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 |
|---|---|---|---|---|
1,380,178 | <blockquote>
<p>Find the equation of a circle that intersects the $y$-axis at the origin and at the point $(0,6)$ and also touches the $x$ axis.</p>
</blockquote>
<p>Okay, so I wasn't sure how to do this so I looked at the answer at the back of the book and decided to graph it. The book says the answer is $x^2+y^2-6... | David Quinn | 187,299 | <p>If I understand your set-up correctly (and I am not sure I do), then you need to rotate the plane back away from the camera by an angle of $85.7$ degrees from its straight-on position.</p>
<p>It would be quicker for you to check this with your camera than for me to enter the solution, and of course I may have misun... |
1,963,572 | <p>Is it possible that a function takes in an equation and pops out another equation by modifying the first one like f(2x+2=0)=3x+3=1. </p>
| Matias Heikkilä | 66,856 | <p>While I can think of no immediate use for a function that takes in an equation and outputs another.. why not? A function $f:A \to B$ is a thing that maps the members of <strong>any</strong> set $A$ to the members of <strong>any</strong> set $B$. As long as the sets $A$ and $B$ are properly defined, there is no probl... |
3,551,183 | <p>Let <span class="math-container">$f$</span> be a function from <span class="math-container">$\mathbb{R}^n$</span> to <span class="math-container">$\mathbb{R}^m$</span>. </p>
<p>In general, mathematicians like to generalize a mathematical concept very very much. </p>
<p>I think they like generalization more than ... | John Bentin | 875 | <p>For a sufficiently smooth function <span class="math-container">$f$</span>, Taylor's theorem in one dimension may be written <span class="math-container">$$f(x+u)=\sum_{k=0}^l\frac1{k!}f^{(k)}(x)u^k +\varepsilon(x,u,l),$$</span>where the “error” term <span class="math-container">$\varepsilon(x,u,l)$</span> is treate... |
3,332,999 | <p>Why are there <span class="math-container">$2^n-1$</span> terms in the inclusion-exclusion formula of <span class="math-container">$n$</span> sets?</p>
<p>An example of what I mean by inclusion-exclusion formula is this:</p>
<p>There are three sets (i.e. <span class="math-container">$n$</span> <span class="math-co... | sebastian | 604,625 | <p>The amount of terms can be calculated as <span class="math-container">$\Sigma_{i = 1}^n \binom{n}{i}$</span>. Using the binomial theorem we get: <span class="math-container">$$\Sigma_{i = 1}^n \binom{n}{i} = \Sigma_{i=0}^n \binom{n}{i}1^i1^{n - i} - 1 = (1 + 1)^n - 1 = 2^n - 1$$</span></p>
|
3,332,999 | <p>Why are there <span class="math-container">$2^n-1$</span> terms in the inclusion-exclusion formula of <span class="math-container">$n$</span> sets?</p>
<p>An example of what I mean by inclusion-exclusion formula is this:</p>
<p>There are three sets (i.e. <span class="math-container">$n$</span> <span class="math-co... | Henno Brandsma | 4,280 | <p>Of the first kind (one set) there are <span class="math-container">$\binom{3}{1}$</span> terms (we use just 1 set from the 3 we have). </p>
<p>Of the second kind, <span class="math-container">$\binom{3}{2}$</span> terms (that many pairs from 3 elements)</p>
<p>Of the last kind <span class="math-container">$\binom{... |
79,012 | <p>Let
$$f(z)=\frac{1}{(z-1)(z-2)}$$
and let
$$R_1=\Bigl\{z\Bigm| 1<|z|<2\Bigr\}\quad\text{ and }\quad R_2=\Bigl\{z\Bigm| |z|>2\Bigr\}.$$</p>
<p>How do you find the Laurent series convergent on $R_1$? Also how do you do it for $R_2$?</p>
<p>I'm having serious trouble with this as I can't see how to expand ... | Sasha | 11,069 | <p><strong>HINT</strong>: For $R_1$
$$
\frac{1}{(z-1)(z-2)} = \frac{1}{(z-1)((z-1)-1)}
$$
And expand in $w=z-1$. For $R_2$:
$$
\frac{1}{(z-1)(z-2)} = \frac{1}{z^2} \frac{1}{\left( 1 - \frac{1}{z}\right) \left( 1 - \frac{2}{z}\right)}
$$
and expand in $w=\frac{1}{z}$.</p>
|
4,470,820 | <p>I shall find the integral by Feynman’s Technique Integration on a particular integral</p>
<p><span class="math-container">$\displaystyle I(a)=\int_{0}^{\pi} \ln (a \cos x+1) d x,\tag*{} $</span>
where <span class="math-container">$-1\leq a \leq 1.$</span>
<span class="math-container">$\displaystyle \begin{aligned}I^... | Quanto | 686,284 | <p>Note that, with <span class="math-container">$r= \frac cb -\sqrt{\frac{c^2}{b^2}-1}$</span>
<span class="math-container">$$\ln (b \cos x+c)=\ln\frac{c+\sqrt{c^2-b^2}}2+\ln\left(1+2r\cos x+r^2\right)
$$</span>
Thus, per <a href="https://math.stackexchange.com/a/3980719/686284"><span class="math-container">$\int_0^\pi... |
624,826 | <p>It is known that in $p\%$ of cases, Joe helps Bill in his game of chess; In all the rest of cases Bill plays alone. When he does receive help, his probability of victory is $0.95$, and without help, $0.75$. It is known that if Bill lost a game , the probability Joe helped him is $3/13$. Calculate $p$. </p>
<p>Im pr... | Shaurya Gupta | 102,986 | <p>$P(\text{lose | help}) = 1 - 0.95 = \displaystyle \frac{1}{20}$<br>
$\displaystyle P(\text{lose | no help}) = 1 - 0.75 = \frac{1}{4} $<br>
$\displaystyle P(\text{help}) = \frac{p}{100}$<br>
$\displaystyle P(\text{help | lose}) = \frac{3}{13}$<br>
$$P(\text{lose}) = P(\text{help}) \times P(\text{lose | help}) + P(\te... |
3,395,500 | <p>I am an undergraduate student currently taking a course in Multivariable Calculus. Long story short, I took a hiatus from doing any sort of serious mathematics for a couple of years, and now that I am back I am finding it difficult to pick up where I left off.</p>
<p>The "textbook" that we are using is more of a co... | user0102 | 322,814 | <p>For convenience, consider that <span class="math-container">$F$</span> is absolutely continuous and strictly monotone. Then the random variable <span class="math-container">$U = F_{X}(x)$</span> has distribution given by
<span class="math-container">\begin{align*}
G_{U}(u) = \textbf{P}(U\leq u) = \textbf{P}(F_{X}(x)... |
3,395,500 | <p>I am an undergraduate student currently taking a course in Multivariable Calculus. Long story short, I took a hiatus from doing any sort of serious mathematics for a couple of years, and now that I am back I am finding it difficult to pick up where I left off.</p>
<p>The "textbook" that we are using is more of a co... | Ekesh Kumar | 614,428 | <p>Yes, it follows a uniform distribution.</p>
<p>We know that <span class="math-container">$F$</span> is non-decreasing, and we can further note that </p>
<p><span class="math-container">$$P(F(X) \leq x) = P(X \in \{y \mid F(y) \leq x\})$$</span></p>
<p>Now set <span class="math-container">$M = \sup\{y \mid F(y) \l... |
722,890 | <p>Ok, so I know that since the numerator has a higher power that long division is needed. So after doing that, the main fraction is $\frac{-6x-36}{x^2 + 12x + 36}$. I think that's right. But my problem is that after you factor the denominator they're both equal to $(x+6)$. So how would you use partial fraction decompo... | lab bhattacharjee | 33,337 | <p>$$\frac{x^3}{x^2+12x+36}=\frac{x^3}{(x+6)^2}=x+A+\frac B{x+6}+\frac C{(x+6)^2}$$</p>
|
3,088,503 | <p>For <span class="math-container">$r\in [0,1)$</span> define
<span class="math-container">$$
I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt.
$$</span>
Numerical experiments hint at <span class="math-container">$I(r) = 0$</span> for all <span class="math-container">$r\in [0,1)$</span> but I can't show ... | Hussein | 486,661 | <p>It seems to me that this result is a particular case (or rather 1D adaptation) of Newton's shell theorem. In 2D, the theorem states that no net gravitational force is perceived by objects located inside a uniform hollow sphere.</p>
<p>Here, consider the integral
<span class="math-container">$$
\vec I(M) = \int_{X\i... |
3,088,503 | <p>For <span class="math-container">$r\in [0,1)$</span> define
<span class="math-container">$$
I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt.
$$</span>
Numerical experiments hint at <span class="math-container">$I(r) = 0$</span> for all <span class="math-container">$r\in [0,1)$</span> but I can't show ... | robjohn | 13,854 | <p>With <span class="math-container">$\gamma$</span> being the counter-clockwise unit circle and <span class="math-container">$\bar\gamma$</span> being the clockwise unit circle,
<span class="math-container">$$
\begin{align}
\int_0^{2\pi}\frac{r-\cos(t)}{1-2r\cos(t)+r^2}\,\mathrm{d}t
&=\int_0^{2\pi}\frac12\left(\fr... |
3,621,135 | <p>Prove that the metric space <span class="math-container">$P[a,b]$</span> of all polynomials with uniform metric <span class="math-container">$d(f,g) = \max|f(t)- g(t)|$</span> for every <span class="math-container">$f, g$</span> belonging to <span class="math-container">$P[a,b]$</span> and <span class="math-containe... | John B | 301,742 | <p>The solutions of <span class="math-container">$x'=x$</span> with <span class="math-container">$x(0)=x_0$</span> are <span class="math-container">$\phi_t(x_0)=e^tx_0$</span> (<span class="math-container">$x(0)=1$</span> is not sufficient for what you want). So it follows from
<span class="math-container">$$
\phi_{x+y... |
2,089,980 | <p>Those are the mechanics:</p>
<ul>
<li>You always draw from 2 to 10 cards.</li>
<li>You only use from A to 10. So 40 cards.</li>
</ul>
<p>What I want to obtain:</p>
<ul>
<li>The % of times you gonna obtain a number repeated.</li>
</ul>
<p>I want to make a graphic with how much % of at least one repeat you obtain ... | Mees de Vries | 75,429 | <p>For example, suppose that $n = 1$ and $m = 2$. Then, $T_{n,m} = 2$. Namely, you can put the ball in box one, or you can put the ball in box two. The boxes are <em>distinct</em>, so these are separate scenarios. However, $S_{n,m} = 1$: you always end up with one box with a ball in it, and one box without any balls. B... |
2,709,977 | <p>let $X$ be a compact Hausdorff space.consider commutative Banach algebra $C(X)$.</p>
<p>let for $x\in X$ is fixed. define$$M_x=\{f:f\in C(X)\ \text{and}\ f(x)=0\}$$.</p>
<p>then $M_x$ is proper two sided ideal. but is $M_x$ maximal ideal in $C(X)$</p>
<p>any hint</p>
| José Carlos Santos | 446,262 | <p>If $(x,y)$ belongs to your set and $t\in[0,1]$, then every point of the form $\bigl((1-t)x,t+(1-t)y\bigr)$ also does, because$$t+(1-t)y-(1-t)x=t+(1-t)(y-x)>0.$$Therefore, your set is path connected, because the map\begin{array}{rccc}\gamma\colon&[0,1]&\longrightarrow&\left\{(x,y)\in\mathbb{R}^2\,\midd... |
3,318,166 | <p>This may be a silly question but I want to make sure my general intuition is correct. I don't think it is possible for an <span class="math-container">$n$</span>-dimensional manifold to live inside <span class="math-container">$\mathbb{R}^m$</span> for <span class="math-container">$m < n$</span> because in the "b... | John Hughes | 114,036 | <p>Your intuition is right, but only if you insist that "live inside" means
that
<span class="math-container">$$
f : M^n \to \Bbb R^m.
$$</span>
the function that puts the manifold into <span class="math-container">$m$</span>-space, is <strong>continuous</strong>. If you allow arbitrary functions, then there's certain... |
3,056,793 | <p>See <a href="https://qr.ae/TUn74K" rel="nofollow noreferrer">What is directional derivative?</a></p>
<blockquote>
<p>It is just the usual derivative i.e. the ONE VARIABLE/ONE DIMENSIONAL
Derivative obtained from breaking down a VECTOR DERIVATIVE.</p>
<p>This example will make it clear. Consider a humming bird moving... | José Carlos Santos | 446,262 | <p>Yes, there is a difference. And a big one. Let us assume that <span class="math-container">$f$</span> is a differentiable map from <span class="math-container">$\mathbb{R}^n$</span> into <span class="math-container">$\mathbb R$</span>.</p>
<p>The <strong>directional derivative</strong> of <span class="math-containe... |
3,056,793 | <p>See <a href="https://qr.ae/TUn74K" rel="nofollow noreferrer">What is directional derivative?</a></p>
<blockquote>
<p>It is just the usual derivative i.e. the ONE VARIABLE/ONE DIMENSIONAL
Derivative obtained from breaking down a VECTOR DERIVATIVE.</p>
<p>This example will make it clear. Consider a humming bird moving... | John Cataldo | 521,790 | <p>If you have a function <span class="math-container">$f=f(x_1,...,x_n)$</span> which depends on <span class="math-container">$n$</span> variables the partial derivative <span class="math-container">$\frac{\partial f}{\partial x_i}$</span> is a directional derivative in the direction of the vector <span class="math-co... |
2,794,717 | <p>I am stuck on the following question.</p>
<p>Prove that $S(n+1, n) = \frac{n(n+1)}{2}$ for any $n \ge 1$</p>
<p>This is where $S(n+1,n)$ is the Stirling number, partitioning a $n+1$-element set into $n$ parts.</p>
<p>What first steps should be taken to try and derive this?</p>
| Supriyo Halder | 438,576 | <p>You have to partition $n+1$ element set in $n$ parts. Now each of the $n$ part is non empty. So, the only possibility the partition case is that, $n-1$ singleton subsets and a subset of $2$ element.So, we have to just focus on the $2$ element set. Now the possible no of $2$ element set is ${n+1\choose 2}$, which is ... |
1,885,438 | <blockquote>
<p>Let $X=\mathbb{R}\times \mathbb{Z}$ and $\tau$ the product topology $\tau_u \times \tau_{CF}$ where $\tau_u$ is the usual topology on $\mathbb{R}$ and $\tau_{CF}$ the cofinite topology on $\mathbb{Z}$. Study the contiuity of $f : (X, \tau)\to (\mathbb{R},\tau_u)$ defined as
$$(t,k)\mapsto t-k$$</p>
... | egreg | 62,967 | <p>Your idea is good, but you shouldn't consider a particular neighborhood of $(0,0)$.</p>
<hr>
<p>Fix a point $(t_0,k_0)\in X$. A basic neighborhood of this point has the form $U=(t_0-\delta,t_0+\delta)\times C$, where $C$ is cofinite and $k_0\in C$, in particular both upper and lower unbounded.</p>
<p>Then
$$
f(U)... |
61,030 | <p><a href="http://reference.wolfram.com/language/ref/Dataset.html"><code>Dataset</code></a> rather conveniently truncates the display of a dataset that has many rows. However, occasionally, I would like to show more of a dataset without having to convert to e.g. a <a href="http://reference.wolfram.com/language/ref/Tab... | gwr | 764 | <p><em>Until scrollbars are implemented...</em></p>
<p>... I have found this little function for simple tabular data to work out fine with everyday work. One can size the viewer window (and probably use the tricks posted so far to extend the range of formatted output?).</p>
<p>Also you can set a specific column (nume... |
61,513 | <p>Hello. I'd like to consider the open unit ball in an infinite dimensional Hilbert space and ask when can we fit infinitely many open balls of radius $r<1$ inside.</p>
<p>For example, when $r=1/(1+\sqrt2)$, we can pick an orthonormal basis $(x_1,...)$ for our Hilbert space and put the centers of the balls at $(1-... | Pietro Majer | 6,101 | <p>The optimality of your configuration can be shown as a plain consequence of the Kirszbraun theorem.</p>
<p>(I happened to ask myself this problem too, and eventually added <a href="http://en.wikipedia.org/wiki/Packing_problem#Spheres_into_a_Euclidean_ball" rel="nofollow">this short section</a> in a wiki article, th... |
108,523 | <p>Is there, similar to the Mehler kernel, a closed formula for the heat kernel of the heat equation associated to the Laplacian
$$ -\sum_j \frac{d^2}{dx_j^2} + 2\sqrt{-1} \sum_j \lambda_j \frac{d}{dx_j} + \sum_{ij} a_{ij}x_ix_j$$
on $\mathbb{R}^n$? Here, the matrix $(a_{ij})$ is supposed to be symmetric and positive d... | Piero D'Ancona | 7,294 | <p>Lars Hörmander did some work on classifying Mehler-type formulas for general quadratic fomrs in $\xi$ and $x$. Take a look at his paper on Math.Zeitschrift 219 (1995) pp.413-449</p>
|
3,144,178 | <blockquote>
<p>Let <span class="math-container">$z=a+ib,w=c+id\in\mathbb{C}. $</span>Prove <span class="math-container">$|z|-|w|\leq|z-w|$</span></p>
</blockquote>
<p>Solution: If <span class="math-container">$|z|-|w|\leq 0$</span> then we are done since <span class="math-container">$|z-w|\geq 0$</span>. Suppose <s... | Barry Cipra | 86,747 | <p>Note that <span class="math-container">$|z|-|w|\le||z|-|w||$</span> from properties of the absolute value. Now</p>
<p><span class="math-container">$$\begin{align}
||z|-|w||\le|z-w|
&\iff(|z|-|w|)^2\le|z-w|^2\\
&\iff|z|^2-2|zw|+|w|^2\le(z-w)\overline{(z-w)}=|z|^2-(z\overline{w}+\overline{z}w)+|w|^2\\
&\i... |
2,072,294 | <p>Suppose I want a card A from a deck. I shuffle a deck and draw a card. The probability that this card is $A$ is $1/52$.</p>
<p>Now, suppose I reshuffle this deck n times and draw a card each time. I intend to find the mean number of times I would draw card $A$. Would I have to model this as a Binomial random variab... | Olivier Oloa | 118,798 | <p>Since $ x \mapsto \Gamma(\cdot)$ is a <em>continuous function</em>, as $h \to 0$, one has
$$
\Gamma(h+\alpha) \sim \Gamma(0+\alpha)=\Gamma(\alpha), \quad \Gamma(h+1)\sim \Gamma(1)=1
$$ giving
$$
\dfrac{\Gamma(h+\alpha)\cdot h}{\Gamma(h+1)}\sim \dfrac{\Gamma(0+\alpha)\cdot h}{\Gamma(0+1)}=\Gamma(\alpha)\cdot h
$$</p>... |
27,968 | <p>What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think</p>
<p>$$ \mathbb R^+ $$</p>
<p>but I believe that that is usually used to denote "all real numbers including infinity".</p>
<p>So is there a standard way to denote the set</p>
<p>$$ \{x \in \mathbb R : x \geq 0\} \; ?$$ ... | Björn Friedrich | 203,412 | <p>The unambiguous notations are: for the <strong>positive-real numbers</strong>
<span class="math-container">$$
\mathbb{R}_{>0} = \left\{ x \in \mathbb{R} \mid x > 0 \right\} \,,
$$</span>
and for the <strong>non-negative-real numbers</strong>
<span class="math-container">$$
\mathbb{R}_{\geq 0} = \left\{ x \in \... |
801,155 | <p>Suppose $f(x)$ is some function with domain [0,1] and $\sum_{n\ge1}f\left(\frac 1n\right)$ converges,
than $a_n=f\left(\frac 1n\right)\to 0$ but does $b_n=f\left(\frac 1{n^2}\right)$ also converge to zero ?
I could not prove the last statement or find any counter-example,
I only guess that since nothing known about ... | Hoot | 127,490 | <p>Well, $\mathbb{A}^1 - \{1\}$ is an affine variety too, with coordinate ring $k[T]_{T-1}$. If what you say about $\varphi^*$ is true then is there any hope of
$$
k[C] \xrightarrow{\varphi^*} k[T] \to k[T]_{T-1}
$$
being an isomorphism?</p>
|
2,950,011 | <p>The difference <span class="math-container">$$\log(2)-\sum_{n=1}^{100}\frac{1}{2^n n}$$</span> is </p>
<p>(1) less than <span class="math-container">$0$</span></p>
<p>(2) greater than <span class="math-container">$1$</span></p>
<p>(3) less than <span class="math-container">$\frac{1}{2^{100}101}$</span></p>
<p>(... | José Carlos Santos | 446,262 | <p>Use the fact that<span class="math-container">\begin{align}\log(2)&=-\log\left(\frac12\right)\\&=\frac12+\frac1{2\times2^2}+\frac1{3\times2^3}+\cdots\\&=\sum_{n=1}^\infty\frac1{n2^n}.\end{align}</span></p>
|
855,665 | <p>suppose that we have following binary digits</p>
<p>$00011001.110 $,we can do following thing</p>
<p>$00011001.110=1\cdot2^4+1\cdot2^3+1\cdot2^0+1\cdot2^{-1}+1\cdot2^{-2}=25.75$ </p>
<p>then what does means?</p>
<p>We then “float” the binary point:</p>
<p>$00011001.110 \Rightarrow 1.1001110 \cdot 2^4 $</p>
<... | Tony | 161,342 | <p>Here is what the "shorten" command does to a polynomial $f(x)</p>
<p>$\bullet$ make a substitution to remove the $x^(n-1)$ term</p>
<p>$\bullet$ scale $x$ by some rational number,</p>
<p>$\bullet$ if $\mbox{deg } (f,x)=2$ then square-free factor the discriminant</p>
<p>I was hoping that through some substituti... |
2,702,833 | <p>Instead of using L'Hospital's rule can this be proved by using the definition of the limit and or the squeeze theorem.</p>
| Mark Viola | 218,419 | <p>In <a href="http://math.stackexchange.com/questions/1589429/how-to-prove-that-logxx-when-x1/1590263#1590263">THIS ANSWER</a>, I showed using only the limit definition of the exponential function and Bernoulli's Inequality that the exponential function satisfies the inequalities</p>
<p>$$1+x\le e^x\le \frac1{1-x}\ta... |
748,588 | <p>Imagine you have two finite alternating series.</p>
<p>$$S_a=a_1-a_2+a_3-a_4+\cdots+a_n$$</p>
<p>$$S_b=b_1-b_2+b_3-b_4+\cdots+ b_n$$</p>
<p><strong>Question:</strong>
If $|a_i|>|b_i|$ is $S_a>S_b$?</p>
| rlartiga | 93,314 | <p>$$S_a=a_1-a_2+a_3-a_4+\cdots+a_n$$</p>
<p>$$S_b=b_1-b_2+b_3-b_4+\cdots+ b_n$$</p>
<p>$$S_b-S_a=\sum_{i=1}^{\lfloor{\frac{n}{2}}\rfloor} b_{2i-1}-a_{2i-1} + \sum_{i=1}^{\lfloor{\frac{n}{2}}\rfloor} a_{2i}-b_{2i} $$</p>
<p>From that one clearly one can make the difference in the odd members small and in the even me... |
4,508,923 | <p>I'm doing exercises on Bonnafe's beautiful book Representations of <span class="math-container">$SL_2(\mathbb{F}_q)$</span>. The first exercise is:</p>
<p>1.1. Let <span class="math-container">$k$</span> be the subfield of <span class="math-container">$\mathbb{F}_{q}$</span> generated by <span class="math-container"... | Jyrki Lahtonen | 11,619 | <p>I might do this as follows, related to the hint in a way. Turning it into a counting argument.</p>
<p>Consider the set
<span class="math-container">$$S=\{z\in\Bbb{F}_{q^2}\mid z^{q+1}=1\}.$$</span>
As <span class="math-container">$\Bbb{F}_{q^2}^*$</span> is cyclic of order <span class="math-container">$q^2-1=(q-1)(q... |
1,963,335 | <p><strong>So when I run newton method for certain values in matlab it gives the following result. Question is how do you tell this is linear or quadratic? Book says Notice how the convergence rate actually looks linear for the first 20 iterations and then converges quadratically? How?</strong> </p>
<p>Problem descrip... | levap | 32,262 | <p>The first method is incorrect (it gives the wrong answer) while the second method results in a correct answer but not really justified by what you wrote.</p>
<p>To see the problem with the first method, consider for example the simple equation
$$ \tag{1} f'(x) \equiv 1. $$</p>
<p>The correct way of solving this e... |
1,221,795 | <blockquote>
<p>Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, </p>
</blockquote>
<p>I want to prove by contradiction, and then find a continuous $g$ such that $g(a) = g(b) = 0$ but $... | user48672 | 138,298 | <p>Counter example:
$f(x)= const$, $g(x)=sin(x) $, $a=0, b=2 \pi$.</p>
<p>Direct computations shows that
$\int_{0}^{2\,\pi }f(x) \; \mathrm{sin}\left( x\right) dx =0$.</p>
<p>So you need more constraints.</p>
|
1,221,795 | <blockquote>
<p>Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, </p>
</blockquote>
<p>I want to prove by contradiction, and then find a continuous $g$ such that $g(a) = g(b) = 0$ but $... | Kyle Raftogianis | 231,457 | <p>I'll rename the $x_0$ you defined to $c \in [a, b]$.</p>
<p>Let $p$ be the partition $(x_0 < x_1 < x_2 < x_3) = (a < c - \frac{\delta}{2} < c + \frac{\delta}{2} < b)$.</p>
<p>On the intervals $[x_0, x_1]$ and $[x_2, x_3]$, either $g$ vanishes or both $f$ and $g$ are positive, so
$$
\inf_{x \in... |
322,365 | <p>Let $\phi:R[x]\rightarrow C$ defined for a fixed $z\in C$ as $\phi(p(x))=p(z)$ where $p\in R[x]$. Then this map is supposed to be surjective. Could someone explain me please why?</p>
| azimut | 61,691 | <p>No.</p>
<p>While it is true that $\chi_A$ is monic of degree $n$, in general there are many such polynomials, and only a single one of them is $\chi_A$.</p>
|
322,365 | <p>Let $\phi:R[x]\rightarrow C$ defined for a fixed $z\in C$ as $\phi(p(x))=p(z)$ where $p\in R[x]$. Then this map is supposed to be surjective. Could someone explain me please why?</p>
| Easy | 60,079 | <p>If I understand correctly, I think you mean for any monic polynomial $p(x)$ of degree $n$ over $F$, there is always a $n$-dimensional matrix corresponding to. This is true as you can always take the corresponding companion matrix.</p>
|
36,778 | <p>I have a pre-processed image, from which I have to get coordinates of some components.</p>
<p><img src="https://i.stack.imgur.com/xg4lA.png" alt="sample image"></p>
<p>There are 7 green components (pigs actually) that I have already isolated from an angry bird screen shot. So what I want is to locate those six com... | bill s | 1,783 | <p>Here's an approach quite similar to what you tried. First, <code>allG</code> is all the green points which are then clustered into the 7 groups called <code>clus</code>. The mean of each group provides an approximation to the central point of the clusters.</p>
<pre><code>img=Import["http://i.stack.imgur.com/uO8QJ.p... |
2,354,218 | <p>On page 5 of Royden, Fitzpatrick's real analysis, the book defined cardinality of the set in the following way:</p>
<blockquote>
<p>Given a set $X$, the relation of equipotence (invertible map) is an equivalence relation on the collection of $2^X$ of all subsets of $X$. The equivalence class of a set with respect... | Bumblebee | 156,886 | <p>Suppose $f\neq 0.$ Then $g(z)=f(z)f(-z)$ is also an entire function. By the Liouville's theorem $g$ must be a constant function. Therefore if $f(z)$ has a zero, then $f(-z)$ has a pole which contradicts the entireness of $f.$</p>
|
78,357 | <p>I would like to know whether a Heyting algebra gives rise to ring in a similar way that a Boolean algebra gives rise to a Boolean ring. In a Boolean algebra $(B,\lor,\land,\lnot,0,1)$ I can define multiplication and addition as follows:</p>
<p>$$a * b = a \land b$$
$$a + b = (a \land \lnot b) \lor (b \land \lnot a)... | Emil Jeřábek | 12,705 | <p>There are no terms that would define a group structure (let alone a ring) in all Heyting algebras.</p>
<p>The first observation is that if $f(x)$ is a term-definable involution, then $f(x)=x$: by inspection of the Rieger–Nishimura lattice, we see that all other unary term-definable $f$ are either constant in Boolea... |
2,068,499 | <p>Related to <a href="https://math.stackexchange.com/questions/2068466/confusion-of-central-limit-theory">Confusion of central limit theory</a>, and the fact that I just finished my first course in (Master's-level) graduate-level probability, which relates to this material.</p>
<p>The Central Limit Theorem states tha... | Community | -1 | <blockquote>
<p>$$\bar{X}_n\overset{d}{\to}\mathcal{N}\left(\mu, \dfrac{\sigma^2}{n}\right)\tag{1}$$
as $n \to \infty$.</p>
</blockquote>
<p>You don't need any deep mathematics to see why (1) is not true. Because the <em>notations</em> in (1) does not make sense at all.</p>
<p>Think about the following example. N... |
2,068,499 | <p>Related to <a href="https://math.stackexchange.com/questions/2068466/confusion-of-central-limit-theory">Confusion of central limit theory</a>, and the fact that I just finished my first course in (Master's-level) graduate-level probability, which relates to this material.</p>
<p>The Central Limit Theorem states tha... | Violapterin | 392,335 | <p>Your notation is sloppy and confused me quite a while. If we put this way: $$
F_{\sqrt{n} (\overline{X}_n -\mu)} \overset{d}{\to} \mathcal{N}(0,\sigma^2)
$$ implies $$
F_{\sqrt{n} (\overline{X}_n -\mu)} -\mathcal{N}(0,\sigma^2) \overset{d}{\to} 0
$$ implies (explained just below) $$
F_{\overline{X}_n -\mu} -\mathcal... |
240,470 | <p>I usually produce graphics with Mathematica and then use Keynote (I am on a Mac running the latest Big Sur OS) for post processing. Until version 12.0.0.0 I would directly highlight the graphics in Mathematica, copy it (Cmd+C) and then paste it (Cmd+V) directly in Keynote; this would produce a perfect vector copy of... | OldTomMorris | 39,714 | <p>I had the same issue but I found a solution (Mathematica v. 12.1.0 with Keynote 7.1) .</p>
<p>In Mathematica you can go to the menu bar and select</p>
<p>"Edit -> Copy as... -> PDF".</p>
<p>Works well and does not rasterize.</p>
|
2,828,070 | <p>If $$\log_{3x+5} (ax^2+8x+3)>2$$ find the interval in which x lies.</p>
<p>Answer is $\left(-\dfrac43 , -\dfrac{23}{22}\right)$. Now hint me how to solve the problem please.</p>
| gt6989b | 16,192 | <p>Your logic implies, either $\mathbb{P}[A] = \mathbb{P}[B]$ or $\mathbb{P}[A \cap B] = 0$. Certainly, if $A \cap B = \emptyset$ the second condition holds, but consider $A \cap B$ being some discrete event out of a continuous probability, so $A \cap B \ne \emptyset$ but $\mathbb{P}[A\cap B] = 0$...</p>
<p>Thus it is... |
3,264,420 | <p>From Convex Optimization by Boyd and Vandenberghe:</p>
<blockquote>
<p>Let <span class="math-container">$T \in \Bbb R^{n \times n}$</span> be nonsingular. Let <span class="math-container">$f: \Bbb R^n
\rightarrow \Bbb R$</span> convex and twice continuously differentiable.</p>
<p>Define <span class="math-c... | Thomas | 89,516 | <p>Let <span class="math-container">$z_i= T_{i,j}y_j$</span>. Than <span class="math-container">$\partial_{y_k} z_i=T_{i,k}$</span></p>
<p><span class="math-container">$\partial_{y_k} f(Ty)=\partial_{z_i} f(z)\partial_{y_k} z_i=\partial_{z_i} f(z)T_{i,k}=T^+_{k,i}\nabla_{i}f(Ty)=[T^+\nabla f(Ty)]_k$</span></p>
<p>s... |
70,198 | <p>For the periodic Lorenz gas Sinai showed that rescaling the trajectory of the tracer particle yields Brownian motion in the limit. Does there exist a similar result for the random Lorenz gas? If not, do people believe that there is such a limit?</p>
<p>By the random Lorenz gas I mean: take circular scatterers dist... | Gjergji Zaimi | 2,384 | <p>I don't think such a theorem has been proved for the random Lorentz gas. First I want to point out that Sinai proved those scaling limit results for the case of (2D) periodic Lorentz gas with <em>finite horizon</em> (finite maximum free path for the particle). The case of periodic Lorentz gas with unbounded horizon ... |
70,198 | <p>For the periodic Lorenz gas Sinai showed that rescaling the trajectory of the tracer particle yields Brownian motion in the limit. Does there exist a similar result for the random Lorenz gas? If not, do people believe that there is such a limit?</p>
<p>By the random Lorenz gas I mean: take circular scatterers dist... | Community | -1 | <p>I have been looking at random (and periodic) Lorentz gas models recently in preparing a review which has just appeared at <a href="https://arxiv.org/abs/1402.7010" rel="nofollow noreferrer">arxiv:1402.7010</a>. As far as I know, there are rigorous results only for the low density (Boltzmann-Grad) limit, and for mod... |
2,676,479 | <p>Say you have $f(x) $ and $g(x)$ and $g(x) = f^{-1}(x) $.</p>
<p>I observed that these two curves need not intersect, for example with $f(x) = e^x$ and $g(x) = \ln x $ never intersecting each other.</p>
<p>I also observed that a function can either have one, two, or three intersections with its inverse, but I was u... | Jobo | 333,553 | <p>I have an easy counterexample:</p>
<p>$$f(x)=10\cos(x)$$
and its inverse,
$$g(x)=\cos^{-1}(x/10)$$</p>
<p>Take a look at the plot on desmos, that should convince you. In fact, you can bump up the 10 to get arbitrarily many intersections! </p>
<p>As a more general answer to your question, the number of intersecti... |
1,401,972 | <p>I have the sequence of functions: $s_n(x)= \displaystyle\sum_{k=0}^n \frac{k^2x}{1+k^4x^2}$ where $x \in [0,1]$</p>
<p>Consider$ [\delta, 1]$ for any $\delta \in (0,1]$</p>
<p>Now $0 \leq \frac{k^2x}{1+k^4x^2}\leq \frac{1}{k^2x} \leq \frac{1}{k^2\delta} $
So for any $x \in [\delta,1]$ The series converges uniform... | Community | -1 | <p>Addressing the question "Why would I need to show it's not continuous on $[0,1]$? Surely it's on $(0,1]$?"</p>
<p>In fact, $s$ <strong>is</strong> continuous on $(0,1]$. To see this, choose $x \in (0,1]$ and $0 < \delta < x$. Since $(s_n)$ converges uniformly on $[\delta, 1]$, the limit function $s$ is contin... |
4,257,947 | <p>Edit: Reframed the problem to find an actual answer! Though I may well be talking to myself at this point. Answer below the original problem.</p>
<hr />
<p>The original context of this question is a ridiculous, brainless mobile game. But the geometric question that arises seems quite interesting--I suspect it has al... | Eric Snyder | 585,349 | <p>In the spirit of trying to continue through with my own progress, here is a partial answer that gets us much closer. First, a picture from GeoGebra. You can <a href="https://www.geogebra.org/calculator/fjujk9ps" rel="nofollow noreferrer">play with it here</a>; points <span class="math-container">$M$</span> and <span... |
24,835 | <p>If we accept the axiom of choice, we take the responsibility of living in a world in which, e.g., a ball in Euclidean 3-space is equiscindable to two isometric copies of itself (Banach-Tarski). So we have a very natural and seemingly harmless axiom (that most matehmaticians are probably willing to accept and use) on... | John Stillwell | 1,587 | <p>I broadly agree that large cardinals don't have paradoxical consequences,
especially at the "everyday world" level. However, large cardinals have
consequences that were initially surprising to set theorists. I think it was a
surprise when Scott proved ( in 1961) that measurable cardinals imply the
existence of non-... |
2,424,061 | <p>A question asks "Is $f(x)=x^2$ an onto function or not?" Here the domain and codomain are not mentioned specifically, so what codomain should I consider?</p>
| Xander Henderson | 468,350 | <p>This is a poorly worded question. In order to define a function, we need three ingredients: the domain, the codomain, and some rule for mapping the domain into the codomain. Without those data, it is impossible to answer the question.</p>
|
1,684,045 | <p>Let $0<2x<\pi$. If $$\sum_{n=-2}^{\infty}\cos^n x=8,$$ then please find $x.$ </p>
<p>I tried $\sum_{n=-2}^{\infty}\cos^n x=\frac{1}{1-\cos x}+\frac{1}{\cos x}+\frac{1}{\cos^2 x}=\frac{1+\cos x}{\cos^2 x \sin^2 x}=8.$ But I cant find $x.$</p>
| mickep | 97,236 | <p>You can continue your calculation, to write
$$
\frac{1+\cos x}{\cos^2x\sin^2x}=\frac{1+\cos x}{\cos^2x(1-\cos^2x)}=\frac{1}{\cos^2x(1-\cos x)}.
$$
Next, let $y=\cos x$. Then you should solve the equation
$$
\frac{1}{y^2(1-y)}=8,\quad\text{i.e.}\quad y^3-y^2+\frac{1}{8}=0.
$$
To solve this equation in $y$, I think yo... |
2,824,996 | <p>I'm investigating iterations using Newton's method for $f(x)=x^5-x+1$.</p>
<p>I'm getting unusual results though. I've found that some starting values will result with a "triple oscillation" of results where iterations will loop between values of approximately $1.0003$, $0.7504$ and $-0.0871$. </p>
<p>However the ... | Ross Millikan | 1,827 | <p>You can get <a href="http://www.wolframalpha.com/input/?i=e%5Ex%3Dx%5E100" rel="nofollow noreferrer">solutions</a> in terms of the Lambert W function, giving $$x = -100 W(-1/100)\\x = -100 W(1/100)\\x = -100 W_{(-1)}(-1/100)$$
or get numeric answers
$$x \approx 1.01015,-0.990147,647.278$$
One dimension root finding ... |
2,216,723 | <p>A player at draw poker holds the seven of spades and the eight, nine,
ten, and ace of diamonds. Aware that all the other players are drawing three cards, he figures that any hand he could win with a flush he could also win with a straight. For which should he draw? </p>
| egreg | 62,967 | <p>More generally, consider the $n\times n$ matrix
$$
M(a)=\begin{bmatrix}
a & a & a & \dots & a \\
a & a & a & \dots & a \\
\vdots & \vdots & \ddots & \dots & \vdots \\
a & a & a & \dots & a
\end{bmatrix}
$$
The characteristic polynomial of $M(a)$ is
$$
\... |
253,495 | <p>It used to be that if I typed <code>==</code> or <code>-></code> these would automatically be rendered as <code>\[Equal]</code> and <code>\[Rule]</code>.</p>
<p>Now, for some reason, this feature has disappeared in 12.3.1.</p>
<p>I would like to have it back but I can't find any relevant option neither in the Set... | hieron | 16,373 | <p>Since I can't test it in MMA 12.3, (my version 12.1) try this></p>
<pre><code>SetOptions[$FrontEnd,
InputAutoReplacements -> {"->" -> "\[Rule]",
":>" -> "\[RuleDelayed]", "<=" -> "\[LessEqual]",
">=" -> &quo... |
44,740 | <p>This question is closely related to MO <a href="https://mathoverflow.net/questions/4347/ffxexpx-1-and-other-functions-just-in-the-middle-between-linear-and-expo">f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential.</a> Consider $e^{e^x-1}$, this is the generating function of the ... | David E Speyer | 297 | <p>This is an attempt to summarize some work related to question 2 which I do not fully understand myself. I am summarizing Sections 1.1 and 1.2 of <a href="http://www.math.toronto.edu/graduate/Dudko-thesis.pdf" rel="nofollow">Dudko's thesis</a>, whose exposition is excellent (including work of earlier authors which he... |
2,565,567 | <p>In a certain Algebra 2 class of 30 students, 22 of them play basketball and 18 of them play baseball. There are 3 students who play neither sport. What is the probability that a student chosen randomly from the class plays both basketball and baseball?</p>
<p>I know how to calculate the probability of students play... | Bram28 | 256,001 | <p>That formula only works if events $A$ (play basketball) and $B$ (play baseball) are independent, but they are not in this case, since out of the $18$ players that play baseball, $13$ play basketball, and hence $P(A|B) = \frac{13}{18} < \frac{22}{30} = P(A)$ (in other words: one who plays basketball is less likely... |
4,315,926 | <p>This question is in the context of a robotics problem. The goal is to track a robot using both its onboard odometry system and a VR system (HTC Vive Pro) using a VR controller mounted to the robot.</p>
<p>What is <strong>known</strong> is the transformation between odometry origin and the robot (measurements <span c... | spaceman | 522,096 | <p>From what you have deduced, we can apply here the <a href="https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula" rel="nofollow noreferrer">Faà di Bruno's formula</a>
<span class="math-container">$$
\begin{align}
I_n(a) &= \frac{(-1)^{n-1}\pi}{(n-1)!} \frac{d^{n-1}}{da^{n-1}} \left(\frac{1}{\sqrt{a^2 - 1}}... |
597,638 | <p>I need to evaluate this integral using calculus of residues: </p>
<p>$$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$</p>
<p>I know I need to consider $\displaystyle \int_0^\infty$$\frac{\ln(z)}{1+z^4}\mathrm{d}z$.<br>
Then the integrand has singularities at at $-e^{i\pi/4}$, $e^{i\pi/4}$, $-e^{3i\pi/4}$, and $e^{... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
318,740 | <p>How to find the projection(the closest point) of point $a$ to half-space $(p,x)\le\alpha$, where $p,x\in\mathbb R^n$,$\alpha \in \mathbb R$? $(p,x)$ is dot product.</p>
| Damien L | 59,825 | <p>1) Find a normal vector $\vec \nu$ to the hyperplane $\mathrm H = \{x|p.x = \alpha\}$ ;</p>
<p>2) Calculate the distance $d$ between $a$ and $\rm H$ ;</p>
<p>3) Then, then point you want is $a - d \vec \nu$.</p>
|
2,201,908 | <ol>
<li>$(P \vee Q)$ & $R$ </li>
<li>$(R$ & $P) \rightarrow S$</li>
<li>$(Q$ & $R) \rightarrow S$ </li>
</ol>
<p>Conclusion is $S$</p>
<p>without using conditional proof, please?</p>
<p>And perhaps if you have time, please show me the conditional proof method or at least which CP you would use.
Thank yo... | quasi | 400,434 | <p>This version is almost the same as my first answer, and it's the same length, but by making use of "Constructive Dilemma", it's perhaps conceptually a little simpler. The proofs are exactly the same for the first $10$ steps.</p>
<p>\begin{align*}
\text{Given:}&&1.&\;\; (P \lor Q) \land R\\[2pt]
&&am... |
533,194 | <p>Do you know how to solve this problem? I have two sets and need to solve $A \cup X = B. $</p>
<p>Thanks a lot for your help.</p>
| amWhy | 9,003 | <p>Note that if in fact there exists a set $B = A \cup X$, then $A\subseteq B$ AND $X\subseteq B$.</p>
<p>If $A$ and $X$ are disjoint, so that $A \cap X = \varnothing,$ then $$A\cup X = B \iff (A\cup X)\setminus A = B\setminus A \iff X = B\setminus A$$</p>
<p>If they are not disjoint, then you we need to know what $... |
1,981,806 | <p>$f(x) = \frac{x}{x^2 + 1}$. How do you find $f^{-1}$?</p>
<p>Is try and solve for y and I get the following:</p>
<p>$y = \frac{x}{x^2 + 1}$</p>
<p>$y(x^2 + 1) = x$</p>
<p>$yx^2 + y = x$</p>
<p>$y = -yx^2 + x$</p>
<p>$y = x(-yx + 1)$</p>
<p>It seems whatever I do I can't isolate the x variable, I just go in c... | Emilio Novati | 187,568 | <p>Solving for $x$ the equation
$$
yx^2-x+y=0
$$
we find
$$
x=\frac{1\pm \sqrt{1-4y^2}}{2y}
$$</p>
<p>this means that the range of the function $f(x)=\frac{x}{x^2+1}$ is the interval $(-1/2,1/2)$ and the function is not invertible because for any value of $y$ in this interval we have two corresponding values of $x$ s... |
13,877 | <p><a href="http://my.unix-center.net/~martian/math%20album/%E6%95%B0%E5%AD%A6%E4%B8%93%E8%BE%91%202%20-%20%E5%86%B0%E5%B2%9B%E7%81%AB%E5%B1%B1/%E7%9B%B4%E5%B8%83%E7%BD%97%E9%99%80%E7%9A%84%E5%B7%A8%E5%B2%A9%20%28On%20twin%20prime%29.html" rel="nofollow">a sequence on twin primes</a> In the diagram, How do the stripes ... | Alex B. | 3,212 | <p>Don't code it! Instead compute the first 4 or so values by hand and see if you understand what is going on. You don't even have to look at the numbers, you just need to see what you are doing when you evaluate the code. E.g. try to say in words what you do when you follow the code. Pretend that you are explaining th... |
3,048,011 | <p>I am learning linear algebra and new to it. I can not solve this problem. I think it has a trick that I don't know.</p>
<blockquote>
<p>for T(a linear map), <span class="math-container">$T:V\rightarrow V$</span> and every <span class="math-container">$v$</span> in <span class="math-container">$V$</span> the <span... | Rebellos | 335,894 | <p><strong>Hint :</strong></p>
<p>Suppose that <span class="math-container">$T,T^2,\dots, T^k$</span> are not linearly independent. Then, there exists <span class="math-container">$\lambda_1, \lambda_2, \dots, \lambda_k \in \mathbb R$</span> not all zero, such that :</p>
<p><span class="math-container">$$\lambda_1T ... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | Djura Marinkov | 361,183 | <p>In the last round you can get $\frac{1+2+3+4+5}{6}$ or lose $p\frac 1 6$, whenever the second is more than the first you should stop. So once you have scored more than 15 you should stop. If you score 15 it doesn't matter if you continue or stop.</p>
|
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | Parcly Taxel | 357,390 | <p>Before deciding whether to stop or roll, suppose you have a non-negative integer number of points $n$.</p>
<blockquote>
<p>How many more rolls should you make to maximise the expected gain over stopping (zero)?</p>
</blockquote>
<p>Suppose that further number of rolls is another non-negative integer $k$. Now con... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | richard1941 | 133,895 | <p>Why is your summation divergent? For n sufficiently large, each term is less than 123456 * (11/12)^n, which certainly converges to something finite. Particularly, look up "gradient series" in a book on engineering economics (time value of money). There you will find that x/(1-x)^2 = x+2x^+3x^3+4x^4+... Or you... |
1,796,056 | <p>I'm reading <a href="https://proofwiki.org/wiki/Non-Trivial_Discrete_Space_is_not_Connected" rel="nofollow noreferrer">this proof</a> that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton <span class="math-container">${x... | Forever Mozart | 21,137 | <p>It is true that finite sets are closed in every T$_1$ space, and thus they are closed in every discrete space. Also by the definition of the discrete topology, $\textit{every}$ subset of the space is open. So suppose $X$ is discrete and has more than one point. Let $x\in X$. Then $\{x\}$ is open. It is also closed... |
4,023,679 | <p>So my understanding is that a delta function <span class="math-container">$\delta(t)$</span> has the value of <span class="math-container">$\infty$</span> at <span class="math-container">$t = 0$</span> and <span class="math-container">$\delta(t) = 0$</span> everywhere else. Essentially, it's just one big spike on a ... | EarthIsNotFlat | 812,897 | <p>You can decompose the terms <span class="math-container">$\frac{z}{(z-n)n}$</span> as <span class="math-container">$\frac{1}{z-n} + \frac{1}{n}$</span>. So it becomes <span class="math-container">$$\sum_{n = 1}^{\infty} \frac{1}{z-n} + \frac{1}{n}$$</span></p>
<p>For <span class="math-container">$z=0$</span>, it ca... |
3,738,646 | <p>I think that a constant sequence is a Cauchy sequence because of the definition of Cauchy sequence:</p>
<p>The sequence <span class="math-container">$\{q_n\}$</span> of rationals is a Cauchy sequence if, for every <span class="math-container">$\epsilon>0$</span>, where <span class="math-container">$\epsilon$</spa... | user376343 | 376,343 | <p>You claime to have proved the boundedness of the sequence.</p>
<p>MONOTONICITY:<br />
Clearly, <span class="math-container">$s_1>s_0=\sqrt2.$</span><br />
For <span class="math-container">$n\geq 1,$</span> et us compute
<span class="math-container">$$\begin{aligned}s_{n+1}-s_n&=\sqrt{2+\sqrt{s_n}}-\sqrt{2+\sq... |
2,934,625 | <p>There is a little thing that I do not understand, about the Fourier transform of a product of functions in <span class="math-container">$L^1$</span> (and only in this space), with the relation <span class="math-container">${\mathcal F}(f g)(\lambda) = \mathcal{F}(f)\star \mathcal{F}(g) (\lambda)$</span> <strong>(and... | René Gy | 130,022 | <p>Hereafter is a proof of the following proposition: </p>
<blockquote>
<p>Let <span class="math-container">$n,m$</span> be natural numbers and let<br>
<span class="math-container">$$f(n):=\sum_{\underset{\underset{ (i,n)=(j,n)=1 } { i+j\le n}} {0\le
i,j \le n }} ij .$$</span> Let <span class="math-container">$p... |
2,809,243 | <p>I want to show that there doesn't exist any $3$-dimensional manifold for which the fundamental group is isomorphic to the direct sum $\Bbb Z\oplus \Bbb Z\oplus\Bbb Z\oplus\Bbb Z$
I have calculated the fundamental group of $K_{3,3}$ graph and $K_5$ graph and if i have done right it was the direct sum of four copies o... | Camilo Arosemena-Serrato | 33,495 | <p>Let us first suppose <span class="math-container">$M$</span> is compact.</p>
<p>Any compact orientable irreducible <span class="math-container">$3$</span>-manifold <span class="math-container">$M$</span> with infinite <span class="math-container">$\pi_1(M)$</span> is a <span class="math-container">$K(\pi_1(M),1)$</s... |
225,367 | <p>Prove that a group of order 36 must have a normal subgroup of order 3 or 9.</p>
<p>Let n2 be the number of 2-Sylow subgroups of G (with |G|=36). Then
n must be 1 or 3.
Let n3 be the number of 3-Sylow subgroups of G. then n3=1 or n3=4
if n3=1 we have 1 3-sylow group of order 9. and it is also a normal group(from sy... | Mikasa | 8,581 | <p>Assume that $G$ has $4$ Sylow $3$-groups, as you noted $n_3=(1+3k)|4, n_3\neq1$. Defining the conjugation action on the set of these $4$ Sylow $3$-groups, we have the induced homomorphism $\phi: G\longrightarrow S_4$. So $\frac{G}{\ker\phi}\hookrightarrow S_4$, but $3^2$ divides |G|=36 and does not divide $|S_4| = 2... |
2,152,736 | <p>With respect to a matrix $A\in\mathbb{R}^{n\times n}$, is there any if and only if conditions that make sure the eigen values of $A$ remain real and distinct? $A$ here is not symmetric but is stable and also there exists a $P$ such that $PAP^{-1}$ is symmetric.
For having real eigen values, $A$ needs to symmetric or... | Laray | 396,534 | <p>You already got a sufficient condition for realness (matrix being hermetian) but a condition for distint-ness is not that easy.
You could use <a href="https://math.stackexchange.com/questions/2102931/gerschgorins-theorem-round-1">Gerschgorin-circles</a> for a start. </p>
<p>This theorem gives you circles in the co... |
2,121,733 | <p>$F(x)=\begin{cases} x-1 & \text{if } x \text{ is even}\\ 2x &\text{if } x \text{ is odd}\end{cases}$</p>
<p>I need to exhibit the left inverse of this function. I know it exists because the function is one-to-one. Now, it's easy for me to switch the $x$'s and $y$'s to get:
$$g(x)=\begin{cases} \frac{x}{2} \... | Sarvesh Ravichandran Iyer | 316,409 | <p>So we are given:
$$
f(x) = \cases {
x-1 & x $\text{ even}$ \\
2x & x $\text { odd}$ \\
}
$$</p>
<p>Now, you can see this function is injective. However, we claim it is surjective, and this is the reason why:
Let $l$ be an integer. </p>
<p>Suppose $l$ is <em>odd</em>, then $l+1$ is even, right? And from th... |
4,458,244 | <p>To factorize <span class="math-container">$xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$</span> I used the fact that <span class="math-container">$x=-y$</span> and <span class="math-container">$y=z$</span> and <span class="math-container">$x=-z$</span> make the expression zero. Hence it factorize to <spa... | user26857 | 121,097 | <p><span class="math-container">$xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)=xy\left(x+y\right)+yz(y+\color{red}{x}-\color{red}x-z)-xz\left(x+z\right)=xy(x+y)+yz(y+\color{red}x)-yz(\color{red}x+z)-xz(x+z)=y(x+y)(x+z)-z(x+z)(x+y)=(x+y)(x+z)(y-z)$</span></p>
|
4,022,066 | <p>I don't quite understand how <span class="math-container">$\dfrac1 x$</span> is unbounded while <span class="math-container">$\dfrac{x}{1+x^2}$</span> is not.</p>
| poetasis | 546,655 | <p>We have
<span class="math-container">$$\quad\lim\limits_{x \to 0}\space (x)=0
\qquad \text{while}\qquad
\lim\limits_{x \to 0}\space (x^n+1)=1\quad$$</span>
so the limit is as follows</p>
<p><span class="math-container">$$
\lim\limits_{x \to 0}\frac{1}{x}=\infty\qquad
\text{while}\qquad
\lim\limits_{x \to 0}\bigg(\... |
3,350,074 | <p>Say, I have a circle </p>
<p><span class="math-container">$$\left(x – \frac 12 \right)^2 + \left( y – \frac 12 \right)^2 = \frac 14$$</span></p>
<p>If I have a point <span class="math-container">$(0,0)$</span> and a point on the circumference of the circle <span class="math-container">$(x_1,y_1)$</span>, how woul... | Martin Argerami | 22,857 | <p>If you want to avoid the empty set, you need to make sure that <span class="math-container">$f(B\cap C)$</span> does not hit all of <span class="math-container">$f(B)\cap f(C)$</span>. So for instance you could take <span class="math-container">$B=\{1,2\}$</span>, <span class="math-container">$C=\{2,3\}$</span>, and... |
4,516,512 | <p>The finite subgroups of <span class="math-container">$\mathrm{GL}_2(\mathbb{C})$</span> are pretty much classified in several expository papers in the web as well as posts on this site. However, I would like to know that how does one show that the Heisenberg group <span class="math-container">$H_3$</span> cannot be ... | Kenta S | 404,616 | <p>The Heisenberg group <span class="math-container">$H_3$</span> is a <span class="math-container">$3$</span>-group, hence supersolvable. By Theorem 16 of Serre <em>Linear Representations of Finite Groups</em>, any irreducible representation of <span class="math-container">$H_3$</span> is induced from a character of a... |
1,201,065 | <p>So we have that $a=1$, $b=0$, $c=1$, so our point reduces to $(t,t^{2}+1)$. Plugging that into the general equation for a line we have $t^{2}+ 1 = mt + b$ where $m$ is the slope and $b$ is the $y$-intercept. So we need this line to intersect $f$ once. Setting the graph of $f$ and the line equal to each other we hav... | Jack | 206,560 | <p>So here is the answer:</p>
<p>The point $(t,t^{2}+1)$ lies on the graph $f(x) = x^{2}+1$ and we need to find the line that passes through this point and intersects with the graph only once. So we can precede as follows:</p>
<p>The line in question is $y = kx+b$. To find the slope we have $ k = \frac{t^{2}+1-y}{t... |
2,321,490 | <p>I know that :</p>
<p><span class="math-container">$$\ln: (0,+\infty )\to \mathbb{R}$$</span>
<span class="math-container">$$\ln(x):=\int_1^x \dfrac{dt}{t}$$</span></p>
<hr>
<p>And :</p>
<p><span class="math-container">$$\ln^{-1}: \mathbb{R} \to \mathbb{R}^+$$</span>
<span class="math-container">$$\ln^{-1}(x):=\e... | Jean Marie | 305,862 | <p>The discovery of (decimal) logarithms by Napier (tables published in 1614) predates by a century or so the discovery of the fact that the area between $1$ and $a$ under the hyperbola with equation $y=\dfrac{1}{x}$ has a value which is proportional to a multiple of the (decimal) logarithm of $a$. It was not until the... |
2,590,359 | <p>I have to find a distance between a plane represented with a triangle and a point in $3D$ space. I searched other questions but the plane is usually represented by a plane equation or a normal, so I am having trouble finding this distance when the plane is represented by three points.</p>
| Community | -1 | <p>Using the cross product, obtains <strong>a</strong> normal vector $n =(a,b,c)$ and then Write the equation of the plan in form
$$(\Bbb P):ax+by+cz+d =0$$</p>
<p>Then </p>
<p>$$d((x_0,y_0,z_0), \Bbb P) =\frac{|ax_0+by_0+ cz_0+d|}{\sqrt{a^2+b^2+c^2}}$$</p>
|
2,525,055 | <p>How can the differential of $\mathbf r\cdot\mathbf r$ be written as $2\mathbf r\cdot d\mathbf r$? That is,</p>
<p>$$d(\mathbf r\cdot\mathbf r)=2\mathbf r\cdot d\mathbf r$$</p>
<p>How? Here, $\mathbf r$ is a vector.</p>
| Trevor Gunn | 437,127 | <p>If $\mathbf{r} = (x_1,\dots,x_n)$ then
$$d(\mathbf{r} \cdot \mathbf{r}) = d\left( \sum_{i = 1}^n x_i^2 \right) = \sum_{i = 1}^n d\left(x_i^2\right) = \sum_{i = 1}^n 2x_idx_i = 2\mathbf{r} \cdot d\mathbf{r}. $$</p>
|
231,027 | <p>I have the following complicated expression:</p>
<pre><code>(3 (Sqrt[6] a^13 + a^12 e0 + Sqrt[6] a^11 e0^2 + a^10 e0^3 + Sqrt[6] a^9 e0^4 -
a^8 e0^5 - Sqrt[6] a^7 e0^6 - a^6 e0^7 + Sqrt[6] a^5 e0^8 - a^4 e0^9 -
Sqrt[6] a^3 e0^10 + a^2 e0^11 -Sqrt[6] a e0^12 + e0^13)) /
(256 e0 (a^6 + Sqrt[6] a^5 e0 + a^4 e0^2... | Ulrich Neumann | 53,677 | <p>Subsitude <code>a-> eps e0</code></p>
<pre><code>expr=expr = (3 (Sqrt[6] a^13 + a^12 e0 + Sqrt[6] a^11 e0^2 + a^10 e0^3 +
Sqrt[6] a^9 e0^4 - a^8 e0^5 - Sqrt[6] a^7 e0^6 - a^6 e0^7 +
Sqrt[6] a^5 e0^8 - a^4 e0^9 - Sqrt[6] a^3 e0^10 + a^2 e0^11 -
Sqrt[6] a e0^12 + e0^13))/(256 e0 (a^6 + Sqrt[6] a^5 e0 + a^4 e0^2 ... |
1,503,369 | <p>Let <span class="math-container">$n$</span> be a Von Nemuan natural number, i.e. the set of all natural numbers less then <span class="math-container">$n$</span>.
Why it holds that <span class="math-container">$\mathscr{P}^-(n)\overset{def}{=}\{u:u\subseteq \{0,...,n-1\}\}$</span>
<span class="math-container">$=\mat... | mookid | 131,738 | <p>Using induction: assume that
$$
\dim U = \dim U_1 + \cdots + \dim U_n \implies { \rm every }\,x\ldots
$$
then, assuming
$$\dim U = \dim U_1 + \cdots + \dim U_{n+1}$$
and defining $V =U_1 + \cdots + U_n $: $U = V + U_{n+1}$.
$$
\dim U \le \dim V + \dim U_{n+1} \le \dim U_1 + \cdots + \dim U_{n} + \dim U_{n+1} = \d... |
54,391 | <p>Consider the following two structure-adding refinements of the fundamental group of a topological space:</p>
<ul>
<li><p>the set $\pi_1(X)$ inherits a quotient topology from the compact-open topology of $X^{S^1}$, under which it is sometimes a topological group. This was discussed <a href="https://mathoverflow.net... | Jeremy Brazas | 5,801 | <p>I don't really have a great feeling for the information captured by the "toposophic" fundamental group so this answer is a bit one-sided towards what I can say about the quotient topology on the fundamental group and what it has to do with covering maps. Maybe you can form something useful from this.</p>
<p>First, ... |
1,672,666 | <p>I'm currently stuck on a proof that's giving me a hard time, not sure where to start:</p>
<p>Prove or disprove: If an integer $q \gt 1$ is composite, then $q$ will have some divisor $d \gt 1$ such that $d \lt \sqrt{q}$</p>
<p>Any ideas on how to approach these problems or any help at all would be greatly appreciat... | Tim Raczkowski | 192,581 | <p>It is not true that a series converges if $\lim a_n=0$. For example, consider
$$\sum_{n=1}^\infty\frac 1n.$$</p>
<p>Clearly, $\lim_{n\to\infty}\frac 1n=0$, but it is easily shown by the integral test that the series diverges.</p>
|
4,145,243 | <p><a href="https://i.stack.imgur.com/1XO2v.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1XO2v.png" alt="enter image description here" /></a></p>
<p>The way I approached the problem is first by solving a system of equations:
<span class="math-container">$$
r_x \frac{dx}{dt} + r_y\frac{dy}{dt} = \f... | rschwieb | 29,335 | <p>The irreducible (i.e. simple) submodules are nonisomorphic because they have distinct annihilators.</p>
<p><span class="math-container">$V_1$</span> is annihilated by <span class="math-container">$0\oplus V_2\oplus V_3\oplus\cdots$</span> and <span class="math-container">$V_2$</span> is annihilated by <span class="m... |
4,610,333 | <p>Show <span class="math-container">$n^{\frac{1}{n}}>(n+1)^{\frac{1}{n+1}}$</span> for <span class="math-container">$n\geq3$</span>.</p>
<p>I took the derivative of <span class="math-container">$x^{1/x}$</span> and it is a decreasing function for <span class="math-container">$x>e$</span>.</p>
<p>I was just tryin... | LL 3.14 | 731,946 | <p>Let <span class="math-container">$n\geq 3$</span>. Then by the binomial formula
<span class="math-container">$$
(1+\tfrac{1}{n})^n = \sum_{k=0}^n \binom{n}{k} \frac{1}{n^k} = \sum_{k=0}^n \frac{1}{k!}\left(1\cdot\frac{n-1}{n}\cdot\dots\frac{n-k-1}{n}\right) < \sum_{k=0}^n \frac{1}{k!} < e < n,
$$</span>
and... |
1,064,534 | <p>$$x^3-3y^2x=-1$$
$$3yx^2 -y^3=1$$</p>
<p>This was the real part and imaginary part on a previous question I asked, instead of the system it was easier to just use polar coordinates to solve, but if this was just a system unrelated to the problem, how would one solve it? I seem to have trouble when the terms are lik... | Saibal | 191,019 | <p>If you want to use complex numbers, here's an easier <em>ad hoc</em> way (I tried to describe how I found the solution step by step):</p>
<p>Note that, the terms involved arise in the expansion of $(x-y)^3$, but the signs are messed up. In particular, we need signs of the terms in the same equation to be same to us... |
1,064,534 | <p>$$x^3-3y^2x=-1$$
$$3yx^2 -y^3=1$$</p>
<p>This was the real part and imaginary part on a previous question I asked, instead of the system it was easier to just use polar coordinates to solve, but if this was just a system unrelated to the problem, how would one solve it? I seem to have trouble when the terms are lik... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
4,357,233 | <p>Question</p>
<blockquote>
<p><span class="math-container">$$\sqrt {x^2+1} =x-1$$</span></p>
</blockquote>
<p>my attempt</p>
<p><span class="math-container">$$ x^2 + 1 = x^2-2x+1 $$</span>
and x=0</p>
<p>by putting the value of 0 = false</p>
| Community | -1 | <p>On squaring an equation you may obtain extraneous solution,therefore you would have to recheck your solutions with original equation.</p>
<p>Also, <span class="math-container">$\sqrt {x^2+1} =x-1,$</span></p>
<p><span class="math-container">$x-1 \geq 0$</span> <span class="math-container">$\implies$</span> <span cla... |
3,104,936 | <p>I was wondering if anyone could explain to me why this holds true?</p>
<p><span class="math-container">$$C\left(n, k\right) = \frac{C\left(n, k-1\right) \cdot C(n - (k - 1), 1)}k$$</span></p>
<p>or</p>
<p><span class="math-container">$$C\left(3, 2\right) = \frac{C\left(3, 1\right) \cdot C(2, 1)}2$$</span></p>
<p... | Mayuresh L | 503,630 | <p>LHS is number of ways to choose <span class="math-container">$k$</span> objects from <span class="math-container">$n$</span> objects.</p>
<p>RHS is nothing but,
now from <span class="math-container">$n$</span> objects first choose <span class="math-container">$k-1$</span> objects and then from remaining <span class... |
4,529,873 | <p>I am currently dealing with the two terms "Gateaux-derivation" and "Frechet-derivation" and would like to know if there is a Frechet differentiable function <span class="math-container">$f$</span> and Gateaux differentiable function <span class="math-container">$g$</span>, so that <span class="ma... | gerw | 58,577 | <p>The composition can even be discontinuous: Consider <span class="math-container">$f \colon \mathbb R^2 \to \mathbb R$</span> defined via
<span class="math-container">$$
f(x,y) = \exp\left(-\frac{(y-x^2)^2}{x^6}\right)$$</span>
for <span class="math-container">$x \ne 0$</span> and <span class="math-container">$f(0,y)... |
35,139 | <p>I've came upon a few methods of how to display "processing" information while <em>Mathematica</em> is running. But, from what I've understood, this only works in between lines of codes. Is it possible to display a "how far is something being processed"-information during a line of code?</p>
<p>For example, I wish t... | Simon Woods | 862 | <p>A simple way to do it would be to partition the big list into chunks, and map over each chunk in turn, monitoring the progress of that outer loop.</p>
<pre><code>data = Range[10^6];
chunks = Partition[data, 1000];
result = Flatten[Table[g /@ chunks[[i]], {i, Length[chunks]}]] ~Monitor~ i;
Short[result]
(* {g[1], g... |
35,139 | <p>I've came upon a few methods of how to display "processing" information while <em>Mathematica</em> is running. But, from what I've understood, this only works in between lines of codes. Is it possible to display a "how far is something being processed"-information during a line of code?</p>
<p>For example, I wish t... | Stefan | 2,448 | <p>Or you can use something like this:</p>
<pre><code>Monitor[
Table[Length@FactorInteger[2^n - 1], {n, 50, 500, 50}],
Grid[{{Text[Style["Integer Factorization :", Darker[Blue, 0.66]]],
ProgressIndicator[n, {50, 500}]},
{Text[Style["Factoring 2^n-1 with n : ", Darker[Blue, 0.66]]], n}},
Alignment ... |
368,332 | <p>Find the value of $\displaystyle\lim_{n \rightarrow \infty} \dfrac 1{n^4}\left[1\left(\sum^n_{k=1}k\right)+2\left(\sum^{n-1}_{k=1}k\right)+3\left(\sum^{n-2}_{k=1}k\right)+\dots+n.1\right]$</p>
<p>Please guide how to proceed in this case .....Thanks..</p>
<p>Here answer is : </p>
<p>The required limit is $\dfrac{1... | Dimitris | 37,229 | <p>$$\displaystyle\lim_{n \rightarrow \infty} \dfrac 1{n^4}\left[1\left(\sum^n_{k=1}k\right)+2\left(\sum^{n-1}_{k=1}k\right)+3\left(\sum^{n-2}_{k=1}k\right)+\dots+n.1\right]=$$
$$\lim\limits_{n\rightarrow\infty}\dfrac{1}{n^4}\left[1\dfrac{n(n+1)}{2}+2\dfrac{(n-1)(n)}{2}+\dots +k\dfrac{(n-k+1)(n-k+2)}{2}+\dots+n\cdot1\r... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.