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 |
|---|---|---|---|---|
1,541,800 | <p>I happened to stumble upon the following matrix:
$$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$</p>
<p>And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial, then:
$$ P(A)=\begin{bmatrix}
P(a) & P'(a) \... | Dac0 | 291,786 | <p>It's a general statement if <span class="math-container">$J_{k}$</span> is a Jordan block and <span class="math-container">$f$</span> a function matrix then
<span class="math-container">\begin{equation}
f(J)=\left(\begin{array}{ccccc}
f(\lambda_{0}) & \frac{f'(\lambda_{0})}{1!} & \frac{f''(\lambda_{0})}{2!} ... |
1,541,800 | <p>I happened to stumble upon the following matrix:
$$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$</p>
<p>And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial, then:
$$ P(A)=\begin{bmatrix}
P(a) & P'(a) \... | WW1 | 88,679 | <p>$$ A =a \mathbb{I}+M $$
where
$$M = \begin{bmatrix}
0 & 1 \\
0 & 0
\end{bmatrix}$$</p>
<p>most relevantly, $M$ is an upper triangular matrix satisfying $ M^n=0;n>1$</p>
<p>$$ A^n=\left(a \mathbb{I}+M \right)^n $$
Since all these matrices ( $ \mathbb{I}$ and $M$ ) commute we can... |
1,541,800 | <p>I happened to stumble upon the following matrix:
$$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$</p>
<p>And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial, then:
$$ P(A)=\begin{bmatrix}
P(a) & P'(a) \... | CR Drost | 42,154 | <p>Just to expand on the comment of @Myself above...</p>
<h1>Adjoining algebraics as matrix maths</h1>
<p>Sometimes in mathematics or computation you can get away with adjoining an algebraic number $\alpha$ to some simpler ring $R$ of numbers like the integers or rationals $\mathbb Q$, and these characterize all of y... |
1,609,945 | <p>Prove that the set of points for which $\arg(z-1)=\arg(z+1) +\pi/4$ is part of the set of points for which $|z-j|=\sqrt 2$ and show which part clearly in a diagram.</p>
<p>Intuition for the geometrical interpretation of this would be really appreciated. </p>
<p>And I was wondering greatest and lowest value for $|z... | OnoL | 65,018 | <p><strong>Hint:</strong></p>
<ol>
<li>Define for each $t\in T$ a function $d^t:S\to \mathbf R_+$ such that $d^t(s)=|s-t|$. Prove that: (i) $d^t$ is continuous for each $t\in T$ and (ii) $d^t$ achieves a minimum $m(t)>0$. </li>
<li>Let $M=\{m(t)\,|\,t\in T\}$. Prove that $\inf M>0$. (<strong>further hint:</stron... |
1,999,352 | <p>so what really is the meaning of a metric space and why is it so important in topology?</p>
| layman | 131,740 | <p>The idea of a mathematical space is that it is a set endowed with some type of structure (and the word structure just means there's more to the definition of the set than just a collection of elements -- there's an additional feature added to the set; we call this feature the structure of the set). Here are some ex... |
2,687,932 | <p>Let $x\in\mathbb{R}$. Prove that $x=-1$ if and only if $x^3+x^2+x+1=0$.
This is a bi-conditional statement, thus to prove it we need to prove:
<a href="https://i.stack.imgur.com/PRPSC.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PRPSC.png" alt="enter image description here"></a></p>
| Surb | 154,545 | <p>$$x=-1\implies x^3+x^2+x+1=(-1)^3+(-1)^2+-1+1=-1+1-1+1=0.$$</p>
|
1,985,849 | <p>I am really stumped by this equation.
$$e^x=x$$
I need to prove that this equation has no real roots. But I have no idea how to start.</p>
<p>If you look at the graphs of $y=e^x$ and $y=x$, you can see that those graphs do not meet anywhere. But I am trying to find an algebraic and rigorous proof. Any help is appre... | Astyx | 377,528 | <p>Study $f : x \mapsto e^x - x$</p>
<p>Its derivative is $x \mapsto e^x-1$ which cancels at $0$, is negative before and positive after, thus $f$ reaches a minimum at $0$, and that minimum is $f(0) = 1 \gt 0$. Thus $f$ never cancels, and there are therefore no real solutions to the equation $e^x -x = 0$ ie $e^x = x$</... |
90,548 | <p>Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra?</p>
<p>I guess in order to narrow it down a bit, I'll phrase it like this: what are some necessary conditions on an algebra for ... | Vladimir Dotsenko | 1,306 | <p>A homological condition that might be useful: in the Hopf case, the Yoneda algebra $Ext_A^\bullet(k,k)$ embeds into the Hochschild cohomology $HH^\bullet(A,A)$, moreover, there is a Gerstenhaber algebra structure on the Yoneda algebra, and this embedding is an embedding of Gerstenhaber algebras. </p>
<p>Reference: ... |
2,941,896 | <p>I want to ask you for following statement</p>
<blockquote>
<p><span class="math-container">$f$</span> is continuous from <span class="math-container">$X$</span> to <span class="math-container">$Y$</span>, where <span class="math-container">$X$</span> is compact Hausdorff, connected and <span class="math-container... | Wuestenfux | 417,848 | <p>Hint: <span class="math-container">$\log(ab) = \log(a) + \log(b)$</span>. This should be enough to simplify the sum on the left-hand side.</p>
|
505,617 | <p>Let $a,b,c$ be nonnegative real numbers such that $a+b+c=3$, Prove that</p>
<blockquote>
<p>$$ \sqrt{\frac{a}{a+3b+5bc}}+\sqrt{\frac{b}{b+3c+5ca}}+\sqrt{\frac{c}{c+3a+5ab}}\geq 1.$$</p>
</blockquote>
<p>This problem is from <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=555716">http:... | Karuppiah Kannappan | 95,107 | <p>We know the one result if sum of two numbers $a$ and $b$ is constant, the max. of $ab$ occurs when both $a$ and $b$ equal. Eg. if $a+b=6$, then $\max\{ab\}=9$. Use this result, we get proof.</p>
|
221,762 | <p>Let $f$ be a newform of level $\Gamma_1(N)$ and character $\chi$ which is not induced by a character mod $N/p$. I learned from <a href="http://wstein.org/books/ribet-stein/main.pdf" rel="nofollow">these notes</a> by Ribet and Stein that $|a_p|=p^{(k-1)/2}$ where $k$ is the weight of $f$. So I wonder</p>
<p>1, the ... | GH from MO | 11,919 | <p>Here is an answer to both questions.</p>
<p><strong>First question.</strong> The quoted result is a special case of Theorem 9.1.10 in the <a href="http://wstein.org/books/ribet-stein/main.pdf" rel="nofollow">Ribet-Stein notes</a>, which in turn is identical to Theorem 3 in <a href="http://gdz.sub.uni-goettingen.de/... |
659,988 | <p>I understood the definition of a $\sigma$-algebra, that its elements are closed under complementation and countable union, but as I am not very good at maths, I could not visualize or understand the intuition behind the meaning of "closed under complementation and countable union". </p>
<p>If we consider the set X... | Spock | 108,632 | <p>Let $X = \{a, b, c, d\}$, a possible sigma algebra on $X$ is $Σ = \{∅, \{a, b\}, \{c, d\}, \{a, b, c, d\}\}$. <br/>
I think this is a good example.</p>
|
894,917 | <p>I'm stuck with the limit $\lim_{n\to\infty} (2^n)\sin(n) $. I've been trying the squeeze theorem but it doesn't seem to work. I can't think of a second way to tackle the problem. Any push in the right direction would be much appreciated. </p>
<blockquote>
<p>Also, please don't just post the answer up because I wa... | Community | -1 | <p>Every interval $[n\pi/2-1/2, n\pi/2+1/2]$, $n$ odd contains an integer. Use this to show some subsequence tends to $\infty$ and some subsequence tends to $-\infty$. -- <a href="https://math.stackexchange.com/questions/894917/limit-of-2n-sinn-as-n-goes-to-infinity#comment1847157_894917">David Mitra</a> </p>
|
2,882,678 | <p>Why are the morphisms of the category of sets functions?
Shouldn't the morphisms take an object in a category and turn it into another object of the category, i.e. map Set to Set. I don't understand how e.g.
$f(x)=x^2$
is a map from a set to a set. </p>
<p>If the morphism had been the image of a set under a functio... | freakish | 340,986 | <blockquote>
<p>Shouldn't the morphisms take an object in a category and turn it into another object of the category</p>
</blockquote>
<p>No, what makes you think so? Are you perhaps confusing <em>morphisms</em> with <em>functors</em>?</p>
<p>Morphism is just an arrow between two objects. Objects in set category ar... |
2,882,678 | <p>Why are the morphisms of the category of sets functions?
Shouldn't the morphisms take an object in a category and turn it into another object of the category, i.e. map Set to Set. I don't understand how e.g.
$f(x)=x^2$
is a map from a set to a set. </p>
<p>If the morphism had been the image of a set under a functio... | red_trumpet | 312,406 | <blockquote>
<p>take an object in a category and turn it into another object of the category</p>
</blockquote>
<p>This is what is called a <a href="https://en.wikipedia.org/wiki/Functor" rel="nofollow noreferrer">functor</a>.</p>
<p>A morphism is (more or less) exactly a generalization of functions between sets. On... |
1,225,551 | <p>I have a vector $\mathbf{x}$ (10 $\times$ 1), a vector $\mathbf{f}(\mathbf{x})$ (10 $\times$ 1) and an invertible square matrix $\mathbf{Z}(\mathbf{x})$ (10 $\times$ 10) which is block diagonal:</p>
<p>$$\mathbf{Z}(\mathbf{x}) = \begin{bmatrix}\mathbf{A} & 0\\\mathbf{C} & \mathbf{B}\end{bmatrix}$$</p>
<p>$... | Astrinus | 229,726 | <p>I've found that the solution is much more simple than I thought.</p>
<p>Consider the whole term $\frac{\partial(\mathbf{Z}^{-1})}{\partial \mathbf{x}}\mathbf{f} = -\mathbf{Z}^{-1}\frac{\partial\mathbf{Z}}{\partial \mathbf{x}}\mathbf{Z}^{-1}\mathbf{f}$, with the vector $\mathbf{f}$ too; the derivative term in the ri... |
194,867 | <p>Let $f(z)$ be a function meromorphic in a simply connected convex domain $D$ (subset of the complex plane with positive area or the whole complex plane) where $z$ is a complex number.</p>
<p>Are there such functions $f(z)$ where $\Re(f(z))$ is periodic in the domain (no periods larger than the domain please :p ) bu... | hmakholm left over Monica | 14,366 | <p>The real part of $z\mapsto iz$ has period $p$ for any $p\in \mathbb R$.</p>
|
4,306,419 | <p>Can I write <span class="math-container">$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}$</span> implies <span class="math-container">$\frac{dx}{dy}=\frac{x^2-y^2}{2yx}$</span> ?</p>
<p>Actually I have been given two differential equations to solve.</p>
<p><span class="math-container">$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}$</span> ... | JJacquelin | 108,514 | <p>If you are not convinced, instead of Eq.<span class="math-container">$(1)$</span> or Eq.<span class="math-container">$(2)$</span> consider the differential equation <span class="math-container">$(3)$</span> in which no <span class="math-container">$\frac{dy}{dx}$</span> or <span class="math-container">$\frac{dx}{dy}... |
1,045,941 | <p>Usually this is just given as a straight up definition in a calculus course. I am wondering how you prove it?
I tried using the limit definition, $$\lim\limits_{h\rightarrow 0} \dfrac{\log(x+h)-\log(x)}{h}$$
but this led to no developments.</p>
| k170 | 161,538 | <p>First note that</p>
<blockquote>
<p>$$ e= \lim\limits_{z\to 0}\left(1+z\right)^{\frac{1}{z}}$$</p>
</blockquote>
<p>So now
$$ \lim\limits_{h\to 0}\frac{\ln(x+h)-\ln (x)}{h} = \lim\limits_{h\to 0}\frac{\ln\left(\frac{x+h}{x}\right)}{h} $$
$$
= \lim\limits_{h\to 0}\frac{1}{h}\ln\left(1+\frac{h}{x}\right) = \lim\li... |
1,045,941 | <p>Usually this is just given as a straight up definition in a calculus course. I am wondering how you prove it?
I tried using the limit definition, $$\lim\limits_{h\rightarrow 0} \dfrac{\log(x+h)-\log(x)}{h}$$
but this led to no developments.</p>
| marty cohen | 13,079 | <p>As I have written here before
(somewhere),
I like using the
functional equation
for logs of
$f(xy) = f(x)+f(y)$.</p>
<p>From this,
$f(xy)-f(x) = f(y)$
or
$f(x)-f(y) = f(x/y)
$.</p>
<p>Getting this into a form
that looks like a derivative,
$f(x+h)-f(x)
=f(1+h/x)
$.</p>
<p>Since $f(1) = 0$
(from $f(1) = f(1)-f(1) =... |
2,583,232 | <p>Prove that $\lim_{n \to \infty} \dfrac{2n^2}{5n^2+1}=\dfrac{2}{5}$</p>
<p>$$\forall \epsilon>0 \ \ :\exists N(\epsilon)\in \mathbb{N} \ \ \text{such that} \ \ \forall n >N(\epsilon) \ \ \text{we have} |\dfrac{-2}{25n^2+5}|<\epsilon$$
$$ |\dfrac{-2}{25n^2+5}|<\epsilon \\ |\dfrac{2}{25n^2+5}|<\epsi... | Stu | 460,772 | <p>Let $\varepsilon >0$</p>
<p>$|\dfrac{-2}{25n^2+5}|<\varepsilon\iff \dfrac{2}{25n^2+5}<\varepsilon$</p>
<p>$\dfrac{2}{25n^2+5}<\dfrac{2}{25n^2}<\varepsilon$ so $n^2>\dfrac{2}{25\varepsilon}\iff n>\dfrac{\sqrt{2}}{5\sqrt{\varepsilon}}$</p>
<p>Then $N(\varepsilon)=\bigg\lfloor\dfrac{\sqrt{2}}{5\... |
3,984,778 | <p>Ackermann's function and all the up-arrow notation is based on exponentiating. We know for a fact that the factorial function grows faster than any power law so why not build an iterative sequence based on factorials? I am thinking of the simple function defined by
<span class="math-container">$$
a(n)=a(n-1)!
$$</sp... | Simply Beautiful Art | 272,831 | <p>To answer the question concerning writing this in for loops, here's some python-like pseudocode to consider:</p>
<pre><code># Some integer to compute a(n).
n: int
# Start with 3,
solution = 3
# and then apply the factorial n-1 times.
for i in [1, 2, ..., n-1]:
# To compute the factorial, start with 1,
factor... |
243,377 | <p>Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t_i>0$ to the left or the right with equal probabilities. </p>
<p>To formalize this, we have $(X_n)_{n\geq 0}$ such that $X_0=0$, $Pr[X_n=t_n]=0.5$ and $Pr[X_n=-t_n]=0.5$ (hence st... | js21 | 21,724 | <p>Yes. Indeed, if $s = \sum_{i \geq 1} t_i^2 <1$, then
$$
\mathbb{P}[ \ \ \forall n, \sum_{i=1}^n X_i \in [-1,1] \ \ ] \geq 1-s > 0.
$$
To see this, note that $M_n = |\sum_{i=1}^n X_i|$ is a nonnegative submartingale, so that Doob's martingale inequality yields
$$
\mathbb{P}[ \max_{1 \leq j \leq n} M_j > 1 ]... |
3,720,012 | <blockquote>
<p>(Feller Vol.1, P.241, Q.35) Let <span class="math-container">$S_n$</span> be the number of successes in <span class="math-container">$n$</span> Bernoulli trials. Prove
<span class="math-container">$$E(|S_n-np|) = 2vq b(v; n, p) $$</span>
where <span class="math-container">$v$</span> is the integer such ... | Pepe Silvia | 418,447 | <p>I'd be tempted to take this hint and see how I can simplify it first. And by simplify I mean get rid of the summation. So we have
<span class="math-container">\begin{align*}
&\sum_{k=0}^{v-1}(np-k)\left(\begin{array}{cc} n \\ k\end{array}\right)p^kq^{n-k} \\
=&np\sum_{k=0}^{v-1}\left(\begin{array}{cc} n \\ k... |
262,773 | <p>In the curve obtained with following</p>
<pre><code> Plot[{1/(3 x*Sqrt[1 - x^2])}, {x, 0, 1}, PlotRange -> {0, 1},
GridLines -> {{0.35, 0.94}, {}}]
</code></pre>
<p>how can one fill the top and bottom with different colors or patterns such that two regions are perfectly visible in a black-n-white printout?<... | kglr | 125 | <pre><code>Plot[{1/(3 x*Sqrt[1 - x^2]), 1/(3 x*Sqrt[1 - x^2])}, {x, 0, 1},
PlotStyle -> Directive[AbsoluteThickness[2], Opacity[1], Black],
PlotRange -> {0, 1},
GridLines -> {{0.35, 0.94}, {}},
Method -> "GridLinesInFront" -> True,
GridLinesStyle -> Directive[Black, Dashed],
Fill... |
2,901,971 | <p>The question is as follows: </p>
<blockquote>
<p>Prove that for every $\beta \in \mathbb{R}$, $\sup(-\infty, \beta) = \beta$.</p>
</blockquote>
<p>The goal of the problem is to prove this without using the $\epsilon-\delta$ method. My professor gave us an idea in class, but for some reason, it isn't really maki... | nonuser | 463,553 | <ol start="4">
<li><p>If $|X|=|Y|=n$ and $\delta \geq n/2$.</p></li>
<li><p>If $|X|=|Y|=n$ and $\varepsilon \geq n^2-n+1$.</p></li>
<li><p>If $|X|=|Y|=n \ge 1$ Suppose that $X=\{x_1,x_2,..,x_n\}$ and $Y=\{y_1,y_2,...,y_n\}$. Two vertices $x_i$ and $y_j$ are adjacent in $G$ if and only if $i+j \ge n+1$.</p></li>
</ol>
|
1,517,086 | <p>I spent a long time trying to find a natural deduction derivation for the formula $\exists x(\exists y A(y) \rightarrow A(x))$, but I always got stuck at some point with free variables in the leaves. Could someone please help me or give me some hints to find a proof. </p>
<p>Thanks.</p>
| Jannik Vierling | 287,757 | <p>The following is a natural deduction proof of the formula <span class="math-container">$\exists x (\exists y A(y) \rightarrow A(x))$</span>
<span class="math-container">$$
\dfrac
{
\dfrac
{
\dfrac
{
\dfrac
{
\dfrac
{
\dfrac
{}
{
[\exis... |
2,413,899 | <p>When I was 7 or 8 years old, I was playing with drawing circles. For some reason, I thought about taking a 90° angle and placing the vertex on the circle and marking where the two sides intersected the circle.</p>
<p><a href="https://i.stack.imgur.com/oskEP.png" rel="nofollow noreferrer"><img src="https://i.stack.i... | Claude Leibovici | 82,404 | <p>If you cannot use special functions, then either numerical integration or approximation would be required.</p>
<p>For example, consider the Taylor expansion built around $x=\frac 12$ (mid point of the integration interval selected in order to tvoid promoting one of the bounds). You would get
$$2^{x^2+x}=2^{3/4}+2... |
745,876 | <p>$$\exists! x : A(x) \Rightarrow \exists x : A(x)$$
Assuming that $A(x)$ is an open sentence. I'm new to abstract mathematics and proofs, so I came here to ask for some simplification. Thanks</p>
| dtldarek | 26,306 | <p>Notation $\exists! x. P(x)$ means "there is a unique $x$ such that $P(x)$". The common way to express this is by adding "$P(y)$ implies $x = y$", that is:</p>
<p>$$\exists x :P(x) \land \Big(\forall y : P(y) \implies (x = y)\Big).$$</p>
<p>However, please check, if this actually is the definition of $\exists!$ you... |
1,792,402 | <p>I am trying to solve <a href="https://en.wikipedia.org/wiki/Mean_value_theorem" rel="nofollow">Mean Value theorem</a> but i ran into a road block trying to solve the question. So the question is:
Assuming</p>
<p>$$ \frac{f(b) − f(a)}{b − a} = f'(c) \quad \text{for some } c \in (a, b)$$</p>
<p>Let $f(x) = \sqrt x$ ... | HighEnergy | 330,386 | <p>The above answers are correct; however, I would like to point out that in lieu of utilizing $$\frac{f(b)-f(a)}{b-a},$$ you utilized $f'(a)$ and $f'(b)$.
Before using <code>MVT</code>, make sure f is continuous on $[a,b]$ and differentiable on $(a,b)$, which they are here.</p>
|
2,513,509 | <p>If $f,g,h$ are functions defined on $\mathbb{R},$ the function $(f\circ g \circ h)$ is even:</p>
<p>i) If $f$ is even.</p>
<p>ii) If $g$ is even.</p>
<p>iii) If $h$ is even.</p>
<p>iv) Only if all the functions $f,g,h$ are even. </p>
<p>Shall I take different examples of functions and see?</p>
<p>Could anyone ... | don-joe | 360,559 | <ul>
<li><p>A good point to start is always to put in some values (n=10, n=100, n=1000, ...) to get a first guess.</p></li>
<li><p>In this particular case, try to look at the limit $1/n = x \rightarrow 0$ and apply L'Hôpital's rule (multiple times)</p></li>
</ul>
|
99,312 | <p>Let C be a geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that</p>
<ul>
<li>$L \cap K' = K$, and</li>
<li>$C(L) \neq \emptyset$?</li>
</ul>
<p>Note that the hypotheses on C are necessary -- the curve x^2 + y^2 = 0, w... | Olivier Benoist | 2,868 | <p>I think this is a consequence of (variants of) Hilbert's irreducibility theorem. Let me explain why. Suppose that $C$ is a geometrically integral curve defined over a number field $K$. Let $K'/K$ be a normal finite extension. We will show that infinitely many points of $C$ are defined over a field disjoint from $K'$... |
1,972,252 | <blockquote>
<p>If $v^TAv = v^TBv$ for all constant vectors $v$ and $A,B$ are matrices of size $n$ by $n$, is it true that $A=B$? </p>
</blockquote>
<p>I have thought about using basis vectors but cannot get system of equations uniquely. Is there a trick here?</p>
| W. mu | 369,495 | <p>The question is the same as: </p>
<blockquote>
<p>Is it true $A=0$, if $v^TAv=0$, for all vectors $v$? </p>
</blockquote>
<p>If $A$ is anti-symmetric, it is not.</p>
|
2,375,023 | <p>So I have to find an interval (in the real numbers) such that it contains all roots of the following function:
$$f(x)=x^5+x^4+x^3+x^2+1$$</p>
<p>I've tried to work with the derivatives of the function but it doesn't give any information about the interval, only how many possible roots the function might have.</p>
| Masacroso | 173,262 | <p>HINT: observe that</p>
<p>$$\lim_{x\to +\infty}f(x)=+\infty,\qquad\lim_{x\to -\infty}f(x)=-\infty$$</p>
<p>and</p>
<p>$$\lim_{x\to \pm\infty}\frac{4x^4}{x^5}=0$$</p>
<p>Thus exists some $\alpha>1$ such that $$|x^5|>|4x^4|>|
x^4+x^3+x^2+1|$$</p>
<p>when $x>\alpha$ or $x<-\alpha$.</p>
|
1,015,264 | <p>This is a worked out example in my book, but I am having a little trouble understanding it:</p>
<p>Consider the system of equations:</p>
<p>$$x'=y+x(1-x^2-y^2)$$
$$y'=-x+y(1-x^2-y^2)$$</p>
<p>The orbits and limit sets of this example can be easily determined by using polar coordinates. (My question: what is the m... | AlexR | 86,940 | <p>Firsly, anything dependent on $x^2+y^2$ (such as $1-x^2-y^2 = 1-(x^2+y^2)$) lends itself to conversion into polar (or spherical or cylindrical) coordinates. We obtain
$$\begin{align*}
r^2 & = x^2+y^2 \\
\Rightarrow \frac d{dt} r^2 & = \frac d{dt} x^2 + \frac d{dt} y^2 \\
\Rightarrow 2rr' & = 2xx' + 2yy' ... |
877,683 | <p><img src="https://www.anonimg.com/img/c0423915f5f5fae8f9790506c36393f8.jpg" alt="">
Based on my last question I learned that this is an envelope of a parabola</p>
<p><a href="https://math.stackexchange.com/questions/874478/what-is-this-geometric-pattern-called">What is this geometric pattern called?</a></p>
<p>But... | Community | -1 | <p>The curve you see is by <a href="http://en.m.wikipedia.org/wiki/B%C3%A9zier_curve#Quadratic_B.C3.A9zier_curves" rel="nofollow">definition</a> a quadratic bézier curve which is always a segment of a parabola. </p>
|
1,031,464 | <p>I am supposed to simplify this:</p>
<p>$$(x^2-1)^2 (x^3+1) (3x^2) + (x^3+1)^2 (x^2-1) (2x)$$</p>
<p>The answer is supposed to be this, but I can not seem to get to it:</p>
<p>$$x(x^2-1)(x^3+1)(5x^3-3x+2)$$</p>
<p>Thanks</p>
| Pauly B | 166,413 | <p>$$(x^2-1)^2 (x^3+1) (3x^2) + (x^3+1)^2 (x^2-1) (2x)$$
Factor out the $(x^2-1)$ term to get
$$(x^2-1)\left((x^2-1) (x^3+1) (3x^2) + (x^3+1)^2(2x)\right)$$
then factor out the $(x^3+1)$ term to get
$$(x^2-1)(x^3+1)\left((x^2-1) (3x^2) + (x^3+1)(2x)\right)$$
then factor out the $x$ term to get
$$x(x^2-1)(x^3+1)\left((x... |
4,232,341 | <p>To prove <span class="math-container">$\lim\limits_{n\to\infty} \left(1+\frac{x}{n}\right)^{n}$</span> exists, we prove that the sequence
<span class="math-container">$$f_n=\left(1+\frac{x}{n}\right)^n$$</span>
is bounded and monotonically increasing toward that bound.</p>
<p><strong>Proof Attempt:</strong></p>
<hr ... | Daniel Sehn Colao | 649,328 | <p>I'll give you the intuition of how it works:</p>
<p>To begin with, Fubini's Theorem states that <span class="math-container">$F(x,y) = \int_c^y \int_a^x f(u,v)dudv = \int_a^x \int_c^y f(u,v)dvdu$</span>, where F is a continuous function over the region R defined by <span class="math-container">$[a,b]\times[c,d]$</sp... |
3,427,794 | <p>Let's see I have the following equation</p>
<p><span class="math-container">$$
x=1
$$</span></p>
<p>I take the derivate of both sides with respect to <span class="math-container">$x$</span>:</p>
<p><span class="math-container">$$
\frac{\partial }{\partial x} x = \frac{\partial }{\partial x}1
$$</span></p>
<p>The... | user | 505,767 | <p>Partial derivative requires a function as argument therefore if we assume, with little a abuse of notation, <span class="math-container">$x$</span> as a function <span class="math-container">$x= x(x,y,z,\ldots)$</span> and we states that <span class="math-container">$x$</span> is a constant function <span class="ma... |
4,184,196 | <blockquote>
<p>Let <span class="math-container">$\hat{f} : \Bbb S^1 \to \Bbb R^2$</span>, <span class="math-container">$\hat{f}(x,y) = (x,y)$</span> and <span class="math-container">$\hat{g} : \Bbb S^1 \to \Bbb R^2$</span>, <span class="math-container">$\hat{g}(x,y) = -(x,y)$</span>. Show that there exists a Homotopy ... | Andrew Paul | 419,177 | <p><span class="math-container">$\hat{f}$</span> and <span class="math-container">$\hat{g}$</span> map from the unit <em>circle</em> (<span class="math-container">$S^1$</span>) to the plane. Intuitively, one can imagine that <span class="math-container">$t\in[0,1]$</span> parameterizes a rotation along the circle. In p... |
2,007,584 | <p>Let $M$ be a smooth manifold and $f:M\rightarrow \mathbb{R}$ be a smooth function such that $f(M)=[0,1]$. Let $1/2$ be a regular value and suppose we consider the open and non-empty set $U:=f^{-1}(\frac{1}{2},\infty)\subset M$. I would like to show that $f^{-1}(\frac{1}{2})$ must coincide with the topological bounda... | Eric Schlarmann | 381,012 | <p>Let $f: (-\epsilon, 1+\epsilon) \to [0,1]$<br/>
$t\mapsto \begin{cases}0 & t\leq 0 \\ \frac{3}{2}t & 0<t<\frac{1}{3}\\
\frac 1 2 & \frac 1 3 \leq t \leq \frac 2 3\\ \frac 3 2 t- \frac 1 2 & 1>t> \frac 2 3 \\ 1 & t\geq 1\end{cases}$</p>
<p>Then the preimage of $\frac 1 2$ is not conta... |
2,553,853 | <p>I know that I am to use the thm. R/A. is a integral domain iff A is prime.
So I want an case where for some a,b in R and ab in A we have a nor b in A.</p>
<p>I thought it had something to do with the conjugate, but I'm lost.</p>
| 57Jimmy | 356,190 | <p>What would be "the conjugate", since you do not know what ring $R$ you are dealing with?</p>
<p>As suggested in the comments, $x^7+1=\underset{a}{(x+1)} \underset{b}{(x^6-x^5+x^4-x^3+x^2-x+1)}$. Neither $a$ nor $b$ are in the ideal $A=(x^7 + 1)$, since all nonzero elements there have degree at least 7, but their pr... |
250,651 | <p>(<a href="https://math.stackexchange.com/questions/1940157/coend-of-mathscrdf-bullet-g-bullet?noredirect=1#comment3982852_1940157">Crosspost</a> from stack)</p>
<p>Given categories $\mathscr{C}$ and $\mathscr{D}$ and functors $F,G: \mathscr{C} \to \mathscr{D}$, we can form a bifunctor
$$\mathscr{D}(F(\bullet), G(\... | HeinrichD | 98,306 | <p>Basic rule in general category theory: Restrict to categories with one object, aka monoids, and see what happens. (Alternatively, restrict to thin categories, aka preorders.) Gain some intuition and generalize this to arbitrary categories.</p>
<p>So here, if $M$ is a monoid, we have two objects $X$ and $Y$ of $\mat... |
629,996 | <p>At <a href="https://math.stackexchange.com/questions/629950/why-i-left-px-in-mathbbz-leftx-right2-mid-p0-right-is-not-a-prin#629954">this question</a> I asked about specific one...</p>
<p>But I think that I don't understand the basic: </p>
<blockquote>
<p>If I have an ideal $I$, what avoid it to be principal? ... | Amzoti | 38,839 | <p>Another approach: Let</p>
<p>$$y = v x$$</p>
<p>The derivatives are:</p>
<p>$$y' = v + x v', ~~y'' = 2 v' + x v''$$</p>
<p>Substitute those into the ODE and solve for $v$ and then $y$.</p>
<p><strong>Update</strong></p>
<p>Upon substitution, we get:</p>
<p>$$x^4 v'' + x^2(2x - 1) v' = 0$$</p>
<p>Let $v' = w$... |
106,560 | <p>Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?</p>
| Minhyong Kim | 1,826 | <p>I would have preferred not to comment seriously on Mochizuki's work before much more thought had gone into the very basics, but judging from the internet activity, there appears to be much interest in this subject, especially from young people. It would obviously be very nice if they were to engage with this circle... |
17,270 | <p>I just joined MathSE and it's beautiful here, except for the fact that some unregistered users ask a question and never come back. Most of the time these questions are trivial, though they still consume answerers' (valuable) time which never gets rewarded. I thought it was okay until I saw someone's profile with the... | Robert Soupe | 149,436 | <p>No, it's not okay, except under very special circumstances.</p>
<p>These are the special circumstances I can think of, in rough order of likelihood:</p>
<ul>
<li>You ask a lot of questions but get no answers at all to any of them.</li>
<li>You ask a lot of questions but get only one answer on each question. The on... |
164,746 | <p>For example</p>
<p><a href="https://i.stack.imgur.com/bFDdw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bFDdw.png" alt="enter image description here"></a></p>
<p>But I want to strip those box edge except the bottom ones like this</p>
<p><a href="https://i.stack.imgur.com/DpzG1.png" rel="nof... | Alexei Boulbitch | 788 | <p>Try this:</p>
<pre><code>Plot3D[-x^2, {x, -1, 1}, {y, -1, 1}, Boxed -> False,
ViewPoint -> {-1.94, -1.94, 1.94}]
</code></pre>
<p><a href="https://i.stack.imgur.com/ghoCf.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ghoCf.jpg" alt="enter image description here"></a></p>
<p>May be it ... |
4,624,695 | <p>I have an exercise with text: With lines</p>
<p><span class="math-container">$$
p_1 \ldots \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-4}{1}, \quad p_2 \ldots \frac{x-5}{2}=\frac{y-1}{-1}=\frac{z-2}{1} .
$$</span>
Determine one plane <span class="math-container">$\pi$</span> with respect to which the lines <span class="mat... | Awnon Bhowmik | 515,900 | <p>By the King's property of integration, we have</p>
<p><span class="math-container">\begin{align}I&=\int_a^b f(x)\,\mathrm dx=\int_a^b f(a+b-x)\,\mathrm dx\end{align}</span></p>
<p>So, we can write
<span class="math-container">$$J=\int_0^{2a} f(x)\,\mathrm dx=\int_0^{2a} f(2a-x)\,\mathrm dx$$</span></p>
<p>Adding... |
2,822,126 | <blockquote>
<p>If $0⩽x⩽y⩽z⩽w⩽u$ and $x+y+z+w+u=1$, prove$$
xw+wz+zy+yu+ux⩽\frac15.
$$</p>
</blockquote>
<p>I have tried using AM-GM, rearrangement, and Cauchy-Schwarz inequalities, but I always end up with squared terms. For example, applying AM-GM to each pair directly gives$$
x^2+y^2+z^2+w^2+u^2 ⩾ xw + wz + zy + ... | HK Lee | 37,116 | <p>(1) Consider $$ \Delta =\bigg\{x=(x_i)\in \mathbb{R}^5\bigg| \sum_i\
x_i=1,\ x_i\geq 0\bigg\}
$$</p>
<p>If $\sigma x =\sum_i\ x_ie_{i+1}$ where $e_i$ is a canonical basis
in $\mathbb{R}^5$, then define $$ f: \Delta\rightarrow \mathbb{R},\
f(x)=x\cdot \sigma (x) $$</p>
<p>Define $H=\bigg\{ v\in \mathbb{R}^n\bigg| \... |
3,105,184 | <p>Three shooters shoot at the same target, each of them shoots just once. The three of them respectively hit the target with a probability of 60%, 70%, and 80%. What is the probability that the shooters will hit the target</p>
<p>a) at least once</p>
<p>b) at least twice</p>
<hr>
<p>I have an approach to this but ... | user247327 | 247,327 | <p>"a) at least once". Writing "H" for a hit and "M" for a miss, and writing hits and misses in the order "ABC" there are 3 possibilities:
HMM: probability is (0.6)(0.3)(0.2)= 0.036
MHM: probability is (0.4)(0.7)(0.2)= 0.056
MMH: probability is (0.4)(0.3)(0.8)= 0.096
The probability of any one of these happening is 0.... |
3,105,184 | <p>Three shooters shoot at the same target, each of them shoots just once. The three of them respectively hit the target with a probability of 60%, 70%, and 80%. What is the probability that the shooters will hit the target</p>
<p>a) at least once</p>
<p>b) at least twice</p>
<hr>
<p>I have an approach to this but ... | awkward | 76,172 | <p>Consider the polynomial
<span class="math-container">$$P(x)=(0.4\, +0.6 x) (0.3\, +0.7 x) (0.2\, +0.8 x)$$</span>
The neat thing about <span class="math-container">$P(x)$</span> is that when expanded it gives all you could possibly want to know about the probabilities of the number of shooters hitting the target:
<s... |
256,138 | <p>I need to generate four positive random values in the range [.1, .6] with (at most) two significant digits to the right of the decimal, and which sum to exactly 1. Here are three attempts that do not work.</p>
<pre><code>x = {.15, .35, .1, .4}; While[Total[x] != 1,
x = Table[Round[RandomReal[{.1, .6}], .010], 4]];... | agei | 82,027 | <p>Strategy:</p>
<ol>
<li>Offset the target value range by 0.1</li>
<li>Generate 4x random values in the range [0-1]</li>
<li>Rebase range based on sum of values generated</li>
<li>Rebase values to reflect offset (0.1) x4 values => 0.6</li>
<li>Round to 2 digits (excepting floating point error)</li>
</ol>
<blockquot... |
239,387 | <p>Someone could explain how to build the smallest field containing to $\sqrt[3]{2}$.</p>
| Asaf Karagila | 622 | <p>Haskell Curry gave an internal definition, but we can also define this field externally:</p>
<p>$$F=\bigcap\{K\mid\Bbb Q\subseteq K, K\text{ is a field}, \sqrt[3]2\in K\}$$</p>
<p>Namely, $F$ is the intersection of all the fields which extend $\Bbb Q$ and contain $\sqrt[3]2$. We know this collection is not empty b... |
2,133,870 | <p>I'm having a little trouble solving this contour integral. I've found the singularities to be at 0, 2, and $1\pm i$. I'm not quite sure how to evaluate the integral when there are two poles on the contour instead of just one. Any help is appreciated.</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
382,603 | <p>Let $F$ be a finite field. Prove that the following are equivalent:</p>
<p>i) $A \subset B$ or $B \subset A$ for each two subgroups $A,B$ of $F^*$.</p>
<p>ii) $\#F^*$ equals 2, 9, a Fermat-prime or $\#F^* -1$ equals a Mersenne prime.</p>
<p>Any ideas for i => ii ? I don't know where to start, except for remarking... | DonAntonio | 31,254 | <p>Jyrki's comment is a huge hint: if $\;\,p,q\,\mid\; |\Bbb F^*|\;$ for two different primes $\,p,q\,$, then there exist cyclic subgroups $\,P,Q\,$ of order $\,p,q\,$ resp., and clearly neither $\,P\subset Q\,$ nor $\,Q\subset P\,$, so $\,|F^*|\,$ is divisible only by one single prime (or it is the trivial group... tr... |
931,748 | <p>I do not understand this solution and this formula and why we are using (1+1)^n...
I need some help to get an idea of what is going on here
Thanks</p>
<p><img src="https://i.stack.imgur.com/S9IL5.png" alt="enter image description here"></p>
| Zubin Mukerjee | 111,946 | <p>Suppose $2\mid (x^2-1)$. Then $2\mid (x-1)(x+1)$. </p>
<p>By <a href="http://en.wikipedia.org/wiki/Euclid's_lemma" rel="nofollow">Euclid's lemma</a>, either $2\mid (x+1)$ <a href="http://en.wiktionary.org/wiki/inclusive_or" rel="nofollow">or</a> $2\mid (x-1)$. </p>
<p>But $x+1$ is even <a href="http://en.wikip... |
931,748 | <p>I do not understand this solution and this formula and why we are using (1+1)^n...
I need some help to get an idea of what is going on here
Thanks</p>
<p><img src="https://i.stack.imgur.com/S9IL5.png" alt="enter image description here"></p>
| Deepak | 151,732 | <p>$x-1$ and $x+1$ are separated by $2$ so they are either both odd or both even. The first case means that the product will not be divisible by two, so the second case must hold true.</p>
<p>Hence $2|(x-1)(x+1) \implies 2|(x+1)$ and $2|(x-1)$ so $4|(x-1)(x+1)$.</p>
|
3,740,647 | <p>I want to load an external Magma file within another Magma file. (Both files are saved in the same directory.) I want to be able to quickly change which external file is being loaded, ideally at the beginning of the file making the load call, so that I can easily run the same code with various inputs.</p>
<p>(The ex... | ev.gal | 682,411 | <p>Here is sample code illustrating the approach proposed by @DavidCraven. (The file <code>realFile.txt</code> contains the Magma commands we desire to load.)</p>
<pre><code>fileNameReal := "realFile.txt";
PrintFile("dummyLoadFile.txt","load \"" cat fileNameReal cat "\";&quo... |
1,465,989 | <p>I'm trying to find the column space of $\begin{bmatrix}a&0\\b&0\\c&0\end{bmatrix}$, which I think is $span\left(~\begin{bmatrix}a\\b\\c\end{bmatrix}~\begin{bmatrix}0\\0\\0\end{bmatrix}~\right)$. Since by definition a span needs to include the null vector, this is redundant. Would it also be correct to ... | Community | -1 | <p>If $\{\vec v_1,\vec v_2, \cdots, \vec v_n\}$ are linearly independent $c_1 \vec v_1+c_2 \vec v_2+\cdots+c_n \vec v_n= \vec 0$ iff $c_1=c_2=\cdots=c_n=0$. Considering $\vec v_n=\vec 0$, we can get $c_1 \vec v_1+c_2 \vec v_2+\cdots+c_n \vec v_n= \vec 0$ by setting $c_1=c_2=\cdots=c_{n-1}=0$ and taking any $c_n \neq 0$... |
1,465,989 | <p>I'm trying to find the column space of $\begin{bmatrix}a&0\\b&0\\c&0\end{bmatrix}$, which I think is $span\left(~\begin{bmatrix}a\\b\\c\end{bmatrix}~\begin{bmatrix}0\\0\\0\end{bmatrix}~\right)$. Since by definition a span needs to include the null vector, this is redundant. Would it also be correct to ... | user514425 | 514,425 | <p>If <span class="math-container">$S=\{v : v=(0,0)\}$</span> we will show that its linearly dependent.
let suppose that S is Linearly independent such that <span class="math-container">$cv=0$</span>.
<span class="math-container">$c$</span> may not be a zero so <span class="math-container">$S$</span> is linearly depen... |
2,983,370 | <p><strong>question:</strong></p>
<p>maximum value of <span class="math-container">$\theta$</span> untill which the approximation <span class="math-container">$\sin\theta\approx \theta$</span> holds to within <span class="math-container">$10\%$</span> error is </p>
<p><span class="math-container">$(a)10^{\circ}$</spa... | Hagen von Eitzen | 39,174 | <p>The next best approximation is
<span class="math-container">$$\sin x \approx x-\frac 16x^3 =x\cdot \left(1-\frac16x^2\right)$$</span>
(and this is an underestimation),
so we look for <span class="math-container">$\frac16x^2\approx 0.1$</span>, <span class="math-container">$x^2\approx 0.6\approx 0.64=0.8^2$</span>. F... |
4,283,063 | <p>For positive real numbers <span class="math-container">$a,b ,c$</span> prove that: <span class="math-container">$a^{b+c}b^{c+a}c^{a+b} ≤ (a^ab^bc^c)^2$</span></p>
<p>My working</p>
<p><span class="math-container">$c^c≥b^c≥b^a$</span></p>
<p><span class="math-container">$(c^c)^2 ≥ b^{a+c}$</span></p>
<p>Similarly
<sp... | Ma Ye | 977,864 | <p>The notation in the question and the description do not agree and I am using the latter.</p>
<p>Consider <span class="math-container">$f(x)=e^x-1$</span>, <span class="math-container">$h(x)=x$</span>. Let <span class="math-container">$g(x)=x$</span> if <span class="math-container">$x$</span> is rational, and <span c... |
4,508,840 | <p>I need to find the summation
<span class="math-container">$$S=\sum_{r=0}^{1010} \binom{1010}r \sum_{k=2r+1}^{2021}\binom{2021}k$$</span></p>
<p>I tried various things like replacing <span class="math-container">$k$</span> by <span class="math-container">$2021-k$</span> and trying to add the 2 summations to a pattern... | drhab | 75,923 | <p>Finding that summation...., too lazy for that. But solving that puzzle you gave in your description is tempting enough to give it a try.</p>
<hr />
<p>First a modulation.</p>
<blockquote>
<p>Let it be that <span class="math-container">$B$</span> throws <span class="math-container">$n$</span> dice and <span class="ma... |
4,068,529 | <p>I need to find <span class="math-container">$63^{63^{63}} \bmod 100$</span>. <br />
This is what I've got so far: <br />
Since <span class="math-container">$\gcd(63,100)=1$</span> we can use Euler's theorem. We have <span class="math-container">$\phi (100)=40$</span> so <span class="math-container">$63^{40} \equiv 1... | tommik | 791,458 | <p>Using properties of Gaussians, Expectation and variance you get that</p>
<p><span class="math-container">$$Y_2\sim N(\alpha+\beta\mu;\beta^2\sigma^2+v^2)$$</span></p>
|
4,068,529 | <p>I need to find <span class="math-container">$63^{63^{63}} \bmod 100$</span>. <br />
This is what I've got so far: <br />
Since <span class="math-container">$\gcd(63,100)=1$</span> we can use Euler's theorem. We have <span class="math-container">$\phi (100)=40$</span> so <span class="math-container">$63^{40} \equiv 1... | RobertTheTutor | 883,326 | <p>If you like, define <span class="math-container">$Y_3 = \alpha + \beta Y_1$</span>. Use properties of expectation. If <span class="math-container">$E[X] = \mu,$</span> then because expectation is a linear sum, <span class="math-container">$E[bX] = b\mu$</span>, and <span class="math-container">$E[X+c] = \mu + c$</... |
4,068,529 | <p>I need to find <span class="math-container">$63^{63^{63}} \bmod 100$</span>. <br />
This is what I've got so far: <br />
Since <span class="math-container">$\gcd(63,100)=1$</span> we can use Euler's theorem. We have <span class="math-container">$\phi (100)=40$</span> so <span class="math-container">$63^{40} \equiv 1... | chuck | 244,783 | <p>The CDF of random variable <span class="math-container">$Y_2$</span> is given by
<span class="math-container">$$
F_{Y_2}(y_2)\equiv P(Y_2\leq y_2)
$$</span>
Therefore,
<span class="math-container">$$
P(Y_2\leq y_2)=Pr(\alpha+\beta Y_1+U\leq y_2)=Pr(\beta Y_1+U\leq y_2-\alpha)=F_{\beta Y_1+U}(y_2-\alpha).
$$</span>
T... |
598,838 | <p><span class="math-container">$11$</span> out of <span class="math-container">$36$</span>? I got this by writing down the number of possible outcomes (<span class="math-container">$36$</span>) and then counting how many of the pairs had a <span class="math-container">$6$</span> in them: <span class="math-container">$... | Newb | 98,587 | <p>That's right. The easier approach would be to calculate the chance of <em>not</em> rolling a $6$ - that's just $\frac56$ for the first die, and $\frac56$ for the second die, so by the product rule (as the events are independent), the probability is $\frac56 \cdot \frac56 = \frac{25}{36}$. </p>
<p>Then the probabili... |
815,963 | <p>I am trying to understand how to do proof by induction for inequalities. The step that I don't fully understand is making an assumption that n=k+1. For equations it is simple. For example:</p>
<blockquote>
<p><strong>Prove that 1+2+3+...+n = $ \frac {n(n+1)}{2} $ is valid for $ i \ge 1 $</strong> </p>
</blockquo... | amWhy | 9,003 | <p>I think you've got the logic wrong with respect to proof by induction. Let's say we are trying to prove a proposition $P(n)$, which might be, for example, $$P(n): \;1 + 2 + 3 +\cdots + n = \frac{n(n+1)}{2}, \;n\geq 1$$</p>
<ul>
<li><p>Base CaseProve that $P(1)$ is true. That is, we test whether the proposition hold... |
423,158 | <p>Maybe I'm too naive in asking this question, but I think it's important and I'd like to know your answer. So, for example I always see that people just write something like "let $f:R\times R\longrightarrow R$ such that $f(x,y)=x^2y+1$", or for example "Let $g:A\longrightarrow A\times \mathbb{N}$ such that for every ... | Dylan Yott | 62,865 | <p>For your first example, such a function exists because you've explicitly told us what $f(x,y)$ should be for each $x$ and $y$. On the other hand, $g$ may or may not exist. It isn't immediately clear what to make $g(x)$ for a given $x$.</p>
|
423,158 | <p>Maybe I'm too naive in asking this question, but I think it's important and I'd like to know your answer. So, for example I always see that people just write something like "let $f:R\times R\longrightarrow R$ such that $f(x,y)=x^2y+1$", or for example "Let $g:A\longrightarrow A\times \mathbb{N}$ such that for every ... | Ittay Weiss | 30,953 | <p>This is a good question. To properly address it one needs to be very precise about what a function is and what it means to have defined/constructed one. So, in the modern era we define a function to be a triple $(A,f,B)$ where $A,B$ are sets and $f\subseteq A\times B$ is a relation from $A$ to $B$. We write $f(x)=y$... |
3,717,172 | <blockquote>
<p>The random variable <span class="math-container">$X$</span> has an exponential distribution function given by density:
<span class="math-container">$$
f_X(x) =
\begin{cases}
e^{-x}, & x\ge 0,\\
0, & x<0.
\end{cases}
$$</span>
Find the distribution and density function of the random variable... | gt6989b | 16,192 | <p><strong>HINT</strong>
<span class="math-container">$$
\begin{split}
F_Y(y)
&= \mathbb{P}[Y \le y] \\
&= \mathbb{P}[\max\{X^2,2-X\} \le y] \\
&= \mathbb{P}[X^2 \le y,2-X \le y] \\
&= \mathbb{P}[|X| \le \sqrt{y},X \ge 2+y] \\
\end{split}
$$</span>
Can you finish and express <span class="math-contai... |
3,326,310 | <p>From Serge Lang's Linear Algebra:</p>
<blockquote>
<p>Let <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span>, <span class="math-container">$x_3$</span> be numbers. Show that:</p>
<p><span class="math-container">$$\begin{vmatrix} 1 & x_1 & x_1^2\\ 1 &x_2 & x_2^2\\ ... | Community | -1 | <p>The general proof is not difficult.</p>
<p>From the definition of a determinant (sum of products), the expansion must be a polynomial in <span class="math-container">$x_1,x_2,\cdots x_n$</span>, of degree <span class="math-container">$0+1+2+\cdots n-1=\dfrac{(n-1)n}2$</span>, and the coefficient of every term is <s... |
369,435 | <p>My task is as in the topic, I've given function $$f(x)=\frac{1}{1+x+x^2+x^3}$$
My solution is following (when $|x|<1$):$$\frac{1}{1+x+x^2+x^3}=\frac{1}{(x+1)+(x^2+1)}=\frac{1}{1-(-x)}\cdot\frac{1}{1-(-x^2)}=$$$$=\sum_{k=0}^{\infty}(-x)^k\cdot \sum_{k=0}^{\infty}(-x^2)^k$$ Now I try to calculate it the following w... | André Nicolas | 6,312 | <p>Let $x\ne 1$. By the usual formula for the sum of a finite geometric series, we have $1+x+x^2+x^3=\frac{1-x^4}{1-x}$. So your expression is equal to $\frac{1-x}{1-x^4}$.</p>
<p>Expand $\frac{1}{1-x^4}$ in a power series, multiply by $1-x$. Unless we are operating purely formaly, we will need to assume that $|x|\lt... |
37,380 | <p>I do have noisy data and want to smooth them by a Savitzky-Golay filter because I want to keep the magnitude of the signal. </p>
<p>a) Is there a ready-to-use Filter available for that? </p>
<p>b) what are appropriate values for m (the half width) and for the coefficients for 3000-4000 data points?</p>
| Joseph | 4,636 | <p>The following code will filter noisy data…</p>
<pre><code>SGKernel[left_?NonNegative, right_?NonNegative, degree_?NonNegative, derivative_? NonNegative] :=
Module[{i, j, k, l, matrix, vector},
matrix = Table[ (* matrix is symmetric *)
l = i + j;
If[l == 0,
left + right + 1,
... |
37,380 | <p>I do have noisy data and want to smooth them by a Savitzky-Golay filter because I want to keep the magnitude of the signal. </p>
<p>a) Is there a ready-to-use Filter available for that? </p>
<p>b) what are appropriate values for m (the half width) and for the coefficients for 3000-4000 data points?</p>
| Frank | 44,130 | <p><br>
I came across the code pasted below a couple of years back and have been using it ever since. It has the advantage that you can apply it also to {x,y} data right away. (BTW, it looks like PH Lundow's code below, last updated back in 2007, is in parts very similar to the one posted by John...)<br>Best,<br>Frank<... |
3,003,982 | <p>By integrating by parts twice, show that <span class="math-container">$I_n$</span>, as defined below for integers <span class="math-container">$n > 1$</span>, has the value shown.</p>
<blockquote>
<p><span class="math-container">$$I_n = \int_0^{\pi / 2} \sin n \theta \cos \theta \,d\theta = \frac{n-\sin(\frac{... | Aleksas Domarkas | 562,074 | <p><span class="math-container">$$\cos{(x)} \sin{\left( n x\right) }=\frac{\sin{\left( \left( n+1\right) x\right) }+\sin{\left( \left( n-1\right) x\right) }}{2}$$</span>
<span class="math-container">$$\int{\left. \cos{(x)} \sin{\left( n x\right) }dx\right.}=-\frac{\cos{\left( \left( n+1\right) x\right) }}{2 \left( n... |
3,003,982 | <p>By integrating by parts twice, show that <span class="math-container">$I_n$</span>, as defined below for integers <span class="math-container">$n > 1$</span>, has the value shown.</p>
<blockquote>
<p><span class="math-container">$$I_n = \int_0^{\pi / 2} \sin n \theta \cos \theta \,d\theta = \frac{n-\sin(\frac{... | Community | -1 | <p><strong>Hint:</strong></p>
<p>There's an escape from the infinite loop. If you observe closely, you will see that after two iterations you get to something like</p>
<p><span class="math-container">$$I=b+aI$$</span></p>
<p>where the constants <span class="math-container">$a,b$</span> are computable.</p>
<p>You na... |
2,523,570 | <p>$15x^2-4x-4$,
I factored it out to this:
$$5x(3x-2)+2(3x+2).$$
But I don’t know what to do next since the twos in the brackets have opposite signs, or is it still possible to factor them out?</p>
| N. F. Taussig | 173,070 | <p>To split the linear term of $15x^2 - 4x - 4$, you must find two numbers with product $15 \cdot (-4) = -60$ and sum $-4$. They are $-10$ and $6$. Hence,
\begin{align*}
15x^2 - 4x - 4 & = 15x^2 - 10x + 6x - 4 && \text{split the linear term}\\
& = 5x(3x - 2) + 2(3x - 2) && \tex... |
2,830,926 | <p>I am implementing a program in C which requires that given 4 points should be arranged such that they form a quadrilateral.(assume no three are collinear)<br>
Currently , I am ordering the points in the order of their slope with respect to origin.<br>
See <a href="https://ibb.co/cfDHeo" rel="nofollow noreferrer">htt... | Graham Kemp | 135,106 | <p>Let $q:=1-p$</p>
<p>$\begin{align}\mathsf M'_X(u) & = \mathcal D_u\left(\dfrac{p}{1-(1-p)e^u}\right)^r\\[1ex] & =p^r \mathcal D_u(1-qe^u)^{-r}\\[1ex] &= -rp^r (1-qe^u)^{-r-1}\mathcal D_u (1-qe^u)\\[1ex] &= rqp^r (1-qe^u)^{-r-1}e^u\\[2ex]\mathsf M''_X(u) &= \mathcal D_u\left(rqp^r (1-qe^u)^{-r-1}... |
2,727,974 | <blockquote>
<p>I am in need of this important inequality $$\log(x+1)\leqslant x$$.</p>
</blockquote>
<p>I understand that $\log(x)\leqslant x$. For $c\in\mathbb{R}$. However is it true that $\log(x+c)\leqslant x$?</p>
<p>It is hard to accept because it seems like $c$ cannot be arbitrary.
I have tried to prove thi... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: write $$x-\log(x+1)\geq 0$$ and define $$f(x)=x-\log(x+1)$$ and use calculus. And $$f'(x)=\frac{x+1-x}{x(x+1)}$$</p>
|
2,727,974 | <blockquote>
<p>I am in need of this important inequality $$\log(x+1)\leqslant x$$.</p>
</blockquote>
<p>I understand that $\log(x)\leqslant x$. For $c\in\mathbb{R}$. However is it true that $\log(x+c)\leqslant x$?</p>
<p>It is hard to accept because it seems like $c$ cannot be arbitrary.
I have tried to prove thi... | Simply Beautiful Art | 272,831 | <p>The derivative argument works. For your specific inequality, note that the derivative argument can be written in terms of integrals:</p>
<p>$$y-1=\int_1^y\frac11{\rm~d}t\ge\int_1^y\frac1t{\rm~d}t=\ln(y)$$</p>
<p>where $y=x+1>1$.</p>
<p>For $c<1$ it follows trivially from above. For $c>1,y>0$, note tha... |
2,174,270 | <p>I have an elliptic curve $E$ over $\mathbb{F}_{7}$ defined by $y^2=x^3+2$ with the point at infinity $\mathcal{O}$</p>
<p>I am given the point $(3,6)$ and need to find the line which intersects with $E$ at <strong>only this point</strong></p>
<p>I am told that this line is $y\equiv (4x+1)\mod 7$</p>
<p>I have ver... | DonAntonio | 31,254 | <p>An idea: </p>
<p>A general line through $\;(3,6)\;$ is of the form $\;y-6=m(x-3)\implies \color{red}{y=mx-(3m-6)}\;$ , and we want this line to intersect $\;E\;$ <strong>only</strong> at the given point, so we need the equation over $\;\Bbb F_7\;$ : $\;(mx-(3m-6))^2=x^3+2\;$, to have <em>one single solution</em> (... |
72,012 | <p>I'm trying to discretize a region with "pointy" boundaries to study dielectric breaking on electrodes with pointy surfaces. So far I've tried 3 types of boundaries, but the meshing functions stall indefinitely. I've let it run for several hours and it doesn't complete. Seems strange that it would take so long</p>
<... | Community | -1 | <p>I would suggest not going through the <code>Plot</code> functions for this. They're designed to produce a good <em>visual</em> representation of the region, which is not necessarily the same as a good representation for doing numerical computation on. Besides, the plot already discretizes the region into a polygon, ... |
2,896,780 | <p>With a trusted 3rd party, running Secret Santa is easy: The 3rd party labels each person $1,\dotsc,n$, and then randomly chooses a derangement from among all possible derangements of $n$ numbers. Person $i$ will then give a gift to the number in position $i$ of the derangement. The trusted 3rd party is responsible... | hmakholm left over Monica | 14,366 | <p>How about this:</p>
<ol>
<li><p>Everybody generates a random private-public key pair.</p></li>
<li><p>Everybody publishes their public key <em>anonymously</em> (see below).</p></li>
<li><p>Collaboratively choose a random seed:</p>
<p>a. Everybody chooses a random seed component.<br>
b. Everybody publishes a hash o... |
195,556 | <p>My math background is very narrow. I've mostly read logic, recursive function theory, and set theory.</p>
<p>In recursive function theory one studies <a href="http://en.wikipedia.org/wiki/Partial_functions">partial functions</a> on the set of natural numbers. </p>
<p>Are there other areas of mathematics in which (... | Qiaochu Yuan | 232 | <p>Sure. For example, <a href="http://en.wikipedia.org/wiki/Meromorphic_function" rel="noreferrer">meromorphic functions</a> in complex analysis are partial functions. <a href="http://en.wikipedia.org/wiki/Multiplicative_inverse" rel="noreferrer">Multiplicative inverse</a> is another partial function (say on a <a href=... |
181,321 | <p>Assuming that <code>a.b=z</code>,where <code>b={b1,b2,b3}</code> and <code>z=x b1+y b2+z b3</code>, obviously <code>a={x,y,z}</code>, how to realize it by mathematica?</p>
| GerardF123 | 58,885 | <p>If I understand correctly, you have a single linear equation with 3 unknowns.</p>
<p>It's not possible to back out the 3 unknown values. You have an equation for a 3D hyperplane with an infinite number of solutions. </p>
|
3,454,395 | <p>So I’m working on this equation <span class="math-container">$z^{10} + 2z^5 + 2 = 0$</span> to find all complex solutions, and I think I managed to solve it, but I can’t find solution manual for it, since it is really old exam task. The thing that makes me uncomfortable with my solution is that, shouldn’t I get just... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$1+z^5=-1\pm i=\sqrt2e^{i(2n\pi+\pi\pm\pi/4)}$$</span> where <span class="math-container">$n$</span> is any integer</p>
<p><span class="math-container">$$z=2^{1/10}\text{exp}\left(\dfrac{i\left(2n\pi+\pi\pm\pi/4\right)}5\right)$$</span>
where <span class="math-container">$0\le n\le4$<... |
3,714,995 | <blockquote>
<p>Using sell method to find the volume of solid generated by revolving the region bounded by <span class="math-container">$$y=\sqrt{x},y=\frac{x-3}{2},y=0$$</span> about <span class="math-container">$x$</span> axis, is (using shell method)</p>
</blockquote>
<p>What I try:</p>
<p><a href="https://i.sta... | Sumanta | 591,889 | <p>Since <span class="math-container">$A$</span> is self-adjoint, all eigenvalues of <span class="math-container">$A$</span> are real. Also, <span class="math-container">$\lambda$</span> is an eigenvalue of <span class="math-container">$A$</span> implies <span class="math-container">$Av=\lambda v$</span> for some non-z... |
33,303 | <p>Is it possible to attach certain pieces of code to certain controls in a Manipulate? For example, consider the following Manipulate</p>
<pre><code>Manipulate[
data = Table[function[x], {x, -Pi*10, Pi*10, Pi/1000}];
ListPlot[{x, data}, PlotRange -> {{start, stop}, Automatic}]
, {function, {Sin, Cos, Tan}}
, ... | rm -rf | 5 | <p>In such cases, you can get better flexibility by switching to a <code>DynamicModule</code> and building up the GUI yourself. Then, you can pull the data generating step out of the plotting dynamic, so that the latter can be manipulated freely without regenerating the data. </p>
<pre><code>DynamicModule[{function = ... |
803,800 | <p>Define a sequence $(a_n)_{n = 1}^{\infty}$ of real numbers by $a_n = \sin(n)$. This sequence is bounded (by $\pm1$), and so by the <a href="http://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem" rel="noreferrer">Bolzano-Weierstrass Theorem</a>, there exists a convergent subsequence.</p>
<p><strong>My que... | Stijn Hanson | 145,554 | <p>This isn't a very analytic way of doing it but consider the continued fraction expansion $\pi = [a_0; a_1, a_2, \ldots]$. Generally we know that, if $p_n/q_n$ are the convergents to $\pi$ then $|q_n\pi - p_n| \leq 1/q$ so the $p_n$ are the closest integers to being multiples of $\pi$ so (with slight abuse of notatio... |
914,440 | <p>By the definition of topology, I feel topology is just a principle to define "open sets" on a space(in other words, just a tool to expand the conception of open sets so that we can get some new forms of open sets.) But I think in the practical cases, we just considered Euclidean space most and the traditional form o... | GWilliams | 169,969 | <p>I think your idea that topology is just to define open sets misses the point. I interpret the point of topology as attempting to generalize the concept of a continuous function primarily, and open sets are often the most direct pedagogical route there. </p>
<p>Further, one often does want to leave euclidean space. ... |
306,744 | <p>So if the definition of continuity is: $\forall$ $\epsilon \gt 0$ $\exists$ $\delta \gt 0:|x-t|\lt \delta \implies |f(x)-f(t)|\lt \epsilon$. However, I get confused when I think of it this way because it's first talking about the $\epsilon$ and then it talks of the $\delta$ condition. Would it be equivalent to say: ... | gnometorule | 21,386 | <p>Let
$$f(x) := \begin{cases} 1, & x\in \mathbb{Q} \\ 0, & x\in \mathbb{R} - \mathbb{Q}, \\ \end{cases}$$
a very discontinuous function. Then $\epsilon := 2$ will do, for any $x, t, \delta >0$ you choose. So this cannot be equivalent. </p>
|
4,272,214 | <h2>The Equation</h2>
<p>How can I analytically show that there are <strong>no real solutions</strong> for <span class="math-container">$\sqrt[3]{x-3}+\sqrt[3]{1-x}=1$</span>?</p>
<h2>My attempt</h2>
<p>With <span class="math-container">$u = -x+2$</span></p>
<p><span class="math-container">$\sqrt[3]{u-1}-\sqrt[3]{u+1}=... | B. Goddard | 362,009 | <p>Certainly for any real <span class="math-container">$x$</span>,</p>
<p><span class="math-container">$$x-3 < x-1$$</span></p>
<p>so</p>
<p><span class="math-container">$$\sqrt[3]{x-3} < \sqrt[3]{x-1}$$</span></p>
<p>since cube root is a monotonically increasing function. So</p>
<p><span class="math-container">$... |
2,971,980 | <p>Show that if <span class="math-container">$0<b<1$</span> it follows that
<span class="math-container">$$\lim_{n\to\infty}b^n=0$$</span>
I have no idea how to express <span class="math-container">$N$</span> in terms of <span class="math-container">$\varepsilon$</span>. I tried using logarithms but I don't see h... | user | 505,767 | <p>Recall that by the definition <span class="math-container">$\lim_{n\to\infty}a_n=L\in \mathbb{R}$</span> means that</p>
<p><span class="math-container">$$\forall \epsilon>0 \quad \exists \bar n \quad \forall n>\bar n \quad |a_n-L|<\epsilon$$</span></p>
<p>In that case, note that for any <span class="math-... |
843,763 | <blockquote>
<p>Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. </p>
</blockquote>
<p>Can anyone give some hint to solve this problem? I tried contradiction but could not reach a proof.</p>
<p>I spend part of the day studying this... | user411953 | 411,953 | <p>$f \colon x \rightarrow x-\lfloor x \rfloor$ </p>
<p>$F=\{f(nx),n \in \Bbb{Z}\}$</p>
<p>$F \neq \emptyset$ and $x \ge 0$ $\exists d=\inf(F)$</p>
<p>We can prove that $d=0$ (suppose a $\gt$ 0 and ...)</p>
<p>hence $\forall$ y $\gt$ 0 $\exists$ n $\in$ $\Bbb{N}$ , $y \gt f(nx) \gt 0$ </p>
<p>$a,b \in [0;1[$
if $b... |
3,481,531 | <p>I was solving a problem where there are 4 points and we need to check if they form a square or not. Now if all 4 points are the same, is it square or not. My initial thought was it is a zero-sided square but wanted to clarify if my understanding is right or not. </p>
| Fred | 380,717 | <p>If <span class="math-container">$a \le b$</span> and <span class="math-container">$c \le d$</span> then we have two cases:</p>
<ol>
<li><p><span class="math-container">$a < b$</span> and <span class="math-container">$c< d$</span>, then <span class="math-container">$[a,b] \times [c,d]$</span> is a rectangle (i... |
3,518,232 | <p>I am given with a Triangle <span class="math-container">$ABC$</span> which is inscribed in circle <span class="math-container">$\omega$</span>. The tangent lines to <span class="math-container">$\omega$</span> at <span class="math-container">$B$</span> and <span class="math-container">$C$</span> meet at <span class=... | Sunaina Pati | 807,973 | <p>Probably a different solution than the previous answer, hence sharing it!</p>
<p><strong>My Proof</strong>: Beautiful problem indeed!</p>
<p>Since <span class="math-container">$T$</span> is the point of intersection of tangent at <span class="math-container">$B$</span> and <span class="math-container">$C$</span> wrt... |
3,152,532 | <p>What is meaning of <span class="math-container">$f^2(x)$</span> ? There seems to be confusion in its interpretation.</p>
<p>Is <span class="math-container">$f^2(x)$</span> same as <span class="math-container">$(f(x))^2$</span> or <span class="math-container">$f\circ f$</span>?</p>
| José Carlos Santos | 446,262 | <p>It depends upon the context. It can be both. But if <span class="math-container">$f$</span> is a function from some set <span class="math-container">$S$</span> into <span class="math-container">$\mathbb R$</span>, then it can only be <span class="math-container">$\bigl(f(x)\bigr)^2$</span>. And if it is a function f... |
3,152,532 | <p>What is meaning of <span class="math-container">$f^2(x)$</span> ? There seems to be confusion in its interpretation.</p>
<p>Is <span class="math-container">$f^2(x)$</span> same as <span class="math-container">$(f(x))^2$</span> or <span class="math-container">$f\circ f$</span>?</p>
| David | 651,991 | <p>I would interpret it as <span class="math-container">$(f(x))^2$</span> whenever it makes sense. If <span class="math-container">$f(x)$</span> is not defined (for example, if it is a vector), then I would understand <span class="math-container">$f\circ f$</span></p>
<p>Anyway, I would make sure the meaning is consis... |
4,369,480 | <p>In's commonly said that in VAE, we use reparameterization trick because "we can't backpropagate through stochastic node"
<a href="https://i.stack.imgur.com/ED72y.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ED72y.png" alt="enter image description here" /></a></p>
<p>It makes sense fro... | Joe | 693,577 | <p>I think you are confused about back propagation. You never need to take the gradient of the input with respect to anything (because there is no layer BEFORE the input), nor do you need to make assumptions about the distribution of the input.</p>
<p>The 'reparametrization trick' makes some assumption about the parame... |
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