blockquote>
<p>Find the maximum value of <span class="math-container">$y/x$</span> if it satisfies <span class="math-container">$(x-5)^2+(y-4)^2=6$</span>.</p>
</blockquote>
<p>Geometrically, this is finding the slope of the tangent from the origin to the circle. Other than solving this equation with <span class="m... | Narasimham | 95,860 | <p>Locate the circle center and draw the displaced circle with its radius including other sides using Pythagoras thm.
Add two angles at max slope tangent point around origin O as shown directly:</p>
<p><a href="https://i.stack.imgur.com/dD8na.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dD8na.png... |
78,478 | <blockquote>
<p>Prove that $\frac{1}{n} \sum_{k=2}^n \frac{1}{\log k}$ converges to $0.$</p>
</blockquote>
<p>Okay, seriously, it's like this question is mocking me. I know it converges to $0$. I can feel it in my blood. I even proved it was Cauchy, but then realized that didn't tell me what the limit <em>was</em... | hmakholm left over Monica | 14,366 | <p>In general, if $a_n\to 0$, then $\frac1n \sum_{k=0}^n a_k \to 0$ too.</p>
<p>(For any $\varepsilon>0$, find an $N$ such that $|a_n|<\varepsilon/2$ for all $n>N$, and then take enough terms beyond $N$ that they dominate whatever the terms <em>before</em> $N$ contribute to the average).</p>
<p>Even more gen... |
352,849 | <p>I have to show that $\lim \limits_{n\rightarrow\infty}\frac{n!}{(2n)!}=0$ </p>
<hr>
<p>I am not sure if correct but i did it like this :
$(2n)!=(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))\cdot (n!)$ so I have $$\displaystyle \frac{1}{(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))}$$ and $$\lim \limits_{... | Elias Costa | 19,266 | <p>If so addressing trivial rigorously I suggest using the notation produtory to fatorial use the formula $n!=\prod_{k=1}^{n}$ .
\begin{align}
0\leq \frac{n!}{(2n)!}
=
&
\frac{\big(\prod_{k=1}^{n}k\big)}{\big(\prod_{k=1}^{2n}k\big)}
\\
=
&
\frac{\big(\prod_{k=1}^{n}k\big)}{\big(\prod_{k=n+1}^{2n}k\big)\big(\pr... |
2,596,213 | <p>I'm having huge troubles with problems like this. I know the following:</p>
<p>$$\frac{\sin{x}}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+O(x^7)$$</p>
<p>and </p>
<p>$$\ln{(1+t)}=t-\frac{t^2}{2}+\frac{t^3}{3}+O(t^4)$$</p>
<p>So</p>
<p>$$\ln{\left(1+\left(-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+O(x... | user | 505,767 | <p>You need to consider</p>
<p>$$\frac{\sin{x}}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+O(x^8)$$</p>
<p>and</p>
<p>$$\ln{(1+t)}=t-\frac{t^2}{2}+\frac{t^3}{3}+O(t^4)$$</p>
<p>then substitute and expand keeping only the terms with order less than $x^8$ thus you don't need to expand all the expression but on... |
51,752 | <p>Can someone give an argument, if possible using only the axioms of set theory, because I'm <strong>very</strong> weak there and have virtually no background, except the usual knowledge of the operation with sets one has to have when doing non-set theoretic non-research mathematics, why $\emptyset \in \emptyset$ or $... | Asaf Karagila | 622 | <p>The empty set $\varnothing$ is the only set which satisfies $\forall x(x\notin y)$ (that means the formula is true if and only if $y=\varnothing$)</p>
<p>There are many ways to define the empty set (the set of all $x$ such that $x\neq x$ - we will use this formula later on) but by the axiom of extensionality it is ... |
184,824 | <p>I have two piecewise function</p>
<pre><code>equ1 = Piecewise[{{0.524324 + 0.0376478x, 0.639464 <= x <= 0.839322}}]
equ2 = Piecewise[{{-0.506432 + 1.48068x, 0.658914 <= x <= 0.77085}}]
</code></pre>
<p>Now, I am trying to solve <code>equ1 = equ2</code>.</p>
<p>Firstly I tried <code>FindRoot</code>: </... | Bill | 18,890 | <p>If you have 100 equations and you want to find the unique solution to the whole set and each of your <code>Piecewise</code> are written in exactly the same form as the two you showed and the solution must lie in the intersection of all those inequalities then perhaps</p>
<pre><code>equ1 = Piecewise[{{0.524324 + 0.0... |
2,012,947 | <p>I'm trying to prove that if f,g are continuous functions, and if E is a dense subset of X $(\text{or } Cl(E) = X)$ and if $f(x)=g(x) \forall x \in E$ then $f(x)=g(x) \forall x \in X$. </p>
<p>I understand that if f,g are continuous, then:</p>
<blockquote>
<p>$\exists \delta_1, \delta_2$ such that $\forall X \in ... | Learnmore | 294,365 | <p>Suppose that $f(a)\neq g(a);a\in X$.</p>
<p>Let $d(f(a),g(a))=r>0$.</p>
<p>Since $f$ is continuous at $a$ so $\exists \delta_1>0$ such that $f(B(a,\delta_1))\subset B(f(a),\frac{r}{3})$.</p>
<p>Since $g$ is continuous at $a$ so $\exists \delta_2>0$ such that $g(B(a,\delta_2))\subset B(g(a),\frac{r}{3})$.... |
3,391,225 | <p>What is the term for a (connected?) set <span class="math-container">$S$</span> of the plane <span class="math-container">$\mathbb{R}^2$</span> such that the intersection of <span class="math-container">$S$</span> with every horizontal line <span class="math-container">$\ell_{b}: y=b$</span> is either empty, or an i... | Pacciu | 8,553 | <p>From where I’m from, such a set is called <em><span class="math-container">$y$</span>-simple domain</em> or <em><span class="math-container">$y$</span>-normal domain</em>.</p>
|
4,394,247 | <p>I know how to represent the sentence “there is exactly one person that is happy”,</p>
<p>∀y∀x((Happy(x)∧Happy(y))→(x=y))</p>
<p>Edit: ∃x∀y(y=x↔Happy(y))
(NOW, I actually know how to represent it)</p>
<p>Where x and y represent a person.</p>
<p>However, my problem is that I can’t figure out how to say “there are exac... | A J | 934,287 | <p>You can extend this simply as: if 4 people are happy then at least one is equal to the other.
<span class="math-container">$\forall x,y,z,w((Happy(x)\land Happy(y) \land Happy(z) \land Happy(w))\rightarrow(x=y \lor x=z \lor x=w \lor y=z \lor y=w \lor z=w)) \land \exists x,y,z (Happy(x)\land Happy(y) \land Happy(z) \... |
869,218 | <p>I am reading Stillwell's <em>Numbers and Geometry</em>. There is an exercise about Egyptian fractions which is the following:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/JiSpw.png" alt="enter image description here"></p>
</blockquote>
<p>I've tried to do it in the following way - Expressing an arbitr... | Deathkamp Drone | 56,720 | <p>Here's one systematic method to see if there are ways of expressing a fraction as a sum of two Egyptian fractions:</p>
<p>$$\begin{align}\frac{4}{5}=\frac{1}{m}+\frac{1}{n} &\Leftrightarrow 4mn=5m+5n \\
&\Leftrightarrow 5m-4mn+5n=0 \\
&\Leftrightarrow \left(2m-\frac{5}{2}\right)\left(\frac{5}{2}-2n\righ... |
869,218 | <p>I am reading Stillwell's <em>Numbers and Geometry</em>. There is an exercise about Egyptian fractions which is the following:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/JiSpw.png" alt="enter image description here"></p>
</blockquote>
<p>I've tried to do it in the following way - Expressing an arbitr... | individ | 128,505 | <p>As I said a decision for 3 terms there. <a href="https://math.stackexchange.com/questions/450280/erd%C3%B6s-straus-conjecture/831870#831870">Erdős-Straus conjecture</a> </p>
<p>As for the answers of two terms, the decision is not always exist.</p>
<p>For the equation: $$\frac{1}{X}+\frac{1}{Y}=\frac{b}{A}$$ </... |
205,926 | <p>I'm trying to understand a proof about density of a subset $X$ in its one-point compactification $Y$.</p>
<p>We can do this proof by contradiction, suppose we don't have $\operatorname{cl}(X) = Y$.
This implies that $\operatorname{cl}(X) = X$. </p>
<p>Why? Can anyone help me?</p>
<p>Thanks</p>
| Idan | 37,998 | <p>Suppose $\operatorname{cl}(X)\not=Y$. We know $X\subseteq \operatorname{cl}(X)$ so we get $\operatorname{cl}(X)=X$ and $\infty \notin \operatorname{cl}(X).$ So by definition of closure, there exists a (wlog, open) neighborhood $U$ of $\infty$ s.t. $U \cap X=\emptyset$. The topology of the extension is defined to be ... |
979,432 | <p>i was recently watching a single variable calculus video of mit 18.01, lecture 23. in that it is said that average height of a point on semicircle with respect to arc length is 2/pi.I have a hard time to understand that point. i understand why average height of point on semi circle with respect to x is pi/4. but i d... | Jim H | 473,669 | <p>The average value with respect to $x$ over interval $[a,b]$ is found by: </p>
<p>Cut the interval into $n$ equal subintervals of length $\Delta x = \frac{b-a}{n}$. Note that $n = \frac{\Delta x}{b-a}$.
On each subinterval, use a single value of $x$ (which I'll call $x_i$).
Sum the values of $f\left( x_i \right)$ an... |
4,540,637 | <blockquote>
<p>Given a line <span class="math-container">$y=kx$</span> on a Cartesian coordinate, I want to find an equation of a parabola, whose base is on that line at point <span class="math-container">$(x_1,y_1)$</span> and passes through point <span class="math-container">$(x_2,y_2)$</span>.</p>
</blockquote>
<p>... | Abel Wong | 1,090,313 | <p><span class="math-container">$x_1$</span> and <span class="math-container">$y_1$</span> make the problem complicate. First, we do a translation to make <span class="math-container">$x_1, y_1$</span> move to origin in <span class="math-container">$x'-y'$</span> coordinate system.</p>
<p>Now, the parabola touch <span ... |
691,734 | <p>Consider the sequence defined recursively by $x_1$=$\sqrt2$ and where $x_n$=$\sqrt2$ + $x_n$$_-$$_1$. </p>
<p>Find a explicit formula for the $n^t$$^h$ term.</p>
<p>I considered using the general equation to find an explicit formula for any term in an arithmetic sequence. a$_n$ = a$_1$ + $d(n-1)$, but I came to no... | Mhenni Benghorbal | 35,472 | <p>Here is an approach.</p>
<p>$$ x_{n+1}-x_{n}=\sqrt{2} \implies \sum_{i=0}^{n-1}( x_{i+1}-x_{i}) = \sqrt{2}\sum_{i=0}^{n-1}1 $$</p>
<p>$$ \implies x_n-x_0=\sqrt{2} n .$$</p>
|
234,340 | <p>Suppose that I have two real-valued matrices $\bf{A}$ and $\bf{B}$. Both matrices are exactly the same size. I multiply both matrices together in a point-by-point fashion similar to the Matlab <code>A .* B</code> operation.</p>
<p>Under what conditions can I approximately separate $\bf{A}$ and $\bf{B}$ using Prin... | Bitwise | 42,051 | <p>It seems to me that you can't separate the matrices from their pointwise multiplication (Hadamard/Schur product) without additional constraints.</p>
<p>Consider some matrix C. Any number in C is decomposable into an infinite number of products of two real numbers... which would give you an infinite number of "perfe... |
3,662,286 | <p>Can you raise the imaginary number i to a power that is an irrational number?</p>
| Lukas Rollier | 737,665 | <p>Take <span class="math-container">$c \in \mathbb{C}$</span> arbitrarily. One might define <span class="math-container">$i^c := e^{\ln(i^c)} = e^{c \ln(i)} = e^{c \cdot \frac{i\pi}{2}}$</span>, which works. There is a catch though: this makes use of the logarithm, which is not nicely defined on the entire complex pla... |
661,269 | <p>Check if $\mathbb{Z}_5/x^2 + 3x + 1$ is a field. Is $(x+2)$ a unit, if so calculate its inverse. </p>
<p>I would say that this quotient ring is not a field, because $<x^2 + 3x + 1>$ is not a maximal ideal, since $x^2 + 3x + 1 = (x+4)^2$ is not irreducible. </p>
<p>However, the result should still be a ring,... | André Nicolas | 6,312 | <p>You are asking for the distribution of $Y=a_1X_1+\cdots +a_n X_n$, where more generally the $X_i$ are independent normal, means $\mu_i$, variances $\sigma_i^2$. The random variable $Y$ has normal distribution, mean $\sum_1^n a_i \mu_i$, variance $\sum_1^n a_i^2 \sigma_i^2$. </p>
|
3,646,911 | <p>Exercise 14.7.4 from Dummit and Foote</p>
<blockquote>
<p>Let <span class="math-container">$K=\mathbb{Q}(\sqrt[n]{a})$</span>, where <span class="math-container">$a\in \mathbb{Q}$</span>, <span class="math-container">$a>0$</span> and suppose <span class="math-container">$[K:\mathbb{Q}]=n$</span>(i.e., <span cl... | nonuser | 463,553 | <p><span class="math-container">$$32=\binom50x^{5}-\binom52x^{3} + \binom52x^{1}-\binom53x^{-1} + \binom54x^{-3}-\binom55x^{-5} $$</span></p>
<p>Or
<span class="math-container">$$32x^5=\binom50x^{10}-\binom52x^{8} + \binom52x^{6}-\binom53x^{4} + \binom54x^{2}-\binom55 $$</span></p>
<p><span class="math-container">$... |
613,940 | <p>Given two parameters $a$ and $b$ (both positive integers), please estimate
the order of growth of the following function:</p>
<p>$$F(t)=\left\{\begin{array}{ll} 1, \, &t\le a \\ F(t-1) + b\cdot F(t-a),&t>a\end{array}\right.$$ </p>
<p>My guess is $\Theta\left(b^{t/a}\right)$. Any answer that might hel... | Peter | 82,961 | <p>How about ${b}^{\frac{t}{a}}$ ? Because if t increases by a, F approximately is multiplied with b ? </p>
|
19,521 | <p>I am trying to integrate a hat function for a project that I am doing and have found a method to do so but I find it sloppy. Currently I have the basis function</p>
<pre><code>\[Psi][z_] := z - Subscript[Z, i]/ \[CapitalDelta]z + 1;
</code></pre>
<p>which I am trying to integrate from $z_{i-1}$ to $z_{i+1}$. I bre... | Xerxes | 5,406 | <p>Short answer: <em>Mathematica</em> has no problem integrating piecewise or hat functions.</p>
<p>Your notation seems to me to be needlessly complex. Why bother to define $Z_i$ when it's just $Z_0+i\Delta z$? Isn't your $\psi$ just <code>1-Abs[z-c]/Δz</code>? However, I'll try to adhere to the spirit of your notatio... |
4,236,878 | <p>Given a symmetric matrix <span class="math-container">$S$</span> and positive definite matrix <span class="math-container">$B$</span>, with <span class="math-container">$S,B \in \mathbb{R}^{n \times n}$</span> can one prove that</p>
<p><span class="math-container">\begin{align*}
\text{tr}((S-B)B) \le -\mu(S) \text{t... | user1551 | 1,551 | <p>If <span class="math-container">$\lambda_\max(S)\le0$</span>, then <span class="math-container">$S$</span> is negative semidefinite and <span class="math-container">$S-B$</span> is negative definite. Therefore <span class="math-container">$\operatorname{tr}((S-B)B)<0$</span>. On the other hand, <span class="math-... |
1,560,050 | <p>I want to solve the homogenous part of a stretched string problem where $y=y(x)$.</p>
<p>$$y'' + y = 0$$</p>
<p>with the boundary conditions such that: $y(0)=y(\pi/2)=0$</p>
<p>The differential equation gives rise to a solution on the form:
$$y = a \cos(x) + b \sin(x)$$</p>
<p>But when applying the boundary con... | Simon S | 21,495 | <p>Usually this problem in physics--with such boundary conditions--has ODE $$y'' + k^2y = 0$$ The problem then turns into finding values of $k$ for which the boundary conditions are met. In this case we find infinitely many discrete $k$, corresponding to the fundamental and the harmonics above:</p>
<p>$$y_k(x) = A_k\s... |
3,872,033 | <p>Currently I meet with the following interesting problem.</p>
<p>Let <span class="math-container">$x_1,\cdots,x_n$</span> be i.i.d standard Gaussian variables. How to calculate the probability distribution of the sum of their absoulte value, i.e., how to calculate
<span class="math-container">$$\mathbb{P}(|x_1|+\cdot... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
134,444 | <p>I have the following code that determines when the second business day of each month is (given a start and end date). I have a few If statements I would like to replace with functional programming.</p>
<pre><code>getAccrualDates[fromDate_List,toDate_List]:=
(
today = fromDate;
projectionDate =toDate;
(*If the proj... | Edmund | 19,542 | <p>You may use the <a href="http://reference.wolfram.com/language/guide/DateAndTime.html" rel="nofollow noreferrer">Date & Time</a> guide functions to greatly simplify your code.</p>
<pre><code>ClearAll[getAccrualDates];
getAccrualDates[start_DateObject, end_DateObject, frequency_, location_] :=
DatePlus[#, locat... |
2,296,544 | <p>Let $\{F_n\}, n\in \mathbb{N}$ be the sequence of Fibonacci numbers such that $F_1=1$, $F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ $\forall n\geq2$.</p>
<p>Define a new sequence $\{S_n\}$ such that $S_n=F_n+1$ $\forall n\in \mathbb{N}$.</p>
<p>Now the question is: For every prime $p$, does there exist an $N\in \mathbb{N}$,... | Robert Z | 299,698 | <p>Hint. The answer is yes. Show that for any prime $p\not=5$,
$$p\;\mbox{divides}\;S_{p^2-3}=F_{p^2-3}+1.$$
See for example Jack D'Aurizio's answer here:<a href="https://math.stackexchange.com/questions/1985541/fibonacci-sequence-problem-prove-that-there-are-infinitely-many-prime-numbers-s/1985568#1985568">Fibonacci ... |
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