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 |
|---|---|---|---|---|
603,469 | <p>Ok, so I'm not very good with these proving by induction thingies. Need a little help please.</p>
<p>How do I prove </p>
<p>$$\sum_{j=0}^n\left(-\frac12\right)^j=\frac{2^{n+1}+(-1)^n}{3\cdot2^n}$$</p>
<p>when $n$ is a non-negative integer.</p>
<p>I got the basis step and such down, but I'm pretty bad with expone... | Brian M. Scott | 12,042 | <p>Your induction hypothesis is that</p>
<p>$$\sum_{j=0}^n\left(-\frac12\right)^j=\frac{2^{n+1}+(-1)^n}{3\cdot2^n}\;,\tag{1}$$</p>
<p>and you want to prove that</p>
<p>$$\sum_{j=0}^{n+1}\left(-\frac12\right)^j=\frac{2^{n+1}+(-1)^{n+1}}{3\cdot2^{n+1}}\;.\tag{2}$$</p>
<p>The natural thing to do is to split the sum in... |
3,460,204 | <p>If <span class="math-container">$X\subseteq \mathbb R^n$</span> and <span class="math-container">$x\in X$</span> is an isolated point, then <span class="math-container">$x$</span> is a cluster point of <span class="math-container">$X^c$</span>.</p>
<p>I don't know if that is true or not, can someone help me, please... | Todd | 730,431 | <p>If <span class="math-container">$x$</span> is a cluster point, then, there is an open set <span class="math-container">$U\subset X^c\cup\{x\}$</span> s.t. <span class="math-container">$U\cap \{x\}=\{x\}$</span>. Let <span class="math-container">$\delta >0$</span> s.t. <span class="math-container">$]x,x+\delta [\s... |
2,553,853 | <p>I know that I am to use the thm. R/A. is a integral domain iff A is prime.
So I want an case where for some a,b in R and ab in A we have a nor b in A.</p>
<p>I thought it had something to do with the conjugate, but I'm lost.</p>
| Ashwin Iyengar | 95,668 | <p>The ring is an integral domain if and only if the ideal is prime. The ideal is prime if and only if the polynomial is irredicible, and you can see that $x^7 +1$ is not either by finding an explicit factorization into smaller degree polynomials, or by noting that if it were irreducible, then $\mathbb R[x]/(x^7+1)$ wo... |
155,237 | <p>The axiom of constructibility $V=L$ leads to some very interesting consequences, one of which is that it becomes possible to give explicit constructions of some of the "weird" results of AC. For instance, in $L$, there is a definable well-ordering of the real numbers (since there is a definable well-ordering of the ... | Andreas Blass | 6,794 | <p>Let me emphasize what Noah de-emphasized (by putting it in parentheses) in his answer: Instead of the "really cheap" idea of the first free ultrafilter in $L$-order, one can use the $L$-order of subsets of $\mathbb N$ to greedily construct an ultrafilter. The $L$-ordering of the subsets of $\mathbb N$ is a $\Delta^1... |
2,492,191 | <p>Let $f(x)=11x^3+15x^2+9x-2$. I want to calculate the remainder of $f(97)/11$.
I know that $97\pmod{11}=9$. But it also doesn't help with calculation of remainder with the help of modular arithmetic. I also cant get to the answer with $f(9)$ with mod of $11$.</p>
<p>What should I do?</p>
| José Carlos Santos | 446,262 | <p>Since $11\equiv0\pmod{11}$, that $15\equiv4\pmod{11}$, and that $9\equiv-2\pmod{11}$, then for any integer $x$, $f(x)\equiv4x^2-2x-2\pmod{11}$. Since $97\equiv-2\pmod{11}$,\begin{align}f(97)&\equiv f(-2)\pmod{11}\\&\equiv4\times(-2)^2-2\times(-2)-2\pmod{11}\\&\equiv7\pmod{11}.\end{align}</p>
|
2,492,191 | <p>Let $f(x)=11x^3+15x^2+9x-2$. I want to calculate the remainder of $f(97)/11$.
I know that $97\pmod{11}=9$. But it also doesn't help with calculation of remainder with the help of modular arithmetic. I also cant get to the answer with $f(9)$ with mod of $11$.</p>
<p>What should I do?</p>
| Shaun | 104,041 | <p>Well, $$
\begin{align}11(97)^3+15(97)^2+9(97)-2 & =4(-2)^2-2(-2)-2 \\
&= 5+2\pmod{11} \\
&=7\pmod{11}\end{align} $$ because $97=-2\pmod{11}$.</p>
|
17,270 | <p>I just joined MathSE and it's beautiful here, except for the fact that some unregistered users ask a question and never come back. Most of the time these questions are trivial, though they still consume answerers' (valuable) time which never gets rewarded. I thought it was okay until I saw someone's profile with the... | Community | -1 | <p>Yes, it is okay to not accept answers. Acceptance is a form of vote: this site has upvotes, downvotes, accept-votes, close-votes, delete-votes, etc (availability depends on reputation). Some users choose to not use some or all of categories of votes: e.g., never downvote, or never accept, or never vote at all. They... |
17,270 | <p>I just joined MathSE and it's beautiful here, except for the fact that some unregistered users ask a question and never come back. Most of the time these questions are trivial, though they still consume answerers' (valuable) time which never gets rewarded. I thought it was okay until I saw someone's profile with the... | Git Gud | 55,235 | <p>I think this question can be interpreted in two ways.</p>
<p>One is <code>Is it morally acceptable/ethical/polite/whatever to rarely accept answers?</code></p>
<p>The other is <code>Do the rules allow people to rarely accept answers?</code></p>
<p>The answer to the second question is 'yes, people are not forced t... |
2,441,660 | <p>Suppose I have $n$ observations, which are all normally distributed with the same mean (which is unknown) but each has a different variance (the different variances could be called $v_1,...,v_n$ for example, which are all assumed to be known).</p>
<p>What is the best estimate of the mean? And further, how does one ... | Ennar | 122,131 | <p>What you did also counts solutions where $x,\,y,\,z$ can be $0$ as well, but that can't happen with dice.</p>
<p>You should do this for $$(x-1)+(y-1) + (z-1) = 9 -3,$$ i.e. $$x'+y'+z' =6$$ because it is now ok to have $x', y'$ or $z'$ to be $0$. </p>
<p>Now we get $\binom{6+3-1}{3-1} = 28.$ But, this is wrong as w... |
2,938,372 | <p>Seems to me like it is. There are only finitely many distinct powers of <span class="math-container">$x$</span> modulo <span class="math-container">$p$</span>, by Fermat's Little Theorem (they are <span class="math-container">$\{1, x, x^2, ..., x^{p-2}\}$</span>), and the coefficient that I choose for each of thes... | Franklin Pezzuti Dyer | 438,055 | <p>Here's the flaw in your reasoning: when you consider <span class="math-container">$x$</span>, the variable of your polynomial, <span class="math-container">$x$</span> is <em>not</em> taken to be a number in <span class="math-container">$\mathbb Z_p$</span>. In fact, <span class="math-container">$x$</span> is not rea... |
1,554,871 | <p>The mean value theorem tells</p>
<p>" If $f:[a,b]\rightarrow \mathbb R$ is continuous and $g$ is an integrable function that does not change sign on $[a, b]$, then there exists $c$ in $(a, b)$ such that</p>
<p>$\int_a^b f(x)g(x)dx=f(c)\int_a^b g(x)dx$."</p>
<p>If $g$ changes its sign, is this theorem still true?<... | levap | 32,262 | <p>A useful theorem states that if $f_n \rightarrow f$ uniformly, then $f$ must be continuous. Since your pointwise limit function is not continuous, the convergence cannot be uniform.</p>
<p>If you don't want to use the theorem, you can check whether $\mathrm{sup}_{x \in \mathbb{R}} |f_n(x) - f(x)| \rightarrow 0$. </... |
123,813 | <p>Let $E$ be an elliptic curve over $\mathbb{Q}$. It is known from Gross and Zagier that if $\textrm{rank}_{\textrm{an}}(E) \leq 1$, then</p>
<p>$$\textrm{rank}(E) \geq \textrm{rank}_{\textrm{an}}(E).$$ </p>
<p>Instead, suppose I know that $\textrm{rank}(E) \leq 1$, are there any (unconditional) inequalities that re... | James Weigandt | 4,872 | <p>The short answer to your question is no. </p>
<p>One of the central open problems related to BSD rank conjecture is to find a way to canonically construct points of infinite order on elliptic curves with analytic rank at least 2. This is difficult because it requires passing from analytic properties of the $L$-func... |
2,040,175 | <p>The following diagram shows 9 distinct points chosen from the sides of a triangle.</p>
<p><a href="https://i.stack.imgur.com/3DUhm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3DUhm.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$(1)$</span> How many line seg... | Community | -1 | <p>By the following arrangement we have $2,3,4$ points on each side of the triangle. There are a total of $9$ points. Select any $3$ from them. Total number of ways are: $\binom{9}{3} = 84$. Number of collinear point selection cases: $\binom{2}{3} +\binom{3}{3} +\binom{4}{3} = 5$ ways. Totally $84-5 = 79$ ways. Hope it... |
216,473 | <p>Let me begin with some background: I used to enjoy mathematics immensely in school, and wanted to pursue higher studies. However, everyone around me at that time told me it was a stupid area (that I should focus on earning as soon as possible), and so instead I opted for an engineering degree. While in college, I us... | glebovg | 36,367 | <p>To be honest I was never good at high school geometry and geometry in general. I am also interested in mathematics and want to pursue a PhD in pure math. I always thought there was something wrong with geometric arguments -- and it turns out there is. Many philosophers and mathematicians explained why they are not s... |
2,908,186 | <p>There's an old-school pocket calculator trick to calculate <span class="math-container">$x^n$</span> on a pocket calculator, where both, <span class="math-container">$x$</span> and <span class="math-container">$n$</span> are real numbers. So, things like <span class="math-container">$\,0.751^{3.2131}$</span> can be ... | Cheerful Parsnip | 2,941 | <p>A standard trick is to calculate the natural logarithm first to get the exponent under control:</p>
<p>$$\log(\lim_{a\to\infty}(n(x^{1/a}-1)+1)^a)=\lim_{a\to\infty}a\log(nx^{1/a}-n+1)$$
Set $u=1/a$.
We get
$$\lim_{u\to 0}\frac{\log (nx^u-n+1)}{u}$$
Use L'Hopital:
$$\lim_{u\to 0}\frac{nx^u\log x}{nx^u-n+1}=n\log x=\... |
2,908,186 | <p>There's an old-school pocket calculator trick to calculate <span class="math-container">$x^n$</span> on a pocket calculator, where both, <span class="math-container">$x$</span> and <span class="math-container">$n$</span> are real numbers. So, things like <span class="math-container">$\,0.751^{3.2131}$</span> can be ... | Barry Cipra | 86,747 | <p>If $x$ (actually $\ln x$) is relatively small, then $x^{1/4096}=e^{(\ln x)/4096}\approx1+(\ln x)/4096$, in which case</p>
<p>$$n(x^{1/4096}-1)+1)\approx1+{n\ln x\over4096}$$</p>
<p>If $n$ is also relatively small, then</p>
<p>$$(n(x^{1/4096}-1)+1)^{4096}\approx\left(1+{n\ln x\over4096}\right)^{4096}\approx e^{n\l... |
256,138 | <p>I need to generate four positive random values in the range [.1, .6] with (at most) two significant digits to the right of the decimal, and which sum to exactly 1. Here are three attempts that do not work.</p>
<pre><code>x = {.15, .35, .1, .4}; While[Total[x] != 1,
x = Table[Round[RandomReal[{.1, .6}], .010], 4]];... | creidhne | 41,569 | <p>Use <a href="https://reference.wolfram.com/language/ref/Normalize.html" rel="nofollow noreferrer">Normalize</a> to find a list of random values between 0.1 and 0.6, which are rounded to two decimal places, and that total to 1.</p>
<pre><code>rr = 0;
While[Total[rr] != 1,
rr = Round[Normalize[RandomReal[{.1, .6}, 4... |
256,138 | <p>I need to generate four positive random values in the range [.1, .6] with (at most) two significant digits to the right of the decimal, and which sum to exactly 1. Here are three attempts that do not work.</p>
<pre><code>x = {.15, .35, .1, .4}; While[Total[x] != 1,
x = Table[Round[RandomReal[{.1, .6}], .010], 4]];... | martin | 9,923 | <pre><code>RandomSample[IntegerPartitions[100,{4},Range[10,60]]/100.,1]
</code></pre>
|
239,387 | <p>Someone could explain how to build the smallest field containing to $\sqrt[3]{2}$.</p>
| Haskell Curry | 39,362 | <p>Consider the set $ F $ of expressions of the form
$$
\left\{ a + b \cdot \sqrt[3]{2} + c \cdot \left( \sqrt[3]{2} \right)^{2} \,\Big|\, (a,b,c) \in \mathbb{Q}^{3} \right\}.
$$
This is the smallest subfield of $ \mathbb{R} $ containing $ \mathbb{Q} $ and $ \sqrt[3]{2} $.</p>
<p>Clearly, $ \mathbb{Q} \cup \{ \sqrt[3]... |
1,438,512 | <p>I was given this problem at school to look at home as a challenge, after spending a good 2 hours on this I can't seem to get further than the last part of the equation. I'd love to see the way to get through 2) before tomorrow's lesson as a head start.</p>
<p>So the problem is as follows:</p>
<p>1) Quadratic Equat... | juantheron | 14,311 | <p>Given $\displaystyle \alpha,\beta$ are the roots of $2x^2+8x+4=0.$ </p>
<p>So$\displaystyle \alpha+\beta = -\frac{8}{2}=-4$ and $\displaystyle \alpha\cdot \beta = \frac{1}{2}.$</p>
<p>Now for Second part, Using $\bullet\; \bf{x^2-(sum \; of \; roots)x+(product\; of \; roots) =0}$</p>
<p>So here $\displaystyle \bf... |
1,791,849 | <p>On the <a href="https://en.wikipedia.org/wiki/Pentagon" rel="nofollow">Wikipedia Page about Pentagons</a>, I noticed a statement in their work saying that $\sqrt{25+10\sqrt{5}}=5\tan(54^{\circ})$ and $\sqrt{5-2\sqrt{5}}=\tan(\frac {\pi}{5})$</p>
<p>My question is: How would you justify that? My goal is to simplify ... | lab bhattacharjee | 33,337 | <p>As $54\cdot5\equiv90\pmod{180},$</p>
<p>if $\tan5x=\infty,5x=180^\circ n+90^\circ$ where $n$ is any integer</p>
<p>$x=36^\circ n+18^\circ$ where $n\equiv0,\pm1,\pm2\pmod5$</p>
<p>Using $\tan5x$ <a href="http://mathworld.wolfram.com/Tangent.html" rel="nofollow">expansion</a>, the roots of $$5t^4-10t^2+1=0$$ are $\... |
3,363,810 | <p>I;m not sure how to go about proving this. I just started learning about it and would appreciate some help.</p>
| Da Mike | 346,084 | <p>When dealing with equalities about sets, a good technique is to show that the left side of the equation is always a subset of the right side, and that the right side is also a subset of the left. Thus, the sets are equal. In your case, you would want to prove that </p>
<p><span class="math-container">$$
X \cup (X \... |
496,178 | <p>Let $t$ be a positive real number. Differentiate the function</p>
<blockquote>
<p>$$g(x)=t^x x^t.$$</p>
</blockquote>
<p>Your answer should be an expression in $x$ and $t$.</p>
<p>came up with the answer </p>
<blockquote>
<p>$$(x/t)+(t/x)\ln(t^x)(x^t)=\ln(t^x)+\ln(x^t)=x\ln t+t\ln x .$$</p>
</blockquote>
<p... | Michael Hoppe | 93,935 | <p>Why not try logarithmic differentiation:</p>
<p>$$\ln\bigl((g(x)\bigr)=x\ln(t)+t\ln(x),$$
thus
$$\frac{g'(x)}{g(x)}=\ln(t)+\frac{t}{x},$$
from which $g'$ is easily onbtained.</p>
<p>Michael</p>
|
931,748 | <p>I do not understand this solution and this formula and why we are using (1+1)^n...
I need some help to get an idea of what is going on here
Thanks</p>
<p><img src="https://i.stack.imgur.com/S9IL5.png" alt="enter image description here"></p>
| Rene Schipperus | 149,912 | <p>One way is that the assumption implies that $x$ must be odd, so $x=2n+1$ and thus $$x^2=4n^2+4n+1$$</p>
|
931,748 | <p>I do not understand this solution and this formula and why we are using (1+1)^n...
I need some help to get an idea of what is going on here
Thanks</p>
<p><img src="https://i.stack.imgur.com/S9IL5.png" alt="enter image description here"></p>
| lab bhattacharjee | 33,337 | <p>$x^2\equiv1\pmod2\iff x\equiv1\pmod2,x=2a+1$ where $a$ is any integer</p>
<p>$$\implies x^2=(2a+1)^2=8\frac{a(a+1)}2+1\equiv1\pmod8$$</p>
|
3,221,652 | <p><span class="math-container">$ a= \frac{2}{5}; f(x) = 10x^3 +6x^2 + x -2 $</span></p>
<p>Have difficulties with advanced polynomials of this type. I can not factor.To find the right root. Here is my solution:</p>
<p><span class="math-container">$f(x) = 10x^3 +6x^2 + x -2 = 10x^3 +6x^2 +1x^2 -1x^2 +x -2 = (5x^2+x)(... | PrincessEev | 597,568 | <p>If you want to show that <span class="math-container">$a$</span> is a root of <span class="math-container">$f$</span>, just plug in <span class="math-container">$a=2/5$</span> for <span class="math-container">$x$</span> in the definition of <span class="math-container">$f$</span>. That is to say, find <span class="m... |
1,901,302 | <p>Let $R$ be a principal ideal ring, meaning that every ideal is generated by one element. Given a subset $A\subseteq R$, it is generally not possible to choose one element $x\in A$ so that $I(\{x\})=I(A)$, i.e. the ideal generated by $A$ need not be generated by a single element of $A$.</p>
<p>As an example one can ... | Elle Najt | 54,092 | <p>PIDs are Noetherian, and Noetherian rings have the acc on ideals. So order some countable subset of A and consider the ideals generated by the first n. </p>
|
2,657,632 | <p>My question involves part (b) of Chapter 11 problem 6.4 in Artin's Algebra textbook.</p>
<blockquote>
<p>In each case, describe the ring obtained from <span class="math-container">$\mathbb{F_2}$</span> by adjoining an element <span class="math-container">$α$</span> satisfying the given relation:</p>
<p>(a) <span cla... | Magdiragdag | 35,584 | <p>The answer depends on how you interpret "adjoining an element $\alpha$ to $F$ satisfying $f(\alpha) = 0$".</p>
<p>The standard interpretation is: consider the algebraic closure $\bar F$ of $F$, take an element $\alpha \in \bar F$ satisfying $f(\alpha) = 0$ and look at the smallest field containing both $F$ and $\al... |
2,983,370 | <p><strong>question:</strong></p>
<p>maximum value of <span class="math-container">$\theta$</span> untill which the approximation <span class="math-container">$\sin\theta\approx \theta$</span> holds to within <span class="math-container">$10\%$</span> error is </p>
<p><span class="math-container">$(a)10^{\circ}$</spa... | Angina Seng | 436,618 | <p>For small theta
<span class="math-container">$$\frac{\sin\theta}\theta\approx1-\frac{\theta^2}6.$$</span>
So we get a <span class="math-container">$10\%$</span> error about where <span class="math-container">$\theta^2/6\approx 0.1$</span>, that is
<span class="math-container">$\theta\approx\sqrt{0.6}\approx0.8$</spa... |
592,805 | <p>How to prove $$\limsup\limits_{n \to \infty} \frac{1}{\sqrt n}B(n) = +\infty$$ using Blumenthal zero-one law, where $(B(t))_{t \geq 0}$ is a Brownian motion?</p>
| saz | 36,150 | <p>The idea of the proof is rather similar to the hints provided by AlexR., but using the time inversion it is easier to see how to apply Blumenthal's 0-1 law.</p>
<p><strong>Hints</strong>:</p>
<ol>
<li>It is widely known that $W_t := t \cdot B_{1/t}$ is a Brownian motion. This implies in particular that $$\limsup_{... |
273,086 | <p>I'd like to deploy a website to display and increment counters that track daily activities.</p>
<p>Here's a minimal working example:</p>
<pre><code>bin=CreateDatabin[]
CloudDeploy[FormPage[{"Pushups" -> {0, 5, 10, 15, 20, 25, 30},
"Water" -> {0, .5, 1, 2}},
Data... | lericr | 84,894 | <p>Let's stick with the Databin idea for now. It will give us the timestamp functionality for free. We can play with the form locally before deploying it.</p>
<ul>
<li><p>Create the databin</p>
<pre><code>exerciseDataBin = CreateDatabin["exercise"]
</code></pre>
</li>
<li><p>Define a function to update the da... |
4,508,840 | <p>I need to find the summation
<span class="math-container">$$S=\sum_{r=0}^{1010} \binom{1010}r \sum_{k=2r+1}^{2021}\binom{2021}k$$</span></p>
<p>I tried various things like replacing <span class="math-container">$k$</span> by <span class="math-container">$2021-k$</span> and trying to add the 2 summations to a pattern... | R. J. Mathar | 805,678 | <p>For <span class="math-container">$N=1010$</span> or others
<span class="math-container">\begin{equation}
S_N\equiv \sum_{r=0}^N\binom{N}{r}\sum_{k=2r+1}^{2N+1}\binom{2N+1}{k}
=
\end{equation}</span>
<span class="math-container">\begin{equation}
\sum_{r=0}^N\binom{N}{r}\sum_{k=2r+1}^{2N+1}\binom{2N+1}{k}
+
\sum_{r=0}... |
1,266,525 | <p>I'm shown part of the function $g(x) = \sqrt{4x-3}$. Is it injective? I said yes as per definition if $f(x) = f(y)$, then $x =y$. Is this right? </p>
<p>Under what criteria is $g(x)$ bijective? For what domain and co domain does $g(x)$ meet this criteria? I'm totally lost on this one! </p>
<p>Find the inverse of $... | wythagoras | 236,048 | <p>A function is bijective if it is surjective and injective. A function is surjective if the range is the codomain, i.e. if every value in the codomain is the output of the function. </p>
<p>An inverse function is the function reflected trough $y=x$. </p>
|
598,838 | <p><span class="math-container">$11$</span> out of <span class="math-container">$36$</span>? I got this by writing down the number of possible outcomes (<span class="math-container">$36$</span>) and then counting how many of the pairs had a <span class="math-container">$6$</span> in them: <span class="math-container">$... | amWhy | 9,003 | <p>Yes indeed, you've got them all. So counting them, we get $11$ of the possible $36$ outcomes of which at least one $6$ is rolled. Now simply express this probability as a fraction!</p>
|
598,838 | <p><span class="math-container">$11$</span> out of <span class="math-container">$36$</span>? I got this by writing down the number of possible outcomes (<span class="math-container">$36$</span>) and then counting how many of the pairs had a <span class="math-container">$6$</span> in them: <span class="math-container">$... | constructor | 158,202 | <p><a href="http://www.wolframalpha.com/input/?i=chance+of+throwing+1+6%27s+with+2+dice" rel="nofollow">http://www.wolframalpha.com/input/?i=chance+of+throwing+1+6%27s+with+2+dice</a></p>
<p>X = # of occurrences of 6 with 2 dice</p>
<p>P(X=1) = P(the first dice has 6, the second hasn't) + P(the second dice has 6, the... |
3,851,650 | <p>We want to prove that the number of sequences <span class="math-container">$(a_1,...,a_{2n})$</span> such that
<span class="math-container">$$
• \text{ every } a_i \text{ is equal to} ±1;\\• a_1 + ··· + a_{2n} = 0;\\• \text{ every partial sum satisfies } a_1 + ··· + a_i > −2
$$</span>
is a Catalan number.</p>
<p... | pipiyun | 825,006 | <p>This question is similar to the paths (0,2) to (2n,2) that do not touch X axis. The number of paths touches at least one point on the axis should be <span class="math-container">$2n \choose n+2$</span>.
The total number of paths should be <span class="math-container">$2n \choose n$</span>.
Then you can get the final... |
2,389,324 | <p>For a continuous Function $f$, prove that:</p>
<p>$$\lim_{x\to0^+}\int_{x}^{2x} \frac{1}{t} f(t) dt = ln(2)f(0)$$ </p>
<p>I have already concluded that since f is continuous, it's therefore integrable.Moreover I assumed there is a function $F$
$$F(x)=\int_{x}^{2x} f(s)ds$$
in order to simplify the limit express... | hamam_Abdallah | 369,188 | <p><strong>hint</strong> </p>
<p>the substitition $$t=ux $$ makes it easier.</p>
<p>$$I=\int_1^2 f (ux)\frac {du}{u} $$</p>
<p>use the second Mean formula and continuity at $x=0$.</p>
<p>$$I=f (c_x)\int_1^2\frac {du}{u} $$</p>
<p>Done.</p>
|
4,563,725 | <p>Is it possible for two non real complex numbers a and b that are squares of each other? (<span class="math-container">$a^2=b$</span> and <span class="math-container">$b^2=a$</span>)?</p>
<p>My answer is not possible because for <span class="math-container">$a^2$</span> to be equal to <span class="math-container">$b$... | Deepak | 151,732 | <p>It is entirely possible. Consider the complex conjugate cube roots of one. <span class="math-container">$\omega = -\frac 12 + \frac{\sqrt 3}{2}i$</span> and <span class="math-container">$\overline \omega = -\frac 12 - \frac{\sqrt 3}{2}i$</span></p>
<p>Note that <span class="math-container">$\omega = (\overline \omeg... |
3,190,601 | <p>Suppose <span class="math-container">$f$</span> is continuous on <span class="math-container">$[-1,1]$</span> and differentiable on <span class="math-container">$(-1,1)$</span>. Find:</p>
<p><span class="math-container">$$\lim_{x\rightarrow{0^{+}}}{\bigg(x\int_x^1{\frac{f(t)}{\sin^2(t)}}\: dt\bigg)}$$</span></p>
<... | zhw. | 228,045 | <p>Hint: L'Hopital is applicable in the case where the denominator <span class="math-container">$\to \infty,$</span> no matter what the numerator is doing.</p>
|
2,523,570 | <p>$15x^2-4x-4$,
I factored it out to this:
$$5x(3x-2)+2(3x+2).$$
But I don’t know what to do next since the twos in the brackets have opposite signs, or is it still possible to factor them out?</p>
| T C Molenaar | 497,768 | <p>It has to be $-2$:
$$15x^2-4x-4 = 5x(3x-2)+2(3x-2) = (5x+2)(3x-2)$$</p>
|
2,523,570 | <p>$15x^2-4x-4$,
I factored it out to this:
$$5x(3x-2)+2(3x+2).$$
But I don’t know what to do next since the twos in the brackets have opposite signs, or is it still possible to factor them out?</p>
| Vassilis Markos | 460,287 | <p>Note that:
$$15x^2-4x-4=15x^2-10x+6x-4=5x(3x-2)+2(3x-2)=(5x+2)(3x-2)$$</p>
<p>You probably missed a sign! ;)</p>
|
1,461,311 | <p>In axiomatic approach to real numbers, that is by defining them to be the complete ordered field, one is expected to prove every theorem and solve every problem by using ultimately only the axioms. I was trying to solve a Spivak's calculus problem that asked to show that the sum of any number of real numbers is stil... | Will R | 254,628 | <p>The Principle of Induction is a property of the natural number system; see, for example, <a href="https://en.wikipedia.org/wiki/Peano_axioms" rel="nofollow">Peano's axioms</a>. Hence, there isn't really any way to "prove" the Principle of Induction from the ordered field axioms, i.e., $(\text{P}1)$-$(\text{P}12)$ in... |
2,965,993 | <p>Suppose you have the surface <span class="math-container">$\xi$</span> defined in <span class="math-container">$\mathbb{R}^3$</span> by the equation:
<span class="math-container">$$ \xi :\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$</span>
For <span class="math-container">$ x \geq 0$</span> , <span clas... | Christian Blatter | 1,303 | <p>Assume for the moment <span class="math-container">$a=b=c=1$</span>. The surface <span class="math-container">$\xi$</span> then is the unit sphere <span class="math-container">$S^2$</span>. A typical point <span class="math-container">${\bf n}$</span> of <span class="math-container">$S^2$</span> in the first octant... |
699,933 | <p>There is a proof of the real case of Cauchy-Schwarz inequality that expands $\|\lambda v - w\|^2 \geq 0 $, gets a quadratic in $\lambda$, and takes the discriminant to get the Cauchy-Schwarz inequality. In trying to do the same thing in the complex case, I ran into some trouble. First, there are proofs <a href="http... | Thusle Gadelankz | 840,795 | <p>I recently revisited this problem and stumbled into another way around this, using a standard linear analysis trick. For the sake of completeness for future readers, I will add it in this answer.</p>
<p>It actually suffices to show that <span class="math-container">$$| \text{Re} \langle x,y \rangle | \leq \lVert x \... |
2,174,270 | <p>I have an elliptic curve $E$ over $\mathbb{F}_{7}$ defined by $y^2=x^3+2$ with the point at infinity $\mathcal{O}$</p>
<p>I am given the point $(3,6)$ and need to find the line which intersects with $E$ at <strong>only this point</strong></p>
<p>I am told that this line is $y\equiv (4x+1)\mod 7$</p>
<p>I have ver... | Travis Willse | 155,629 | <p>If the line $L$ intersects $E$ only at the given point $P$ (and the point $\mathcal O$ at infinity), then the intersection of $L$ and $E$ at $P$ must have multiplicity $2$, and so $L$ is the tangent line to $E$ at $P$. For an algebraic plane curve $\{p(x, y) = 0\}$, the tangent line at $P = (a, b)$ is
$$p_x(a, b)(x ... |
1,900,788 | <blockquote>
<p>The probability of rain in Greg's area on Tuesday is $0.3$. The probability that Greg's teacher will give him a pop quiz in Tuesday is $0.2$. The events occur independently of each other. What is the probability of neither events occur?</p>
</blockquote>
<p>My approach: </p>
<p>Probability of rain o... | JMoravitz | 179,297 | <p>Note the following two things:</p>
<blockquote>
<p><strong>Principle of inclusion-exclusion (2-set case)</strong>
<span class="math-container">$$Pr(A\cup B) = Pr(A)+Pr(B)-Pr(A\cap B)$$</span></p>
<p><strong>Definition of independent events</strong></p>
<p>The following are equivalent statements</p>
<ul>
<li><p><span... |
2,596,457 | <ul>
<li>If $\lim _{n\rightarrow \infty }a_{n}=a$ then $\left\{a_{n}:n\in\mathbb{N}\right\} \cup \left\{ a\right\}$ is compact.</li>
</ul>
<p><strong>I couldn't do anything. Can you give a hint?</strong></p>
<p>Note: in the question $a_n\in\mathbb{R}$.</p>
| Gödel | 467,121 | <p><strong>Hint:</strong> Prove that $\{a_n:n\in\mathbb{N}\}\cup\{a\}$ is closed and bounded</p>
|
2,596,457 | <ul>
<li>If $\lim _{n\rightarrow \infty }a_{n}=a$ then $\left\{a_{n}:n\in\mathbb{N}\right\} \cup \left\{ a\right\}$ is compact.</li>
</ul>
<p><strong>I couldn't do anything. Can you give a hint?</strong></p>
<p>Note: in the question $a_n\in\mathbb{R}$.</p>
| Kenny Lau | 328,173 | <p>Let $X$ be an open cover of $\{ a_n \mid n \in \Bbb N \} \cup \{a\}$.</p>
<p>Then, there is an open set containing $a$, called $X_a$.</p>
<p>Since $\displaystyle \lim_{n \to \infty} a_n = a$, $X_a$ must contain all but finitely many members of $\{ a_n \mid n \in \Bbb N \}$.</p>
<p>Then, pick the open sets contain... |
2,896,780 | <p>With a trusted 3rd party, running Secret Santa is easy: The 3rd party labels each person $1,\dotsc,n$, and then randomly chooses a derangement from among all possible derangements of $n$ numbers. Person $i$ will then give a gift to the number in position $i$ of the derangement. The trusted 3rd party is responsible... | Misha Lavrov | 383,078 | <p>Here's one possible strategy. Number the people $1, 2, \dots, n$. They do the following things, in order. (Whenever someone chooses a random number, I assume it's uniformly random from some fixed range $[1,N]$, say $[1,2n]$, excluding all values they've seen before.)</p>
<ol>
<li>Person $1$ picks a random number $x... |
3,946,974 | <blockquote>
<p>To prove:<span class="math-container">$$\text{If } A_1\subseteq A_2 \subseteq ... \subseteq A_n\text{ ,then } \bigcup_{i=1}^n A_i = A_n$$</span> using the axioms of ZFC Set Theory.</p>
</blockquote>
<p>Honestly, this statement is <strong>very obvious</strong>, but I do not want to take that for granted.... | Infinity_hunter | 826,797 | <p>This is regarding your second edit. First of all denoting <span class="math-container">$A_1 \subseteq A_2 .... \subseteq A$</span> is not correct when the index set <span class="math-container">$I$</span> is uncountable and moreover you don't know the elements of <span class="math-container">$I$</span>.</p>
<p>Take ... |
934,353 | <p>I am a high school student in Calculus, and we are finishing learning basic limits. I am reviewing for a big test tomorrow, and I could do all of the problems correctly except this one.</p>
<p>I have no idea how to solve the problem this problem correctly. I looked up the answer online, but I can't figure out how t... | Clinton Bradford | 167,198 | <p>As you did, convert to $\dfrac{1-\sqrt{1+x}}{x\sqrt{1+x}}$. (If you substitute, you still get $0$ in the denominator, so there is a little more work to be done.)</p>
<p>Then use the method of "algebraic conjugates":</p>
<p>$$\dfrac{1-\sqrt{1+x}}{x\sqrt{1+x}} \cdot \dfrac {1+\sqrt{1+x}}{1+\sqrt{1+x}} = \dfrac{1-(1+... |
732,334 | <p>How can we solve this equation?
$x^4-8x^3+24x^2-32x+16=0.$ </p>
| wendy.krieger | 78,024 | <p>You could factorise it, in the manner of $(x-2)^4=0$. I saw those factors immediately.</p>
<p>One process is to note that $16$ has divisors, and one can try various combinations of this such that the sum gives eight.</p>
<p>Possibilities include $2, 2, 2, 2$ and $4, 4, 1, -1$. However, one can not produce the se... |
803,800 | <p>Define a sequence $(a_n)_{n = 1}^{\infty}$ of real numbers by $a_n = \sin(n)$. This sequence is bounded (by $\pm1$), and so by the <a href="http://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem" rel="noreferrer">Bolzano-Weierstrass Theorem</a>, there exists a convergent subsequence.</p>
<p><strong>My que... | Dominic Michaelis | 62,278 | <p>I am not very confident that it converges, because this is how well you can approximate $10^k \cdot \pi$ with an integer. But as the difference will be infinite many times greater than $0.9$ and as you get bounds you see that this sequence is divergent.</p>
<p>I think this is more a question what is explicit for yo... |
47,561 | <p>The Hilbert matrix is the square matrix given by</p>
<p>$$H_{ij}=\frac{1}{i+j-1}$$</p>
<p>Wikipedia states that its inverse is given by</p>
<p>$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-i}}{{i+j-2}\choose{i-1}}^2$$</p>
<p>It follows that the entries in the inverse matrix are all i... | Johann Cigler | 5,585 | <p>A simple proof is in the paper <a href="http://arxiv.org/abs/math/0609283" rel="noreferrer">"Fibonacci numbers and orthogonal polynomials"</a>
by Christian Berg.</p>
|
1,091,653 | <p>Is this correct ?</p>
<p>$$
\frac{d}{dt} \left( \int_0^t \phi(t)dt \right) = \phi(t)
$$</p>
<p>If not, how can I recover $$ \phi(t) $$ knowing only $$ \int_0^t \phi(t)dt $$ ?</p>
| drhab | 75,923 | <p>The expression $\int_{0}^{t}\phi\left(t\right){\rm d}t$ makes no
sense to me. It cannot be that $t$ is a variable and a constant at
the same time. I presume that you mean $\int_{0}^{t}\phi\left(x\right){\rm d}x$ </p>
<p>For $t>0$ prescribe $\phi(t)$ by $t\mapsto 0$ if $t\neq 1$ and $t\mapsto c$ otherwise, where ... |
164,328 | <p>What are good introductory textbooks available on Cohomology of Groups?</p>
| Zurab Silagadze | 32,389 | <p>At the undergraduate level, some elements of group cohomology is explained in John Moody's book "Groups for Undergraduates" <a href="http://rads.stackoverflow.com/amzn/click/9810221053" rel="nofollow">http://www.amazon.com/Groups-Undergraduates-John-Atwell-Moody/dp/9810221053</a> At more advanced (but still undergr... |
1,553,440 | <p>Let , $g(x)=f(x)+f(1-x)$ and $f'(x)<0$ for all $x\in (0,1)$. Then , $g$ is monotone increasing in </p>
<p>(A) $(1/2,1)$.</p>
<p>(B) $(0,1/2)$</p>
<p>(C) $(0,1/2)\cup (1/2,1)$.</p>
<p>(D) none.</p>
<p>We have , $g'(x)=f'(x)-f'(1-x)$. Now $g'(x)>0$ if $f'(x)>f'(1-x)$. As $f'(x)<0$ so , $f$ is monotone... | Hamit | 277,958 | <p>Can u conclude this inequality???
$$f(\frac{2}{8})+f(\frac{6}{8})<f(\frac{1}{8})+f(\frac{7}{8}) $$
So $D$ is the answer.</p>
|
2,965,756 | <p>I want to intuitively say tht the answer is yes, but if it so happens that <span class="math-container">$|\mathbf{a}|=|\mathbf{b}|cos(\theta)$</span>, where <span class="math-container">$\theta$</span> is the angle between the two vectors, then the equation will be satisfied without the two vectors being the same.</... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$a$</span> is the zero vector and <span class="math-container">$b$</span> is not zero you get a contradiction to your statement. </p>
|
263,983 | <p>Is there a method to measure <strong>theoretic time load</strong> (hope +theoretic memory load) of an evaluation?</p>
<p>They should not change - <strong>always generate same value after pressing Enter key</strong>(=evaluation).</p>
<p>Hope they are even same in different environment(=different computers).</p>
<p>Fo... | Syed | 81,355 | <p>Mathematica runs on general-purpose operating systems (Windows/Linux/MacOS). On such systems, a multitude of services (daemons) are constantly running in the background. These services load the system differently at different times. The OS alone determines when programs get access to the requested resource (CPU, Mem... |
2,491,881 | <p>I'm trying to factorise $(4 + 3i)z^2 + 26iz + (-4+3i)$.</p>
<p>I tried to use the quadratic formula to factorise $(4 + 3i)z^2 + 26iz + (-4+3i)$, where $b = 26i$, $a = (4 + 3i)$, $c = (-4+3i)$. This gives us the roots $z = \dfrac{-4i - 12}{25}$ and $z =-4i - 3$.</p>
<p>But $\left( z + \dfrac{4i + 12}{25} \right) \l... | DanielWainfleet | 254,665 | <p>Let $A=\{n: x_n\in X\}$ and $B=\{n: x_n\in Y\}.$ At least one of $A,B$ is an infinite set. </p>
<p>If $A$ is infinite let $A=\{f(n):N\in\Bbb N\}$ where $f(n)<f(n+1).$ The sequence $\sigma=(x_{f(n)})_{n\in \Bbb N}$ is a sub-sequence of $(x_n)_{n\in\Bbb N}.$ </p>
<p>All of the entries of $\sigma$ belong to $X$ s... |
774,332 | <p>I have a radial Schrödinger equation for a particle in Coulomb potential:</p>
<p>$$i\partial_t f(r,t)=-\frac1{r^2}\partial_r\left(r^2\partial_r f(r,t)\right)-\frac2rf(r,t)$$</p>
<p>with initial condition</p>
<p>$$f(r,0)=e^{-r^2}$$</p>
<p>and boundary conditions</p>
<p>$$\begin{cases}
|f(0,t)|<\infty\\
|f(\in... | Dmoreno | 121,008 | <p>Regarding the regularity condition when applying finite differences, some authors make the following approach:</p>
<p>Consider the RHS of the equation at $r \to 0 $ and compute the limit with the help of L'Hôpital's rule:</p>
<p>$$\lim_{r\to 0} \left(- \frac{1}{r^2} \partial_r (r^2 f_r) - \frac{2}{r} f \right) = ... |
774,332 | <p>I have a radial Schrödinger equation for a particle in Coulomb potential:</p>
<p>$$i\partial_t f(r,t)=-\frac1{r^2}\partial_r\left(r^2\partial_r f(r,t)\right)-\frac2rf(r,t)$$</p>
<p>with initial condition</p>
<p>$$f(r,0)=e^{-r^2}$$</p>
<p>and boundary conditions</p>
<p>$$\begin{cases}
|f(0,t)|<\infty\\
|f(\in... | Ruslan | 64,206 | <p>Another approach to set a regularity condition is to make a change of variables to switch the type of boundary condition. For this equation, if $f(r)$ is regular at $r=0$, then $\frac{\text{d}}{\text{d}\rho}f(\rho^2)=0$ at $\rho=0$.</p>
<p>We can switch to $g(\rho)=f(\rho^2)$ and rewrite the equation for $g(\rho)$:... |
3,343,550 | <p>Show that if <span class="math-container">$(a,15)=1$</span>, then <span class="math-container">$a^4\equiv1 \mod 15$</span>, so that we do not have primitive roots of <span class="math-container">$15$</span>
Please help me with this problem.</p>
| Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Use the <em>Chinese remainder theorem</em> to check that if <span class="math-container">$a$</span> is coprime with <span class="math-container">$15$</span>, i.e. prime to <span class="math-container">$3$</span> and to <span class="math-container">$5$</span>, the order of <span class="... |
2,319,766 | <p>Lets say ,I have 100 numbers(1 to 100).I have to create various combinations of 10 numbers out of these 100 numbers such that no two combinations have more than 5 numbers in common given a particular number can be used max three times.
E.g.</p>
<ol>
<li>Combination 1: 1,2,3,4,5,6,7,8,9,10</li>
<li>Combination 2: ... | John | 7,163 | <p>Here's a hint on applying <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)" rel="noreferrer">stars and bars</a> to your problem.</p>
<p>The guy has $14$ animals of some combination of type. The animals are the "stars":</p>
<p>$$\star\star\star\star\star\star\star\star\star\star\star\star\star\... |
3,029,585 | <p>I have been stumped for a few days on this...It would be great if anyone can point me to enlightenment :)</p>
<p>Here's what I have tried. Let <span class="math-container">$Y = X^3$</span>, where X is a standard normal distribution with mean 0 and variance 1. Then</p>
<p><span class="math-container">$P(Y \leq y) =... | Hagen von Eitzen | 39,174 | <p>Any element <span class="math-container">$\alpha$</span> of <span class="math-container">$F$</span> is a rational expression in the numbers adjoined. As such, it can involve only <em>finitely</em> many of the <span class="math-container">$\sqrt[2^k]3$</span>. If in such an expression, <span class="math-container">$\... |
3,481,531 | <p>I was solving a problem where there are 4 points and we need to check if they form a square or not. Now if all 4 points are the same, is it square or not. My initial thought was it is a zero-sided square but wanted to clarify if my understanding is right or not. </p>
| Z Ahmed | 671,540 | <p>A point say (h,k) is a point circle:of zero radius.
<span class="math-container">$$(x-h)^2+(y-k)^2=0$$</span> or also a point-square <span class="math-container">$$|x-h|+|y-k|=0.$$</span></p>
|
3,518,232 | <p>I am given with a Triangle <span class="math-container">$ABC$</span> which is inscribed in circle <span class="math-container">$\omega$</span>. The tangent lines to <span class="math-container">$\omega$</span> at <span class="math-container">$B$</span> and <span class="math-container">$C$</span> meet at <span class=... | dan_fulea | 550,003 | <p>Here is a rewritten version of the OP with a picture. The order of "inventing" the points (in the solution using inversion) is the same as the order of the "inverted" points in a lemma. The lemma is simple, extracts the essence and gives the bridge for the solution. (Writing the lemma is not so simple, points have t... |
1,059,989 | <p>I mean are there examples of problems that have been proven to be undecidable, in the sense that it would not be possible to devise a deterministic computer program that outputs a solution for an instance of the problem. And yet human mathematicians have come up with such a solution.</p>
| Matt Samuel | 187,867 | <p>It is common to find that undecidable problems often have a solution in certain special cases even though there is no general solution. The halting problem is an excellent example; certainly it is possible to prove that (most) programs written by humans intended for a practical purpose do eventually halt.</p>
|
2,893,388 | <p>My textbook is confusing me a little. Here is a worked example from my textbook:</p>
<blockquote>
<p>Line <span class="math-container">$l$</span> has the equation <span class="math-container">$\begin{pmatrix}3\\ -1\\ 0\end{pmatrix}+\lambda \begin{pmatrix}1\\ -1\\ 1\end{pmatrix}$</span> and point <span class="math-c... | Simon Terrington | 302,396 | <p>OK so if $(X, \tau)$ has a countable basis then there may still be some bases that are not countable. If we let $X$ be the real line and $\tau$ the usual set of open sets then $\tau$ is a basis for the reals, as is the set of all open intervals, but the set of all open intervals with rational end points is a countab... |
52,677 | <p>Why people have to find quadratic formula,isn't that the formula cannot solve a polynomial with 2 and 1/2 degree?
and just curious, how many roots does a polynomial with 2 and 1/2 degree have and how to solve them all(by formula)?</p>
| André Nicolas | 6,312 | <p>A <strong>polynomial</strong> $P(x)$ in one indeterminate with coefficients in a ring $A$ is an expression of the form
$$a_nx^n+a_1x^{n-1}+ \cdots + a_0$$
where $n$ is an integer $\ge 0$ and the <em>coefficients</em> $a_0$, $a_1$, $\dots$, $a_n$ are elements of the ring $A$.</p>
<p>Familiar examples are polynomials... |
1,250,459 | <p>Let $A\in M_{m \times n}(\mathbb{R})$, $x\in \mathbb{R}^n$ and $b,y\in \mathbb{R}^m$. Show that if $Ax=b$ and $A^ty=0_{\mathbb{R}^m}$, then $\langle b,y\rangle=0$. Also make a geometric interpretation.</p>
<p>I think I may have something to do with overdetermined / underdetermined system, but do not know how to pro... | TravisJ | 212,738 | <p>I (personally) think of closed sets the way I explained here: <a href="https://math.stackexchange.com/questions/1185719/how-would-i-explain-an-open-set/1185749#1185749">How would I explain an 'open set'</a></p>
<p>One way to think of closed sets is they are the complement of an open set. (In many instances... |
1,434,486 | <p>In case the question didn't display in the title correctly: How do we bound the function $\frac{-1}{2}\sum_{m=n}^{\infty}\frac{1}{m^{2}}$?</p>
<p>I think a way I can do this is to show that $$\sum_{m=n}^{\infty}\frac{1}{m^{2}} < \frac{1}{n^{2}} + \int_{n}^{\infty}\frac{1}{x^{2}}$$</p>
<p>But I'm a little unsure... | André Nicolas | 6,312 | <p>Estimating via the integral works well. Another way is to note that the tail (if we forget about the $-\frac{1}{2}$ in front) is less than
$$\frac{1}{(n-1)(n)}+\frac{1}{(n)(n+1)}+\frac{1}{(n+1)(n+2)}+\cdots.$$</p>
<p>But $\frac{1}{(n-1)(n)}=\frac{1}{n-1}-\frac{1}{n}$ and the next term is $\frac{1}{n}-\frac{1}{n+1}$... |
1,434,486 | <p>In case the question didn't display in the title correctly: How do we bound the function $\frac{-1}{2}\sum_{m=n}^{\infty}\frac{1}{m^{2}}$?</p>
<p>I think a way I can do this is to show that $$\sum_{m=n}^{\infty}\frac{1}{m^{2}} < \frac{1}{n^{2}} + \int_{n}^{\infty}\frac{1}{x^{2}}$$</p>
<p>But I'm a little unsure... | DanielWainfleet | 254,665 | <p>The process is this: For every $m>n$ and for $m-1 \leq x <m$ let $f(x)=1/m$. And let $g(x)=1/x$ for all $x >0$. Then for all $m>n$ we have $f(x) \le g(x)$ with equality only when $x$ is an integer greater than $n $.Therefore $$\sum_{m=n}^{\infty} 1/m^2=1/n^2 + \sum_{m=n+1}^{\infty} 1/m^2=$$ $$=1/n^2 +\s... |
3,860,623 | <p>I'm trying to prove
<span class="math-container">$$\forall z\in\mathbb C-\{-1\},\ \left|\frac{z-1}{z+1}\right|=\sqrt2\iff\left|z+3\right|=\sqrt8$$</span>
thus showing that the solutions to <span class="math-container">$\left|(z-1)/(z+1)\right|=\sqrt2$</span> form the circle of center <span class="math-container">$-3... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$\frac{z-1}{z+1}=\sqrt 2 e^{i\theta} \implies z=\frac{1+\sqrt 2 e^{i\theta}}{1-\sqrt 2 e^{i\theta}}$$</span></p>
<p>and <span class="math-container">$$z+3= \frac{4-2\sqrt 2 e^{i\theta}}{1-\sqrt 2 e^{i\theta}} \implies |z+3|^2=\frac{4-2\sqrt 2 e^{i\theta}}{1-\sqrt 2 e... |
398,409 | <p>I was asked to help someone with this problem, and I don't really know the answer why. But I thought I'd still try.</p>
<p>$$\lim_{t \to 10} \frac{t^2 - 100}{t+1} \cos\left( \frac{1}{10-t} \right)+ 100$$</p>
<p>The problem lies with the cos term. What can I do with the cos term to remove divide by 0 ? </p>
<p>I f... | Benjamin Dickman | 37,122 | <p><strong>Incomplete Answer:</strong>
At least for the cube root case, this - like the square root case - follows from a theorem of Katok (2007). You can also get nice approximations in both the square root case and cube root case.</p>
<p>For more details on the above, see (<a href="http://www.ccsenet.org/journal/ind... |
398,409 | <p>I was asked to help someone with this problem, and I don't really know the answer why. But I thought I'd still try.</p>
<p>$$\lim_{t \to 10} \frac{t^2 - 100}{t+1} \cos\left( \frac{1}{10-t} \right)+ 100$$</p>
<p>The problem lies with the cos term. What can I do with the cos term to remove divide by 0 ? </p>
<p>I f... | Lubin | 17,760 | <p>Let me start from @Cantlog’s response, which points out that the difficulty is in knowing when $1+pa$ is an $m$-th power. The answer depends strongly on the divisibility of $m$ by $p$. In particular, if $m$ is indivisible by $p$, then $1+pa$ is <em>always</em> an $m$-th power. Indeed, as @Jyrki points out, $(1+pa)^z... |
1,108,918 | <p>Given two functions $f$ and $g$ whose derivatives $f'$ and $g'$ satisfy : $f'(x)=g(x), g'(x)=-f(x),f(0)=0, g(0)=1$ for all $x$ in an interval $J$. $~~~\cdots(A)$</p>
<p>(a) Prove that $f^2(x)+g^2(x)=1 ~\forall~x \in J $</p>
<p>(b) Let $F$ and $G$ be another pair of functions in $J$ which satisfy the given conditio... | Martin R | 42,969 | <p>You have already defined
$$
h(x)=[f(x)-F(x)]^2+[g(x)-G(x)]^2
$$
You only need to differentiate this equation:</p>
<p>$$
\begin{align}
h'(x) &= 2 [f(x)-F(x)][f'(x)-F'(x)] + 2 [g(x)-G(x)][g'(x)-G'(x)] \\
&= 2 [f(x)-F(x)][g(x)-G(x)] - 2 [g(x)-G(x)][f(x)-F(x)] \\
&= 0
\end{align}
$$</p>
<p>so that $h$ is... |
3,652,879 | <p>If <span class="math-container">$\langle x_n\rangle $</span> is a sequence of positive real numbers such that <span class="math-container">$$x_{(n+2)}=\frac{(x_{n+1}+ x_{n})}{2}$$</span> for all <span class="math-container">$n \in \mathbb{N},\ $</span> let <span class="math-container">$x_1 <x_2$</span></p>
<p>th... | Hrishabh | 776,474 | <p>Suppose that <span class="math-container">$x_n <= x_2$</span> for all n less than or equal to k then <span class="math-container">$x_{k+1}=\frac{{x_k}+x_{k-1}}{2} <= \frac{2x_2}{2}=x_2$</span> so by induction sequence is bounded.Now suppose <span class="math-container">$x_{n-1} <= x_{n}$</span> for all n le... |
1,540,412 | <p>Let x be a (right) eigenvector of A corresponding to an eigenvalue λ and let y be a left eigenvector of A corresponding to a different eigenvalue µ, where λ $\neq$ µ. Show that x∗y = 0. Hint : Ax = λx and y'A = µy'</p>
| Aaron | 9,863 | <p><strong>Hint:</strong> Calculate $y'Ax$ two different ways, and relate the answer to $y'x$.</p>
|
2,834,219 | <p>I'd like to numerically evaluate the following integral using computer software but it has a singularity at $x=1$:</p>
<p>\begin{equation}
\int_1^{\infty} \frac{x}{1-x^4} dx
\end{equation}</p>
<p>I was thinking of a variable transformation, rewriting the expression, or something of a kind. One of my attempts was t... | José Carlos Santos | 446,262 | <p>You can't do that, because$$\frac x{x-x^4}=\frac1{1-x}\times\frac x{1+x+x^2+x^3}$$and therefore near $1$ this functions behaves as $\frac1{4(1-x)}$. And if $a>1$, the integral$$\int_1^a\frac1{4(1-x)}\,\mathrm dx$$diverges.</p>
|
323,971 | <p>I know basic things about cardinality (I'm only in High School) like that since $\mathbb{Q}$ is countable, its cardinality is $\aleph_0$. Also that the cardinality of $\mathbb{R}$ is $2^{\aleph_0}$.</p>
<blockquote>
<p>Are there any direct applications of these numbers outside of theoretical math?</p>
</blockquot... | Asaf Karagila | 622 | <p>The answer is an obvious <em>no</em>. For two main reasons:</p>
<ol>
<li><p>With the exception of occasional naive approach to sets, there is little to no use of set theory outside theoretical mathematics. So any application would be indirect and purely coincidental.</p></li>
<li><p>Applied mathematics is not conce... |
368,800 | <p>Let <span class="math-container">$G$</span> be a finite group, let <span class="math-container">$X$</span> be a locally compact Hausdorff space, and let <span class="math-container">$G$</span> act freely on <span class="math-container">$X$</span>. It is well-known that the canonical quotient map <span class="math-co... | Mark Grant | 8,103 | <p>There is a general cohomological lower bound for the Schwarz genus of a map <span class="math-container">$p:E\to B$</span>. Namely, if there are cohomology classes <span class="math-container">$x_1,\ldots , x_k\in H^*(B)$</span> such that <span class="math-container">$0=p^*(x_i)\in H^*(E)$</span> for all <span class... |
1,344,883 | <p>I had an idea that passes by declaring a new type of <strong>computer variable</strong> (like Integer, Double, etc.) <strong>that represents a statistical probability distribution (PDF)</strong>, for that I would need to define the basic operations; sum, multiplication, inverse and negation.</p>
<p>The problem is t... | Tryss | 216,059 | <p>This can be rewritten as</p>
<p>$$ xe^x = 1$$</p>
<p>And here, there is no "simple" answer, you need to introduce the
<a href="https://en.wikipedia.org/wiki/Lambert_W_function#Applications" rel="nofollow">Lambert W function</a>.</p>
|
1,752,506 | <p>Question: $ \sqrt{x^2 + 1} + \frac{8}{\sqrt{x^2 + 1}} = \sqrt{x^2 + 9}$</p>
<p>My solution: $(x^2 + 1) + 8 = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) - \sqrt{x^2 + 9} \sqrt{x^2 + 1} = 0$</p>
<p>$=> \sqrt{x^2 + 9} (\sqrt{x^2 + 9} - \sqrt{x^... | Community | -1 | <p>The equation says$$ \sqrt{x^2 + 1} + \frac{8}{\sqrt{x^2 + 1}}=\frac{x^2 + 9}{\sqrt{x^2 + 1}} = \sqrt{x^2 + 9}.$$</p>
<p>So either $x^2+9=0$ and $x=\pm3i$, or $\sqrt{x^2+9}=\sqrt{x^2+1}$, which is impossible (by squaring).</p>
|
246,571 | <p>How can I calculate the following limit epsilon-delta definition?</p>
<p>$$\lim_{x \to 0} \left(\frac{\sin(ax)}{x}\right)$$</p>
<p>Edited the equation, sorry...</p>
| Brusko651 | 50,608 | <p>This is easy if you know the power series of the sine function.</p>
<p><span class="math-container">$$\sin(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} +\cdots$$</span></p>
<p>Then <span class="math-container">$$\sin(ax) = a x - \frac{a^3 x^3}{6} +... |
2,906,797 | <p>I want to express this polynomial as a product of linear factors:</p>
<p>$x^5 + x^3 + 8x^2 + 8$</p>
<p>I noticed that $\pm$i were roots just looking at it, so two factors must be $(x- i)$ and $(x + i)$, but I'm not sure how I would know what the remaining polynomial would be. For real roots, I would usually just d... | Mohammad Riazi-Kermani | 514,496 | <p>If you divide $$ x^5 + x^3 + 8x^2 + 8$$ by $$(x-i)(x+i) = x^2+1$$ you will get $$x^3+8$$ which factors as $$(x^3+8) = (x+2)(x^2-2x+4)$$ which has a solution of $x=-2$</p>
<p>Now use quadratic formula to solve $x^2-2x+4=0$ to find other roots and factor if you wish.</p>
|
1,187,142 | <p>Let a(n) be an arithmetic sequence and, as usual let $A(n)=a(1) + a(2) + \dots + a(n).$ If $A(5) = 50$ and $A(20) = 650$, find $A(15).$</p>
<p>I'm not so sure how to solve that.
So </p>
| Joffan | 206,402 | <p>Using $d$ to denote the common difference between terms, we can see that the terms from $a_6$ to $a_{10}$ are each exactly $5d$ more than the terms from $a_1$ to $a_5$. </p>
<p>This means that
$$\begin{align}A(10) &= 2A(5)+25d \\
\text{Similarly, }A(15) &= 3A(5)+75d \\
\text{ and } A(20) &=4A(5) + 150d... |
890,988 | <p>My book states that: </p>
<p>$f$ is a quasiconcave function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: </p>
<p>$f(x) \geq f(y) \implies f(tx + (1 - t)y) \geq f(y)$ </p>
<p>$f$ is a quasiconvex function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: </p>
<p>$f(y) \leq f(x) \implies f(tx + (1 - t)y) \le... | Community | -1 | <p>Nothing is wrong. Every monotone function is both quasiconvex and quasiconcave.</p>
<p>Indeed, the definition of quasiconvexity amounts to saying that on every closed interval, the function attains its maximum at an endpoint. Same for quasiconcavity, except replace maximum with minimum. Every monotone function has... |
890,988 | <p>My book states that: </p>
<p>$f$ is a quasiconcave function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: </p>
<p>$f(x) \geq f(y) \implies f(tx + (1 - t)y) \geq f(y)$ </p>
<p>$f$ is a quasiconvex function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: </p>
<p>$f(y) \leq f(x) \implies f(tx + (1 - t)y) \le... | user160518 | 451,930 | <p>I see incorrect answers on this topic everywhere and nobody is correcting them.
In the case of z= f(x,y)= xy, the graph that you draw ( rectangular hyperbola) is a level curve and not the function of the graph. The graph is originally in 3-D. To check its properties ( mainly quasi-concavity and quasi- convexity), l... |
890,988 | <p>My book states that: </p>
<p>$f$ is a quasiconcave function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: </p>
<p>$f(x) \geq f(y) \implies f(tx + (1 - t)y) \geq f(y)$ </p>
<p>$f$ is a quasiconvex function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: </p>
<p>$f(y) \leq f(x) \implies f(tx + (1 - t)y) \le... | nrazzak | 737,658 | <p>The function <span class="math-container">$f(x,y) = xy$</span> is neither quasiconcave nor quasiconvex and you can show that by solving the bordered Hessian.</p>
<p>For a formal definition, let <span class="math-container">$f(x,y)$</span> be a function with continuous partial derivatives and continuous cross partial... |
760,654 | <p>If $\lambda$ is the eigenvalue of matrix $A$,what is the eigenvalue of $A^TA$?I have no clue about it. Can anyone help with that?</p>
| Klaas van Aarsen | 134,550 | <p>Generally, you won't be able to say much about them.</p>
<p>However, if $A$ is for instance real symmetric, it is diagonalizable with real eigenvalues, meaning there is an orthogonal matrix B (that is with $B^{-1}=B^T$) and a diagonal matrix D such that:
$$A = B D B^T$$
$$A^TA = (B D B^T)^T B D B^T = B D^T B^T B D ... |
410,663 | <p>Can anyone in detail explain the procedure for finding the harmonic conjugate of the function $$u(x,y) = x^{2} - y\cdot (y+1)$$</p>
<p>I am new to this and I would like to know. </p>
| Pedro | 23,350 | <p>OK, so $$u_x(x,y)=2x$$ and $$u_y(x,y)=-2y-1$$</p>
<p>This means that $v_y(x,y)=2x$ and $v_x(x,y)=2y+1$. Integrating w.r.t to $y$ and $x$ respectively you get that $$v(x,y)=2xy+h(x)+C$$ $$v(x,y)=2xy+x+f(y)$$ thus $v(x,y)=2xy+x+C$, $C$ some constant.</p>
|
182,301 | <p>I have asked a question <a href="https://mathematica.stackexchange.com/questions/182247/constructing-possibilities-of-a-generating-function">here</a>.
I want to reproduce the coefficients of a generating function of the form:
<span class="math-container">$$(1 + x)^2 (1 + x + x^2 + x^3+\cdots+x^n)^{n-1}$$</span>
It ... | Ulrich Neumann | 53,677 | <p>Perhaps I didn't understand your question, but </p>
<pre><code>Table[ CoefficientList[(1 + x)^2 (Sum[x^i, {i, 0, n}])^(n - 1),x] , {n, 1, 5}]
</code></pre>
<p>evaluates the list of coefficients you're looking for.</p>
|
182,301 | <p>I have asked a question <a href="https://mathematica.stackexchange.com/questions/182247/constructing-possibilities-of-a-generating-function">here</a>.
I want to reproduce the coefficients of a generating function of the form:
<span class="math-container">$$(1 + x)^2 (1 + x + x^2 + x^3+\cdots+x^n)^{n-1}$$</span>
It ... | Daniel Lichtblau | 51 | <p>One can automate the generating construction readily enough using <code>Product</code> and <code>Sum</code>. Getting total degrees and then the subsets of balues that give them is a bit more work. I show one method below.</p>
<pre><code>gen[n_] := Module[
{vars, x, t, genFunc, coeffs},
vars = Array[x, n + 1, 0]... |
2,973,825 | <p>when you are checking to see if a sum of say <span class="math-container">$k^2$</span> from <span class="math-container">$k=1$</span> to to <span class="math-container">$k=n$</span> is equal to a sum of <span class="math-container">$(k+1)^2$</span> from <span class="math-container">$k=0$</span> to <span class="math-... | LeoDucas | 214,422 | <p>It's a cute one. Let's just assume the cards are numbered <span class="math-container">$\{1 \dots 52\}$</span>. We draw cards <span class="math-container">$a_1 < a_2 < a_3 < a_4 < a_5$</span>, and hand out in this order <span class="math-container">$b_1, b_2, b_3, b_4$</span>. The way we order this set <... |
869,358 | <p>This is a seemingly simple induction question that has me confused about perhaps my understanding of how to apply induction</p>
<p>the question;</p>
<p>$$\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\ \le\ 2-\frac{1}{n},\ \forall\ n \ge1.$$</p>
<p>this true for $n=1$, so assume the expression is true for $n\le k$. which pr... | Marco Flores | 164,575 | <p>What you really need is $2 − \frac{1}{k} + \frac{1}{(k+1)^2} \leq 2 − \frac{1}{(k+1)}$,</p>
|
1,545,945 | <p>Here my question :Perform the following arithmetic operations for the numbers in the bases indicated and write out answers in base 5 notation:
a) 244 + 132 (base 5).
b) 11101 × 111 (base 2).
c) F7 – B6 (base 16).
this is the first time I hear of(Base 5 notation)
can anyone help please ?</p>
| Lubin | 17,760 | <p>It’s valuable to have a little experience and skill in such computations. The idea is to think carefully of what you’re doing when you do decimal computations with paper and pencil, then carry the principles over to the unfamiliar situation.</p>
<p>To add $244_5$ and $132_5$, you start with the units digit, and add... |
1,545,945 | <p>Here my question :Perform the following arithmetic operations for the numbers in the bases indicated and write out answers in base 5 notation:
a) 244 + 132 (base 5).
b) 11101 × 111 (base 2).
c) F7 – B6 (base 16).
this is the first time I hear of(Base 5 notation)
can anyone help please ?</p>
| John Molokach | 90,422 | <p>You can think of different number bases like a variation of the base 10 decimal system we already use. I'll explain by working out the first problem you list and then leave the others to you.</p>
<p>In base 5, you would count starting at zero and have the following numbers:</p>
<p>$$\{0,1,2,3,4,10,11,12,13,14,20,... |
378,228 | <p>This is probably known, but I have not located a reference.</p>
<p>Let <span class="math-container">$P$</span> be the convex hull of <span class="math-container">$k$</span> points in <span class="math-container">$\mathbb R^n$</span> with rational coordinates. Consider the Euclidean square norm function <span class="... | Fedor Petrov | 4,312 | <p>Yes, this is true in general case. Let <span class="math-container">$P={\rm conv} M$</span> for a finite set <span class="math-container">$M\subset \mathbb{Q}^n$</span>, and let <span class="math-container">$x_0$</span> be a minimizer of <span class="math-container">$\|x\|^2$</span> over <span class="math-container"... |
2,647,868 | <p>I'm very confused at the following question:</p>
<blockquote>
<p>Find the basis for the image and a basis of the kernel for the following matrix:
$\begin{bmatrix} 7 & 0 & 7 \\ 2 & 3 & 8 \\ 9 & 0 & 9 \\ 5 & 6 & 17 \end{bmatrix}$</p>
</blockquote>
<p>I just don't know how to do an... | gbox | 103,441 | <p>$Im(T)$ is the column span.</p>
<p>For vectors to be a basis they need to be: </p>
<ol>
<li>linear independent </li>
<li>span the subspace</li>
</ol>
<p>So we are left to check for linear dependence $$\begin{pmatrix} 7 & 0 & 7 \\ 2 & 3 & 8 \\ 9 & 0 & 9 \\ 5 & 6 & 17 \end{pmatrix}\a... |
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