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 |
|---|---|---|---|---|
1,397,646 | <blockquote>
<p>Given $f_n:= e^{-n(nx-1)^2} $, any suggested approaches for showing that $\displaystyle \lim_{n \to \infty}\int^1_0 f_n(x)dx = 0$? </p>
</blockquote>
<p>I have already shown that $f_n \to 0$ pointwise (and not uniformly) on $[0,1]$ and have been trying to show that $$ \lim_{n \to \infty}\int^1_0 f_n(... | Claude Leibovici | 82,404 | <p>In the same spirit as Cameron Williams's answer, assuming you know the error function, $$ \int e^{-n(nx-1)^2}\,dx=\frac{\sqrt{\pi } }{2 n^{3/2}}\text{erf}\left(\sqrt{n} (n x-1)\right)$$ which makes $$ \int_0^1 e^{-n(nx-1)^2}\,dx=\frac{\sqrt{\pi } }{2 n^{3/2}}\Big(\text{erf}\left(\sqrt{n}\right)+\text{erf}\left((n-1... |
1,634 | <p>I am not too certain what these two properties mean geometrically. It sounds very vaguely to me that finite type corresponds to some sort of "finite dimensionality", while finite corresponds to "ramified cover". Is there any way to make this precise? Or can anyone elaborate on the geometric meaning of it?</p>
| David E Speyer | 297 | <p><a href="https://mathoverflow.net/questions/1634/finite-type-finite-morphism/1643#1643">Peter-arndt</a> gives exactly the right intuition. I will add a cautionary example: It is not literally true that finite fibers implies finite. For example, consider
Spec k[x, y]/(xy-1) ---> Spec k[x]. The way I think about this... |
758,466 | <blockquote>
<p>A fair 6-sided die is tossed 8 times. The sequence of 8 results is recorded to form an 8-digit number. For example if the tosses give {3, 5, 4, 2, 1, 1, 6, 5}, the resultant number is $35421165$. What is the probability that the number formed is a multiple of 8.</p>
</blockquote>
<p>I solved this by ... | Zhuli | 141,184 | <p>We only care about the last three digits obviously, so the possible sums of just those three die are $100a + 10b + c$, where $a$, $b$, and $c$ are all between $1$ and $6$, inclusive.</p>
<p>We check the divisibility by taking this modulo $8$. If the result is equivalent to $0$, then it is divisible.</p>
<p>Setting... |
758,466 | <blockquote>
<p>A fair 6-sided die is tossed 8 times. The sequence of 8 results is recorded to form an 8-digit number. For example if the tosses give {3, 5, 4, 2, 1, 1, 6, 5}, the resultant number is $35421165$. What is the probability that the number formed is a multiple of 8.</p>
</blockquote>
<p>I solved this by ... | Neat Math | 843,178 | <p>Let's use Zhuli's notation.</p>
<p><span class="math-container">$c$</span> has a probability of <span class="math-container">$\frac{1}{2}$</span> to be even.</p>
<p>Now if <span class="math-container">$\frac{c}{2}$</span> is even (i.e., <span class="math-container">$4|c$</span>), then <span class="math-container">$4... |
1,470,244 | <p>Prove that the sum of the moduli of the roots of $x^4-5x^3+6x^2-5x+1=0$ is $4$.<br></p>
<hr>
<p>I tried various methods to find its roots,but no luck.Is there any method to solve it without actually finding the roots of the equation.Please help me.<br>
Thanks.</p>
| implicati0n | 204,739 | <p>I'll give you an answer which is the application of a simple method for finding the solutions of symmetric polynomials:</p>
<p>Because $0$ is not a root, divide by $x^2$. Then use the substitution
$$
t=x+\frac{1}{x}.
$$</p>
<p>This yields</p>
<p>$$
t^2-5t+4=0,
$$</p>
<p>which you can solve in a simple matter an... |
4,070,810 | <p>I'm trying to find the number of subgroups of <span class="math-container">$\mathbb{Z}^n$</span> such that the quotient is a <span class="math-container">$\bf{cyclic}$</span> group of order <span class="math-container">$p^k$</span> (<span class="math-container">$p$</span> a prime).</p>
<p>I guess I don't know enough... | Limestone | 866,771 | <p>Observe that, <span class="math-container">$\triangle AEP\sim \triangle BEQ$</span> and hence <span class="math-container">$\angle APE=\angle BQE$</span>. From here, it is seen that quadrilateral <span class="math-container">$PCQE$</span> is cyclic.</p>
<p>Thus, <span class="math-container">$\angle CPQ=\angle CEQ=... |
1,002,191 | <p>I have a problem:</p>
<blockquote>
<p>A family has six children, A,B,C,D,E,F. The ages of these six children are added together to make an integer less than $32$ where none of the children are less than one year old.</p>
</blockquote>
<p>What is the chances of guessing the ages of all of the children.</p>
<p>I ... | André Nicolas | 6,312 | <p>We want to solve $x_1+\cdots+x_7=32$, where all the $x_i$ are $\ge 1$. For $x_7\ge 1$, which means $x_1+\cdots +x_6\lt 32$.</p>
<p>If you use $31$, then since $x_7\ge 1$, the ages of the real children would be forced to add up to less than $31$.</p>
<p><strong>Remark:</strong> This answers the Stars and Bars quest... |
306,337 | <p>I have a question.
Is this integral improper?
$$\int_0^\infty \frac{5x}{e^x-e^{-x}} \, dx = \int_0^a \frac{5x}{e^{x}-e^{-x}} \, dx+ \int_a^\infty \frac{5x}{e^x-e^{-x}} \, dx$$</p>
<p>Why is $\displaystyle\int_0^a \frac{5x}{e^{x}-e^{-x}}dx$ an improper integral at point $x=0$? And $\displaystyle\int_a^\infty \frac{5... | Jack Zega | 62,659 | <p>Maybe you can try a Taylor series expansion. Make the lower limit of your first integral on the RHS $L$ such that $0 < L \ll 1$, then let $L < a \ll 1$. You can do a Taylor series expansion of the denominator which will be $2x + O(x^3)$. You can cancel a factor of $x$ giving the integrand $(5/2)\cdot(1+O(x^2))... |
2,820,842 | <p>I am trying to prove that if $Q$ is an algebraic extension of $F$
($F$ is a field), if $R$ is a subring of $Q$, and we have
$F\subseteq R$, then $R$ is a subfield of $Q$.</p>
<p>My first approach was to show every element has an inverse. But there was no result.</p>
<p>Any help would be great.</p>
| nonuser | 463,553 | <p>Since $p\mid a-b$ we have $a\equiv _p b$, so</p>
<p>$$a^p-b^p = \underbrace{(a-b)}_{m\cdot p^n}(\underbrace{a^{p-1}+a^{p-2}b+...+b^{p-1}}_{\equiv p\cdot a^{p-1}}) = m\cdot p^n\cdot p \cdot k$$</p>
<p>where $a^{p-1}+a^{p-2}b+...+b^{p-1} =p\cdot k$ and obviously $p\not{|}\; mk$ </p>
|
167,536 | <p>It has been years since I took calculus and I am trying to figure out if it is possible to calculate the derivative of a mutli variable equation and what the result would be. This equation is from the video game Diablo 3 that represents your damage based on your critical chance and critical damage. I am trying to de... | Argon | 27,624 | <p>There are two variables in $z$, $x$ and $y$. Taking the <a href="http://en.wikipedia.org/wiki/Partial_derivative" rel="nofollow">partial derivative</a> in respect to $x$ (treating $y$ like a constant), we get:</p>
<p>$$\frac{\partial z}{\partial x}=y$$</p>
<p>similarly, the partial derivative in terms of $y$ is</... |
167,536 | <p>It has been years since I took calculus and I am trying to figure out if it is possible to calculate the derivative of a mutli variable equation and what the result would be. This equation is from the video game Diablo 3 that represents your damage based on your critical chance and critical damage. I am trying to de... | Micah | 30,836 | <p>As the other answer says, your partial derivatives are
$$\frac{\partial z}{\partial x} = y, \quad \frac{\partial z}{\partial y} = x \, . $$</p>
<p>What does this mean in terms of actual gameplay? It means that, if you currently have stats $x=x_0$, $y=y_0$, you should (approximately) value small changes in $x$ by mu... |
47,188 | <p>I am an amateur mathematician, and I had an idea which I worked out a bit and sent to an expert. He urged me to write it up for publication. So I did, and put it on arXiv. There were a couple of rounds of constructive criticism, after which he tells me he thinks it ought to go a "top" journal (his phrase, and he wen... | Community | -1 | <p>First of all, congratulations for making it as far as you have. I have a similar story, in 1986 I met with Stephen Wolfram and discussed with him my research. Wolfram was very supportive and even offered to edit and publish my research if I would just write it up. Then Wolfram left academia, forcing me to reach out ... |
737,212 | <p>Please how can I easily remember the following trig identities:
$$
\sin(\;\pi-x)=\phantom{-}\sin x\quad \color{red}{\text{ and }}\quad \cos(\;\pi-x)=-\cos x\\
\sin(\;\pi+x)=-\sin x\quad \color{red}{\text{ and }}\quad \cos(\;\pi+x)=-\cos x\\
\sin(\frac\pi2-x)=\phantom{-}\cos x\quad \color{red}{\text{ and }}\quad \co... | Blue | 409 | <p>I <a href="http://daylateanddollarshort.com/bloog/proof-without-words-angle-sum-and-difference-formulas/" rel="noreferrer">created the images</a> that "inspired" <a href="http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities" rel="noreferrer">the illustration of the Angle-... |
737,212 | <p>Please how can I easily remember the following trig identities:
$$
\sin(\;\pi-x)=\phantom{-}\sin x\quad \color{red}{\text{ and }}\quad \cos(\;\pi-x)=-\cos x\\
\sin(\;\pi+x)=-\sin x\quad \color{red}{\text{ and }}\quad \cos(\;\pi+x)=-\cos x\\
\sin(\frac\pi2-x)=\phantom{-}\cos x\quad \color{red}{\text{ and }}\quad \co... | colormegone | 71,645 | <p>Attempting to memorize all the relations you'll see in trigonometry is a daunting task, at least until you've worked with them for some time. For the sort of identities you're talking about, it can be helpful to go back to the unit circle and picture the angles there.</p>
<p><img src="https://i.stack.imgur.com/Fay... |
40,463 | <p>If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a linkage
(e.g., a rectangle) whose edges are geodesics of fixed length,
and whose vertices are joints, and again ask if it ... | Pierre Dehornoy | 9,248 | <p>This might be a cheat, but if you consider a linkage turning once around the meridian of a torus, it might be not flexible. Actually, a linkage homeomorphic to a circle that minimizes length in its homology class is not flexible.</p>
|
3,127,795 | <p>So I’m struggling to understand how to find the angle in this circle, we’ve recently learnt about trigonometry and like finding the area of a circle and all that but I can’t seem to remember which formula I have to use to find this angle. Can anyone lend a helping hand?
<img src="https://i.stack.imgur.com/V2Sm7.jp... | Jonathan Levy | 379,205 | <p>You have an isosceles triangle with two sides as the radius of 7. This is because they tell you O is the center. Then you have <span class="math-container">$\sin(\theta/2)=5/7$</span>. So the angle is <span class="math-container">$\sin^{-1}(5/7)*2$</span></p>
|
3,752,167 | <p>Freshman question, really, but the more I think about it, the more I doubt.</p>
<p>Suppose that two sets belong to the same equivalence class. Are they in effect interchangeable? (I understand that there is no axiom of `interchangeability' in the definition of an equivalence relation.)</p>
<p>For example, consider t... | Daniel | 391,594 | <p>I may be just rephrasing some of the answers, but I like to think that an equivalence relation is simply a problem specific notion of equality; the relation tries to summarize all the important properties that are relevant to the problem in question. In this sense, if any two objects are equivalent they can be used ... |
2,009,134 | <p>Suppose $$\frac{{{{\sin }^4}(\alpha )}}{a} + \frac{{{{\cos }^4}(\alpha )}}{b} = \frac{1}{{a + b}}$$ for some $a,b\ne 0$. </p>
<p>Why does $$\frac{{{{\sin }^8}(\alpha )}}{{{a^3}}} + \frac{{{{\cos }^8}(\alpha )}}{{{b^3}}} = \frac{1}{{{{(a + b)}^3}}}$$</p>
| Yuriy S | 269,624 | <p>There is a very direct way to solve this. First, use the given equation to find, for example, $\sin^2 \alpha$:</p>
<p>$$x=\sin^2 \alpha$$</p>
<p>$$\frac{1}{a} x^2+\frac{1}{b}(1-x)^2=\frac{1}{a+b}$$</p>
<p>Solving this easy quadratic, we get a single root:</p>
<p>$$x=\frac{a}{a+b}=\sin^2 \alpha$$</p>
<p>It foll... |
749,097 | <p>In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this <em>dream product rule</em> true? There are of course trivial examples, and also many instances where the equality is true at a handful of points. Less o... | David | 119,775 | <p>There are many cases where this will happen. By simple algebra, the relation $(uv)'=u'v'$ can be written as
$$\Bigl(\frac{u'}{u}-1\Bigr)\Bigl(\frac{v'}{v}-1\Bigr)=1$$
and this leads to
$$\ln u(t)=\int\frac{v'}{v'-v}\,dt\ .$$
You will have to make sure that the integral exists and gives a suitable result, but this s... |
209,728 | <p>I'm attempting to calculate the gradient of a function defined by two variables.
After following a tutorial I found, I have the following code: </p>
<pre><code>f[x_, y_] = (2 x^4 - x^2 + 3 x - 7 y + 1)/(e^(2 x^2/3 + 3 y^2/4));
(*t is gradient of f*)
t[{x_, y_}] := {
D[f[x, y], x],
D[f[x, y], y]
};
(*
Trying to use... | H. Khan | 67,854 | <p>Via @kglr: </p>
<pre><code>ClearAll[t];
t[{x_, y_}] = {
D[f[x, y], x],
D[f[x, y], y]
};
NestList[t, {.51, -.445}, n]
</code></pre>
|
268,482 | <p>One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was proven in the paper. </p>
<p>The work was reported on a few seminars, an... | Gil Kalai | 1,532 | <p>Yes, there are quite a few papers of this kind. Of course, if proving the implication to known theorems or between the various conjectures is interesting and challenging then this is "ordinary". If the paper contains only a statement of the conjectures then this is indeed more special. There are journals which have ... |
268,482 | <p>One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was proven in the paper. </p>
<p>The work was reported on a few seminars, an... | Piyush Grover | 30,684 | <p>Depending upon the content of the paper, you may look at the new Arnold Mathematical Journal (<a href="http://www.springer.com/mathematics/journal/40598" rel="noreferrer">http://www.springer.com/mathematics/journal/40598</a>).</p>
<p>From their site, "Problems, objectives, work in progress. Most scholarly publicati... |
268,482 | <p>One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was proven in the paper. </p>
<p>The work was reported on a few seminars, an... | Ofir Gorodetsky | 31,469 | <p>Supplementing Carlo's answer, I also want to recommend the journal 'Mathematics of Computation'. Several important number-theoretic conjectures were first stated there, supported with numerical data. So if some computation was involved in providing evidence for your conjectures, it seems like a suitable choice.</p>
... |
2,407,183 | <blockquote>
<p>Calculate $\int_{\lambda} dz/(z^2-1)^2$, where $\lambda$ is the path
in $\mathbb{R^2}-\{1,-1\}$ plotted below:</p>
<p><a href="https://i.stack.imgur.com/T1uFH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/T1uFH.png" alt="enter image description here"></a></p>
</blockquote>
... | Simply Beautiful Art | 272,831 | <p>I noticed that the integrand is faily quadratic between integer values, hence</p>
<p>$$-I=\underbrace{\int_0^2\frac{\sin(\pi x)}{\ln(x)}~\mathrm dx}_{\approx-3.64061894952}+\sum_{n=2}^\infty\int_n^{n+1}\frac{\sin(\pi x)}{\ln(x)}~\mathrm dx$$</p>
<p>$$\small\frac{\sin(\pi x)}{\ln(x)}=\frac{(-1)^{n+1}}{t^2\ln^3(t)}\... |
423,479 | <p>I know that $$\lim_{n\rightarrow \infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)}=1,$$ but I'm interested in the exact behaviour of </p>
<p>$$a_n =1- \left( \frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)} \right) ^2$$</p>
<p>particularily compared to $$b_n = \frac{1}{4n}$$</p>
<p>I haven't studied asympto... | Start wearing purple | 73,025 | <p>Using <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation#Stirling.27s_formula_for_the_gamma_function">Stirling formula</a> you can show that
$$\frac{\Gamma(x+\frac12)}{\sqrt{x}\,\Gamma(x)}=1-\frac{1}{8x}+\frac{1}{128x^2}+O(x^{-3})$$
as $x\rightarrow \infty$. In principle one can obtain as many terms in... |
2,161,463 | <p>I am in the final stages of a proof and need help. I have simplified my starting expression expression down to $\dfrac{v\Gamma(-1+\frac{v}{2})}{2\Gamma(\frac{v}{2})}$</p>
<p>I know the above expression is to equal $\frac{v}{v-2}$</p>
<p>I am having ahard time getting there.</p>
<p>I know $\Gamma(n) = (n-1)!$</p>
... | Kenny Wong | 301,805 | <p>Use $\Gamma(z+1) = z \Gamma(z)$.</p>
<p>See <a href="https://en.wikipedia.org/wiki/Gamma_function#Main_definition" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Gamma_function#Main_definition</a>.</p>
|
357,138 | <blockquote>
<p>If $a_1,a_2,\dotsc,a_n $ are positive real numbers, then prove that</p>
</blockquote>
<p>$$\lim_{x \to \infty} \left[\frac {a_1^{1/x}+a_2^{1/x}+.....+a_n^{1/x}}{n}\right]^{nx}=a_1 a_2 \dotsb a_n.$$ </p>
<p>My Attempt: </p>
<p>Let $P=\lim_{x \to \infty} \left[\dfrac {a_1^{\frac{1}{x}}+a_2^{\frac {1... | learner | 33,640 | <p>Let $$\begin{align}P &=\lim_{x \to \infty} \Big[\dfrac {a_1^{\dfrac{1}{x}}+a_2^{\dfrac {1}{x}}+.....+a_n^{\dfrac {1}{x}}}{n}\Big]^{nx}\\ \implies
\ln P &=\lim_{x \to \infty} \ln \Big[\dfrac {a_1^{\dfrac{1}{x}}+a_2^{\dfrac {1}{x}}+.....+a_n^{\dfrac {1}{x}}}{n}\Big]^{nx} \\&=\lim_{x \to \infty} nx \ln \... |
478,523 | <p>I'm trying to reason through whether $\int_{x=2}^\infty \frac{1}{xe^x} dx$ converges.</p>
<p>Intuitively, It would seem that since $\int_{x=2}^\infty \frac{1}{x} dx$ diverges, then multiplying the denominator by something to make it smaller would make it converge.</p>
<p>If I apply a Taylor expansion to the $\fra... | Mhenni Benghorbal | 35,472 | <p>Just notice that,</p>
<p>$$ x \geq 2 \implies \frac{1}{x}\leq \frac{1}{2}\implies \frac{1}{xe^{x}}\leq \frac{1}{2e^{x}}= \frac 1 2 e^{-x}. $$</p>
<p>So you can compare with the function $ \frac 1 2 e^{-x} $.</p>
|
815,418 | <p>Ok, so I've been playing around with radical graphs and such lately, and I discovered that if the </p>
<pre><code>nth x = √(1st x √ 2nd x ... √nth x);
</code></pre>
<p>Then</p>
<p>$$\text{the "infinith" } x = x$$</p>
<p>Example: </p>
<p>$$\sqrt{4\sqrt{4\sqrt{4\sqrt{4\ldots}}}}=4$$<br>
Try it yourself, type calc... | Emily | 31,475 | <p>Suppose $y = \sqrt{x\sqrt{x\sqrt{x\cdots}}}$.</p>
<p>Multiply both sides by $x$ and take the square root:</p>
<p>$$\sqrt{xy} = \sqrt{x\sqrt{x\sqrt{x\cdots}}} = y$$</p>
<p>Therefore, $\sqrt{xy} = y$, and solving we have $xy = y^2 \implies x = y$.</p>
|
803,608 | <p>I believe that the statement is True, and this is my argument:</p>
<p>Since there exists an element $((1-\sqrt{2})/(1-(\sqrt{2})^2)\in\mathbb{Q}[\sqrt{2}]$ such that their products gives $1$ (multiplicative identity), therefore ( $1+\sqrt{2}$ ) is a unit. </p>
<p>A $unit$ is an element $a\in R|ab=1$ where as $b\in... | Bill Dubuque | 242 | <p>Probably this exercise is a warmup for the proof that $\,\Bbb Q[\sqrt{2}]\,$ is a field. The key idea is simple: to invert $\,w = 1+\sqrt{2}\,$ simply <em>rationalize the denominator</em> of $\,\dfrac{1}{1+\sqrt 2},\,$ i.e. multiply the numerator and denominator by the conjugate $\,w' = 1-\sqrt{2}\,$ to force the de... |
4,077,223 | <p><span class="math-container">$\exists x \phi(x)$</span> in mathematics means there is something <span class="math-container">$c$</span>, s.t. <span class="math-container">$\phi(c)$</span> holds.
What is the formulation of that same idea as a inference rule, an axiom, or something else in a proof system in FOL?</p>... | Mauro ALLEGRANZA | 108,274 | <p>In the <a href="https://iep.utm.edu/nat-ded/#H7" rel="nofollow noreferrer">Natural Deduction</a> proof system we have a pair of inference rules for the existential (as well as the universal) quantifier:</p>
<blockquote>
<p><span class="math-container">$(\exists \text I) \ \varphi [x/a] \vdash \exists x \varphi$</spa... |
4,077,223 | <p><span class="math-container">$\exists x \phi(x)$</span> in mathematics means there is something <span class="math-container">$c$</span>, s.t. <span class="math-container">$\phi(c)$</span> holds.
What is the formulation of that same idea as a inference rule, an axiom, or something else in a proof system in FOL?</p>... | supinf | 168,859 | <p>There are different ways to introduce <span class="math-container">$\exists$</span> in a FOL proof system.</p>
<p>I think the most common approach is that <span class="math-container">$\exists$</span> is defined via <span class="math-container">$\forall$</span>:</p>
<p><span class="math-container">$$\exists x \phi(x... |
438,818 | <p>I am looking for all two consecutive integers $A$ and $A+1$, which can be represented as sums of two squares
$A=a^2+b^2$ and $A+1=c^2+d^2$, $a,b,c,d>0$.</p>
| Gerry Myerson | 8,269 | <p>Not a complete answer --- I doubt there is one --- but $$(n^2-n)^2+(n^2-n)^2,(n^2-2n)^2+(n^2-1)^2,(n^2-n-1)^2+(n^2-n+1)^2$$ gives three consecutive numbers, each a sum of two non-zero squares. This example is taken from <a href="http://www.math.ksu.edu/~cochrane/research/sumsquare.pdf" rel="nofollow">Cochrane and Dr... |
2,214,231 | <blockquote>
<p>Find the derivative of $\tan^3[\sin(2x^2-17)]$. </p>
</blockquote>
<p>Sorry if my question is a little too specific but I am confused on this trig equation. After completing the derivative I was wondering why does the $3tan^2$ not distribute to $sec^2$? Is there a rule for this? How come the exponent... | Alex Robinson | 366,219 | <p>Consider x = 0, at this point ln(o) is undefined, but you are really solving for ln(x) = 1, and the only thing that fits the bill is taking x to be e, which gives your solution as the coordinate (e,0)</p>
|
20,802 | <p>Look at the following example:</p>
<p>Which picture has four apples?</p>
<p>A<a href="https://i.stack.imgur.com/Tpm46.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Tpm46.png" alt="enter image description here" /></a></p>
<hr />
<p>B <a href="https://i.stack.imgur.com/AOv29.png" rel="noreferrer"><img src=... | Aeryk | 401 | <p>Perhaps "shows" instead of "has". If you asked me to show you 4 apples, I can't think of a logical argument in favor of me grabbing 5 apples and smiling smugly.</p>
|
20,802 | <p>Look at the following example:</p>
<p>Which picture has four apples?</p>
<p>A<a href="https://i.stack.imgur.com/Tpm46.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Tpm46.png" alt="enter image description here" /></a></p>
<hr />
<p>B <a href="https://i.stack.imgur.com/AOv29.png" rel="noreferrer"><img src=... | Mark Walker | 15,960 | <p>This was not the question:</p>
<p>"Which pictures have four apples?"</p>
<p>This test is a multiple choice question that does not appear to have other than the 4 unique choices: a, b, c, d.</p>
<p>There is no all of the above or b-d choice which can be selected.</p>
<p>If the test was a fill in the blank w... |
3,580,258 | <p>Hi: The definition I'll use is this: Let <span class="math-container">$F$</span> be an abelian group and <span class="math-container">$X$</span> a subset of <span class="math-container">$F$</span>. Then <span class="math-container">$F$</span> is a free abelian group on <span class="math-container">$X$</span> if for ... | Shaun | 104,041 | <p>The <span class="math-container">$X$</span> in the free abelian group definition is a collection of formal symbols. They are assumed to have the minimum properties necessary for <span class="math-container">$\langle X\rangle$</span> to be an abelian group. That is what it means to be "free" in this context.</p>
<p>... |
546,239 | <p>Myself and my Math teacher are at a disagreement in to what the proper method of solving the question <em>In how many ways can four students be chosen from a group of 12 students?</em> is.</p>
<p>The question comes straight out of a Math revision sheet from a Math book distributed under the national curriculum. The... | Benjamin Dickman | 37,122 | <p>You have 12 choices for the first student chosen, 11 choices for the next, then 10, then 9. However, this over-counts everything by a factor of 4! (the number of ways in which four objects can be arranged with regard to order).</p>
<p>Thus, the answer is</p>
<p>$$\frac{12\cdot 11 \cdot 10 \cdot 9}{4!} = 495$$</p>
|
2,203,770 | <p>If I am asked to find all values of $z$ such that $z^3=-8i$, what is the best method to go about that?</p>
<p>I have the following formula:
$$z^{\frac{1}{n}}=r^\frac{1}{n}\left[\cos\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)\right]$$</p>
<p>for $k=0,\pm1, \pm2... | haqnatural | 247,767 | <p>Short way write it as $$z^{ 3 }=-8i\\ z^{ 3 }=8{ e }^{ i\frac { 3\pi }{ 2 } }\\ z=\sqrt [ 3 ]{ 8 } { e }^{ i\left( \frac { \frac { 3\pi }{ 2 } +2\pi k }{ 3 } \right) }=2{ e }^{ i\left( \frac { 3\pi +4\pi k }{ 6 } \right) },k=0,1,2\quad $$</p>
|
2,203,770 | <p>If I am asked to find all values of $z$ such that $z^3=-8i$, what is the best method to go about that?</p>
<p>I have the following formula:
$$z^{\frac{1}{n}}=r^\frac{1}{n}\left[\cos\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)\right]$$</p>
<p>for $k=0,\pm1, \pm2... | Joe | 107,639 | <p>The problem with you solution is the following: you didn't wrote properly the modulus and the exponential part of $w:=-8i$.</p>
<p>Clearly
$$
r=|w|=8\\
-i=\exp{\left(\frac{3}{2}\pi i\right)}=\exp{\left(\frac{3}{2}\pi i+2ki\pi\right)}\;,\;\;k\in\Bbb Z
$$
so that
$$
z^3=r\exp[{i\pi\left(3/2+2k\right)}]\\
\Longrightar... |
199,199 | <p>Suppose a box contains 5 white balls and 5 black balls.</p>
<p>If you want to extract a ball and then another:</p>
<p>What is the probability of getting a black ball and then a black one?</p>
<p>I think that this is the answer:</p>
<p>Let $A:$ get a black ball in the first extraction, $B:$ get a black ball in th... | wnvl | 23,122 | <p>If the first ball is not replaced after extraction, both probabilities are the same.</p>
<p>$P(\text{first ball black and second ball black})$</p>
<p>$=P(\text{first ball black}) \cdot P(\text{second ball black } | \text{ first ball black})$</p>
<p>$=\frac{5}{10}\cdot\frac{4}{9}$</p>
<p>$=\frac{2}{9}$</p>
|
3,474,847 | <p><strong>Prop:</strong> The union of two open sets is open. </p>
<p><strong>Pf:</strong> Let <span class="math-container">$(X,d)$</span> be a metric space, and <span class="math-container">$U$</span> and <span class="math-container">$V$</span> are non empty and open sets such that <span class="math-container">$U\sub... | Fred | 380,717 | <p>With all due respect, your proof is chaotic. For <span class="math-container">$x \in X$</span> and <span class="math-container">$r>0$</span> let us denote by <span class="math-container">$B_r(x)$</span> the set</p>
<p><span class="math-container">$$B_r(x)= \{y \in X: d(y,x) <r\}.$$</span></p>
<p>If <span cla... |
3,606,255 | <p>I want to ask you something about the following quadratic inequality: <span class="math-container">$$9x^2+12x+4\le 0.$$</span> Let us find the "=0" points. Here we have <span class="math-container">$9x^2+12x+4=9(x+\frac{2}{3})^2\ge0$</span> (we can factor this by calculating the discriminant <span class="math-contai... | Matteo | 686,644 | <p>In this case there is no need to find the intersection with <span class="math-container">$y=0$</span>, because you re considering the inequality:
<span class="math-container">$$9(x+\frac{2}{3})^2\ge0$$</span>
which is always true for every <span class="math-container">$x$</span> in <span class="math-container">$R$</... |
1,351,881 | <p>This argument comes up once every while on Lambda the Ultimate. I want to know where the flaw is.</p>
<p><em>Take a countable number of TMs all generating different bitstreams. Construct a Cantor TM which runs every nth TM upto the nth bit and outputs the reversed bit.</em></p>
<p><em>Now I have established there ... | Marco | 247,273 | <p>There are only a countable number of constructible reals. The Cantor TM only proves that any enumeration can be extended with an extra element. Since it didn't start of with a bijection between the natural numbers and the reals, the construction doesn't prove that that bijection doesn't exist.</p>
<p>(I answered my... |
1,351,881 | <p>This argument comes up once every while on Lambda the Ultimate. I want to know where the flaw is.</p>
<p><em>Take a countable number of TMs all generating different bitstreams. Construct a Cantor TM which runs every nth TM upto the nth bit and outputs the reversed bit.</em></p>
<p><em>Now I have established there ... | Russell Easterly | 51,106 | <p>There are several problems with this argument. If you want to use contradiction to prove the set of all TM's is uncountable you must first assume there is an enumeration of all TM's. This "proof" only assumes we have a countably infinite set of TM's with unique output. There are a lot of such sets. One such is the s... |
2,606,999 | <p>$f: X \to Y$ prove that if $A \subseteq X$, then A is a subset of the pre-image of the image of $A$, which is shown by these symbols respectively: $f^{-1}[f[A]]$</p>
<p>This is my proof:</p>
<p>If $A \nsubseteq f^{-1}[f[A]]$, then there exists "$x$" as an element of $A$, such that $f^{-1}[f[A]] \neq x$
which is a ... | egreg | 62,967 | <p>You might prove it by contradiction like in your attempt. However
$$
A\not\subseteq f^{-1}[f[A]]
$$
means that</p>
<blockquote>
<p>there exists $x\in A$ such that $x\notin f^{-1}[f[A]]$</p>
</blockquote>
<p><em>There's a big difference between “$\notin$” and “$\ne$”.</em></p>
<blockquote>
<p>Since $x\notin f^... |
1,506,763 | <p>This is not true for infinite measures (<a href="https://math.stackexchange.com/questions/342039/pointwise-convergence-but-not-in-measure">Pointwise convergence, but not in measure</a>). Is it true for a finite measure? Namely, let a finite (probability) measure $\mu(\cdot)$. Does a point-wise convergence of $\mu-$... | Reveillark | 122,262 | <p>If $f_n \overset{a.e.}{\longrightarrow} f$ and the space has finite measure, by Egoroff's theorem $f_n$ converges to $f$ almost uniformly, and therefore converges in measure to $f$.</p>
|
1,506,763 | <p>This is not true for infinite measures (<a href="https://math.stackexchange.com/questions/342039/pointwise-convergence-but-not-in-measure">Pointwise convergence, but not in measure</a>). Is it true for a finite measure? Namely, let a finite (probability) measure $\mu(\cdot)$. Does a point-wise convergence of $\mu-$... | shalop | 224,467 | <p>Yeah this is true. You can use Egoroff, but a direct proof using only DCT is not difficult either.</p>
<p>Let $\epsilon>0$. Define $A_n:=\{x: |f(x)-f_n(x)|>\epsilon\}$. Since $f_n \to f$ pointwise, it follows that $1_{A_n} \to 0$ pointwise. Then notice that all of the functions $1_{A_n}$ are bounded above by ... |
1,515,417 | <p>I understand the idea that some infinities are "bigger" than other infinities. The example I understand is that all real numbers between 0 and 1 would not be able to "fit" on an infinite list.</p>
<p>I have to show whether these sets are countable or uncountable. If countable, how would you enumerate the set? If un... | ncmathsadist | 4,154 | <p>Set 1 is very countable; it can be placed into 1-1 correspondence with the integers. Set 2 is not countable. It is extremally disconnected and its intersection with $[0,1]$ is homeomorphic to a Cantor set.</p>
|
335,258 | <p>Find the domain of the function:
$$f(x)= \sqrt{x^2 - 4x - 45}$$</p>
<p>I'm just guessing here; how about if I square everything and then put it in the graphing calculator?
Thanks,
Lauri</p>
| amWhy | 9,003 | <p>The function $f(x)$ is not defined when $$x^2 - 4x - 45 \lt 0,\tag{1}$$ as the square root function is defined in the reals for non-negative reals only. The only valid "input" for the square root function is non-negative real numbers.</p>
<p>If a function $f(x)$ is not defined on an interval(s) = $I \subseteq \math... |
3,901,613 | <p>Determine <span class="math-container">$\int_Axy\space\space d(x,y)$</span>, when <span class="math-container">$A$</span> is a closed set, bounded by the curve <span class="math-container">$y=x-1$</span>, and the parabel <span class="math-container">$y^2 = 2x+6.$</span></p>
<p>I believe we only need to know the rang... | Z Ahmed | 671,540 | <p>Take log on base 2, then <span class="math-container">$$16^x \log_2 2=2^x \log_2 16. \implies 16^x= 4. 2^x$ \implies 2^{4x}=2^{x+2} \implies 4x=x+2 \implies x=2/3.$$</span></p>
|
3,901,613 | <p>Determine <span class="math-container">$\int_Axy\space\space d(x,y)$</span>, when <span class="math-container">$A$</span> is a closed set, bounded by the curve <span class="math-container">$y=x-1$</span>, and the parabel <span class="math-container">$y^2 = 2x+6.$</span></p>
<p>I believe we only need to know the rang... | user2661923 | 464,411 | <p>Alternative approach</p>
<p><span class="math-container">$$16^{2^x} = [2^4]^{2^x} = 2^{4 \times 2^x}.$$</span></p>
<p>Since this is equal to <span class="math-container">$2^{16^x}$</span> you have</p>
<p><span class="math-container">$$16^x = 4 \times 2^x \implies 8^x = 4 \implies x = (2/3).$$</span></p>
|
167,440 | <blockquote>
<p>Let $\Bbb F$ be a field of characteristic $p\gt 0$ and $f(x)=x^{p^n}-c \in\Bbb F[x]$ where $n$ is a positive integer. If $c \notin \{a^p:a\in \Bbb F \}$, show that $f$ is irreducible in $\Bbb F[x]$.</p>
</blockquote>
<p>I recently started studying Field theory, so I don't know How to approach this pr... | Georges Elencwajg | 3,217 | <p>Let $\gamma\in \mathbb F^{alg}$ be a root of $f(x)=x^{p^n}-c \in\Bbb F[x]$ in an algebraic closure of $\mathbb F$.<br>
Thus we have $f(x)=x^{p^n}-c=(x-\gamma)^{p^n}$.<br>
On the other hand let $m(x)\in \Bbb F[x]$ be the minimal polynomial of $\gamma$ over $\Bbb F$. </p>
<p>If $n(x)$ is an irreducible monic facto... |
3,317,088 | <blockquote>
<p>Let
<span class="math-container">\begin{align*}
\mathcal{P} : &\min_{x}~& (x + a)^2 \\
& s.t. & x + b \leq 0
\end{align*}</span></p>
</blockquote>
<p>where <span class="math-container">$ a, b \in \mathbb{R} $</span>. I would like to find <span class="math-contain... | Alireza Shamsian | 573,454 | <p>That's the point! As <span class="math-container">$R$</span> is a domain, we conclude <span class="math-container">$\langle0\rangle$</span> is a prime ideal of <span class="math-container">$R$</span>. So <span class="math-container">$\dim R=r$</span> means that there is a maximal chain like <span class="math-contain... |
30,402 | <p>The envelope of parabolic trajectories from a common launch point is itself a parabola.
In the U.S. soon many will have a chance to observe this fact directly, as the 4th of July is traditionally celebrated with fireworks.</p>
<p>If the launch point is the origin, and the trajectory starts off at angle $\theta$ and... | Andrey Rekalo | 5,371 | <ul>
<li><p>E. Torricelli, who was the last Galileo's secretary, suggested a purely geometrical method to find the envelope in his <em>De motu Proiectorum</em>. He also coined the term `parabola of safety'. Apparently it was the first example of computation of an envelope. The method is briefly described in <a href="h... |
1,119,524 | <p>Find $\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3}$</p>
<p>My professor showed us a few ways to compute the limit</p>
<p>1) Factor the numerator</p>
<p>$\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3} = \lim_{x \rightarrow 3} (x + 3) = 6$</p>
<p>2)</p>
<p>$\lim_{x \rightarrow 3} \frac{f(x) - f(3)}{x - 3} = f'(3) =... | davidlowryduda | 9,754 | <p>Let's get you started.</p>
<p>Suppose without loss of generality that $\mu_1 \leq \mu_2$. Then we have that
$$ \mu = x_1\mu_1 + x_2\mu_2 \leq (x_1 + x_2)\mu_2 = \mu_2.$$
Each other subcomponent of the proof looks extremely similar to this line.</p>
|
1,119,524 | <p>Find $\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3}$</p>
<p>My professor showed us a few ways to compute the limit</p>
<p>1) Factor the numerator</p>
<p>$\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3} = \lim_{x \rightarrow 3} (x + 3) = 6$</p>
<p>2)</p>
<p>$\lim_{x \rightarrow 3} \frac{f(x) - f(3)}{x - 3} = f'(3) =... | Brian M. Scott | 12,042 | <p>HINT: Without loss of generality assume that $\mu_1\le\mu_2$. Then</p>
<p>$$\mu=x_1\mu_1+x_2\mu_2=x_1\mu_1+x_2\big(\mu_1+(\mu_2-\mu_1)\big)=(x_1+x_2)\mu_1+\ldots$$</p>
<p>Finish this off, and it gives you half of what you want, and a similar idea applied to $\mu_2$ gets you the other hald.</p>
<p>Then try to appl... |
141,112 | <p>I'm trying to put the following equation in determinant form:
$12h^3 - 6ah^2 + ha^2 - V = 0$, where $h, a, V$ are variables (this is a volume for a pyramid frustum with $1:3$ slope, $h$ is the height and $a$ is the side of the base, $V$ is the volume). </p>
<p>The purpose of identifying the determinant is to constr... | user326210 | 326,210 | <p>You can prove, using Warmus's criteria, that the function can't be nomographed as it's currently written. Roughly speaking, it has four independent <span class="math-container">$h$</span> terms <span class="math-container">$(h^0, h^1, h^2, h^3)$</span> that can't be reduced further, but you can only directly nomogr... |
4,215,824 | <p>Is there a formal definition of "almost always less than" or "almost always greater than"? I think one could define it using probabilities but not sure how to go about it. If one could show the following, then I think you could say <span class="math-container">$X$</span> is almost always less tha... | User5678 | 632,875 | <p>Yes, a more typical terminology would be to say “almost surely (a.s.) less than <em>value</em>”</p>
<p><span class="math-container">$$X < c\;a.s \iff P(X<c)=1$$</span></p>
<p>Now, if the above holds <em>and</em> <span class="math-container">$\{\omega \in \Omega:X(\omega)\geq c\} = \emptyset$</span> then we can... |
1,932,599 | <p>Find the solution of the differential equation that satisfies the given initial condition <span class="math-container">$$~y'=\frac{x~y~\sin x}{y+1}, ~~~~~~y(0)=1~$$</span></p>
<p>When I integrate this function I get
<span class="math-container">$$y+\ln(y)= -x\cos x + \sin x + C.$$</span></p>
<p>Have I integrated ... | ComplexF | 369,196 | <p>What you can do is to use the so-called <em>Separation of variables</em> (see for example <a href="https://en.wikipedia.org/wiki/Separation_of_variables" rel="nofollow">https://en.wikipedia.org/wiki/Separation_of_variables</a>):</p>
<ol>
<li>Write $y' = \frac{dy}{dx}$</li>
<li>Do the following transformation: $\fra... |
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | Wuestenfux | 417,848 | <p>Well, <span class="math-container">$\frac{a+n}{b+n} = \frac{\frac{a}{n}+1}{\frac{b}{n}+1}$</span>. So if <span class="math-container">$n\rightarrow \infty$</span>, then <span class="math-container">$\frac{a}{n}\rightarrow 0$</span> and <span class="math-container">$\frac{b}{n}\rightarrow 0$</span>. Thus <span class=... |
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | José Carlos Santos | 446,262 | <p>You should start by thinking about particular cases. For instance, <span class="math-container">$\dfrac{3+2}{7+2}=\dfrac59$</span>, which is indeed closer to <span class="math-container">$1$</span> than <span class="math-container">$\dfrac37$</span>.</p>
<p>Anyway, note that, if <span class="math-container">$a<b... |
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | Steven Alexis Gregory | 75,410 | <p>Suppose <span class="math-container">$a,b,n \in \mathbb Q$</span>, <span class="math-container">$0 < a < b$</span> and <span class="math-container">$n > 0$</span>.</p>
<p><span class="math-container">$$\color{red}{\dfrac ab}
= \dfrac{a(b+n)}{b(b+n)}
= \dfrac{ab+an}{b(b+n)}
\color{red}{<} \dfrac{a... |
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | Milan | 503,397 | <p><span class="math-container">$$ f(x)=\frac{a+x}{b+x} $$</span> <span class="math-container">$ b > 0$</span></p>
<p><span class="math-container">$$\lim_{x→ ∞} \frac{a+x}{b+x}=1$$</span>
<span class="math-container">$$ f'(x)= \frac{b+x-a-x}{(b+x)^2}=\frac{b-a}{(b+x)^2}$$</span></p>
<p>We can conclude if <span cla... |
3,194,249 | <p>If we are given <span class="math-container">$d$</span> an integer, then I've seen it written that the floor of <span class="math-container">$\sqrt{d^2-1}=d-1$</span>, but how is this found?</p>
<p>It's not immediately obvious, at least not to me.It makes me feel there must be an algorithm that can be used . </p>
... | timon92 | 210,525 | <p>It is because <span class="math-container">$$(d-1)^2 \le d^2-1 < d^2,$$</span> so <span class="math-container">$$d-1 \le \sqrt{d^2-1} < d,$$</span> so <span class="math-container">$$\lfloor \sqrt{d^2-1}\rfloor = d-1.$$</span></p>
|
2,259,099 | <blockquote>
<p><em><strong>Theorem 29.2.</strong></em> Let <span class="math-container">$X$</span> be a Hausdorff space. Then <span class="math-container">$X$</span> is locally compact if and only if given <span class="math-container">$x \in X$</span>, and given a neighborhood <span class="math-container">$U$</span> o... | Chickenmancer | 385,781 | <p>The subspace $\overline{V}$ is compact since $Y$ is a compact space, and since closed subspaces of compact spaces are compact, you have the result.</p>
|
2,081,612 | <p>I'd like to evaluate the following integral:</p>
<p>$$\int \frac{\cos^2 x}{1+\tan x}dx$$</p>
<p>I tried integration by substitution, but I was not able to proceed. </p>
| lab bhattacharjee | 33,337 | <p>$$\dfrac{2\cos^2x}{1+\tan x}=\dfrac2{(1+\tan x)(1+\tan^2x)} =\dfrac{1+\tan^2x+1-\tan^2x}{(1+\tan x)(1+\tan^2x)}$$</p>
<p>$$=\dfrac1{1+\tan x}+\dfrac{1-\tan x}{1+\tan^2x}$$</p>
<p>$$=\dfrac{\cos x}{\cos x+\sin x}+\cos^2x(1-\tan x)$$</p>
<p>$\cos^2x(1-\tan x)$ can be managed easily.</p>
<p>Now $\dfrac{2\cos x}{\co... |
6,929 | <p>For example, I have posted quite a few questions regarding mathematics in general, because I want to learn more about the field and I just want to make sure I get an answer or an opinion from someone who has good knowledge about my question. I tag it with the advice tag and I have seen, more than once, that people w... | Tom Oldfield | 45,760 | <p>I think the problem is one of the kind of advice being sought. From the FAQ:</p>
<p>"You should only ask practical, answerable questions based on actual problems that you face."</p>
<p>All questions asked on the main site should be definitively answerable, in the sense that they should not be open to interpretatio... |
978,399 | <p>A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by 2^20, giving me a rough approximation of ln (5). I can't for the life of my figure out why this is.</p>
<p>Also: How can... | Travis Willse | 155,629 | <p>There is no expression for $\ln$, which is a transcendental function, in terms of radical functions and the usual arithmetic operations, whose combinations are, by contrast, algebraic functions.</p>
<p>You can, however, produce arbitrarily good approximations of $\log$ (or any other continuous function for that mat... |
1,721,345 | <p>Found a game related to knight's tour <a href="http://www.emathhelp.net/math-games-and-logic-puzzles/knight-score/" rel="nofollow" title="Knight's Score">Knight's Score</a></p>
<p>The basic idea is to gather as much points as possible, avoiding some cells.</p>
<p>What is the math of this?
Can algorithm be cons... | Robert Israel | 8,508 | <p>As Hagen noted, because there are only finitely many cells and you can't visit a cell more than once, the game is finite. Thus there is a "brute-force" algorithm to find an optimal path by examining all possible paths; of course, depending on the particular instance, the optimal value may turn out to be positive, n... |
27,666 | <p>I have reading a book myself; more than 90 % of the book is ok to me; but I have some problems on it. Is it ok to ask my questions in a related chat-room? Because I believe that they are so easy, and it seems shameful to me to ask such an absurd question?</p>
| hardmath | 3,111 | <p>Yes, one of the functions of the chatrooms is to allow discussion of problems in a "chatty" fashion that is not appropriate to the Q&A style of main Math.SE. Back and forth discussions often start in the Comment section of a Question and get migrated to Chat.</p>
|
2,736,286 | <p>Just wondering if I am anywhere near the correct answer.</p>
<p>We have the set {z $\in \mathbb{C} : |z - 1| < |z + i|$}</p>
<p>So we let $0 < \delta < |z + i| - |z - 1|$ then use the triangle inequality to prove $\delta + |z - 1| < |z + i|$.</p>
<p>Thanks!</p>
| Mohammad Riazi-Kermani | 514,496 | <p>So far so good.</p>
<p>Now you consider the open ball around your $z$ with radius $\delta /2$ and show that it is included in the region $|z-1|< |z+i|$ using the triangle inequality. </p>
|
2,913,156 | <p>I'm working through the definitions given in <strong>2.18</strong> in Baby Rudin, and I was wondering how to prove that <span class="math-container">$E = [0,1] \cap \Bbb{Q}$</span> is closed in <span class="math-container">$\Bbb{Q}$</span>.</p>
<p>Clearly, the set <span class="math-container">$[0,1] \cap \Bbb{R}$</... | Peter Szilas | 408,605 | <p>A bit formal :</p>
<p>Baby Rudin: Theorem 2.30.</p>
<p>Suppose $Y \subset X$. $A$ subset $E$ of $Y$ is open relative to $Y$</p>
<p>$\iff$</p>
<p>$E=Y \cap G$ for some open subset $G$ of $X$.</p>
<p>$B= [0,1]$ is closed in $\mathbb{R}$.</p>
<p>$B^c:= \mathbb{R}$\ $[0,1]$ is open in $\mathbb{R}$. </p>
<p>Hence ... |
897,955 | <p>For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof</p>
<p>I'm following a book and it just uses this, it doesn't say anything about the function, so I've not assumed it's one-to-one. I've spent to much time on this (as in it should be 4 lines for both ways at most) so I hope you ... | Ishfaaq | 109,161 | <p>Well the proofs seem okay. But it can be considerably shortened. I'm sure somewhere in the book you use, you will find the following definitions, </p>
<p>$$ f^{-1}(A) = \{ x \ | \ f(x) \in A \} $$
$$ f(B) = \{f(x) \ | \ x \in B\} $$</p>
<p>$\implies$</p>
<p>$x \in U \implies f(x) \in f(U) \implies f(x) \in V$ fr... |
54,856 | <p>I'm trying to simplify an expression with sums of <code>ArcTan</code>, for which I've written a transformation rule that works for simple sums, so that <code>ArcTan[a] + ArcTan[b] -> ArcTan[(a+b)/(1-ab)]</code>. However, it fails for terms of the type <code>5 ArcTan[a] + ArcTan[b]</code>. One way to simplify this... | Michael | 18,486 | <p>How about this?</p>
<pre><code>ArcTan[Simplify[TrigExpand[Tan[5 ArcTan[a] + ArcTan[b]]]]]
</code></pre>
<p><img src="https://i.stack.imgur.com/CC08o.png" alt="ArcTan Simplify"></p>
<p>Could save yourself some work!</p>
<h1>Explanation:</h1>
<p>The trick here is to understand the identity you mentioned is about ... |
1,578,378 | <p>Why does $$\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$$</p>
<p>This is used in a derivation of the Fourier coefficients. I see why they flip them but I don't understand why this is true and when we are allowed to do such a thing.</p>
<p>Thanks... | Disintegrating By Parts | 112,478 | <p>If $f$ is square integrable on $[-L,L]$, then
$$
\lim_{N\rightarrow\infty}\int_{-L}^{L}\left|f(x)-a_0-\sum_{n=1}^{N}\{a_n\sin(n\pi x/L)+b_n\cos(n\pi x/L)\}\right|^2dx =0,
$$
where $a_n$, $b_n$ are the Fourier coefficients. That's enough to say that, for all square integrable $g$ on $[-L,L]$, the following... |
1,270,107 | <p>How many real roots does the below equation have?</p>
<p>\begin{equation*}
\frac{x^{2000}}{2001}+2\sqrt{3}x^2-2\sqrt{5}x+\sqrt{3}=0
\end{equation*}</p>
<p>A) 0 B) 11 C) 12 D) 1 E) None of these</p>
<p>I could not come up with anything.</p>
<p>(Turkish Math Olympiads 2001)</p>
| Clement C. | 75,808 | <p><strong>Hint:</strong> $2\sqrt{3} x^2 - 2\sqrt{5} x + \sqrt{3}$ is positive and has no roots.</p>
|
489,528 | <p>I need to check for convergence here. Why is every example completely different and nothing which I've learned from the examples before helps me in the next? Hate it!</p>
<p>$$ \sum_{n\geq1} \frac{n^{n-1}}{n!} $$</p>
<p>so I tried the ratio test and I got</p>
<p>$$ \frac{1}{1+\frac{1}{n}} $$
which converges to $... | Nick Peterson | 81,839 | <p><strong>Hint:</strong> Note that you can write
$$
\frac{n^{n-1}}{n!}=\frac{n\cdot n\cdots n}{2\cdot3\cdots n}
$$
where the numerator contains $n-1$ copies. This can be written as
$$
\frac{n^{n-1}}{n!}=\frac{n}{2}\cdot\frac{n}{3}\cdots\frac{n}{n}.
$$
Can this possibly tend to $0$ as $n\rightarrow\infty$?</p>
|
489,528 | <p>I need to check for convergence here. Why is every example completely different and nothing which I've learned from the examples before helps me in the next? Hate it!</p>
<p>$$ \sum_{n\geq1} \frac{n^{n-1}}{n!} $$</p>
<p>so I tried the ratio test and I got</p>
<p>$$ \frac{1}{1+\frac{1}{n}} $$
which converges to $... | DonAntonio | 31,254 | <p>Ratio test:</p>
<p>$$\frac{(n+1)^n}{(n+1)!}\frac{n!}{n^{n-1}}=\left(1+\frac1n\right)^n\frac n{n+1}\xrightarrow[n\to\infty]{}e>1$$</p>
<p>so the series...</p>
<p><strong>Added on request:</strong></p>
<p>Since</p>
<p>$$\frac{k!}{(k-1)!}=k\;,\;\;\frac1{n^{n-1}}=\frac n{n^n}\;,\;\;\text{we get}$$</p>
<p>$$\fra... |
207,382 | <p>I was interested in exploiting the Terminal` package, as demonstrated in <a href="https://mathematica.stackexchange.com/questions/13961/plotting-graphics-as-ascii-plots">this post</a>, to show some ASCII plots straight through the terminal of my Raspberry Pi Model 3 B+. However, it doesn't seem that I can get a plot... | bobthechemist | 7,167 | <p><strong>Note</strong> After looking through the referenced python code, it stands to reason that all of the tools are available from within Mathematica. I have yet to fully debug this sub-answer. You can take advantage of <code>$DisplayFunction</code> to estimate what <code>Terminal</code> did:</p>
<pre><code>$Di... |
3,034,374 | <p>Two cyclists start from the same place to ride in the same direction.A starts at noon with a speed of 8km/hr and B starts at 2pm with a speed of 10km/hr.At what times A and B will be 5km apart ?
My thought process:
As A starts early at 12 so it will have already covered 16km(8*2).
so S relative=V relative*t or say 1... | AgentS | 168,854 | <p>I'll give an alternative, which might look a bit less tedious.</p>
<p>Assume <span class="math-container">$t=0$</span> is at 12 noon.<br>
Since <span class="math-container">$A$</span> starts at <span class="math-container">$t=0$</span> and goes at a speed of 8, we can express the distance traveled by him as
<span ... |
2,944,397 | <blockquote>
<p>In one plane <span class="math-container">$Oxy$</span>, given the circle <span class="math-container">$(\!{\rm C}\!): x^{2}+ y^{2}- 2x+ 4y- 4= 0$</span> around point <span class="math-container">$O$</span> for <span class="math-container">$60^{\circ}$</span> and it maps in the circle <span class="math... | Community | -1 | <p>We have <span class="math-container">$x^{2}+ y^{2}- 2x+ 4y- 4= 0 \therefore {x}'^{2}- (\!2\sqrt{3}+ 1\!){x}'+ {y}'^{2}- (\!\sqrt{3}- 2\!){y}'- 4= 0$</span>. Using twice
<span class="math-container">$$ax^{\,2}+ bx+ c= \frac{(\!2ax+ b\!)^{2}+ 4ac- b^{2}}{4a}\,\therefore\,\frac{1}{4}(\!-\,2{x}'+ 2\sqrt{3}+ 1\!)^{2}+ \f... |
2,260,975 | <p>Problem: Let $\mu : \mathcal{B}(\mathbb{R}^k) \rightarrow \mathbb{R}$ be a nonnegative and finitely additive set function with $\mu(\mathbb{R}^k) < \infty $.
Suppose that
\begin{equation}
\mu(A) = \sup \{ \mu(K) : K \subset A, K \text{ compact}\}
\end{equation}
for each $A \in \mathcal{B}(\mathbb{R}^k)$.
Then $... | Veridian Dynamics | 408,632 | <p>This is enough:</p>
<p><strong>Claim</strong>: Let $(K_{n})_{k\in\mathbb{N}}$ a decreasing sequence of compacts subsets such that $\bigcap_{n\in\mathbb{N}}K_{n}=\emptyset$. Then, $\lim_{m\to+\infty}\mu(K_{m})=0$.</p>
<p>If not, there is some $\delta>0$ such that $\mu(K_{n})>\delta$, for all $n$. In particula... |
2,900,014 | <p>How would solve for $a$ in this equation without using an approximation ?
is it possible?</p>
<p>where $x>0$ and $0<a<\infty$</p>
<p>$x=\Sigma _{i=1}^{n} i^a$</p>
<p>for example $120=\Sigma _{i=1}^{6} i^a$ what is $a$ in this equation?</p>
| Claude Leibovici | 82,404 | <p>If you want solve for $a$ the equation $$x=\sum_{i=1}^{n} i^a=H_n^{(-a)}$$ you will need a numerical method (as already said in answers) and, for that, you would need some initial estimate.</p>
<p>To get such an estimate, you could consider the asymptotics
$$H_n^{(-a)}=n^a \left(\frac{n}{a+1}+\frac{1}{2}+\frac{a}{1... |
4,295,212 | <p>I have a statics problem in which my approach differs from the suggested solution, so I was wondering if anyone can point out my flawed logic. Below is the question.</p>
<p>Two identical uniform rods <span class="math-container">$ab$</span> and <span class="math-container">$bc$</span>, each of length <span class="m... | YNK | 587,353 | <p><a href="https://i.stack.imgur.com/R6M7c.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/R6M7c.png" alt="RodsinEquilibrium" /></a></p>
<p>It is true that the tension force <span class="math-container">$T$</span> in the string is a single force. But, this tension in the string exerts two equal forc... |
2,349,476 | <p>Suppose $n$ and $k$ are positive integers. Are there any known conditions on $n$ and $k$ such that the polynomial $x^{k+1}-x^k-n$ has rational or integer roots?</p>
| Bob Krueger | 228,620 | <p>It is a well-known fact (attribution?) that the rational roots of a polynomial with leading coefficient $a$ and constant term $b$ must be of the form $\pm \frac{p}{q}$, where $p$ divides $b$ and $q$ divides $a$. In your equation, the only possible rational roots are plus/minus factors of $n$. Let $m$ divide $n$ be a... |
244 | <p>I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a Hilbert scheme to be smooth?</p>
| Daniel Erman | 4 | <p>I don't know of many global conditions for a Hilbert scheme to be smooth/singular. Ben's answer probably gives the most interesting example of a smooth Hilbert scheme, namely the Hilbert scheme of n points on a smooth surface.</p>
<p>Here are two more examples of smooth Hilbert schemes.</p>
<p>1) The Hilbert sch... |
3,807,504 | <p>Suppose that <span class="math-container">$z=f(x,y)$</span> is given. where <span class="math-container">$f(x,y)=x^3+y^3$</span>.
I am a bit confused about the notation of <span class="math-container">$z_x$</span> and <span class="math-container">$f_x$</span>.</p>
<p>A lot of people use the below notation for <span ... | Isaac Browne | 429,987 | <p>Probably the best way of going about this one is to write it as an infinite sum, as there does not appear to be an easy closed form.</p>
<p>So, what we can do is write
<span class="math-container">$$\int_0^R \frac{bx}{x^a-b}dx = -\int_0^R \frac{x}{1-x^a/b}dx = -\int_0^R \sum_{n=0}^\infty x\left(\frac{x^a}{b}\right)^... |
310,513 | <p>I haven't found this yet and I'm somehow not sure if my idea is correct.</p>
<p>The Problem: Let $X$ be a separable Banach-Space, let $x_k\to x$ weakly and such that for every $\lambda_k \to \lambda$ weakly-* there holds $\lambda_k(x_k)\to \lambda(x)$. Then $x_k\to x$ strongly (in the norm of $X$).</p>
<p>My idea ... | Harald Hanche-Olsen | 23,290 | <p>I think your proof looks just fine. There is nothing to add and nothing to subtract. Oh, except you need to take a subsequence to get the first inequality. But you must already have intended that, or else why did you say <em>further</em> subsequence? Hahn–Banach and compactness are powerful tools. You should not be ... |
2,672,497 | <p>$$\lim _{n\to \infty }\sum _{k=1}^n\frac{1}{n+k+\frac{k}{n^2}}$$ I unsuccessfully tried to find two different Riemann Sums converging to the same value close to the given sum so I could use the Squeeze Theorem. Is there any other way to solve this?</p>
| 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... |
19,290 | <pre><code>MouseAppearance[Plot[Sin[x], {x, 0, 5}], "LinkHand"]
</code></pre>
<p><img src="https://i.stack.imgur.com/XjHbo.png" alt="hand"></p>
<p>Unfortunately I cannot get a screen grab to show the link hand but it is a cartoon hand with a finger pointing. In most/many graphics packages there is also an open hand (... | kglr | 125 | <p>On Windows (Vista 64bit), you can find several hand icons in native cursor format <a href="http://reference.wolfram.com/mathematica/ref/format/CUR.html" rel="nofollow noreferrer"><code>CUR</code></a> in the directory <code>C:\Windows\Cursors</code> </p>
<pre><code> {hand1, hand2, hand3, hand4, hand5} = {Import["C:\... |
19,290 | <pre><code>MouseAppearance[Plot[Sin[x], {x, 0, 5}], "LinkHand"]
</code></pre>
<p><img src="https://i.stack.imgur.com/XjHbo.png" alt="hand"></p>
<p>Unfortunately I cannot get a screen grab to show the link hand but it is a cartoon hand with a finger pointing. In most/many graphics packages there is also an open hand (... | John Fultz | 309 | <p>There is an undocumented "Hand" named cursor that should work on all platforms.</p>
<pre><code>MouseAppearance[Plot[Sin[x], {x, 0, 6}], "Hand"]
</code></pre>
<p>Unfortunately, there isn't anything that resembles a grabbed hand. Here are some other undocumented ones. Nothing here is secret, but most of it is comp... |
3,118,238 | <p>I know there is a pretty simply way to check if a 2D point is inside a circle, I wanted to know how to do there same but in the 3rd dimension. The variables are the point's X, Y, Z and the sphere's 3D point and radius.</p>
<p>Let's say we know that X and Y is inside the circle, can we rotate it 90 degrees and prete... | PackSciences | 588,260 | <p>If a point <span class="math-container">$M$</span> of cartesian coordinates <span class="math-container">$x_0,y_0,z_0$</span> is inside a circle of center <span class="math-container">$x_c,y_c,z_c$</span> of radius <span class="math-container">$R$</span>, it means that <span class="math-container">$\sqrt{(x_0-x_c)^2... |
1,018,672 | <blockquote>
<p><span class="math-container">$$\int_0^{\infty} \frac{1}{x^3-1}dx$$</span></p>
</blockquote>
<p>What I did:</p>
<p><span class="math-container">$$\lim_{\epsilon\to0}\int_0^{1-\epsilon} \frac{1}{x^3-1}dx+\lim_{\epsilon\to0}\int_{1+\epsilon}^{\infty} \frac{1}{x^3-1}dx$$</span></p>
<hr />
<p><span class="ma... | Ron Gordon | 53,268 | <p>We can in fact evaluate the Cauchy principal value of the integral as follows (which is what I think you were trying to do).</p>
<p>Consider the following contour integral:</p>
<p>$$\oint_C dz \frac{\log{z}}{z^3-1}$$</p>
<p>$C$ is a modified keyhole contour about the positive real axis of outer radius $R$ and inn... |
1,056,045 | <blockquote>
<p>Show that </p>
<p>$2^{3n}-1$ is divisble by $7$ for all $n$ $\in \mathbb N$</p>
</blockquote>
<p>I'm not really sure how to get started on this problem, but here is what I have done so far:</p>
<p>Base case $n(1)$:</p>
<p>$\frac{2^{3(1)}-1}{7} = \frac{8-1}{7} = \frac{7}{7}$</p>
<p>But not sur... | barak manos | 131,263 | <p>For simplicity, please note that $2^{3n}=8^n$.</p>
<hr>
<p>First, show that this is true for $n=1$:</p>
<ul>
<li>$\frac{8^1-1}{7}=1\in\mathbb{N}$</li>
</ul>
<p>Second, assume that this is true for $n$:</p>
<ul>
<li>$\frac{8^n-1}{7}=k\in\mathbb{N}$</li>
</ul>
<p>Third, prove that this is true for $n+1$:</p>
<u... |
3,744,610 | <p>The following puzzle was introduced by the psychologist Peter Wason in 1966, and is one of the
most famous subject tests in the psychology of reasoning.</p>
<blockquote>
<p>Four cards are placed on the table in front of you. You are told
(truthfully) that each has a letter printed on one side and a digit on
the othe... | Shubham Johri | 551,962 | <p>The statements <span class="math-container">$(1)~V\implies O$</span> and <span class="math-container">$(2)~V'\cup O$</span> are equivalent. See the truth table:<span class="math-container">$$\begin{matrix}V&O&(1)&(2)\\0&0&1&1\\0&1&1&1\\1&0&0&0\\1&1&1&1\end{... |
541,926 | <p>I've been wrecking my brain with this problem and I really hope you can help me. You see I have a triangle that is either an isosceles or equilateral or right and I have to find a way to:
1)Convert it to a right one by moving one of its vertices,
2)Convert it to an isosceles one by moving one of its vertices,
3)Conv... | Han de Bruijn | 96,057 | <p>Perhaps your question can be answered by a standard piece of Finite Element
theory. Here comes.<P>Let's consider the simplest non-trivial finite element shape in two dimensions:
the linear triangle. Function behaviour is approximated inside such a triangle
by a <I>linear</I> interpolation between the function values... |
362 | <p>The Math Stack Exchange chat room has had some heated arguments/suspensions in the past and I was wondering if there were any plans for the newly opened Math Overflow chat room to have its own moderators to provide the required special attention to prevent similar problem situations?</p>
| Community | -1 | <p>The site <a href="http://mathoverflow.net">http://mathoverflow.net</a> is <em>not</em> a new site; it exists since more than three and a half years (end of september it will be four). Only it joined the SE-network recently. Then like now it has moderators (the same as before). Who they are can be seen on the already... |
1,326,816 | <p>How to find the Maclaurin series of the function $$f(x)=\frac{1}{(9-x^2)^2}$$
I guess we are gonna use derivatives but i have no idea how the final answer should be formed.</p>
| JimmyK4542 | 155,509 | <p>Start with the familiar geometric series: $\dfrac{1}{1-y} = \displaystyle\sum_{n = 0}^{\infty}y^n$.</p>
<p>Next, differentiate both sides to get this series: $\dfrac{1}{(1-y)^2} = \displaystyle\sum_{n = 0}^{\infty}ny^{n-1}$.</p>
<p>Now, what can you plug in for $y$ to make $\dfrac{1}{(1-y)^2}$ more like $\dfrac{1}... |
2,485,447 | <p>My attempt:</p>
<p>3x≡1 mod 7 (1)</p>
<p>4x≡1 mod 9 (2)</p>
<p>Multiply (1) by 5</p>
<p>Multiply (2) by 7</p>
<p>x≡5 mod 7</p>
<p>x≡7 mod 9</p>
<p>So x≡9k+7</p>
<p>9k+7=5(mod7)</p>
<p>k=5(mod7)</p>
<p>k=7j+5</p>
<p>x=9(7j+5)+7</p>
<p>=63j+52</p>
<p>x≡52(mod63)</p>
| Ozzy | 473,926 | <p>Summing two equations you get $7x=2$, or $0=2$ in mod 7. So no solutions exist.</p>
|
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